Elements of Physical Oceanography: A derivative of the Encyclopedia of Ocean Sciences

Elements of Physical Oceanography Editor-in-Chief John H. Steele Marine Policy Center, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA Editors Steve A. Thorpe National Oceanography Centre, University of Southampton, Southampton, UK School of Ocean Sciences, Bangor University, Menai Bridge, Anglesey, UK Karl K. Turekian Yale University, Department of Geology and Geophysics, New Haven, Connecticut, USA Subject Area Volumes from the Second Edition Climate & Oceans edited by Karl K. Turekian Elements of Physical Oceanography edited by Steve A. Thorpe Marine Biology edited by John H. Steele Marine Chemistry & Geochemistry edited by Karl K. Turekian Marine Ecological Processes edited by John H. Steele Marine Geology & Geophysics edited by Karl K. Turekian Marine Policy & Economics guest edited by Porter Hoagland, Marine Policy Center, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts Measurement Techniques, Sensors & Platforms edited by Steve A. Thorpe Ocean Currents edited by Steve A. Thorpe

ENCYLOPEDIA OF

OCEAN SCIENCES: ELEMENTS OF PHYSICAL OCEANOGRAPHY Editor

STEVE A. THORPE

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CONTENTS Elements of Physical Oceanography: Introduction

ix

SURFACE WAVES, TIDES, AND SEA LEVEL Surface Gravity and Capillary Waves Wave Generation by Wind Rogue Waves

W K Melville

3

J A T Bye, A V Babanin

12

K Dysthe, H E Krogstad, P Mu¨ller

Waves on Beaches Wave Energy

R A Holman

29

M E McCormick, D R B Kraemer

Whitecaps and Foam

41 J Gemmrich

47

D C Chapman, G S Giese

Tsunami

55

P L-F Liu

Storm Surges

62

R A Flather

Coastal Trapped Waves Tides

37

E C Monahan

Breaking Waves and Near-Surface Turbulence Seiches

18

76

J M Huthnance

87

D T Pugh

Tidal Energy

95

A M Gorlov

Sea Level Change

103

J A Church, J M Gregory

Sea Level Variations Over Geological Time

109

M A Kominz

115

THE AIR-SEA INTERFACE Heat and Momentum Fluxes at the Sea Surface

P K Taylor

127

Sea Surface Exchanges of Momentum, Heat, and Fresh Water Determined by Satellite Evaporation and Humidity

K Katsaros

Freshwater Transport and Climate Air–Sea Gas Exchange

L Yu

135 145

S Wijffels

152

B Ja¨hne

160

Air–Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-methane Hydrocarbons, Organo-halogens J W Dacey, H J Zemmelink

170

Air–Sea Transfer: N2O, NO, CH4, CO

176

Gas Exchange in Estuaries

C S Law

M I Scranton, M A de Angelis

Penetrating Shortwave Radiation Radiative Transfer in the Ocean

C A Paulson, W S Pegau

Bubbles

W Alpers

D K Woolf

192

C D Mobley

Atmospheric Transport and Deposition of Particulate Material to the Oceans R Arimoto Surface Films

184

198 J M Prospero, 208 218 221

v

vi

CONTENTS

BOUNDARY LAYERS: THE UPPER OCEAN BOUNDARY LAYER Upper Ocean Vertical Structure

J Sprintall, M F Cronin

Wind- and Buoyancy-Forced Upper Ocean

229

M F Cronin, J Sprintall

237

Upper Ocean Space and Time Variability

D L Rudnick

246

Upper Ocean Mean Horizontal Structure

M Tomczak

252

Upper Ocean Structure: Responses to Strong Atmospheric Forcing Events Upper Ocean Mixing Processes

L K Shay

J N Moum, W D Smyth

Langmuir Circulation and Instability

S Leibovich

Upper Ocean Heat and Freshwater Budgets

P J Minnett

262 281 288 297

BOUNDARY LAYERS: THE BENTHIC BOUNDARY LAYER Turbulence in the Benthic Boundary Layer Benthic Boundary Layer Effects

R Lueck

D J Wildish

311 317

BOUNDARY LAYERS: UNDER-ICE BOUNDARY LAYER Under-Ice Boundary Layer Ice–Ocean Interaction

M G McPhee, J H Morison J H Morison, M McPhee

327 335

INTERNAL WAVES Internal Waves Internal Tides

C Garrett

349

R D Ray

357

PROCESSES OF DIAPYCNAL MIXING Three-dimensional (3d) Turbulence

W D Smyth, J N Moum

Laboratory Studies of Turbulent Mixing Internal Tidal Mixing

W Munk

Estimates of Mixing

385

A C Naveira Garabato

C H Gibson

Open Ocean Convection Deep Convection

J R N Lazier

Differential Diffusion

414 422

R W Schmitt

431

A E Gargett

Dispersion and Diffusion in the Deep Ocean

396 406

A Soloviev, B Klinger

Double-Diffusive Convection

375 381

J M Klymak, J D Nash

Energetics of Ocean Mixing Fossil Turbulence

J A Whitehead

367

440 R W Schmitt, J R Ledwell

448

HORIZONTAL DISPERSION, TRANSPORT, AND OCEAN PROPERTIES Vortical Modes Intrusions

E L Kunze

D L Hebert

459 464

CONTENTS Dispersion in Shallow Seas

J T Holt, R Proctor

Dispersion from Hydrothermal Vents Nepheloid Layers

K R Helfrich

I N McCave

Heat Transport and Climate

469 475 484

H L Bryden

El Nin˜o Southern Oscillation (ENSO) North Atlantic Oscillation (NAO) Water Types And Water Masses

vii

495

K E Trenberth J W Hurrell

W J Emery

Neutral Surfaces and the Equation of State

T J McDougall, D R Jackett

502 515 523 532

ICE Sea Ice: Overview

W F Weeks

541

Sea Ice Dynamics

M Leppa¨ranta

550

Sea Ice Polynyas

P Wadhams

561

S Martin

579

PROCESSES IN COASTAL AND SHELF SEAS Beaches, Physical Processes Affecting Shelf Sea and Slope Sea Fronts

A D Short

587

J Sharples, J H Simpson

598

APPENDICES Appendix 1. SI Units and Some Equivalences

611

Appendix 6. The Beaufort Wind Scale and Seastate

614

INDEX

617

ELEMENTS OF PHYSICAL OCEANOGRAPHY: INTRODUCTION Physical Oceanography is one of the several different, if not entirely distinct, sciences of the ocean. It is concerned with kinematics and dynamics, fluxes and stress, waves, tides, flows and mixing. These factors all impinge in fundamental ways on (and gain understanding from) the other companion sciences, marine biology, geochemistry and geology, the ocean flows advecting and diffusing the dissolved solutes and particulate matter – living or inanimate – in the water, and largely controlling their distributions and movements. This volume is a selection of articles from the Encyclopedia of Ocean Sciences about ocean physics, air-sea transfers, waves, mixing, ice (a topic of rapidly growing interest) or, more generally, about the processes of transfer of properties such as heat, salinity, momentum and dissolved gases, within and into the ocean. It provides a general source of reference to the present state of knowledge of some aspects of the subject. The articles are arranged in eight sections in an order and with connections that might be found useful in teaching courses on ocean physics at undergraduate or postgraduate level. The volume does not, however, follow the conventional order of textbooks or courses in physical oceanography which commonly begin with a discussion of the properties of seawater and their effect on density (a discussion which is difficult to make attractive to students whose excitement and interest are more easily aroused by dynamical processes which are visible and might be used to recreational advantage, such as waves — surfing and sailing being popular sports) before going on to set up relationships to describe the behaviour of fluids. Nor does the volume include a very important component of a physical oceanography course – that of ocean currents and circulation, the form they take in different oceans and seas and how they are driven – a subject that has its own separate volume in this series: Ocean Currents. The order of sections initially follows a progress downwards from the sea surface. The first section is about surface waves and other changes in the level of the sea surface. Included are articles on the ‘rogue’ waves that cause damage to ships, and tsunamis that have resulted in severe damage and loss of life in recent years. The tides are perhaps the oldest marine subjects of scientific observation, enquiry and speculation, beginning with the Ancient Greeks (Cartwright, 1999). Articles on tides and tidal energy are included in this section as well as the climate-related topic of sea level change. This is followed by a section in which are described the exchanges of heat, momentum and gases between the atmosphere and the ocean, topics of particular current importance because of the part played by the ocean in the composition of the atmosphere and in climate change. The presence of surface films may restrict the exchange of gases, whilst bubbles created by breaking waves enhance the transfer. Much of the mixing and turbulent dissipation of kinetic energy within the ocean takes place in its boundary layers. Three different boundary layers are identified as sub-topics in the third section, the upper ocean, the under-ice, and the benthic boundary layers. (It appears logical to include the benthic boundary layer with its companions although the order of descent through the ocean is interrupted.) Mixing in the three differs. The upper ocean boundary layer is strongly affected by the wind, and the consequent waves and Langmuir circulation, as well as by the buoyancy changes resulting from solar radiation, and heat and freshwater fluxes. The mixing in the benthic boundary layer is driven largely by the stress on the seabed, very little by geothermal heat (or buoyancy) flux. Like the others, the under-ice boundary layer is also driven by stress, but also by buoyancy resulting from freezing, salt rejection or ice melting. Internal waves and tides are the subject of the short fourth section and, because in breaking they contribute to mixing, they provide an introduction to the subject addressed in the following section, processes of diapycnal mixing, including turbulent mixing, convection, double diffusive convection and diffusion. There is presently keen debate about how the deep ocean is mixed, and consequently this is a subject of active and developing investigation. It is central to the study of ocean physics, relating the sources of energy to the mixing and circulation of the oceans (Wunsch and Ferrari, 2004). Articles on the vitally important subjects of horizontal dispersion and the lateral transport of heat around the World are included in the sixth section. (The oceans transport about as much heat from the equatorial to the arctic regions as does the atmosphere!) The major interannual oscillations of the Pacific and Atlantic Oceans, ENSO and NAO, the former characterized by sea-surface temperature fluctuations and the latter by

ix

x

ELEMENTS OF PHYSICAL OCEANOGRAPHY: INTRODUCTION

differences in sea level pressure are, included in this section, as are two articles-on water masses and neutral surfaces-that relate indirectly to dispersion and to the consequent density of seawater. The final two sections are (returning to the sea surface) about ice and polynyas, and processes in the coastal and shelf seas. Because of space limitations, it has been necessary to be selective and to omit some important topics that would be included in a thoroughly comprehensive account (or perhaps in a broad-ranging taught course). Most notably these are an account of some of the instruments used in measurement, such as current meters, floats, CTD and turbulence sensors. These are described in articles in the full Encyclopedia and in the companion special topic volume on ‘Measurement Techniques, Platforms and Sensors.’ Some related topics, e.g., models of ocean circulation, are found in the full Encyclopedia. It is regretted that there are also no articles about the development of sedimentary waves, ripples and dunes or about the important relationship between turbulence and particles of sediment or algae. The author of each article is an expert in his or her field. They are all distinguished researchers who have given time to write concisely and lucidly about their subjects, and the Editors are indebted to them all for the time given and the care taken in preparing these accounts. The articles in this volume would not have been produced without the considerable help of the several members of the Encyclopedia’s Editorial Advisory Board listed below. Each provided advice and suggestions about the content and authorship of particular subject areas covered in the Encyclopedia. In addition to thanking the authors of the articles in this volume, the Editors wish to thank the members of the Editorial Board for the time they gave to identify and encourage authors, to read and comment on (and sometimes to suggest improvements to) the written articles, and to make this venture possible.

Editorial Advisory Board Members who helped in the production of this volume Garry Bass, Ken Brink, Robert Duce, John Gould, Ann Gargett, Chris Garrett, Peter Liss, Nick McCave, Dennis McGillicuddy, Ken Melville, Jim Moum, Colin Summerhayes, Stewart Turner, Bob Weller and James Yoder. Steve A. Thorpe Editor

REFERENCES Cartwright DE (1999) Tides: A Scientific History, p. 292. Cambridge: Cambridge University Press. Wunsch C and Ferrari R (2004) Vertical mixing, energy and the general circulation of the oceans. Annual Review of Fluid Mechanics 36: 281--314.

SURFACE WAVES, TIDES, AND SEA LEVEL

SURFACE GRAVITY AND CAPILLARY WAVES W. K. Melville, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, USA

surface, from local to global scales, depends on the surface wave field.

Copyright & 2001 Elsevier Ltd.

Basic Formulations

Introduction Ocean surface waves are the most common oceanographic phenomena that are known to the casual observer. They can at once be the source of inspiration and primal fear. It is remarkable that the complex, random wave field of a storm-lashed sea can be studied and modeled using well-developed theoretical concepts. Many of these concepts are based on linear or weakly nonlinear approximations to the full nonlinear dynamics of ocean waves. Early contributors to these theories included such luminaries as Cauchy, Poisson, Stokes, Lagrange, Airy, Kelvin and Rayleigh. Many of the current challenges in the study of ocean surface waves are related to nonlinear processes which are not yet well understood. These include dynamical coupling between the atmosphere and the ocean, wave–wave interactions, and wave breaking. For the purposes of this article, surface waves are considered to extend from low frequency swell from distant storms at periods of 10 s or more and wavelengths of hundreds of meters, to capillary waves with wavelengths of millimeters and frequencies of O(10) Hz. In between are wind waves with lengths of O(1–100) m and periods of O(1–10) s. Figure 1 shows a spectrum of surface waves measured from the Research Platform FLIP off the coast of Oregon. The spectrum, F, shows the distribution of energy in the wave field as a function of frequency. The wind wave peak at approximately 0.13 Hz is well separated from the swell peak at approximately 0.06 Hz. Ocean surface waves play an important role in air– sea interaction. Momentum from the wind goes into both surface waves and currents. Ultimately the waves are dissipated either by viscosity or breaking, giving up their momentum to currents. Surface waves affect upper-ocean mixing through both wave breaking and their role in the generation of Langmuir circulations. This breaking and mixing influences the temperature of the ocean surface and thus the thermodynamics of air–sea interaction. Surface waves impose significant structural loads on ships and other structures. Remote sensing of the ocean

The dynamics and kinematics of surface waves are described by solutions of the Navier–Stokes equations for an incompressible viscous fluid, with appropriate boundary and initial conditions. Surface waves of the scale described here are usually generated by the wind, so the complete problem would include the dynamics of both the water and the air above. However, the density of the air is approximately 800 times smaller than that of the water, so many aspects of surface wave kinematics and dynamics may be considered without invoking dynamical coupling with the air above. The influence of viscosity is represented by the Reynolds number of the flow, Re ¼ UL=m, where U is a characteristic velocity, L a characteristic length scale, and n ¼ m=r is the kinematic viscosity, where m is the viscosity and r the density of the fluid. The Reynolds number is the ratio of inertial forces to viscous forces in the fluid and if Re 441, the effects of viscosity are often confined to thin boundary layers, with the interior of the fluid remaining essentially inviscid ðn ¼ 0Þ. (This assumes a homogeneous fluid. In contrast, internal waves in a continuously stratified fluid are rotational since they introduce baroclinic generation of vorticity in the interior of the fluid). Denoting the fluid velocity by u ¼ ðu; v; wÞ, the vorticity of the flow is given by z ¼ r  u. If z ¼ 0, the flow is said to be irrotational. From Kelvin’s circulation theorem, the irrotational flow of an incompressible ðr:u ¼ 0Þ inviscid fluid will remain irrotational as the flow evolves. The essential features of surface waves may be considered in the context of incompressible irrotational flows. For an irrotational flow, u ¼ rf where the scalar f is a velocity potential. Then, by virtue of incompressibility, f satisfies Laplace’s equation r2 f ¼ 0

½1

We denote the surface by z ¼ ðx; y; tÞ, where ðx; yÞ are the horizontal coordinates and t is time. The kinematic condition at the impermeable bottom at z ¼ h; is one of no flow through the boundary: @f ¼0 @z

at

z ¼ h

½2

3

4

SURFACE GRAVITY AND CAPILLARY WAVES

16 _1

)

Swell peak

Φ (m2 Hz

12

Wind sea peak Pitch _ roll

8 4

Heave

0

0

0.02

0.04

0.06

0.08

0.1 f (Hz)

(A)

0.12

0.14

0.16

0.18

0.2

Φ (m2 Hz

_1

)

10 0 _2

10

_4

10

_6

10

_2

10

_1

100

10

f (Hz)

(B)

Figure 1 (A) Surface displacement spectrum measured with an electromechanical wave gauge from the Research Platform FLIP in 8 m s1 winds off the coast of Oregon. Note the wind-wave peak at 0.13 Hz, the swell at 0.06 Hz and the heave and pitch and roll of FLIP at 0.04 and 0.02 Hz respectively. (B) An extension of (A) with logarithmic spectral scale, note that from the wind sea peak to approximately 1 Hz the spectrum has a slope like f4, common in wind-wave spectra. (Reproduced with permission from Felizardo FC and Melville WK (1995). Correlations between ambient noise and the ocean surface wave field. Journal of Physical Oceanography 25: 513–532.)

There are two boundary conditions at z ¼ Z: @Z @Z @Z þu þv ¼w @t @x @y

½3

@f 1 2 þ u þ gZ ¼ ðpa  pÞ=r @t 2

½4

The first is a kinematic condition which is equivalent to imposing the condition that elements of fluid at the surface remain at the surface. The second is a dynamical condition, a Bernoulli equation, which is equivalent to stating that the pressure p at z ¼ Z , an infinitesimal distance beneath the surface, is just a constant atmospheric pressure, pa , plus a contribution from surface tension. The effect of gravity is to impose a restoring force tending to bring the surface back to z ¼ 0. The effect of surface tension is to reduce the curvature of the surface. Although this formulation of surface waves is considerably simplified already, there are profound difficulties in predicting the evolution of surface waves based on these equations. Although Laplace’s equation is linear, the surface boundary conditions are nonlinear and apply on a surface whose specification is a part of the solution. Our ability to accurately

predict the evolution of nonlinear waves is limited and largely dependent on numerical techniques. The usual approach is to linearize the boundary conditions about z ¼ 0.

Linear Waves Simple harmonic surface waves are characterized by an amplitude a, half the distance between the crests and the troughs, and a wavenumber vector k with 7k7 ¼ k ¼ 2p=l, where l is the wavelength. The surface displacement, (unless otherwise stated, the real part of complex expressions is taken) Z ¼ aeiðk:xstÞ

½5

where s ¼ 2p=T is the radian frequency and T is the wave period. Then ak is a measure of the slope of the waves, and if akoo1, the surface boundary conditions can be linearized about z ¼ 0. Following linearization, the boundary conditions become @Z ¼w @t

½6

SURFACE GRAVITY AND CAPILLARY WAVES

at z ¼ 0

½7

where the linearized Laplace pressure is  2  @ Z @2Z pa  p  ¼ G þ @x2 @y2

½8

where G is the surface tension coefficient. Substituting for Z and satisfying Laplace’s equation and the boundary conditions at z ¼ 0 and h gives ig0 acoshkðz þ hÞ f¼ s coshkh



s2 ¼ g0 k tanh kh

½10

  g0 ¼ g 1 þ Gk2 =r

½11

and

Equations relating the frequency and wavenumber, s ¼ sðkÞ, are known as dispersion relations, and for linear waves provide a fundamental description of the wave kinematics. The phase speed, 

1=2 g0 tanh kh k

½12

is the speed at which lines of constant phase (e.g., wave crests) move. For waves propagating in the x-direction, the velocity field is u¼

g0 ak cosh ðz þ hÞ iðkxstÞ e s cosh kh

ðu; v; w; pÞ ¼

½9

where

c ¼ s=k ¼

so that there is no vertical motion, just a uniform sloshing backwards and forwards in the horizontal plane in phase with the surface displacement Z. The phase speed c ¼ ðg0 hÞ1=2 , is independent of the wavenumber. Such waves are said to be nondispersive. Waves propagating towards shore eventually attain this condition, and, as the depth tends to zero, nonlinear effects become important as ak increases. For very deep water, kh441,  g0 k ig0 k 0 kz ; 0;  ; rg e Z s s

½18

so that the water particles execute circular motions that decay exponentially with depth. The horizontal motion is in phase with the surface displacement, and the phase speed of the waves c ¼ ðg0 =kÞ

1=2

¼

hg k

1 þ Gk2 =rg

i1=2

½19

These deep-water waves are dispersive; that is, the phase speed is a function of the wavenumber as shown in Figure 2. The influence of surface tension relative to gravity is determined by the value of the dimensionless parameter S ¼ Gk2 =rg. When S ¼ 1, the wavelength l ¼ 1:7cm and the phase speed is a minimum at c ¼ 23 cm s1 . When S441, surface tension is the dominant restoring force, the wavelength is less than 1.7 cm, and the phase speed increases as the wavelength decreases. When Soo1, gravity is the dominant restoring force, the

½13 10

½14

ig0 ak sinh ðz þ hÞ iðkxstÞ e w¼ s cosh kh

½15

_

v¼0

c (m s 1)

  @f G @2Z @2Z þ þ gZ ¼ @t r @x2 @y2

5

1/2

c = (g/k + Γk / ρ)

1

and the pressure p ¼ rg0 Z

cosh ðz þ hÞ cosh kh

½16

The velocity decays with depth away from the surface, and, to leading order, elements of fluid execute elliptical orbits as the waves propagate. For shallow water, khoo1,  0  gk 0 ; 0; 0; rg Z ðu; v; w; pÞ ¼ s

½17

0.1 0.001

0.01

0.1

1

10

100

λ (m)

Figure 2 The phase speed of surface gravity-capillary waves as a function of wavelength l. A minimum phase speed of 23 cm s1 occurs for l ¼ 0.017 m. Shorter waves approach pure capillary waves, whereas longer waves become pure gravity waves. Note that there are both capillary and gravity waves for a given phase speed. This is the basis of the generation of parasitic capillary waves on the forward face of steep gravity waves.

6

SURFACE GRAVITY AND CAPILLARY WAVES

wavelength is greater than 1.7 cm, and the phase speed increases as the wavelength increases.

so waves appear at the front of the group and disappear at the rear of the group as it propagates. For shallow water gravity waves, khoo1, cg ¼ c.

The Group Velocity Using the superposition principle over a continuum of wavenumbers a general disturbance (in two spatial dimensions) can be represented by Zðx; tÞ ¼

Z

N

aðkÞeiðkxstÞ dk

½20

N

where, as above, only the real part of the integral is taken. Assuming the disturbance is confined to wavenumbers in the neighbourhood of ko , and expanding sðkÞ about ko gives sðkÞ ¼ sðko Þ þ ðk  ko Þ

ds þy dk k¼k0

½21

whence Zðx; tÞ 6e

iðko xsðko ÞtÞ

Second Order Quantities The energy density (per horizontal surface area) of surface waves is 1 E ¼ rg0 a2 2

½26

being the sum of the kinetic and potential energies. In the case of gravity waves, the potential energy results from the displacement of the surface about its equilibrium horizontal position. For capillary waves, the potential energy arises from the stretching of the surface against the restoring force of surface tension. The mean momentum density M is given by 1 E M ¼ rsa2 coth khe ¼ e 2 c

Z

N

aðkÞeiðkko Þðxcg tÞ dk þ y

½22

  ds cg ¼   dk

½23

N

where

k¼ko

is the group velocity. Eqn [22] demonstrates that the modulation of the pure harmonic wave propagates at the group velocity. This implies that an isolated packet of waves centered around the wavenumber ko will propagate at the speed cg , so that an observer wishing to follow waves of the same length must travel at the group velocity. Since the energy density is proportional to a2 (see below), it is also the speed at which the energy propagates. These properties of the group velocity apply to linear waves, and more subtle effects may become important at large slopes. In general, cg ac. For deep-water gravity waves, 1 1g1=2 cg ¼ c ¼ 2 2 k

½24

so the wave group travels at half the phase speed, with waves appearing at the rear of a group propagating forward and disappearing at the front of the group. For deep-water capillary waves, s2 ¼ Gk3 =r;

c ¼ ðGk=rÞ1=2 ;

3 cg ¼ c 2

½25

½27

where the unit vector e ¼ k=k. To leading order, linear gravity waves transfer energy without transporting mass; however, there is a second order mass transport associated with surface waves. In a Lagrangian description of the flow it can be shown that for irrotational inviscid wave motion the mean horizontal Lagrangian velocity (Stokes drift) of a particle of fluid originally at z ¼ zo is ul ¼ ska2

cosh2kðz þ hÞ 2sinh2 kh

e

½28

which reduces to ðakÞ2 ce2kzo e when kh441. This second order velocity arises from the fact that the orbits of the particles of fluid are not closed. Integrating eqn [28] over the depth it can be shown that this mean Lagrangian velocity accounts for the wave momentum M in the Eulerian description. The Stokes drift is important for representing scalar transport near the ocean surface, but this transport is likely to be significantly enhanced by the intermittent larger velocities associated with wave breaking. Longer waves, or swell, from distant storms can travel great distances. An extreme example is the propagation of swell along great circle routes from storms in the Southern Ocean to the coast of California. For waves to travel so far, the effects of dissipation must be small. In deep water, where the wave motions have decayed away to negligible levels at depth, the contributions to the dissipation come from the thin surface boundary layer and the rate of

7

SURFACE GRAVITY AND CAPILLARY WAVES

strain of the irrotational motions in the bulk of the fluid. It can be shown that the integral is dominated by the latter contributions, and the timescale for the decay of the wave energy is just  te ¼ 

1 dE E dt

1

 1 ¼ 4k2

½29

2

or s=8pnk wave periods. This gives negligible dissipation for long-period swell in deep water over scales of the ocean basins. More realistic models of wave dissipation must take into account breaking and near surface turbulence which is sometimes parameterized as a ‘super viscosity’ or eddy viscosity, several orders of magnitude greater than the molecular value. When waves propagate into shallow water, the dominant dissipation may occur in the bottom boundary layer. Eqn [27] shows that dissipation of wave energy is concomitant with a reduction in wave momentum, but since momentum is conserved, the reduction of wave momentum is accompanied by a transfer of momentum from waves to currents. That is, net dissipative processes in the wave field lead to the generation of currents.

Waves on Currents: Action Conservation Waves propagating in varying currents may exchange energy with the current, thus modifying the waves. Perhaps the most dramatic examples of this effect come when waves propagating against a current become larger and steeper. Examples occur off the east coast of South Africa as waves from the Southern Ocean meet the Aghulas Current; as North Atlantic storms meet the northward flowing Gulf Stream, or at the mouths of estuaries as shoreward propagating waves meet the ebb tide. For currents U ¼ ðU; VÞ that only change slowly on the scale of the wavelength, and a surface displacement of the form Z ¼ aðx; y; tÞeiyðx;y;tÞ

½30

where a is the slowly varying amplitude and y is the phase. The absolute local frequency ¼ @y=@t, and the x- and y-components of the local wavenumber are given by k ¼ @y=@x; l ¼ @y=@y: The frequency seen by an observer moving with the current U is 

@y þ U:ry  @t

 ½31

which is equal to the intrinsic frequency s. Thus s ¼ o  U:k

½32

which is just the Doppler relationship. We also have, @k þ ro ¼ 0; @t

½33

which can be interpreted as the conservation of wave crests, where k is the spatial density of crests and o the wave flux. The velocity of a wave packet along rays is dxi @s ¼ Ui þ ¼ Ui þ cgi dt @xi

½34

which is simply the vector sum of the local current and the group velocity in a fluid at rest. Furthermore, refraction is governed by @Uj @s dki ¼ kj  dt @xi @xi

½35

where the first term on the right represents refraction due to the current and the second is due to gradients in the waveguide, such as changes in the depth. It is this latter term which results in waves, propagating from deep water towards a beach, refracting so that they propagate normal to shore. For steady currents, the absolute frequency is constant along rays but the intrinsic frequency may vary, and the dynamics lead to a remarkable and quite general result for linear waves. If E is the energy density then the quantity A ¼ E=s, the wave action, is conserved: 

@A @  Ui þ cgi A ¼ 0 þ @t @xi

½36

In other words, the variations in the intrinsic frequency s and the energy density E, are such as to conserve the quotient. This theory permits the prediction of the change of wave properties as they propagate into varying currents and water depths. For example, in the case of waves approaching an increasing counter current, the waves will move to shorter wavelengths (higher k), larger amplitudes, and hence greater slopes, ak. As the speed of the adverse current approaches the group velocity, the waves will be ‘blocked’ and be unable to propagate further. In this simplest theory, a singularity occurs with the wave slope becoming infinite, but higher order effects lead to reflection of the waves and the same blocking effect. This theory also forms the basis of models of long-wave–short-wave

8

SURFACE GRAVITY AND CAPILLARY WAVES

interaction that are important for wind-wave generation and the interpretation of remote sensing measurements of the ocean surface, including the remote sensing of long nonlinear internal waves.

however, instabilities of the two-dimensional soliton solutions, and the effects of higher-order nonlinearities, random phase and amplitude fluctuations in real wave fields give pause to the applicability of these idealized theoretical results.

Nonlinear Effects The nonlinearity of surface waves is represented by the wave slope, ak. For typical gravity waves at the ocean surface the average slope may be Oð102  101 Þ; small, but not negligibly so. Nonlinear effects may be weak and can be described as a perturbation to the linear wave theory, using the slope as an expansion parameter. This approach, pioneered by Stokes in the mid-nineteenth century, showed that for uniform approach deep-water gravity waves,   s2 ¼ gk 1 þ a2 k2 þ y ;

½37

and Z ¼ a cos y þ

1 2 a k cos 2y þ y 2

½38

Weakly nonlinear gravity waves have a phase speed greater than linear waves of the same wavelength. The effect of the higher harmonics on the shape of the waves leads to a vertical asymmetry with sharper crests and flatter troughs. The largest such uniform wave train has a slope of ak ¼ 0:446 a phase speed of 1.11c, and a discontinuity in slope at the crest containing an included angle of 1201. This limiting form has sometimes been used as the basis for the models of wave breaking; however, uniform wave trains are unstable to side-band instabilities at significantly lower slopes, and it is unlikely that this limiting form is ever achieved in the ocean. With the assumption of both weak nonlinearity and weak dispersion (or small bandwidth, dk/ kooo1), it may be shown that if Zðx; y; tÞ ¼ Re ½Aðx; y; tÞeiðko xso tÞ 

½39

where so ¼ sðko Þ and Re means that the real part is taken, then the complex wave envelope Aðx; y; tÞ satisfies a nonlinear Schro¨dinger equation or one of its variants. Solutions of the nonlinear Schro¨dinger equation for initial conditions that decay sufficiently rapidly in space evolve into a series of envelope solitons and a dispersive tail. Solitons propagate as waves of permanent form and survive interactions with other solitons with just a change of phase. Attempts have been made to describe ocean surface waves as fields of interacting envelope solitons;

Resonant Interactions Modeling the generation, propagation, interaction, and dissipation of wind-generated surface waves is of great importance for a variety of scientific, commercial and social reasons. A rigorous theoretical foundation for all components of this problem does not yet exist, but there is a rational theory for weakly nonlinear wave–wave interactions. For linear waves freely propagating away from a storm, the spectral content at any later time is explicitly defined by the initial storm conditions. For a nonlinear wave field, wave–wave interactions can lead to the generation of wavenumbers different from those comprising the initial disturbance. For surface gravity waves, these nonlinear effects lead to the generation of waves of lower and higher wavenumber with time. The timescale for this evolution in a random homogeneous wave field is of the order of (ak)4 times a characteristic wave period; slow, but significant over the life of a storm. The foundation of weakly nonlinear interactions between surface waves is the resonant interaction between waves satisfying the linear dispersion relationship. It is a simple consequence of quadratic nonlinearity that pairs of interacting waves lead to the generation of waves having sum and difference frequencies relative to the original waves. Thus k3 ¼ 7k1 7k2 ; s3 ¼ 7s1 7s2

½40

If in addition, si ði ¼ 1; 2; 3Þ satisfies the dispersion relationship, then the interaction is resonant. In the case of surface waves, the nonlinearities arise from the surface boundary conditions, and resonant triads are possible for gravity capillary waves, and gravity waves in water of intermediate depth. For deep-water gravity waves, cubic nonlinearity is required before resonance occurs between a quartet of wave components: k1 7k2 7k3 7k4 ¼ 0; s1 7s2 7s3 7s4 y ¼ 0;

si ¼ ðgki Þ1=2

½41

These quartet interactions comprise the basis of nonlinear wave–wave interactions in operational models of surface gravity waves. Exact resonance is not required, since even with detuning significant energy transfer can occur across the spectrum. The

SURFACE GRAVITY AND CAPILLARY WAVES

formal basis of these theories may be cast as problems of multiple spatial and temporal scales, and higher-order interactions should be considered as these scales increase, and the wave slope increases.

Parasitic Capillary Waves The longer gravity waves are the dominant waves at the ocean surface, but recent developments in air–sea interaction and remote sensing, have placed increasing importance on the shorter gravity-capillary waves. Measurements of gravity-capillary waves at sea are very difficult to make and much of the detailed knowledge is based on laboratory experiments and theoretical models. Laboratory measurements suggest that the initial generation of waves at the sea surface occurs in the gravity-capillary wave range, initially at wavelengths of O(1) cm. As the waves grow and the fetch increases, the dominant waves, those at the peak of the spectrum, move into the gravity-wave range. A simple estimate of the effects of surface tension based on the surface tension parameter S using the gravity

9

wavenumber k would suggest that they are unimportant, but as the wave slope increases and the curvature at the crest increases, the contribution of the Laplace pressure near the crest increases. A consequence is that so-called parasitic capillary waves may be generated on the forward face of the gravity wave (Figure 3). The source of these parasitic waves can be represented as a perturbation to the underlying gravity wave caused by the localized Laplace pressure component at the crest. This is analogous to the ‘fish-line’ problem of Rayleigh, who showed that due to the differences in the group velocities, capillary waves are found ahead of, and gravity waves behind, a localized source in a stream. In this context the capillary waves are considered to be steady relative to the crest. The possibility of the direct resonant generation of capillary waves by perturbations moving at or near the phase speed of longer gravity waves is implied by the form of the dispersion curve in Figure 2. Free surfaces of large curvature, as in parasitic capillary waves, are not irrotational and so the effects of viscosity in transporting vorticity and dissipating energy must be accounted for. Theoretical and

(A)

(B)

(C)

(D)

Figure 3 (A)–(D) Evolution of a gravity wave towards breaking in the laboratory. Note the generation of parasitic capillary waves on the forward face of the crest. (Reproduced with permission from Duncan JH et al. (1994) The formation of a spilling breaker. Physics of Fluids 6: S2.)

10

SURFACE GRAVITY AND CAPILLARY WAVES

numerical studies show that the viscous dissipation of the longer gravity waves is enhanced by one to two orders of magnitude by the presence of parasitic capillary waves. These studies also show that the observed high wavenumber cut-off in the surface wave spectrum that has been observed at wavelengths of approximately O(103–102) m can be explained by the properties of the spectrum of parasitic capillary waves bound to short steep gravity waves.

(A)

(B)

Wave Breaking Although weak resonant and near-resonant interactions of weakly nonlinear waves occur over slow timescales, breaking is a fast process, lasting for times comparable to the wave period. However, the turbulence and mixing due to breaking may last for a considerable time after the event. Breaking, which is a transient, two-phase, turbulent, free-surface flow, is the least understood of the surface wave processes. The energy and momentum lost from the wave field in breaking are available to generate turbulence and surface currents, respectively. The air entrained by breaking may, through the associated buoyancy force on the bubbles, be dynamically significant over times comparable to the wave period as the larger bubbles rise and escape through the surface. The sound generated with the breakup of the air into bubbles is perhaps the dominant source of high frequency sound in the ocean, and may be used diagnostically to characterize certain aspects of air–sea interaction. Figure 4 shows examples of breaking waves in a North Atlantic storm. Since direct measurements of breaking in the field are so difficult, much of our understanding of breaking comes from laboratory experiments and simple modeling. For example, laboratory experiments and similarity arguments suggest that the rate of energy loss per unit length of the breaking crest of a wave of phase speed c is proportional to rg1 c5 , with a proportionality factor that depends on the wave slope, and perhaps other parameters. Attempts are underway to combine such simple modeling along with field measurements of the statistics of breaking fronts to give an estimate of the distribution of dissipation across the wave spectrum. Recent developments in the measurement and modeling of breaking using optical, acoustical microwave and numerical techniques hold the promise of significant progress in the next decade.

See also Breaking Waves and Near-Surface Turbulence. Heat and Momentum Fluxes at the Sea Surface. Internal Waves. Langmuir Circulation and Instability. Surface Films. Wave Energy. Wave Generation by Wind. Whitecaps and Foam. (C) Figure 4 Waves in a storm in the North Atlantic in December 1993 in which winds were gusting up to 50–60 knots and wave heights of 12–15 m were reported. Breaking waves are (A) large, (B) intermediate and (C) small scale. (Photographs by E. Terrill and W.K. Melville; reproduced with permission from Melville, (1996).)

Further Reading Komen GJ, Cavaleri L, Donelan M, et al. (1994) Dynamics and Modelling of Ocean Waves. Cambridge: Cambridge University Press.

SURFACE GRAVITY AND CAPILLARY WAVES

Lamb H (1945) Hydrodynamics. New York: Dover Publications. LeBlond PH and Mysak LA (1978) Waves in the Ocean. Amsterdam: Elsevier. Lighthill J (1978) Waves in Fluids. Cambridge: Cambridge University Press. Mei CC (1983) The Applied Dynamics of Ocean Surface Waves. New York: John Wiley. Melville WK (1996) The role of wave breaking in air–sea interaction. Annual Review of Fluid Mechanics 28: 279--321.

11

Phillips OM (1977) The Dynamics of the Upper Ocean. Cambridge: Cambridge University Press. Whitham GB (1974) Linear and Nonlinear Waves. New York: John Wiley. Yuen HC and Lake BM (1980) Instability of waves on deep water. Annual Review of Fluid Mechanics 12: 303--334.

WAVE GENERATION BY WIND J. A. T. Bye, The University of Melbourne, Melbourne, VIC, Australia A. V. Babanin, Swinburne University of Technology, Melbourne, VIC, Australia & 2009 Elsevier Ltd. All rights reserved.

Introduction The prime focus in this article is on ocean waves (which have always captured the scientific imagination), although results from wind-wave tank studies are also introduced wherever appropriate. Growth mechanisms fall naturally into three phases: (a) the onset of waves on a calm sea surface, (b) mature growth in the confused sea state under moderate winds, and (c) sea-spray-dominated wave environments under very high wind speeds. Of these three phases, (b) has the greatest general importance, and numerous practical formulas have been developed over the years to represent its properties. Figure 1 illustrates the sea state which occurs at the top end of phase (b) in a strong gale (wind speed c. 25 ms  1, Beaufort force 9). An important consideration is that wave generation by wind involves three main physical processes: (1) direct input from the wind, (2) nonlinear transfer between wavenumbers, and (3) wave dissipation. This article is specifically dedicated to (1); however, we briefly review (2) and (3) below. Nonlinear interactions within the wave system can only be neglected for infinitesimal waves. To a first approximation, the wind wave can be regarded as almost sinusoidal with negligible steepness (i.e., linear), but its very weak mean nonlinearity (i.e., finite steepness and deviation of its shape from the sinusoid) is generally believed to control the evolution of the wave

Figure 1 The sea state during a strong gale.

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field. Theoretical models of the air–sea boundary layer indicate that the input of momentum from the wind is centered in the short gravity waves. The wind pumps energy mostly into short (high-frequency) and slowly moving waves of the wave field which then transfer this energy across the continuous spectrum of waves of all scales mainly toward longer (lower-frequency) components, which may be traveling at speeds close to the wind speed, thus allowing them to grow into the dominant waves of frequencies close to the peak frequency of the wave (energy) spectrum. The transfer of energy toward shorter (higher-frequency) waves where it is dissipated occurs at a much less significant rate. Wave breaking is the major player in the third important mechanism, which drives wave evolution – wave energy dissipation. The Southern Ocean has the greatest potential for wave growth due to the never ceasing progression of intense storm systems over vast expanses of sea surface, unimpeded by land masses. Yet, wave models (http://www.knmi.nl/waveatlas/) indicate that the significant wave height (the average crest-to-trough height of the one-third highest waves) rarely goes beyond 10 m. The process, which controls the wave growth, is the dissipation by wave breaking, and to a lesser extent radiation of wave energy away from the storm centers, and into the adjacent seas.

Theories of Wave Growth Phase (a): The Onset of Waves

We consider firstly the initial generation of waves over a flat water surface, independently of the simultaneous generation of a surface drift current. The key theoretical result is that the initial wavelength which can be excited on the air–water interface is a wave of wavelength 17 mm, which is the capillary gravity wave of minimum phase speed 230 mm s  1, controlled by gravity and surface tension. The classical Kelvin–Helmholtz analysis completed in 1871, which relies on random natural disturbances present on the water surface, shows that this wave can only be excited by a velocity shear across the sea surface exceeding 6.5 m s  1. Observations, however, show that waves are generated at much lower wind speeds, of order 1–2 m s  1. In order to resolve this dilemma, another mechanism was proposed by Phillips in 1957. It takes into account the turbulent structure of wind flow. Turbulent pressure pulsations in the air create infinitesimal hollows and ridges in the water surface, which, once the

WAVE GENERATION BY WIND

pressure pulsation is removed, may start propagating as free waves (similarly to the waves from a thrown stone). If the phase speed of such free waves is the same as the advection speed of the pressure pulsations by the wind, a resonant coupling can occur which will then lead these waves to grow beyond the infinitesimal stage. The first wave to be generated as the wind speed increases is likely to be the wave of minimum phase speed, propagating at an angle to the wind direction. Laboratory observations indicate that at slightly higher wind speeds, wave growth results from a shear flow instability mechanism. These two processes acting in the open ocean give rise to cat’s paws, which are groups of capillary-gravity wavelets (ripples) generated by wind gusts. These results are applicable for clean water surfaces. In the presence of surfactants (surface-active agents), which lower the surface tension, ripple growth is inhibited, and at a sufficiently high surfactant concentration it may be totally suppressed. Phytoplankton are a major source of surfactants that produce surface films, and hence slicks, which are regions of relatively smooth sea surface. Phase (b): Mature Growth

Once the finite-height waves exist, other and much more efficient processes take over the air–sea interaction. Jeffreys in 1924 and 1925 pioneered the analytical research of the wind input to the existing waves by employing effects of the wave-induced pressure pulsations in the air. Potential theory predicts such pressure fluctuations to be in antiphase with the waves, which results in zero average momentum/ energy flux. Jeffreys hypothesised a wind-sheltering effect due to presence of the waves which causes a

13

shift of the induced pressure maximum toward the windward wave face and brings about positive flux from the wind to the waves. The original theory of Jeffreys was based on an assumed phenomenon of the air-flow separation over wave crests. Experiments conducted between 1930 and 1950 with wind blown over solid waves found such an effect to be small and the theory fell into a long disrepute. Jeffreys’ sheltering ideas are now coming back, with both experimental and theoretical evidence lending support to his qualitative conclusions. The period from 1957 until the beginning of the new century was dominated by the Miles theory (MT) of wave generation. This linear and quasi-laminar theory, originally suggested by Miles, was later modified by Janssen to allow for feedback changes of the airflow due to growing wind-wave seas. MT regards the air turbulence to be important only in forming the mean boundary-layer wind profile. In such a profile, a critical height exists where the wind speed equals the phase speed of the waves (Figure 2). Wave-induced air motion at this height leads to waterslope-coherent air-pressure perturbations at the water surface and hence to energy transfer to the waves. MT however fails to comprehensively describe known features of the air–sea interaction. For example, for adverse winds the critical height does not exist and therefore no wind-wave energy transfer is expected, but attenuation of waves by such winds is observed. Therefore, a number of nonlinear and fully turbulent alternatives have been developed over the past 40 years. One of the most consistent fully turbulent approaches is the two-layer theory first suggested by Townsend, and advanced by Belcher and Hunt (TBH). TBH revives the sheltering idea in a new form: by considering perturbations of the turbulent shear

U(Z ) − c z x

Direction of wave propagation

Figure 2 Mean streamlines in the turbulent flow over waves according to the MT, in a frame of reference moving with the wave. The critical layer occurs at the height (Z) where the wave speed (C) equals the wind speed (U(Z)). Reproduced from Phillips OM (1966) The Dynamics of the Upper Ocean, figure 4.3. Cambridge, UK: Cambridge University Press, with permission from Cambridge University Press.

14

WAVE GENERATION BY WIND

stresses, which are asymmetric along the wave profile. While still in need of experimental verification, particularly for realistic non-monochromatic threedimensional wave fields, this theory has been extensively and successfully utilized in phase-resolvent numerical simulations of the air–sea interaction by Makin and Kudryavtsev. TBH and similar theories attract serious attention because the nature of the air–sea interface is often nonlinear and always fully turbulent. Air–sea interaction is also superimposed by a variety of physical phenomena, which alter the wave growth. Wave breaking appears to cause air-flow separation, which brings the ideas of Jeffreys back in their original form; and gustiness and nonstationarity of the wind, the presence of swell and wave groups, nonlinearity of wave shapes, modulation of surface roughness by the longer waves have all been found to cause either a reduction or an enhancement of the wind-wave input. These processes of active wave generation give rise to the windsea in which a simple measure of the sea state, relevant to wave growth, is the wave age (c=u ) where c is the wave speed of the dominant waves, and u is the friction velocity in the air (the square root of the wind stress divided by the air density). The age of the windsea increases with fetch (the distance from the coast over which the wind is blowing), and the windsea becomes ‘fully developed’, that is, the energy flux from the wind and the dissipation flux are in balance, at a wave age of about 35. Empirical relations for the properties of the fully developed sea in terms of the wind speed (U) at 10 m (approximately the height of the bridge on large ships) given by Toba are: Hs ¼ 0.30U2/g and Ts ¼ 8.6U/g in which Ts ( ¼ 2pc/g) is the significant wave period and g is the acceleration of gravity. As the fetch increases, Hs and Ts both increase toward their fully developed values, and the wave spectrum spreads to lower frequencies. Older seas of wave age greater than 35 can also exist after the wind has moderated. The observations of the velocity structure in the atmospheric boundary layer by Hristov, Miller, and Friehe have shown directly the existence of the MT critical layer mechanism for fast-moving waves of wave age about 30. It is not yet known whether it operates for younger wave age, where a quasilaminar theory may not be appropriate.

the two fluids. In recent times, it has been realized that this model is inadequate, especially at very high wind speeds. The link between the two phases is the breaking wave. In moderate winds (less than about 25 m s  1) the sea state is characterized by whitecapping due to the production of foam in a roller on the wave crests, and also foam streaks on the sea surface (Figure 1), whereas at very high wind speeds (greater than about 30 m s  1, Beaufort force 12) the air is filled with foam. This transition arises from the structure of the breaking waves. In moderate winds, the roller remains attached to the parent wave and dissipates by the formation of foam streaks down its forward face, the trailing face of the wave remaining almost foam free. In this situation the airflow separates over the troughs and reattaches at the crests of the wave, producing Jeffreys-like phase shifts between the pressure and the underlying wave surface which enhance the energy flux to the wave. At very high wind speeds, on the other hand, the foam detaches from the wave crests, and is jetted forward into the air where it disperses vertically and horizontally before returning to the water surface. This process implies a return of momentum to the atmosphere, and hence the sea surface drag coefficient (which is an overall measure of the efficiency of momentum transfer from the atmosphere to the ocean both to waves and turbulence), which has been rising in phase (b), becomes ‘capped’ and possibly even reduces in phase (c). The all-pervasive presence of spray in extreme winds has prompted the anecdotal statement that ‘‘in hurricane conditions the air is too thick to breathe and too thin to swim in.’’ In summary, at very high wind speeds, the airflow effectively streams over the wave elements, which are reduced to acting as sources of spray. The spray then stabilizes the wind profile, and caps the sea surface drag coefficient, and interestingly, this feedback most likely allows the hurricanes to exist in the first place. This analysis has been greatly stimulated by the dropwindsonde observations of Powell, Vickery, and Reinhold in which wind profiles in hurricanes were measured for the first time, and also subsequently by experiments in high-wind-speed wind-wave tanks.

Experiments and Observations Phase (c): Very High Wind-Speed Wave Environments

The processes discussed in the previous two subsections are all grounded in two-layer fluid dynamics in which there exists a sharp interface between

Direct Measurements of Wave Growth Rates

The wind-to-wave energy input, which for each wave component is proportional to the time average of the product of the sea surface slope and sea surface atmospheric pressure, is the only source function,

WAVE GENERATION BY WIND

responsible for wave development, which can so far be measured directly, although this is an extremely difficult experimental task, and only a handful of attempts have been undertaken. The principal theoretical difficulty is that the sea surface atmospheric pressure must be estimated by extrapolating downward from the measurement level. The pressure pulsations of interest are of the order of 10  5  10  4 of the mean atmospheric pressure and therefore require very sensitive probes. The surface-coherent oscillations are superposed, at the same frequencies, by random turbulent fluctuations, which are tens and hundreds of times greater in magnitude. This implies that a sophisticated data analysis is required to separate the signal buried in the noise. The wave-induced pressure decays rapidly away from the wavy surface and thus, particularly for short wave scales, it has to be sensed very close to the surface, below the wave crests of dominant waves. At the same time, the air-pressure probes have to stay dry. The last requirement leads either to measurements being conducted above the crests, which limits the estimates to the amplification of the dominant waves only, or to the use of a wavefollowing technique. The latter has a limited capability beyond the laboratory conditions and involves further complications due to multiple corrections needed to recover the signal contaminated by air motion in the tubes connecting the pressure probes with pressure transducers. The first field experiment of the kind, conducted by Snyder and others in 1981, resulted in a parameterization of wind input across the wave spectrum, which has been frequently used until now. Most of these measurements, however, were taken by stationary wave probes above the wave crests, and the winds involved were very light, mostly around 4 m s  1. Waves at such winds are known not to break, and this fact implies an air–sea energy balance, very different from that at moderate and strong winds. Therefore, extrapolation of these results into normal wave conditions has to be exercised with great caution. Another field experiment was conducted by Hsiao and Shemdin in 1983. It used a wave-following technology and thus was able to obtain a spectral set of measurements somewhat beyond the dominant wave scales. This study produced a parameterization in which the growth rates were very low. Its drawback comes from the fact, that in the majority of circumstances the measured waves were ‘quite old’, half of the records being above the limit for the fully developed windsea. For such waves, the growth rates are expected to be very small if not zero, and given the measurement and analysis errors, the

15

interpretation of the low growth values becomes quite uncertain. On the other hand, a set of precision wavefollowing measurements conducted by Donelan in 1999 in a wind-wave tank where the waves were very young (c/uE1), produced a growth rate 2.5 times that of Hsiao and Shemdin’s, and also demonstrated a very significant wave attenuation rate by the adverse wind. The differences between these two data sets stimulated the latest campaign undertaken by Donelan and others in 2006. The Lake George experiment in Australia employed precision laboratory instruments in a field site. The site was chosen such that it provided a variety of wind-wave conditions, including very strongly wind-forced and very steep waves normally unavailable for measuring in the open ocean. The results revealed some new properties of the air–sea interaction, in which wave growth rates merged with previous results at moderate winds, but deviated significantly in strong wind conditions with continually breaking steep waves, in which full flow separation, that is, detachment of the streamlines of the airflow at the wave crest and reattachment well up the windward face of the preceding wave, occurred leading to a reduction of the wind input. This reduction means that as the winds become stronger the wind-to-wave input will keep growing, but the growth rates will be reduced compared to simple extrapolations to extreme conditions of the input measured at moderate winds. This behavior, which is consistent with that in very high wind speeds in the open ocean, did not appear to be associated with spray production. It is worth mentioning that one of the key properties of the wind input – its directional distribution – has never been measured. It was assumed to be a cosine function by Plant, but no data on the wind input directional distribution are available. Such measurements cannot be made adequately in a windwave tank, and are a formidable task in the field where a spatial array of wave-following pressure probes would have to be operated. Directional wave input distribution, nevertheless, is an integral part of any wave forecast model and therefore this problem remains a major challenge for the experimentalists.

Reverse Momentum Transfer

Nonlinear interactions transfer energy to the longer, faster-propagating waves, which after leaving the region of generation are known as swell. The swell may travel at a speed greater than the local wind speed, and even propagate in the opposite direction

16

WAVE GENERATION BY WIND

to the wind, leading to the possibility of reverse momentum transfer from the waves to the wind. The direct effect of waves propagating faster than the wind has been measured in a wind-wave tank by Donelan; however, when the results were applied to swell propagating in the ocean, the damping effect was found to be much too large. This is well known to surfers, who rely on the arrival of swells from distant storms: their propagation across the Pacific Ocean (over a distance of c. 10 000 km) was measured in a classical campaign conducted by Snodgrass and others in 1963. Reverse momentum transfer has also been observed in wind profiles. In Lake Ontario, while a swell was running against a very light wind, the wind speed increased downward (toward the sea surface) due to the propagation of the swell, rather than the normal decrease. This is a clear example of reverse momentum transfer arising from the presence of a wave train of nonlocal origin. Reverse momentum transfer, however, is a ubiquitous process in windseas in which part of the wind input is returned to the atmosphere by the dissipation process, especially the injection of spray.

Numerical Modeling of the Wind Input Over the past few decades, numerical modeling of ocean waves has developed into a largely independent field of study. Two different kinds of models have been used to study the wind input. Historically, spectral models based on known physics were the first. Their progress is described in great detail in the book by Komen and others. Given the uncertainties of such predictions due to simultaneous action of the multiple wave dynamics processes, the capacity of such models to scrutinize the wind input function is limited. For example, very high quality synoptic analyses of weather systems are necessary in order to discriminate between the various coupling mechanisms for wave growth by comparing observational wave data from wave buoys with the predictions of coupled wind-wave simulations. In these models the formulation of the wave energy dissipation is based on tuning the total energy balance. Csanady, in a lucid textbook on air–sea interaction, notes that in a fetch-limited windsea only about 6% of the momentum transferred from the wind to the water supports the downwind growth of the dominant waves, the remainder being accounted for locally by the dissipation stress, that is, the rate of loss of momentum from the wave field to the ocean. The phase-resolvent models are another kind of numerical simulations of air–sea interaction, which

reproduce wind input and wave evolution in physical rather than wavenumber space. Such models solve the basic fully nonlinear equations of fluid mechanics explicitly and recent advances in numerical techniques allow us to reproduce the water surface, airflow, and wave motion with potentially absolute precision and unlimited temporal and spatial resolution. Use of such models to forecast waves globally is obviously not feasible, but they now constitute a very effective tool for dedicated studies of wind–wave interaction. The interested reader is referred to recent research by Makin and Kudryavtsev, and by Chalikov and Sheinin.

Conclusions It is clear from this article that there are still many tasks ahead to fully understand wave generation by wind. The nonlocal aspects of wave generation by wind are a particularly challenging topic. Contemporary interest lies with our climate system. The interface between the atmosphere and the ocean is vital in this regard. This holistic view calls urgently for further study, especially of extreme events in which major momentum transfers occur, affecting the land through the initiation of hurricanes, and the sea through mixing below the wave boundary layer into the deep ocean.

See also Breaking Waves and Near-Surface Turbulence. Surface Gravity and Capillary Waves. Tsunami. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading Belcher SE and Hunt JCR (1998) Turbulent flow over hills and waves. Annual Review of Fluid Mechanics 30: 507--538. Bye JAT and Jenkins AD (2006) Drag coefficient reduction at very high wind speeds. Journal of Geophysical Research 111: C03024 (doi:10.1029/2005JC003114). Chalikov D and Sheinin D (2005) Modeling extreme waves based on equations of potential flow with a free surface. Journal of Computational Physics 210: 247--273. Csanady GT (2001) Air–Sea Interaction Laws and Mechanisms, 239pp. Cambridge, UK: Cambridge University Press. Donelan MA (1999) Wind-induced growth and attenuation of laboratory waves. In: Sajjadi SG, Thomas NH, and Hunt JCR (eds.) Wind-Over-Wave Couplings: Perspectives and Prospects, pp. 183--194. Oxford, UK: Clarendon.

WAVE GENERATION BY WIND

Donelan MA, Babanin AV, Young IR, and Banner ML (2006) Wave follower measurements of the wind input spectral function. Part 2: Parameterization of the wind input. Journal of Physical Oceanography 36: 1672--1688. Hristov T, Friehe C, and Miller S (2003) Dynamical coupling of wind and ocean waves through waveinduced air flow. Nature 422: 55--58. Jones ISF and Toba Y (eds.) (2001) Wind Stress over the Ocean, 307pp. Cambridge, UK: Cambridge University Press. Komen GI, Cavaleri L, Donelan M, Hasselmann K, Hasselmann S, and Janssen PAEM (1994) Dynamics and Modelling of Ocean Waves, 532pp. Cambridge, UK: Cambridge University Press. Kudryavtsev VN and Makin VK (2007) Aerodynamic roughness of the sea surface at high winds. BoundaryLayer Meteorology 125: 289--303. Makin VK and Kudryavtsev VN (2003) Wind-over-waves coupling. In: Sajjadi SG and Hunt LJ (eds.) Wind Over Waves II: Forecasting and Fundamentals of Applications, pp. 46--56. Chichester: Horwood Publishing.

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Phillips OM (1966) The Dynamics of the Upper Ocean. Cambridge, UK: Cambridge University Press. Powell MD, Vickery PJ, and Reinhold TA (2003) Reduced drag coefficient for high wind speeds in tropical cyclones. Nature 422: 279--283. Snodgrass FE, Groves GW, Hasselmann KF, Miller GR, Munk WH, and Powers WH (1966) Propagation of ocean swell across the Pacific. Philosophical Transactions of the Royal Society of London 259: 431--497. Toba Y (1972) Local balance in the air–sea boundary processes. Part I: On the growth process of wind waves. Journal of the Oceanographical Society of Japan 28: 15--26. Young IR (1999) Wind Generated Ocean Waves, 288pp. Oxford, UK: Elsevier.

Relevant Website http://www.knmi.nl/waveatlas – The KNMI/ERA-40 Wave Atlas.

ROGUE WAVES Rogue waves are not mariners’ tales. They have been observed and documented, most succinctly from oil platforms. Two well-studied examples are the Draupner ‘New Year’s Wave’ and the Gorm platform waves discussed below (Figure 2). With their sometimes catastrophic impact the motivation for investigating rogue waves is clear, and the scientific community has studied the topic for some time, more intensely since 2000. Despite these efforts, there is no generally accepted explanation or theory for the occurrence of rogue waves. There is even no consensus of how to define a rogue wave. Some of the inherent difficulties are related to the random nature on ocean waves: a wave recording will show waves of different sizes and shapes. In discussing rogue waves one introduces the notion of ‘one wave’, which is the recorded elevation over one wave period, containing one crest and one trough. One also distinguishes between the wave height H (the distance from trough to crest) and the

K. Dysthe, University of Bergen, Bergen, Norway H. E. Krogstad, NTNU, Trondheim, Norway P. Mu¨ller, University of Hawaii, Honolulu, HI, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The terms ‘rogue’ or ‘freak’ waves have long been used in the maritime community for waves that are much higher than expected, given the surrounding sea conditions. For the seafarer these unexpected waves represent a frightening and often life-threatening experience. There are many accounts of such waves hitting passenger and container ships, oil tankers, fishing boats, and offshore and coastal structures, sometimes with catastrophic consequences. It is believed that more than 22 supercarriers were lost to rogue waves between 1969 and 1994 (Figure 1).

Norse Variant

Anita

Christinaki

Marina di Equa

Tito Campanella

Artemis

Mar. 1973

Mar. 1973

Feb. 1994

Dec. 1981

Jan. 1984

Dec. 1980

Deaths: 29

Deaths: 32

Deaths: 28

Deaths: 20

Deaths: 27

Deaths: 0

Silvia Ossa

Arctic Ocean

Sandalion

Arctic Ocean

Nov. 1980

Oct. 1976

60°

Deaths: 37

Skipper 1 Apr. 1987

North Atlantic Ocean

North

North America Pacific Ocean Tropic of Cancer

North Pacific 30°N Ocean

Equator

0°

South America

Tropic of Capricorn

Mar. 1991

South Pacific Ocean

Indian Ocean South Atlantic Ocean

Australia

The World

Deaths: 24

Deaths: 30

30°S

60°

Antonis Demades Feb. 1970 Deaths: 0 Antparos Jan. 1981 Deaths: 31

Antarctica 150° 120° 90° 60° 30° W 0° 30° E 60° 90° 120° 150° 180°

Alborada Jul. 1984

Deaths: 0

Asia

Africa

Deaths: 0 Mezada

Europe

Testarossa

Rhodain Sailor

Golden Pine

Mar. 1973

Dec. 1982

Jan. 1981

Deaths: 30

Deaths: 5

Deaths: 25

Bolivar Maru Jan. 1969 Deaths: 31

Arctic Career

Chandragupta

Derbyshire

Dinav

Onomichi Maru

Jun. 1985

Jan. 1978

Dec. 1980

Dec. 1980

Dec. 1980

Deaths: 28

Deaths: 69

Deaths: 44

Deaths: 35

Deaths: 0

Figure 1 Locations of 22 supercarriers assumed to be lost after collisions with rogue waves between 1969 and 1994. & C. Kharif and E. Pelinovsky. Used with permission.

18

ROGUE WAVES

Gorm

15

15

10

10

5

5

0

0

−5

−5 0

50

Draupner

20

Height (m)

Height (m)

20

100

150

Time (s)

19

0

50

100 Time (s)

150

200

Figure 2 Two examples of rogue waves. ‘Gorm’ is one of the abnormal waves recorded at the Gorm field in the North Sea on 17 Nov.1984. The wave that stands out has a crest height of 11 m, which exceeds the significant wave height of 5 m by a factor of 2.2. ‘Draupner’ is the ‘New Year Wave’ recorded at the Draupner platform in the North Sea 1 Jan. 1995. The crest height is about 18.5 m and exceeds the significant wave height of 11.8 m by a factor of 1.54. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

crest height Zcr (the distance from mean sea level to crest). Early wave statistics from the 1950s suggests that the most probable maximum wave height, HM, in a wave record containing N waves is given by HM C

Hs pffiffiffiffiffiffiffiffiffiffiffiffi 2ln N 2

½1

where Hs is the significant wave height defined as four times the standard deviation of the surface elevation. (The old definition of significant wave height as the mean of the one-third largest waves, H1/3, is approximately 5% lower than Hs.) Thus, as the duration of the wave record increases, the expected maximum wave height increases as well, although quite slowly. In a constant sea state with a mean wave period of 10 s, eqn [1] predicts that the most probable maximum wave height reaches 2Hs after 8.3 h, whereas 2.5Hs needs an observation period of c. 1 month. One question is whether observed rogue waves are just rare or extreme events within standard statistical models, or whether they are due to some exceptional physical conditions not contained in these models. The above numbers demonstrate the challenge in discriminating between these two options: a wave that might be tagged as exceptional in a short wave record might turn out to be consistent with standard statistics in a longer record. In any case, rogue waves stick out of the sea states they appear in, as seen in Figure 2. The operational approach is to call a wave a rogue wave whenever the wave height, H, exceeds a certain threshold related to the sea state. This article follows this practice and uses the generally accepted criterion H=Hs > 2

½2

A wave observation far beyond any reasonable statistical expectation will be called an abnormal rogue wave.

Surface Gravity Waves Rogue waves are surface gravity waves. Linear waves in deep water are characterized by the dispersion relation o2 ¼ gk

½3

which relates the frequency o of the waves to their wavenumber, k, and the gravitational acceleration, g. In shallower water, the dispersion relation is o2 ¼ gk tanh(kd), where d is the water depth; the phase speed (for deep-water waves) is given by cp ¼ o/k ¼ g/o; and the group velocity (with which the energy travels) is given by cg ¼ @o/qk ¼ g/2o. Surface waves are generated by the wind, first as short ripples. Given sufficient time and distance offshore, longer and longer waves will dominate. The wave spectrum describes the distribution of wave energy with frequency and direction. As the wind continues to blow, the peak of the spectrum moves to lower frequencies and therefore faster phase speeds, until, in a fully developed sea, the waves at the peak have a phase speed close to the wind speed. The evolution of the spectrum is driven by the energy input from the wind, energy redistribution by nonlinear wave–wave interactions, and dissipation by wave breaking. Numerical models developed to predict the spectral evolution are used in wave forecasting. The spectral models predict the average partition of

20

ROGUE WAVES

energy among the different waves, from which statistical quantities like Hs can be found. They do not deal with the wave phases and therefore cannot give information about individual waves, let alone rogue waves.

Physical Mechanisms A rogue wave represents a very high concentration of wave energy. The energy of a wave of height H ¼ 2Hs is roughly a factor of 10 times larger than the average energy of the surrounding waves. The most important mechanisms capable of concentrating energy appear to be superposition and spatial, dispersive, and nonlinear focusing. Superposition

At any given location of the ocean, waves meet with varying wavelengths and directions. Occasionally several waves add up constructively to produce a much larger wave. This is a rough interpretation of the standard linear model which considers the surface to be a superposition of independent waves (see below). So-called second-order models only slightly modify this picture, while higher-order models take into account the weak resonant interactions among the waves.

focusing of wave energy in particular places and to extreme waves. Wave–current interaction. The simplest example is a wave propagating from still water into an opposing current. A wave of phase speed cp in still water can be completely blocked by an opposing current of only cp/4. Although storm waves with cpB15 m s  1 are not stopped, they will be retarded and their wavelengths reduced, when running against a current. The flux of wave action (energy divided by intrinsic frequency) is conserved implying that the current puts energy into the waves as it squeezes them. For intense current jets like parts of the Aguhlas current off the SE coast of Africa, the situation may be more serious. When storm waves or swell are going in the opposite direction to the jet, they may be trapped where the jet is widening or at meanders, as shown by ray tracing in Figure 3. The trapped waves are refracted toward the jet center, causing the oscillating paths with reflection (or caustics) near the edges as seen in the figure. At reflection, the wave amplitude is significantly amplified compared to its value at the jet center. If waves from neighboring rays add up constructively around reflection, the result may be a rogue wave. Dispersive Focusing

Spatial Focusing

Spatial focusing can be achieved by the refraction of waves by variable bottom topography or currents. Topographic focusing. As waves propagate into shallower water and their wavelength becomes comparable to the water depth, the waves refract, align their crests with the topography, and steepen. Along irregular coastlines, this might lead to (a)

Gravity waves are dispersive with phase and group velocities being inversely proportional to the frequency, that is, long waves traveling faster than short waves. This fact is utilized for producing a large wave at a given position d in a wave tank by creating a short wave train where the frequency decreases with time as o(t) ¼ o0  gt/2d, a so-called chirp (Figure 4). It has been shown that a chirped wave

(b)

Figure 3 (a) Rays of waves getting trapped when moving upstream into a widening current jet (green streamlines). (b) Trapping of waves at a current meander.

ROGUE WAVES

21

groups, producing some very large waves having a maximum surface elevation Zmax much larger than the initial amplitude a of the wave train. Simulations show enhancement factors Zmax/a between 3 and 4, while wave tank experiments show somewhat smaller values. Although impressive wave focusing can be achieved through this nonlinear effects, the initial states from which they develop are rather special and are not likely to occur spontaneously in a storm-generated wave field. For narrow-band random waves, however, it has been shown, using the nonlinear Schro¨dinger equation (NLS), that the BF instability persists provided the relative bandwidth d ¼ Do/o satisfies the criterion dos Figure 4 A chirped wave group, with the short waves in front of the long ones, contracts to a largewave. Then, the longer waves overtake the short ones giving a mirror image of the initial situation.

train that produces strong focusing in the absence of other waves will still do so when a random wave field is added. The chirp may actually be dwarfed by the random waves so that it remains invisible until it focuses. The dispersive focusing is basically a linear effect and occurs even in a linear Gaussian sea in those rare circumstances in which waves moving in the same direction happen to have the contrived phase relations necessary to form a chirped wave train. Physical mechanisms able to produce such phase relations and chirped wave trains have not been identified for the ocean.

where s is the steepness defined as s ¼ ka¯ and a¯ is the rms value of the amplitude. The ratio s/d is called the Benjamin–Feir index (BFI). Theory and simulations in one horizontal dimension have shown that wave spectra with BFI41 are unstable and develop on the timescale of s  2 wave periods toward marginal stability. While the instability develops, there is an increase in the population of extreme waves. This effect has also been verified experimentally in a wave flume. As will be pointed out below, three-dimensional, two horizontal dimensions and time simulations and recent wave basin experiments indicate that the effect only appears for very long-crested waves.

Statistics of Large Waves The Gaussian Sea

The Gaussian sea (or standard linear model) considers a random superposition of independent waves

Nonlinear Focusing

A regular unidirectional wave train of frequency o and amplitude a is known to be unstable to modulations. For deep-water waves, sidebands at o7Do will grow provided that Do pffiffiffi o 2ka o

½4

where a is the amplitude and k the wavenumber. This is the Benjamin–Feir (BF) instability. As the instability develops, the wave train disintegrates into wave groups on a timescale of (ka)  2 wave periods. The evolution of the instability has been studied both experimentally and by numerical simulations, starting with a regular wave train, seeded with sidebands at o7Do satisfying [4]. As the groups are formed, further focusing takes place within the

½5

Zðx; y; tÞ ¼

N X

an sinðkxn x þ kyn y  on t þ yn Þ ½6

n¼1

with amplitude an, horizontal wavenumber vector kn ¼ (kxn, kyn), frequency on ¼ o(kn) given by the dispersion relation, and random phases yn. Within limits, it does not matter from which distributions amplitude, wavenumber vector, and phase are drawn, the central limit theorem assures a Gaussian distribution of the surface elevation: 1 2 2 PðZÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi eZ =2s 2 2ps

½7

with zero mean and standard deviation s. Modern wave statistics and observations have modified and refined the standard linear model, taking into account nonlinearity to second order in the steepness s.

22

ROGUE WAVES

100

Simulation Tayfun Gaussian

PDF

10–2

10–4

10–6

10–8 –6

–4

–2

0 

2

4

6

Figure 5 Typical probability density function (PDF) of the sea surface elevation Z (scaled by the standard deviation s) as simulated by a third-order numerical model (black), compared to second-order theory (red) and linear Gaussian theory (blue). Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

In Figure 5, a second-order modification of [7] is compared to data from 3-D nonlinear simulations. It is seen that the modified distribution breaks the symmetry of the Gaussian, having lower probability of a deep trough than a high crest. Single Point Extremes

For realistic wave fields, state-of-the-art wave statistics expresses the wave height exceedance probabilities in terms of Weibull distributions:  a x ½8 PðH > xHs Þ ¼ exp  b where the free parameters a and b are found to vary only slightly with the sea state (characterized by the average wave steepness and the directional spread), at least for simple wind seas. Numerical simulations and observations suggest that the probability distribution for the maximum wave height within a record of N waves can be obtained by assuming the waves to be independent. It then follows from eqn [8] that the most probable maximum wave height within a record of N waves is HM ¼ Hs ðbln NÞ1=a

½9

The expression [1] of the so-called narrow-band linear model is recovered for a ¼ 2 and b ¼ 1/2. The exceedance probabilities of the crest height Zcr are also given by Weibull distributions, where the parameters a and b are now, however, found to vary significantly with some of the sea state parameters. Figure 6 shows the probability of exceedance of the wave and crest heights for some of the currently used expressions. Forristall’s model for crest heights, based on numerical simulations that include secondorder nonlinear effects, shows good agreement with

observed data. The same is true for Næss’ model, N, for wave heights, based on a Gaussian sea with a typical wave spectrum. Observe that whereas the Gaussian model (G) severely underpredicts the probability of observing large crests, it gives reliable predictions for the wave height (N). Also note that the tail of Forristall’s crest height distribution depends strongly on wave steepness. In the introduction we defined a rogue wave by the criterion H/Hs42. According to Næss’ model this corresponds to an exceedance probability of about 10  4. The same exceedance probability is obtained for the model F1 if one chooses for the crest height the criterion Zcr =Hs > 1:25

½10

Both these rogue wave criteria are used interchangeably. We call a rogue wave an abnormal rogue wave when it cannot plausibly be explained by stateof-the-art wave statistics. As mentioned above, this depends to some extent on the size of the data set. Nevertheless, if one observes a wave with a height or crest exceedance probability more than 2 orders of magnitude below the expected probability, this will indeed be exceptional and indicative of an abnormal rogue wave. Space–Time Extremes

Whereas the extreme value theory of wave records taken at single points has been the subject of extensive research, the corresponding theory for spatial data is less developed. One problem is that the concept of neighboring maxima and minima and hence the concept of wave height is not well defined in a 2-D field. There exist, however, accurate asymptotic

ROGUE WAVES

23

10−2

Probability of exceedance

10−3 10−4 10−5 10−6

F1

G

FH

F2

N

10−7 10−8 10−9 1

1.2

1.4

1.6

1.8

2 cr /Hs, H /Hs

2.2

2.4

2.6

2.8

3

Figure 6 Probability of exceedance for crest heights (left) and wave heights (right). G, Linear Gaussian model; F1, Forristall’s second-order model for medium wave steepness; F2, Forristall’s second-order model for high wave steepness; N, Næss’ wave height model for Gaussian seas and typical wind wave spectra; FH, Forristall’s empirical wave height model based on buoy data from the Mexican gulf. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

expressions for the maximum crest height for multidimensional Gaussian fields. The following simple example illustrates that even Gaussian theory predicts very high crests when maxima are considered over an extended spatial area. Consider a storm over an area 100 km  100 km and lasting for 6 h. With a mean wave period of 10 s, we expect a mean wavelength lpE200 m. For a directional spread of about 201 one further expects a mean crest length lcE450 m. If we define lplc as the characteristic area of one wave, then there are at each instant of time about 105 waves within the 2-D storm area. By the Gaussian theory we then have that the expected maximum crest height:

•

over the storm area at a fixed time is EðZmax Þspace ¼ 1:32Hs

•

at a fixed location over a 6-h period is EðZmax Þtime ¼ 1:02Hs

•

½11

½12

demonstrates that even the conservative Gaussian theory predicts waves with much larger crest heights when the spatial dimensions of the wave field are taken into account. The Shape of Large Waves

Rogue waves have been described as walls of water or pyramidal waves, surrounded by holes in the ocean. Apart from a few research stereoimaging systems, there exist no operational instruments that directly measure the height of the surface over a sufficiently large area for extensive time periods. Thus, the 2-D shape of the surface elevation, Z, of large waves has mainly to be inferred from numerical simulations and analytic methods. A useful tool is the so-called Slepian model representation (SMR) of a stationary Gaussian stochastic surface. Consider a wave with a high maximum at x ¼ 0 at a fixed instant of time. According to the Slepian theory, the surface Z(x) around the maximum where rZ ¼ 0 may be written as

and over the storm area during the 6-h period is EðZmax Þspaceþtime ¼ 1:69Hs

½13

This last value is quite high, and far into what would be considered to be a rogue wave. This example

ZðxÞ ¼ Zð0Þ

rðxÞ þ DðxÞ rð0Þ

½14

where r(x) is the covariance function and the residual process D(x) is Gaussian with zero mean.

24

ROGUE WAVES

6

6 4

4 z

z

2 2

0 0

–2 –4 –10

5 –5 0 Wave propagation direction

10

–2 –10

–5 0 5 Wave crest direction

10

Figure 7 Averaged and scaled surface profile in the wave propagation and crest directions at an extreme wave crest obtained from large scale third-order simulations (full curve). The dashed curves is the scaled spatial covariance function. The horizontal distance is scaled by the wavelength at the spectral peak divided by 2p. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

The approximation Z(x)BZ(0)r(x)/r(0) is only reasonable in a region where D(x) is small, which typically surrounds the maximum out to about one wave/ crest length. Thus, for a Gaussian surface the average wave profile around a very high crest is that of the scaled covariance function of the wave field. Due to the symmetry of the Gaussian [7], the probability distribution of the crest height Zcr is identical to that of the trough depth Ztr and the average shape of a deep trough is the mirror image of the one of a high crest. The inclusion of nonlinearity breaks this symmetry, as has already been demonstrated in Figure 5. The average shape of an extreme wave can also be expected to change. A comparison of the Slepian model with data from large-scale third-order simulations is shown in Figure 7. It is seen that the simulated crest becomes more narrow and the neighboring troughs less deep. The ratio R between the extreme crest height and the nearest trough depth is 1.5 and 2.3, respectively. Observations of extreme waves indicate that R is scattered around a mean value of 2.2. It follows from the above discussion that on an average large wave events typically occur in short groups. As the waves pass through a group envelope like in Figure 8, they exhibit various shapes.

Experiments and Observations Wave Tank Experiments

Controlled experiments in wave tanks have long been used to study the effect of waves on vessels and structures. Most of this work deals with unidirectional waves which are forced to violent breaking or extreme crest heights through dispersive focusing. Three-dimensional wave basins can carry out similar experiments using spatial focusing.

Figure 8 During half a wave period, as the waves move trough the group envelope, a large wave event may be seen as a large wave crest (black), a large wave height (green), or a deep trough (red) (here seen in the group velocity frame).

Wave tanks are essential for testing vessels and structures in extreme and violent conditions. Modern wave tank facilities are able to reconstruct accurately rogue wave profiles from field observations and to record the response of ships and structures to these waves. However, most of the field wave data are point observations of 3-D waves, in contrast to the unidirectional reconstructions in the wave tank. Field Measurements

Instrumentation able to measure individual wave properties consists of laser and radar altimeters, buoys, and subsurface instruments (pressure gauges and surface tracking acoustic devices). Although some instruments, in particular buoys and radars, give consistent results for the wave height, the data deviate considerably when measuring the wave crests. When measured from subsurface instrumentation and buoys, the crest wave statistics is below the Gaussian theory, whereas narrow beam radar and, in particular, laser altimeters invariably show crest heights above the Gaussian theory. The differences are explained by the lateral motion of buoys tending to avoid high crests, and the inherent area averaging that occurs for pressure gage measurements and radars with broad footprints. On the other hand, it is often suggested that laser recordings

ROGUE WAVES

are sensitive to sea spray, thus overpredicting the real crest height. Good wave measurements from fixed installations tend to confirm second-order wave theory, although some care needs to be taken for platform interference. Instrumental and other errors pose a challenge when searching for exceptional waves in wave records, where erroneous spikes are prone to be mistaken for rogue waves. Frigg data The wave elevation measurements with a Plessey wave radar at the Frigg oil field in the northern North Sea (depth 100 m) is a long-term quality-checked data set containing 10 000 time series each of 20-min duration where Hs42 m. The criteria H42Hs and Zc41.25Hs yield 79 and 74 rogue waves, confirming that these criteria have roughly the same probability of exceedance. The set contains a total of 1.6 million waves and suggests that the probability for H to exceed 2Hs is about 5  10  5. This is quite close to the probability inferred from curve N in Figure 6, and a similar result applies to the maximum crest height. The conclusion is therefore that the Frigg data, although containing some rather extreme height and crest ratios, do not really show any abnormal rogue waves deviating from the theory. Moreover, the ratio of the crest height to the trough depth is scattered around 2.1 for the rogue waves, in accordance with nonlinear numerical simulations. Gorm data The Frigg findings are in strong contrast to the much-referred-to data set from the

25

Gorm field in the central North Sea at a depth of 40 m. These data, which were collected with a radar altimeter for more than 12 years, are indeed astonishing. The crest and wave height distributions show two completely different populations of waves: a normal population adhering to current wave statistics, and what could be denoted an abnormal population. This is clearly seen in Figure 9, which shows the empirical distribution function of Hmax/Hs (found for each of the 20-min time series). The Næss model for typical storm spectra is included as a dashed curve. The corresponding plot for the crest height looks similar. The data contain 24 waves where H/Hs ranges from 2.2 to 2.94 (the most extreme case is seen in Figure 2). Perhaps coincidentally, the deviation from the Næss’ theory happens around the rogue wave criterion. It should be observed, however, that the significant wave height is only 2–4 m for the most extreme waves. The Draupner Incident The Draupner New Year wave occurred on January 1995 and is shown in Figure 2. It was recorded by a laser instrument at an unmanned satellite platform, and minor damage on a temporary deck below the main platform deck supports the reading. This wave record has been extensively discussed in the scientific literature, and independent ship observations have confirmed that the weather situation was extreme. The crest height is 18.5 m above the mean water level and the wave height is 26 m. Although the crest and wave height

3 2. 8

−log(−log(1−F 1/N ))

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1

1

1.2

1.4

1.6 1.8 H max /Hs

2

2.2

2.4 2.6 2.8

3

Figure 9 Empirical distribution function of Hmax/Hs for B5000 20-min records from the Gorm field. The filled circles are representative points, whereas the open circles represent individual records. The dashed straight line is Næss’ wave height model. Points falling below the straight line indicate a larger frequency of occurrence. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

26

ROGUE WAVES

values are slightly below the so-called 100-year values for the Draupner site, the wave was indeed quite unexpected for the observed HsI12 m. Satellite and Radar Measurements of Rogue Waves

Space-borne synthetic aperture radar (SAR) is currently the only instrument that has the capacity of observing wave fields over large spatial areas. The satellites scan the world’s oceans daily, although on sparse tracks. SAR measures the intensity of the backscattered signal (which depends on the amplitude and slope of the scattering waves) and the Doppler shift (which depends on the waves’ orbital velocity). Recently, there have been attempts to infer the surface elevation from SAR data, and the possibility of globally measuring rogue waves from a satellite has caught the media and the space organizations with excitement. However, the algorithms converting the radar backscatter signal to surface elevation have not yet been validated and it is unclear whether the results obtained so far are correct and whether the method is viable. Marine radars, situated onshore, on platforms, or on ships, meet with the same problems.

interactions suffice. Solutions of MNLS compare favorably both with tank experiments and fully nonlinear 3-D simulations over a time horizon of (ops2)  1, provided Do=op ¼ Oðs1=2 Þ. Figure 10 shows three cases initiated with a Joint North Sea Wave Observation Project (JONSWAP) spectrum and different angular distributions. The model allows for internal spectral energy transfer and the spectrum changes on the timescale (ops2)  1, most pronounced for the long-crested case C (BFIZ1 for all cases). The large computational domain, containing c. 104 waves, admits calculation of probability distributions at any time during the evolution process. Figure 11 shows the probability of exceedance of the scaled crest height for the three cases of Figure 10, at times 25, 50, and 100 peak periods (Tp). The cases A and B show little change and fit the theoretical second-order statistics quite well. The long-crested case C however, has a significant increase in the occurrence of large waves during the early development (25Tp) of the spectral instability. This last result has been confirmed by experiments in wave flumes and represents presently a hot topic in rogue wave research.

Numerical Simulations

Conclusions

Large-scale simulations of random ocean waves have so far not been feasible without simplifications of the full nonlinear equations. Although fully nonlinear simulations are gradually becoming available, major simulation tools still apply approximate versions of the full equations. One example of an approximate equation is the modified nonlinear Schro¨dinger equation (MNLS). It assumes a narrow spectral band Do around a peak frequency op and describes the evolution of the wave field for small steepness s, were the cubic wave–wave

Rogue waves are surface gravity waves and operationally defined as waves which satisfy one or both of the criteria [2] and [10]. Their unexpectedness causes a special danger to ships and to offshore and coastal structures. Current wave statistics predicts the exceedance probability of extremely high waves as functions of the sea state. Although most rogue wave observations seem to be consistent with the statistical predictions, there is a small group of rogue wave observations falling outside the predictions. No generally accepted

(a)

(b)

Short-crested

(c)

Medium

Long-crested

Figure 10 Simulated surfaces of a large-scale third-order simulation. Approximately 2% of the computational domain is shown at an early stage of the simulation for a short-(A), medium-(B), and long-(C) crested case. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 40 & 2008 by Annual Reviews.

ROGUE WAVES

100

Case A: t = 25Tp

100

Case A: t = 50Tp

100

10−1

10−1

10−1

10−2

10−2

10−2

10−3

10−3

10−3

10−4

100

0 1 2 3 4 5 6 Case B: t = 25Tp

10−4

100

0 1 2 3 4 5 6 Case B: t = 50Tp

10−4

100

10−1

10−1

10−1

10−2

10−2

10−2

10−3

10−3

10−3

10−4

100

0 1 2 3 4 5 6 Case C: t = 25Tp

10−4

100

0 1 2 3 4 5 6 Case C: t = 50Tp

10−4

100

10−1

10−1

10−1

10−2

10−2

10−2

10−3

10−3

10−3

10−4

0 1 2 3 4 5 6

10−4

0 1 2 3 4 5 6

10−4

27

Case A: t = 100Tp

0 1 2 3 4 5 6 Case B: t = 100Tp

0 1 2 3 4 5 6 Case C: t = 100Tp

0 1 2 3 4 5 6

Figure 11 Simulated probability of exceedance (along vertical axes) of the crest height Zc (scaled by the standard deviation s along horizontal axes) for the cases A, B, and C at times and 25, 50, and 100 peak periods (Tp). Solid line: simulations; dotted line: Gaussian theory; dashed line: second-order theory.

explanation or theory for the occurrence of such abnormal rogue waves has so far been given. There seems, however, to be a consensus among researchers that occurrences of unexpected and dangerous waves in coastal waters is mostly caused by focusing due to refraction by bottom topography or current gradients. This explanation might also be valid for the extreme waves observed in intense ocean currents like the Agulhas Current off the eastern coast of South Africa. In the open ocean away from strong current gradients there is also growing evidence for the occurrence of abnormal rogue waves, though reasonable doubt about the reliability of some of the measurements and observations is still warranted. Progress is expected to come from a combination of more reliable

measurements, extensive field observations, careful statistical analyses, and numerical simulations. Tank experiments are of vital importance for the engineering community but their controlled condition might not pay proper tribute to the uncontrollable nature of rogue waves in the world’s oceans.

See also Breaking Waves and Near-Surface Turbulence. Surface Gravity and Capillary Waves.

Further Reading Dysthe K, Krogstad H, and Mu¨ller P (2008) Oceanic rogue waves. Annual Review of Fluid Mechanics 40: 287--310.

28

ROGUE WAVES

Kharif C and Pelinovski E (2003) Physical mechanisms of the rogue wave phenomenon. European Journal of Mechanics B/Fluids 22: 603--634. Lavrenov I (2003) Wind-Waves in Oceans. Dynamics and Numerical Simulation. New York: Springer. Mu¨ller P and Henderson D (eds.) (2005) Rogue Waves. Proceedings, 14th ‘Aha Huliko’a Hawaiian Winter Workshop, Special Publication, 193pp. University of Hawaii at Manoa, School of Ocean and Earth Science and Technology.

Olagnon M and Athanassoulis G (eds.) (2000) Rogue Waves 2000: Proceedings of a Workshop in Brest, France, 29–30 Nov. 2000. Plouzane´, France: Editions IFREMER. Olagnon M and Prevosto M (eds.) (2004) Rogue Waves 2004: Proceedings of a Workshop in Brest, France, 20– 22 Oct. 2004. Plouzane´, France: Editions IFREMER. Tucker MJ and Pitt EG (2001) Ocean Engineering Book Series, Vol. 5: Waves in Ocean Engineering, 521pp. Amsterdam: Elsevier.

WAVES ON BEACHES

Offshore

Incident waves

Beach Topography

Shoaling

Wave motions are one of the most familiar of oceanographic phenomena. The waves that we see on beaches were originally generated by ocean winds and storms, sometimes at long distances from their final destination. In fact, groups of waves, generated by large storms, have been tracked from the Southern Ocean near Australia all the way to Alaska. Open ocean waves can be thought of as simple sinusoids that are superimposed to yield a realistic sea. Waves entering the nearshore, called incident waves, can have wave periods (T, the time between consecutive passages of wave crests) ranging from 2 to 20 s, with 10 s a typical value. Wave heights (H, the vertical distance from the trough to peak of a wave) can exceed 10 m, but are typically 1 m, representing an energy density, of 1250 J m2 (r is the density of sea water, g is the acceleration of gravity) and a flux of power impinging on the coast of about 10 kW per meter of coastline. Although this is a substantial amount of power, it is not enough to make broad commercial exploitation of wave power economical at the time of writing. Of interest in this section are the dynamics of waves once they progress into the shallow beach environment such that the ocean bottom begins to restrict the water motions. Most people are familiar with refraction (the turning of waves toward the beach), wave breaking, and swash (the back and forth motion of the water’s edge), but are less familiar with the other types of fluid motion that are generated near the beach. Figure 1 illustrates schematically the evolution of ocean wave energy as it moves from deep water (top of the figure) through progressively shallower water toward the beach (bottom of the figure). Offshore, most energy lies in waves of roughly 10 s period (middle of the figure). However, the processes of shoaling distribute that energy to both higher frequencies (right half of the figure) and lower frequencies (left half) including mean flows. In general, these processes can be distinguished as those that

Cross-shore location

Introduction

Shoaling

Infragravity

High frequency

Wind waves

Beach Topography

Break point

Copyright & 2001 Elsevier Ltd.

occur offshore of the surf zone (where waves become overly steep and break) and those that occur within the surf zone. The axes of Figure 1, cross-shore position and frequency, are two of several variables that can be used to structure a discussion of near-shore fluid dynamics. Other important distinctions that will be made include whether the incident waves are monochromatic (single frequency) versus random (including a range of frequencies), depth-averaged versus depth-dependent, longshore uniform (requiring consideration of only one horizontal dimension, 1HD) versus long shore variable (2HD), and linear versus nonlinear.

Breaking

'Mean' flows

Shoreline

R. A. Holman, Oregon State University, College of Oceanic and Atmospheric Sciences, Corvallis, OR, USA

Infragravity

Wind waves

High frequency

Turbulence

Far infragravity Swash

Reflection

0.001

0.01

0.1

1.0

10

Frequency (Hz) Figure 1 Schematic of important near-shore processes showing how the incident wave energy that drives the system evolves as the waves progress from offshore to the shoreline (top to bottom of figure). Wave evolution is grouped into processes occurring seaward of the breakpoint (labeled ‘shoaling’) and those within the surf zone (denoted ‘breaking’). In both cases, energy is spread to lower (left) and higher (right) frequencies. The beach topography provides the bottom boundary condition for flow, so is important to wave processes. In turn, the waves move sediment, slowly changing the topography. Wind and tides may be important in some settings, but are not shown here.

29

30

WAVES ON BEACHES

Table 1

Kinematic relationships of linear wavesa

The Dynamics of Incident Waves Much of the early progress in understanding nearshore waves was based on the examination of a monochromatic wave train, propagating onto a long shore uniform beach (1HD). Many observable properties can be explained in terms of a few principles including conservation of wave crests, of momentum, and of energy. Most dynamics are depthaveraged. Table 1 lists a number of properties of monochromatic ocean waves, in the linear limit of infinitesimal wave amplitude (known as linear, or Airy wave theory). The general expressions (center column) contain complicated forms that can be substantially simplified for both shallow (depths less than 1/20 of the deep water wavelength, L0 ) and deep (depths greater than 1/2 L0 ) water limits. The speed of wave propagation is known as the celerity or phase speed, c, to distinguish it from the velocity of the actual water particles. In deep water, c depends only on the wave period (independent of depth). However, as the wave enters shallower water, the wavelength decreases and the phase speed becomes slower (contrary to common belief, this is not a result of bottom friction). An interesting consequence of the slowing is wave refraction, the turning of waves toward the coast. For a wave approaching the coast at any angle, the end in shallower water will always progress more slowly than

the deeper end. By propagating faster, the deeper end will begin to catch up to the shallow end, effectively turning the wave toward the beach (refraction). In shallow water, the general expression for celerity, c ¼ ðghÞ1=2 , depends only on depth so that waves of all periods propagate at the same speed. The energy density of Airy waves (energy per unit area) is the sum of kinetic and potential energy components and depends only on the square of the wave height (Table 1). Perhaps of more interest is the rate at which this energy is propagated by the wave train, known as the wave power, P, or wave energy flux. In deep water, wave energy progresses at half the speed of wave phase (individual wave crests will out-run the energy packet), whereas in shallow water energy travels at the same speed as wave phase and the flux depends only on H 2 h1=2 . Offshore of the surf zone, wave energy is conserved since there is no breaking dissipation and energy loss through bottom friction has been shown to be negligible except over very wide flat seas. Thus, as the depth, h, decreases, the wave height, H, must increase to conserve H 2 h1=2 . This is a phenomenon familiar to beachgoers as the looming up of a wave just before breaking. The combined result of shoaling is reduced wavelength and increased wave height, hence waves that become increasingly steep and may become unstable and break. One criterion for breaking is that

WAVES ON BEACHES

the increasing water particle velocities (u in Table 1) exceed the decreasing wave phase speed such that the water leaps ahead of the wave in a curling or plunging breaker. From the relationships in Table 1, it can be found that this occurs. when g, the wave height to depth ratio ðg ¼ H=hÞ exceeds a value of 2. Of course, for waves that have steepened to the point of breaking, the approximations of infinitesimal waves, inherent in Airy wave theory, are badly violated. However, observations show that g does reach a limiting value of approximately 1 for monochromatic waves and 0.4 for a random wave field. As waves continue to break across the surf zone, the wave height decreases in a way that g is approximately maintained, and the wave field is said to be saturated (cannot get any larger). The above-saturation condition implies that wave heights will be zero at the shoreline and there will be no swash, in contradiction to common observation. Instead, it can be shown that very small amplitude waves, incident on a sloping beach, will not break unless their shoreline amplitude, as , exceeds a value determined by s2 as rk gb2

½1

where k is an O(1) constant. For larger amplitudes, the wave amplitude at any cross-shore position is the sum of a standing wave contribution of this maximum value plus a dissipative residual that obeys the saturation relationship. The ratio of terms on the left-hand side of eqn.[1] is important to a wide range of nearshore phenomena and is often re-written as the Iribarren number, x0 ¼

b ðHs =L0 Þ1=2

½2

where the measure of wave amplitude is replaced by the offshore significant wave height,1 Hs. This form clarifies the importance of beach steepness, made dynamically important by comparing it to the wave steepness, Hs =L0 . For very large values of x0 , the beach acts as a wall and is reflective to incident

1

Although the peak to trough vertical distance for monochromatic waves is a unique and hence sensible measure of wave height, for random waves this scale is a statistical quantity, representing a distribution. A single measure, often chosen to represent the random wave Reld, is the significant wave height, Hs, defined as the average height of the largest one-third of the waves. This statistic was chosen historically as best representing the value that would be visually estimated by a semitrained observer. It is usually calculated as four times the standard deviation of the sea surface times series.

31

waves (the non-breaking case from eqn. [1]). For smaller values, the presence of the sloping beach takes on increasing importance as the waves begin to break as the plunging breakers that surfers like, where water is thrown ahead of the wave and the advancing crest resembles a tube. Still smaller beach steepnesses (and x0 ) are associated with spilling breakers in which a volume of frothy turbulence is pushed along with the advancing wave front.

Radiation Stress: the Forcing of Mean Flows and Set-up The above discussion is based on the assumption of linearity, strictly true only for waves of infinitesimal amplitude. Because the dynamics are linear, energy in a wave of some particular period, say 10 s, will always be at that same period. In fact, once wave amplitude is no longer negligible, there are a number of nonlinear interactions that may transfer energy to other frequencies, for example to drive currents (zero frequency). Nonlinear terms describe the action of a wave motion on itself and arise in the momentum equation from the advective terms, uru, and from the integrated effect of the pressure term. For waves, the time-averaged effect of these terms can easily be calculated and expressed in terms of the radiation stress, S, defined as the excess momentum flux due to the presence of waves. Since a rate of change of momentum is the equivalent of a force by Newton’s second law, radiation stress allows us to understand the time-averaged force exerted by waves on the water column through which they propagate. A spatial gradient in radiation stress, for instance a larger flux of momentum entering a particular location than exiting, would then force a current. Radiation stress is a tensor such that Sij is the flux of i-directed momentum in the j-direction. For waves in shallow water, approaching the coast at an angle y, the components of the radiation stress tensor are " S¼

Sxx

Sxy

Syx

Syy

#

" ¼E

ðcos2 y þ 1=2Þ cos y sin y cos y sin y ðsin2 y þ 1=2Þ

# ½3

where x is the cross-shore distance measured positive to seaward from the shoreline and y is the long-shore distance measured in a right-hand coordinate system with z positive upward from the still water level. For a wave propagating straight toward the beach (y ¼ 0) Sxx ¼ 3=2E is the shoreward flux of shoreward-directed momentum. The increase of wave height (hence energy, E) associated with shoaling outside the surf zone must be accompanied by an

32

WAVES ON BEACHES

increasing shoreward flux of momentum (radiation stress). This gradient, in turn, provides an offshore thrust, pushing water away from the break point and yielding a lowering of mean sea level called setdown. As the waves start to break and decrease in height through the surf zone, the decreasing radiation stress pushes water against the shore until an opposing pressure gradient balances the radiation stress gradient. The resulting set-up at the shoreline, Z, ¯ a contributor to coastal erosion and flooding, is found to depend on the offshore significant wave height, Hs , as Z¯ max ¼ KHs x0

½4

where K is found empirically to be 0.45. If waves approach the beach at an angle, they also carry with them a shoreward flux of longshore-directed momentum, Syx . Cross-shore gradients in this quantity, due to the breaking of waves in the surf zone, provide a net long shore force that accelerates a long shore current, V¯ along the beach until the forcing just balances bottom friction. If the cross-shore structure of V¯ is solved for, a discontinuity is evident at the seaward limit of the surf zone, where the radiation stress forcing jumps from zero (seaward of the break point) to a large value (where the wave just begin to break). This discontinuity is an artifact of the fact that every wave breaks at exactly the same location for an assumed monochromatic wave forcing, and must be artificially smoothed by an assumed horizontal mixing for this case. However, a natural random wave field consists of an ensemble of waves with (for linear waves) a Rayleigh distribution of heights. Depthlimited breaking of such a wave field will be spread over a region from offshore, where a few largest waves break, to onshore where the smallest waves finally begin to dissipate. The spatially distributed nature of these contributions to the average radiation stress provides a natural smoothing, often obviating the need for additional horizontal smoothing.

Nonlinear Incident Waves The above discussion dwelt on the nonlinear transfer of energy from incident waves to mean flows. Nonlinearities will also transfer energy to higher frequencies, yielding a transformation of incident wave shape from sinusoidal to peaky and skewed forms. The Ursell number, ðH=LÞðL=hÞ3 , measures the strength of the nonlinearity. For monochromatic incident waves, this evolution was often modeled in terms of an ordered Stokes expansion of the wave form to produce a series of harmonics (multiples of the incident

wave frequency) that are locked to the incident wave. For waves with Ursell number of O(1), propagating in depths that are not large compared to the wave height, higher order theories must be used to model the finite amplitude dynamics. For a random sea under such theories, the total evolution of the spectrum must be found by summing the spectral evolution equations for all possible Fourier pairs (in other words, all frequencies in the sea can and will interact with all other frequencies). Such approaches are very successful in predicting the evolving shape and nonlinear statistics (important for driving sediment transport) for natural random wave fields outside the surf zone.

Vertically Dependent Processes Depth-independent models are successful in reproducing many nearshore fluid processes but cannot explain several important phenomena, for example undertow, offshore-directed currents that exist in the lower part of the water column under breaking waves. The primary cause of depth dependence arises from wave-breaking processes. When waves break, the organized orbital motions break down, either through the plunge of a curling jet of water thrown ahead of the advancing wave or as a turbulent foamy mass (called a roller) carried on the advancing crest. Both processes originate at the surface but drive turbulence and bubbles into the upper part of the water column. The transfer of momentum from wave motions to mean currents described by radiation stress gradients above does not account for the existence of an intermediate repository, the active turbulence of the roller, that decays slowly as it is carried with the progressing wave. This time delay causes a shift of the forcing of longshore currents, such that a current jet will occur landward of the location expected from study of the breaking locations of incident waves. The other consequence of the vertical dependence of the momentum transfer is that the shoreward thrust provided by wave breaking is concentrated near the surface. Set-up, the upward slope of sea level against the shore, will balance the depth averaged wave forcing. However, due to the vertical structure of the forcing, shoreward flows are driven near the top of the water column and a balancing return flow, the undertow, occurs in the lower water column. Undertow strengths can reach 1 m s1.

2HD Flows – Circulation All of the previous discussion was based on the assumption that all processes were long-shore uniform

WAVES ON BEACHES

(1 HD) so that no long-shore gradients existed. It is rare in nature to have perfect long shore uniformity. Most commonly, some variability (often strong) exists in the underlying bathymetry. This can lead to refractive focusing (the concentration of wave energy by refraction of waves onto a shallower area) and the forcing of long-shore gradients in wave height, hence of setup. Since setup is simply a pressure head, longshore gradients will drive long-shore currents toward low points where the converging water will turn seaward in a jet called a rip current. It is possible to develop long-shore gradients in wave height (hence rips) in the absence of long-shore variations in bathymetry. Interactions between two elements of the wave field (either two incident wave trains from different directions or an incident wave and an edge wave, defined below) can force rip currents if the interacting trains always occur with a fixed relative phase.

Infragravity Waves and Edge Waves There is a further, very important consequence of the fact that natural wave fields are not monochromatic, but instead are random. For random waves, wave height is no longer constant but varies from wave to wave. Usually these variations are in the form of groups of five to eight waves, with heights gradually increasing then decreasing again. This observation is known as surf beat and is particularly familiar to surfers. A consequence of these slow variations is that the radiation stress of the waves is no longer constant, but also fluctuates with wave group timescales and forces flows (and waves) in the near-shore with corresponding wave periods. These waves have periods of 30–300 s and are called infragravity waves, in analogy to infrared light being lower frequency than its visible light counterpart. The direct forcing of infragravity motions described above can be thought of as a time-varying setup, with the largest waves in a group forcing shoreward flows that pile up in setup, followed by seaward flow as the setup gradients dominate over the weaker forcing of the small waves. If the modulations of the incident wave group are long shore uniform, this result of this setup disturbance will simply be a free (but low frequency) wave motion that propagates out to sea. However, in the normal case of wave groups with longshore (as well as time) variability, we can think of the response by tracing rays as the setup disturbance tries to propagate away. Rays that travel offshore at an angle to the beach will refract away from the beach normal (essentially the opposite of incident wave refraction, discussed

33

earlier). For rays starting at a sufficiently steep angle to the normal, refraction can completely turn the rays such that they re-approach and reflect from the shore in a repeating way and the energy is trapped within the shallow region of the beach. These trapped motions are called edge waves because the wave motions are trapped in the near-shore wave guide. (Any region wherein wave celerity is a minimum can similarly trap energy by refraction and is known as a wave guide. The deep ocean sound channel is a wellknown example and allows propagation of trapped acoustic energy across entire ocean basins.) Motions that do not completely refract and thus are lost to the wave guide are called leaky modes. The requirement that wave rays start at a sufficiently steep angle to be trapped by refraction can be expressed in terms of the long-shore component of wavenumber, ky . For large ky (waves with a large angle to the normal), rays will be trapped in edge waves whereas small ky motions will be leaky modes. The cutoff between these is s2 =g. In the same sense that waves that slosh in a bathtub occur as a discrete set of modes that exactly fit into the tub, edge waves occur in a set of modes that exactly fit between reflection at the shoreline and an exponentially decaying tail offshore. The detailed form of the waves depends on the details of the bathymetry causing the refraction. However, for the example of a plane beach of slope bðh ¼ xtanbÞ, the cross-shore forms of the lowest four modes (mode numbers, n ¼ 0, 1, 2, 3), are shown in Figure 2 and are given in Table 2. The existence of edge waves as a resonant mode of wave energy transmission in the near-shore has several impacts. First, the dispersion relation provides a selection for particular scales. For example, Figure 3 shows a spectrum of infragravity wave energy collected at Duck, North Carolina, USA. The concentration of energy into very clear, preferred scales is striking and has led to suggestions that edge waves may be responsible for the generation of sand bars with corresponding scales. Second, because edgewave energy is trapped in the near-shore, it can build to substantial levels even in the presence of weak, incremental forcing. Moreover, because edge-wave energy is large at the shoreline where the incident waves have decayed to their minimum due to breaking, edge waves may feasibly be the dominant fluid-forcing pattern on near-shore sediments in these regions. The magnitudes of infragravity energy (including edge waves and leaky modes) have been found to depend on the relative beach steepness as expressed by the Iribarren number (eqn. [2]). For steep beaches (high x0 ), very little infragravity energy can be

34

WAVES ON BEACHES

1.0

Table 3 types

Magnitude of infragravity waves on different beach

 (x)

0.5

n=2

n=0

0

n=1

n=3

_ 0.5 0

10

20

30

40

50

60

70

 2x / g tan

aLeast squares regression slope between the significant swash height (computed from the infragravity band energy only) and the offshore significant wave height. Reproduced from Howd PA, Oltman-Shay J, and Holman RA (1991) Wave variance partitioning in the trough of a barred beach. Journal of Geophysical Research 96 (C7), 12781} 12795.)

Figure 2 Cross-shore structure of edge waves. Only the lowest four mode numbers of the larger set are shown. The mode number, n, describes the number of zero crossings of the modes. So, for example, a mode 1 edge wave always has a low offshore, opposite a shoreline high, and visa versa. Edge waves propagate along the beach.

representative beach locations, with mean values of x0 and of m, the linear regression slope between the measured significant swash magnitude,2Rs , in the infragravity band, and the offshore significant wave height, Hs .

Table 2 The average abundance of the refractory elements in the Earth’s crust, and their degree of enrichment, relative to aluminum, in the oceans

Shear Waves

generated and the beaches are termed reflective due to the high reflection coefficient for the incident waves. However, for low-sloping beaches (small x0 ), infragravity energy can be dominant, especially compared to the highly dissipated incident waves. On the Oregon Coast of the USA, for example, swash spectra have been analyzed in which 99% of the variation is at infragravity timescales (making beachcombing an energetic activity). Table 3 lists five

Up until the mid-1980s long shore currents were viewed as mean flows whose dynamics were readily described as in the above sections. However, field data from the Field Research Facility in Duck, North Carolina, provided surprising evidence that as longshore currents accelerated on a beach with a welldeveloped sand bar, the resulting current was not steady but instead developed slow fluctuations in strength and a meandering pattern in space. Typical wave periods of these wave are hundreds of seconds and long-shore wavelengths are just hundreds of meters (Figure 3). These very low frequencies are called far infragravity waves, in analogy to the relationship of far infrared to infrared optical frequencies. However, the wavelengths are several orders of magnitude shorter than that which would be expected for gravity waves (e.g., edge waves or leaky modes) of similar periods. These meanders have been named shear waves and arise due to an instability of strong currents, similar to the instability of a rising column of smoke. The name comes from the dependence of the dynamics (described briefly below) on the shear of the longshore current (the cross-shore gradient of the longshore current). A jet-like current with large shear, such as might develop on a barred beach where the wave forcing is concentrated over and near the bar,

2 Swash oscillations are commonly expressed in terms of their vertical component.

WAVES ON BEACHES

1

2

2

1

0.050 0

0 0.045 0.040

Frequency (Hz)

0.035 0.030 0.025 0.020 0.015 0.010 0.005

Southward

Northward

0.025

0.020

0.015

0.010

0.005

0

_ 0.005

_ 0.010

_ 0.015

_ 0.020

_ 0.025

0

_1

coastal currents, flowing along a continental shelf, have been shown to develop similar instabilities although with very larges scales. Similarly, under wave motions, the bottom boundary layer, matching the moving wave oscillations of the water column interior with zero velocity at the fixed boundary, is also unstable. It can be shown that a necessary condition for such an instability is the presence of an inflection point in the velocity profile (the spatial curvature of the current field changes sign), a requirement satisfied by long shore currents on a beach. In that case, crossshore perturbations will extract energy from the mean long-shore current at a rate that depends on the strength of the current shear. Thus, these perturbations will grow to become first wave-like meanders, then if friction is not strong, to become a field of turbulent eddies. The extent of this evolution (wave-like versus eddy-like) is not yet known for natural beach environments. Shear waves can clearly form an important component of the near-shore current field.) Root mean square (RMS) velocity fluctuations can reach 35 cm s1 in both cross-shore and long-shore components of flow. This corresponds to an RMS swing of the current of 70 cm s1, with many oscillations much larger.

50

40

30

20

10

0

Cyclic alongshore wavenumber (m )

35

% Power

Figure 3 The spectrum of low frequency wave motions, as measured by current meters sampling the long shore component of velocity, from Duck, North Carolina, USA. The vertical axis corresponds to frequency and the horizontal axis to along shore component of wavenumber. Positive wavenumbers describe wave motions propagating along the beach to the south. Surprisingly, wave energy is not spread broadly in frequency– wavenumber space, but concentrates on specific ridges. Black lines, indicating the theoretical dispersion lines for edge waves (mode numbers marked at figure top), provide a good match to much of the data at low frequencies with some offset at higher frequency associated with Doppler shifting by the long-shore current. The concentration of low-frequency energy angling up to the right corresponds to shear waves. (After Oltman-Shay J and Guza RT (1987) Infragravity edge wave observations on two California beaches. Journal of Physical Oceanography 17(5): 644–663.)

can develop strong shear waves. In contrast, for a broad, featureless planar beach, the shear of any generated long-shore current will be weak so that shear wave energy may be undetectable. This explains why shear waves were not discovered until field experiments were carried out on barred beaches. The instability by which shear waves are generated has a number of other analogs in nature. Large-scale

Conclusions As ocean waves propagate into the shoaling waters of the nearshore, they undergo a wide range of changes. Most people are familiar with the refraction, shoaling and eventual breaking of waves in a near-shore surf zone. However, this same energy can drive strong secondary flows. Wave breaking pushes water shoreward, yielding a super-elevation at the shoreline that can accentuate flooding and erosion. Waves arriving at an angle to the beach will drive strong currents along the beach that can transport large amounts of sediment. Often these currents form circulation cells, with strong rip currents spaced along the beach. Natural waves occur in groups, with heights that vary. The breaking of these fluctuating groups drives waves and currents at the same modulation timescale, called infragravity waves. These can be trapped in the nearshore by refraction as edge waves. Even long shore currents can develop instabilities called shear waves that drive meter-per-second fluctuations in the current strength with timescales of several minutes. The apparent physics that dominates different beaches around the world often appears to vary. For

36

WAVES ON BEACHES

example, on low-sloping energetic beaches, infragravity energy often dominates the surf zone, whereas shear waves can be very important on barred beaches. In fact, the physics is unchanging in these environments, with only the observable manifestations of that physics changing. The unification of these diverse observations through parameters such as the Iribarren number is an important goal for future research.

Nomenclature E H Hs L L0 P Rs RMS S T V¯ X a as c cg g h k

wave energy density wave height significant wave height wavelength deep water wave length wave power or energy flux significant swash height root mean square statistic radiation stress (wave momentum flux) wave period mean longshore current distance coordinate in the direction of wave propagation wave amplitude wave amplitude at the shoreline wave celerity, or phase velocity wave group velocity acceleration of gravity water depth wavenumber (inverse of wavelength)

n m n u v x y z b g Z theta x0 r s j Z¯ max r

ratio of group velocity to celerity ratio of infragravity swash height to offshore wave height edge wave mode number water particle velocity under waves long-shore component of wave particle velocity cross-shore position coordinate long-shore position coordinate vertical coordinate beach slope ratio of wave height to local depth for breaking waves sea surface elevation angle of incidence of waves relative to normal Iribarren number density of water radialfrequency2p=T cross-shore structure function for edge waves mean set-up at the shoreline gradient operator

See also Beaches, Physical Processes Affecting. Breaking Waves and Near-Surface Turbulence. Coastal Trapped Waves. Sea Level Change. Surface Gravity and Capillary Waves. Wave Generation by Wind.

WAVE ENERGY M. E. McCormick and D. R. B. Kraemer, The Johns Hopkins University, Civil Engineering Department, Baltimore, MD, USA Copyright & 2001 Elsevier Ltd.

Introduction In the last half of the twentieth century, humankind finally realized that fossil fuel resources are finite and that use of those fuels has environmental consequences. These realizations have prompted the search for other energy resources that are both renewable and environmentally ‘friendly’. One such resource is the ocean wind wave. This is a form of solar energy in that the sun is partly responsible for the winds that generate water waves. The exploitation of water waves has been a goal for thousands of years. Until recent times, however, only sporadic efforts were made, and these were generally directed at a specific function. In the 1960s, Yoshio Masuda, the ‘renaissance man’ of wave energy conversion, came up with a scheme to convert the energy of water waves into electricity by using a floating pneumatic device. Originally, the Masuda system was used to power remote navigation aides, such as buoys. One such buoy system was purchased by the US Coast Guard which, in turn, requested an analysis of the performance of the system. The results of that analysis were reported by McCormick (1974). This was the first of a long list of theoretical

and experimental studies of the pneumatic and other wave energy conversion systems. (For summaries of some of the works, see the Further Reading section.) The most recent collective type of publication is that edited by Nicholls (1999), written under the joint sponsorship of the Engineering Committee on Oceanic Resources (ECOR) and the Japan Marine Science and Technology Center (JAMSTEC). In the late 1970s and early 1980s, JAMSTEC co-sponsored a full-scale trial of a floating, offshore pneumatic system called the Kaimei. The 80 m long, 10 m wide Kaimei (Figures 1 and 2) was designed to produce approximately 1.25 MW of electricity while operating in the Sea of Japan. This power was to be produced by 10 pneumatic turbo-generators. Eight of these (produced in Japan) utilized a unidirectional turbine. The other two utilized bi-directional turbines designed by Wells in the UK and McCormick in the USA. Unfortunately, the designed electrical power production was never attained by the system. The Wells turbine was found to be the most effective for wave energy conversion, and is now being used to power fixed pneumatic systems in the Azores, in India, and on the island of Islay off of the coast of Scotland. Most of the published works resulting from research, development, and demonstration efforts are directed at the production of electrical energy. However, Hicks et al. (1988) described a wave energy conversion technique that could be used to produce potable water from ocean salt water. This technique had been developed earlier by Pleass and Hicks. Their work resulted in a commercial system

Wave-induced air flow Air turbine Capture Chambers

Oscillating water column

Figure 1 Schematic diagram of the Kaimei wave energy conversion system, consisting of 10 capture chambers and 10 pneumaticelectric generating systems.

37

38

WAVE ENERGY

Figure 4 The McCabe wave pump located 500 m off the coast of Kilbaha, County Clare, Ireland.

Figure 2 The Kaimei deployed in the Sea of Japan.

called the Del Buoy, and inspired the efforts of others to apply the McCabe wave pump (Figures 3 and 4) to the production of potable water. The high-pressure pumps located between the barges pump sea water through a reverse osmosis (RO) desalination system located on the shore. The first deployment of the McCabe wave pump occurred in 1996 in the Shannon River, western Ireland. A second deployment of the system is expected in the spring of the year 2000 at the same location.

Wave Power: Resource and Exploitation A mathematical expression for the power of water waves is obtained from the linear wave theory. Simply put, the expression is based on the waves having a sinusoidal profile, as sketched in Figure 5. The wave power (energy flux) expression is: 1 P ¼ rgH 2 bcG 8

½1

where r is the mass density of salt water (approximately 1030 kg m3), g is the gravitational acceleration

(9.81 m s2), H is the wave height in meters, b is the wave crest width of interest in meters, and the vector cG is called the group velocity. In deep water, defined as water depth (h) greater than half of the wave length (l), the group velocity (in m s1) is approximately: c gT 7cG 7C C 2 4p

½2

where c is the wave celerity (the actual speed of the wave), and T is the period of the wave in seconds. In shallow water, defined as where hrl=20, the group velocity is approximated by: pffiffiffiffiffiffi ½3 7cG 7CcC gh Consider an average wave approaching the central Atlantic states of the contiguous United States. The average wave height and period of waves in deep water are approximately 1 m and 7 s, respectively. For this wave, the wave power per crest width is: 7P7 1 ¼ rg2 H 2 TC6:90ðkW=mÞ b 32p

½4

from eqn[1] combined with the expression in eqn [2]. Pumps

c  Power barge Pitching motion Damping plate Water intake

H

To RO unit

Figure 3 Sketch of the McCabe wave pump.

Figure 5 Wave notation.

WAVE ENERGY

The percentage of this power that can be captured depends on both the width of the capturing system and the frequency (or period) characteristics of the system. Each wave power system has one fundamental frequency (fn). If the inverse of that frequency (1/fn ¼ Tn) is the same as the average wave period, then the system is in resonance with the average wave, and the maximum amount of wave power will be extracted by the wave power system. This, then, is the design goal, i.e. to design the system to resonate with the design wave. When resonance is achieved, then another phenomenon occurs which is of benefit to the system: i.e. resonant focusing, where diffraction draws energy toward the system. For a single degree of freedom wave power system, such as the heaving buoy sketched in Figure 6, the wave power absorbed by the system comes from a crest width equal to the width of the system (B) plus an additional width equal to the wavelength divided by 2p. Hence, in deep water, the total power available to the single degree of freedom system operating in the average wave is: 7P7 ¼

  1 l rg2 H 2 T B þ 32p p

½5

(A)

To water pump

To power grid

Flotation collar of diameter B Oscillating water column

Water intake (B) Figure 6 Floating oscillating water column wave energy converter, designed to produce electrical power for either potable water production or electricity. (A) Plan view; (B) elevation.

39

where the wavelength in deep water is approximately: lC

gT 2 2p

½6

Thus, for the aforementioned average wave, the deep-water wavelength is about 76.5 m. Consider an ideal 1 m diameter heaving system operating in the 1 m, 7 s average wave. For this wave, the wave power captured by the system is 6:90ð1 þ 76:5=2pÞ kW, or approximately 91 kW. If bus-bar conversion efficiency is 50%, then about 45.5 kW will be supplied to the power grid for consumption. In the contiguous United States, each citizen requires about 1 kW, on average, at any time. Hence, this system would supply 87.5 citizens. To use the same system coupled with a RO desalinator to supply potable water, the value of the osmotic pressure of the desalinator’s membranes is required. This value is 23 atmospheres or approximately 23 bars. From fluid mechanics, the power is equal to the volume rate of flow in the system multiplied by the back pressure. Hence, the 1 m diameter system would supply approximately 5 (US) gallons of salt water per second to the RO unit. Half of this flow would become product (potable) water, while the other half would be brine waste. This ideal system would supply 2.5 gallons per second (about 225 000 gallons per day) of potable water. Each US citizen residing in the contiguous United States uses about 60 gallons per day, on average. So, the wavepowered desalination system would satisfy the daily potable water needs of approximately 3700 citizens. The numbers presented in the last two paragraphs illustrate the potential of wave energy conversion. The electrical and water producing systems described are ideal. In actuality, the waves in the sea are random in nature. The system, then, must be tuned to some design wave, such as that having an average wave period.

Economics of Wave Power Conversion The economics of ocean wave energy conversion vary, depending on both the product (electricity or potable water) and the location. To illustrate this, consider the following two cases. First, on Lord Howe Island in the South Pacific, the cost of electrical energy is about 45 (US) cents per kilo-Watt hour (kWh). Electricity produced by wave energy conversion would cost about 15 cents/kWh. Hence, for such an application, wave energy conversion would be extremely cost-effective. On the other

40

WAVE ENERGY

hand, in San Diego, California, the energy cost is about 13 cents/kWh, making wave energy conversion cost-ineffective. For the production of potable water, the McCabe wave pump, coupled with a RO system, will produce potable water at approximately US $1.10 per cubic meter (265 US gallons). On some remote islands, the cost of potable water is approximately $4.00 per gallon. On the coast of Saudi Arabia, on the Arabian Sea, the cost of potable water is $3.10 per cubic meter. Therefore, these locations, wave-powered desalination systems are very costeffective.

Concluding remarks The reader is encouraged to consult the Further Reading section for more information on wave energy conversion. There are many activities presently underway in this area of technology. These can be found on the Internet by searching the world wide web for wave energy conversion.

See also Coastal Trapped Waves. Internal Waves. Seiches. Storm Surges. Surface Gravity and Capillary Waves. Tides. Tsunami. Wave Generation by Wind. Waves on Beaches.

Further Reading Count B (ed.) (1980) Power from Sea Waves. London: Academic Press. Hicks D, Pleass CM, and Mitcheson G (1988) Delbuoy Wave-Powered Seawater Desalination System. Proceedings of OCEANS’88, US Department of Energy, pp. 1049–1055. McCormick ME (1981) Ocean Wave Energy Conversion. New York: Wiley-Interscience. McCormick ME (1974) An analysis of power generating buoys. Journal of Hydronautics 8: 77--82. McCormick ME, McCabe RP, and Kraemer DRB (1999) Utilization of a hinged-barge wave energy conversion system. International Journal of Power Energy Systems 19: 11--16. McCormick ME and Murtagh JF (1992) Large-Scale Experimental Study of the McCabe Wave Pump. US Naval Academy Report EW-3-92, January. McCormick M, Murtagh J, and McCabe P (1998) LargeScale Experimental Study of the McCabe Wave Pump. European Wave Energy Conference (European Union), Patras, Greece, Paper H1. Nicholls HB (1999) Workshop Group on Wave Energy Conversion. Engineering Committee on Oceanic Resources (ECOR), St Johns, Newfoundland, Canada. Ross D (1998) Power from the Waves. Oxford: Oxford University Press. Shaw R (1982) Wave Energy. A Design Challenge. Chichester, UK: Ellis Horwood.

WHITECAPS AND FOAM E. C. Monahan, University of Connecticut at Avery Point, Groton, CT, USA Copyright & 2001 Elsevier Ltd.

Introduction Oceanic whitecaps and sea foam are, respectively, the transient and semipermanent bubble aggregates that are found on the surface of the ocean when certain meteorological conditions prevail. These features are of sufficient size to be detectable by eye, and an individual whitecap or foam patch can readily be recorded using standard low-resolution photographic or video systems. When they are present in sufficient number on the sea surface they alter the general visual albedo, and microwave emissivity, of that surface, thus rendering their collective presence detectable by various satellite-borne instruments. Almost all the bubbles that make up these structures were initially produced at the sea surface by breaking waves, and to understand the presence and distribution of whitecaps and foam patches it is necessary to first consider the genesis, and fate within the oceanic surface layer, of these bubbles. It will become apparent from the discussions contained in the following sections that the bubbles whose presence in great numbers is signaled by the appearance of whitecaps play a major role in the air–sea exchange of gases that are important in establishing our climate, and in the production of the sea-salt aerosol that contributes to the pool of cloud condensation nuclei in the atmosphere over the ocean. These same bubbles facilitate the sea-to-air transfer of heat and moisture, and scavenge from the bulk sea water and carry to the ocean surface various surfactant organic, and adhering inorganic, materials.

Spilling Wave Crests: Stage A Whitecaps When a wave breaks in the more typical spilling mode, and even more so when a wave collapses in a plunging fashion, great numbers of bubbles are formed and constrained initially to a relatively small volume of water, typically extending beneath the surface a distance no greater than the height of the source wave and having lateral dimensions of only a few meters at most. Although these intense bubble

clouds, often called alpha plumes, are individually often of convoluted shape, the concentration of bubbles in these alpha-plumes tends to decrease exponentially with depth, with an e-folding, or scale, depth that increases modestly from less than a meter to several meters, as the sea state increases in response to increasing wind speeds. The concentration of bubbles within an alpha-plume that has just been formed can be so great that the aggregate fraction of the water volume occupied by these bubbles, the void fraction in the terminology of the underwater acoustician, reaches 20% or even 30%. The size spectrum of the bubbles within such a plume is very broad, the bubbles present having radii ranging from several micrometers up to almost 10 mm (see Figure 1). Although there is no clear consensus on where the peak in the alpha-plume bubble number density spectrum lies, many authors would contend that it falls at a bubble radius of 50 mm or less. It has been suggested that the amplitude of this spectrum then falls off with increasing bubble radius in such a fashion that for over perhaps a decade of radius the total volume of the bubbles falling within a unit increment of size remains almost constant. It is thought that at even larger bubble radii this spectrum ‘rolls off’ even more rapidly, with less and less air being contained in those bubbles that fall in the larger and larger size ‘bins’, but there are as yet insufficient unambiguous observations to verify this contention. The manifestation on the sea surface of an alphaplume, the stage A whitecap, is the most readily detected category of whitecap or foam patch. Although bubbles on the surface in a stage A whitecap typically burst within a second of having arrived at the air– water interface, there is often a certain momentary ‘packing’ of bubbles, both vertically and laterally, on this surface, which results in this category of whitecap being truly white, with an albedo of about 0.5 which does not vary significantly over the entire visible portion of the electromagnetic spectrum. Since the visible albedo of the sea surface away from whitecaps is often 0.03–0.08, the average albedo of this surface will be noticeably increased when even a small fraction of the ocean surface is covered by spilling wave crests. Many of the satellite-borne passive microwave radiometers detect the electromagnetic emissions from the sea surface at wavelengths on the order of 10 mm. At such wavelengths a stage A whitecap is an almost perfect emitter, what in optics would be deemed a ‘black body’, while the rest of the sea surface at these wavelengths has a

41

42

WHITECAPS AND FOAM

microwave emissivity on only 30% or 40%. Thus it only requires a small fraction of the ocean surface to be covered by stage A whitecaps for there to be a measurable increase in the apparent microwave brightness temperature of this surface. An observer located within an alpha-plume would observe, once the downward movement associated with the spilling event had been dissipated, a high level of small scale turbulence, and, superimposed on top of the rapid random motions caused by this turbulence, a clear upward movement of the larger bubbles. The reduction of gravitational potential energy associated with the upward motion of these big bubbles frees energy that then contributes to the mixing and turbulence within the plume, and this enhanced mixing, which extends to the very surface of the stage A whitecap, greatly increases the effective air–sea gas transfer coefficient, or ‘piston velocity’, associated with this whitecap, as compared to the gas transfer coefficient associated with the wind ruffled but whitecap-less adjacent portions of the ocean surface. These upward moving bubbles drag water along with them, and the resulting upward, buoyant, flow often induces two-dimensional, horizontal divergence at the surface; factors which also enhance the air–sea exchange of gases. Further, for gases that diffuse slowly through water, the fact that each bubble is a gas vacuole traveling from the body of the water to the air–sea interface can be an important consideration. These large bubbles, with their large cross-sectional areas and rapid rise velocities, are also important in the scavenging and transport to the sea surface of the various surfaceactive materials that are often present in high concentrations in the oceanic mixed layer.

109

A 8

10

BD

107

B x

105

_

μm 1)

106

∂C/∂R (m

_3

J

x

A2

C

104

D 103

102

x

101

E

100 10

20

50

100

200 500 1000 2000 5000

Bubble radius (μm)

Figure 1 The number of bubbles per cubic meter of sea water, per micrometer bubble radius increment, as a function of bubble radius, as to be expected in (A) the alpha plume beneath a stage A whitecap, (B) the beta plume beneath a stage B whitecap, (C and D) in various portions of a gamma plume, and (E) the background, near-surface bubble layer. See Monahan and Van Pattern (1989) for further details.

Decaying Foam Patches: Stage B Whitecaps Within seconds of a wave ceasing to break, the associated stage A whitecap has been transformed in a decaying foam patch, a stage B whitecap. As a consequence of the intense turbulence present in the alpha plume that had been present beneath the stage A whitecap, the initial lateral extent of the stage B whitecap (and of the top of the beta-plume which is located beneath it) is typically considerably greater than that of a stage A whitecap, some would contend upwards of ten times greater. The greatest discrepancies in size between parent stage A whitecaps and the initial daughter stage B whitecaps occur in those cases where the wave crest spills persistently, or episodically, as it moves along over the sea surface leaving in its wake a long stage B whitecap, or a

WHITECAPS AND FOAM

0.1

B4 B2

A2 B3

0.01

A3 B1

A1 WA and WB

series of decaying foam patches with short distances between them. As was the case with the stage A whitecap, most of the bubbles that come to the surface in a stage B whitecap burst within a second of their arrival at the interface, and thus a stage B whitecap owes its existence, as did its precursor stage A whitecap, to the continuing arrival at the surface of new bubbles from the dependent bubble plume or cloud. The concentration of bubbles within the associated beta-plume is much smaller than it was in the alpha-plume that preceded it, for three reasons: (1) the plume has been diffused over a greater volume of sea water; (2) many of the large bubbles that were present in the precursor alpha plume have by now reached the sea surface and burst; and (3) most of the very smallest bubbles, those with radii of only a few micrometers, have gone into solution (see Figure 1). (The very smallest bubbles can dissolve even when the oceanic surface layer is saturated with respect to nitrogen and oxygen, because at a depth of even a meter they are subjected to significant additional hydrostatic pressure, and because with their small radii they experience a marked increase in internal pressure due to the influence of surface tension.) The stage B whitecap decays by being torn into tattered foam patches by the turbulence of the surface layer, and by having these ever and ever smaller patches fading as the supply of bubbles from the associated portions of the beta-plume becomes exhausted. The cumulative effect of these factors is that the visually resolvable macroscopic area of a stage B whitecap decreases exponentially with time, with a characteristic e-folding time of 3–4 s. A stage B whitecap appears to the eye as a group of irregularly shaped pale blue, or green, areas clustered on the ocean surface. The visible albedo of a stage B whitecap is initially intermediate between that of a stage A whitecap and that of the ruffled sea surface, but within a few seconds its albedo approaches the low value associated with the wave-roughened sea surface. As a consequence of the relatively larger initial area of stage B whitecaps as compared to stage A whitecaps, and on account of the fact that the characteristic lifetime of a stage B whitecap is considerably greater than that of a stage A one, at any instant the fraction of the ocean surface covered by stage B whitecaps is typically at least an order of magnitude greater than the fraction covered by stage A whitecaps (see Figure 2). The beta plume beneath each stage B whitecap is relatively rich in bubbles of intermediate size, with some investigators suggesting that the bubble number density spectrum for this plume has a peak at a bubble radius of about 50 mm (see Figure 1). When bubbles of this size burst at the surface in a whitecap

43

0.001 A3

A2

0.0001

0.00001 1

2

5

10

20

50

_

U (m s 1) Figure 2 The fraction of the ocean surface covered by stage A (curves A1–A3) and stage B (curves B1–B4) whitecaps as a function of 10 m-elevation wind speed. See Monahan and Van Pattern (1989) for further details.

they inject into the atmosphere droplets of several micrometers radius, called jet droplets, which contribute to the sea-to-air transfer of moisture and latent, and often sensible, heat. The rupture of the upper, exposed, hemisphere of these bubbles when they burst on the sea surface also produces smaller droplets, called film droplets, which constitute a

44

WHITECAPS AND FOAM

significant fraction of the cloud condensation nuclei in the maritime troposphere. The largest bubbles produced by a breaking wave, most of which reach the surface in the stage A whitecap, are even more effective at generating these film droplets when they burst. Because bubbles are relatively scarce within stage B whitecaps, these sea surface manifestations of beta plumes have visual albedos and microwave emissivities not greatly different from those of the adjacent, wind ruffled, surface, and are thus much more difficult to detect by remote sensing than are stage A whitecaps.

Wind-Dependence of Oceanic Whitecap Coverage The frequency of wave breaking, and the average intensity of the individual breaking wave, both increase with increasing wind speed. The combined effect of these two factors is that the fraction of the sea surface covered at any moment by spilling wave crests, i.e. by stage A whitecaps, increases rapidly with strengthening wind speed. This can be seen from Figure 2, where the curves labeled A1, A2, etc. are summary descriptions of the dependence of stage A whitecap coverage on 10 m-elevation wind speed. Curve A1, describing the most comprehensive set of stage A whitecap observations (actually a combination of four such sets), is described by eqn [1]. WA ¼ 3:16  107 U3:2

½1

where WA is the fraction of the sea surface covered at any instant by spilling wave crests and U is the 10 melevation wind speed expressed in meters per second. Understandably, the fraction of the sea surface covered instantaneously by decaying foam patches, i.e. by stage B whitecaps, shows a similar strong dependence on wind speed. This can be seen from the steep slopes of the curves B1, B2, etc., on the log–log plot in Figure 2. Eqn [2] defines curve B1, which is a summary description of extensive observations, from both the Atlantic and Pacific Oceans, of stage B whitecap coverage made by several investigators. WB ¼ 3:84  106 U3:41

½2

Here WB represents the instantaneous fraction of the sea surface covered by decaying foam patches, and U is again the 10 m-elevation wind speed. The fact that for both categories of whitecap the fraction of the ocean surface occupied by these features varies with the wind speed raised to something slightly more than the third power, is consistent with the

contention that whitecap coverage varies with the friction velocity (see Heat and Momentum Fluxes at the Sea Surface and Wave Generation by Wind) raised to the third power. It should be stressed that although whitecap coverage, both stage A and stage B, is most sensitive to wind speed, it also varies with the thermal stability of the lower marine atmospheric boundary layer, and with wind duration and fetch. Any factor that influences sea state will also affect whitecap coverage. For near-neutral atmospheric stability, oceanic whitecap coverage begins to be noticed when the 10 m-elevation wind speed reaches 3 or 4 m s1. (There is not a distinct threshold for the onset of whitecapping at a wind speed of 7 m s1 as was contended in some of the early literature on this subject.) Since whitecap coverage, particularly stage A coverage, is readily detectable from space, and given that whitecap coverage is very sensitive to wind speed, it is apparent that satellite observations of whitecap coverage can be routinely used to infer over-water wind speeds.

Stabilized Sea Foam Many of the first bubbles to rise to the sea surface after a breaking wave has entrained air, not only scavenge organic material from the upper meter or so of the sea but also, as they reach the air–sea interface, accrue some of the organic material that is often found on that surface (not necessarily in the form of coherent slicks). As a consequence of accreting on their surface considerable dissolved, and other, organic material, such ‘early rising’ bubbles may become stabilized, and hence they may not break immediately, but rather persist on the ocean surface for protracted periods. If such a bubble has managed to coat its entire upper hemisphere with such surfactant material, the markedly reduced surface tension of its film ‘cap’ that results from this circumstance may enable this bubble to persist indefinitely at the air–water interface. Such bubbles are certainly present at the sea surface long enough to be winnowed into windrows, those distinctive, essentially downwind, foam and seaweed streaks that appears on the sea surface when a strong wind has been blowing consistently. Often organized convective motions are present in the upper layer of the ocean. Such Langmuir cells have associated with them lines of horizontal, two-dimensional, surface convergence and divergence, oriented for the most part downwind. When such Langmuir cells are present, stabilized bubbles will be drawn into the convergence zones, and since they are buoyant, they will

WHITECAPS AND FOAM

remain to form fairly uniformly spaced foam lines on the sea surface marking the locations of such zones (Figure 3). It should be noted that the ‘late arriving’ bubbles, representing the vast majority of the bubbles rising within any alpha-plume, do not persist on the air–sea interface for more than a second or so, even when the surface waters are quite organically rich. The ability of bubbles to effectively scavenge surfactant organic matter from the bulk sea water and transport this material to the sea surface provides what has been described as an ‘organic memory’ to the upper mixed layer of the ocean. The more bubbles that have been injected into the upper layer of

45

the ocean by breaking waves in the recent past, the more organic material has been brought to the sea surface and remains there. Although wave action is ‘a two way street’, in that the same waves which upon breaking produce the bubbles that carry organic material to the air–water interface also stir and mix the surface layer, none-the-less the net effect of high sea states is to alter the partition of organic matter between the bulk fluid and the interface in favor of the interface. This can be inferred from the observation that as a high wind event persists, more and more foam lines, containing more and more stabilized bubbles, appear on the ocean surface. In stormy conditions, such foam, or spume, can be

_1

d

10

m

s

in

W

Windrows

SW SF

m

A

10

B 

10

m

LC

10 m

90˚ Rotation





  Figure 3 A view looking obliquely down at the sea surface showing stage A and stage B whitecaps, foam and spume lines, and simultaneously a view looking obliquely up toward the same sea surface showing the alpha- and beta-plumes associated with these whitecaps, the gamma-plumes, and the near-surface bubble layer. The influence on these features of a classical Langmuir circulation, which is indicated by arrows, is depicted. A, Stage A whitecap; B, Stage B whitecap; SF, stabilized foam; SW, seaweed; LC, Langmuir circulation; a, plume of stage A whitecap; b, plume of stage B whitecap; g, old (microbubble) plume; Z, background bubble layer; y bubble curtain. From Monahan and Lu, 1990.

46

WHITECAPS AND FOAM

blown off the crests of waves, along with quite large drops of water, called ‘spume drops’, adding further to the indeterminacy that often prevails in such circumstances regarding the actual location of the air– water interface. Not only do the above-mentioned Langmuir cells advent buoyant stabilized bubbles into the surface convergence zones, these same cells are believed to move the residual, long-lasting, gamma bubble plumes (those left after the dissipation of the beta plumes) into these same zones. (Alpha bubble plumes have readily detectable stage A whitecaps as their sea surface signatures, and the location of the betaplumes into which these alpha-plumes decay can be determined from the position on the sea surface of their associated stage B whitecaps, but the large, diffuse, bubble-poor gamma-plumes into which the beta-plumes decay, have no apparent surface manifestation.) The influence of Langmuir cells on stabilized sea surface foam, on gamma-plumes, and on the near surface layer that contains an ever sparser concentration of small bubbles, is depicted in Figure 3.

Global Implications As can be seen from the curves in Figure 2, even at quite high wind speeds such as 15 m s1 (33.5 miles h1), only a small fraction of the sea surface is covered by stage B whitecaps (0.04 or 4%), and an even smaller fraction of that surface is covered by stage A whitecaps (0.002 or 0.2%). Yet the total area of all the world’s oceans is very great (3.61  1014 m2), and as a consequence the total area of the global ocean covered by whitecaps at any instant is considerable. If a wind speed of 7 m s1 is taken as a representative value, then at any instant some 7.0  1010 m2, i.e. some 70 000 km2, of stage A whitecap area is present on the surface of the global ocean. Following from this, and including such additional information as the terminal rise velocity of bubbles, it can be deduced that some 7.2  1011 m2, i.e. some 720 000 km2 of individual bubble surface area are destroyed each second in all the stage A whitecaps present on the surface of all the oceans, and an equal area of bubble surface is being generated in the same interval. The vast amount of bubble surface area destroyed each second on the surface of all the world’s oceans, and the great volume of water (some 2.5  1011 m3) swept by all the bubbles that burst on

the sea surface each second, have profound implications for the global rate of air–sea exchange of moisture, heat and gases. An additional preliminary calculation following along these lines, suggests that all the bubbles breaking on the sea surface each year collect some 2 Gt of carbon during their rise to the ocean surface.

See also Heat and Momentum Fluxes at the Sea Surface. Wave Generation by Wind.

Further Reading Andreas EL, Edson JB, Monahan EC, Rouault MP, and Smith SD (1995) The spray contribution to net evaporation from the sea review of recent progress. Boundary-Layer Meteorology 72: 3--52. Blanchard DC (1963) The electrification of the atmosphere by particles from bubbles in the sea. Progress in Oceanography 1: 73--202. Bortkovskii RS (1987) Air–Sea Exchange of Heat and Moisture During Storms, revised English edition. Dordrecht: D. Reidel [Kluwer]. Liss PS and Duce RA (eds.) (1997) The Sea Surface and Global Change. Cambridge: Cambridge University Press. Monahan EC and Lu M (1990) Acoustically relevant bubble assemblages and their dependence on meteorological parameters. IEEE Journal of Oceanic Engineering 15: 340--349. Monahan EC and MacNiocaill G (eds.) (1986) Oceanic Whitecaps, and Their Role in Air–Sea Exchange Processes. Dordrecht: D. Reidel [Kluwer]. Monahan EC and O’Muircheartiaigh IG (1980) Optimal power-law description of oceanic whitecap coverage dependence on wind speed. Journal of Physical Oceanography 10: 2094--2099. Monahan EC and O’Muircheartiaigh IG (1986) Whitecaps and the passive remote sensing of the ocean surface. International Journal of Remote Sensing 7: 627--642. Monahan EC and Van Patten MA (eds.) (1989) Climate and Health Implications of Bubble-Mediated Sea–Air Exchange. Groton: Connecticut Sea Grant College Program. Thorpe SA (1982) On the clouds of bubbles formed by breaking wind waves in deep water, and their role in air–sea gas transfer. philosophical Transactions of the Royal Society [London] A304: 155-210.

BREAKING WAVES AND NEAR-SURFACE TURBULENCE J. Gemmrich, University of Victoria, Victoria, BC, Canada & 2009 Elsevier Ltd. All rights reserved.

Introduction Most readers will associate wave breaking with breaking surf at shallow beaches. This article, however, deals with breaking wind waves in deep water where wave and turbulence fields are not affected by the presence of the seafloor. Deep-water surface waves are sometimes compared to a gearbox linking the atmosphere to the oceans. In this analogy, breaking waves would indicate a high gear. They play a dominant role in many upper ocean processes, such as momentum transfer from wind to ocean currents, dissipation of wave energy, entrainment of air bubbles, disruption of surface films, and the generation of sea spray and aerosols, besides being a source of ambient noise. Wave breaking causes enhanced turbulent kinetic energy levels in the near-surface layer and thus governs turbulent transport of heat, gases, and particles in the near-surface zone. The forces exerted by breaking waves on ships and offshore structures are up to 10 times larger than for nonbreakers, and especially larger breaking waves pose significant danger to seafarers. Turbulence generated by breaking waves is very intermittent and coexists with turbulence generated by other sources such as shear stress, convection, internal waves, and Langmuir circulation. Presently, it is not understood how wave-induced turbulence interacts with this background turbulence, and these complex interactions are not discussed in any detail in this article.

Breaking Waves Wave breaking occurs on a wide range of scales and strength. At low wind speeds, the breaking of short wind waves of centimeter to meter wave lengths commonly does not generate any visible air entrainment and is called microbreaking. These small breakers disrupt the molecular boundary layer and play an important role in air–sea gas exchange. Microbreakers break up the cool surface skin of the

ocean, a process best detected with infrared sensors. At moderate to high wind speed, breaking waves start to generate small air bubbles which can be observed as whitecaps. The onset of whitecap generation is associated with a minimum wind speed of order 5 m s1, although no absolute threshold wind speed applies universally. Depending on the whitecap generation mechanism, the waves are labeled spilling breakers or plunging breakers. The most dramatic form of breaking is found in plunging breakers, where the waves overturn and a sheet of moving water plunges down at some distance forward from the wave crest, creating a large air cavity as well as smaller bubbles surrounding the intruding jet. This type of breaker is most common in shallow water and on beaches, but has also been observed in open ocean conditions of wind waves propagating against swell. In deep water nearly all breakers are spilling breakers in which a turbulent current at the wave crest entrains air, leading to a whitecap at the forward edge of the crest. Initial air fractions in breaking waves can be as high as 70–80%, but very rapidly decrease to o10%. In spilling breakers these high air fractions are limited to a shallow depth of O(0.1 m). At depths below 1 m, air fractions of order 103, decaying within one wave period, to order 105 are observed. It is apparent to the casual observer that wave breaking activity and whitecap coverage increase with increasing wind speed. However, this is only an indirect dependence and vastly different breaking rates are observed on different occasions with similar wind speeds but varied other factors such as fetch, wave age, or underlying currents. It has been long known that waves on the ocean often form groups consisting of roughly four to eight individual waves with wave crests near the center of the group being much higher than waves at the beginning or end. This group structure also affects the breaking occurrence. Observing the coastal ocean from a high vantage point or a low-altitude airplane, one finds that successive dominant whitecaps are often separated by one wave length but the period between the onsets of successive breakers is equal to two wave periods. This is because waves tend to break near the center of the group where their steepness is at a maximum and in deep water, the wave group propagates at half the phase speed of the individual waves. Therefore, one wave period after the breaking

47

48

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

onset, the breaking crest has propagated forward of the group center (its speed relative to the group being half its phase speed), so that its amplitude and thus its steepness are reduced. However, after two wave periods, when the group has propagated by one wave length, the next crest has reached the center of the group and is likely to break, leading to the observed regularity (see Figure 1). This process is most prevalent in narrow-banded, nearly unidirectional wave fields, most commonly observed in coastal waters or on larger lakes. The larger directional spreading and broader spectral shape of open-ocean wave fields often obscure the idealistic breaker regularity described above. Recent theoretical and observational studies suggest that the onset of breaking is determined by the redistribution of wave energy due to nonlinear wave–wave interactions rather than by direct wind forcing. These nonlinear wave hydrodynamics are reflected in the shape of the wave energy spectrum S(o), where o is the wave frequency. In particular, the wave saturation sðoÞ ¼ 2g2 o5 SðoÞ, where g is the gravitational acceleration, provides a good indicator of breaking activity. It also illustrates that wave breaking cannot be ascribed to individual wave properties but rather to the complex interplay of many wave components. Observations suggest that a certain wave saturation must be exceeded for breaking to set in, and the breaking rate increases

t=0 Wave breaking at x = 0

with increasing wave saturation. The exact form of this threshold behavior is not yet well established. It seems impossible to define precise threshold conditions for the onset of breaking of individual waves within a random wave spectrum. Theory predicts that waves break when the Lagrangian downward acceleration  @ 2Z/@t2 of the fluid at the crest exceeds a portion of the gravitational acceleration, @ 2 Z=@t2 > ag, where a ¼ 0.5 for the limiting Stokes wave but ao0.39 for the so-called almost steepest waves, and Z(t) is the surface elevation record at a fixed point. This threshold mechanism has been the basis for several theoretical studies of wave breaking; however, it is not conclusively supported by observations, where wave breaking has been observed at accelerations below the threshold value. Similarly, the theoretical maximum steepness of the limiting Stokes wave, ak ¼ p/7, where a and k are wave amplitude and wave number, respectively, is hardly ever observed in the field, and extensive observations reveal that breaking and nonbreaking waves cannot be separated on the basis of local steepness alone. Wave breaking may occur at all wave scales. However, the spectral distribution of breakers depends on the wave development and the most common breakers are associated with waves of higher frequencies than op, the frequency at the peak of the energy spectrum. In fact, only in young wave fields,

t=T No wave breaking

t = 2T Wave breaking at x = 

Figure 1 Idealized wave breaking periodicity. Waves tend to break in the center of wave groups. Thus, breakers are separated by one wave length l, and the period between successive breakers equals two wave periods T.

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

where the wave saturation at the peak is sufficiently large, are breaking waves observed at all scales, including the dominant waves. As the wave field develops the distribution of breaking scales narrows and its peak shifts further away from the dominant wave frequency toward higher frequencies. The total breaking rate, defined as the number of breaking waves of all frequencies passing a fixed location, depends on the distribution of breaking scales and thus on the shape of the wave saturation s(o). Limiting the breaking spectrum to breaking waves that result in visible whitecaps, one typically finds about 50–100 breakers per hour for open-ocean conditions and 12-m s  1 wind speed.

Turbulence beneath Breaking Waves Direct measurement of the fine-scale velocity field in the ocean and especially in the near-surface layer is extremely challenging. Surface waves are a source of enhanced turbulence. However, they also create the major difficulties in near-surface velocity measurements. Typical turbulent velocity fluctuations are O(10  2–10  1 m s  1) and thus are 10–100 times smaller than the wave-related velocities. In terms of kinetic energy, the wave motion contains 2–4 orders of magnitude higher energy levels than the turbulent motion. Furthermore, the nonlinear advection associated with the wave orbital motion modulates the turbulent flow observed at a fixed mooring and affects the dissipation estimates obtained from single point velocity records that rely on Taylor’s hypothesis of frozen turbulence. Nevertheless, significant progress in measuring wave-induced turbulence under natural conditions has been made, starting around the mid-1980s. However, most detailed information stems from controlled laboratory experiments, although the breaking characteristics in these studies are often very different to those of natural breaking waves. Focused superposition of dispersive mechanically generated waves leads to well-defined wave breaking even in the absence of wind forcing. This setup allows repeatable turbulence measurements. Turbulence beneath these breaking waves is seen to spread downward; within the first two wave periods this spreading is approximately a linear function of time and occurs more slowly thereafter. The final spreading depth of roughly twice the wave height is reached after four wave periods and by then about 90% of the energy lost by the breaking wave has been dissipated. Thereafter, the remaining decaying turbulence spreads only slightly further and may persist for tens of wave periods but can only be detected in an otherwise rather calm environment.

49

Behind the breaking crest, vortices of size comparable to the wave height are generated. These rotors may play an important role in mixing gases and pollutants such as small oil droplets. Nearly half of the energy lost from the breaking wave is associated with the entrainment of air bubbles, although part of it will be converted into turbulence kinetic energy (TKE) as larger bubbles rise through the water column. Under natural conditions, turbulence is commonly characterized by the dissipation rate of TKE e, which may be inferred from the turbulence velocity shear @u/@z or rate of strain @u/@x, or from wavenumber velocity spectra S(k). Thus, two fundamentally different approaches exist in oceanic turbulence measurements: (1) observation of the velocity shear or the rate of velocity strain, and (2) observation of the velocity field in space or time. In isotropic turbulence, the rate of dissipation is related to the rate of strain or the turbulence shear by 

   @u 2 15 @u 2 ¼ n e ¼ 15n @x 2 @z where n is the kinematic viscosity of the fluid, u the horizontal velocity component, and x and z are the horizontal and vertical coordinates, respectively. These relations are the basis for the pioneering studies of near-surface turbulence measurements made with towed hot-film anemometers, electromagnetic current meters, and the common microstructure profilers utilizing airfoil shear probes. The second class of turbulence measurements makes use of Kolmogorov’s inertial subrange hypothesis; within a subrange of the wavenumber band the onedimensional wavenumber spectrum S(k) ¼ Ae2/3k  5/3 has a universal form which depends only on the energy dissipation, where k is the wavenumber and A is a universal constant. This simple relationship between energy dissipation and wavenumber spectra allows the estimation of e from velocity measurements. Due to recent advances in sonar technology it is now possible to resolve instantaneous velocity profiles at spatial scales of a few millimeters and temporal resolution of a tenth of a second. These scales are suitable for turbulence measurements in the upper ocean. A common reference level for turbulence studies in boundary layers is the flow along a rigid wall. In this classic reference case, often labeled wall layer or constant stress layer, the velocity profile is logarithmic and the turbulent stress in the inner boundary layer t ¼ ru2 is nearly constant, where r is the fluid density and u the friction velocity. The TKE

50

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

dissipation per unit mass is given by e ¼ u3 ðkzÞ1, with k ¼ 0.4 being the von Ka´rma´n constant. Many studies show that turbulence in the ocean surface layer is enhanced compared to turbulence in a constant stress layer and there is strong evidence that the turbulence enhancement is due to breaking waves. The magnitude as well as the depth dependence of the time-averaged TKE dissipation in the near-surface layer of a wind-driven ocean departs significantly from the classic constant-stress-layer form. Observations indicate that the surface layer may be divided into three regimes. In the top layer wave breaking directly injects TKE down to a depth zb. In this injection layer, dissipation is highest and most likely depth-independent. Below this layer, the waveinduced turbulence diffuses downward and dissipates, as has been also demonstrated in the laboratory experiments. In this diffusive region, the decay of turbulence with depth is stronger than the wall-layer dependence epz  1. However, the exact depth dependence of the wave-induced turbulence is not well established and profiles consistent with epzn, with n in the range 4 to 2, as well as exponential profiles, epe  z, have been observed. Some open ocean observations under strong wind forcing and significant swell revealed enhanced dissipation values but depth dependence consistent with wall-layer scaling. Further down in the water column at a depth zt, sufficiently far from the air–sea interface, the contribution of waves becomes small compared to local shear production, and turbulence properties are well described by the constant stress layer scaling. There is, as yet, no conclusive observational evidence for the vertical extension of the different regimes. In particular, the depth of direct TKE injection dominates the total dissipation. Thus, zb is a crucial parameter in turbulence closure models, where it is implemented as the surface mixing length, which will also affect the mean profiles of tracers such as salt or heat. Mean dissipation profiles are commonly referenced to the mean water surface, and the oscillation of the sea surface poses a challenge for observation and interpretation of near-surface turbulence. Mooring or tower-based observations are limited to observations below the troughs, and depth is referenced to the mean still water line. Surface-following measurements from floats or ships are, in principle, also suitable to monitor the region above the troughs and depth reference is made with respect to the instantaneous surface. For example, in waves of 0.5-m wave height, a nominal 1-m depth observations from a tower is equivalent to a surface-referenced depth varying from 0.75 m at the location of the wave trough to 1.25 m in the crest region. The same

observation from a float would be converted to depth values ranging from 1.25 m at the trough to 0.75 m at the crest, if referenced to the still water line (see Figure 2). Due to the strong depth dependence of dissipation, the choice of coordinate systems affects the resultant dissipation profile. Microstructure profilers operated in a rising mode are capable of observing turbulence up to the sea surface. However, enhanced dissipation levels associated with wave breaking are very intermittent, and the profiling frequency is too low to adequately resolve these events. Despite the observational challenges, quality turbulence data in the aquatic near-surface layer in the presence of breaking waves have been collected starting in the mid-1980s. (Pioneering studies started in the 1960s.) The general consensus is that the enhancement of average dissipation en ¼ e=ðu3 ðkzÞ1 Þ in the diffusive layer is of order 10–100, where e is taken as the mean over a few minutes, and the depth of the wave enhanced layer (zt) is confined to a depth corresponding to a few times the significant wave height, that is 2–6 m typically. However, under severe storm conditions, the extent of the wave enhanced layer could be more than 10 m. Bubble clouds generated by wave breaking have been observed to such depths, but it is not known to what extents wave turbulence or coherent structures such as Langmuir circulation are the responsible bubble transport mechanisms. The depth of the injection layer zb is not well established. Estimates range from zb ¼ O(0.1 Hs) to zbXHs, or about 0.2–1 m. Instantaneous dissipation levels can be much higher than seen in the mean profiles. Beneath an active breaking wave, turbulence enhancements en ¼ O(104) have been observed. These high values persist only for a few seconds. Turbulence beneath individual breaking waves decays as eptm, where observations indicate m E 4 in the diffusion layer and m E 7.6 in the injection layer. Approximately five wave periods after the onset of breaking, the turbulence levels have decayed to the background level of wall-layer flows. Thus, sufficiently fast sampled dissipation measurements reveal the coexistence of two distinct contributions, a wide distribution centered on constant-stress-layer turbulence levels (log(en)E0) and a smaller and narrower distribution representing breaking waves and centered on log(en)X2. The broader distribution of lower enhancement rates is associated with periods between breaking events and is broadly consistent with a wall-layer flow. The largest turbulence levels, occurring beneath the actively breaking crest, play an important role in the breakup of air cavities and thus determine the initial bubble size distribution.

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

Tower observation reference: mean water line

51

Float observation reference: mean water line

c

c

t t

Tower observation reference: free surface

Float observation reference: free surface

c

c

t

t

Figure 2 Perceived depth of a measurement, depending on observation platform and choice of surface reference.

The balance between surface tension gw and turbulent pressure forces leads to the so-called Hinze scale aH, which describes the resulting bubble radius: aH ¼ Aðgw =rÞ3=5 e2=5 , where A is a constant in the range 0.36–0.5. Field observations yielded aHC10  3m, but more observations are required to establish a possible range of these initial bubble sizes. In spilling breakers, wave breaking occurs on the wave crest and turbulence levels have decayed significantly by the time the succeeding trough is reached. Therefore, turbulence levels in the crest region are larger than in the trough region and more than half of the energy is dissipated above the mean water line.

Wave breaking is a very intermittent phenomenon and the resulting turbulence fields are very patchy. Therefore, long time series or a suite of several sensors are required to obtain reliable statistics and sound estimates of the contribution of wave breaking to upper ocean processes. Alternatively, it was suggested in 1985 that the scale and strength of breaking may be characterized by the length of the breaking crest and its propagation speed. This opens up the possibility of remote sensing of the integral contribution of processes associated with wave breaking. The central quantity in this concept is the spectral density function L(c). It is defined in a way that L(c)dc describes the average total length of breaking

52

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

wave crests (perpendicular to the wave propagation), per unit area, that have speeds in the range c to c þ dc. Within a given surface area there might be several breaking crests at any given time, many of them that are only breaking along a fraction of the total crest lengths. To determine L(c), breaking crests that propagate at a similar speed are combined, the total lengths of these breaking crests are added up, and then the sum is divided by the area of the observed surface patch. The passage rate of breaking crests propagating at speed c past a fixed point is cL(c). The Rfractional surface turnover rate per unit time is R ¼ cL(c)dc, which can also be interpreted as the breaking frequency at a fixed point, that is the number of breakers passing a fixed location per unit time. Furthermore, the fourth and fifth moment of this spectral density function may be related to the dynamics of wave breaking. These relationships are based on similarity scaling of breakers and were confirmed in wave tank experiments. Therefore a further challenge is the proper scaling of the laboratory experiments to wave scales observed under natural conditions. Quasi-steady breakers can be generated by towing a submerged hydrofoil along a test channel. These experiments established that the rate of energy loss per unit length of breaking crest is proportional to c5. Therefore, the wave energy dissipation due to the breaking of waves of scale corresponding to phase speed c is e(c)dc ¼ brg  1 c5L(c)dc, where b is an unknown, nondimensional proportionality factor, originally assumed to be constant. However, b might depend on nondimensional expressions of, for example, the wave-scale or the wave-field nonlinearity. The total energy dissiassociated with whitecaps is E ¼ brg  1 Rpation 5 c L(c)dc. Momentum and energy are related by M ¼ Ec  1 and the spectrally resolved momentum flux from breaking waves to currents is m(c)dc ¼ brg  1c4L(c)dc. The total momentum flux from the field to currents is given as M ¼ brg  1 Rwave 4 c L(c)dc. Evaluation of this integral over all scales (phase speeds c) of breaking waves, including microbreakers, and assuming no wave growth in space yields M ¼ tw , where tw is the atmospheric momentum flux supported by the form drag of the waves. This momentum flux balance might prove to be a key relation in estimating the proportionality factor b. However, so far observations of the breaking crest density L(c) in the ocean do not adequately resolve the small-scale waves and microbreakers of the breaking spectrum. In a wind-driven sea, breaking waves provide the strongest contribution to near-surface turbulence. Other effects of surface waves are increased

dissipation levels prior to the onset of air entrainment. This prebreaking turbulence is consistent with wave–turbulence interaction in a rotational wave field. Increased near-surface Reynolds stresses due to the Stokes drift of nonbreaking waves may increase horizontal transports. Interaction of the Stokes drift with the wind-driven current may trigger Langmuir circulation, conceptually described as counterrotating cells aligned in the wind direction. Langmuir cells are large eddies and may be an important part of the turbulence field as well as influencing the turbulence generated by other processes such as breaking waves. Little is known how small-scale wave-enhanced turbulence affects lateral dispersion; however, it is likely that this process is dominated by Langmuir turbulence.

Conclusion Especially in mid- to high latitudes breaking waves are a ubiquitous feature of the open ocean. The last two to three decades have seen increased study of breaking deep water waves and new insight has been gained from laboratory experiments and field observations as well as through theoretical wave modeling. Earlier attempts to relate breaking to geometrical or kinematic features of individual wave crests are slowly being replaced by the concept of nonlinear hydrodynamics of the wave field leading to wave breaking. Wave breaking plays an important role in many processes of air–sea interaction, and the waveinduced turbulence is a relevant quantity in assessing its contributions. Near-surface turbulence observations show a mean dissipation enhancement of 1–2 orders of magnitude due to the effect of wave breaking. However, the detailed structure of the turbulence field and the length scales involved are still not resolved conclusively. Instantaneous dissipation levels are up to 4 orders larger than dissipation levels in a wall-layer flow. These initial high turbulence levels decay rapidly, but are likely to play a defining role in the breakup of air bubbles and thus in the setup of the bubble size distribution. Turbulence levels are highest beneath the wave crest and there is a need for more observations with adequate sampling of the region above the trough line. The concept of relating breaking wave kinematics and dynamics to whitecap properties that can be observed remotely, say with aerial video imagery or subsurface acoustical tracking, is intriguing and holds the promise of new observational insight in wave breaking processes. However, any quantitative assessments of energy dissipation and momentum fluxes based on this concept directly depend on the

BREAKING WAVES AND NEAR-SURFACE TURBULENCE

proportionality factor b. Currently, only very limited data exist and estimates of b are inconclusive; in fact, it is not even established that b is constant.

Nomenclature a aH c E g k m(c) M R

S(o) u u x z zb zt gw e en e(c) k L(c) n r s(o) t tw o op

wave amplitude Hinze scale wave phase speed total energy dissipation associated with wave breaking gravitational acceleration wave number spectrally resolved momentum flux from breaking waves to currents total momentum flux associated with wave breaking fractional surface turnover rate per unit time; equivalent to breaking frequency at a fixed location wave energy spectrum horizontal velocity component friction velocity horizontal coordinate vertical coordinate injection layer depth depth of enhanced wave-induced turbulence surface tension dissipation rate of turbulent kinetic energy enhancement of average dissipation spectrally resolved energy dissipation by breaking waves von Ka´rma´n constant spectral density function of breaking crest lengths kinematic viscosity fluid density wave saturation turbulent stress atmospheric momentum flux supported by the form drag of the waves wave frequency frequency of the peak of the wave energy spectrum

See also Air–Sea Gas Exchange. Bubbles. Estimates of Mixing. Langmuir Circulation and Instability. Rogue Waves. Surface Gravity and Capillary Waves. Turbulence in the Benthic Boundary Layer. Wave Generation by Wind.

53

Further Reading Banner ML (2005) Rougue waves and wave breaking – how are these phenomena related? In: Mu¨ller P and Henderson D (eds.) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http://www.soest.hawaii.edu/Pub Services/2005pdfs/Banner.pdf (accessed Feb. 2008). Banner ML and Peregrine DH (1993) Wave breaking in deep water. Annual Review of Fluid Mechanics 25: 373--397. Baschek B (2005) Wave-current action in tidal fronts. In: Mu¨ller P and Henderson D (eds.) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http:// www.soest.hawaii.edu/PubServices/2005pdfs/Baschek.pdf (accessed Feb. 2008). Colbo K and Li M (1999) Parameterizing particle dispersion in Langmuir circulation. Journal of Geophysical Research 104: 26059--26068. Donelan MA and Magnusson AK (2005) The role of focusing in generating rogue wave conditions. In: Mu¨ller P and Henderson D (eds.) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http:// www.soest.hawaii.edu/PubServices/2005pdfs/donelan.pdf (accessed Feb. 2008). Garrett C, Li M, and Farmer DM (2000) The connection between bubble size spectra and energy dissipation rates in the upper ocean. Journal of Physical Oceanography 30: 2163--2171. Gemmrich J (2005) A practical look at wave-breaking criteria. In: Mu¨ller P and Henderson D (eds.) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http://www.soest.hawaii.edu/PubServices/2005pdfs/ Gemmrich.pdf (accessed Feb. 2008). Gemmrich JR and Farmer DM (1999) Observations of the scale and occurrence of breaking surface waves. Journal of Physical Oceanography 29: 2595--2606. Gemmrich JR and Farmer DM (2004) Near surface turbulence in the presence of breaking waves. Journal of Physical Oceanography 34: 1067--1086. Holthuijsen LH and Herbers THC (1986) Statistics of breaking waves observed as whitecaps in the open sea. Journal of Physical Oceanography 16: 290--297. Melville WK (1996) The role of surface-wave breaking in air–sea interaction. Annual Review of Fluid Mechanics 26: 279--321. Melville WK and Matusov P (2002) Distribution of breaking waves at the ocean surface. Nature 417: 58--62. Mu¨ller P and Henderson D (eds.) (2005) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http:// www.soest.hawaii.edu/PubServices/2005pdfs/TOC2005. html (accessed Feb. 2008). Phillips OM (1985) Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. Journal of Fluid Mechanics 156: 505--531. Rapp R and Melville WK (1990) Laboratory measurements of deep water breaking waves. Philosophical Transactions of the Royal Society of London A 331: 735--780. Song J-B and Banner ML (2002) On determining the onset and strength of breaking for deep water waves.

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BREAKING WAVES AND NEAR-SURFACE TURBULENCE

Part 1: Unforced irrotational wave groups. Journal of Physical Oceanography 32: 2541--2558. Sullivan PP, McWilliams JC and Melville WK (2005) Surface waves and ocean mixing: Insights from numerical simulations. In: Mu¨ller P and Henderson D (eds.) Proceedings ‘Aha Huliko’ a Hawaiian Winter Workshop, Jan. 2005. http://www.soest.hawaii.edu/PubServices/2005 pdfs/Sullivan.pdf (accessed Feb. 2008).

Terray EA, Donelan MA, Agrawal YC, et al. (1996) Estimates of kinetic energy dissipation under breaking waves. Journal of Physical Oceanography 26: 792--807. Thorpe SA (1995) Dynamical processes of transfer at the sea surface. Progress in Oceanography 35: 315--352. Thorpe SA (2005) The Turbulent Ocean. Cambridge, UK: Cambridge University Press.

SEICHES D. C. Chapman, Woods Hole Oceanographic Institution, Woods Hole, MA, USA G. S. Giese, Woods Hole Oceanographic Institution, Woods Hole, MA, USA Copyright & 2001 Elsevier Ltd.

History In 1781, J. L. Lagrange found that the propagation velocity of a ‘long’ water wave (one whose wavelength is long compared to the water depth h) is given by ðghÞ1=2, where g is gravitational acceleration. Merian showed in 1828 that such a wave, reflecting back and forth from the ends of a closed rectangular basin of length L, produces a standing wave with a period T, given by

Introduction Seiches are resonant oscillations, or ‘normal modes’, of lakes and coastal waters; that is, they are standing waves with unique frequencies, ‘eigenfrequencies’, imposed by the dimensions of the basins in which they occur. For example, the basic behavior of a seiche in a rectangular basin is depicted in Figure 1. Each panel shows a snapshot of sea level and currents every quarter-period through one seiche cycle. Water moves back and forth across the basin in a periodic oscillation, alternately raising and lowering sea level at the basin sides. Sea level pivots about a ‘node’ in the middle of the basin at which the sea level never changes. Currents are maximum at the center (beneath the node) when the sea level is horizontal, and they vanish when the sea level is at its extremes. Seiches can be excited by many diverse environmental phenomena such as seismic disturbances, internal and surface gravity waves (including other normal modes of adjoining basins), winds, and atmospheric pressure disturbances. Once excited, seiches are noticeable under ordinary conditions because of the periodic changes in water level or currents associated with them (Figure 1). At some locations and times, such sea-level oscillations and currents produce hazardous or even destructive conditions. Notable examples are the catastrophic seiches of Nagasaki Harbor in Japan that are locally known as ‘abiki’, and those of Ciutadella Harbor on Menorca Island in Spain, called ‘rissaga’. At both locations extreme seiche-produced sea-level oscillations greater than 3 m have been reported. Although seiches in most harbors do not reach such heights, the currents associated with them can still be dangerous, and for this reason the study of coastal and harbor seiches and their causes is of practical significance to harbor management and design. In this article we place emphasis on marine seiches, especially those in coastal and harbor waters.

T¼

2L

½1

nðghÞ1=2

 (x, t )

t=0

h

T 4

T 2

3T 4

T

x=0

x=L

Figure 1 Diagram of a mode-one seiche oscillation in a closed basin through one period T . Panels show the sea surface and currents (arrows) each quarter-period. The basin length is L. The undisturbed water depth is h, and the deviation from this depth is denoted by =mi. Note the node at x ¼ L=2=mn where the sea surface never moves (i.e. =mi ¼ 0=mn).

55

56

SEICHES

where n ¼ 1; 2; 3; y is the number of nodes of the wave (Figure 1) and designates the ‘harmonic mode’ of the oscillation.Eqn [1] is known as Merian’s formula. F. A. Forel, between 1869 and 1895, applied Merian’s formula with much success to Swiss lakes, in particular Lake Geneva, the oscillations of which had long been recognized by local inhabitants who referred to them as ‘seiches’, apparently from the Latin word ‘siccus’ meaning ‘dry’. Forel’s seiche studies were of great interest to scientists around the world and by the turn of the century many were contributing to descriptive and theoretical aspects of the phenomenon. Perhaps most noteworthy was G. Chrystal who, in 1904 and 1905, developed a comprehensive analytical theory of free oscillations in closed basins of complex form. By the end of the nineteenth century, it was widely recognized that seiches also occurred in open basins, such as harbors and coastal bays, either as lateral oscillations reflecting from side-to-side across the basin, or more frequently, as longitudinal oscillations between the basin head and mouth. Longitudinal harbor oscillations are dynamically equivalent to lake seiches with a node at the open mouth (Figure 2), and a modified version of Merian’s formula for such open basins gives the period as T¼

4L 1=2

ð2n  1ÞðghÞ

½2

where n ¼ 1; 2; 3; y. The dynamics leading to both eqns [1] and [2] are discussed below. Interest in coastal seiches was fanned by the development of highly accurate mechanical tide recorders (see Tides) which frequently revealed surprisingly regular higher-frequency or ‘secondary’ oscillations in addition to ordinary tides. Even greater motivation was provided by F. Omori’s observation, reported in 1900, that the periods of destructive sea waves (see Tsunami) in harbors were often the same as those of the ordinary ‘secondary waves’ in those same harbors. This led directly to a major field, laboratory, and theoretical study that was carried out in Japan from 1903 through 1906 by K. Honda, T. Terada, Y. Yoshida, and D. Isitani, who concluded that coastal bays can be likened to a series of resonators, all excited by the same sea with its many frequencies of motion, but each oscillating at its own particular frequencies – the seiche frequencies. Most twentieth century seiche research can be traced back to problems or processes recognized in the important work of Honda and his colleagues. In particular, it became widely accepted that most coastal and harbor seiching is forced by open sea

 (x, t )

t=0

h

T 4

T 2

3T 4

T

x=0

x=L

Figure 2 Diagram of a seiche oscillation in a partially open basin through one period T . Panels show the sea surface and currents (arrows) each quarter-period. The basin length is L with the open end at x ¼ L. The undisturbed water depth is h, and the deviation from this depth is denoted by =mi. Note that the node (where =mi ¼ 0=mn) is located at the open end.

processes. A. Defant developed numerical modeling techniques which modern computers have made very efficient, and B. W. Wilson applied the theory to ocean engineering problems. Many others have made major contributions during the twentieth century.

Dynamics The dynamics of seiches are easiest to understand by considering several idealized situations with simplified physics. More complex geometries and physics have been considered in seiche studies, but the basic features developed here apply qualitatively to those studies. In a basin in which the water depth is much smaller than the basin length, fluid motions may be described by the depth-averaged velocity u and the deviation of the sea surface from its resting position

SEICHES

Z. Changes in these quantities are related through momentum and mass conservation equations: qu qZ ¼ g  ru qt qx

½3

qZ qu þh ¼0 qt qx

½4

in which h is the fluid depth at rest, r is a coefficient of frictional damping, g is gravitational acceleration, x is the horizontal distance and t is time (see Figure 1). Nonlinear and rotation effects have been neglected, and only motions in the x direction are considered. Eqn [3] states that the fluid velocity changes in response to the pressure gradient introduced by the tilting of the sea surface, and is retarded by frictional processes. Eqn [4] states that the seasurface changes in response to convergences and divergences in the horizontal velocity field; that is, where fluid accumulates qu=qxo0 the sea surface must rise, and vice versa. Eqns [3] and [4] can be combined to form a single equation for either Z or u, each having the same form. For example, q2 u qu q2 u þ r  gh 2 ¼ 0 2 qt qt qx

½5

Closed Basins

The simplest seiche occurs in a closed basin with no connection to a larger body of water, such as a lake or even a soup bowl. Figure 1 shows a closed basin with constant depth h and vertical sidewalls. At the sides of the basin, the velocity must vanish because fluid cannot flow through the walls, so u ¼ 0 at x ¼ 0 and L. Solutions of eqn [5] that satisfy these conditions and oscillate in time with frequency o are u ¼ u0 ert=2 cosðotÞsin

np  x L

½6

where u0 is the maximum current, n ¼ 1; 2; 3; y, and o ¼ ½ghðnp=LÞ2  r2 =41=2 . The corresponding sea-surface elevation is Z¼

i np  u0 oL rt=2 h r e x ½7 sinðotÞ  cosðotÞ cos gnp 2o L

Eqns [6] and [7] represent the normal modes or seiches of the basin. The integer n defines the harmonic mode of the seiche and corresponds to the number of velocity maxima and sea-level nodes (locations where sea level does not change) which occur where cosðnpx=LÞ ¼ 0.

57

The spatial structure of the lowest or fundamental mode seiche ðn ¼ 1Þ is shown schematically in Figure 1 through one period and was described above. Sea level rises and falls at each sidewall, pivoting about the node at x ¼ L=2. The velocity vanishes at the sidewalls and reaches a maximum at the node. Sea level and velocity are almost 901 out of phase; the velocity is zero everywhere when the sea level has its maximum displacement, whereas the velocity is maximum when the sea level is horizontal. The effect of friction is to cause a gradual exponential decay or damping of the oscillations and a slight decrease in seiche frequency with a shift in phase between u and Z. If friction is weak (small r), the seiche may oscillate through many periods before fully dissipating. In this case, the frequency is close to the undamped value, ðghÞ1=2 np=L with period ðT ¼ 2p=oÞ given by Merian’s formula,eqn [1]. If friction is sufficiently strong (very large r), the seiche may fully dissipate without oscillating at all. This occurs when r > 2ðghÞ1=2 np=L, for which the frequency o becomes imaginary. The speed of a surface gravity wave in this basin is ðghÞ1=2 , so the period of the fundamental seiche ðn ¼ 1Þ is equivalent to the time it takes a surface gravity wave to travel across the basin and back. Thus, the seiche may be thought of as a surface gravity wave that repeatedly travels back and forth across the basin, perfectly reflecting off the sidewalls and creating a standing wave pattern. Partially Open Basins

Seiches may also occur in basins that are connected to larger bodies of water at some part of the basin boundary (Figure 2). For example, harbors and inlets are open to the continental shelf at their mouths. The continental shelf itself can also be considered a partially open basin in that the shallow shelf is connected to the deep ocean at the shelf edge. The effect of the opening can be understood by considering the seiche in terms of surface gravity waves. A gravity wave propagates from the opening to the solid boundary where it reflects perfectly and travels back toward the opening. However, on reaching the opening it is not totally reflected. Some of the wave energy escapes from the basin into the larger body of water, thereby reducing the amplitude of the reflected wave. The reflected portion of the wave again propagates toward the closed sidewall and reflects back toward the open side. Each reflection from the open side reduces the energy in the oscillation, essentially acting like the frictional effects described above. This loss of energy due to the radiation of waves into the deep basin is called ‘radiation damping.’ Its effect is to

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produce a decaying response in the partially open basin, similar to frictional decay. In general, a wider basin mouth produces greater radiation damping, and hence a weaker resonant seiche response to any forcing. Conversely, a narrower mouth reduces radiation damping and hence increases the amplification of fundamental mode seiches, theoretically becoming infinite as the basin mouth vanishes. However, seiches are typically forced through the basin mouth (see below), so a narrow mouth is expected to limit the forcing and yield a decreased seiche response. J. Miles and W. Munk pointed out this apparent contradiction in 1961 and referred to it as the ‘harbor paradox.’ Later reports raised a number of questions concerning the validity of the harbor paradox, among them the fact that frictional damping, which would increase with a narrowing of the mouth, was not included in its formulation. The seiche modes of an idealized partially open basin (Figure 2) can be found by solving eqn [5] subject to a prescribed periodic sea-level oscillation at the open side; Z ¼ Z0 cosðstÞ at x ¼ L where s is the frequency of oscillation. For simplicity, friction is neglected by setting r ¼ 0. The response in the basin is cosðkxÞ Z ¼ Z0 cosðstÞ cosðkLÞ u ¼ Z0 ðg=hÞ1=2 sinðstÞ

sinðkxÞ cosðkLÞ

½8

½9

where the wavenumber k is related to the frequency by s ¼ ðghÞ1=2 k. The response is similar to that in the closed basin, with the velocity and sea level again 901 out of phase. The spatial structure ðkÞ is now determined by the forcing frequency. Notice that both the velocity and sea-level amplitudes are inversely proportional to cosðkLÞ, which implies that the response will approach infinity (resonance) when cosðkLÞ ¼ 0. This occurs when k ¼ ðnp  p=2Þ=L, or equivalently when s ¼ ðghÞ1=2 ðnp  p=2Þ=L where n ¼ 1; 2; 3y is any integer. These resonances correspond to the fundamental seiche modes for the partially open basin. They are sometimes called ‘quarter-wave resonances’ because their spatial structure consists of odd multiples of quarter wavelengths with a node at the open side of the basin ðx ¼ LÞ. The first mode ðn ¼ 1Þ contains one-quarter wavelength inside the basin (as in Figure 2), so its total wavelength is equal to four times the basin width L, and its period is T ¼ 2p=s ¼ 4L=ðghÞ1=2 . Other modes have periods given by eqn [2]. Despite the fact that these modes decay in time owing to radiation damping, they are expected to be

the dominant motions in the basin because their amplitudes are potentially so large. That is, if the forcing consists of many frequencies simultaneously, those closest to the seiche frequencies will cause the largest response and will remain after the response at other frequencies has decayed. Furthermore, higher modes ðn  2Þ have shorter length scales and higher frequencies, so they are more likely to be dissipated by frictional forces, leaving the first mode to dominate the response. This is much like the ringing of a bell. A single strike of the hammer excites vibrations at many frequencies, yet the fundamental resonant frequency is the one that is heard. Finally, the enormous amplification of the resonant response means that a small-amplitude forcing of the basin can excite a much larger response in the basin. Observations in the laboratory as well as nature reveal that seiches have somewhat longer periods than those calculated for the equivalent idealized open basins discussed above. This increase is similar to that which would be produced by an extension in the basin length, L, and it results from the fact that the water at the basin mouth has inertia and therefore is disturbed by, and participates in, the oscillation. This ‘mouth correction’, which was described by Lord Rayleigh in 1878 with respect to air vibrations, increases with the ratio of mouth width to basin length. In the case of a fully open square harbor, the actual period is approximately one-third greater than in the idealized case. In nature, forcing often consists of multiple frequencies within a narrow range or ‘band’. In this case, the response depends on the relative strength of the forcing in the narrow band and the resonant response at the seiche frequency closest to the dominant band. If the response at the dominant forcing frequency is stronger than the response at the resonant frequency, then oscillations will occur primarily at the forcing frequency. For example, the forcing frequency s in eqn any seiche frequency, and the energy in the forcing at the resonant seiche frequency may be so small that it is not amplified enough to overwhelm the response at the dominant forcing frequency. In this case, the observed oscillations, sometimes referred to as ‘forced seiches’, will have frequencies different from the ‘free seiche’ frequencies discussed above.

Generating Mechanisms and Observations Seiches in harbors and coastal regions may be directly generated by a variety of forces, some of which are depicted in Figure 3: (1) atmospheric pressure

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59

Atmospheric pressure fluctuations

Wind stress

Surface gravity waves

Internal gravity waves

Seismic activity

Figure 3 Sketch of various forcing mechanisms that are known to excite harbor and coastal seiches. Arrows for atmospheric pressure fluctuations, surface gravity waves, and internal gravity waves indicate propagation. Arrows for wind stress and seismic activity indicate direction of forced motions.

fluctuations; (2) surface wind stress (see Storm Surges); (3) surface gravity waves caused by seismic activity (see Tsunami); (4) surface gravity waves formed by wind (see Wave Generation by Wind); and (5) internal gravity waves (see Internal Tides and Internal Tides). It should be kept in mind that each of these forcing mechanisms can also generate or enhance other forcing mechanisms, thereby indirectly causing seiches. Thus, precise identification of the cause of seiching at any particular harbor or coast can be difficult. To be effective the forcing must cause a change in the volume of water in the basin, and hence the sea level, which is usually accomplished by a change in the inflow or outflow at the open side of the basin. The amplitude of the seiche response depends on both the form and the time dependence of the forcing. The first mode seiche typically has a period somewhere between a few minutes and an hour or so, so the forcing must have some energy near this period to generate large seiches. As an example, a sudden increase in atmospheric pressure over a harbor could force an outflow of water, thus lowering the harbor sea level. When the atmospheric pressure returns to normal, the harbor rapidly refills, initiating harbor seiching. However, although atmospheric pressure may change rapidly enough to match seiche frequencies, the magnitude of such high frequency fluctuations typically produces sea-level changes of only a few centimeters, so direct forcing is unlikely though not unknown.

Several examples of direct forcing of fundamental and higher-mode free oscillations of shelves, bays, and harbors by atmospheric pressure fluctuations have been described. In the case of the observations at Table Bay Harbor, in Cape Town, South Africa, it was noted that ‘a necessary ingredient ywas found to be that the pressure waves approach y from the direction of the open sea.’ In lakes, seiches are frequently generated by relaxation of direct wind stress, and since wind stress acting on a harbor can easily produce an outflow of water, this might seem to be a significant generation mechanism in coastal waters as well. However, strong winds rarely change rapidly enough to initiate harbor seiching directly. That is, the typical timescales for changes in strong winds are too long to match the seiche mode periods. Nevertheless, wind relaxation seiches have been observed in fiords and long bays such as Buzzards Bay in Massachusetts, USA. Tsunamis are rare, but they consist of large surface gravity waves that can generate enormous inflows into coastal regions, causing strong seiches, especially in large harbors. The resulting seiches, which may be a mix of free and forced oscillations, were a major motivation for early harbor seiche research as noted earlier (see Tsunami). Direct generation of seiches by local seismic disturbances (as distinct from forcing by seismically generated tsunamis) is well established but very unusual. For example, the great Alaskan earthquake of 1964 produced remarkable

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seiches in the bayous along the Gulf of Mexico coast of Louisiana, USA. Similar phenomena are the sometimes very destructive oscillations excited by sudden slides of earth and glacier debris into highlatitude fiords and bays. Most wind-generated surface gravity waves tend to occur at higher frequencies than seiches, so they are not effective as direct forcing mechanisms. However, in some exposed coastal locations, windgenerated swells combine to form oscillations, called ‘infragravity’ waves, with periods of minutes. These low-frequency surface waves are a well-known agent for excitation of seiches in small basins with periods less than about 10 minutes. Noteworthy are observations of 2–6 min seiches in Duncan Basin of Table Bay Harbor, Cape Town, South Africa, that occur at times of stormy weather. In 1993, similar short period seiches were reported in Barbers Point Harbor at Oahu, Hawaii, and their relationship to local swell and infragravity waves was demonstrated (see Surface Gravity and Capillary Waves). Perhaps the most effective way of directly exciting harbor and coastal seiches is by internal gravity waves. These internal waves can have large amplitudes and their frequency content often includes seiche frequencies. Furthermore, internal gravity

waves are capable of traveling long distances in the ocean before delivering their energy to a harbor or coastline. In recent years this mechanism has been suggested as an explanation for the frequently reported and sometimes hazardous harbor seiches with periods in the range of 10–100 min. There is little or no evidence of a seismic origin for these seiches, and their frequency does not match that of ordinary ocean wind-generated surface waves. Their forcing has often been ascribed to meteorologically produced long surface waves. For example, it has been suggested that the ‘abiki’ of Nagasaki Harbor and the ‘Marrobbio’ in the Strait of Sicily may be forced by the passage of large low-pressure atmospheric fronts. In 1996, evidence was found that the hazardous 10minute ‘rissaga’ of Ciutadella Harbor, Spain, and offshore normal modes are similarly excited by surface waves generated by atmospheric pressure oscillations, and it was proposed that the term ‘meteorological tsunamis’ be applied to all such seiche events. However, attributing the cause of remotely generated harbor seiches to meteorologically forced surface waves does not account for observations that such seiches are frequently associated with ocean tides. In 1908 it was noted that in many cases harbor

0.4

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Figure 4 An example of harbor seiches from Puerto Princesa at Palawan Island in the Philippines. The tidal signal has been removed from this sea-level record to accentuate the bursts of 75-min harbor seiches. The seiches are excited by the arrival at the harbor mouth of internal wave packets produced by strong tidal current flow across a shallow sill some 450 km away.

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seiche activity occurs at specific tidal phases. In the 1980s, a clear association was found between tidal and seiche amplitudes and it was suggested that tidegenerated internal waves could be a significant agent for excitation of coastal and harbor seiches. In 1990, a study of the fundamental theoretical questions concerning transfer of momentum from internal waves to seiche modes and the wide frequency gap between tides and harbor seiches indicated that the high-frequency energy content of tide-generated internal solitary waves is sufficient to account for the energy of the recorded seiches, and a dynamical model for the generating process was published. Observations at Palawan Island in the Philippines have demonstrated that harbor seiches can be forced by tide-generated internal waves and, as might be expected, there was also a strong dependency between seiche activity and water column density stratification. Periods of maximum seiche activity are associated with periods of strong tides, an example of which is given in Figure 4, which shows an 8-day sea-level record from Puerto Princesa at Palawan Island with the tidal signal removed. Bursts of 75min harbor seiches are excited by the arrival at the harbor mouth of internal wave packets produced by strong tidal current flow across a shallow sill some 450 km away. The internal wave packets require 2.5 days to reach the harbor, producing a similar delay between tidal and seiche patterns. As an illustration, note the change in seiche activity from a diurnal to a semidiurnal pattern that is evident in Figure 4. A similar shift in tidal current patterns occurred at the internal wave generation site several days earlier. More recent observations at Ciutadella Harbor in Spain point to a second process producing internal wave-generated seiches. Often the largest seiche events at that harbor occur under a specific set of conditions – seasonal warming of the sea surface and extremely small tides – which combine to produce very stable conditions in the upper water column. It has been suggested that under those conditions, meteorological processes can produce internal waves

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by inducing flow over shallow topography and that these meteorologically produced internal waves are responsible for the observed seiche activity.

See also Internal Tides. Internal Waves. Storm Surges. Surface Gravity and Capillary Waves. Tides. Tsunami. Wave Generation by Wind.

Further Reading Chapman DC and Giese GS (1990) A model for the generation of coastal seiches by deep-sea internal waves. Journal of Physical Oceanography 20: 1459--1467. Chrystal G (1905) On the hydrodynamic theory of seiches. Transactions of the Royal Society of Edinburgh 41: 599--649. Defant A (1961) Physical Oceanography, vol 2. New York: Pergamon Press. Forel FA (1892) Le Leman (Collected Papers), 2 vols. Lausanne, Switzerland: Rouge. Giese GS, Chapman DC, Collins MG, Encarnacion R, and Jacinto G (1998) The coupling between harbor seiches at Palawan Island and Sulu Sea internal solitons. Journal of Physical Oceanography 28: 2418--2426. Honda K, Terada T, Yoshida Y, and Isitani D (1908) Secondary undulations of oceanic tides. Journal of the College of Science, Imperial University, Tokyo 24: 1--113. Korgen BJ (1995) Seiches. American Scientist. 83: 330– 341. Miles JW (1974) Harbor seiching. Annual Review of Fluid Mechanics 6: 17--35. Okihiro M, Guza RT, and Seymour RT (1993) Excitation of seiche observed in a small harbor. Journal of Geophysical Research 98: 18 201--18 211. Rabinovich AB and Monserrat S (1996) Meteorological tsunamis near the Balearic and Kuril islands: descriptive and statistical analysis. Natural Hazards 13: 55--90. Wilson BW (1972) Seiches. Advances in Hydroscience 8: 1--94.

TSUNAMI P. L.-F. Liu, Cornell University, Ithaca, NY, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction Tsunami is a Japanese word that is made of two characters: tsu and nami. The character tsu means harbor, while the character nami means wave. Therefore, the original word tsunami describes large wave oscillations inside a harbor during a ‘tsunami’ event. In the past, tsunami is often referred to as ‘tidal wave’, which is a misnomer. Tides, featuring the rising and falling of water level in the ocean in a daily, monthly, and yearly cycle, are caused by gravitational influences of the moon, sun, and planets. Tsunamis are not generated by this kind of gravitational forces and are unrelated to the tides, although the tidal level does influence a tsunami striking a coastal area. The phenomenon we call a tsunami is a series of water waves of extremely long wavelength and long period, generated in an ocean by a geophysical disturbance that displaces the water within a short period of time. Waves are formed as the displaced water mass, which acts under the influence of gravity, attempts to regain its equilibrium. Tsunamis are primarily associated with submarine earthquakes in oceanic and coastal regions. However, landslides, volcanic eruptions, and even impacts of objects from outer space (such as meteorites, asteroids, and comets) can also trigger tsunamis. Tsunamis are usually characterized as shallowwater waves or long waves, which are different from wind-generated waves, the waves many of us have observed on a beach. Wind waves of 5–20-s period (T ¼ time interval between two successive wave crests or troughs) have wavelengths (l ¼ T2(g/2p) distance between two successive wave crests or troughs) of c. 40–620 m. On the other hand, a tsunami can have a wave period in the range of 10 min to 1 h and a wavelength in excess of 200 km in a deep ocean basin. A wave is characterized as a shallowwater wave when the water depth is less than 5% of the wavelength. The forward and backward water motion under the shallow-water wave is felted throughout the entire water column. The shallow water wave is also sensitive to the change of water depth. For instance, the speed (celerity) of a shallowwater wave is equal to the square root of the product

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of the gravitational acceleration (9.81 m s  2) and the water depth. Since the average water depth in the Pacific Ocean is 5 km, a tsunami can travel at a speed of about 800 km h  1 (500 mi h  1), which is almost the same as the speed of a jet airplane. A tsunami can move from the West Coast of South America to the East Coast of Japan in less than 1 day. The initial amplitude of a tsunami in the vicinity of a source region is usually quite small, typically only a meter or less, in comparison with the wavelength. In general, as the tsunami propagates into the open ocean, the amplitude of tsunami will decrease for the wave energy is spread over a much larger area. In the open ocean, it is very difficult to detect a tsunami from aboard a ship because the water level will rise only slightly over a period of 10 min to hours. Since the rate at which a wave loses its energy is inversely proportional to its wavelength, a tsunami will lose little energy as it propagates. Hence in the open ocean, a tsunami will travel at high speeds and over great transoceanic distances with little energy loss. As a tsunami propagates into shallower waters near the coast, it undergoes a rapid transformation. Because the energy loss remains insignificant, the total energy flux of the tsunami, which is proportional to the product of the square of the wave amplitude and the speed of the tsunami, remains constant. Therefore, the speed of the tsunami decreases as it enters shallower water and the height of the tsunami grows. Because of this ‘shoaling’ effect, a tsunami that was imperceptible in the open ocean may grow to be several meters or more in height. When a tsunami finally reaches the shore, it may appear as a rapid rising or falling water, a series of breaking waves, or even a bore. Reefs, bays, entrances to rivers, undersea features, including vegetations, and the slope of the beach all play a role modifying the tsunami as it approaches the shore. Tsunamis rarely become great, towering breaking waves. Sometimes the tsunami may break far offshore. Or it may form into a bore, which is a steplike wave with a steep breaking front, as the tsunami moves into a shallow bay or river. Figure 1 shows the incoming 1946 tsunami at Hilo, Hawaii. The water level on shore can rise by several meters. In extreme cases, water level can rise to more than 20 m for tsunamis of distant origin and over 30 m for tsunami close to the earthquake’s epicenter. The first wave may not always be the largest in the series of waves. In some cases, the water level will fall significantly first, exposing the bottom of a bay

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Figure 1 1946 tsunami at Hilo, Hawaii (Pacific Tsunami Museum). Wave height may be judged from the height of the trees.

or a beach, and then a large positive wave follows. The destructive pattern of a tsunami is also difficult to predict. One coastal area may see no damaging wave activity, while in a neighboring area destructive waves can be large and violent. The flooding of an area can extend inland by 500 m or more, covering large expanses of land with water and debris. Tsunamis may reach a maximum vertical height onshore above sea level, called a runup height, of 30 m. Since scientists still cannot predict accurately when earthquakes, landslides, or volcano eruptions will occur, they cannot determine exactly when a tsunami will be generated. But, with the aid of historical records of tsunamis and numerical models, scientists can get an idea as to where they are most likely to be generated. Past tsunami height measurements and computer modeling can also help to forecast future tsunami impact and flooding limits at specific coastal areas.

Historical and Recent Tsunamis Tsunamis have been observed and recorded since ancient times, especially in Japan and the Mediterranean areas. The earliest recorded tsunami occurred in 2000 BC off the coast of Syria. The oldest reference of tsunami record can be traced back to the sixteenth century in the United States. During the last century, more than 100 tsunamis have been observed in the United States alone. Among them, the 1946 Alaskan tsunami, the 1960 Chilean tsunami, and the 1964 Alaskan tsunami were the three most destructive tsunamis in the US history. The 1946 Aleutian earthquake (Mw ¼ 7.3) generated catastrophic tsunamis that attacked the Hawaiian Islands after traveling about 5 h and killed 159 people.

(The magnitude of an earthquake is defined by the seismic moment, M0 (dyn cm), which is determined from the seismic data recorded worldwide. Converting the seismic moment into a logarithmic scale, we define Mw ¼ (1/1.5)log10M0  10.7.) The reported property damage reached $26 million. The 1960 Chilean tsunami waves struck the Hawaiian Islands after 14 h, traveling across the Pacific Ocean from the Chilean coast. They caused devastating damage not only along the Chilean coast (more than 1000 people were killed and the total property damage from the combined effects of the earthquake and tsunami was estimated as $417 million) but also at Hilo, Hawaii, where 61 deaths and $23.5 million in property damage occurred (see Figure 2). The 1964 Alaskan tsunami triggered by the Prince William Sound earthquake (Mw ¼ 8.4), which was recorded as one of the largest earthquakes in the North American continent, caused the most destructive damage in Alaska’s history. The tsunami killed 106 people and the total damage amounted to $84 million in Alaska. Within less than a year between September 1992 and July 1993, three large undersea earthquakes strike the Pacific Ocean area, causing devastating tsunamis. On 2 September 1992, an earthquake of magnitude 7.0 occurred c. 100 km off the Nicaraguan coast. The maximum runup height was recorded as 10 m and 168 people died in this event. A few months later, another strong earthquake (Mw ¼ 7.5) attacked the Flores Island and surrounding area in Indonesia on 12 December 1992. It was reported that more than 1000 people were killed in the town of Maumere alone and two-thirds of the population of Babi Island were swept away by the tsunami. The maximum runup was estimated as 26 m. The final toll of this Flores earthquake stood at 1712 deaths and more than 2000 injures. Exactly

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Figure 2 The tsunami of 1960 killed 61 people in Hilo, destroyed 537 buildings, and damages totaled over $23 million.

7 months later, on 12 July 1993, the third strong earthquake (Mw ¼ 7.8) occurred near the Hokkaido Island in Japan (Hokkaido Tsunami Survey Group 1993). Within 3–5 min, a large tsunami engulfed the Okushiri coastline and the central west of Hokkaido, impinging extensive property damages, especially on the southern tip of Okushiri Island in the town of Aonae. The runup heights on the Okushiri Island were thoroughly surveyed and they varied between 15 and 30 m over a 20-km stretch of the southern part of the island, with several 10-m spots on the northern part of the island. It was also reported that although the runup heights on the west coast of Hokkaido are not large (less than 10 m), damage was extensive in several towns. The epicenters of these three earthquakes were all located near residential coastal areas. Therefore, the damage caused by subsequent tsunamis was unusually large. On 17 July 1998, an earthquake occurred in the Sandaun Province of northwestern Papua New Guinea, about 65 km northwest of the port city of Aitape. The earthquake magnitude was estimated as Mw ¼ 7.0. About 20 min after the first shock, Warapo and Arop villages were completely destroyed by tsunamis. The death toll was at over 2000 and many of them drowned in the Sissano Lagoon behind the Arop villages. The surveyed maximum runup height was 15 m, which is much higher than the predicted value based on the seismic information. It has been suggested that the Papua New Guinea tsunami could be caused by a submarine landslide. The most devastating tsunamis in recent history occurred in the Indian Ocean on 26 December 2004.

An earthquake of Mw ¼ 9.0 occurred off the west coast of northern Sumatra. Large tsunamis were generated, severely damaging coastal communities in countries around the Indian Ocean, including Indonesia, Thailand, Sri Lanka, and India. The estimated tsunami death toll ranged from 156 000 to 178 000 across 11 nations, with additional 26 500–142 000 missing, most of them presumed dead.

Tsunami Generation Mechanisms Tsunamigentic Earthquakes

Most tsunamis are the results of submarine earthquakes. The majority of earthquakes can be explained in terms of plate tectonics. The basic concept is that the outermost part the Earth consists of several large and fairly stable slabs of solid and relatively rigid rock, called plates (see Figure 3). These plates are constantly moving (very slowly), and rub against one another along the plate boundaries, which are also called faults. Consequently, stress and strain build up along these faults, and eventually they become too great to bear and the plates move abruptly so as to release the stress and strain, creating an earthquake. Most of tsunamigentic earthquakes occur in subduction zones around the Pacific Ocean rim, where the dense crust of the ocean floor dives beneath the edge of the lighter continental crust and sinks down into Earth’s mantle. These subduction zones include the west coasts of North and South America, the coasts of East Asia (especially Japan), and many Pacific island chains (Figure 3). There are

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different types of faults along subduction margins. The interplate fault usually accommodates a large relative motion between two tectonic plates and the overlying plate is typically pushed upward. This upward push is impulsive; it occurs very quickly, in a

Ridge axis divergent boundary

few seconds. The ocean water surface responds immediately to the upward movement of the seafloor and the ocean surface profile usually mimics the seafloor displacement (see Figure 4). The interplate fault in a subduction zone has been responsible for

Subduction zone Convergent boundary

Transform

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Zone of extension with continents

Uncertain plate boundary

Figure 3 Major tectonic plates that make up the Earth’s crust.

(a)

(b)

Stuck S ubdu cting

(c)

Overriding plate

Slow distortion

p late

(d) Earthquake starts tsunami

Tsunami waves spread

Stuck area ruptures, releasing energy in an earthquake Figure 4 Sketches of the tsunami generation mechanism caused by a submarine earthquake. An oceanic plate subducts under an overriding plate (a). The overriding plate deforms due to the relative motion and the friction between two tectonic plates (b). The stuck area ruptures, releasing energy in an earthquake (c). Tsunami waves are generated due to the vertical seafloor displacement (d).

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most of the largest tsunamis in the twentieth century. For example, the 1952 Kamchatka, 1957 Aleutian, 1960 Chile, 1964 Alaska, and 2004 Sumatra earthquakes all generated damaging tsunamis not only in the region near the earthquake epicenter, but also on faraway shores. For most of the interplate fault ruptures, the resulting seafloor displacement can be estimated based on the dislocation theory. Using the linear elastic theory, analytical solutions can be derived from the mean dislocation field on the fault. Several parameters defining the geometry and strength of the fault rupture need to be specified. First of all, the mean fault slip, D, is calculated from the seismic moment M0 as follows: M0 ¼ mDS

estimation, the fault plane can be approximated as a rectangle with length L and width W. The aspect ratio L/W could vary from 2 to 8. To find the static displacement of the seafloor, we need to assign the focal depth d, measuring the depth of the upper rim of the fault plane, the dip angle d, and the slip angle l of the dislocation on the fault plane measured from the horizontal axis (see Figure 5). For an oblique slip on a dipping fault, the slip vector can be decomposed into dip-slip and strike-slip components. In general, the magnitude of the vertical displacement is less for the strike-slip component than for the dip-slip component. The closed form expressions for vertical seafloor displacement caused by a slip along a rectangular fault are given by Mansinha and Smylie. For more realistic fault models, nonuniform stressstrength fields (i.e., faults with various kinds of barriers, asperities, etc.) are expected, so that the actual seafloor displacement may be very complicated compared with the smooth seafloor displacement computed from the mean dislocation field on the fault. As an example, the vertical seafloor displacement caused by the 1964 Alaska earthquake is sketched in Figure 6. Although several numerical models have considered geometrically complex faults, complex slip distributions, and elastic layers of variable thickness, they are not yet disseminated in

½1

din

al)

where S is the rupture area and m is the rigidity of the Earth at the source, which has a range of 6–7  1011 dyn cm  2 for interplate earthquakes. The seismic moment, M0, is determined from the seismic data recorded worldwide and is usually reported as the Harvard Centroid-Moment-Tensor (CMT) solution within a few minutes of the first earthquake tremor. The rupture area is usually estimated from the aftershock data. However, for a rough

To

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 Foot block

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 Strike angle  Dip angle 

 Slip angle L

X OY parrallel to the horizontal Earth surface; OZ pointing upward;  is the azimuth of OX measuring clockwise from the latitudinal Figure 5 A sketch of fault plane parameters.

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NW

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m Horizontal SW displacements

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Figure 6 A sketch of 1964 Alaska earthquake generated vertical seafloor displacement (G. Plafker, 2006).

tsunami research. One of the reasons is that our knowledge in source parameters, inhomogeneity, and nonuniform slip distribution is too incomplete to justify using such a complex model. Certain earthquakes referred to as tsunami earthquakes have slow faulting motion and very long rupture duration (more than several minutes). These earthquakes occur along the shallow part of the interplate thrust or de´collement near the trench (the wedge portion of the thin crust above the interface of the continental crust and the ocean plate). The wedge portion consisting of thick deformable sediments with low rigidity, and the steepening of rupture surface in shallow depth all favor the large displacement of the crust and possibility of generating a large tsunami. Because of the extreme heterogeneity, accurate modeling is difficult, resulting in large uncertainty in estimated seafloor displacement. Landslides and Other Generation Mechanisms

There are occasions when the secondary effects of earthquakes, such as landslide and submarine slump, may be responsible for the generation of tsunamis. These tsunamis are sometimes disastrous and have gained increasing attentions in recent years. Landslides are generated when slopes or sediment deposits become too steep and they fail to remain in equilibrium and motionless. Once the unstable conditions are present, slope failure can be triggered by storm, earthquakes, rains, or merely continued

deposition of materials on the slope. Alternative mechanisms of sediment instability range between soft sediment deformations in turbidities, to rotational slumps in cohesive sediments. Certain environments are particularly susceptible to the production of landslides. River delta and steep underwater slopes above submarine canyons are likely sites for landslide-generated tsunamis. At the time of the 1964 Alaska earthquake, numerous locally landslide-generated tsunamis with devastating effects were observed. On 29 November 1975, a landslide was triggered by a 7.2 magnitude earthquake along the southeast coast of Hawaii. A 60-km stretch of Kilauea’s south coast subsided 3.5 m and moved seaward 8 m. This landslide generated a local tsunami with a maximum runup height of 16 m at Keauhou. Historically, there have been several tsunamis whose magnitudes were simply too large to be attributed to the coseismic seafloor movement and landslides have been suggested as an alternative cause. The 1946 Aleutian tsunami and the 1998 Papua New Guinea tsunami are two significant examples. In terms of tsunami generation mechanisms, two significant differences exist between submarine landslide and coseismic seafloor deformation. First, the duration of a landslide is much longer and is in the order of magnitude of several minutes or longer. Hence the time history of the seafloor movement will affect the characteristics of the generated wave and needs to be included in the model. Second, the

68

TSUNAMI

effective size of the landslide region is usually much smaller than the coseismic seafloor deformation zone. Consequently, the typical wavelength of the tsunamis generated by a submarine landslide is also shorter, that is, c. 1–10 km. Therefore, in some cases, the shallow-water (long-wave) assumption might not be valid for landslide-generated tsunamis. Although they are rare, the violent geological activities associated with volcanic eruptions can also generate tsunamis. There are three types of tsunamigeneration mechanism associated with a volcanic eruption. First, the pyroclastic flows, which are mixtures of gas, rocks, and lava, can move rapidly off an island and into an ocean, their impact displacing seawater and producing a tsunami. The second mechanism is the submarine volcanic explosion, which occurs when cool seawater encounters hot volcanic magma. The third mechanism is due to the collapse of a submarine volcanic caldera. The collapse may happen when the magma beneath a volcano is withdrawn back deeper into the Earth, and the sudden subsidence of the volcanic edifice displaces water and produces a tsunami. Furthermore, the large masses of rock that accumulate on the sides of volcanoes may suddenly slide down the slope into the sea, producing tsunamis. For example, in 1792, a large mass of the mountain slided into Ariake Bay in Shimabara on Kyushu Island, Japan, and generated tsunamis that reached a height of 10 m in some places, killing a large number of people. In the following sections, our discussions will focus on submarine earthquake-generated tsunamis and their coastal effects.

Modeling of Tsunami Generation, Propagation, and Coastal Inundation To mitigate tsunami hazards, the highest priority is to identify the high-tsunami-risk zone and to educate the citizen, living in and near the risk zone, about the proper behaviors in the event of an earthquake and tsunami attack. For a distant tsunami, a reliable warning system, which predicts the arrival time as well as the inundation area accurately, can save many lives. On the other hand, in the event of a nearfield tsunami, the emergency evacuation plan must be activated as soon as the earth shaking is felt. This is only possible, if a predetermined evacuation/ inundation map is available. These maps should be produced based on the historical tsunami events and the estimated ‘worst scenarios’ or the ‘design tsunamis’. To produce realistic and reliable inundation maps, it is essential to use a numerical model that calculates accurately tsunami propagation from

a source region to the coastal areas of concern and the subsequent tsunami runup and inundation. Numerical simulations of tsunami have made great progress in the last 50 years. This progress is made possible by the advancement of seismology and by the development of the high-speed computer. Several tsunami models are being used in the National Tsunami Hazard Mitigation Program, sponsored by the National Oceanic and Atmospheric Administration (NOAA), in partnership with the US Geological Survey (USGS), the Federal Emergency Management Agency (FEMA), to produce tsunami inundation and evacuation maps for the states of Alaska, California, Hawaii, Oregon, and Washington. Tsunami Generation and Propagation in an Open Ocean

The rupture speed of fault plane during earthquake is usually much faster than that of the tsunami. For instance, the fault line of the 2004 Sumatra earthquake was estimated as 1200-km long and the rupture process lasted for about 10 min. Therefore, the rupture speed was c. 2–3 km s  1, which is considered as a relatively slow rupture speed and is still about 1 order of magnitude faster than the speed of tsunami (0.17 km s  1 in a typical water depth of 3 km). Since the compressibility of water is negligible, the initial free surface response to the seafloor deformation due to fault plane rupture is instantaneous. In other words, in terms of the tsunami propagation timescale, the initial free surface profile can be approximated as having the same shape as the seafloor deformation at the end of rupture, which can be obtained by the methods described in the previous section. As illustrated in Figure 6, the typical cross-sectional free surface profile, perpendicular to the fault line, has an N shape with a depression on the landward side and an elevation on the ocean side. If the fault plane is elongated, that is, L4 4W, the free surface profile is almost uniform in the longitudinal (fault line) direction and the generated tsunamis will propagate primarily in the direction perpendicular to the fault line. The wavelength is generally characterized by the width of the fault plane, W. The measure of tsunami wave dispersion is represented by the depth-to-wavelength ratio, that is, m2 ¼ h/l, while the nonlinearity is characterized by the amplitude-to-depth ratio, that is, e ¼ A/h. A tsunami generated in an open ocean or on a continental shelf could have an initial wavelength of several tens to hundreds of kilometers. The initial tsunami wave height may be on the order of magnitude of several meters. For example, the 2004 Indian Ocean tsunami

TSUNAMI

had a typical wavelength of 200 km in the Indian Ocean basin with an amplitude of 1 m. The water depth varies from several hundreds of meters on the continental shelf to several kilometers in the open ocean. It is quite obvious that during the early stage of tsunami propagation both the nonlinear and frequency dispersion effects are small and can be ignored. This is particularly true for the 2004 Indian tsunami. The bottom frictional force and Coriolis force have even smaller effects and can be also neglected in the generation area. Therefore, the linear shallow water (LSW) equations are adequate equations describing the initial stage of tsunami generation and propagation. As a tsunami propagates over an open ocean, wave energy is spread out into a larger area. In general, the tsunami wave height decreases and the nonlinearity remains weak. However, the importance of the frequency dispersion begins to accumulate as the tsunami travels a long distance. Theoretically, one can estimate that the frequency dispersion becomes important when a tsunami propagates for a long time: sffiffiffi  h l 3 ½2 t4 4td ¼ g h or over a long distance: x4 4xd ¼ td

pffiffiffiffiffiffi l3 gh ¼ 2 h

½3

In the case of the 2004 Indian Ocean tsunami, tdE700 h and xdE5  105 km. In other words, the frequency dispersion effect will only become important when tsunamis have gone around the Earth several times. Obviously, for a tsunami with much shorter wavelength, for example, lE20 km, this distance becomes relatively short, that is, xdE5  102 km, and can be reached quite easily. Therefore, in modeling transoceanic tsunami propagation, frequency dispersion might need to be considered if the initial wavelength is short. However, nonlinearity is seldom a factor in the deep ocean and only becomes significant when the tsunami enters coastal region. The LSW equations can be written in terms of a spherical coordinate system as:

@z 1 @P @ @h þ þ ðcosjQÞ ¼  ½4 @t Rcosj @c @j @t @P gh @z þ ¼0 @t Rcosj @c

½5

@Q gh @z þ ¼0 @t R @j

½6

69

where (c,j) denote the longitude and latitude of the Earth, R is the Earth’s radius, z is free surface elevation, P and Q the volume fluxes (P ¼ hu and Q ¼ hv, with u and v being the depth-averaged velocities in longitude and latitude direction, respectively), and h the water depth. Equation [4] represents the depth-integrated continuity equation, and the time rate of change of water depth has been included. When the fault plane rupture is approximated as an instantaneous process and the initial free surface profile is prescribed, the water depth remains timeinvariant during tsunami propagation and the righthand side becomes zero in eqn [4]. The 2004 Indian Ocean tsunami provided an opportunity to verify the validity of LSW equations for modeling tsunami propagation in an open ocean. For the first time in history, satellite altimetry measurements of sea surface elevation captured the Indian Ocean tsunami. About 2 h after the earthquake occurred, two NASA/French Space Agency joint mission satellites, Jason-1 and TOPEX/Poseidon, passed over the Indian Ocean from southwest to northeast ( Jason-1 passed the equator at 02:55:24UTC on 26 December 2004 and TOPEX/ Poseidon passed the equator at 03:01:57UTC on 26 December 2004) (see Figure 7). These two altimetry satellites measured sea surface elevation with accuracy better than 4.2 cm. Using the numerical model COMCOT (Cornell Multi-grid Coupled Tsunami Model), numerical simulations of tsunami propagation over the Indian Ocean with various fault plane models, including a transient seafloor movement model, have been carried out. The LSW equation model predicts accurately the arrival time of the leading wave and is insensitive of the fault plane models used. However, to predict the trailing waves, the spatial variation of seafloor deformation needs to be taken into consideration. In Figure 8, comparisons between LSW results with an optimized fault plane model and Jason-1/TOPEX measurements are shown. The excellent agreement between the numerical results and satellite data provides a direct evidence for the validity of the LSW modeling of tsunami propagation in deep ocean.

Coastal Effects – Inundation and Tsunami Forces

Nonlinearity and bottom friction become significant as a tsunami enters the coastal zone, especially during the runup phase. The nonlinear shallow water (NLSW) equations can be used to model certain aspects of coastal effects of a tsunami attack. Using the same notations as those in eqns [4]–[6], the NLSW

70

TSUNAMI

20

15

0.8

10

0.4 0

0.2

TOP

−10

EX

−0.2 −0.4

−1

0

−5

Jason

Latitude (deg)

0.6 5

−0.6

−15

−0.8 70

75

80

85 90 Longitude (deg)

95

100

105

Figure 7 Satellite tracks for TOPEX and Jason-1. The colors indicate the numerically simulated free surface elevation in meter at 2 h after the earthquake struck.

equations in the Cartesian coordinates are @z @P @Q þ þ ¼0 @t @x @y

½7

    @P @ P2 @ PQ @z þ þ þ gH þ tx H ¼ 0 @t @x H @y H @x

½8

    @Q @ PQ @ Q2 @z þ gH þ ty H ¼ 0 þ þ @t @x H @y H @y

½9

The bottom frictional stresses are expressed as tx ¼

gn2 PðP2 þ Q2 Þ1=2 H 10=3

½10

ty ¼

gn2 QðP2 þ Q2 Þ1=2 H 10=3

½11

where n is the Manning’s relative roughness coefficient. For flows over a sandy beach, the typical value for the Manning’s n is 0.02. Using a modified leapfrog finite difference scheme in a nested grid system, COMCOT is capable of solving both LSW and NLSW equations simultaneously in different regions. For the nested grid system, the inner (finer) grid adopts a smaller grid size and time step compared to its adjacent outer (larger) grid. At the beginning of a time step, along the interface of two different grids, the volume flux, P and Q, which is product of water depth and depthaveraged velocity, is interpolated from the outer (larger) grids into its inner (finer) grids. And at the end of this time step, the calculated water surface elevations, z, at the inner finer grids are averaged to update those values of the larger grids overlapping the finer grids, which are used to compute the volume fluxes at next time step in the outer grids. With this procedure, COMCOT can capture near-shore

TSUNAMI

(b)

1 Jason-1 Model

Water surface elevation (m)

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

−5

0

5 10 Latitude (deg)

Water surface elevation (m)

(a)

Model vs. TOPEX

1

Model TOPEX

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

15

71

−5

0

5 10 Latitude (deg)

15

Figure 8 Comparisons between optimized fault model results and Jason-1 measurements (a)/TOPEX measurements (b).

(a)

Grids with dx = 36.7 m

(b)

5.6

5.6 Ulee Lheue BANDA ACEH

5.55 Latitude (deg)

5.55

5.5 Lampuuk Lhoknga

5.45

5.5

5.45 Leupung

5.4

Inundated area

5.4

Dry land Ocean

95.15

95.2

95.25

95.3

95.35

95.4

95.15

95.2

95.25 95.3 Longitude (deg)

95.35

95.4

Figure 9 Calculated inundation areas (a) and overlaid with QUICKBIRD image (b) in Banda Aceh, Indonesia.

features of a tsunami with a higher spatial and temporal resolution and at the same time can still keep a high computational efficiency. To estimate the inundation area caused by a tsunami, COMCOT adopts a simple moving boundary scheme. The shoreline is defined as the interface between a wet grid and its adjacent dry grids. Along the shoreline, the volume flux is assigned to be zero. Once the water surface elevation at the wet grid is higher than the land elevation in its adjacent dry grid, the shoreline is moved by one grid toward the dry grid and the volume flux is no longer zero and need to be calculated by the governing equations. COMCOT, coupled up to three levels of grids, has been used to calculate the runup and inundation areas at Trincomalee Bay (Sri Lanka) and Banda Aceh (Indonesia). Some of the numerical results for Banda Aceh are shown here.

The calculated inundation area in Banda Aceh is shown in Figure 9. The flooded area is marked in blue, the dry land region is rendered in green, and the white area is ocean region. The calculated inundation area is also overlaid with a satellite image taken by QUICKBIRD in Figure 9(b). In the overlaid image, the thick red line indicates the inundation line based on the numerical simulation. In the satellite image, the dark green color (vegetation) indicates areas not affected by the tsunami and the area shaded by semitransparent red color shows flooded regions by this tsunami. Obviously, the calculated inundation area matches reasonably well with the satellite image in the neighborhood of Lhoknga and the western part of Banda Aceh. However, in the region of eastern Banda Aceh, the simulations significantly underestimate the inundation area. However, in general, the agreement

TSUNAMI

between the numerical simulation and the satellite observation is surprisingly good. In Figure 10, the tsunami wave heights in Banda Aceh are also compared with the field measurements by two Japan survey teams. On the coast between Lhoknga and Leupung, where the maximum height

5.52

5.52

5.5

5.5

5.48

5.48

is measured more than 30 m, the numerical results match very well with the field measurements. However, beyond Lhoknga to the north, the numerical results, in general, are only half of the measurements, except in middle regions between Lhoknga and Lampuuk.

15

Survey by Tsuji et al.

Tsunami heights (m)

72

Survey by Shibayama et al.

5.46 5.44

5.46

5.4

5.38 5.36 40

Survey by Tsuji et al. Survey by Shibayama et al. Numerical result Nearest numerical result

30 20 10 Tsunami heights (m)

5.38

0

0 95.24

95.26

95.28

95.3 95.32 Longitude (deg)

95.34

95.36

95.38

95.36

95.38

Survey by Tsuji et al.

Leupung

5.4

5

5.62

5.42

5.42

10

Lhoknga

5.44

5.36 95.2

Flooded area Dry land Ocean

Latitude (deg)

Latitude (deg)

Latitude (deg)

Lampuuk

Survey by Tsuji et al. Survey by Shibayama et al. Numerical result Nearest numerical result

5.6

Survey by Shibayama et al.

Flooded area

5.58

Dry land Ocean

Ulee Lheue

5.56 5.54 95.24

BANDA ACEH

95.26

95.28

95.22 95.24 95.26 95.28 Longitude (deg)

95.32 95.3 Longitude (deg)

95.34

Figure 10 Tsunami heights on eastern and northern coast of Banda Aceh, Indonesia. The field survey measurements are from Tsuji et al. (2005) and Shibayama et al. (2005).

120.0° E 135.0° E

150.0° E 165.0° E

180.0° E

165.0° W 150.0° W 135.0° W 120.0° W 105.0° W 90.0° W

75.0° W

60.0° W

45.0° W

60.0° N

45.0° N

30.0° N

15.0° N

15.0° S

30.0° S

45.0° S

Figure 11 The locations of the existing and planned Deep-Ocean Assessment and Reporting of Tsunamis (DART) system in the Pacific Ocean (NOAA magazine, 17 Apr. 2006).

TSUNAMI

Tsunami Hazard Mitigation The ultimate goal of the tsunami hazard mitigation effort is to minimize casualties and property damages. This goal can be met, only if an effective tsunami early warning system is established and a proper coastal management policy is practiced. Tsunami Early Warning System

The great historical tsunamis, such as the 1960 Chilean tsunami and the 1964 tsunami generated near Prince William Sound in Alaska, prompted the US government to develop an early warning system in the Pacific Ocean. The Japanese government has

Bidirectional communication and control

Iridium satellite

also developed a tsunami early warning system for the entire coastal community around Japan. The essential information needed for an effective early warning system is the accurate prediction of arrival time and wave height of a forecasted tsunami at a specific location. Obviously, the accuracy of these predictions relies on the information of the initial water surface displacement near the source region, which is primarily determined by the seismic data. In many historical events, including the 2004 Indian Ocean tsunami, evidences have shown that accurate seismic data could not be verified until those events were over. To delineate the source region problem, in the United States, several federal agencies and states

DART II System Optional met sensors

Optional sensor mast Iridium and GPS antennas

Wind Barometric pressure Sea surface temperature and conductivity Air temperature/ relative humidity

Lifting handle 2.0 m

Electronic systems and batteries

Tsunami warning center

Surface buoy 2.5-m diameter 4000-kg displacement Swivel 1.8 m

Acoustic transducers (two each) Tsunameter

25-mm chain (3.5m) Signal flag

Glass ball flotation

Bidirectional acoustic telemetry

13-mm polyester

~75 m

25-mm nylon 22-mm nylon

19-mm nylon Acoustic transducer Acoustic release CPU Batteries Sensor Anchor 325 kg

Figure 12 A sketch of the second-generation DART (II) system.

73

13-mm chain (5 m) Anchor 3100 kg

1000 − 6000 m

74

TSUNAMI

Newport, Oregon Highway 101 This map is intended for emergency planning purposes only

Yaquina Bay

These models can simulate a ‘design tsunami’ approaching a coastline, and they can predict which areas are most at risk to being flooded. The tsunami inundation maps are an integral part of the overall strategy to reduce future loss of life and property. Emergency managers and local governments of the threatened communities use these and similar maps to guide evacuation planning. As an example, the tsunami inundation map (Figure 13) for the coastal city of Newport (Oregon) was created using the results from a numerical simulation using a design tsunami. The areas shown in orange are locations that were flooded in the numerical simulation

Acknowledgment Highway 101

Figure 13 Tsunami inundation map for the coastal city of Newport, Oregon.

have joined together to create a warning system that involves the use of deep-ocean tsunami sensors to detect the presence of a tsunami. These deep-ocean sensors have been deployed at different locations in the Pacific Ocean before the 2004 Indian Ocean tsunami. After the 2004 Indian Ocean tsunami, several additional sensors have been installed and many more are being planned (see Figure 11). The sensor system includes a pressure gauge that records and transmits the surface wave signals instantaneously to the surface buoy, which sends the information to a warning center via Iridium satellite (Figure 12). In the event of a tsunami, the information obtained by the pressure gauge array can be used as input data for modeling the propagation and evolution of a tsunami. Although there have been no large Pacific-wide tsunamis since the inception of the warning system, warnings have been issued for smaller tsunamis, a few of which were hardly noticeable. This tends to give citizens a lazy attitude toward a tsunami warning, which would be fatal if the wave was large. Therefore, it is very important to keep people in a danger areas educated of tsunami hazards. Coastal Inundation Map

Using numerical modeling, hazards in areas vulnerable to tsunamis can be assessed, without the area ever having experienced a devastating tsunami.

The work reported here has been supported by National Science Foundation with grants to Cornell University.

See also Heat and Momentum Fluxes at the Sea Surface. Sea Level Variations Over Geological Time. Wave Generation by Wind. Waves on Beaches.

Further Reading Geist EL (1998) Local tsunami and earthquake source parameters. Advances in Geophysics 39: 117--209. Hokkaido Tsunami Survey Group (1993) Tsunami devastates Japanese coastal region. EOS Transactions of the American Geophysical Union 74: 417--432. Kajiura K (1981) Tsunami energy in relation to parameters of the earthquake fault model. Bulletin of the Earthquake Research Institute, University of Tokyo 56: 415--440. Kajiura K and Shuto N (1990) Tsunamis. In: Le Me´haute´ B and Hanes DM (eds.) The Sea: Ocean Engineering Science, pp. 395--420. New York: Wiley. Kanamori H (1972) Mechanism of tsunami earthquakes. Physics and Earth Planetary Interactions 6: 346--359. Kawata Y, Benson BC, Borrero J, et al. (1999) Tsunami in Papua New Guinea was as intense as first thought. EOS Transactions of the American Geophysical Union 80: 101, 104--105. Keating BH and Mcguire WJ (2000) Island edifice failures and associated hazards. Special Issue: Landslides and Tsunamis. Pure and Applied Geophysics 157: 899--955. Liu PL-F, Lynett P, Fernando H, et al. (2005) Observations by the International Tsunami Survey Team in Sri Lanka. Science 308: 1595. Lynett PJ, Borrero J, Liu PL-F, and Synolakis CE (2003) Field survey and numerical simulations: A review of the

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1998 Papua New Guinea tsunami. Pure and Applied Geophysics 160: 2119--2146. Mansinha L and Smylie DE (1971) The displacement fields of inclined faults. Bulletin of Seismological Society of America 61: 1433--1440. Satake K, Bourgeois J, Abe K, et al. (1993) Tsunami field survey of the 1992 Nicaragua earthquake. EOS Transactions of the American Geophysical Union 74: 156--157. Shibayama T, Okayasu A, Sasaki J, et al. (2005) The December 26, 2004 Sumatra Earthquake Tsunami, Tsunami Field Survey in Banda Aceh of Indonesia. http://www.drs.dpri.kyoto-u.ac.jp/sumatra/indonesia-ynu/ indonesia_survey_ynu_e.html (accessed Feb. 2008). Synolakis CE, Bardet J-P, Borrero JC, et al. (2002) The slump origin of the 1998 Papua New Guinea tsunami. Proceedings of Royal Society of London, Series A 458: 763--789.

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Tsuji Y, Matsutomi H, Tanioka Y, et al. (2005) Distribution of the Tsunami Heights of the 2004 Sumatra Tsunami in Banda Aceh measured by the Tsunami Survey Team. http://www.eri.u-tokyo.ac.jp/namegaya/sumatera/ surveylog/eindex.htm (accessed Feb. 2008). von Huene R, Bourgois J, Miller J, and Pautot G (1989) A large tsunamigetic landslide and debris flow along the Peru trench. Journal of Geophysical Research 94: 1703--1714. Wang X and Liu PL-F (2006) An analysis of 2004 Sumatra earthquake fault plane mechanisms and Indian Ocean tsunami. Journal of Hydraulics Research 44(2): 147--154. Yeh HH, Imamura F, Synolakis CE, Tsuji Y, Liu PL-F, and Shi S (1993) The Flores Island tsunamis. EOS Transactions of the American Geophysical Union 74: 369--373.

STORM SURGES R. A. Flather, Bidston Observatory, Proudman Oceanographic Laboratory, Bidston Hill, Prenton, UK

mitigate their destructive effects are therefore of vital concern.

Copyright & 2001 Elsevier Ltd.

Storm Surge Equations Introduction and Definitions Storm surges are changes in water level generated by atmospheric forcing; specifically by the drag of the wind on the sea surface and by variations in the surface atmospheric pressure associated with storms. They last for periods ranging from a few hours to 2 or 3 days and have large spatial scales compared with the water depth. They can raise or lower the water level in extreme cases by several meters; a raising of level being referred to as a ‘positive’ surge, and a lowering as a ‘negative’ surge. Storm surges are superimposed on the normal astronomical tides generated by variations in the gravitational attraction of the moon and sun. The storm surge component can be derived from a time-series of sea levels recorded by a tide gauge using: surgeresidual ¼ ðobservedsealevelÞ ðpredictedtidelevelÞ

½1

producing a time-series of surge elevations. Figure 1 shows an example. Sometimes, the term ‘storm surge’ is used for the sea level (including the tidal component) during a storm event. It is important to be clear about the usage of the term and its significance to avoid confusion. Storms also generate surface wind waves that have periods of order seconds and wavelengths, away from the coast, comparable with or less than the water depth. Positive storm surges combined with high tides and wind waves can cause coastal floods, which, in terms of the loss of life and damage, are probably the most destructive natural hazards of geophysical origin. Where the tidal range is large, the timing of the surge relative to high water is critical and a large surge at low tide may go unnoticed. Negative surges reduce water depth and can be a threat to navigation. Associated storm surge currents, superimposed on tidal and wave-generated flows, can also contribute to extremes of current and bed stress responsible for coastal erosion. A proper understanding of storm surges, the ability to predict them and measures to

76

Most storm surge theory and modeling is based on depth-averaged hydrodynamic equations applicable to both tides and storm surges and including nonlinear terms responsible for their interaction. In vector form, these can be written: @z þ r  ðDqÞ ¼ 0 @t

½2

@q 1 þ q  rq  f k  q ¼ grðz  z¯ Þ  rpa @t r ½3 1 2 þ ðts  tb Þ þ Ar q rD where t is time; z the sea surface elevation; z¯ the equilibrium tide; q the depth-mean current; ss the wind stress on the sea surface; sb the bottom stress; pa atmospheric pressure on the sea surface; D the total water depth (D ¼ h þ z, where h is the undisturbed depth); r the density of sea water, assumed to be uniform; g the acceleration due to gravity; f the Coriolis parameter (¼ 2o sinj, where o is the angular speed of rotation of the Earth and j is the latitude); k a unit vector in the vertical; and A the coefficient of horizontal viscosity. Eqn [2] is the continuity equation expressing conservation of volume. Eqn [3] equates the accelerations (left-hand side) to the force per unit mass (right-hand side). In this formulation, bottom stress, sb is related to the current, q, using a quadratic law: tb ¼ krqjqj

½4

where k is a friction parameter (B0.002). Similarly, the wind stress, ss , is related to W, the wind velocity at a height of 10 m above the surface, also using a quadratic law: ts ¼ cD ra WjWj

½5

where ra is the density of air and cD a drag coefficient. Measurements in the atmospheric boundary layer suggest that cD increases with wind speed, W, accounting for changes in surface roughness associated with wind waves. A typical form due to J. Wu is:

STORM SURGES

77

6

5

4

Elevation (m)

3

2 1 0 _1 _2 _3 0

6

12 18 12 Nov 1970

0

6

12 13 Nov 1970

18

0

Figure 1 Water level (dashed line), predicted tide (line with J), and the surge residual (continuous line) at Sandwip Island, Bangladesh, during the catastrophic storm surge of 12–13 November 1970 (times are GMT).

103 cD ¼ 0:8 þ 0:065W

½6

Alternatively, from dimensional analysis, H. Charnock obtained gz0 =u2 ¼ a, where z0 is the aerodynamic roughness length associated with the surface wavefield, u is the friction velocity ðu2 ¼ ts =ra Þ, and a is the Charnock constant. So, the roughness varies linearly with surface wind stress. Assuming a logarithmic variation of wind speed with height z above the surface, WðzÞ ¼ ðu =kÞ ln(z=z0 ), where k is von Ka´rma´n’s constant. It follows that for z ¼ 10 m:    2 cD ¼ ð1=kÞln gz= acD W 2

½7

Estimates of a range from 0.012 to 0.035.

Generation and Dynamics of Storm Surges The forcing terms in eqn [3] which give rise to storm surges are those representing wind stress and the horizontal gradient of surface atmospheric pressure. Very simple solutions describe the basic mechanisms. The sea responds to atmospheric pressure variations by adjusting sea level such that, at depth, pressure in the water is uniform, the hydrostatic approximation. Assuming in eqn [3] that q ¼ 0 and ss ¼ 0, then

grz þ ð1=rÞrpa ¼ 0, so rgz þ pa ¼ constant . This gives the ‘inverse barometer effect’ whereby a decrease in atmospheric pressure of 1 hPa produces an increase in sea level of approximately 1 cm. Wind stress produces water level variations on the scale of the storm. Eqns [5] and [6] imply that the strongest winds are most important since effectively ts pW 3. Both pressure and wind effects are present in all storm surges, but their relative importance varies with location. Since wind stress is divided by D whereas rpa is not, it follows that wind forcing increases in importance in shallower water. Consequently, pressure forcing dominates in the deep ocean whereas wind forcing dominates in shallow coastal seas. Major destructive storm surges occur when extreme storm winds act over extensive areas of shallow water. As well as the obvious wind set-up, with the component of wind stress directed towards the coast balanced by a surface elevation gradient, winds parallel to shore can also generate surges at higher latitudes. Wind stress parallel to the coast with the coast on its right will drive a longshore current, limited by bottom friction. Geostrophic balance gives (in the Northern Hemisphere) a surface gradient raising levels at the coast (see Tides). Amplification of surges may be caused by the funneling effect of a converging coastline or estuary and by a resonant response; e.g. if the wind forcing

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travels at the same velocity as the storm wave, or matches the natural period of oscillation of a gulf, producing a seiche. As a storm moves away, surges generated in one area may propagate as free waves, contributing as externally generated components to surges in another area. Generally, away from the forcing center, the response of the ocean consists of longshore propagating coastally trapped waves. Examples include external surges in the North Sea (see Figure 2), which are generated west and north of Scotland and propagate anticlockwise round the basin like the diurnal tide; approximately as a Kelvin wave. Low mode continental shelf waves have been identified in surges on the west coast of Norway, in the Middle Atlantic Bight of the US, in the East China Sea, and on the north-west shelf of Australia. Currents associated with low mode continental shelf waves generated by a tropical cyclone crossing the north-west shelf of Australia have been observed and explained by numerical modeling. Edge waves can also be generated by cyclones travelling parallel to the coast in the opposite direction to shelf waves. From eqns [3] and [4], bottom stress (which dissipates surges) also depends on water depth and is non-linear; the current including contributions from tide and surge, q ¼ qT þ qs. Consequently, dissipation of surges is stronger in shallow water and where tidal currents are also strong. In deeper water and where tides are weak, free motions can persist for long times or propagate long distances. For example, the Adriatic has relatively small tides and seiches excited by storms can persist for many days. In areas with substantial tides and shallow water, non-linear dynamical processes are important, resulting in interactions between the tide and storm surge such that both components are modified. The main contribution arises from bottom stress, but time-dependent water depth, D ¼ h þ zðtÞ, can also be significant (e.g. ts =ðrDÞ will be smaller at high tide than at low tide). An important consequence is that the linear superposition of surge and tide without accounting for their interaction gives substantial errors in estimating water level. For example, for surges propagating southwards in the North Sea into the Thames Estuary, surge maxima tend to occur on the rising tide rather than at high water (Figure 3). Interactions also occur between the tide–surge motion and surface wind waves (see below).

Areas Affected by Storm Surges Major storm surges are created by mid-latitude storms and by tropical cyclones (also called hurricanes and typhoons) which generally occur in

geographically separated areas and differ in their scale. Mid-latitude storms are relatively large and evolve slowly enough to allow accurate predictions of their wind and pressure fields from atmospheric forecast models. In tropical cyclones, the strongest winds occur within a few tens of kilometers of the storm center and so are poorly resolved by routine weather prediction models. Their evolution is also rapid and much more difficult to predict. Consequently prediction and mitigation of the effects of storm surges is further advanced for mid-latitude storms than for tropical cyclones. Tropical cyclones derive energy from the warm surface waters of the ocean and develop only where the sea surface temperature (SST) exceeds 26.51C. Since their generation is dependent on the effect of the local vertical component of the Earth’s rotation, they do not develop within 51 of the equator. Figure 4 shows the main cyclone tracks. Areas affected include: the continental shelf surrounding the Gulf of Mexico and on the east coast of the US (by hurricanes); much of east Asia including Vietnam, China, the Philippines and Japan (by typhoons); the Bay of Bengal, in particular its shallow north-east corner, and northern coasts of Australia (by tropical cyclones). Areas affected by mid-latitude storms include the North Sea, the Adriatic, and the Patagonian Shelf. Inland seas and large lakes, including the Great Lakes, Lake Okeechobee (Florida), and Lake Biwa (Japan) also experience surges. The greatest loss of life due to storm surges has occurred in the northern Bay of Bengal and Meghna Estuary of Bangladesh. A wide and shallow continental shelf bounded by extensive areas of low-lying poorly protected land is impacted by tropical cyclones. Cyclone-generated storm surges on 12–13 November 1970 and 29–30 April 1991 (Figure 5) killed approximately 250 000 and 140 000 people, respectively, in Bangladesh. A severe storm in the North Sea on 31 January–1 February 1953 generated a large storm surge, which coincided with a spring tide to cause catastrophic floods in the Netherlands (Figure 6) and south-east England, killing approximately 2000 people. Subsequent government enquiries resulted in the ‘Delta Plan’ to improve coastal defences in Holland, led to the setting up of coastal flood warning authorities, and accelerated research into storm surge dynamics. The city of Venice, in Italy, suffers frequent ‘acqua alta’ which flood the city, disrupting its life and accelerating the disintegration of the unique historic buildings. In 1969 Hurricane Camille created a surge in the Gulf of Mexico which rose to 7 m above mean sea level, causing more than 100 deaths and about 1

Figure 2 Propagation of an external surge in the North Sea from a numerical model simulation.

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Phase of tide, in hours after H.T Figure 3 Frequency distribution relative to the time of tidal high water of positive and negative surges at Lerwick (northern North Sea) and Southend (Thames Estuary). The phase distribution at Lerwick is random, whereas due to tide–surge interaction most surge peaks at Southend occur on the rising tide (re-plotted from Prandle D and Wolf J, 1978, The interaction of surge and tide in the North Sea and River Thames. Geophys. J.R. Astr. Soc., 55: 203–216, by permission of the Royal Astronomical Society).

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Figure 4 Tropical cyclone tracks (from Murty, 1984, reproduced by permission of the Department of Fisheries and Oceans, Canada).

STORM SURGES

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Figure 5 Cyclone tracks (dashed lines) and maximum computed surge elevation measured in meters in the northern Bay of Bengal during the cyclones of (A) 1970 and (B) 1991 from numerical model simulations (contour interval 0.5 m). (Re-plotted from Flather, 1994, by permission of the American Meteorological Society.)

billion dollars worth of damage. An earlier cyclone in the region, in 1900, flooded the island of Galveston, Texas, with the loss of 6000 lives.

Storm Surge Prediction Early research on storm surges was based on analysis of observations and solution of simplified – usually linearized – equations for surges in idealized channels, rectangular gulfs, and basins with uniform depth. The first self-recording tide gauge was installed in 1832 at Sheerness in the Thames Estuary, England, so datasets for analysis were available from an early stage. Interest was stimulated by events such as the 1953 storm surge in the North Sea, which highlighted the need for forecasts.

First prediction methods were based on empirical formulae derived by correlating storm surge elevation with atmospheric pressure, wind speed and direction and, where appropriate, observed storm surges from a location ‘upstream’. Long time-series of observations are required to establish reliable correlations. Where such observations existed, e.g. in the North Sea, the methods were quite successful. From the 1960s, developments in computing and numerical techniques made it possible to simulate and predict storm surges by solving discrete approximations to the governing equations (eqns [1] and [2]). The earliest and simplest methods, pioneered in Europe by W. Hansen and N.S. Heaps and in the USA by R.O. Reid and C.P. Jelesnianski, used a time-stepping approach based on finite difference approximations

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Figure 6 Breached dyke in the Netherlands after the 1953 North Sea storm surge. (Reproduced by permission of RIKZ, Ministry of Public Works, The Netherlands.)

on a regular grid. Surge–tide interaction could be accounted for by solving the non-linear equations and including tide. Effects of inundation could also be included by allowing for moving boundaries; water levels computed with a fixed coast can be O(10%) higher than those with flooding of the land allowed. Recent developments have revolutionized surge modeling and prediction. Among these, coordinate transformations, curvilinear coordinates and grid nesting allow better fitting of coastal boundaries and enhanced resolution in critical areas. A simple example is the use of polar coordinates in the SLOSH (Sea, Lake and Overland Surges from Hurricanes) model focusing on vulnerable sections of the US east coast. Finite element methods with even greater flexibility in resolution (e.g. Figure 7) have also been used in surge computations in recent years. There has also been increasing use of three-dimensional (3-D) models in storm surge studies. Their main advantage is that they provide information on the vertical structure of currents and, in particular, allow the bottom stress to be related to flow near the seabed. This means that in a 3-D formulation the bottom stress need not oppose the direction of the depth mean flow and hence of the water transport. Higher surge estimates result in some cases. In the last decades many countries have established and now operate model-based flood warning systems. Although finite element methods and 3-D models have been developed and are used extensively for research, most operational models are still based on depth-averaged finite-difference formulations. A key requirement for accurate surge forecasts is accurate specification of the surface wind stress. Surface wind and pressure fields from numerical

weather prediction (NWP) models are generally used for mid-latitude storms. Even here, resolution of small atmospheric features can be important, so preferably NWP data at a resolution comparable with that of the surge model should be used. For tropical cyclones, the position of maximum winds at landfall is critical, but prediction of track and evolution (change in intensity, etc.) is problematic. Presently, simple models are often used based on basic parameters: pc , the central pressure; Wm , the maximum sustained 10 m wind speed; R, the radius to maximum winds; and the velocity, V, of movement of the cyclone’s eye. Assuming a pressure profile, e.g. that due to G.J. Holland: h i ½8 pa ðrÞ ¼ pc þ Dpexp ðR=rÞB where r is the radial distance from the cyclone center, Dp the pressure deficit (difference between the ambient and central pressures), and B is a ‘peakedness’ factor typically 1.0oBo2.5. Wind fields can then be estimated using further assumptions and approximations. First, the gradient or cyclostrophic wind can be calculated as a function of r. An empirical factor (B0.8) reduces this to W, the 10 m wind. A contribution from the motion of the storm (maybe 50% of V) can be added, introducing asymmetry to the wind field, and finally to account for frictional effects in the atmospheric boundary layer, wind vectors may be turned inwards by a cross-isobar angle of 101–251. Such procedures are rather crude, so that cyclone surges computed using the resulting winds are unlikely to be very accurate. Simple vertically integrated models of the atmospheric boundary layer have been used to compute winds from a pressure

STORM SURGES

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Figure 7 A finite element grid for storm surge calculations on the east coast of the USA, the Gulf of Mexico, and Caribbean (from Blain CA, Westerink JJ and Leuttich RL (1994), The influence of domain size on the response characteristics of a hurricane storm surge model. J. Geophys. Res., 99(C9) 18467–18479. Reproduced by permission of the American Geophysical Union.).

distribution such as eqn [8], providing a more consistent approach. In reality, cyclones interact with the ocean. They generate wind waves, which modify the sea surface roughness, z0 , and hence the wind stress generating the surge. Wind- and wave-generated turbulence mixes the surface water changing its temperature and so modifies the flux of heat from which the cyclone derives its energy. Progress requires improved understanding of air–sea exchanges at extreme wind speeds and high resolution coupled atmosphere– ocean models.

Interactions with Wind Waves As mentioned above, observations suggested that Charnock’s awas not constant but depended on

water depth and ‘wave age’, a measure of the state of development of waves. Young waves are steeper and propagate more slowly, relative to the wind speed, than fully developed waves and so are aerodynamically rougher enhancing the surface stress. These effects can be incorporated in a drag coefficient which is a function of wave age, wave height, and water depth and agrees well with published datasets over the whole range of wave ages. Further research, considering the effects of waves on airflow in the atmospheric boundary layer, led P.A.E.M. Janssen to propose a wave-induced stress enhancing the effective roughness. Application of this theory requires the dynamical coupling of surge and wave models such that friction velocity and roughness determine and are determined by the

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waves. Mastenbroek et al. obtained improved agreement with observed surges on the Dutch coast by including the wave-induced stress in a model experiment (Figure 8). However, they also found that the same improvement could be obtained by a small increase in the standard drag coefficient. In shallow water, wave orbital velocities also reach the seabed. The bottom stress acting on surge and tide is therefore affected by turbulence introduced at the seabed in the wave boundary layer. With simplifying assumptions, models describing these effects have been developed and can be used in storm surge modeling. Experiments using both 2-D and 3-D surge models have been carried out. 3-D modeling of surges in the Irish Sea using representative waves shows significant effects on surge peaks and improved agreement with observations. Bed stresses are much enhanced in shallow water. Because the processes depend on the nature of the bed, a more complete treatment should take account of details of bed types. Non-linear interactions give rise to a wave-induced mean flow and a change in mean water depth (wave set-up and set-down). The former has contributions from a mean momentum density produced by a non-zero mean flow in the surface layer (above the trough level of the waves), and from wave breaking. Set-up and set-down arise from the ‘radiation stress’, which is defined as the excess momentum flux

due to the waves (see Waves on Beaches). Mastenbroek et al. showed that the radiation stress has a relatively small influence on the calculated water levels in the North Sea but cannot be neglected in all cases. It is important where depth-induced changes in the waves, as shoaling or breaking, dominate over propagation and generation, i.e. in coastal areas. The effects should be included in the momentum equations of the surge model. Although ultimately coupled models with a consistent treatment of exchanges between atmosphere and ocean and at the seabed remain a goal, it appears that with the present state of understanding the benefits may be small compared with other inherent uncertainties. In particular, accurate definition of the wind field itself and details of bed types (rippled or smooth, etc.) are not readily available.

Data Assimilation Data assimilation plays an increasing role, making optimum use of real-time observations to improve the accuracy of initial data in forecast models. Bode and Hardy (see Further Reading section) reviewed two approaches, involving solution of adjoint equations and Kalman filtering. The Dutch operational system has used Kalman filtering since 1992, incorporating real-time tide gauge data from the east

Hoek van Holland

Vlissingen

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Figure 8 Computed surge elevations during 13–16 February 1989 with (dashed line) and without (dotted) wave stress compared with observations (continuous line). (Re-plotted from Mastenbroek et al., 1994 by permission of the American Geophysical Union.)

STORM SURGES

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Forecast lead time (h) Figure 9 Variation of forecast errors in surge elevation with forecast lead-time from the Dutch operational model with (continuous line) and without (dashed line) assimilation of tide gauge data. (Replotted from Flather, 2000, by permission of Elsevier Science BV.)

coast of Britain. Accuracy of predictions (Figure 9) is improved for the first 10–12 h of the forecast.

Related Issues Extremes

Statistical analysis of storm surges to derive estimates of extremes is important for the design of coastal defenses and safety of offshore structures. This requires long time- series of surge elevation derived from observations of sea level where available or, increasingly, from model hindcasts covering O(50 years) forced by meteorological analyses.

susceptible to tropical cyclones could also be extended. Research is in progress to assess and quantify some of these effects, e.g. with tide–surge models forced by outputs from climate GCMs. An important issue is that of distinguishing climate-induced change from the natural inter-annual and decadal variability in storminess and hence surge extremes.

See also Tides. Waves on Beaches.

Further Reading Climate Change Effects

Climate change will result in a rise in sea level and possible changes in storm tracks, storm intensity and frequency, collectively referred to as ‘storminess’. Changes in water depth with rising mean sea level (MSL) will modify the dynamics of tides and surges, increasing wavelengths and modifying the generation, propagation, and dissipation of storm surges. Increased water depth implies a small reduction in the effective wind stress forcing, suggesting smaller surges. However, effects of increased storminess may offset this. It has been suggested, for example, that increased temperatures in some regions could raise sea surface temperatures resulting in more intense and more frequent tropical cyclones. Regions

Bode L and Hardy TA (1997) Progress and recent developments in storm surge modelling. Journal of Hydraulic Engineering 123(4): 315--331. Flather RA (1994) A storm surge prediction model for the northern Bay of Bengal with application to the cyclone disaster in April 1991. Journal of Physical Oceanography 24: 172--190. Flather RA (2000) Existing operational oceanography. Coastal Engineering 41: 13--40. Heaps NS (1967) Storm surges. In: Barnes H (ed.) Oceanography and Marine Biology Annual Review 5, pp. 11--47. London: Allen and Unwin. Mastenbroek C, Burgers G, and Janssen PAEM (1993) The dynamical coupling of a wave model and a storm surge model through the atmospheric boundary layer. Journal of Physical Oceanography 23: 1856--1866.

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Murty TS (1984) Storm surges – meteorological ocean tides. Canadian Bulletin of Fisheries and Aquatic Sciences 212: 1--897. Murty TS, Flather RA, and Henry RF (1986) The storm surge problem in the Bay of Bengal. Progress in Oceanography 16: 195--233.

Pugh DT (1987) Tides, Surges, and Mean Sea Level. Chichester: John Wiley Sons. World Meteorological Organisation (1978) Present Techniques of Tropical Storm Surge Prediction. Report Report 13. Marine science affairs, WMO No. 500. Geneva, Switzerland.

COASTAL TRAPPED WAVES J. M. Huthnance, CCMS Proudman Oceanographic Laboratory, Wirral, UK Copyright & 2001 Elsevier Ltd.

Introduction Many shelf seas are dominated by shelf-wide motions that vary from day to day. Oceanic tides contribute large coastal sea-level variations and (on broad shelves) large currents. Atmospheric pressure and (especially) winds generate storm surges; strong currents and large changes of sea level. Other phenomena on these scales are wind-forced upwelling, along-slope currents and poleward undercurrents common on the eastern sides of oceans, responses to oceanic eddies, and alongshore pressure gradients. All these responses depend on natural waves that travel along or across the continental shelf and slope. These waves, which have scales of about one to several days and tens to hundreds of kilometers according to the width of the continental shelf and slope, are the subject of this article. Also included are ‘Kelvin’ waves, also coastally trapped, that travel cyclonically around ocean basins but with typical scales of thousands of kilometers both alongshore and for offshore decrease of properties. The waves have been widely observed through their association with the above phenomena. In fact they have been identified along coastlines of various orientations and all continents in both the Northern and Southern Hemispheres. Typically, the identification involves separating forced motion from the accompanying free waves. The ‘lowest’ mode with simplest structure (see below) has been most often identified; its peak coastal elevation is relatively easily measured. More complex forms need additional offshore measurements (usually of currents) for identification. This has been done (for example) off Oregon, the Middle Atlantic Bight and New South Wales (Australia). Observations substantiate many of the features described in the following sections.

Formulation Analysis is based on Boussinesq momentum and continuity equations for an incompressible sea of near-uniform density between a gently-sloping

seafloor z ¼ hðxÞ and a free surface z ¼ Zðx; tÞ where the surface elevation Z ¼ 0 for the sea at rest. Cartesian coordinates x  ðx; yÞ; z (vertically up) rotate with a vertical component f =2. The motion, velocity components ðu; v; wÞ, is assumed to be nearly horizontal and in hydrostatic balance. (These assumptions are almost always made for analysis on these scales; they are probably not necessary but certainly simplify the analysis.) At the surface, pressure and stress match atmospheric forcing (for free waves). There is no component of flow into the seabed (generalizing to zero onshore transport uh at the coast); u-0 far from the coast (the trapping condition) or is specified by forcing.

Straight Unstratified Shelf This is the simplest context. Taking x offshore (and y alongshore; Figure 1) the depth is hðxÞ. Uniformity along shelf suggests wave solutions fuðxÞ; vðxÞ; ZðxÞg expðiky þ istÞ. For positive wave frequency, s, k > 0 corresponds to propagation in  y, with the coast on the right (‘forward’ in the Northern Hemisphere). Then the momentum equations give u; v in terms of Z satisfying 0

ðhZ0 Þ þKZ ¼ 0

½1

where KðaÞ  kfh0 =s þ ðs2  f 2 Þ=g  k2 h (uniform f ); primes (0 ) denote cross-shelf differentiation q=qx. The boundary conditions become hðsZ0 þ fkZÞ-0ðx-0Þ; Z-0ðx-NÞ

½2

Free wave modes are represented by eigensolutions of eqns. [1] and [2]. Successive modes with more offshore nodes correspond to large positive K and arise in two ways. The term ðs2  f 2 Þ=g  k2 h in K represents the gravity wave mechanism, modified by rotation; it increases with frequency s. For kf > 0 it gives rise to the ‘Kelvin wave’ – the mode with simplest offshore form; decay but no zeros of elevation. Other forms depending on this term (kf o0 and/or nodes of elevation offshore) are termed edge waves and discussed elsewhere. If there is no slope or boundary, K equals this term alone and plane inertiogravity waves are solutions of eqn. [1]. The term kfh0 =s in K increases with decreasing s if everywhere h increases offshore and kf > 0. It represents the following potential vorticity (angular

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y z t

as

Co


0 (Northern Hemisphere).

Shallow Time t Deep

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Figure 2 Topographic wave mechanism. m displacement, velocity, sr relative vorticity (Northern Hemisphere).

momentum) restoring mechanism. If fluid is displaced into shallower water, it spreads laterally to conserve volume and therefore spins more slowly in total; taking account of the earth’s rotation, it acquires anticyclonic relative vorticity. ‘Forwards’ along the slope, the resulting up-slope velocity implies an up-slope displacement in time. Hence the upslope displacement propagates ‘forwards’ along the slope. (Behind it, the anticyclonic relative vorticity implies down-slope flow restoring the fluid location from its previous up-slope displacement.) This sequence is depicted in Figure 2. These modes are referred to as continental shelf waves. The mode forms and frequencies are known for several analytic models, e.g., level, uniformly sloping, exponential concave and convex shelves bordering an ocean of uniform depth. Numerical solutions for the waveforms and dispersion relations sðkÞ are easily found for any depth profile hðxÞ. For any monotonic profile hðxÞ the following have been proved. Phase propagates ‘forwards’ for all modes with so7f 7. Waves forms with 1, 2,y nodes offshore have frequencies 7f 7 > s1 > s2 > y defined for all k (subject to kf > 0). The Kelvin wave

frequency s0 ðkÞ > s1 ðkÞ is likewise defined for all kðkf > 0Þ and passes smoothly through 7f 7 to s0 > 7f 7 for large enough k. Edge waves with 0 (if kf o0), 1, 2,y nodes in the offshore form have increasing frequencies s > 7f 7; however, low wavenumbers (and frequencies) are excluded where the dispersion curves break the trapping criterion 0oKðNÞ  ðs2  f 2 Þ=g  k2 hðNÞ (Figure 3). Besides these properties, the following features are typical. Bounded h0 =h ensures a maximum sM in sðkÞ; near s1M, mode 1 velocity tends to be maximal near the shelf edge, and polarized anticyclonically; the nearest approach to inertial motion in this topographic context; here the group velocity qs=qk of energy propagation (in y) reverses through zero. If k-N, then sn -f =ð2n þ 1Þ for the n-node continental shelf wave which becomes concentrated over the ‘beach’ at the coast. As s, k-0, the shelf wave and Kelvin wave speeds s=k approach constant (maximum) values and u=s; v; n approach constant forms so that cross-slope velocities tend to zero. Variables v, Z and Z0 are in phase or antiphase, v and Z0 being near the geostrophic balance fv ¼ gZ0 ; u is 901 out of phase. Typically, Kelvin-wave currents in shallow shelf waters are polarized cyclonically but first-mode continental shelf wave currents are anticyclonic. Continental shelf wave-forms depend on the shape rather than the horizontal scale L of the depth profile. Phase speeds scale as fL and ðu; v; ZÞ scale as ðsU=f ; U; fUL=gÞ where the velocity scale U may be typically 0.1 m s1. Kelvin wave forms depend more on the depth; usually the phase speed is just less than ½ghðNÞ1=2 and ðu; v; ZÞ scale as ðsZL=h; ðg=hÞ1=2 Z; ZÞ where Z is typically 0.1 to 1 m. Quantitative results depend on the strength of forcing and accurate profile modeling; numerical calculations should be used for real shelves. The typical maximum in sðkÞ and associated reversal of group velocity qs=qk appears to be of practical significance. Shelf waves with frequency near the maximum (for some mode) appear in several observations, e.g., North Carolina sea levels, Scottish and Vancouver Island diurnal tides, wind-driven flow north of Scotland and over Rockall Bank. There may be a bias in seeking motion correlated with local forcing, i.e., responses with nonpropagating energy. Figure 4 shows modeled rotary currents over the shelf edge, continental shelf waves near the maximum frequency with slow energy propagation, as a response to impulsive wind forcing.

Other Geometry Continental shelf waves exist in more general contexts than a straight shelf, as identification in nature

COASTAL TRAPPED WAVES

by continuity. The boundary conditions for no flow through the bottom, zero pressure at the surface and trapping become

testifies. Analyses also verify their possibility in rectangular and circular basins. Perfect trapping around islands is only possible if so7f 7; then results are qualitatively as for a straight shelf except that wavelength around the island (and hence frequency) is quantized. For a broad shelf (distant coast), with the continental slope regarded as a scarp, again only waves in so7f 7 are trapped and results are qualitatively as for a straight shelf. An exception is the lowest mode, a ‘double’ Kelvin wave decaying to both sides. A seamount is again similar but introduces the same quantization as an island. A ridge comprises two scarps back-to-back. Each has its set of waves propagating ‘forwards’ (relative to the local slope) in so7f 7; a double Kelvin wave is associated with any net depth difference. Edge waves also propagate in both senses for s=7f 7 large enough to make K40 (see eqn. [1]). Similarly, a trench has sets of waves appropriate to each side. All cases hðxÞ and radial geometry hðrÞ can be treated numerically in the same way as a straight monotone profile.

  qh=qxðqp=qx þ fkp=sÞ þ f 2  s2 N 2 qp=qz ¼ 0 ðz ¼ hÞ

½4

qp=qz þ N 2 p=g ¼ 0 ðz ¼ 0Þ p-0ðx-NÞ

cos½N ðz þ hÞ=cn  exp ½ist þ isy=cn  fx=cn 

½3



f

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=f 1

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O

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where cn ¼ ðghÞ1=2 ðn ¼ 0Þ; cn ¼ Nh=np ðn ¼ 1; 2; yÞ. Similar solutions, with vertical structure distributed roughly as N, exist for any NðzÞ > 0. Propagation is ‘forwards’ (cyclonic around the deep sea; anticyclonic around a cylindrical island) but depends only on density stratification. For a sloping bottom hðxÞ with offshore scale L (shelf width) and depth scale H, the parameter S  N 2 H 2 =f 2 L2 indicates the importance of stratification. For small S, the solutions in so7f 7 are depthindependent Kelvin and continental shelf waves. As S increases, wave speeds s=k increase and nodes of u; v; p in the ðx; zÞ cross-section tilt outwards from the vertical towards horizontal. Correspondingly, seasonal changes have been observed in the vertical structure and offshore scale of currents on the Oregon shelf (for example) and off Vancouver Island, where stratification increases the offshore decay scale of upper-level currents. Quite moderate S may imply

We consider the simplest context, a straight shelf with rest state density r0 ðzÞ; let N 2  g=r0 dr0 =dz. A wave form fuðx; zÞ; vðx; zÞ; wðx; zÞ; rðx; zÞ; pðx; zÞg expðiky þ istÞ is posed. Hydrostatic balance, density, and momentum equations give r; w; u and v, respectively, in terms of p; then

0 1

½5

A flat bottom (uniform h) with uniform N25g/h admits the simplest Kelvin wave and (for n  0) internal Kelvin wave solutions

Stratification

    q2 p=qx2 þ f 2  s2 q=qz N2 qpqz  k2 p ¼ 0

89

k

Figure 3 Qualitative dispersion diagram. —— trapped waves, - - - - nearly trapped waves, - - - s2 ¼ f 2 þ k 2 ghðNÞ.

COASTAL TRAPPED WAVES

Norway

90

1200GMT 5/1/76

Scotland

Figure 4 Modeled continental shelf waves around Scotland at 1200Z, 5 January 1976 after impulsive wind forcing near Shetland on 3 January 1976. Dashed line shows 200 m depth contour. (Reproduced with permission from Huthnance JM (1995) Circulation, exchange and water masses at the ocean margin: the role of physical processes at the shelf edge. Progress in Oceanography 35: 353–431.)

s monotonic increasing in k, contrasting with the maximum in sðkÞ common among unstratified modes. For large S, the modes in so7f 7 become internal Kelvin-like waves with x replaced by x  h1 ðzÞ; S-N corresponds to a shelf width L much less than the internal deformation scale NH=f ; the slope is ‘seen’ only as a coastal wall. Internal Kelvin waves have been observed in the Great Lakes and around Bermuda, where the bottom slope is steep. Similarly, records off Peru show an offshore scale B70 km, greater than the shelf width, because cn =f is large near the equator. For s > 7f 7 and nonzero S, trapping is imperfect. However, there are frequencies sðkÞ at which waves are almost trapped, radiate energy only slowly or respond with maximal amplitude to sustained forcing. These sðkÞ appear to correspond to dispersion curves in so7f 7 but there is still some uncertainty about the role of these waves; they may need some oceanic forcing. Bottom-trapped waves are an idealized form with motion everywhere parallel to a plane sloping seafloor (in uniform N 2 ); they decay away from the seafloor. They may propagate for so7f 7 or s > 7f 7 and up or down the slope, but always with a

component ‘forwards’ along the slope. If this phase propagation direction is f relative to the along-slope direction, then the velocity being transverse is at angle f to the slope and the frequency is s ¼ Nqh=qx cosf. In so7f 7 the general coastal trapped wave form tends for large k to bottomtrapped waves confined near the seafloor maximum of Nqh=qxo7f 7; then s ¼ Nqh=qx. If the maximum Nqh=qxo7f 7, then s increases to 7f 7 as k increases, a qualitative difference from unstratified behavior; formally there is a smooth transition at s ¼ 7f 7. Bottom-trapped wave identification may be difficult (despite the apparent prevalence of near-bottom currents), requiring knowledge of the local slope and stratification. There is evidence from continental slopes off the eastern USA and NW Africa.

Friction Friction causes cross-shelf phase shifts and significant damping of coastal trapped waves. The depth-integrated alongshore momentum balance R for idealized uniform conditions ðqp=qy ¼ 0; udz ¼ 0Þ is qv=qt þ rv=h ¼ t=rh suggesting that the flow v lags the forcing stress t less for low frequency, shallow

COASTAL TRAPPED WAVES

water, and large friction r. For example, nearshore currents lag the wind less than currents in deeper water offshore. Damping rates may be estimated as r=h ¼ Oð0:003 Uh1 Þ, i.e., a decay time less than 4 days for a typical current U ¼ 0.1 m s1 and depth h ¼ 100 m. In this estimate of friction, U should represent all currents present, e.g., tidal currents can provide strong damping. This decay time converts in to a decay distance cg h=r for a wave with energy propagation speed cg . Such decay distances are largest (hundreds to a thousand kilometers or more) for long waves with ‘forward’ energy propagation; much less for (short) waves with ‘backward’ energy propagation.

Mean Flows Mean currents are significant in many places where continental shelf waves have been observed, e.g., adjacent to the Florida current. Waves with phase speeds of a few meters per second or less may be significantly affected by boundary currents of comparable speed, or by vorticity of order f. By linear theory, advection in a uniform mean current V is essentially trivial, but could reverse the propagation of slower waves (higher modes or short wavelengths). Shear V 0  dV=dx modifies the background potential vorticity to PðxÞ  ðf þ V 0 Þ=h; then the gradient of P (rather than f =h) underlies continental shelf wave propagation. ‘Barotropic’ instability is possible if V is strong enough; necessary conditions are P0 ðxs Þ ¼ 0 (some xs ) and P0 ½Vðxs Þ  V > 0 for some x; the growth rate is bounded by max 7V 0 =27. Gulf Stream meanders have been interpreted as barotropically unstable shelf waves from Blake Plateau. In a stratified context, these effects of mean flow V may still apply. Additionally, vertical shear qV=qz is associated with horizontal density gradients: f qV=qz ¼ gr1 0 qr=qx

½7

and associated ‘baroclinic’ instability extracting gravitational potential energy. Two-layer models represent qr=qx by a sloping interface; the varying layer depths are another source of gradients qP=qx in each layer. Thus baroclinic instability may occur even if the current and total depth are uniform. More generally, slow flows over a gently sloping bottom are unstable only if qP=qx has both signs in the system. Such a two-layer channel model predicts instability at peak-energy frequencies in Shelikov Strait, Alaska (for example) and a corresponding wavelength roughly matching that observed. However, we caution that two-layer models may unduly

91

segregate internal Kelvin and continental shelf wave types, say, exaggerating the multiplicity of wave forms and scope for instability. In continuous stratification, the equivalent potential vorticity gradient qf =qx þ q2 V=qx2 þ f 2 ðN 2 qV=qzÞ=qz may support waves in the interior. Density contours rising coastward in association with a shelf edge surface jet modify the fastest continental shelf wave to an inshore ‘frontal-trapped’ form. The bottom slope also supports waves. Over a uniform bottom slope, additional bottom features can couple and destabilize the interior and bottom modes, even if Vðx; zÞ is otherwise stable. However, the bottom slope stabilizes bottom-intensified waves under an intermediate uniformly stratified layer.

Non-linear Effects These mirror typical nonlinear effects for waves. For example, each part of the nonlinear Kelvin wave form moves with the local speed v þ ½gðh þ ZÞ1=2 ; crests (Z and v positive) gain on troughs (Z and v negative) and wave fronts steepen. Similarly, for internal Kelvin waves between an upper layer, depth h, and a deep lower layer (density difference Dr) the local speed is ðghDr=rÞ1=2 everywhere; troughs gain on crests. Dispersion limits this nonlinear steepening; the associated shorter wavelengths propagate more slowly; and the steepening is left behind by the wave. For small amplitudes and long waves, steepening and dispersion are small and can balance in permanentform sech2 ðky þ stÞ solutions. Steepening may be important for internal Kelvin waves, but for typical continental shelf wave amplitudes (near linear) it takes weeks or months, longer than likely frictional decay times. Mean currents may be forced via frictional contributions to the time-averaged nonlinear convective derivatives in the momentum equations. The mean flows, scale h1 h0 f 1 uˆ2 , are typically confined close to the coast or the shelf break (uˆ denotes on–offshore excursion in the waves). Another mechanism is wave-induced form drag over an irregular bottom, giving a biased response to variable forcing; a ‘forward’ flow along the shelf. For example, low-frequency sinusoidal wind forcing may give a mean current up to a maximum fraction ð2pÞ1 of the value under a steady wind, i.e., some centimeters per second, principally in shallower shelf waters. Three-way interactions between coastal-trapped waves can occur if s3 ¼ s1 7s2 , k3 ¼ k1 7k2 , possible for particular combinations according to the shape of the dispersion curve. Typical timescales for energy exchange are many days; effects may be masked by frictional decay. Near a group velocity of

92

COASTAL TRAPPED WAVES

zero, there may be more response to a range of energy inputs.

Alongshore Variations If changes in the stratification and continental shelf form are small in one wavelength, then individual wave modes conserve a longshore energy flux; local wave forms are as for a uniform shelf. Thus Kelvin wave amplitudes increase as f 1=2 and are confined closer to the coast at higher latitudes. Energy flux conservation implies a large amplitude increase if waves of frequency s approach a shelf region where the maximum (sM ) for their particular mode is near s, as for Scottish and Vancouver Island diurnal tides; the energy tends to ‘pile up.’ If shelf variations cause sM to fall well below s, then the waves are totally reflected, with large amplitudes near where sM ¼ s. Poleward-propagating waves experience changing conditions. Near the equator, f is small, S (effective stratification) is large and internal Kelvin-like waves are expected, as off Peru; as f increases poleward, waves evolve to less stratified forms, more like continental shelf waves. (Variations of f are special in supporting offshore energy leakage to Rossby waves in the ocean.) Small irregularities in the shelf (lateral, vertical scales eðgHÞ1=2 =f ; eH) generally cause Oðe2 Þ effects, but OðeÞ nearby and in phase shifts after depth changes. Scattering occurs, preferentially to adjacent wave modes and (if unstratified) the highest mode at the incident frequency (having near-zero group velocity). However, long waves, LW , on long topographic variations (as above; LT ) adopt the appropriate local form; scattering is slow unless LW BLT . If the depth profile has a self-similar form h½ðx  cðyÞÞ=LðyÞ then long (44L) continental shelf waves propagate with changes of amplitude but no scattering or change of form, provided that c and L also vary slowly ðc=c0 ; L=L0 44LÞ. Likewise, there is no scattering if the depth is hðxÞ where r2 xðx; yÞ ¼ 0, representing approximately uniform topographic convexity. In these cases, stronger currents and shorter wavelengths are implied on narrow sections of shelf. Abrupt features are apt to give the strongest scattering, substantial local changes or eddies on the flow. Scattering is the means of slope–current adjustment to a changed depth profile. However, all energy must remain trapped in so7f 7; even in s > 7f 7 special interior angles p=ð2n þ 1Þ can give perfect Kelvin wave energy transmission (for example). Successive reflections in a finite shelf (embayment) may synthesize near-resonant waves with small energy leakage. A complete barrier across the shelf implies reflection into (short, slow) waves of

opposite group velocity. There is a considerable literature of particular calculations. However, it is difficult to generalize, because of the several nonscattering cases interspersed among those with strong scattering.

Generation and Role of Coastaltrapped Waves Oceanic motion may impinge on the continental shelf. Notably at the equator, waves travel eastward to the coast and divide to travel north and south. In general, oceanic motions accommodate to the presence of the coast and shelf; at a wall, by internal Kelvin waves (vertical structure modes); for more realistic shelf profiles, by coastal trapped waves. Oceanic signals tend to be seen at the coast if alongshelf scale >wave-decay distance or if the feature is shallower than the shelf-water depth. Natural modes of the ocean are significantly affected by continental shelves. Modes depending on f =h gradients have increased frequencies and forms concentrated over topography. Numerical models have shown 13 modes with periods between 30 and 80 hours, each mode being localized over one shelf area. Atmospheric pressure forcing the sea surface can be effective in driving Kelvin and edge waves, especially if there is some match of speed and scale, more likely in shallower (shelf) seas. Longshore wind stress s is believed to be the most effective means of generating coastal trapped waves. Within the forcing region, the flow tends to match the wind field; when or where the forcing ceases, the wave travels onwards (‘forwards’) and is then most recognizable. A simple view of this forcing is that s accelerates the alongshore transport hu. A more sophisticated view is that s induces a cross-shelf surface transport 7s7=rf ; coastal blocking induces a compensating return flow beneath, which is acted upon by the Coriolis force to give the same accelerating alongshore transport. A typical stress 0.1 N m2 for 105 s (B 1 day) accelerates 100 m water to 0.1 ms1. Winds blowing across depth contours may be comparably effective if the coast is distant. Other generation mechanisms include scattering (of alongshore flow, especially) by shelf irregularities as above, variable river runoff, and a co-oscillating sea. Shelf-sea motion is often dominated by tides and responses to wind forcing. On a narrow shelf, oceanic elevation signals penetrate more readily to the coast as the higher-mode decay distances are short; the tide is represented primarily by a Kelvin wave spanning the ocean and shelf. Model fits to semidiurnal measurements, showing a dominant

COASTAL TRAPPED WAVES

1983 31 Aug

1984 19 Nov

10 Oct

93

29 Dec

18 Mar

7 Feb

50

Longshelf velocity (cm/s)

_ 50

Hindcast

f25/1000

Observed less eddy mode 50

_ 50

f22/125

50

_ 50

f21/125 0

20

40

60

80

100 120 Days

140

160

180

200

220

Figure 5 Comparison of hindcast along-shelf currents using three coastal-trapped wave modes with measured currents in a crossshelf section after band-pass filtering and removing an eddy mode. (Reproduced with permission from Church JA, White NJ, Clarke AJ, Freeland HJ and Smith RL (1986) Coastal-trapped waves on the East Australian continental shelf. Part II: model verification. Journal of Physical Oceanography 16: 1945–1957.)

Kelvin wave, have been made off California, Scotland and north-west Africa, for example. Several areas at higher latitudes show dominant continental shelf wave contributions to diurnal tidal currents, e.g., west of Scotland, Vancouver Island, Yermak Plateau. On a wide shelf, there is correspondingly greater scope for wind-driven elevations and currents. The extensive forcing scale, typically greater than the shelf width, induces flow with minimal structure (typically no reversals across the shelf) corresponding to the lowest-mode continental shelf wave with maximum elevation signal at the coast. In stratified conditions, the upwelling or downwelling response also corresponds to a wave (or waves). As a wave travels, its amplitude is continually incremented by local forcing. A model based on this approach was used to make the hindcast of measured currents on the south-east Australian shelf shown in Figure 5. At any fixed position, the motion results from local forcing and from arriving waves, bringing the influence of forcing (e.g., upwelling) ‘forwards’ from the ‘backward’ direction. In the Peruvian upwelling regime, for example, variable currents are not well correlated with local winds but include internal Kelvin-like features coming from nearer the equator. This is hardly compatible with (common) simplifications of a zero alongshore pressure

gradient. Moreover, the waves carry the influence of assumed ‘backward’ boundary conditions far into a model. The same applies for steady flow. Friction introduces a ‘forward’ decay distance for a coastaltrapped wave; this distance has a definite low-frequency limit. Currents decay over these distances according to their structure as a wave combination. Thus alongshore evolution or adjustment of flow (however forced) is affected by coastal-trapped waves whose properties should guide model design.

Summary This article considers waves extending across the continental shelf and/or slope and having periods of the order of one day or longer. Their phase propagation is generally cyclonic, with the coast to the right in the Northern Hemisphere, a sense denoted ‘forward’; cross-slope displacements change watercolumn depth and relative vorticity, causing crossslope movement of adjacent water columns. At short-scales, energy propagation can be in the opposite ‘backward’ sense. Strict trapping occurs only for periods longer than half a pendulum day; shorterperiod waves leak energy to the deep ocean, albeit only slowly for some forms. The waves travel faster in stratified seas and on broad shelf-slope profiles;

94

COASTAL TRAPPED WAVES

speeds can be affected, even reversed, by along-shelf flows and reverses of bottom slope. Large amplitudes and abrupt alongshore changes in topography cause distortion and transfers between wave modes. The waves form a basis for the behavior (response to forcing, propagation) of shelf and slope motion on scales of days and the shelf width. Hence, they are important in shelf and slope–sea responses to forcing by tides, winds (e.g., upwelling), density gradients, and oceanic features. Their propagation (distance before decay) implies nonlocal response (over a comparable distance), especially in the ‘forward’ direction.

See also Internal Tides. Internal Waves. Storm Surges. Tides. Vortical Modes.

Further Reading Brink KH (1991) Coastal trapped waves and wind-driven currents over the continental shelf. Annual Review of Fluid Mechanics 23: 389--412. Brink KH and Chapman DC (1985) Programs for computing properties of coastal-trapped waves and wind-driven motions over the continental shelf and slope. Woods Hole Oceanographic Institution, Technical Report 85–17, 2nd edn, 87-24. Dale AC and Sherwin TJ (1996) The extension of baroclinic coastal-trapped wave theory to superinertial frequencies. Journal of Physical Oceanography 26: 2305--2315. Huthnance JM, Mysak LA and Wang D-P (1986) Coastal trapped waves. In: Mooers CNK (ed.) Baroclinic Processes on Continental Shelves. Coastal and Estuarine Sciences, 3, pp. 1–18. Washington DC. American Geophysical Union. LeBlond PH and Mysak LA (1978) Waves in the Ocean. New York: Elsevier.

TIDES D. T. Pugh, University of Southampton, Southampton, UK Copyright & 2001 Elsevier Ltd.

Introduction Even the most casual coastal visitor is familiar with marine tides. Slightly more critical observers have noted from early history, relationships between the movements of the moon and sun, and with the phases of the moon. Several plausible and implausible explanations for the links were advanced by ancient civilizations. Apart from basic curiosity, interest in tides was also driven by the seafarer’s need for safe and effective navigation, and by the practical interest of all those who worked along the shore. Our understanding of the physical processes which relate the astronomy with the complicated patterns observed in the regular tidal water movements is now well advanced, and accurate tidal predictions are routine. Numerical models of the ocean responses to gravitational tidal forces allow computations of levels both on- and offshore, and satellite altimetry leads to detailed maps of ocean tides that confirm these. The budgets and flux of tidal energy from the earth–moon dynamics through to final dissipation in a wide range of detailed marine processes has been an active area of research in recent years. For the future, there are difficult challenges in understanding the importance of these processes for many complicated coastal and open ocean phenomena.

The two main tidal features of any sea-level record are the range (measured as the height between successive high and low levels) and the period (the time between one high (or low) level and the next high (or low) level).Spring tides are semidiurnal tides of increased range, which occur approximately twice a month near the time when the moon is either new or full. Neap tides are the semidiurnal tides of small range which occur between spring tides near the time of the first and last lunar quarter. The tidal responses of the ocean to the forcing of the moon and the sun are very complicated and tides vary greatly from one site to another. Tidal currents, often called tidal streams, have similar variations from place to place. Semidiurnal, mixed, and diurnal currents occur; they usually have the same characteristics as the local tidal changes in sea level, but this is not always so. For example, the currents in the Singapore Strait are often diurnal in character, but the elevations are semidiurnal. It is important to make a distinction between the popular use of the word ‘tide’ to signify any change of sea level, and the more specific use of the word to mean only regular, periodic variations. We define tides as periodic movements which are directly related in amplitude and phase to some periodic geophysical force. The dominant geophysical forcing function is the variation of the gravitational field on the surface of the earth, caused by the regular movements of the Moon–Earth and Earth–Sun systems. Movements due to these gravitational forces are termed gravitational tides. This is to distinguish them from the smaller movements due to regular meteorological forces which are called eithermeteorological or more usually radiational tides.

Gravitational Potential Tidal Patterns Modern tidal theory began when Newton(1642– 1727) applied his formulation of the Law of Gravitational Attraction: that two bodies attract each other with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. He was able to show why there are two tides for each lunar transit. He also showed why the half-monthly springto-neap cycle occurred, why once-daily tides are a maximum when the Moon is furthest from the plane of the equator, and why equinoctial tides are larger than those at the solstices.

The essential elements of a physical understanding of tide dynamics are contained in Newton’s Laws of Motion and in the principle of Conservation of Mass. For tidal analysis the basics are Newton’s Laws of Motion and the Law of Gravitational Attraction. The Law of Gravitational Attraction states that for two particles of masses m1 and m2, separated by a distance r the mutual attraction is: F¼G

m1 m2 r2

½1

G is the universal gravitational constant.

95

96

TIDES

Use is made of the concept of the gravitational potential of a body; gravitational potential is the work which must be done against the force of attraction to remove a particle of unit mass to an infinite distance from the body. The potential at P on the Earth’s surface (Figure 1) due to the moon is: Gm Op ¼  MP

½2

This definition of gravitational potential, involving a negative sign, is the one normally adopted in physics, but there is an alternative convention often used in geodesy, which treats the potential in the above equation as positive. The advantage of the geodetic convention is that an increase in potential on the surface of the earth will result in an increase of the level of the free water surface. Potential has units of L2T2. The advantage of working with gravitational potential is that it is a scalar property, which allows simpler mathematical manipulation; in particular, the vector, gravitational force on a particle of unit mass is given by  grad (Op). Applying the cosine law to DOPM in Figure 1 MP2 ¼ a2 þ r2  2ar cos f

a a2 ‘MP ¼ r 1  2 cosf þ 2 r r

½3

1=2 ½4

P1 ¼ cos f

P3 ¼



1=2 Gm a a2 1  2 cosf þ 2 r r r

OP ¼ 

 1 5cos3 f  3cos f 2

½9

 1 a2  Gm 3 3cos2 f  1 r 2

  qOp ¼ 2gDl cos2 f  13 qa

½11

horizontally in the direction of increasing f. ½5

Gm a a2 1 þ P1 ðcosfÞ þ 2 P2 ðcosfÞ OP ¼  r r r

a2 þ 2 P3 ðcosfÞ þ y r

qOp ¼ gDl sin2f adf

½12

For the Moon: ½6

P M



½10

The force on the unit mass at P may be resolved into two components as functions of f:

which may be expanded:

O

½8

The tidal forces represented by the terms in this potential are calculated from their spatial gradients  grad (Pn). The first term in the equation is constant (except for variations in r) and so produces no force. The second term produces a uniform force parallel to OM because differentiating with respect to (a cos f) yields a gradient of potential which provides the force necessary to produce the acceleration in the earth’s orbit towards the center of mass of the Moon–Earth system. The third term is the major tide-producing term. For most purposes the fourth term may be neglected, as may all higher terms. The effective tide-generating potential is therefore written as:



a

½7

 1 3cos2 f  1 2

P2 ¼

vertically upwards : 

Hence we have: Op ¼ 

The terms in Pn(cos f) are the Legendre Polynomials:

r Moon

Earth

Figure 1 The general position of the point P on the Earth’s surface, defined by the angle f.

Dl ¼

  3 ml a 3 2me Rl

½13

m1 is the lunar mass and me is the Earth mass. R1, the lunar distance, replaces r. The resulting forces are shown in Figure 2. To generalize in three dimensions, the lunar angle f must be expressed in suitable astronomical variables. These are chosen to be declination of the Moon north or south of the equator, the north–south latitude of P, fp, and the hour angle of the moon, which is the difference in longitude between the meridian of P and the meridian of the sublunar point V on the Earth’s surface.

TIDES

97

mass, distance, and declination substituted for lunar parameters. The ratio of the two tidal amplitudes is:   ms R l 3 m1 Rs

(A)

(B)

Figure 2 The tide-producing forces at the Earth’s surface, due to the Moon. (A) The vertical forces, showing an outward pull at the equator and a smaller downward pull at the poles. (B) The horizontal forces, which are directed away from the poles towards the equator, with a maximum value at 451 latitude.

The Equilibrium Tide An equilibrium tide can be computed from eqn [10] by replacing cos2f by the full astronomical expression in terms of d1, fp and the hour angle C1. The equilibrium tide is defined as the elevation of the sea surface that would be in equilibrium with the tidal forces if the Earth were covered with water and the response is instantaneous. It serves as an important reference system for tidal analysis. It has three coefficients which characterize the three main species of tides: (1) the long period species; (2) the diurnal species at a frequency of one cycle per day (cos C); and (3) the semidiurnal species at two cycles per day(cos 2C). The equilibrium tide due to the sun is expressed in a form analogous to the lunar tide, but with solar

For the semidiurnal lunar tide at the equator when the lunar declination is zero, the equilibrium tidal amplitude is 0.27 m. For the sun it is 0.13 m. The solar amplitudes are smaller by a factor of 0.46 thanthose of the lunar tide, but the essential details are the same. The maximum diurnal tidal ranges occur when the lunar declination is greatest. The ranges become very small when the declination is zero. This is because the effect of declination is to produce an asymmetry between the two high- and the two low-water levels observed as a point P rotates on the earth within the two tidal bulges. The fortnightly spring/neap modulation of semidiurnal tidal amplitudes is due to the various combinations of the separate lunar and solar semidiurnal tides. At times of spring tides the lunar and solar forces combine together, but at neap tides the lunar and solar forces are out of phase and tend to cancel. In practice, the observed spring tides lag the maximum of the tidal forces, usually by one or two days due to the inertia of the oceans and energy losses. This delay is traditionally called the age of the tide. The observed ocean tides are normally much larger than the equilibrium tide because of the dynamic response of the ocean to the tidal forces. But the observed tides do have their energy at the samefrequencies (or periods) as the equilibrium tide. This forms the basis of tidal analysis.

Tidal Analysis Tidal analysis of data collected by observations of sea levels and currents has two purposes. First, a goodanalysis provides the basis for predicting tides at future times, a valuable aid for shipping and other operations. Secondly, the results of analyses can be mapped and interpreted scientifically in terms of the hydrodynamics of the seas and their responses to tidal forcing. In tidal analysis the aim is to produce significant time-stable tidal parameters which describe the tidal regime at the place of observation. These parameters are often termed tidal constants on the assumption that the responses of the oceans and seas to tidal forces do not change with time. A good tidal analysis seeks to represent the data by afew significant stable numbers which mean something physically. In general, the longer the period of data included in the analysis, the greater the number of

98

TIDES

constants which can be independently determined. If possible, an analysis should give some idea of the confidence which should be attributed to each tidal constant determined. The close relationship between the movement of the moon and sun, and the observed tides, make the lunar and solar coordinates a natural starting point for any analysis scheme. Three basic methods of tidal analysis have been developed. The first, which is now generally of only historical interest, the non-harmonic method, relates high and low water times and heights directly to the phases of the moon and other astronomical parameters. The second method which is generally used for predictions and for scientific work, harmonic analysis, treats the observed tides as the sum of a finite number of harmonic constituents with angular speeds determined from the astronomical arguments. The third method developsthe concept, widely used in electronic engineering, of a frequency-dependent response of a system to a driving mechanism. For tides, the driving mechanism is the equilibrium potential. The latter twomethods are special applications of the general formalisms of time series analysis. Analyses of changing sea levels (scalar quantities) are obviously easier than those of currents (vectors), which can be analysed by resolving into two components. Harmonic Analysis

The basis of harmonic analysis is the assumption that the tidal variations can be represented by a finite number N of harmonic terms of the form: Hn cosðsn t  gn Þ where Hn is the amplitude, gn is the phase lag on the equilibrium tide at Greenwich and sn is the angular speed. The angular speeds sn are determined by an expansion of the equilibrium tide into harmonic terms. The speeds of these terms are found to have the general form: on ¼ ia o1 þ ib o2 þ ic o3 þ ðo4 ; o5 ; o6 termsÞ

½14

where the values of o1 to o6 are the angular speeds related to astronomical parameters and the coefficients, ia to ic are small integers (normally 0, 1 or 2) (Table 1). The phase lags gn are defined relative to the phase of the corresponding term in the harmonic expansion of the equilibrium tide. Full harmonic analysis of the equilibrium tide shows the grouping of tidal terms into species (1;

Table 1 The basic astronomical periods which modulate the tidal forces

Mean solar day (msd) Mean lunar day Sidereal month Tropical year Moon’s perigee Regression of Moon’s nodes Perihelion

Period

Symbol

1.0000 msd 1.0351 msd 27.3217 msd 365.24222 msd 8.85 years 18.61 years 20 942 years

o0 o1 o2 o3 o4 o5 o6

diurnal, semidiurnal y), groups (o2; monthly) and constituents (o3; annual). Response Analysis

The basic ideas involved in response analysis are common to many activities. A system, sometimes called a ‘black box’, is subjected to an external stimulus or input. The output from a system depends on the input and the system response to that input. The response of the system may be evaluated by comparing the input and output functions at various forcing frequencies. These ideas are common in many different contexts, including mechanical engineering, financial modeling and electronics. In tidal analysis the input is the equilibrium tidal potential. The tidal variations measured at a particular site may be considered as the output from the system. The system is the ocean, and we seek to describe its response to gravitational forces. This ‘response’ treatment has the conceptual advantage of clearly separating the astronomy (the input) from the oceanography (the black box). The basic response analysis assumes a linear system, but weak nonlinear interactions can be allowed for with extra terms.

Tidal Dynamics The equilibrium tide consists of two symmetrical tidal bulges directly opposite the moon or sun. Semidiurnal tidal ranges would reach their maximum value of about 0.5 m at equatorial latitudes. The individual high water bulges would track around the earth, moving from east to west in steady progression. These theoretical characteristics are clearly not those of the observed tides. The observed tides in the main oceans have much larger mean ranges, of about 1 m, but there are considerable variations. Times of tidal high water vary in a geographical pattern which bears norelationship to the simple ideas of a double bulge. The tides spread from the oceans onto the surrounding

TIDES

continental shelves, where even larger ranges areobserved. In some shelf seas the spring tidal range may exceed 10 m: the Bay of Fundy, the Bristol Channel and the Argentine Shelf are well-known examples. Laplace (1749–1827) advanced the basic mathematical solutions for tidal waves on a rotating earth. More generally, the reasons for these complicated ocean responses to tidal forcing may be summarized as follows. 1. Movements of water on the surface of the earth must obey the physical laws represented by the hydrodynamic equations of continuity and momentum balance; this means that they must propagate as long waves. Any propagation of a wave, east to west around the earth, is impeded by the north–south continental boundaries. 2. Long waves travel at a speed that is related to the water depth; oceans are too shallow for this to match the tracking of the moon. 3. The various ocean basins have their individual natural modes of oscillation which influence their response to the tide-generating forces. There are many resonant frequencies. However, the whole global ocean system seems to be near to resonance at semidiurnal tidal frequencies, as the observed semidiurnal tides are generally much bigger than the diurnal tides. 4. Water movements are affected by the rotation of the earth. The tendency for water movement to maintain a uniform direction in absolute spacemeans that it performs a curved path in the rotating frame of reference within which our observations are made. 5. The solid earth responds elastically to the imposed gravitational tidal forces, and to the ocean tidal loading. The redistribution of water mass during the tidal cycle affects the gravitational field. Long-Wave Characteristic, No Rotation

Provided that wave amplitudes are small compared with the depth, and that the depth is small compared with the wavelength, then the speed for the wave propagation is: c ¼ ðgDÞ1=2

u ¼ zðg=DÞ1=2

on the value of g and the water depth; any disturbance which consists of a number of separate harmonic constituents will not change its shape as it propagates – this is nondispersive propagation. Waves at tidal periods are long waves, even in the deep ocean, and so their propagation is nondispersive. In the real ocean, tides cannot propagate endlessly as progressive waves. They undergo reflection at sudden changes of depth and at the coastal boundaries. Standing Waves and Resonance

Two progressive waves traveling in opposite directions result in a wave motion, called a standing wave. This can happen where a wave is perfectly reflected at a barrier. Systems which are forced by oscillations close to their natural period have large amplitude responses. The responses of oceans and many seas are close to semidiurnal resonance. In nature, the forced resonant oscillations cannot grow indefinitely because friction limits the response. Because of energy losses, tidal waves are not perfectly reflected at the head of a basin, which means that the reflected wave is smaller than the ingoing wave. It is easy to show that this is equivalent to a progressive wave superimposed on a standing wave with the progressive wave carrying energy to the head of the basin. Standing waves cannot transmit energy because they consist of two progressive waves of equal amplitude traveling in opposite directions. Long Waves on a Rotating Earth

A long progressive wave traveling in a channel on a rotating Earth behaves differently from a wave traveling along a nonrotating channel. The geostrophic forces that affect the motion in a rotating system, cause a deflection of the currents towards the right of the direction of motion in the Northern Hemisphere. The build-up of water on the right of the channel gives rise to a pressure gradient across the channel, which in turn develops until at equilibrium it balances the geostrophic force. The resulting Kelvin wave is described mathematically: 

½15

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Long waves have the special property that the speed c is independent of the frequency, and depends only

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  g 1=2 fy zðyÞ ½17 ; u ð yÞ ¼ c D

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f ¼ [(gD)1/2/f], which depends on the latitude and the water depth. This scale is called the Rossby radius of deformation. At a distance y ¼ c/f from the boundary the amplitude has fallen to 0.37 Ho. At 451N in water of 4000 m depth the Rossby radius is 1900 km, but in water 50 m deep this is reduced to 215 km. Kelvin waves are not the only solution to the hydrodynamic equations on a rotating Earth: a more general form, called Poincare´ waves, gives amplitudes which vary sinusoidally rather than exponentially in the direction transverse to the direction of wave propagation. The case of a standing-wave oscillation on a rotating Earth is of special interest in tidal studies. Away from the reflecting boundary, tidal waves can be represented by two Kelvin waves traveling in opposite directions. The wave rotates about a nodal point, which is called an amphidrome (Figure 3). The cotidal lines all radiate outwards from the amphidrome and the co-amplitude lines form a set of nearly concentric circles around the center at the amphidrome, at which the amplitude is zero. The amplitude is greatest around the boundaries of the basin.

Ocean Tides Dynamically there are two essentially different types of tidal regime; in the wide and relatively deep ocean basins the observed tides are generated directly by

the external gravitational forces; in the shelf seas the tides are driven by co-oscillation with the oceanic tides. The ocean response to the gravitational forcing may be described in terms of a forced linear oscillator, with weak energy dissipation. A global chart of the principal lunar semidiurnal tidal constituent M2 shows a complicated pattern of amphidromic systems. As a general rule these conform to the expected behavior for Kelvin wave propagation, with anticlockwise rotation in the Northern Hemisphere, and clockwise rotation in the Southern Hemisphere. For example, in the Atlantic Ocean the mostfully developed semidiurnal amphidrome is located near 501N, 391W. The tidal waves appear to travel around the position in a form which approximates to a Kelvin wave, from Portugal along the edge of the north-west European continental shelf towards Iceland, and thence west and south past Greenland to Newfoundland. There is a considerable leakage of energy to the surrounding continental shelves and to the Arctic Ocean, so the wave reflected in a southerly direction, is weaker than the wave traveling northwards along the European coast. The patterns of tidal waves on the continental shelf are scaled down as the wave speeds are reduced. In the very shallow water depths (typically less than 20 m) there are strong tidal currents and substantial energy losses due to bottom friction. Tidal waves are

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TIDES

strongly influenced by linear Kelvin wave dynamics and by basin resonances. Energy is propagated to the shallow regions where it is dissipated.

Energy Fluxes and Budgets The energy lost through tidal friction gradually slows down the rate of rotation of the earth, increasing the length of the day by one second in 41 000 years. Angular momentum of the earth–moon system is conserved by the moon moving away from the earth at 3.7 mm per year. The total rate of tidal energy dissipation due to the M2 tide can be calculated rather exactly from the astronomic observations at 2.50 7 0.05 TW, of which 0.1 TW is dissipated in the solid Earth. The total lunar dissipation is 3.0 TW, and the total due to both sun and moon is 4.0 TW. For comparison the geothermal heat loss is 30 TW, and the 1995 total installed global electric capacity was 2.9 TW. Solar radiation input isfive orders of magnitude greater. Most of the 2.4 TW of M2 energy lost in the ocean is due to the work against bottom friction which opposes tidal currents. Because the friction increases approximately as the square of current speed, and the energy loses as the cube, tidal energy loses are concentrated in a few shelf areas of strong tidal currents. Notable among these are the north-west European Shelf, the Patagonian Shelf, the Yellow Sea, the Timor and Arafura Seas, Hudson Bay, Baffin Bay, and the Amazon Shelf. It now appears that up to 25% (1 TW) of the tidal energy may be dissipated by internal tidal waves in the deep ocean, where the dissipation processes contribute to vertical mixing and the breakdown of stratification. Again, energy losses may be concentrated in a few areas, for example where the rough topography of midocean ridges and islandarcs create favorable conditions. One of the main areas of tidal research is increasingly concentrated on gaining a better understanding of the many nonlinear ocean processes that are driven by this cascading tidal energy. Some examples, outlined below, are considered in more detail elsewhere in this Encyclopedia.

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Generation of tidal fronts in shelf seas, where the buoyancy forces due to tidal mixing compete with the buoyancy fluxes due to surface heating; the ratio of the water depth divided by the cube of the tidal currentis a good indicator of the balance between the two factors, and fronts form along lines where this ratio reaches a critical value. River discharges to shallow seas near the mouth of rivers. The local input of freshwater buoyancy may be comparable to the buoyancy

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input fromsummer surface warming. These regions are called ROFIs (regions of freshwater influence). Spring–neap variations in the energy of tidal mixing strongly influence the circulation in these regions. The mixing and dispersion of pollutants often driven by the turbulence generated by tides. Maximum turbulent energy occurs some hours after the time of maximum tidal currents. Sediment processes of erosion and deposition are often controlled by varying tidal currents, particularly over a spring–neap cycle. The phase delay in suspended sediment concentration after maximum currents may be related to the phase lag in the turbulent energy, which has importantconsequences for sediment deposition and distribution. Residual circulation is partly driven by nonlinear responses to tidal currents in shallow water. Tidal flows also induce residual circulation around sandbanks because of the depth variations. In the Northern Hemisphere this circulation is observed to be in a clockwise sense. Near headlands and islands which impede tidal currents, residual eddies can cause marked asymmetry between the time and strength of the tidal ebb and flow currents. Tidal currents influence biological breeding patterns, migration, and recruitment. Some types of fish have adapted to changing tidal currents to assist in their migration: they lie dormant on the seabed when the currents are not favorable. Tidal mixing in shallow seas promotes productivity by returning nutrients to surface waters where light is available. Tidal fronts are known to be areas of high productivity. The most obvious example of tidal influence on biological processes is the zonation of species found at different levels along rocky shorelines. Evolution of sedimentary shores due to the dynamic equilibrium between waves, tides and other processes along sedimentary coasts which resultsin a wide range of features such as lagoons, sandbars, channels, and islands. These are very complicated processes which are still difficult to understandand model. Tidal amphidrome movements. The tidal amphidromes as shown in Figure 4 only fall along the center line of the channel if the incoming tidal Kelvin wave is perfectly reflected. In reality, the reflected wave is weaker, and the amphidromes are displaced towards the side of the sea along which the outgoing tidal wave travels. Proportionately more energy is removed at spring tides than at neap tides, so the amphidromes can

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Figure 4 Map of the principal lunar semidiurnal tide produced by computer. Ocean tides observed by satellite and in situ now agree very closely with computer modeled tides. The dark areas show regions of high tida lamplitude. Note the convergence of cophase lines at amphidromes. In the Northern Hemisphere the tides normally progress in an anticlockwise sense around the amphidrome; in the Southern Hemisphere the progression is usually clockwise.

move by several tens of kilometers during thespring–neap cycle. Nonlinear processes on the basic M2 tide generate a series of higher harmonics, M4, M6y, with corresponding terms such as MS4 for spring–neap interactions. A better understanding of the significance of the amplitudes and phases of these terms in the analyses of shallow-water tides and tidal phenomena will be an important tool in advancing our overall knowledge of the influence of tides on a wide range of ocean processes.

See also Beaches, Physical Processes Affecting. Coastal Trapped Waves. Dispersion in Shallow Seas. Internal Tidal Mixing. Internal Waves. Sea Level Change. Tidal Energy. Upper Ocean Vertical Structure. Waves on Beaches.

Further Reading Cartwright DE (1999) Tides – a Scientific History. Cambridge: Cambridge University Press. Garrett C and Maas LRM (1993) Tides and their effects. Oceanus 36(1): 27--37. Parker BB (ed.) (1991) Tidal Hydrodynamics. New York: John Wiley. Prandle D (1997) Tidal characteristics of suspended sediment concentrations. Journal of Hydraulic Engineering 123: 341--350. Pugh DT (1987) Tides, Surges and Mean Sea Level. Chichester: John Wiley. Ray RD and Woodworth PL (eds.) (1997) Special issue on tidal science in honour of David E Cartwright. Progress in Oceanography 40. Simpson JH (1998) Tidal processes in shelf seas. In: Brink KH and Robinson AR (eds.) The Sea, Vol. 10. New York: John Wiley. Wilhelm H, Zurn W, and Wenzel HG (eds.) (1997) TidalPhenomena: Lecture Notes in Earth Sciences 66. Berlin: Springer-Verlag.

TIDAL ENERGY A. M. Gorlov, Northeastern University, Boston, Massachusetts, USA Copyright & 2001 Elsevier Ltd.

Introduction Gravitational forces between the moon, the sun and the earth cause the rhythmic rising and lowering of ocean waters around the world that results in Tide Waves. The moon exerts more than twice as great a force on the tides as the sun due to its much closer position to the earth. As a result, the tide closely follows the moon during its rotation around the earth, creating diurnal tide and ebb cycles at any particular ocean surface. The amplitude or height of the tide wave is very small in the open ocean where it measures several centimeters in the center of the wave distributed over hundreds of kilometers. However, the tide can increase dramatically when it reaches continental shelves, bringing huge masses of water into narrow bays and river estuaries along a coastline. For instance, the tides in the Bay of Fundy in Canada are the greatest in the world, with amplitude between 16 and 17 meters near shore. High tides close to these figures can be observed at many other sites worldwide, such as the Bristol Channel in England, the Kimberly coast of Australia, and the Okhotsk Sea of Russia. Table 1 contains ranges of amplitude for some locations with large tides. On most coasts tidal fluctuation consists of two floods and two ebbs, with a semidiurnal period of about 12 hours and 25 minutes. However, there are some coasts where tides are twice as long (diurnal tides) or are mixed, with a diurnal inequality, but are still diurnal or semidiurnal in period. The magnitude of tides changes during each lunar month. The

highest tides, called spring tides, occur when the moon, earth and sun are positioned close to a straight line (moon syzygy). The lowest tides, called neap tides, occur when the earth, moon and sun are at right angles to each other (moon quadrature). Isaac Newton formulated the phenomenon first as follows: ‘The ocean must flow twice and ebb twice, each day, and the highest water occurs at the third hour after the approach of the luminaries to the meridian of the place’. The first tide tables with accurate prediction of tidal amplitudes were published by the British Admiralty in 1833. However, information about tide fluctuations was available long before that time from a fourteenth century British atlas, for example. Rising and receding tides along a shoreline area can be explained in the following way. A low height tide wave of hundreds of kilometers in diameter runs on the ocean surface under the moon, following its rotation around the earth, until the wave hits a continental shore. The water mass moved by the moon’s gravitational pull fills narrow bays and river estuaries where it has no way to escape and spread over the ocean. This leads to interference of waves and accumulation of water inside these bays and estuaries, resulting in dramatic rises of the water level (tide cycle). The tide starts receding as the moon continues its travel further over the land, away from the ocean, reducing its gravitational influence on the ocean waters (ebb cycle). The above explanation is rather schematic since only the moon’s gravitation has been taken into account as the major factor influencing tide fluctuations. Other factors, which affect the tide range are the sun’s pull, the centrifugal force resulting from the earth’s rotation and, in some cases, local resonance of the gulfs, bays or estuaries.

Energy of Tides Table 1

Highest tides (tide ranges) of the global ocean

Country

Site

Tide range (m)

Canada England France France Argentina Russia Russia

Bay of Fundy Severn Estuary Port of Ganville La Rance Puerto Rio Gallegos Bay of Mezen (White Sea) Penzhinskaya Guba (Sea of Okhotsk)

16.2 14.5 14.7 13.5 13.3 10.0 13.4

The energy of the tide wave contains two components, namely, potential and kinetic. The potential energy is the work done in lifting the mass of water above the ocean surface. This energy can be calculated as: E ¼ grA

Z

zdz ¼ 0:5grAh2 ;

where E is the energy, g is acceleration of gravity, r is the seawater density, which equals its mass per unit

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volume, A is the sea area under consideration, z is a vertical coordinate of the ocean surface and h is the tide amplitude. Taking an average (gr) ¼ 10.15 kN m3 for seawater, one can obtain for a tide cycle per square meter of ocean surface: E ¼ 1:4h2 ; watt-hour or E ¼ 5:04h2 ; kilojoule The kinetic energy T of the water mass m is its capacity to do work by virtue of its velocity V. It is defined by T ¼ 0.5 m V2. The total tide energy equals the sum of its potential and kinetic energy components. Knowledge of the potential energy of the tide is important for designing conventional tidal power plants using water dams for creating artificial upstream water heads. Such power plants exploit the potential energy of vertical rise and fall of the water. In contrast, the kinetic energy of the tide has to be known in order to design floating or other types of tidal power plants which harness energy from tidal currents or horizontal water flows induced by tides. They do not involve installation of water dams.

Extracting Tidal Energy: Traditional Approach People used the phenomenon of tides and tidal currents long before the Christian era. The earliest navigators, for example, needed to know periodical tide fluctuations as well as where and when they could use or would be confronted with a strong tidal current. There are remnants of small tidal hydromechanical installations built in the Middle Ages around the world for water pumping, watermills and other applications. Some of these devices were exploited until recent times. For example, large tidal waterwheels were used for pumping sewage in Hamburg, Germany up to the nineteenth century. The city of London used huge tidal wheels, installed under London Bridge in 1580, for 250 years to supply fresh water to the city. However, the serious study and design of industrial-size tidal power plants for exploiting tidal energy only began in the twentieth century with the rapid growth of the electric industry. Electrification of all aspects of modern civilization has led to the development of various converters for transferring natural potential energy sources into electric power. Along with fossil fuel power systems and nuclear reactors, which create huge new environmental pollution problems, clean renewable energy sources have attracted scientists

and engineers to exploit these resources for the production of electric power. Tidal energy, in particular, is one of the best available renewable energy sources. In contrast to other clean sources, such as wind, solar, geothermal etc., tidal energy can be predicted for centuries ahead from the point of view of time and magnitude. However, this energy source, like wind and solar energy is distributed over large areas, which presents a difficult problem for collecting it. Besides that, complex conventional tidal power installations, which include massive dams in the open ocean, can hardly compete economically with fossil fuel (thermal) power plants, which use cheap oil or coal, presently available in abundance. These thermal power plants are currently the principal component of world electric energy production. Nevertheless, the reserves of oil and coal are limited and rapidly dwindling. Besides, oil and coal cause enormous atmospheric pollution both from emission of green house gases and from their impurities such as sulfur in the fuel. Nuclear power plants produce accumulating nuclear wastes that degrade very slowly, creating hazardous problems for future generations. Tidal energy is clean and not depleting. These features make it an important energy source for global power production in the near future. To achieve this goal, the tidal energy industry has to develop a new generation of efficient, low cost and environmentally friendly apparatus for power extraction from free or ultra-low head water flow. Four large-scale tidal power plants currently exist. All of them were constructed after World War II. They are the La Rance Plant (France, 1967), the Kislaya Guba Plant (Russia, 1968), the Annapolis Plant (Canada, 1984), and the Jiangxia Plant (China, 1985). The main characteristics of these tidal power plants are given in Table 2. The La Rance plant is shown in Figure 1. All existing tidal power plants use the same design that is accepted for construction of conventional river hydropower stations. The three principal structural and mechanical elements of this designare: a water dam across the flow, which creates an artificial water basin and builds up a water head for operation of hydraulic turbines; a number of turbines coupled with electric generators installed at the lowest point of the dam; and hydraulic gates in the dam to control the water flow in and out of the water basin behind the dam. Sluice locks are also used for navigation when necessary. The turbines convert the potential energy of the water mass accumulated on either side of the dam into electric energy during the tide. The tidal power plant can be designed for operation either by double or single action. Double action means that the turbines work in both water

TIDAL ENERGY Table 2

105

Extant large tidal power plants

Country

Site

Installed power (MW)

Basin area (km2)

Mean tide (m)

France Russia Canada China

La Rance Kislaya Guba Annapolis Jiangxia

240 0.4 18 3.9

22 1.1 15 1.4

8.55 2.3 6.4 5.08

flows, i.e. during the tide when the water flows through the turbines, filling the basin, and then, during the ebb, when the water flows back into the ocean draining the basin. In single-action systems, the turbines work only during the ebb cycle. In this case, the water gates are kept open during the tide, allowing the water to fill the basin. Then the gates close, developing the water head, and turbines start

operating in the water flow from the basin back into the ocean during the ebb. Advantages of the double-action method are that it closely models the natural phenomenon of the tide, has least effect on the environment and, in some cases, has higher power efficiency. However, this method requires more complicated and expensive reversible turbines and electrical equipment. The

Figure 1 Aerial view of the La Rance Tidal Power Plant (Source: Electricite´de France).

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single action method is simpler, and requires less expensive turbines. The negative aspects of the single action method are its greater potential for harm to the environment by developing a higher water head and causing accumulation of sediments in the basin. Nevertheless, both methods have been used in practice. For example, the La Rance and the Kislaya Guba tidal power plants operate under the double-action scheme, whereas the Annapolis plant uses a single-action method. One of the principal parameters of a conventional hydropower plant is its power output P (energy per unit time) as a function of the water flow rate Q (volume per time) through the turbines and the water head h (difference between upstream and downstream water levels). Instantaneous power P can be defined by the expression: P ¼ 9.81 Qh, kW, where Q is in m3s1, h is in meters and 9.81 is the product (rg) for fresh water, which has mass density r ¼ 1000 kg m3 and g ¼ 9.81 m s2. The (rg) component has to be corrected for applications in salt water due to its different density (see above). The average annual power production of a conventional tidal power plant with dams can be calculated by taking into account some other geophysical and hydraulic factors, such as the effective basin area, tidal fluctuations, etc. Tables 2 and 3 contain some characteristics of existing tidal power plants as well as prospects for further development of traditional power systems in various countries using dams and artificial water basins described above.

Extracting Tidal Energy: Non-traditional Approach As mentioned earlier, all existing tidal power plants have been built using the conventional design developed for river power stations with water dams as

Table 3

their principal component. This traditional river scheme has a poor ecological reputation because the dams block fish migration, destroying their population, and damage the environment by flooding and swamping adjacent lands. Flooding is not an issue for tidal power stations because the water level in the basin cannot be higher than the natural tide. However, blocking migration of fish and other ocean inhabitants by dams may represent a serious environmental problem. In addition, even the highest average global tides, such as in the Bay of Fundy, are small compared with the water heads used in conventional river power plants where they are measured in tens or even hundreds of meters. The relatively low water head in tidal power plants creates a difficult technical problem for designers. The fact is that the very efficient, mostly propeller-type hydraulic turbines developed for high river dams are inefficient, complicated and very expensive for lowhead tidal power application. These environmental and economic factors have forced scientists and engineers to look for a new approach to exploitation of tidal energy that does not require massive ocean dams and the creation of high water heads. The key component of such an approach is using new unconventional turbines, which can efficiently extract the kinetic energy from a free unconstrained tidal current without any dams. One such turbine, the Helical Turbine, is shown in Figure 2. This cross-flow turbine was developed in 1994. The turbine consists of one or more long helical blades that run along a cylindrical surface like a screwthread, having a so-called airfoil or ‘airplane wing’ profile. The blades provide a reaction thrust that can rotate the turbine faster than the water flow itself. The turbine shaft (axis of rotation) must be perpendicular to the water current, and the turbine can be positioned either horizontally or vertically. Due to its axial symmetry, the turbine always

Some potential sites for tidal power installations (traditional approach)

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Site

Potential power (MW)

Basin area (km2)

USA USA Russia Russia UK UK Argentina Korea Australia Australia

Passamaquoddy Cook Inlet Mezen Tugur Severn Mersey San Jose Carolim Bay Secure Walcott

400 Up to 18 000 15 000 6790 6000 700 7000 480 570 1750

300 3100 2640 1080 490 60 780 90 130 260

Mean tide (m) 5.5 4.35 5.66 5.38 8.3 8.4 6.0 4.7 8.4 8.4

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the water head for traditional design or unconstrained water current Pw , i.e. Z ¼ Pt /Pw. The maximum power of the Uldolmok tidal project shown in Figure 3 is about 90 MW calculated using the above approach for V ¼ 12 knots, A ¼ 2100 m2 and Z ¼ 0.35. Along with the floating power farm projects with helical turbines described, there are proposals to use large-diameter propellers installed on the ocean floor to harness kinetic energy of tides as well as other ocean currents. These propellers are, in general, similar to the well known turbines used for wind farms.

Helical Turbine

Utilizing Electric Energy from Tidal Power Plants Waterproof chamber for generator and data collectors

Ports

Figure 2 Double-helix turbine with electric generator for underwater installation.

develops unidirectional rotation, even in reversible tidal currents. This is a very important advantage, which simplifies design and allows exploitation of the double-action tidal power plants. A pictorial view of a floating tidal power plant with a number of vertically aligned triple-helix turbines is shown in Figure 3. This project has been proposed for the Uldolmok Strait in Korea, where a very strong reversible tidal current with flows up to12 knots (about 6 m s1) changes direction four times a day. The following expression can be used for calculating the combined turbine power of a floating tidal plant (power extracted by all turbines from a free, unconstrained tidal current): Pt ¼ 0.5ZrAV3, where Pt is the turbine power in kilowatts, Z is the turbine efficiency (Z ¼ 0.35 in most tests of the triple-helix turbine in free flow), r is the mass water density, A is the total effective frontal area of the turbines in m2 (cross-section of the flow where the turbines are installed) and V is the tidal current velocity in m s1. Note, that the power of a free water current through a cross-flow area A is Pw ¼ 0.5rAV3. The turbine efficiency Z, also called power coefficient, is the ratio of the turbine power output Pt to the power of either

A serious issue that must be addressed is how and where to use the electric power generated by extracting energy from the tides. Tides are cyclical by their nature, and the corresponding power output of a tidal power plant does not always coincide with the peak of human activity. In countries with a well-developed power industry, tidal power plants can be a part of the general power distribution system. However, power from a tidal plant would then have to be transmitted a long distance because locations of high tides are usually far away from industrial and urban centers. An attractive future option is to utilize the tidal power in situ for year-round production of hydrogen fuel by electrolysis of the water. The hydrogen, liquefied or stored by another method, can be transported anywhere to be used either as a fuel instead of oil or gasoline or in various fuel cell energy systems. Fuel cells convert hydrogen energy directly into electricity without combustion or moving parts, which is then used, for instance, in electric cars. Many scientists and engineers consider such a development as a future new industrial revolution. However, in order to realize this idea worldwide, clean hydrogen fuel would need to be also available everywhere. At present most hydrogen is produced from natural gases and fossil fuels, which emit greenhouse gases into the atmosphere and harm the global ecosystem. From this point of view, production of hydrogen by water electrolysis using tidal energy is one of the best ways to develop clean hydrogen fuel by a clean method. Thus, tidal energy can be used in the future to help develop a new era of clean industries, for example, to clean up the automotive industry, as well as other energy-consuming areas of human activity.

Conclusion Tides play a very important role in the formation of global climate as well as the ecosystems for ocean

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Electric generators sit above the water

Figure 3 Artist rendition of the floating tidal power plant with vertical triple-helix turbines for Uldolmok Strait (Korean Peninsula).

habitants. At the same time, tides are a substantial potential source of clean renewable energy for future human generations. Depleting oil reserves, the emission of greenhouse gases by burning coal, oil and other fossil fuels, as well as the accumulation of nuclear waste from nuclear reactors will inevitably force people to replace most of our traditional energy sources with renewable energy in the future. Tidal energy is one of the best candidates for this approaching revolution. Development of new, efficient, low-cost and environmentally friendly hydraulic energy converters suited to free-flow waters, such as triple-helix turbines, can make tidal energy available worldwide. This type of machine, moreover, can be used not only for multi-megawatt tidalpower farms but also for mini-power stations with turbines generating a few kilowatts. Such power stations can provide clean energy to small communities or even

individual households located near continental shorelines, straits or on remote islands with strong tidal currents.

See also Tides.

Further Reading Bernshtein LB (ed.) (1996) Tidal Power Plants. Seoul: Korea Ocean Research and Development Institute (KORDI). Gorlov AM (1998) Turbines with a twist. In: Kitzinger U and Frankel EG (eds.) Macro-Engineering and the Earth: World Projects for the Year 2000 and Beyond, pp. 1--36. Chichester: Horwood Publishing. Charlier RH (1982) Tidal Energy. New York: Van Nostrand Reinhold.

SEA LEVEL CHANGE J. A. Church, Antarctic CRC and CSIRO Marine Research, Tasmania, Australia J. M. Gregory, Hadley Centre, Berkshire, UK Copyright & 2001 Elsevier Ltd.

Introduction Sea-level changes on a wide range of time and space scales. Here we consider changes in mean sea level, that is, sea level averaged over a sufficient period of time to remove fluctuations associated with surface waves, tides, and individual storm surge events. We focus principally on changes in sea level over the last hundred years or so and on how it might change over the next one hundred years. However, to understand these changes we need to consider what has happened since the last glacial maximum 20 000 years ago. We also consider the longer-term implications of changes in the earth’s climate arising from changes in atmospheric greenhouse gas concentrations. Changes in mean sea level can be measured with respect to the nearby land (relative sea level) or a fixed reference frame. Relative sea level, which changes as either the height of the ocean surface or the height of the land changes, can be measured by a coastal tide gauge. The world ocean, which has an average depth of about 3800 m, contains over 97% of the earth’s water. The Antarctic ice sheet, the Greenland ice sheet, and the hundred thousand nonpolar glaciers/ ice caps, presently contain water sufficient to raise sea level by 61 m, 7 m, and 0.5 m respectively if they were entirely melted. Ground water stored shallower than 4000 m depth is equivalent to about 25 m (12 m stored shallower than 750 m) of sea-level change. Lakes and rivers hold the equivalent of less than 1 m, while the atmosphere accounts for only about 0.04 m. On the time-scales of millions of years, continental drift and sedimentation change the volume of the ocean basins, and hence affect sea level. A major influence is the volume of mid-ocean ridges, which is related to the arrangement of the continental plates and the rate of sea floor spreading. Sea level also changes when mass is exchanged between any of the terrestrial, ice, or atmospheric reservoirs and the ocean. During glacial times (ice ages), water is removed from the ocean and stored in large ice sheets in high-latitude regions. Variations in the surface loading of the earth’s crust by water and

ice change the shape of the earth as a result of the elastic response of the lithosphere and viscous flow of material in the earth’s mantle and thus change the level of the land and relative sea level. These changes in the distribution of mass alter the gravitational field of the earth, thus changing sea level. Relative sea level can also be affected by local tectonic activities as well as by the land sinking when ground water is extracted or sedimentation increases. Sea water density is a function of temperature. As a result, sea level will change if the ocean’s temperature varies (as a result of thermal expansion) without any change in mass.

Sea-Level Changes Since the Last Glacial Maximum On timescales of thousands to hundreds of thousands of years, the most important processes affecting sea-level are those associated with the growth and decay of the ice sheets through glacial–interglacial cycles. These are also relevant to current and future sea level rise because they are the cause of ongoing land movements (as a result of changing surface loads and the resultant small changes in the shape of the earth – postglacial rebound) and ongoing changes in the ice sheets. Sea-level variations during a glacial cycle exceed 100 m in amplitude, with rates of up to tens of millimetres per year during periods of rapid decay of the ice sheets (Figure 1). At the last glacial maximum (about 21 000 years ago), sea level was more than 120 m below current levels. The largest contribution to this sea-level lowering was the additional ice that formed the North American (Laurentide) and European (Fennoscandian) ice sheets. In addition, the Antarctic ice sheet was larger than at present and there were smaller ice sheets in presently ice-free areas.

Observed Recent Sea-Level Change Long-term relative sea-level changes have been inferred from the geological records, such as radiocarbon dates of shorelines displaced from present day sea level, and information from corals and sediment cores. Today, the most common method of measuring sea level relative to a local datum is by tide gauges at coastal and island sites. A global data set is maintained by the Permanent Service for Mean Sea Level (PSMSL). During the 1990s, sea level has been measured globally with satellites.

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Tide-gauge Observations

Unfortunately, determination of global-averaged sealevel rise is severely limited by the small number of gauges (mostly in Europe and North America) with long records (up to several hundred years, Figure 2). To correct for vertical land motions, some sea-level change estimates have used geological data, whereas others have used rates of present-day vertical land movement calculated from models of postglacial rebound. A widely accepted estimate of the current rate of global-average sea-level rise is about 1.8 mm y1. This estimate is based on a set of 24 long tide-gauge

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records, corrected for land movements resulting from deglaciation. However, other analyses produce different results. For example, recent analyses suggest that sea-level change in the British Isles, the North Sea region and Fennoscandia has been about 1 mm y1 during the past century. The various assessments of the global-average rate of sea-level change over the past century are not all consistent within stated uncertainties, indicating further sources of error. The treatment of vertical land movements remains a source of potential inconsistency, perhaps amounting to 0.5 mm y1. Other sources of error include variability over periods of years and longer and any spatial distribution in regional sea level rise (perhaps several tenths of a millimeter per year). Comparison of the rates of sea-level rise over the last 100 years (1.0–2.0 mm y1) and over the last two millennia (0.1–0.0 mm y1) suggests the rate has accelerated fairly recently. From the few very long tide-gauge records (Figure 2), it appears that an acceleration of about 0.3–0.9 mm y1 per century occurred over the nineteenth and twentieth century. However, there is little indication that sea-level rise accelerated during the twentieth century. Altimeter Observations

Following the advent of high-quality satellite radar altimeter missions in the 1990s, near-global and homogeneous measurement of sea level is possible, thereby overcoming the inhomogeneous spatial sampling from coastal and island tide gauges. However, clarifying rates of global sea-level change requires continuous satellite operations over many

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years and careful control of biases within and between missions. To date, the TOPEX/POSEIDON satellite-altimeter mission, with its (near) global coverage from 661N to 661S (almost all of the ice-free oceans) from late 1992 to the present, has proved to be of most value in producing direct estimates of sea-level change. The present data allow global-average sea level to be estimated to a precision of several millimeters every 10 days, with the absolute accuracy limited by systematic errors. The most recent estimates of global-average sea level rise based on the short (since 1992) TOPEX/POSEIDON time series range from 2.1 mm y1 to 3.1 mm y1. The alimeter record for the 1990s indicates a rate of sea-level rise above the average for the twentieth century. It is not yet clear if this is a result of an increase in the rate of sea-level rise, systematic differences between the tide-gauge and altimeter data sets or the shortness of the record.

Processes Determining Present Rates of Sea-Level Change The major factors determining sea-level change during the twentieth and twenty-first century are ocean thermal expansion, the melting of nonpolar glaciers and ice caps, variation in the mass of the Antarctic and Greenland ice sheets, and changes in terrestrial storage. Projections of climate change caused by human activity rely principally on detailed computer models referred to as atmosphere–ocean general circulation models (AOGCMs). These simulate the global threedimensional behavior of the ocean and atmosphere by numerical solution of equations representing the underlying physics. For simulations of the next hundred years, future atmospheric concentrations of gases that may affect the climate (especially carbon dioxide from combustion of fossil fuels) are estimated on the basis of assumptions about future population growth, economic growth, and technological change. AOGCM experiments indicate that the global-average temperature may rise by 1.4– 5.81C between 1990 and 2100, but there is a great deal of regional and seasonal variation in the predicted changes in temperature, sea level, precipitation, winds, and other parameters. Ocean Thermal Expansion

The broad pattern of sea level is maintained by surface winds, air–sea fluxes of heat and fresh water (precipitation, evaporation, and fresh water runoff from the land), and internal ocean dynamics. Mean sea level

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varies on seasonal and longer timescales. A particularly striking example of local sea-level variations occurs in the Pacific Ocean during El Nin˜o events. When the trade winds abate, warm water moves eastward along the equator, rapidly raising sea level in the east and lowering it in the west by about 20 cm. As the ocean warms, its density decreases. Thus, even at constant mass, the volume of the ocean increases. This thermal expansion is larger at higher temperatures and is one of the main contributors to recent and future sea-level change. Salinity changes within the ocean also have a significant impact on the local density, and thus on local sea level, but have little effect on the global-average sea level. The rate of global temperature rise depends strongly on the rate at which heat is moved from the ocean surface layers into the deep ocean; if the ocean absorbs heat more readily, climate change is retarded but sea level rises more rapidly. Therefore, timedependent climate change simulation requires a model that represents the sequestration of heat in the ocean and the evolution of temperature as a function of depth. The large heat capacity of the ocean means that there will be considerable delay before the full effects of surface warming are felt throughout the depth of the ocean. As a result, the ocean will not be in equilibrium and global-average sea level will continue to rise for centuries after atmospheric greenhouse gas concentrations have stabilized. The geographical distribution of sea-level change may take many decades to arrive at its final state. While the evidence is still somewhat fragmentary, and in some cases contradictory, observations indicate ocean warming and thus thermal expansion, particularly in the subtropical gyres, at rates resulting in sea-level rise of order 1 mm y1. The observations are mostly over the last few decades, but some observations date back to early in the twentieth century. The evidence is most convincing for the subtropical gyre of the North Atlantic, for which the longest temperature records (up to 73 years) and most complete oceanographic data sets exist. However, the pattern also extends into the South Atlantic and the Pacific and Indian oceans. The only areas of substantial ocean cooling are the subpolar gyres of the North Atlantic and perhaps the North Pacific. To date, the only estimate of a global average rate of sealevel rise from thermal expansion is 0.55 mm y1. The warming in the Pacific and Indian Oceans is confined to the main thermocline (mostly the upper 1 km) of the subtropical gyres. This contrasts with the North Atlantic, where the warming is also seen at greater depths. AOGCM simulations of sea level suggest that during the twentieth century the average rate

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In the past decade, estimates of the regional totals of the area and volume of glaciers have been improved. However, there are continuous mass balance records longer than 20 years for only about 40 glaciers worldwide. Owing to the paucity of measurements, the changes in mass balance are estimated as a function of climate. On the global average, increased precipitation during the twenty-first century is estimated to offset only 5% of the increased ablation resulting from warmer temperatures, although it might be significant in particular localities. (For instance, while glaciers in most parts of the world have had negative mass balance in the past 20 years, southern Scandinavian glaciers have been advancing, largely because of increases in precipitation.) A detailed computation of transient response also requires allowance for the contracting area of glaciers. Recent estimates of glacier mass balance, based on both observations and model studies, indicate a contribution to global-average sea level of 0.2 to 0.4 mm y1 during the twentieth century. The model results shown in Figure 3 indicate an average rate of 0.1 to 0.3 mm y1. Greenland and Antarctic Ice Sheets

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of change due to thermal expansion was of the order of 0.3–0.8 mm y1 (Figure 3). The rate rises to 0.6–1.1 mm y1 in recent decades, similar to the observational estimates of ocean thermal expansion.

Nonpolar Glaciers and Ice Caps

Nonpolar glaciers and ice caps are rather sensitive to climate change, and rapid changes in their mass contribute significantly to sea-level change. Glaciers gain mass by accumulating snow, and lose mass (ablation) by melting at the surface or base. Net accumulation occurs at higher altitude, net ablation at lower altitude. Ice may also be removed by discharge into a floating ice shelf and/or by direct calving of icebergs into the sea.

A small fractional change in the volume of the Greenland and Antarctic ice sheets would have a significant effect on sea level. The average annual solid precipitation falling onto the ice sheets is equivalent to 6.5 mm of sea level, but this input is approximately balanced by loss from melting and iceberg calving. In the Antarctic, temperatures are so low that surface melting is negligible, and the ice sheet loses mass mainly by ice discharge into floating ice shelves, which melt at their underside and eventually break up to form icebergs. In Greenland, summer temperatures are high enough to cause widespread surface melting, which accounts for about half of the ice loss, the remainder being discharged as icebergs or into small ice shelves. The surface mass balance plays the dominant role in sea-level changes on a century timescale, because changes in ice discharge generally involve response times of the order of 102 to 104 years. In view of these long timescales, it is unlikely that the ice sheets have completely adjusted to the transition from the previous glacial conditions. Their present contribution to sea-level change may therefore include a term related to this ongoing adjustment, in addition to the effects of climate change over the last hundred years. The current rate of change of volume of the polar ice sheets can be assessed by estimating the individual mass balance terms or by monitoring

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Changes in Terrestrial Storage

Changes in terrestrial storage include reductions in the volumes of some of the world’s lakes (e.g., the Caspian and Aral seas), ground water extraction in excess of natural recharge, more water being impounded in reservoirs (with some seeping into aquifers), and possibly changes in surface runoff. Order-of-magnitude evaluations of these terms are uncertain but suggest that each of the contributions could be several tenths of millimeter per year, with a small net effect (Figure 3). If dam building continues at the same rate as in the last 50 years of the twentieth century, there may be a tendency to reduce sea-level rise. Changes in volumes of lakes and rivers will make only a negligible contribution. Permafrost currently occupies about 25% of land area in the northern hemisphere. Climate warming leads to some thawing of permafrost, with partial runoff into the ocean. The contribution to sea level in the twentieth century is probably less than 5 mm.

Projected Sea-Level Changes for the Twenty-first Century Detailed projections of changes in sea level derived from AOCGM results are given in material listed as Further Reading. The major components are thermal expansion of the ocean (a few tens of centimeters), melting of nonpolar glaciers (about 10–20 cm), melting of Greenland ice sheet (several centimeters), and increased storage in the Antarctic (several centimeters). After allowance for the continuing changes in the ice sheets since the last glacial maximum and the

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surface elevation changes directly (such as by airborne and satellite altimetry during the 1990s). However, these techniques give results with large uncertainties. Indirect methods (including numerical modeling of ice-sheets, observed sea-level changes over the last few millennia, and changes in the earth’s rotation parameters) give narrower bounds, suggesting that the present contribution of the ice sheets to sea level is a few tenths of a millimeter per year at most. Calculations suggest that, over the next hundred years, surface melting is likely to remain negligible in Antarctica. However, projected increases in precipitation would result in a net negative sea-level contribution from the Antarctic ice sheet. On the other hand, in Greenland, surface melting is projected to increase at a rate more than enough to offset changes in precipitation, resulting in a positive contribution to sea-level rise.

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melting of permafrost (but not including changes in terrestrial storage), total projected sea-level rise during the twenty-first century is currently estimated to be between about 9 and 88 cm (Figure 4).

Regional Sea-Level Change Estimates of the regional distribution of sea-level rise are available from several AOGCMs. Our confidence in these distributions is low because there is little similarity between model results. However, models agree on the qualitative conclusion that the range of regional variation is substantial compared with the global-average sea-level rise. One common feature is that nearly all models predict less than average sealevel rise in the Southern Ocean. The most serious impacts of sea-level change on coastal communities and ecosystems will occur during the exceptionally high water levels known as storm surges produced by low air pressure or driving winds. As well as changing mean sea level, climate change could also affect the frequency and severity of these meteorological conditions, making storm surges more or less severe at particular locations.

Longer-term Changes Even if greenhouse gas concentrations were to be stabilized, sea level would continue to rise for several hundred years. After 500 years, sea-level rise from thermal expansion could be about 0.3–2 m but may be only half of its eventual level. Glaciers presently contain the equivalent of 0.5 m of sea level. If the CO2 levels projected for 2100 AD were sustained, there would be further reductions in glacier mass.

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Ice sheets will continue to react to climatic change during the present millennium. The Greenland ice sheet is particularly vulnerable. Some models suggest that with a warming of several degrees no ice sheet could be sustained on Greenland. Complete melting of the ice sheet would take at least a thousand years and probably longer. Most of the ice in Antarctica forms the East Antarctic Ice Sheet, which would disintegrate only if extreme warming took place, beyond what is currently thought possible. The West Antarctic Ice Sheet (WAIS) has attracted special attention because it contains enough ice to raise sea level by 6 m and because of suggestions that instabilities associated with its being grounded below sea level may result in rapid ice discharge when the surrounding ice shelves are weakened. However, there is now general agreement that major loss of grounded ice, and accelerated sea-level rise, is very unlikely during the twenty-first century. The contribution of this ice sheet to sea level change will probably not exceed 3 m over the next millennium.

Summary On timescales of decades to tens of thousands of years, sea-level change results from exchanges of mass between the polar ice sheets, the nonpolar glaciers, terrestrial water storage, and the ocean. Sea level also changes if the density of the ocean changes (as a result of changing temperature) even though there is no change in mass. During the last century, sea level is estimated to have risen by 10–20 cm, as a result of combination of thermal expansion of the ocean as its temperature rose and increased mass of the ocean from melting glaciers and ice sheets. Over the twenty-first century, sea level is expected to rise as a result of anthropogenic climate change. The main contributors to this rise are expected to be thermal expansion of the ocean and the partial melting of nonpolar glaciers and the Greenland ice sheet. Increased precipitation in Antarctica is

expected to offset some of the rise from other contributions. Changes in terrestrial storage are uncertain but may also partially offset rises from other contributions. After allowance for the continuing changes in the ice sheets since the last glacial maximum, the total projected sea-level rise over the twenty-first century is currently estimated to be between about 9 and 88 cm.

See also El Nin˜o Southern Oscillation (Enso). Sea Level Variations Over Geological Time.

Further Reading Church JA, Gregory JM, Huybrechts P, et al. (2001) Changes in sea level. In: Houghton JT (ed.) Climate Change 2001; The Scientific Basis. Cambridge: Cambridge University Press. Douglas BC, Keaney M, and Leatherman SP (eds.) (2000) Sea Level Rise: History and Consequences, 232 pp. San Diego: Academic Press. Fleming K, Johnston P, Zwartz D, et al. Refining the eustatic sea-level curve since the Last Glacial Maximum using far- and intermediate-field sites. Earth and Planetary Science Letters 163: 327–342. Lambeck K (1998) Northern European Stage 3 ice sheet and shoreline reconstructions: Preliminary results. News 5, Stage 3 Project, Godwin Institute for Quaternary Research, 9 pp. Peltier WR (1998) Postglacial variations in the level of the sea: implications for climate dynamics and solid-earth geophysics. Review of Geophysics 36: 603--689. Summerfield MA (1991) Global Geomorphology. Harlowe: Longman. Warrick RA, Barrow EM, and Wigley TML (1993) Climate and Sea Level Change: Observations, Projections and Implications. Cambridge: Cambridge University Press. WWW pages of the Permanent Service for Mean Sea Level, http://www.pol.ac.uk/psmsl/

SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME M. A. Kominz, Western Michigan University, Kalamazoo, MI, USA Copyright & 2001 Elsevier Ltd.

Introduction Sea level changes have occurred throughout Earth history. The magnitudes and timing of sea level changes are extremely variable. They provide considerable insight into the tectonic and climatic history of the Earth, but remain difficult to determine with accuracy. Sea level, where the world oceans intersect the continents, is hardly fixed, as anyone who has stood on the shore for 6 hours or more can attest. But the ever-changing tidal flows are small compared with longer-term fluctuations that have occurred in Earth history. How much has sea level changed? How long did it take? How do we know? What does it tell us about the history of the Earth? In order to answer these questions, we need to consider a basic question: what causes sea level to change? Locally, sea level may change if tectonic forces cause the land to move up or down. However, this article will focus on global changes in sea level. Thus, the variations in sea level must be due to one of two possibilities: (1) changes in the volume of water in the oceans or (2) changes in the volume of the ocean basins.

Sea Level Change due to Volume of Water in the Ocean Basin The two main reservoirs of water on Earth are the oceans (currently about 97% of all water) and glaciers (currently about 2.7%). Not surprisingly, for at least the last three billion years, the main variable controlling the volume of water filling the ocean basins has been the amount of water present in glaciers on the continents. For example, about 20 000 years ago, great ice sheets covered northern North America and Europe. The volume of ice in these glaciers removed enough water from the oceans to expose most continental shelves. Since then there has been a sea level rise (actually from about 20 000 to about 11 000 years ago) of about 120 m (Figure 1A). A number of methods have been used to establish the magnitude and timing of this sea level change. Dredging on the continental shelves reveals human

activity near the present shelf-slope boundary. These data suggest that sea level was much lower a relatively short time ago. Study of ancient corals shows that coral species which today live only in very shallow water are now over 100 m deep. The carbonate skeletons of the coral, which once thrived in the shallow waters of the tropics, yield a detailed picture of the timing of sea level rise, and, thus, the melting of the glaciers. Carbon-14, a radioactive isotope formed by carbon-12 interacting with highenergy solar radiation in Earth’s atmosphere allows us to determine the age of Earth materials, which are about 30 thousand years old. This is just the most recent of many, large changes in sea level caused by glaciers, (Figure 1B). These variations in climate and subsequent sea level changes have been tied to quasi-periodic variations in the Earth’s orbit and the tilt of the Earth’s spin axis. The record of sea level change can be estimated by observing the stable isotope, oxygen-18 in the tests (shells) of dead organisms. When marine microorganisms build their tests from the calcium, carbon, and oxygen present in sea water they incorporate both the abundant oxygen-16 and the rare oxygen18 isotopes. Water in the atmosphere generally has a lower oxygen-18 to oxygen-16 ratio because evaporation of the lighter isotope requires less energy. As a result, the snow that eventually forms the glaciers is depleted in oxygen-18, leaving the ocean proportionately oxygen-18-enriched. When the microorganisms die, their tests sink to the seafloor to become part of the deep marine sedimentary record. The oxygen-18 to oxygen-16 ratio present in the fossil tests has been calibrated to the sea level change, which occurred from 20 000 to 11 000 years ago, allowing the magnitude of sea level change from older times to be estimated. This technique does have uncertainties. Unfortunately, the amount of oxygen18 which organisms incorporate in their tests is affected not only by the amount of oxygen-18 present but also by the temperature and salinity of the water. For example, the organisms take up less oxygen-18 in warmer waters. Thus, during glacial times, the tests are even more enriched in oxygen-18, and any oxygen isotope record reveals a combined record of changing local temperature and salinity in addition to the record of global glaciation. Moving back in time through the Cenozoic (zero to 65 Ma), paleoceanographic data remain excellent due to relatively continuous sedimentation on the ocean floor (as compared with shallow marine and

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terrestrial sedimentation). Oxygen-18 in the fossil shells suggests a general cooling for about the last 50 million years. Two rapid increases in the oxygen-18 to oxygen-16 ratio about 12.5 Ma and about 28 Ma are observed (Figure 1C). The formation of the Greenland Ice Sheet and the Antarctic Ice Sheet are assumed to be the cause of these rapid isotope shifts. Where oxygen-18 data have been collected with a resolution finer than 20 000 years, high-frequency variations are seen which are presumed to correspond to a combination of temperature change and glacial growth and decay. We hypothesize that the magnitudes of these high-frequency sea level changes were considerably less in the earlier part of the Cenozoic than those observed over the last million years. This is because considerably less ice was involved. Although large continental glaciers are not common in Earth history they are known to have been present during a number of extended periods (‘ice house’ climate, in contrast to ‘greenhouse’ or warm climate conditions). Ample evidence of glaciation is found in the continental sedimentary record. In particular, there is evidence of glaciation from about 2.7 to 2.1 billion years ago. Additionally, a long period of glaciation occurred shortly before the first fossils of multicellular organisms, from about 1

billion to 540 million years ago. Some scientists now believe that during this glaciation, the entire Earth froze over, generating a ‘snowball earth’. Such conditions would have caused a large sea level fall. Evidence of large continental glaciers are also seen in Ordovician to Silurian rocks (B420 to 450 Ma), in Devonian rocks (B380 to 390 Ma), and in Carboniferous to Permian rocks (B350 to 270 Ma). If these glaciations were caused by similar mechanisms to those envisioned for the Plio-Pleistocene (Figure 1B), then predictable, high-frequency, periodic growth and retreat of the glaciers should be observed in strata which form the geologic record. This is certainly the case for the Carboniferous through Permian glaciation. In the central United States, UK, and Europe, the sedimentary rocks have a distinctly cyclic character. They cycle in repetitive vertical successions of marine deposits, near-shore deposits, often including coals, into fluvial sedimentary rocks. The deposition of marine rocks over large areas, which had only recently been nonmarine, suggests very large-scale sea level changes. When the duration of the entire record is taken into account, periodicities of about 100 and 400 thousand years are suggested for these large sea level changes. This is consistent with an origin due to a response to changes in the eccentricity of the Earth’s orbit. Higher-frequency cyclicity

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associated with the tilt of the spin axis and precession of the equinox is more difficult to prove, but has been suggested by detailed observations. It is fair to say that large-scale (10 to 4100 m), relatively high-frequency (20 000–400 000 years; often termed ‘Milankovitch scale’) variations in sea level occurred during intervals of time when continental glaciers were present on Earth (ice house climate). This indicates that the variations of Earth’s orbit and the tilt of its spin axis played a major role in controlling the climate. During the rest of Earth history, when glaciation was not a dominant climatic force (greenhouse climate), sea level changes corresponding to Earth’s orbit did occur. In this case, the mechanism for changing the volume of water in the ocean basins is much less clear. There is no geological record of continental ice sheets in many portions of Earth history. These time periods are generally called ‘greenhouse’ climates. However, there is ample evidence of Milankovitch scale variations during these periods. In shallow marine sediments, evidence of orbitally driven sea level changes has been observed in Cambrian and Cretaceous age sediments. The magnitudes of sea level change required (perhaps 5–20 m) are far less than have been observed during glacial climates. A possible source for these variations could be variations in average ocean-water temperature. Water expands as it is heated. If ocean bottom-water sources were equatorial rather than polar, as they are today, bottom-water temperatures of about 21C today might have been about 161C in the past. This would generate a sea level change of about 11 m. Other causes of sea level change during greenhouse periods have been postulated to be a result of variations in the magnitude of water trapped in inland lakes and seas, and variations in volumes of alpine glaciers. Deep marine sediments of Cretaceous age also show fluctuations between oxygenated and anoxic conditions. It is possible that these variations were generated when global sea level change restricted flow from the rest of the world’s ocean to a young ocean basin. In a more recent example, tectonics caused a restriction at the Straits of Gibraltar. In that case, evaporation generated extreme sea level changes and restricted their entrance into the Mediterranean region.

Sea Level Change due to Changing Volume of the Ocean Basin Tectonics is thought to be the main driving force of long-term (Z50 million years) sea level change. Plate tectonics changes the shape and/or the areal extent of the ocean basins.

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Plate tectonics is constantly reshaping surface features of the Earth while the amount of water present has been stable for about the last four billion years. The reshaping changes the total area taken up by oceans over time. When a supercontinent forms, subduction of one continent beneath another decreases Earth’s ratio of continental to oceanic area, generating a sea level fall. In a current example, the continental plate including India is diving under Asia to generate the Tibetan Plateau and the Himalayan Mountains. This has probably generated a sea level fall of about 70 m over the last 50 million years. The process of continental breakup has the opposite effect. The continents are stretched, generating passive margins and increasing the ratio of continental to oceanic area on a global scale (Figure 2A). This results in a sea level rise. Increments of sea level rise resulting from continental breakup over the last 200 million years amount to about 100 m of sea level rise. Some bathymetric features within the oceans are large enough to generate significant changes in sea level as they change size and shape. The largest physiographic feature on Earth is the mid-ocean ridge system, with a length of about 60 000 km and a width of 500–2000 km. New ocean crust and lithosphere are generated along rifts in the center of these ridges. The ocean crust is increasingly old, cold, and dense away from the rift. It is the heat of ocean lithosphere formation that actually generates this feature. Thus, rifting of continents forms new ridges, increasing the proportionate area of young, shallow, ocean floor to older, deeper ocean floor (Figure 2B). Additionally, the width of the ridge is a function of the rates at which the plates are moving apart. Fast spreading ridges (e.g. the East Pacific Rise) are very broad while slow spreading ridges (e.g. the North Atlantic Ridge) are quite narrow. If the average spreading rates for all ridges decreases, the average volume taken up by ocean ridges would decrease. In this case, the volume of the ocean basin available for water would increase and a sea level fall would occur. Finally, entire ridges may be removed in the process of subduction, generating fairly rapid sea level fall. Scientists have made quantitative estimates of sea level change due to changing ocean ridge volumes. Since ridge volume is dependent on the age of the ocean floor, where the age of the ocean floor is known, ridge volumes can be estimated. Seafloor magnetic anomalies are used to estimate the age of the ocean floor, and thus, spreading histories of the oceans 256. The oldest ocean crust is about 200 million years old. Older oceanic crust has been subducted. Thus, it is not surprising that quantitative estimates of sea level change due to ridge volumes are increasingly uncertain and cannot be calculated

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SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

Shallow ocean

Fast spreading rates generate broad ridges

Continent splitting to form two continents

Deep ocean

(A)

(D)

New ridge Slow spreading rates generate narrow ridges

Deep ocean

(B)

Shallow ocean

Young ridge Older ridge

(E) Large igneous provinces

Shallow ocean (C)

Deep ocean (F)

Figure 2 Diagrams showing a few of the factors which affect the ocean volume. (A) Early breakup of a large continent increases the area of continental crust by generating passive margins, causing sea level to rise. (B) Shortly after breakup a new ocean is formed with very young ocean crust. This young crust must be replacing relatively old crust via subduction, generating additional sea level rise. (C) The average age of the ocean between the continents becomes older so that young, shallow ocean crust is replaced with older, deeper crust so that sea level falls. (D) Fast spreading rates are associated with relatively high sea level. (E) Relatively slow spreading ridges (solid lines in ocean) take up less volume in the oceans than high spreading rate ridges (dashed lines in ocean), resulting in relatively low sea level. (F) Emplacement of large igneous provinces generates oceanic plateaus, displaces ocean water, and causes a sea level rise.

SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

before about 90 million years. Sea level is estimated to have fallen about 230 m (7120 m) due to ridge volume changes in the last 80 million years. Large igneous provinces (LIPs) are occasionally intruded into the oceans, forming large oceanic plateaus (see Igneous Provinces). The volcanism associated with LIPs tends to occur over a relatively short period of time, causing a rapid sea level rise. However, these features subside slowly as the lithosphere cools, generating a slow increase in ocean volume, and a long-term sea level fall. The largest marine LIP, the Ontong Java Plateau, was emplaced in the Pacific Ocean between about 120 and 115 Ma (Figure 1D). Over that interval it may have generated a sea level rise of around 50 m. In summary, over the last 200 million years, longterm sea level change (Figure 1D) can be largely attributed to tectonics. Continental crust expanded by extension as the supercontinents Gondwana and Laurasia split to form the continents we see today. This process began about 200 Ma when North America separated from Africa and continues with the East African Rift system and the formation of the Red Sea. The generation of large oceans occurred early in this period and there was an overall rise in sea level from about 200 to about 90 million years. New continental crust, new mid-ocean ridges, and very fast spreading rates were responsible for the long-term rise (Figure 1D). Subsequently, a significant decrease in spreading rates, a reduction in the total length of mid-ocean ridges, and continent– continent collision coupled with an increase in glacial ice (Figure 1C) have resulted in a large-scale sea level fall (Figure 1D). Late Cretaceous volcanism associated with the Ontong Java Plateau, a large igneous province (see Igneous Provinces), generated a significant sea level rise, while subsequent cooling has enhanced the 90 million year sea level fall. Estimates of sea level change from changing ocean shape remain quite uncertain. Magnitudes and timing of stretching associated with continental breakup, estimates of shortening during continental assembly, volumes of large igneous provinces, and volumes of mid-ocean ridges improve as data are gathered. However, the exact configuration of past continents and oceans can only be a mystery due to the recycling character of plate tectonics.

Sea Level Change Estimated from Observations on the Continents Long-term Sea Level Change

Estimates of sea level change are also made from sedimentary strata deposited on the continents. This

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is actually an excellent place to obtain observations of sea level change not only because past sea level has been much higher than it is now, but also because in many places the continents have subsequently uplifted. That is, in the past they were below sea level, but now they are well above it. For example, studies of 500–400 million year old sedimentary rocks which are now uplifted in the Rocky Mountains and the Appalachian Mountains indicate that there was a rise and fall of sea level with an estimated magnitude of 200–400 m. This example also exemplifies the main problem with using the continental sedimentary record to estimate sea level change. The continents are not fixed and move vertically in response to tectonic driving forces. Thus, any indicator of sea level change on the continents is an indicator of relative sea level change. Obtaining a global signature from these observations remains extremely problematic. Additionally, the continental sedimentary record contains long periods of nondeposition, which results in a spotty record of Earth history. Nonetheless a great deal of information about sea level change has been obtained and is summarized here. The most straightforward source of information about past sea level change is the location of the strand line (the beach) on a stable continental craton (a part of the continent, which was not involved in local tectonics). Ideally, its present height is that of sea level at the time of deposition. There are two problems encountered with this approach. Unfortunately, the nature of land–ocean interaction at their point of contact is such that those sediments are rarely preserved. Where they can be observed, there is considerable controversy over which elements have moved, the continents or sea level. However, data from the past 100 million years tend to be consistent with calculations derived from estimates of ocean volume change. This is not saying a lot since uncertainties are very large (see above). Continental hypsography (cumulative area versus height) coupled with the areal extent of preserved marine sediments has been used to estimate past sea level. In this case only an average result can be obtained, because marine sediments spanning a time interval (generally 5–10 million years) have been used. Again, uncertainties are large, but results are consistent with calculations derived from estimates of ocean volume change. Backstripping is an analytical tool, which has been used to estimate sea level change. In this technique, the vertical succession of sedimentary layers is progressively decompacted and unloaded (Figure 3A). The resulting hole is a combination of the subsidence generated by tectonics and by sea level change

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SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

Observed sediment thickness

Measured section

T1

T5

First reduction R1

Decompacted sediment thickness, S*

T1

T1

T4 T0

T0 T3

T0 T2

T2

T2

T2

T1 Paleowater depth

T0

Equivalent basin for sediment thickness S*

(A)

(B)

T0

T1

T2

T3

T4

T5

Depth

R1 Theoretical thermal subsidence

(C)

Height

Sea level change and nonthermal tectonics

(D)

T0

T1

T2

T3

T4

T5

Figure 3 Diagrams depicting the backstripping method for obtaining sea level estimates in a thermally subsiding basin. (A) A stratigraphic section is measured either from exposed sedimentary rocks or from drilling. These data include lithologies, ages, and porosity. Note that the oldest strata are always at the base of the section (T0). (B) Porosity data are used to estimate the thickness that each sediment section would have had at the time of deposition (S*). They are also used to obtain sediment density so that the sediments can be unloaded to determine how deep the basin would have been in the absence of the sediment load (R1). This calculation also requires an estimate of the paleo-water depth (the water depth at the time of deposition). (C) A plot of R1 versus time is compared (by least-squares fit) to theoretical tectonic subsidence in a thermal setting. (D) The difference between R1 and thermal subsidence yields a quantitative estimate of sea level change if other, nonthermal tectonics, did not occur at this location.

SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

(Figure 3B). If the tectonic portion can be established then an estimate of sea level change can be determined (Figure 3C). This method is generally used in basins generated by the cooling of a thermal anomaly (e.g. passive margins). In these basins, the tectonic signature is predictable (exponential decay) and can be calibrated to the well-known subsidence of the mid-ocean ridge. The backstripping method has been applied to sedimentary strata drilled from passive continental margins of both the east coast of North America and the west coast of Africa. Again, estimates of sea level suggest a rise of about 100–300 m from about 200– 110 Ma followed by a fall to the present level (Figure 1D). Young interior basins, such as the Paris 1071

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Basin, yield similar results. Older, thermally driven basins have also been analyzed. This was the method used to determine the (approximately 200 m) sea level rise and fall associated with the breakup of a Pre-Cambrian supercontinent in earliest Phanerozoic time. Million Year Scale Sea Level Change

In addition to long-term changes in sea level there is evidence of fluctuations that are considerably shorter than the 50–100 million year variations discussed above, but longer than those caused by orbital variations (r0.4 million years). These variations appear to be dominated by durations which last either tens 1073

1072

Two-way travel time (s)

0

1

(A)

Ew9009 line 1002

0

20 km

Relative sea leavel change

Relative time

Two-way travel time 0.0 0

10

20 km

HST

0.5

LST

TST

HST

TST LST

1.0 (B)

(C)

(D)

Figure 4 Example of the sequence stratigraphic approach to estimates of sea level change. (A) Multichannel seismic data (gray) from the Baltimore Canyon Trough, offshore New Jersey, USA (Miller et al., 1998). Black lines are interpretations traced on the seismic data. Thick dark lines indicate third-order Miocene-aged (5–23 Ma) sequence boundaries. They are identified by truncation of the finer black lines. Upside-down deltas indicate a significant break in slope associated with each identified sequence boundary. Labeled vertical lines (1071–1073) show the locations of Ocean Drilling Project wells, used to help date the sequences. The rectangle in the center is analyzed in greater detail. (B) Detailed interpretation of a single third-order sequence from (A). Upside-down deltas indicate a significant break in slope associated with each of the detailed sediment packages. Stippled fill indicates the low stand systems tract (LST) associated with this sequence. The gray packages are the transgressive systems tract (TST), and the overlying sediments are the high stand systems tract (HST). (C) Relationships between detailed sediment packages (in B) are used to establish a chronostratigraphy (time framework). Youngest sediment is at the top. Each observed seismic reflection is interpreted as a time horizon, and each is assigned equal duration. Horizontal distance is the same as in (A) and (B). A change in sediment type is indicated at the break in slope from coarser near-shore sediments (stippled pattern) to finer, offshore sediments (parallel, sloping lines). Sedimentation may be present offshore but at very low rates. LST, TST, and HST as in (B). (D) Relative sea level change is obtained by assuming a consistent depth relation at the change in slope indicated in (B). Age control is from the chronostratigraphy indicated in (C). Time gets younger to the right. The vertical scale is in two-way travel time, and would require conversion to depth for a final estimate of the magnitude of sea level change. LST, TST, and HST as in (B). Note that in (B), (C), and (D), higher frequency cycles (probably fourth-order) are present within this (third-order) sequence. Tracing and interpretations are from the author’s graduate level quantitative stratigraphy class project (1998, Western Michigan University).

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SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

of millions of years or a half to three million years. These sea level variations are sometimes termed second- and third-order sea level change, respectively. There is considerable debate concerning the source of these sea level fluctuations. They have been attributed to tectonics and changing ocean basin volumes, to the growth and decay of glaciers, or to continental uplift and subsidence, which is independent of global sea level change. As noted above, the tectonic record of subsidence and uplift is intertwined with the stratigraphic record of global sea level change on the continents. Synchronicity of observations of sea level change on a global scale would lead most geoscientists to suggest that these signals were caused by global sea level change. However, at present, it is nearly impossible to globally determine the age equivalency of events which occur during intervals as short as a half to two million years. These data limitations are the main reason for the heated controversy over third-order sea level. Quantitative estimates of second-order sea level variations are equally difficult to obtain. Although the debate is not as heated, these somewhat longerterm variations are not much larger than the thirdorder variations so that the interference of the two signals makes definition of the beginning, ending and/or magnitude of second-order sea level change

equally problematic. Recognizing that our understanding of second- and third-order (million year scale) sea level fluctuations is limited, a brief review of that limited knowledge follows. Sequence stratigraphy is an analytical method of interpreting sedimentary strata that has been used to investigate second- and third-order relative sea level changes. This paradigm requires a vertical succession of sedimentary strata which is analyzed in at least a two-dimensional, basinal setting. Packages of sedimentary strata, separated by unconformities, are observed and interpreted mainly in terms of their internal geometries (e.g. Figure 4). The unconformities are assumed to have been generated by relative sea level fall, and thus, reflect either global sea level or local or regional tectonics. This method of stratigraphic analysis has been instrumental in hydrocarbon exploration since its introduction in the late 1970s. One of the bulwarks of this approach is the ‘global sea level curve’ most recently published by Haq et al. (1987). This curve is a compilation of relative sea level curves generated from sequence stratigraphic analysis in basins around the world. While sequence stratigraphy is capable of estimating relative heights of relative sea level, it does not estimate absolute magnitudes. Absolute dating requires isotope data or correlation via fossil data into the

200

150

Sea level (meters)

100 Third-Order ΔSL backstripping

50 0 _ 50 _ 100

Third-Order ΔSL sequence stratigraphy shifted down by 100 m

_ 150 _ 200 60

50

40

30

20

10

0

Time (million years before present) Figure 5 Million year scale sea level fluctuations. Estimates from sequence stratigraphy (Haq et al., 1987; solid curve) have been shifted down by 100 m to allow comparison with estimates of sea level from backstripping (Kominz et al., 1998; dashed curve). Where sediments are present, the backstripping results, with uncertainty ranges, are indicated by gray fill. Between backstrip observations, lack of preserved sediment is presumed to have been a result of sea level fall. The Berggren et al. (1995) biostratigraphic timescale was used.

SEA LEVEL VARIATIONS OVER GEOLOGICAL TIME

chronostratigraphic timescale. However, the two-dimensional nature of the data allows for good to excellent relative age control. Backstripping has been used, on a considerably more limited basis, in an attempt to determine million year scale sea level change. This approach is rarely applied because it requires very detailed, quantitative, estimates of sediment ages, paleo-environments and compaction in a thermal tectonic setting. A promising area of research is the application of this method to coastal plain boreholes from the mid-Atlantic coast of North America. Here an intensive Ocean Drilling Project survey is underway which is providing sufficiently detailed data for this type of analysis. Initial results suggest that magnitudes of million year scale sea level change are roughly one-half to one-third that reported by Haq et al. However, in glacial times, the timing of the cycles was quite consistent with those of this ‘global sea level curve’ derived by application of sequence stratigraphy (Figure 5). Thus, it seems reasonable to conclude that, at least during glacial times, global, third-order sea level changes did occur.

Summary Sea level changes are either a response to changing ocean volume or to changes in the volume of water contained in the ocean. The timing of sea level change ranges from tens of thousands of years to over 100 million years. Magnitudes also vary significantly but may have been as great as 200 m or more. Estimates of sea level change currently suffer from significant ranges of uncertainty, both in magnitude and in timing. However, scientists are converging on consistent estimates of sea level changes by using very different data and analytical approaches.

Further Reading Allen PA and Allen JR (1990) Basin Analysis: Principless, Applications. Oxford: Blackwell Scientific Publications. Berggren WA, Kent DV, Swisher CC, and Aubry MP (1995) A revised Cenozoic geochronology and chronostratigraphy. In: Berggren WA, Kent DV, and

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Hardenbol J (eds.) Geochronology, Time Scales and Global Stratigraphic Correlations: A Unified Temporal Framework for an Historical Geology, SEPM Special Publication No. 54, 131--212. Bond GC (1979) Evidence of some uplifts of large magnitude in continental platforms. Tectonophysics 61: 285--305. Coffin MF and Eldholm O (1994) Large igneous provinces: crustal structure, dimensions, and external consequences. Reviews of Geophysics 32: 1--36. Crowley TJ and North GR (1991) Paleoclimatology. Oxford Monographs on Geology and Geophysics, no. 18 Fairbanks RG (1989) A 17,000-year glacio-eustatic sea level record: influence of glacial melting rates on the Younger Dryas event and deep-ocean circulation. Nature 6250: 637--642. Hallam A (1992) Phanerozoic Sea-Level Changes. NY: Columbia University Press. Haq BU, Hardenbol J, and Vail PR (1987) Chronology of fluctuating sea levels since the Triassic (250 million years ago to present). Science 235: 1156--1167. Harrison CGA (1990) Long term eustasy and epeirogeny in continents. In: Sea-Level Change, pp. 141--158. Washington, DC: US National Research Council Geophysics Study Committee. Hauffman PF and Schrag DP (2000) Snowball Earth. Scientific American 282: 68--75. Kominz MA, Miller KG, and Browning JV (1998) Longterm and short term global Cenozoic sea-level estimates. Geology 26: 311--314. Miall AD (1997) The Geology of Stratigraphic Sequences. Berlin: Springer-Verlag. Miller KG, Fairbanks RG, and Mountain GS (1987) Tertiary oxygen isotope synthesis, sea level history, and continental margin erosion. Paleoceanography 2: 1--19. Miller KG, Mountain GS, Browning J, et al. (1998) Cenozoic global sea level, sequences, and the New Jersey transect; results from coastal plain and continental slope drilling. Reviews of Geophysics 36: 569--601. Sahagian DL (1988) Ocean temperature-induced change in lithospheric thermal structure: a mechanism for longterm eustatic sea level change. Journal of Geology 96: 254--261. Wilgus CK, Hastings BS, Kendall CG St C et al. (1988) Sea Level Changes: An Integrated Approach. Special Publication no. 42. Society of Economic Paleontologists and Mineralogists.

THE AIR-SEA INTERFACE

HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE P. K. Taylor, Southampton Oceanography Centre, Southampton, UK Copyright & 2001 Elsevier Ltd.

Introduction The maintenance of the earth’s climate depends on a balance between the absorption of heat from the sun and the loss of heat through radiative cooling to space. For each 100 W of the sun’s radiative energy entering the atmosphere nearly 40 W is absorbed by the ocean – about twice that adsorbed in the atmosphere and three times that falling on land surfaces. Much of this oceanic heat is transferred back to the atmosphere by the local sea to air heat flux. The geographical variation of this atmospheric heating drives the weather systems and their associated winds. The wind transfers momentum to the sea causing waves and the wind-driven currents. Major ocean currents transport heat polewards and at higher latitudes the sea to air heat flux significantly ameliorates the climate. Thus the heat and momentum fluxes through the ocean surface form a crucial component of the earth’s climate system. The total heat transfer through the ocean surface, the net heat flux, is a combination of several components. The heat from the sun is the short-wave radiative flux (wavelength 0.3–3 mm). Around noon on a sunny day this flux may reach about 1000 W m2 but, when averaged over 24 h, a typical value is 100–300 W m2 varying with latitude and season. Part of this flux is reflected from the sea surface – about 6% depending on the solar elevation and the sea state. Most of the remaining short-wave flux is absorbed in the upper few meters of the ocean. In calm weather, with winds less than about 3 m s1, a shallow layer may be formed during the day in which the sea is warmed by a few degrees Celsius (a ‘diurnal thermocline’). However, under stronger winds or at night the absorbed heat becomes mixed down through several tens of metres. Thus, in contrast to land areas, the typical day to night variation in sea surface sea and air temperatures is small, o11C. Both the sea and the sky emit and absorb long-wave radiative energy (wavelength 3–50 mm). Because, under most circumstances, the radiative

temperature of the sky is colder than that of the sea, the downward long-wave flux is usually smaller than the upward flux. Hence the net long-wave flux acts to cool the surface, typically by 30–80 W m2 depending on cloud cover. The turbulent fluxes of sensible and latent heat also typically transfer heat from sea to air. The sensible heat flux is the transfer of heat caused by difference in temperature between the sea and the air. Over much of the ocean this flux cools the sea by perhaps 10–20 W m2. However, where cold wintertime continental air flows over warm ocean currents, for example the Gulf Stream region off the eastern seaboard of North America, the sensible heat flux may reach 100 W m2. Conversely warm winds blowing over a colder ocean region may result in a small sensible heat flux into the ocean – a frequent occurrence over the summertime North Pacific Ocean. The evaporation of water vapor from the sea surface causes the latent heat flux. This is the latent heat of vaporization which is carried by the water vapor and only released to warm the atmosphere when the vapor condenses to form clouds. Usually this flux is significantly greater than the sensible heat flux, being on average 100 W m2 or more over large areas of the ocean. Over regions such as the Gulf Stream latent heat fluxes of several hundred W m2 are observed. In foggy conditions with the air warmer than the sea, the latent heat flux can transfer heat from air to sea. In summertime over the infamous fog-shrouded Grand Banks off Newfoundland the mean monthly latent heat transfer is directed into the ocean, but this is an exceptional case.

Measuring the Fluxes The standard instruments for determining the radiative fluxes measure the voltage generated by a thermopile which is exposed to the incident radiation. Typically the incoming short-wave radiation is measured by a pyranometer which is mounted in gimbals for use on a ship or buoy (Figure 1). For better accuracy the direct and scattered components should be determined separately but, apart from at the Baseline Surface Radiation Network stations which are predominantly situated on land, at present this is rarely done. The reflected short-wave radiation is normally determined from the sun’s elevation

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HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

Figure 1 A pyranometer used for measuring short-wave radiation. The thermopile is covered by two transparent domes. (Photograph courtesy of Southampton Oceanography Centre.)

and lookup tables based on the results of previous experiments. The pyrgeometer used to determine the long-wave radiation is similar to the pyranometer but uses a coated dome to filter out, as far as possible, the effects of the short-wave heating. Because the air close to the sea surface is normally near to the sea temperature, the use of gimbals is less important. However, a clear sky view is required and a number of correction terms have to be calculated for the temperature of the dome and any short-wave leakage. Again, only the downward component is normally measured; the upwards component is calculated from knowledge of the sea temperature and emissivity of the sea surface. The turbulent fluxes may be measured in the nearsurface atmosphere using the eddy correlation method. If upward moving air in an eddy is on average warmer and moister than the downward moving air, then there is an upwards flux of sensible heat and water vapor and hence also an upward latent heat flux. Similarly the momentum flux, or wind stress, may be determined from the correlation between the horizontal and vertical wind fluctuations. Since a large range of eddy sizes may contribute to the flux, fast response sensors capable of sampling at 10 Hz or more must be exposed for periods of the order of 30 min for each flux determination. Three-component ultrasonic anemometers (Figure 2) are relatively robust and, by also determining the speed of sound, can provide an estimate of the sonic temperature flux, a function of the heat and moisture fluxes. The sensors used for determining the fluctuations in temperature and humidity have previously tended to be fragile and prone to contamination by salt particles which are ever-present in the marine atmosphere. However, improved sonic thermometry, and new techniques for water vapor measurement, such as microwave

Figure 2 The sensing head of a three-component ultrasonic anemometer. The wind components are determined from the different times taken for sound pulses to travel in either direction between the six ceramic transducers. (Photograph courtesy of Southampton Oceanography Centre.)

refractometry or differential infrared absorption instruments, are now becoming available. Despite these improvements in instrumentation, obtaining accurate eddy correlation measurements over the sea remains very difficult. If the instrumentation is mounted on a buoy or ship the six components of the wave-induced motion of the measurement platform must be measured and removed from the signal. The distortion both of the turbulence and the mean wind by ship, buoy or fixed tower must be minimized and, as far as possible, corrected for. Thus eddy correlation measurements are not routinely obtained over the ocean, rather they are used in special air–sea interaction experiments to calibrate other less direct methods of flux estimation. For example, in the inertial dissipation method, fluctuations of the wind, temperature, or humidity at a few Hertz are measured and related (through turbulence theory) to the fluxes. This method is less sensitive to flow distortion or platform motion, but relies on various assumptions about the formation and dissipation of turbulent quantities, which may not be valid under some conditions. It has been implemented on a semi-routine basis on some research ships to increase the range of available flux data.

HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

The most commonly used method of flux estimation is variously referred to as the bulk (aerodynamic) formulae. These formulae relate the difference between the value of temperature, humidity or wind (‘x’ in [1]) at some measurement height, z, and the value assumed to exist at the sea surface – respectively the sea surface temperature, 98% saturation humidity (to allow for salinity effects), and zero wind (or any nonwind-induced water current). Thus the flux Fx of some quantity x is: Fx ¼ rUz Cxz ðxz  x0 Þ

½1

where r is the air density, and Uz the wind speed at the measurement height. While appearing intuitively correct (for example, blowing over a hot drink will cool it faster) these formulae can also be derived from turbulence theory. The value for the transfer coefficient, Cxz, characterizes both the surface roughness applicable to x and the relationship between Fx and the vertical profile of x. This varies with the atmospheric stability, which itself depends on the momentum, sensible heat, and water vapor fluxes, as well as the measurement height. Thus, although it may appear simple, Eqn [1] must be solved by iteration, initialized using the equivalent neutral value of Cxz at some standard height (normally 10 m), Cx10n. Typical neutral values (determined using eddy correlation or inertial dissipation data) are shown in Table 1. Many research problems remain. For example: CD10n is expected to depend on the state of development of the wave field, but can this be successfully characterized by the ratio of the predominant wave speed to the wind speed (the wave age), or by the wave height and steepness, or is a spectral representation of the wave field required? What are the effects of waves propagating from other regions (i.e., swell waves)? What is the behavior of CD10n in low wind speed conditions? Furthermore CE10n and CH10n are relatively poorly defined by the available experimental data, and recent bulk algorithms have used theoretical models of the ocean surface (known as surface renewal theory) Table 1

129

to predict these quantities from the momentum roughness length.

Sources of Flux Data Until recent years the only source of data for flux calculation routinely available from widespread regions of the world’s oceans was the weather reports from merchant ships. Organized as part of the World Weather Watch system of the World Meteorological Organisation, these ‘Voluntary Observing Ships (VOS)’ are asked to return coded weather messages at 00 00, 06 00, 12 00, and 18 00 h GMT daily, also recording the observation (with further details) in the ship’s weather logbook. The very basic set of instruments provided will normally include a barometer and a means of measuring air temperature and humidity – typically wet and dry bulb thermometers mounted in a hand swung sling psychrometer or a fixed, louvered ‘Stevenson’ screen. Sea temperature is obtained using a thermometer and an insulated bucket, or by reading the temperature gauge for the engine cooling water intake. Depending on which country recruited the VOS an anemometer and wind vane might be provided, or the ship’s officers might be asked to estimate the wind velocity from observations of the sea state using a tabulated ‘Beaufort scale’. Because of the problems of adequately siting an anemometer and maintaining its calibration, these visual estimates are not necessarily inferior to anemometer-based values. Thus the VOS weather reports include all the variables needed for calculating the turbulent fluxes using the bulk formulae. However, in many cases the accuracy of the data is limited both by the instrumentation and its siting. In particular, a large ship can induce significant changes in the local temperature and wind flow, since the VOS are not equipped with radiometers. The short-wave and long-wave fluxes must be estimated from the observer’s estimate of the cloud amount plus (as appropriate) the solar elevation, or the sea and air temperature and

Typical values (with estimated uncertainties) for the transfer coefficientsa

Flux

Transfer coefficients

Typical values

Momentum

Drag coefficient CD10n(  1000) Stanton no., UH10n Dalton no., UE10n

¼ 0.61 (70.05) þ 0.063 (70.005) U10n (U10n43 m s1) ¼ 0.61 þ 0.57/U10no3 m s1 1.1 (70.2)  103 1.2 (70.1)  103

Sensible heat Latent heat

a Neither the low wind speed formula for CD10n, nor the wind speed below which it should be applied, are well defined by the available, very scattered, experimental data. It should be taken simply as an indication that, at low wind speeds, the surface roughness increases as the wind speed decreases due to the dominance of viscous effects.

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HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

humidity. The unavoidable observational errors and the crude form of the radiative flux formulae imply that large numbers of reports are needed, and correction schemes must be applied, before satisfactory flux estimates can be obtained. While there are presently nearly 7000 VOS, the ships tend to be concentrated in the main shipping lanes. Thus whilst coverage in most of the North Atlantic and North Pacific is adequate to provide monthly mean flux values, elsewhere data is mainly restricted to relatively narrow, major trade routes. For most of the southern hemisphere the VOS data is only capable of providing useful values if averaged over several years, and reports from the Southern Ocean are very few indeed. These shortcomings of VOS-derived fluxes must be borne in mind when studying the flux distribution maps presented below. Satellite-borne sensors offer the potential to overcome these sampling problems. They are of two types, passive sensors which measure the radiation emitted from the sea surface and the intervening atmosphere at visible, infrared, or microwave frequencies, and active sensors which transmit microwave radiation and measure the returned signal. Unfortunately these remotely sensed data do not allow all of the flux components to be adequately estimated. Sea surface temperature has been routinely determined using visible and infrared radiometers since about 1980. Potential errors due, for example, to changes in atmospheric aerosols following volcanic eruptions, mean that these data must be continually checked against ship and buoy data. Algorithms have been developed to estimate the net surface short-wave radiation from top of the atmosphere values; those for estimating the net surface long wave are less successful. The surface wind velocity can be determined to good accuracy by active scatterometer sensors by measuring the microwave radiation backscattered from the sea surface. Unfortunately scatterometers are relatively costly to operate, since they demand significant power from the spacecraft and, to date, few have been flown. The determination of near-surface air temperature and humidity from satellite is hindered by the relatively coarse vertical resolution of the retrieved data. A problem is that the radiation emitted by the near-surface air is dominated by that originating from the sea surface. Statistically based algorithms for determining the near-surface humidity have been successfully demonstrated. More recently neural network techniques have been applied to retrieving both air temperature and humidity; however, at present there is no routinely available product. Thus the satellite flux products for which useful accuracy has been demonstrated are presently limited to momentum, short-wave radiation, and latent heat flux.

Numerical weather prediction (NWP) models (as used in weather forecasting centers) estimate values of the air–sea fluxes as a necessary part of their calculations. Since these models assimilate most of the available data from the World Weather Watch system, including satellite data, radiosonde profiles, and surface observations, it might be expected that NWP models represent the best source of flux data. However, there are other problems. The vertical resolution of these models is relatively poor and many of the near-surface processes which affect the fluxes have to be represented in terms of larger-scale parameters. Improvements to these models are normally judged on the resulting quality of the weather forecasts, not on the accuracy of the surface fluxes; sometimes these may become worse. Indeed, the continual introduction of model changes results in time discontinuities in the output variables. This makes the determination of interannual variations difficult. Because of this, NWP centres such as the European Centre for Medium Range Weather Forecasting (ECMWF) and the US National Centers for Environmental Prediction (NCEP) have reanalyzed the past weather and have gone back several decades. The surface fluxes from these reanalyses are receiving much study. Those presently available appear less accurate than fluxes derived from VOS data in regions where there are many VOS reports; in sparsely sampled regions the model fluxes may be more accurate. There are particular weaknesses in the shortwave radiation and latent heat fluxes. New reanalyses are planned and efforts are being made to improve the flux estimates; eventually these reanalyses will provide the best source of flux data for many purposes.

Regional and Seasonal Variation of the Momentum Flux The main features of the wind regimes over the global oceans have long been recognized and descriptions are available in many books on marine meteorology (see Further Reading). The major features of the wind stress variability derived from ship observations from the period 1980–93 will be summarized here, using plots for January and July to illustrate the seasonal variation. The distribution of the heat fluxes will be discussed in the next section. In northern hemisphere winter (Figure 3A) large wind stresses due to the strong midlatitude westerly winds are obvious in the North Atlantic and the North Pacific west of Japan. To the south of these regions the extratropical high pressure zones result in low wind stress values, south of these is the north-

HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

131

Figure 3 Monthly vector mean wind stress (N m2) for (A) January and (B) July calculated from Voluntary Observing Ship weather reports for the period 1980–93. (Adapted with permission from Josey SA, Kent EC and Taylor PK (1998) The Southampton Oceanography Centre (SOC) Ocean–Atmosphere Heat, Momentum and Freshwater Flux Atlas. SOC Report no. 6.)

east trade wind belt. The Inter-Tropical Convergence Zone (ITCZ) with very light winds is close to the equator in both oceans. In the summertime southern hemisphere the south-east trade wind belt is less well marked. The extratropical high pressure regions are extensive but, despite it being summer, high winds and significant wind stress exist in the midlatitude southern ocean. The north-east monsoon dominates the wind patterns in the Indian Ocean and the South

China Sea (where it is particularly strong). The ITCZ is a diffuse region south of the equator with relatively strong south-east trade winds in the eastern Indian Ocean. In northern hemisphere summer (Figure 3B) the wind stresses in the midlatitude westerlies are very much decreased. Both the north-east and the southeast trade wind zones are evident respectively to the north and south of the ITCZ. This is predominantly

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HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

north of the equator. The south-east trades are particularly strong in the Indian Ocean and feed into a very strong south-westerly monsoon flow in the Arabian Sea. The ship data indicate very strong winds in the Southern Ocean south west of Australia. These are also evident in satellite scatterometer data, which suggest that the winds in the Pacific sector of the Southern Ocean, while still strong, are somewhat less than those in the Indian Ocean sector. In contrast the ship data appear to show very light winds. The reason is that in wintertime there are practically no VOS observations in the far south Pacific. The analysis technique used to fill in the data gaps has, for want of other information, spread the light winds of the extratropical high pressure region farther south than is realistic; a good example of the care needed in interpreting the flux maps available in many atlases.

Regional and Seasonal Variation of the Heat Fluxes The global distribution of the mean annual net heat flux is shown in Figure 4A. The accuracy and method of determination of such flux distributions will be discussed further below, here they will be used to give a qualitative description. Averaged over the year the ocean is heated in equatorial regions and loses heat in higher latitudes, particularly in the North Atlantic. However, this mean distribution is somewhat misleading, as the plots for January (Figure 4B) and July (Figure 4C) illustrate. The ocean loses heat over most of the extratropical winter hemisphere and gains heat in the extratropical summer hemisphere and in the tropics throughout the year. The relative magnitude of the individual flux components is illustrated in Figure 5 for three representative sites in the North Atlantic Ocean. At the Gulf Stream site (Figure 5A) the large cooling in winter dominates the incoming solar radiation in the annual mean. However, even at this site the mean monthly short-wave flux in summer is greater than the cooling. Indeed the effect of the longer daylight periods increases the mean short-wave radiation to values similar to or larger than those observed in equatorial regions (Figure 5C). The midlatitude site (Figure 5B) is typical of large areas of the ocean. The ocean cools in winter and warms in summer, in each case by around 100 W m2. The annual mean flux is small – around 10 W m2 – but cannot be neglected because of the very large ocean areas involved. At this site, and generally over the ocean, this annual balance is between the sum of the latent heat flux and net long-wave flux which cool the ocean, and the net short-wave heating. Only in very cold air flows, as

over the Gulf Stream in winter, is the sensible heat flux significant. As regards the interannual variation of the surface fluxes, the major large-scale feature over the global ocean is the El Nin˜o-Southern Oscillation system in the equatorial Pacific Ocean. The changes in the net heat flux under El Nin˜o conditions are around 40 W m2 in the eastern equatorial Pacific. For extratropical and midlatitude regions the interannual variability of the summertime net heat flux is typically about 20–30 W m2, being dominated by the variations in latent heat flux. In winter the typical variability increases to about 30–40 W m2, although in particular areas (such as over the Gulf Stream) variations of up to 100 W m2 can occur. The major spatial pattern of interannual variability in the North Atlantic is known as the North Atlantic Oscillation (NAO). This represents a measure of the degree to which mobile depressions, or alternatively near stationary high pressure systems, occur in the midlatitude westerly zone.

Accuracy of Flux Estimates It has been shown that, although the individual flux components are of the order of hundreds of W m2, the net heat flux and its interannual variability over much of the world ocean is around tens of W m2. Furthermore it can be shown that a flux of 10 W m2 over 1 year would, if stored in the top 500 m of the ocean, heat that entire layer by about 0.151C. Temperature changes on a decadal time scale are at most a few tenths of a degree, so the global mean budget must balance to better than a few W m2. For these various reasons there is a need to measure the flux components, which vary on many time and space scales, to an accuracy of a few W m2. Given the available data sources and methods of determining the fluxes described in the previous sections, it is not surprising that this level of accuracy cannot be achieved at present. To take an example, in calculating the flux maps shown in Figure 4 from VOS data many corrections were applied to the VOS observations to attempt to remove biases caused by the methods of observation. For example, air temperature measurements were corrected for the heat island caused by the ship heating up in sunny, low wind conditions. The wind speeds were adjusted depending on the anemometer heights on different ships. Corrections were applied to sea temperatures calculated from engine room intake data. Despite these and other corrections, the global annual mean flux showed about 30 W m2 excess heating of the ocean. Previous climatologies

HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

133

Figure 4 Variation of the net heat flux over the ocean, positive values indicate heat entering the ocean: (A) annual mean, (B) January monthly mean, (C) July monthly mean. (Adapted with permission from Josey SA, Kent EC and Taylor PK (1998) The Southampton Oceanography Centre (SOC) Ocean–Atmosphere Heat, Momentum and Freshwater Flux Atlas. SOC Report no. 6.)

calculated from ship data had shown similar biases and the fluxes had been adjusted to remove the bias, or to make the fluxes compatible with estimates of the meridional heat transport in the ocean. However,

comparison of the unadjusted flux data with accurate data from air–sea interaction buoys showed good agreement between the two. This suggests that adjusting the fluxes globally is not correct and that

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HEAT AND MOMENTUM FLUXES AT THE SEA SURFACE

600

600

500

500

400

400

300

300

200

200

100

100

0 Annual (A)

January 40°N, 60°W

July

0 Annual (B)

600

January

July

40°N, 20°W SW radn. LW radn.

500

Latent Heat Sensible Heat

400 300 200 100 0 Annual (C)

January 0°N, 20°W

July

Figure 5 Mean heat fluxes at three typical sites in the North Atlantic for the annual mean, and the January and July monthly means. In each case the left-hand column shows the fluxes which act to cool the ocean while the right-hand column shows the solar heating. (A) Gulf Stream site (401N, 601W), (B) midlatitude site (401N, 201W), (C) equatorial site (01N, 201W).

regional flux adjustments are required; however, the exact form of these corrections is presently not shown. In the future, computer models are expected to provide a major advance in flux estimation. Recently coupled numerical models of the ocean and of the atmosphere have been run for many simulated years during which the modeled climate has not drifted. This suggests that the air–sea fluxes calculated by the models are in balance with the simulated oceanic and atmospheric heat transports. However, it does not imply that the presently estimated flux values are realistic. Errors in the short-wave and latent heat fluxes may compensate one another; indeed in a typical simulation the sea surface temperature stabilized to a value which was, over large regions of the ocean, a few degrees different from that which is observed. Nevertheless the estimation of flux values using climate or NWP models is a rapidly developing field and improvements will doubtless occur in the next few years. There will be a continued need for in situ and satellite data for assimilation into the models and for model development and verification. However, it seems very likely that in future the most accurate routine source of the air–sea flux data will be from numerical models of the coupled ocean–atmosphere system.

See also El Nin˜o Southern Oscillation (Enso). Evaporation and Humidity. Freshwater Transport and Climate. Heat Transport and Climate. North Atlantic Oscillation (Nao). Upper Ocean Heat and Freshwater Budgets. Wave Energy. Wave Generation by Wind. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading Browning KA and Gurney RJ (eds.) (1999) Global Energy and Water Cycles. Cambridge: Cambridge University Press. Dobson F, Hasse L, and Davis R (eds.) (1980) Air–Sea Interaction, Instruments and Methods. New York: Plenum Press. Kraus EB and Businger JA (1994) Atmosphere–Ocean Interaction, 2nd edn. New York: Oxford University Press. Meteorological Office (1978) Meteorology for Mariners, 3rd edn. London: HMSO. Stull RB (1988) An Introduction to Boundary Layer Meteorology. Dordrecht: Kluwer Academic. Wells N (1997) The Atmosphere and Ocean: A Physical Introduction, 2nd edn. London: Taylor and Francis.

SEA SURFACE EXCHANGES OF MOMENTUM, HEAT, AND FRESH WATER DETERMINED BY SATELLITE L. Yu, Woods Hole Oceanographic Institution, Woods Hole, MA, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The ocean and the atmosphere communicate through the interfacial exchanges of heat, fresh water, and momentum. While the transfer of the momentum from the atmosphere to the ocean by wind stress is the most important forcing of the ocean circulation, the heat and water exchanges affect the horizontal and vertical temperature gradients of the lower atmosphere and the upper ocean, which, in turn, modify wind and ocean currents and maintain the equilibrium of the climate system. The sea surface exchanges are the fundamental processes of the coupled atmosphere–ocean system. An accurate knowledge of the flux variability is critical to our understanding and prediction of the changes of global weather and climate. The heat exchanges include four processes: the short-wave radiation (QSW) from the sun, the outgoing long-wave radiation (QLW) from the sea surface, the sensible heat transfer (QSH) resulting from air–sea temperature differences, and the latent heat transfer (QLH) carried by evaporation of sea surface water. Evaporation releases both energy and water vapor to the atmosphere, and thus links the global energy cycle to the global water cycle. The oceans are the key element of the water cycle, because the oceans contain 96% of the Earth’s water, experience 86% of planetary evaporation, and receive 78% of planetary precipitation. The amount of air–sea exchange is called sea surface (or air–sea) flux. Direct flux measurements by ships and buoys are very limited. Our present knowledge of the global sea surface flux distribution stems primarily from bulk parametrizations of the fluxes as functions of surface meteorological variables that can be more easily measured (e.g., wind speed, temperature, humidity, cloud cover, precipitation, etc.). Before the advent of satellite remote sensing, marine surface weather reports collected from voluntary observing ships (VOSs) were the

backbone for constructing the climatological state of the global flux fields. Over the past two decades, satellite remote sensing has become a mature technology for remotely sensing key air–sea variables. With continuous global spatial coverage, consistent quality, and high temporal sampling, satellite measurements not only allow the construction of air–sea fluxes at near-real time with unprecedented quality but most importantly, also offer the unique opportunity to view the global ocean synoptically as an entity.

Flux Estimation Using Satellite Observations Sea Surface Wind Stress

The Seasat-A satellite scatterometer, launched in June 1978, was the first mission to demonstrate that ocean surface wind vectors (both speed and direction) could be remotely sensed by active radar backscatter from measuring surface roughness. Scatterometer detects the loss of intensity of transmitted microwave energy from that returned by the ocean surface. Microwaves are scattered by winddriven capillary waves on the ocean surface, and the fraction of energy returned to the satellite (backscatter) depends on both the magnitude of the wind stress and the wind direction relative to the direction of the radar beam (azimuth angle). By using a transfer function or an empirical algorithm, the backscatter measurements are converted to wind vectors. It is true that scatterometers measure the effects of small-scale roughness caused by surface stress, but the retrieval algorithms produce surface wind, not wind stress, because there are no adequate surface-stress ‘ground truths’ to calibrate the algorithms. The wind retrievals are calibrated to the equivalent neutral-stability wind at a reference height of 10 m above the local-mean sea surface. This is the 10-m wind that would be associated with the observed surface stress if the atmospheric boundary layer were neutrally stratified. The 10-m equivalent neutral wind speeds differ from the 10-m wind speeds measured by anemometers, and these differences are a function of atmospheric stratification and are normally in the order of 0.2 m s–1. To compute

135

136

SEA SURFACE EXCHANGES

the surface wind stress, t, the conventional bulk formulation is then employed: t ¼ ðtx ; ty Þ ¼ rcd Wðu; vÞ

½1

where tx and ty are the zonal and meridional components of the wind stress; W, u, and v are the scatterometer-estimated wind speed at 10 m and its zonal component (eastward) and meridional component (northward), respectively. The density of surface air is given by r and is approximately equal to 1.225 kg m  3, and cd is a neutral 10-m drag coefficient. Scatterometer instruments are typically deployed on sun-synchronous near-polar-orbiting satellites that pass over the equator at approximately the same local times each day. These satellites orbit at an altitude of approximately 800 km and are commonly known as Polar Orbiting Environmental Satellites (POES). There have been six scatterometer sensors aboard POES since the early 1990s. The major characteristics of all scatterometers are summarized in Table 1. The first European Remote Sensing (ERS-1) satellite was launched by the European Space Agency (ESA) in August 1991. An identical instrument aboard the successor ERS-2 became operational in 1995, but failed in 2001. In August 1996, the National Aeronautics and Space Administration (NASA) began a joint mission with the National Space Development Agency (NASDA) of Japan to maintain continuous scatterometer missions beyond ERS satellites. The joint effort led to the launch of the NASA scatterometer (NSCAT) aboard the first Japanese Advanced Earth Observing Satellite (ADEOS-I). The ERA scatterometers differ from the NASA scatterometers in that the former operate on the C band (B5 GHz), while the latter use the Ku band (B14 GHz). For radio frequency band, rain attenuation increases as the signal frequency increases. Compared to C-band satellites, the higher frequencies of Ku band are more vulnerable to signal quality problems caused by rainfall. However, Ku-band satellites have the advantage of being more sensitive to wind variation at low winds and of covering more area. Rain has three effects on backscatter measurements. It attenuates the radar signal, introduces volume scattering, and changes the properties of the sea surface and consequently the properties of microwave signals scattered from the sea surface. When the backscatter from the sea surface is low, the additional volume scattering from rain will lead to an overestimation of the low wind speed actually present. Conversely, when the backscatter is high,

attenuation by rain will reduce the signal causing an underestimation of the wind speed. Under rain-free conditions, scatterometer-derived wind estimates are accurate within 1 m s  1 for speed and 201 for direction. For low (less than 3 m s  1) and high winds (greater than 20 m s  1), the uncertainties are generally larger. Most problems with low wind retrievals are due to the weak backscatter signal that is easily confounded by noise. The low signal/noise ratio complicates the ambiguity removal processing in selecting the best wind vector from the set of ambiguous wind vectors. Ambiguity removal is over 99% effective for wind speed of 8 m s–1 and higher. Extreme high winds are mostly associated with storm events. Scatterometer-derived high winds are found to be underestimated due largely to deficiencies of the empirical scatterometer algorithms. These algorithms are calibrated against a subset of ocean buoys – although the buoy winds are accurate and serve as surface wind truth, few of them have high-wind observations. NSCAT worked flawlessly, but the spacecraft (ADEOS-I) that hosted it demised prematurely in June 1997 after only 9 months of operation. A replacement mission called QuikSCAT was rapidly developed and launched in July 1999. To date, QuikSCAT remains in operation, far outlasting the expected 2–3-year mission life expectancy. QuikSCAT carries a Ku-band scatterometer named SeaWinds, which has accuracy characteristics similar to NSCAT but with improved coverage. The instrument measures vector winds over a swath of 1800 km with a nominal spatial resolution of 25 km. The improved sampling size allows approximately 93% of the ocean surface to be sampled on a daily basis as opposed to 2 days by NSCAT and 4 days by the ERS instruments. A second similar-version SeaWinds instrument was placed on the ADEOS-II mission in December 2002. However, after only a few months of operation, it followed the unfortunate path of NSCAT and failed in October 2003 due – once again – to power loss. The Advanced Scatterometer (ASCAT) launched by ESA/EUMETSAT in March 2007 is the most recent satellite designed primarily for the global measurement of sea surface wind vectors. ASCAT is flown on the first of three METOP satellites. Each METOP has a design lifetime of 5 years and thus, with overlap, the series has a planned duration of 14 years. ASCAT is similar to ERS-1/2 in configuration except that it has increased coverage, with two 500-km swaths (one on each side of the spacecraft nadir track). The data collected by scatterometers on various missions have constituted a record of ocean vector winds for more than a decade, starting in August 1992. These satellite winds provide synoptic global

Table 1

Major characteristics of the spaceborne scatterometers

Characteristics

Operational frequency Spatial resolution Scan characteristics

Daily coverage Period in service

Scatterometer SeaSat-A

ERS-1

ERS-2

NSCAT

SeaWinds on QuikSCAT

SeaWinds on ADEOS II

ASCAT

Ku band

C band

C band

Ku band

Ku band

Ku band

C band

14.6 GHz 50 km  50 km with 100-km spacing Two-sided, double 500 km swaths separated by a 450 km nadir gap

5.255 GHz 50 km  50 km

5.255 GHz 50 km  50 km

13.995 GHz 25 km  25 km

13.402 GHz 25 km  25 km

13.402 GHz 25 km  6 km

5.255 GHz 25 km  25 km

One-sided, single 500-km swath

One-sided, single 500-km swath

Two-sided, double 600-km swaths separated by a 329-km nadir gap

Conical scan, one wide swath of 1800 km

Conical scan, one wide swath of 1800 km

Variable Jul. 1978–Oct. 1978

41% Aug. 1991–May. 1997

41% May. 1995–Jan. 2001

77% Sep. 1996–Jun. 1997

93% Jun. 1999–current

93% Dec. 2002–Oct. 2003

Two-sided, double 500-km swaths separated by a 700-km nadir gap 60% Mar. 2007–current

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SEA SURFACE EXCHANGES

view from the vantage point of space, and provide excellent coverage in regions, such as the southern oceans, that are poorly sampled by the conventional observing network. Scatterometers have been shown to be the only means of delivering observations at adequate ranges of temporal and spatial scales and at adequate accuracy for understanding ocean– atmosphere interactions and global climate changes, and for improving climate predictions on synoptic, seasonal, and interannual timescales. Surface Radiative Fluxes

Direct estimates of surface short-wave (SW) and long-wave (LW) fluxes that resolve synoptic to regional variability over the globe have only become possible with the advent of satellite in the past two decades. The surface radiation is a strong function of clouds. Low, thick clouds reflect large amounts of solar radiation and tend to cool the surface of the Earth. High, thin clouds transmit incoming solar radiation, but at the same time, they absorb the outgoing LW radiation emitted by the Earth and radiate it back downward. The portion of radiation, acting as an effective ‘greenhouse gas’, adds to the SW energy from the sun and causes an additional warming of the surface of the Earth. For a given cloud, its effect on the surface radiation depends on several factors, including the cloud’s altitude, size, and the particles that form the cloud. At present, the radiative heat fluxes at the Earth’s surface are estimated from top-of-the-atmosphere (TOA) SW and LW radiance measurements in conjunction with radiative transfer models. Satellite radiance measurements are provided by two types of radiometers: scanning radiometers and nonscanning wide-field-of-view radiometers. Scanning radiometers view radiance from a single direction and must estimate the hemispheric emission or reflection. Nonscanning radiometers view the entire hemisphere of radiation with a roughly 1000-km field of view. The first flight of an Earth Radiation Budget Experiment (ERBE) instrument in 1984 included both a scanning radiometer and a set of nonscanning radiometers. These instruments obtain good measurements of TOA radiative variables including insolation, albedo, and absorbed radiation. To estimate surface radiation fluxes, however, more accurate information on clouds is needed. To determine the physical properties of clouds from satellite measurements, the International Satellite Cloud Climatology Project (ISCCP) was established in 1983. ISCCP pioneered the cross-calibration, analysis, and merger of measurements from the international constellation of operational weather satellites. Using

geostationary satellite measurements with polar orbiter measurements as supplemental when there are no geostationary measurements, the ISCCP cloudretrieval algorithm includes the conversion of radiance measurements to cloud scenes and the inference of cloud properties from the radiance values. Radiance thresholds are applied to obtain cloud fractions for low, middle, and high clouds based on radiance computed from models using observed temperature and climatological lapse rates. In addition to the global cloud analysis, ISCCP also produces radiative fluxes (up, down, and net) at the Earth’s surface that parallels the effort undertaken by the Global Energy and Water Cycle Experiment – Surface Radiation Budget (GEWEX-SRB) project. The two projects use the same ISCCP cloud information but different ancillary data sources and different radiative transfer codes. They both compute the radiation fluxes for clear and cloudy skies to estimate the cloud effect on radiative energy transfer. Both have a 3-h resolution, but ISCCP fluxes are produced on a 280-km equal-area (EQ) global grid while GEWEX-SRB fluxes are on a 11  11 global grid. The two sets of fluxes have reasonable agreement with each other on the long-term mean basis, as suggested by the comparison of the global annual surface radiation budget in Table 2. The total net radiation differs by about 5 W m  2, due mostly to the SW component. However, when compared with ground-based observations, the uncertainty of these fluxes is about 10–15 W m  2. The main cause is the uncertainties in surface and near-surface atmospheric properties such as surface skin temperature, surface air and near-surface-layer temperatures and humidity, aerosols, etc. Further improvement requires improved retrievals of these properties. In the late 1990s, the Clouds and the Earth’s Radiant Energy System (CERES) experiment was Table 2 Annual surface radiation budget (in W m  2) over global oceans. Uncertainty estimates are based on the standard error of monthly anomalies Data 21-year mean 1984– 2004

ISCCP (Zhang et al., 2004) GEWEX-SRB (Gupta et al., 2006)

Parameter

SW Net downward

LW Net downward

SW þ LW Net downward

173.279.2

 46.979.2

126.3711.0

167.2713.9

 46.375.5

120.9711.9

SEA SURFACE EXCHANGES

developed by NASA’s Earth Observing System (EOS) not only to measure TOA radiative fluxes but also to determine radiative fluxes within the atmosphere and at the surface, by using simultaneous measurements of complete cloud properties from other EOS instruments such as the moderate-resolution imaging spectroradiometer (MODIS). CERES instruments were launched aboard the Tropical Rainfall Measuring Mission (TRMM) in November 1997, on the EOS Terra satellite in December 1999, and on the EOS Aqua spacecraft in 2002. There is no doubt that the EOS era satellite observations will lead to great improvement in estimating cloud properties and surface radiation budget with sufficient simultaneity and accuracy. Sea Surface Turbulent Heat Fluxes

Latent and sensible heat fluxes are the primary mechanism by which the ocean transfers much of the absorbed solar radiation back to the atmosphere. The two fluxes cannot be directly observed by space sensors, but can be estimated from wind speed and sea–air humidity/temperature differences using the following bulk parametrizations: QLH ¼ rLe ce Wðqs  qa Þ

½2

QSH ¼ rcp ch WðTs  Ta Þ

½3

where Le is the latent heat of vaporization and is a function of sea surface temperature (SST, Ts) expressed as Le ¼ (2.501  0.002 37  Ts)  1.06. cp is the specific heat capacity of air at constant pressure; ce and ch are the stability- and height-dependent turbulent exchange coefficients for latent and sensible heat, respectively. Ta/qa are the temperature/ specific humidity at a reference height of 2 m above the sea surface. qs is the saturation humidity at Ts, and is multiplied by 0.98 to take into account the reduction in vapor pressure caused by salt water. The two variables, Ts and W, in eqns [2] and [3] are retrieved from satellites, and so qs is known. The remote sensing of Ts is based on techniques by which spaceborne infrared and microwave radiometers detect thermally emitted radiation from the ocean surface. Infrared radiometers like the five-channel advanced very high resolution radiometer (AVHRR) utilize the wavelength bands at 3.5–4 and 10–12 mm that have a high transmission of the cloud-free atmosphere. The disadvantage is that clouds are opaque to infrared radiation and can effectively mask radiation from the ocean surface, and this affects the temporal resolution. Although the AVHRR satellite orbits the Earth 14 times each day from 833 km

139

above its surface and each pass of the satellite provides a 2399-km-wide swath, it usually takes 1 or 2 weeks, depending on the actual cloud coverage, to obtain a complete global coverage. Clouds, on the other hand, have little effect on the microwave radiometers so that microwave Ts retrievals can be made under complete cloud cover except for raining conditions. The TRMM microwave imager (TMI) launched in 1997 has a full suite of channels ranging from 10.7 to 85 GHz and was the first satellite sensor capable of accurately measuring SST through clouds. The low-inclination equitorial orbit, however, limits the TMI’s coverage only up to c. 381 latitude. Following TMI, the first polar-orbiting microwave radiometer capable of measuring global through-cloud SST was made possible by the NASDA’s advanced microwave scanning radiometer (AMSR) flown aboard the NASA’s EOS Aqua mission in 2002. While SST can be measured in both infrared and microwave regions, the near-surface wind speed can only be retrieved in the microwave region. The reason is that the emissivity of the ocean’s surface at wavelengths of around 11 mm is so high that it is not sensitive to changes in the wind-induced sea surface roughness or humidity fluctuations in the lower atmosphere. Microwave wind speed retrievals are provided by the special sensor microwave/imager (SSM/I) that has been flown on a series of polarorbiting operational spacecrafts of the Defense Meteorological Space Program (DMSP) since July 1987. SSM/I has a wide swath (B 1400 km) and a coverage of 82% of the Earth’s surface within 1 day. But unlike scatterometers, SSM/I is a passive microwave sensor and cannot provide information on the wind direction. This is not a problem for the computation in eqns [2] and [3] that requires only wind speed observations. In fact, the high space-time resolution and good global coverage of SSM/I has made it serving as a primary database for computing the climate mean and variability of the oceanic latent and sensible heat fluxes over the past B20-year period. At present, wind speed measurements with good accuracy are also available from several NASA satellite platforms, including TMI and AMSR. The most difficult problem for the satellite-based flux estimation is the retrieval of the air humidity and temperature, qa and Ta, at a level of several meters above the surface. This problem is inherent to all spaceborne passive radiometers, because the measured radiation emanates from relatively thick atmospheric layers rather than from single levels. One common practice to extract satellite qa is to relate qa to the observed column integrated water vapor (IWV, also referred to as the total precipitable water) from SSM/I. Using IWV as a proxy for qa is based on

140

SEA SURFACE EXCHANGES

several observational findings that on monthly timescales the vertical distribution of water vapor is coherent throughout the entire atmospheric column. The approach, however, produces large systematic biases of over 2 g kg  1 in the Tropics, as well as in the mid- and high latitudes during summertime. This is caused by the effect of the water vapor convergence that is difficult to assess in regions where the surface air is nearly saturated but the total IWV is small. Under such situations, the IWV cannot reflect the actual vertical and horizontal humidity variations in the atmosphere. Various remedies have been proposed to improve the qa–IWV relation and to make it applicable on synoptic and shorter timescales. There are methods of including additional geophysical variables, replacing IWV with the IWV in the lower 500 m of the planetary layer, and/or using empirical orthogonal functions (EOFs). Although overall improvements were achieved, the accuracy remains poor due to the lack of detailed information on the atmospheric humidity profiles. Retrieving Ta from satellite observations is even more challenging. Unlike humidity, there is no coherent vertical structure of temperature in the atmosphere. Satellite temperature sounding radiometers offer little help, as they generally are designed for retrieval in broad vertical layers. The sounder’s low information content in the lower atmosphere does not enable the retrieval of near-surface air temperature with sufficient accuracy. Different methods have been tested to derive Ta from the inferred qa, but all showed limited success. Because of the difficulties in determining qa and Ta, latent and sensible fluxes estimated from satellite measurements have large uncertainties. Three methods have been tested for obtaining better qa and Ta to improve the estimates of latent and sensible fluxes. The first approach is to enhance the information on the temperature and moisture in the lower troposphere. This is achieved by combining SSM/I data with additional microwave sounder data that come from the instruments like the advanced microwave sounding unit (AMSU-A) and microwave humidity sounder (MHS) flown aboard the National Oceanic and Atmospheric Administration (NOAA) polar-orbiting satellites, and the special sensor microwave temperature sounder (SSM/T) and (SSM/T-2) on the DMSP satellites. Although the sounders do not directly provide shallow surface measurements, detailed profile information provided by the sounders can help to remove variability in total column measurements not associated with the surface. The second approach is to capitalize the progress made in numerical weather prediction models that assimilate sounder observations into the physically based system. The qa and Ta estimates

from the models contain less ambiguity associated with the vertical integration and large spatial averaging of the various parameters, though they are subject to systematic bias due to model’s subgrid parametrizations. The third approach is to obtain a better estimation of qa and Ta through an optimal combination of satellite retrievals with the model outputs, which has been experimented by the Objectively Analyzed air–sea Fluxes (OAFlux) project at the Woods Hole Oceanographic Institution (WHOI). The effort has led to improved daily estimates of global air–sea latent and sensible fluxes. Freshwater Flux

The freshwater flux is the difference between precipitation (rain) and evaporation. Evaporation releases both water vapor and latent heat to the atmosphere. Once latent heat fluxes are estimated, the sea surface evaporation (E) can be computed using the following relation: E ¼ QLH =rw Le

½4

where QLH denotes latent heat flux and rw is the density of seawater. Spaceborne sensors cannot directly observe the actual precipitation reaching the Earth’s surface, but they can measure other variables that may be highly correlated with surface rainfall. These include variations in infrared and microwave brightness temperatures, as well as visible and near-infrared albedo. Infrared techniques are based on the premise that rainfall at the surface is related to cloud-top properties observed from space. Visible/infrared observations supplement the infrared imagery with visible imagery during daytime to help eliminate thin cirrus clouds, which are cold in the infrared imagery and are sometimes misinterpreted as raining using infrared data alone. Visible/infrared sensors have the advantage of providing good space and time sampling, but have difficulty capturing the rain from warmtopped clouds. By comparison, microwave (MW) estimates are more physically based and more accurate although time and space resolutions are not as good. The principle of MW techniques is that rainfall at the surface is related to microwave emission from rain drops (low-frequency channels) and microwave scattering from ice (high-frequency channels). While the primary visible/infrared data sources are the operational geostationary satellites, microwave observations are available from SSM/I, the NOAA AMSUB, and the TRMM spacecraft. TRMM opened up a new era of estimating not only surface rainfall but also rain profiles. TRMM is

SEA SURFACE EXCHANGES

equipped with the first spaceborne precipitation radar (PR) along with a microwave radiometer (TMI) and a visible/infrared radiometer (VIRS). Coincident measurements from the three sensors are complementary. PR provides detailed vertical rain profiles across a 215-km-wide strip. TMI (a fivefrequency conical scanning radiometer) though has less vertical and horizontal fidelity in rain-resolving capability, and it features a swath width of 760 km. The VIRS on TRMM adds cloud-top temperatures and structures to complement the description of the two microwave sensors. While direct precipitation information from VIRS is less reliable than that obtained by the microwave sensors, VIRS serves an important role as a bridge between the high-quality but infrequent observations from TMI and PR and the more available data and longer time series data available from the geostationary visible/infrared satellite platforms. The TRMM satellite focuses on the rain variability over the tropical and subtropical regions due to the low inclination. An improved instrument, AMSR, has extended TRMM rainfall measurements to higher latitudes. AMSR is currently aboard the Aqua satellite and is planned by the Global Precipitation Measurement (GPM) mission to be launched in 2009. Combining rainfall estimates from visible/infrared with microwave measurements is being undertaken by the Global Climatology Project (GPCP) to produce global precipitation analyses from 1979 and continuing.

Rain GPCP, TRMM

Evaporation SSMI, TMI, AMSR

141

Summary and Applications The satellite sensor systems developed in the past two decades have provided unprecedented observations of geophysical parameters in the lower atmosphere and upper oceans. The combination of measurements from multiple satellite platforms has demonstrated the capability of estimating sea surface heat, fresh water, and momentum fluxes with sufficient accuracy and resolution. These air–sea flux data sets, together with satellite retrievals of ocean surface topography, temperature, and salinity (Figure 1), establish a complete satellite-based observational infrastructure for fully monitoring the ocean’s response to the changes in air–sea physical forcing. Atmosphere and the ocean are nonlinear turbulent fluids, and their interactions are nonlinear scaledependent, with processes at one scale affecting processes at other scales. The synergy of various satellite-based products makes it especially advantageous to study the complex scale interactions between the atmosphere and the ocean. One clear example is the satellite monitoring of the development of the El Nin˜o–Southern Oscillation (ENSO) in 1997– 98. ENSO is the largest source of interannual variability in the global climate system. The phenomenon is characterized by the appearance of extensive warm surface water over the central and eastern tropical Pacific Ocean at a frequency of c. 3–7 years. The 1997–98 El Nin˜o was one of the most severe events experienced during the twentieth century. During the

Short- and Iong-wave radiation ERBE, CERES ISCCP, GEWEX-SRB

Vector wind Seasat-A, ERS, NSCAT, SeaWinds on QuikSCAT and ADEOS-II, ASCAT

Sensible heat Water vapor and latent heat

Sea surface salinity Aquarius, SMOS

Sea surface temperature AVHRR, TMI, AMSR

Sea level TOPEX/Poseidon, JASON

Figure 1 Schematic diagram of the physical exchange processes at the air–sea interface and the upper ocean responses, with corresponding sensor names shown in red.

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peak of the event in December 1997 (Figure 2), the SST in the eastern equatorial Pacific was more than 5 1C above normal, and the warming was accompanied by excessive precipitation and large net heat transfer from the ocean to the atmosphere. The 1997–98 event was also the best observed thanks largely to the expanded satellite-observing capability. One of the major observational findings was the role of synoptic westerly wind bursts (WWBs) in the onset of El Nin˜o. Figure 3 presents the evolution of zonal wind from NSCAT scatterometer combined with SSM/I-derived wind product, sea surface height (SSH) from TOPEX altimetry, and SST from AVHRR imagery in 1996–98. The appearance of the anomalous warming in the eastern basin in February 1997 coincided with the arrival of the downwelling Kelvin waves generated by the WWB of December 1996 in the western Pacific. A series of subsequent WWB-induced Kelvin waves further enhanced the eastern warming, and fueled

Degrees N

40

18 December 1996

30 20 10 0 –10 –20 100

Degrees N

TOPEX SSH anom (cm)

Wind (m s−1)

120

140 160 Degrees E

180

27 December 1996 40 30 20 10 0 –10 –20 100

the El Nin˜o development. The positive feedback between synoptic WWB and the interannual SST warming in making an El Nin˜o is clearly indicated by satellite observations. On the other hand, the synoptic WWB events were the result of the development of equatorial twin cyclones under the influence of northerly cold surges from East Asia/western North Pacific. NSCAT made the first complete recording of the compelling connection between nearequatorial wind events and mid-latitude atmospheric transient forcing. Clearly, the synergy of various satellite products offers consistent global patterns that facilitate the mapping of the correlations between various processes and the construction of the teleconnection pattern between weather and climate anomalies in one region and those in another. The satellite observing system will complement the in situ ground observations and play an increasingly important role in understanding the cause of global climate changes

120

140 160 Degrees E

180

SST anom (°C)

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120 160 –160 –120 –80 120 160 –160 –120 –80 120 160 –160 –120 –80 –12 –8 –4

0

4

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12

–32 –24 –16 –8 0

8 16 24 32

–2 –1

0

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4

Figure 2 (First column) An example of the scatterometer observations of the generation of the tropical cyclones in the western tropical Pacific under the influence of northerly cold surges from East Asia/western North Pacific. The effect of westerly wind bursts on the development of El Nin˜o is illustrated in the evolution of the equatorial sea level observed from TOPEX/Poseidon altimetry and SST from AVHRR. The second to fourth columns show longitude (horizontally) and time (vertically, increasing downwards). The series of westerly wind bursts (second column, SSM/I wind analysis by Atlas et al. (1996)) excited a series of downwelling Kelvin waves that propagated eastward along the equator (third column), suppressed the thermocline, and led to the sea surface warming in the eastern equatorial Pacific (fourth column).

SEA SURFACE EXCHANGES

qs QLH QSH Ta

Ocean temp. (°C) –3

–2

–1

0

1

2

3

4

5

Ts u v W r rw t tx ty

143

specific humidity at the sea surface latent heat flux sensible heat flux temperature at a reference height above the sea surface temperature at the sea surface zonal component of the wind speed meridional component of the wind speed wind speed density of surface air density of sea water wind stress zonal component of the wind stress meridional component of the wind stress

See also P − E (mm d–1) –10

–5

0

5

10

15

El Nin˜o Southern Oscillation (Enso). Evaporation and Humidity. Heat and Momentum Fluxes at the Sea Surface. Heat Transport and Climate. Upper Ocean Heat and Freshwater Budgets. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading

Net heat loss (W m–2) –80 –60 –40 –20

0

20

40

60

80 100 120

Figure 3 Satellite-derived global ocean temperature from AVHRR (top), precipitation minus evaporation from GPCP and WHOI OAFlux, respectively (middle), and net heat loss (QLH þ QSH þ QLW  QSW) from the ocean (bottom) during the El Nin˜o in Dec. 1997. The latent and sensible heat fluxes QLH þ QSH are provided by WHOI OAFlux, and the short- and long-wave radiative fluxes are by ISCCP.

and in improving the model skills on predicting weather and climate variability.

Nomenclature cd ce ch cp E Le qa

drag coefficient turbulent exchange coefficient for latent heat turbulent exchange coefficient for sensible heat specific heat capacity of air at constant pressure evaporation latent heat of vaporization specific humidity at a reference height above the sea surface

Adler RF, Huffman GJ, Chang A, et al. (2003) The Version 2 Global Precipitation Climatology Project (GPCP) monthly precipitation analysis (1979–present). Journal of Hydrometeorology 4: 1147--1167. Atlas R, Hoffman RN, Bloom SC, Jusem JC, and Ardizzone J (1996) A multiyear global surface wind velocity dataset using SSM/I wind observations. Bulletin of the American Meteorological Society 77: 869--882. Bentamy A, Katsaros KB, Mestas-Nun˜ez AM, et al. (2003) Satellite estimates of wind speed and latent heat flux over the global oceans. Journal of Climate 16: 637--656. Chou S-H, Nelkin E, Ardizzone J, Atlas RM, and Shie C-L (2003) Surface turbulent heat and momentum fluxes over global oceans based on the Goddard satellite retrievals, version 2 (GSSTF2). Journal of Climate 16: 3256--3273. Gupta SK, Ritchey NA, Wilber AC, Whitlock CH, Gibson GG, and Stackhouse RW, Jr. (1999) A climatology of surface radiation budget derived from satellite data. Journal of Climate 12: 2691--2710. Liu WT and Katsaros KB (2001) Air–sea flux from satellite data. In: Siedler G, Church J, and Gould J (eds.) Ocean Circulation and Climate, pp. 173--179. New York: Academic Press. Kubota M, Iwasaka N, Kizu S, Konda M, and Kutsuwada K (2002) Japanese Ocean Flux Data Sets with Use of Remote Sensing Observations (J-OFURO). Journal of Oceanography 58: 213--225. Wentz FJ, Gentemann C, Smith D, and Chelton D (2000) Satellite measurements of sea surface temperature through clouds. Science 288: 847--850.

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Yu L and Weller RA (2007) Objectively analyzed air–sea heat fluxes (OAFlux) for the global oceans. Bulletin of the American Meteorological Society 88: 527--539. Zhang Y-C, Rossow WB, Lacis AA, Oinas V, and Mishchenko MI (2004) Calculation of radiative fluxes from the surface to top of atmosphere based on ISCCP and other global data sets: Refinements of the radiative transfer model and the input data. Journal of Geophysical Research 109: D19105 (doi:10.1029/ 2003JD004457).

Relevant Websites http://winds.jpl.nasa.gov – Measuring Ocean Winds from Space. http://www.gewex.org – The Global Energy and Water Cycle Experiment (GEWEX).

http://precip.gsfc.nasa.gov – The Global Precipitation Climatology Project. http://isccp.giss.nasa.gov – The International Cloud Climatology Project. http://oaflux.whoi.edu – The Objectively Analyzed air–sea Fluxes project. http://www.ssmi.com – The Remote Sensing Systems Research Company http://www.gfdi.fsu.edu – The SEAFLUX Project, Geophysical Fluid Dynamics Institute. http://eosweb.larc.nasa.gov – The Surface Radiation Budget Data, Atmospheric Science Data Center.

EVAPORATION AND HUMIDITY K. Katsaros, Atlantic Oceanographic and Meteorological Laboratory, NOAA, Miami, FL, USA Copyright & 2001 Elsevier Ltd.

Introduction Evaporation from the sea and humidity in the air above the surface are two important and related aspects of the phenomena of air–sea interaction. In fact, most subsections of the subject of air–sea interaction are related to evaporation. The processes that control the flux of water vapor from sea to air are similar to those for momentum and sensible heat; in many contexts, the energy transfer associated with evaporation, the latent heat flux, is of greatest interest. The latter is simply the internal energy carried from the sea to the air during evaporation by water molecules. The profile of water vapor content is logarithmic in the outer layer, from a few centimeters to approximately 30 m above the sea, as it is for wind speed and air temperature under neutrally stratified conditions. The molecular transfer rate of water vapor in air is slow and controls the flux only in the lowest millimeter. Turbulent eddies dominate the vertical exchange beyond this laminar layer. Modifications to the efficiency of the turbulent transfer occur due to positive and negative buoyancy forces. The relative importance of mechanical shear-generated turbulence and density-driven (buoyancy) fluxes was formulated in the 1940s, the Monin-Obukhov theory, and the field developed rapidly into the 1960s. New technologies, such as the sonic anemometer and Lyman-alpha hygrometer, were developed, which allowed direct measurements of turbulent fluxes. Furthermore, several collaborative international field experiments were undertaken. A famous one is the ‘Kansas’ experiment, whose data were used to formulate modern versions of the ‘flux profile’ relations, i.e., the relationship between the profile in the atmosphere of a variable such as humidity, and the associated turbulent flux of water vapor and its dependence on atmospheric stratification. The density of air depends both on its temperature and on the concentration of water vapor. Recent improvements in measurement techniques and the ability to measure and correct for the motion of a ship or aircraft in three dimensions have allowed more direct measurements of evaporation over the ocean. The fundamentals of turbulent transfer in the

atmosphere will not be discussed here, only the special situations that are of interest for evaporation and humidity. As the water molecules leave the sea, they remove heat and leave behind an increase in the concentration of sea salts. Evaporation, therefore, changes the density of salt water, which has consequences for water mass formation and general oceanic circulation. This article will focus on how humidity varies in the atmosphere, on the processes of evaporation, and how it is modified by the other phenomena discussed under the heading of air–sea interaction. All processes occurring at the air–sea interface interact and modify each other, so that none are simple and linear and most result in feedback on the phenomenon itself. The role of wind, temperature, humidity, wave breaking, spray, and bubbles will be broached and some fundamental concepts and equations presented. Methods of direct measurements and estimation using in situ mean measurements and satellite measurements will be discussed. Subjects requiring further research are also explored.

History/Definitions and Nomenclature Many ways of measuring and defining the quantity of the invisible gas, water vapor, in the air have developed over the years. The common ones have been gathered together in Table 1, which gives their name, definition, SI units, and some further explanations. These quantitative definitions are all convertable one into another. The web-bulb temperature may seem rather anachronistic and is completely dependent on a rather crude measurement technique, but it is still a fundamental and dependable measure of the quantity of water vapor present in the air. Evaporation or turbulent transfer of water vapor in the air was first modeled in analogy with down– gradient transfer by molecular conduction in solids. The conductivity was replaced by an ‘Austaush’ coefficient, Ae, or eddy diffusion coefficient, leading to the expression: E ¼ Ae r

@ q¯ @z

½1

where E is the evaporation rate, r the air density, q¯ is mean atmospheric humidity, and z represents the vertical coordinate. Assuming no advection, steady state, and no accumulation of water vapor in the surface layer of the atmosphere (referred to as ‘the

145

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Table 1

Measures of humidity

Nomenclature

Units (SI)

Definition

Absolute humidity Specific humidity

kg m3 g kg1

Mixing ratio

g kg1

Saturation humidity

Any of the above units

Relative humidity (RH) Vapor pressure Dew point temperature

% hPa (or mb) 1K, 1C

Wet-bulb temperature

1K, 1C

Amount of water vapor in the volume of associated moist air The mass of water per unit mass of moist air (or equivalently in the same volume) The ratio of the mass of water as vapor to the mass of dry air in the same volume Can be given in terms of all three units and refers to the maximum amount the air can hold at its current temperature in terms of absolute or specific humidity, corresponds to 100% relative humidity Percent of saturation humidity that is actually in the air The partial pressure of the water vapor in the air The temperature at which dew would form based on the actual amount of water vapor in the air. Dew point depression compared to actual temperature is a measure of the ‘dryness’ of the air This is a temperature obtained by the wetted thermometer of the pair of thermometers used in a psychrometera (see Measurements chapter)

a A psychrometer is a measuring device consisting of two thermometers (mercury in glass or electronic), where one thermometer is covered with a wick wetted with distilled water. The device is aspirated with environmental air (at an air speed of at least 3 m s1). The evaporation of the distilled water cools the air passing over the wet wick, causing a lowering of the wet thermometer’s temperature, which is dependent on the humidity in the air.

constant flux layer’), the Ae is a function of z as the turbulence scales increase away from the air–sea interface and the gradient is a decreasing function of height, z, as the distance from the source of water vapor, the sea surface, increases. Determining E by measuring the gradient of q has not proved to be a good method because of the difficulties of obtaining differences of q accurately enough and in knowing the exact heights of the measurements well enough (say from a ship or a buoy on the ocean). The Ae must also be determined, which would require measurements of the intensity of the turbulent exchange in some fashion. The socalled direct method for evaluating the vapor flux in the atmosphere requires high frequency measurements. This method has been refined during the past 35 years or so, and has produced very good results for the turbulent flux of momentum (the wind stress). Fewer projects have been successful in measuring vapor flux over the ocean, because the humidity sensors are easily corrupted by the presence of spray or miniscule salt particles on the devices, which being hygroscopic, modify the local humidity. Evaporation, E, can be measured directly today by obtaining the integration over all scales of the turbulent flux, namely, the correlation between the deviations from the mean of vertical velocity (w0 ) and humidity (q0 ) at height (z) within the constant flux layer. This correlation, resulting from the averaging of the vapor conservation equation (in analogy to the Reynolds stress term in the Navier–Stokes equation) can be measured directly, if sensors are available that resolve all relevant scales of fluctuations.

The correlation equation is rw  q ¼ r¯ w ¯ q¯ þ rw ¯ 0 q0 ;

½2

where w and q are the instantaneous values and the overbar indicates the time-averaged means. The product of the averages is zero since w ¼ 0. Much discussion and experimentation has gone into determining the time required to obtain a stable mean value of the eddy flux rq ¯ 0 w0 . For the correlation term 0 0 rw ¯ q to represent the total vertical flux, there has to be a spectral gap between high and low frequencies of fluctuations, and the assumption of steady state and horizontal homogeneity must hold. The required averaging time is of the order of 20 min to 1 h. Another commonly used method, the indirect or inertial dissipation method, also requires high frequency sensing devices, but relies on the balance between production and destruction of turbulence to be in steady state. The dissipation is related to the spectral amplitude of turbulent fluctuations in the inertial subrange, where the fluctuations are broken down from large-scale eddies to smaller and smaller scales, which happens in a similar fashion regardless of scale of the eddies responsible for the production of turbulence in the atmospheric boundary layer. The magnitude of the spectrum in the inertial subrange is, therefore, a measure of the total energy of the turbulence and can be interpreted in terms of the turbulent flux of water vapor. The advantage of this method over the eddy correlation method is that it is less dependent on the corrections for flow distortion and motion of the ship or the buoy platform, but it

EVAPORATION AND HUMIDITY

requires corrections for atmospheric stratification and other predetermined coefficients. It would not give the true flux if the production of turbulence was changing, as it does in changing sea states. Most of the time, the direct flux is not measured by either the direct or the indirect method; we resort to a parameterization of the flux in terms of so-called ‘bulk’ quantities. The bulk formula has been found from field experiments where the total evaporation E has been measured directly together with mean values of q and wind speed, U, at one height, z ¼ a (usually referred to as 10 m by adjusting for the logarithmic vertical gradient), and the known sea surface temperature. E ¼ rw0 q0 ¼ r¯  CEa Ua ðqs  qa Þ

½3

where qs is the saturation specific humidity at the air– sea interface, a function of sea surface temperature (SST). Air in contact with a water surface is assumed to be saturated. Above sea water the saturated air has 98% of the value of water vapor density at saturation over a freshwater surface, due to the effects of the dissolved salts in the sea. CEa is the exchange

147

coefficient for water vapor evaluated for the height a. Experiments have shown CEa to be almost constant at 1.1–1.2  103 for Uo18 m s1, for neutral stratification, i.e. no positive or negative buoyancy forces acting and at a height of 10 m, written as CE10N. However, measurements show large variability in CE10N which may be due to the effects of sea state, such as sheltering in the wave troughs for large waves and increased evaporation due to spray droplets formed in highly forced seas with breaking waves. Results from a field experiment, the Humidity Exchange Over the Sea (HEXOS) experiment in the North Sea, are shown in Figure 1. Its purpose was to address the question of what happens to evaporation or water (vapor) flux at high wind speeds. However, the wind only reached 18 m s1 and the measurements showed only weak, if any, effects of the spray. Theories suggest that the effects will be stronger above 25 m s1. More direct measurements are still required before these issues can be settled, especially for wind speeds 420 m s1 (see Further Reading and the section on meteorological sensors for mean measurements for a discussion of the difficulties of making measurements over the sea at high wind speeds).

3

3

10 CEN

2

1

0 0

5

10

15

20

25

_

U10N (m s 1) Figure 1 Vapor flux exchange coefficients from two simultaneous measurement sets: the University of Washington (crosses) and Bedford Institute of Oceanography (squares) data. Thick dashed line is the average value, 1.12  103, for 170 data points. Thin dashed lines indicate standard deviations (from DeCosmo et al., 1996).

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Clausius–Clapeyron Equation The Clausius–Clapeyron equation relates the latent heat of evaporation to the work required to expand a unit mass of liquid water into a unit mass of water as vapor. The latent heat of evaporation is a function of absolute temperature. The Clausius–Clapeyron equation expresses the dependence of atmospheric saturation vapor pressure on temperature. It is a fundamental concept for understanding the role of evaporation in air–sea interaction on the large scale, as well as for gaining insight into the process of evaporation from the sea (or Earth’s) surface on the small scale. Note first of all that the Clausius–Clapeyron equation is highly non-linear, viz: d ln rv DHvap ¼ dT RT 2

½4

where pv is the vapor pressure, T is absolute temperature (1K), and DHvap is the value of the latent heat of evaporation, R is the gas constant for water vapor ¼ 461.53 J kg1 1K1. The dependence of vapor pressure on temperature is presented in a simplified form as:   T0 ðP a Þ es ¼ 610:8 exp 19:85 1  T

½5

where es is vapor pressure in pascals, T0 is a reference temperature set to 01C ¼ 273.16 1K, and T is the actual

temperature in 1K which is accurate to 2% below 301C (Figure 2). Figure 2 displays the saturation vapor pressure and the pressure of atmospheric water vapor for 60% relative humidity. On the right-hand side of the figure, the ordinate gives the equivalent specific humidity values (for a near surface total atmospheric pressure of 1000 hpa). This figure illustrates that the atmosphere can hold vastly larger amounts of water as vapor at temperatures above 201C than at temperatures below 101C. For constant relative humidity, say 60%, the difference in specific humidity or vapor pressure in the air compared with the amount at the air–sea interface, if the sea is at the same temperature as the air, is about three times at 301C what it would be at 101C. Therefore, evaporation is driven much more strongly at tropical latitudes compared with high latitudes (cold sea and air) for the same mean wind and relative humidity as illustrated by eqn [4] and Figure 2.

Tropical Conditions of Humidity By far, most of the water leaving the Earth’s surface evaporates from the tropical oceans and jungles, providing the accompanying latent heat as the fuel that drives the atmospheric ‘heat engines,’ namely, thunderstorms and tropical cyclones. Such extreme and violent storms depend for their generation on the enormous release of latent heat in clouds to create the vertical motion and compensating horizontal

Vapor pressure vs temperature 100

Vapor pressure (hPa)

80

60

40

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0 0

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25

30

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Temperature (˚C) 60% RH

Saturation /100% RH

Figure 2 Vapor pressure (hPa) as a function of temperature for two values of relative humidity, 60% and 100%.

45

EVAPORATION AND HUMIDITY

accelerated inflows. Tropical cyclones do not form over oceanic regions with temperatures o261C, and temperature increases of only 11 or 21C sharply enhance the possibility of formation.

149

the hydrologic cycle of evaporation and precipitation on the atmosphere, the continents would have more extreme climates and be less habitable.

Sublimation–Deposition Latitudinal and Regional Variations The Clausius–Clapeyron equation holds the secrets to the role of water vapor for both weather and climate. Warm moist air flowing north holds large quantities of water. As the air cools by vertical motion, contact with cold currents, and loss of heat by infrared radiation, the air reaches saturation and either clouds, storms and rain form, or fog (over cold surfaces) and stratus clouds. The warmer and moister the original air, the larger the possible rainfall and the larger the release of latent heat. Latitudinal, regional, and seasonal variations in evaporation and atmospheric humidity are all related to the source of heat for evaporation (upper ocean heat content) and the capacity of the air to hold water at its actual temperature. Many other processes such as the dynamics behind convergence patterns and the development of atmospheric frontal zones contribute to the variability of the associated weather.

Vertical Structure of Humidity The fact that the source of moisture is the ocean, lakes, and moist ground explains the vertical structure of the moisture field. Lenses of moist air can form aloft. However, when clouds evaporate at high elevations where atmospheric temperature is low, the absolute amounts of water vapor are also low for that reason. Thus, when the surface air is continually mixed in the atmospheric boundary layer with drier air, being entrained from the free atmosphere across the boundary layer inversion, it usually has a relative humidity less than 100% of what it could hold at its actual temperature. The exceptions are fog, clouds, or heavy rain, where the air has close to 100% relative humidity. The process of exchange between the moist boundary layer air and the upper atmosphere allows evaporation to continue. Deep convection in the inter-tropical convergence zone brings moist air up throughout the whole of the troposphere, even over-shooting into the stratosphere. Moisture that does not rain out locally is available for transport poleward. The heat released in these clouds modifies the temperature of the air. Similarly, over the warm western boundary currents, such as the Gulf Stream, Kuroshio, and Arghulas Currents, substantial evaporation and warming of the atmosphere takes place. Without the modifying effects of

The processes of water molecules leaving solid ice and condensing on it are called sublimation and deposition, respectively. These processes occur over the ice-covered polar regions of the ocean. In the cold regions, this flux is much less than that from open leads in the sea ice due to the warm liquid water, even at 01C. At an ice surface, water vapor saturation is less than over a water surface at the same temperature. This simple fact has consequences for the hydrologic cycle, because in a cloud consisting of a mixture of ice and liquid water particles, the vapor condenses on the ice crystals and the droplets evaporate. This process is important in the initial growth of ice particles in clouds until they become large enough to fall and grow by coalescence of droplets or other ice crystals encountered in their fall. Similar differences in water vapor occur for salty drops, and the vapor pressure over a droplet also depends on the curvature (radius) of the drop. Thus, particle size distribution in clouds and in spray over the ocean are always changing due to exchange of water vapor. For drops to become large enough to rain out, a coalescencetype growth process must typically be at work, since growth by condensation is rather slow.

Sources of Data Very few direct measurements of the flux of water vapor are available over the ocean at any one time. The mean quantities (U, qa , SST) needed to evaluate the bulk formula are reported regularly from voluntary observing ships (VOS) and from a few moored buoys. However, most of such buoys do not measure surface humidity, only a small number in the North Atlantic and tropical Pacific Oceans do so. The VOS observations are confined to shipping lanes, which leaves a huge void in the information available from the Southern Hemisphere. Alternative estimates of surface humidity and the water vapor flux include satellite methods and the surface fluxes produced in global numerical models, in particular, the re-analysis projects of the US Weather Service’s National Center for Environmental Prediction (NCEP) and the European Center for Medium Range Weather Forecasts (ECMWF). The satellite method has large statistical uncertainty and, thus, requires weekly to monthly averages for obtaining reasonable accuracy (730 W m2 and 715 W m2 for

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the weekly and monthly latent heat flux). Therefore, these data are most useful for climatological estimates and for checking the numerical models’ results.

Estimation of Evaporation by Satellite Data The estimation of evaporation/latent heat flux from the ocean using satellite data also relies on the bulk formula. The computation of latent heat flux by the bulk aerodynamic method requires SST, wind speed (U10N ), and humidity at a level within the surface layer qa , as seen in eqn [3]. Therefore, evaluation of the three variables from space is required. Over the ocean, U10N and SST have been directly retrieved from satellite data, but qa has not. A method of estimating qa and latent heat flux from the ocean using microwave radiometer data from satellites was proposed in the 1980s. It is based on an empirical relation between the integrated water vapor W (measured by spaceborne microwave radiometers) and qa on a monthly timescale. The physical rationale is that the vertical distribution of water vapor through the whole depth of the atmosphere is coherent for periods longer than a week. The relation does not work well at synoptic and shorter timescales and also fails in some regions during summer. Modification of this method by including additional geophysical parameters has been proposed with some overall improvement, but the inherent limitation is the lack of information about the vertical distribution of q near the surface. Two possible improvements in E retrieval include obtaining information on the vertical structures of humidity distribution and deriving a direct relation between E and the brightness temperatures (TB) measured by a radiometer. Recent developments provide an algorithm for direct retrieval of boundary layer water vapor from radiances observed by the Special Sensor Microwave/Imager (SSM/I) on operational satellites in the Defense Meteorological Satellite Program since 1987. This sensor has four frequencies, 19.35, 22, 37, and 85.5 GHz, all except the 22 GHz operated at both horizontal and vertical polarizations. The 22 GHz channel at vertical polarization is in the center of a weak water vapor absorption line without saturation, even at high atmospheric humidity. The measurements are only possible over the oceans, because the oceans act as a relatively uniform reflecting background. Over land, the signals from the ground overwhelm the water vapor information. Because all the three geophysical parameters, U10N , W, and SST, can be retrieved from the radiances at the frequencies measured by the older

microwave radiometer, launched in 1978 and operating to 1985 – the Scanning Multichannel Microwave Radiometer (SMMR) on Nimbus-7 (similar to SSM/I, but with 10.6 and 6.6 GHz channels as well, and no 85 GHz channels) – the feasibility of retrieving E directly from the measured radiances was also demonstrated. SMMR measures at 10 channels, but only six channels were identified as significantly useful in estimating E. SSM/I, the operational microwave radiometer that followed SMMR, lacks the low-frequency channels which are sensitive to SST, making direct retrieval of E from TB unfeasible. The microwave imager (TMI) on the Tropical Rainfall Measuring Mission (TRMM), launched in 1998, includes low-frequency measurements sensitive to SST and could, therefore, allow direct estimates of evaporation rates. Figure 3 gives an example of global monthly mean values of humidity obtained solely with satellite data from SSM/I. To calculate qs, gridded data of sea surface temperature can also be used, such as those provided operationally by the US National Weather Service based on infrared observations from the Advanced Very High Resolution Radiometer (AVHRR) on operational polar-orbiting satellites. The exact coincident timing is not so important for SST, since SST varies slowly due to the large heat capacity of water, and this method can only provide useful accuracies when averages are taken over 5 days to a week. Wind speed is best obtained from scatterometers, rather than from the microwave radiometer, in regions of heavy cloud or rain, since scatterometers (which are active radars) penetrate clouds more effectively. Scatterometers have been launched in recent times by the European Space Agency (ESA) and the US National Aeronautic and Space Administration (NASA) (the European Remote Sensing Satellites 1 and 2 in 1991 and 1995, the NASA scatterometer, NSCAT, on a Japanese short-lived satellite in 1996, and the QuikSCAT satellite in 1999).

Future Directions and Conclusions Evaporation has been measured only up to wind speeds of 18 m s1. The models appear to converge on the importance of the role of sea spray in evaporation, indicating that its significance grows beyond about 20 m s1. However, the source function of spray droplets as a function of wind speed or wave breaking has not been measured, nor are techniques for measuring evaporation in the presence of droplets well–developed, whether for rain or sea spray. Since evaporation and the latent heat play such important roles in tropical cyclones and many other weather

EVAPORATION AND HUMIDITY

_ 50

0

50

100

150

200

151

250

Figure 3 Global distribution of monthly mean latent heat flux in W m2 for September 1987. (Reproduced with permission from Schulz et al., 1997.)

phenomena, as well as in oceanic circulation, there is great motivation for getting this important energy and mass flux term right. The bulk model is likely to be the main method used for estimating evaporation for some time to come. Development of more direct satellite methods and validating them should be an objective for climatological purposes. Progress in the past 30 years has brought the estimate of evaporation on a global scale to useful accuracy.

Further Reading Bentamy A, Queffeulou P, Quilfen Y, and Katsaros KB (1999) Ocean surface wind fields estimated from satellite active and passive microwave instruments. Institute of Electrical and Electronic Engineers, Transactions, Geoscience Remote Sensing 37: 2469--2486. Businger JA, Wyngaard JC, lzumi Y, and Bradley EF (1971) Flux-profile relationships in the atmospheric surface layer. Journal of Atmospheric Science 28: 181--189. DeCosmo J, Katsaros KB, Smith SD, et al. (1996) Air–sea exchange of water vapor and sensible heat: The Humidity Exchange Over the Sea (HEXOS) results. Journal of Geophysical Research 101: 12001--12016. Dobson F, Hasse L, and Davies R (eds.) (1980) Instruments and Methods in Air–sea Interaction. New York: Plenum Publishing. Donelan MA (1990) Air–sea Interaction. In: LeMehaute B and Hanes DM (eds.) The Sea, Vol. 9, pp. 239--292. New York: John Wiley. Esbensen SK, Chelton DB, Vickers D, and Sun J (1993) An analysis of errors in Special Sensor Microwave Imager

evaporation estimates over the global oceans. Journal of Geophysical Research 98: 7081--7101. Geernaert GL (ed.) (1999) Air–sea Exchange Physics, Chemistry and Dynamics. Dordrecht: Kluwer Academic Publishers. Geernaert GL and Plant WJ (eds.) (1990) Surface Waves and Fluxes, Vol. 2. Dordrecht: Kluwer Academic Publishers. Katsaros KB, Smith SD, and Oost WA (1987) HEXOS – Humidity Exchange Over the Sea: A program for research on water vapor and droplet fluxes from sea to air at moderate to high wind speeds. Bulletin of the American Meteoroloical Society 68: 466--476. Kraus EB and Businger JA (eds.) (1994) Atmosphere– Ocean Interaction 2nd ed. New York: Oxford University Press. Liu WT and Katsaros KB (2001) Air–sea fluxes from satellite data. In: Siedler G, Church J, and Gould J (eds.) Ocean Circulation and Climate. Academic Press Liu WT, Tang W, and Wentz FJ (1992) Precipitable water and surface humidity over global oceans from SSM/I and ECMWF. Journal of Geophysical Research 97: 2251--2264. Makin VK (1998) Air–sea exchange of heat in the presence of wind waves and spray. Journal of Geophysical Research 103: 1137--1152. Schneider SH (ed.) (1996) Encyclopedia of Climate and Weather. New York: Oxford University Press. Schulz J, Meywerk J, Ewald S, and Schlu¨ssel P (1997) Evaluation of satellite-derived latent heat fluxes. Journal of Climate 10: 2782--2795. Smith SD (1988) Coefficients for sea surface wind stress, heat flux, and wind profiles as a function of wind speed and temperature. Journal of Geophysical Research 93: 15467--15472.

FRESHWATER TRANSPORT AND CLIMATE S. Wijffels, CSIRO Marine Research, Tasmania, Australia Copyright & 2001 Elsevier Ltd.

Introduction The ocean is the largest reservoir of water on the planet, consisting of 96% of the total available surface water (Figure 1) and it covers 75% of the earth’s surface. It is no surprise then that the majority of water cycling through the atmosphere derives from the ocean: about 12.2 Sv (1 Sverdrup ¼ 1  106 m3 s1) precipitates over the oceans compared to only 3.5 Sv over land (Figure 1), while 13.5 Sv evaporates from the oceans compared to 2.2 Sv of evapotranspiration over land. Total runoff into the global oceans is around 1.3–1.5 Sv, which must be balanced in the long term by a slightly higher total evaporation than precipitation over the oceans. In the following the term transport is used to refer to processes within the ocean and atmosphere, while the term flux is used for processes between these media. At any point on the ocean’s surface, the total fresh water flux is often a small residual between the two nearly equal and opposite fluxes of precipitation and evaporation. These fluxes between the ocean and atmosphere have quite different spatial patterns, and together create a rich structure in the mean annual fresh water flux at the ocean surface (Figure 2). The main sources of atmospheric water vapour are the

Global Water Reservoirs and Fluxes Atmosphere 16 EvapoTranspiration 2.2 Sv

Land 59 000

Evaporation 13.5 Sv

Precipitation 3.5 Sv

Rivers 1.3 Sv

3

Precipitation 12.2 Sv

Oceans 1 400 000

3

6

3

Reservoirs in 10 km , Fluxes in 10 m /s (=Sv) Figure 1 Global water reservoirs and fluxes. (Adapted with permission from Schmitt, 1995.)

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subtropical oceans under the atmospheric highpressure belts; the main atmospheric water sinks are the tropical convergence zones (particularly the eastern Indian Ocean/western Pacific) and the polar oceans. Since runoff is a relatively small component of the global fluxes, the transport of moisture within the atmosphere from surface sources to sinks is compensated by an equal and opposite ocean transport of fresh water. The atmosphere’s poleward transport of moisture also carries latent heat, and this heat transport comprises as much as 1.5 PW (1.5  1015 W) of the total of 4 PW of poleward atmospheric energy transport. The compensating ocean fresh water transport is therefore a fundamental parameter in the planetary energy budget. Surface fresh water fluxes impact on the oceans in several ways. They change the salinity of surface waters, imprinting them with properties characteristic of their formation regions. For example, intermediate waters formed in the subpolar regions where excess precipitation occurs are traceable far from their source regions because of their low salinity. Fresh water inputs from runoff or excess precipitation can also profoundly influence local air–sea interaction: the formation of fresh light surface layers can suppress convective mixing and thus isolate warmer, saltier deep waters from atmospheric cooling. Such mechanisms are observed in ocean models that display strong sensitivity to fresh water forcing, especially at high latitudes where deep convective mixing occurs that renews the near-bottom waters in the ocean. Changes in high-latitude fresh water forcing of the Atlantic are suspected to have changed the global thermohaline circulation in the past. The sensitivity of ocean models to changes in fresh water forcing is also manifest in the slowing down of the modeled global ocean thermohaline circulation when high-latitude precipitation increases under greenhouse-gas forcing. Primarily because of the difficulty of measuring rainfall rates over the oceans, estimates of surface fresh water fluxes have been, to date, too uncertain to constrain model behavior well. In the realm of atmospheric modeling, the lack of a reliable benchmark against which to compare the models’ moisture transport is a difficulty – the differences among models are often smaller than those among observations. Atmospheric models often overestimate the poleward transport of moisture compared to estimates based on direct observations. However, these observations are mostly made over the land, with

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Figure 2 (A) Average mean annual fresh water flux out of the ocean (mm y1). The zero line is dotted. (B) Standard deviation of 10 estimates of the above showing where the uncertainties are largest. The contour interval is 250 mm y1 in both (A) and (B). (Reproduced with permission from Wijffels, 2001.)

very few made over the oceans that cover the majority of the surface area of the planet. When forced with observed air–sea fresh water fluxes, ocean models can drift off to unrealistic states, and it has been difficult to distinguish the

cause: either inaccurate model physics or errors in the forcing fields. To avoid this problem (and because fresh water forcing has been considered less important compared to thermal forcing) the practice of forcing the model’s surface salinity field back to an

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observed surface salinity field has become the convention in ocean modeling. Such a flux formulation for salinity is physically unjustifiable. The resulting surface flux and salinity fields in the model are unrealistic. This prevents the use of salinity (the next most commonly observed quantity after temperature), to identify and correct physical errors in ocean models.

Methods of Fresh Water Flux and Transport Estimation In the past, most estimates of ocean fresh water transport derive from surface observations (ship and island) of rainfall rates and parameters (such as wind speed and temperature) used to estimate evaporation. The resulting surface fresh water flux fields can be integrated over the surface area of ocean basins, runoff from the continents added, and the result compared with estimates of ocean fresh water transport based on ocean measurements at specific locations. Surface Fluxes

In recent years, many new estimates of both atmospheric and surface moisture fluxes have appeared. They are based on the development of data-assimilating atmospheric general circulation models and the availability of new satellite datasets that can be used to deduce evaporation, precipitation, and the net moisture content of the atmospheric column. Estimates based on data-assimilating atmospheric models have several difficulties, as described by Trenberth and Guillemont in 1999. First, the model output often does not obey total mass conservation making budget calculations difficult. Second, there is a lack of atmospheric profile data over the oceans; the assimilation of scant island station data into these models produces ‘bulls eyes’ in the surface flux fields, indicating differences between the models and observations. Estimates of evaporation at the surface rely on empirical relations between the flux and parameters based on either radiometric data measured from satellites or marine meteorological measurements such as wind speed, relative humidity, and sea surface temperature. Though constantly improving, these flux formulas, which are required to apply under all conditions, suffer small biases. When accumulated over large areas such as ocean basins, these flux biases can dominate the totals. The accuracy of the ship-based measurements can also be poor and vary between vessels. Precipitation is

particularly challenging to estimate over the ocean because it is sporadic in both time and space. Here, satellite estimates may be the only way to progress, but these also rely on empirical algorithms that require ‘tuning.’ The range (as measured by the standard deviation) of current estimates of the mean annual surface fresh water flux (Figure 2B) is globally about 250 mm y1, which if integrated over the surface area of the Pacific Ocean north of 301S adds up to 1 Sv of fresh water transport, which is as large as the natural transport. The largest uncertainty occurs over the tropics and the area affected by mid-latitude storm tracks, as well as a region in the Southeast Pacific off Chile. These are all regions where precipitation is high, thus confirming that the main uncertainty in the total water flux derives from precipitation estimates. Direct Estimates of Ocean Fresh Water Transport

Ocean fresh water transports can be directly estimated in the same way as those of heat: by examining the flux budgets of volumes of ocean enclosed by long hydrographic lines. The technique is reliant on being able to determine the steady-state portion of the velocity and the salinity field. For a volume of ocean enclosed by a hydrographic section, salt conservation applies in the steady state, as the transport of salt through the atmosphere and in runoff is negligible: Z Z

r Su dx dz ¼ TIS

½1

where r is the in situ density, S the salinity (e.g., 0.035), v the cross-track velocity (into the volume) and x the along-track distance. TSI represents the total salt transport associated with the interbasin exchange, such as the flow through Bering Strait or the Indonesian Throughflow. Mass conservation is written as Z Z

ru dx dz þ ½P  E þ R ¼ TIM

½2

where E, P and R are the net fluxes into the surface of the ocean volume of, respectively, evaporation, precipitation, and runoff, and TM I the interbasin mass transport. The fresh water part of the above total mass transport is just Z Z

ruð1  SÞ dx dz þ ½P  E þ R ¼ TIM  TIS

½3

The P  E þ R eqn [3] is not a useful approach because the mass transport across an ocean section

FRESHWATER TRANSPORT AND CLIMATE

has uncertainties that are larger than the fresh water transport. However, the errors in the total salt and mass transports across a section are strongly correlated, and these errors can be largely canceled through defining an areal average salinity and its deviation for the section: RR ¯S ¼ R R Sdx dz; S0 ¼ S  S¯ dx dz

½4

For simplicity, we also assume that the interbasin transport of salt occurs at a known salinity, SI, so that the associated salt transport is just TSI ¼ SI  TM T. Combining eqns [1], [2] and [4], the surface fresh water flux can now be written as a simple product of the salinity deviation and the velocity field: ½P  E þ R ¼

TIM S0I 

RR

ru S0 dx dz

S¯

½5

Here the first term on the right is referred to as the ‘leakage’ term associated with the total cross-section transport (the interbasin exchange) and the second term is due to correlations of salinity and velocity across the section, which effect a fresh water transport. In practice, v is found from density and windstress measurements using the geostrophic and Ekman assumptions, and often inverse techniques, while S is directly measured along ocean sections. Deriving error estimates for the product of eqn [5] is challenging, since the statistics of the vS0 term are not well known. Simple scaling arguments were used by Wijffels to show that the expected uncertainty in the direct transport estimates based on eqn [5] might be 0.17 Sv outside of the tropics, but as large as 0.3 Sv in the tropics because of uncertainties in the near-surface wind-driven component. Going beyond the simple scaling argument requires simultaneous time-series of both velocity and salinity over basin scales, measurements that are not likely to be available in the short term. Comparison of Direct and Indirect Transport Estimates

Runoff from the continents must be added to the surface E  P fluxes integrated over the ocean basins in order to predict the ocean transport of fresh water. Despite attempts to catalog the runoff of major rivers, there are few global estimates of runoff. Here Baumgartner and Riechel’s 1975 compilation is utilized, which roughly agrees with the runoff deduced from recent atmospheric analyses. The lack of global runoff datasets makes assessing the errors in the runoff fluxes difficult and adds uncertainty to estimates of ocean fresh water transport.

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All three major ocean basins exchange large amounts of sea water through linking passages: the Southern Ocean, the Indonesian Archipelago, and Bering Strait. As these sea water exchanges are much larger than the fresh water exchanges through the atmosphere, it is simpler to present only the divergent part of the ocean fresh water transport (in contrast to the full transports reported by Wijffels and colleagues in 1992). This is equivalent to removing an unknown constant equal to the Pacific– Indian Throughflow for the Indian and South Pacific Oceans, and the Bering Strait flow in the North Pacific and Atlantic. Only the divergence part of the fresh water transport relative to the entrances of the Bering and Throughflow straits (South of Mindinao in the Philippines) will be presented below. While the size and salinity of the Bering Strait flow are relatively well known, those of the Pacific–Indian Throughflow are not. Hence, investigators have had to make assumptions about them in order to generate an estimate of the fresh water divergence over the South Pacific and Indian Oceans – that is, to calculate the ‘leakage’ term in eqn [5]. As direct estimates for the long-term average Throughflow range between 5 and 10 Sv, it remains a large source of uncertainty in the freshwater budgets of the South Pacific and Indian Oceans.

Basin Balances Most direct transport estimates derive from singlesection or regional analyses of long hydrographic lines, many of which were completed during the World Ocean Circulation Experiment during the 1990s. To date, few truly global syntheses have been made and so we report Wijffels’ year 2000 compilation. The divergent part of the ocean fresh water transport in the three major ocean basins is shown in Figure 3. According to the surface flux estimates, the Indian Ocean north of 301S undergoes net evaporation; that is, the ocean circulation must import fresh water to the Indian basin, from which the atmosphere exports it. Only two latitudes are currently constrained by direct ocean transport estimates in this basin: 321S and 181S, shown by the location of the vertical bars in Figure 3A. Over the large evaporative zone between latitudes 151 and 401S, the indirect transport estimates are fairly consistent (they have similar slopes) and are also in reasonable agreement with the direct estimates. It is in the regions of high precipitation north of 101S and south of 401S that the transport curves diverge, confirming again the large differences between estimates of precipitation over

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FRESHWATER TRANSPORT AND CLIMATE

the ocean. The reader may note in particular the 0.5 Sv variability in the net fresh water divergence north of 181S, where the monsoons are active. The Atlantic Ocean is the best-covered by direct transport estimates (Figure 3B), which are remarkably consistent, except for those at 241N, which are from three occupations of a trans-ocean section spanning 30 years. Nearly all of the major transport maxima are delineated by the direct estimates. Again, the indirect estimates diverge most strongly over regions of high precipitation in the tropics and polar regions. When integrated over the Atlantic between 401S and Bering Strait (we have included the entire Arctic Ocean in the Atlantic), the indirect transport divergences range between 1.0 and 0.0 Sv, while, surprisingly, the direct estimates indicate very little net fresh water divergence over the basin. This difference could be due to an underestimate of runoff to the Arctic/Atlantic as well as an underestimate of P  E to the basin. Problems with biases in the indirect transport estimates are even more pronounced in the Pacific Ocean owing to its huge size (Figure 3C). Indirect transport estimates vary wildly over the South Pacific, where in situ atmospheric data and marine observations are very scarce. Here again, despite the different assumptions made to close the ocean mass balance, Throughflow sizes and different data sets, the direct ocean fresh water transport estimates are quite consistent, and show much less scatter than the indirect estimates. Remarkably, despite a 25 years’ difference between section occupations, two direct estimates of the fresh water divergence made near 301S are indistinguishable.

Interbasin Exchange

Figure 3 The divergent part of the ocean fresh water transport (Sv) in each ocean basin. Indirect estimates based on surface flux climatologies and atmospheric analyses are shown as gray continuous lines while direct ocean estimates at discrete latitudes are shown in black with error bars. The ‘classical’ climatology of Baumgartner and Reichel used by Wijffels et al. in 1992 is shown as the thicker gray line. (A) Indian Ocean relative to 301N; (B) Atlantic/Arctic relative to Bering Strait; and (C) Pacific transport relative to Bering Strait and the Throughflow channels off Mindanao. (Adapted with permission from Wijffels, 2001.)

One of the first attempts to deduce the exchange of fresh water between ocean basins was made by Baumgartner and Reichel. Lacking ‘control’ points for the nondivergent part, they assumed zero fresh water transport across the Atlantic equator, and could thus integrate runoff and surface fluxes to deduce the ocean transport. Using new estimates of transport through Bering Strait, Wijffels et al. in 1992 also used the Baumgartner and Reichel climatology to predict the ocean fresh water transport. They deduced that the Pacific received an excess of precipitation and runoff over evaporation of 0.5 Sv, which was then redistributed through the Indonesian Throughflow, Bering Strait, and Southern Ocean to the more evaporative Atlantic and Indian Oceans. The new direct ocean estimates indicate a quite different interocean fresh water exchange. Figure 3C

FRESHWATER TRANSPORT AND CLIMATE

shows that the fresh water divergence over the Pacific between Bering Strait and 301S is near zero: there is a net balance of evaporation, precipitation, and runoff over that basin. Direct transport estimates for the Atlantic/Arctic also suggest a net divergence of fresh water that is much smaller than previously thought. The new direct estimate of a 0.24 Sv convergence between Bering Strait and 301S in the Atlantic is roughly half that predicted by Baumgartner and Reichel (Figure 3B), while the direct estimates at 401S indicate almost no net divergence over the Atlantic/Arctic. The Indian Ocean direct transport estimates, however, remain consonant with net excess evaporation over precipitation over that basin (Figure 3A). Since the Pacific Ocean and Atlantic/Arctic Oceans cannot be the source of the excess ocean fresh water required to supply the Indian deficit, only one possibility remains: excess precipitation and ice melt over the Southern Ocean. The newly available direct estimates imply a fresh water source of about 0.5 Sv south of 301S, highlighting the importance of the Southern Ocean in the global ocean fresh water balance. Mechanisms of Ocean Fresh Water Transport

In 1981 Stommel and Csanady pointed out that ocean heat and fresh water transport is related to the rates of conversion of water from one temperature– salinity class to another. They went further and attempted to model this process with salinity as a simple function of temperature. Recent analyses of fresh water transports across ocean sections and in general circulation models showed this assumption to be wrong, though the underlying idea remains powerful, as it links surface fluxes to water mass inventories and exchanges in temperature–salinity space. Stommel and Csanady’s approach has also been recast in terms of density classes of water, which expresses the competition between surface fluxes and interior ocean mixing in controlling exchange between density classes. The challenge in analysing ocean sections will be in distinguishing the water mass conversion at the surface from that due to internal mixing. Use of the fresh water fluxes will be critical. How and which elements of the circulation achieve the ocean fresh water transport is also of great interest. The definition of a tracer transport mechanism across an ocean section is still, however, somewhat ad hoc. In their pioneering work in 1982, Hall and Bryden chose to form zonal averages (and deviations) of velocity and properties on pressure surfaces. They termed the resulting products the

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‘overturning’ component of tracer transport, while the residual (associated with the correlation of velocity and tracer at a pressure level) was termed the ‘horizontal’ or gyre component. A similar decomposition can be carried out within density layers and, as density is largely determined by temperature, more closely relates back to Stommel and Csanady’s suggestion. Fresh water divergence is also achieved by the interbasin flows, the ‘leakage’ term in eqn [5]. Unfortunately, few detailed decompositions of the ocean fresh water transport across hydrographic lines are available; those that are, however, reveal interesting mechanisms and cases where different circulation components can provide canceling fresh water transports. For example, at 101N in the Pacific, the small net fresh water divergence over the Pacific relative to Bering Strait is due to a balance between three major mechanisms: (1) net export of very fresh water through Bering Strait to the Arctic; (2) a northward fresh water transport by a shallow meridional circulation where the northward Ekman transport in the upper 100 m is fresh and there is a compensating salty southward thermocline flow (100–300 m); (3) southward fresh water transport is achieved by a 300–450 m deep horizontal gyre where salty South Pacific waters flow north in the eastern Pacific and fresh intermediate water flows south in the western Pacific. The deep and bottom water circulations in the low-latitude Pacific achieve little net fresh water transport. In the Indian Ocean a large net evaporation of 0.31 Sv is estimated to occur north of 321S. Three mechanisms act to import this fresh water to the ocean basin: (1) a leakage term associated with the inflow of fresh Indonesian Archipelago waters that are evaporated and leave across 321S as salty thermocline waters; (2) upwelling of deep and intermediate waters and their export as saltier thermocline waters; and (3) horizontal inflow of fresh Antarctic intermediate water in the east that leaves as saltier intermediate water in the west. This latter transport mechanism was also found to be an important fresh water transport mechanism at 321S in the Pacific. It is likely that the recirculation of subtropical mode and intermediate waters between the Southern Ocean and the Southern Hemisphere subtropical gyres may be the single most important mechanism for balancing the large net flux (0.57 Sv) received by the ocean south of 301S from the atmosphere and ice flows. The ability of ocean general circulation models to reproduce the estimated fresh water transports and their mechanisms will be a stringent test of the models’ realism. In ocean-only models, surface-flux

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FRESHWATER TRANSPORT AND CLIMATE

forcing will determine the net equilibrium transports (unless the problematic relaxation boundary conditions are used), but internal model physics will determine how this transport is achieved. It is also noteworthy that the fresh water transports effected by the subtropical gyres through the vS0 term in eqn [5] is not accounted for in simple box models of the global thermohaline circulation, which allow only a single salinity and temperature to represent the major water mass pools. Such models must fold this upper-ocean gyre transport into the deep-water component of the global thermohaline circulation, confusing the role of fresh water forcing as a control on the circulation. More detailed studies of how the ocean transports fresh water are required to isolate the relative roles of the shallow wind-driven gyres and the deep circulation in balancing the surface forcing.

Global Budgets Direct ocean transport estimates are available in all ocean basins at five latitude bands, which allows the total global meridional fresh water transport to be examined. Since few of the major rivers flow meridionally, the zonally integrated meridional ocean transport of fresh water is largely equal and opposite to that in the atmosphere. Therefore, these estimates can be compared with direct estimates of atmospheric moisture transport or those produced by atmospheric general circulation models (Figure 4). Based on a comparison with Oort and Piexoto’s 1983 global atmospheric estimates, international model intercomparison studies concluded that most atmospheric models overestimate the poleward transport of moisture. Direct ocean measurements are still too sparse to shed conclusive light on this issue. In the northern hemisphere high latitudes, the direct estimates agree better with Oort and Piexoto’s than those from Atmospheric Model Intercomparison Project (AMIP), while in the tropics and southern hemisphere the opposite is true. To more usefully constrain the total meridional moisture transport in the atmosphere, more direct ocean fresh water transport estimates are needed as well as a better estimate of their errors – those shown in Figure 4 are based on simple scale arguments and so are rather conservative.

Future Directions Despite many new estimates of surface fresh water fluxes over the oceans having been made, their use in assessing atmospheric models and forcing ocean

Figure 4 Various estimates of the total ocean meridional fresh water transport (Sv). Estimates based on ocean data are shown as black circles; thin lines are indirect estimates based on two recent surface flux climatologies with the Baumgartner and Reichel’s 1975 continental runoff added. Gray shaded is the interquartile range for the atmospheric models participating in AMIP. Oort and Piexoto’s 1983 direct atmospheric estimate is marked as x-x. The two surface flux climatologies are as follows. SOC: Josey SA, Kent EC, Oakley D and Taylor PK (1996) A new global air-sea heat and momentum flux climatology. International WOCE Newsletter 24: 3–5. COADS: da Silva AM, Young C and Levitus S (1994) Atlas of Surface Marine Data, vol. 1: Algorithms and procedures. NOAA Atlas NESDIS 6. (Reproduced with permission from Wijffels, 2001.)

models will depend on the accuracy of their basinwide integrals. These can only be assessed by fresh water transport estimates from ocean data. Direct ocean fresh water transports are not as well reported or analyzed as their companion heat transports. While most estimates are fairly consonant with each other and with error estimates based on simple scaling arguments, others are quite anomalous. Without a detailed breakdown of the mechanisms making up these transports, tracking down the source of these differences is next to impossible. There may be enormous potential in the idea of ‘tuning’ surface flux products by using direct ocean estimates to remove flux biases. This might in turn lead to products that are accurate enough to directly force ocean climate models with confidence, and thus allow meaningful use of salinity as an ocean model diagnostic. Monitoring for changes in ocean fresh water storage may also now be feasible with the availability of salinity sensors that are stable over long deployments on floats and buoys. Estimates of ocean fresh water transport will remain reliant on transport-resolving temperature– salinity sections, until such time as data-assimilating

FRESHWATER TRANSPORT AND CLIMATE

ocean models are sufficiently accurate to capture the essential ocean fresh water transport mechanisms.

See also Deep Convection. Heat and Momentum Fluxes at the Sea Surface. Heat Transport and Climate. Upper Ocean Heat and Freshwater Budgets.

Further Reading Baumgartner A and Reichel E (1975) The World Water Balance. New York: Elsevier. Hall MM and Bryden HL (1982) Direct estimates and mechanisms of ocean heat transport. Deep-Sea Research 29: 339--359. Oort AH and Peixoto’s JP (1983) Global angular momentum and energy balance requirements from observations. Advances in Geophysics 25: 355--490.

159

Schmitt R (1995) The ocean component of the global water cycle. Review of Geophysics supplement: 1395--1409. Schmitt R (1999) The ocean’s response to the freshwater cycle. In: Browning KA and Gurney RJ (eds.) Global Energy and Water Cycles. Cambridge: Cambridge University Press. Stommel HM and Csanady GT (1980) A relation between the TS curve and global heat and atmospheric water transports. Journal of Geophysical Research 85: 495--501. Trenberth KE and Guillemot C (1999) Estimating evaporation-minus-precipitation as a residual of the atmospheric water budget. In: Browning KA and Gurney RJ (eds.) Global Energy and Water Cycles. Cambridge: Cambridge University Press. Wijffels SE, Schmitt RW, Bryden HL, and Stigebrandt A (1992) Transport of freshwater by the oceans. Journal of Physical Oceanography 22: 155--162. Wijffels SE (2001) Ocean transport of freshwater. In: Church J, Gould J, and Siedler G (eds.) Ocean Circulation and Climate. London: Academic Press.

AIR–SEA GAS EXCHANGE a wavy water surface are still not known. A number of new imaging techniques are described which give direct insight into the transfer processes and promise to trigger substantial theoretical progress in the near future.

B. Ja¨hne, University of Heidelberg, Heidelberg, Germany & 2009 Elsevier Ltd. All rights reserved.

Introduction

Theory

The exchange of inert and sparingly soluble gases, including carbon dioxide, methane, and oxygen, between the atmosphere and oceans is controlled by a 20–200-mm-thick boundary layer at the top of the ocean. The hydrodynamics in this layer is significantly different from boundary layers at rigid walls since the orbital motion of the waves is of the same order as the velocities in the viscous boundary layer. Laboratory and field measurements show that wind waves and surfactants significantly influence the gastransfer process. Because of limited experimental techniques, the details of the mechanisms and the structure of the turbulence in the boundary layer at

Mass Boundary Layers

Table 1

jc ¼ ðD þ Kc ðzÞÞrc

½1

Diffusion coefficients for various gases and volatile chemical species in deionized water and in some cases in seawater

Species

Heat 3 Heb,c 4 He 4 Hea Ne Kr Xe 222 Rnb H2 H2a CH4 CO2 DMSb CH3Brb F12b (CCl2F2) F11b (CCl3F) SF6b a

The transfer of gases and volatile chemical species between the atmosphere and oceans is driven by a concentration difference and the transport by molecular and turbulent motion. Both types of transport processes can be characterized by ‘diffusion coefficients’, denoted by D and Kc, respectively (Table 1). The resulting flux density jc is proportional to the diffusion coefficient and the concentration gradient. Thus,

Molecular mass

3.02 4.00 20.18 83.80 131.30 222.00 2.02 16.04 44.01 62.13 94.94 120.91 137.37 146.05

A (10  5 cm2 s  1)

379.2 941 818 886 1608 6393 9007 15 877 3338 1981 3047 5019 2000 3800 4100 3400 2900

Ea (kJ mol  1)

2.375 11.70 11.70 12.02 14.84 20.20 21.61 23.26 16.06 14.93 18.36 19.51 18.10 19.10 20.50 20.00 19.30

s(Fit) (%)

2.1 2.1 1.8 3.5 1.6 3.5 11 1.6 4.3 2.7 1.3

Diffusion coefficient (10  5 cm2 s  1) 5 1C

15 1C

25 1C

35 1C

135.80 5.97 5.10 4.86 2.61 1.02 0.77 0.68 3.17 3.05 1.12 1.07 0.80 0.98 0.58 0.60 0.69

140.72 7.12 6.30 5.88 3.28 1.41 1.12 0.96 4.10 3.97 1.48 1.45 1.05 1.31 0.79 0.81 0.92

145.48 8.39 7.22 7.02 4.16 1.84 1.47 1.34 5.13 4.91 1.84 1.91 1.35 1.71 1.05 1.07 1.20

150.08 9.77 8.48 8.03 4.82 2.40 1.94 1.81 6.23 5.70 2.43 2.43 1.71 2.20 1.37 1.38 1.55

In seawater. Values of diffusion coefficients from fit, not measured values. c Set 15% higher than 4He. Columns 3 and 4 contain the parameters for the fit of the diffusion coefficient: D ¼ A exp[  Ea/(RT )], the last four columns the diffusion coefficients for 5, 15, 25, and 35 1C. Data collected from Ja¨hne B, Heinz G, and Dietrich W (1987) Measurement of the diffusion coefficients of sparingly soluble gases in water. Journal of Geophysical Research 92: 10767–10776; and King DB, De Bryun WJ, Zheng M, and Saltzman ES (1995) Uncertainties in the molecular diffusion coefficient of gases in water for use in the estimation of air–sea exchange. In: Ja¨hne B and Monahan E (eds.) Air–Water Gas Transfer, pp. 13–22. Hanau: Aeon. b

160

161

AIR–SEA GAS EXCHANGE

In a stationary homogeneous case and without sinks and sources by chemical reactions, the flux density j is in vertical direction and constant. Then integration of [1] yields vertical concentration profiles Z 0

Zr

1 dz D þ Kc ðzÞ

The molecular diffusion coefficient is proportional to the velocity of the molecules and the free length between collisions. The same concept can be applied to turbulent diffusion coefficients. Far away from the interface, the free length (called ‘mixing length’) is set proportional to the distance from the interface and the turbulent diffusion coefficient Kc for mass transfer is Kc ¼

k u z Sct

Air-side mass boundary layer (100–1000 μm)

½2 Kc < D

Ca

C Water C w s =  Ca s surface Aqueous mass boundary layer (20–200 μm)

s

Viscous boundary layer (600–2000 μm)

Water phase

zr

Cw

Reference level b

z

½3

where k ¼ 0.41 is the von Ka´rma´n constant, u , the friction velocity, a measure for the velocity fluctuations in a turbulent flow, and Sct ¼ Km/Kc the turbulent Schmidt number. Closer to the interface, the turbulent diffusion coefficients are decreasing even faster. Once a critical length scale l is reached, the Reynolds number Re ¼ u l=v (n is the kinematic viscosity, the molecular diffusion coefficient for momentum) becomes small enough so that turbulent motion is attenuated by viscosity. The degree of attenuation depends on the properties of the interface. At a smooth solid wall, Kcpz3, at a free water interface it could be in the range between Kcpz3 and Kcpz2 depending on surface conditions. Boundary layers are formed on both sides of the interface (Figure 1). When the turbulent diffusivity becomes equal to the kinematic viscosity, the edge of the ‘viscous boundary layer’ is reached. As the name implies, this layer is dominated by viscous dissipation and the velocity profile becomes linear because of a constant diffusivity. The edge of the ‘mass boundary layer’ is reached when the turbulent diffusivity becomes equal to the molecular diffusivity. The relative thickness of both boundary layers depends on the dimensionless ratio Sc ¼ v/D (Schmidt number). The viscous and mass boundary layers are of about the same thickness in the air, because values of D for various gaseous species and momentum are about the same (Scair is 0.56 for H2O, 0.63 for heat, and 0.83 for CO2). In the liquid phase the situation is completely different. With Schmidt numbers in the range from 100 to 3000 (Figure 2 and Table 2), molecular diffusion for a dissolved volatile chemical species is two to three orders of magnitude slower than diffusion of momentum. Thus the mass boundary layer

z∼

Kc < υ

Figure 1 Schematic graph of the mass boundary layers at a gas–liquid interface for a tracer with a solubility a ¼ 3. 1

102

10

103

104

0

105

0

Air-side control

H2O

Atrazine

30

0 Pentachlorophenol

Momentum

DDT

103

3)

Sm

Transition zone

n= ce (

2/ (n =

SO2 (pH 60

½11

This equation establishes the basic analogy between momentum transfer and gas exchange. The transfer coefficient is proportional to the friction velocity in water, which describes the shear stress (tangential force per unit area) t ¼ rw u2w applied by the wind field at the water surface. Assuming stress continuity at the water surface, the friction velocity in water is related to the friction velocity in air by  uw ¼ ua

ra rw

1=2 ½12

The friction velocity in air, ua , can further be linked via the drag coefficient to the wind speed UR at a reference height: cD ¼ ðua =UR Þ2 . Depending on the roughness of the sea surface, the drag coefficient has values between 0.8 and 2.4  10  3. In this way the gas exchange rate is directly linked to the wind speed. The gas exchange further depends on the chemical species and the water temperature via the Schmidt number. Gas Exchange at Rough and Wavy Water Surfaces

A free water surface is neither solid nor is it smooth as soon as short wind waves are generated. On a free water surface velocity fluctuations are possible. Thus, there can be convergence or divergence zone at the surface; surface elements may be dilated or contracted. At a clean water surface dilation or

164

AIR–SEA GAS EXCHANGE

contraction of a surface element does not cause restoring forces, because surface tension only tries to minimize the total free surface area, which is not changed by this process. As a consequence of this hydrodynamic boundary condition, the turbulent diffusivity normal to the interface can now increase with the distance squared from the interface, Kcpz2. Then 1 kw ¼ uw Sc1=2 b

½13

where b is a dimensionless constant. In comparison to the smooth case in [11], the exponent n of the Schmidt number drops from  2/3 to  1/2. This increases the transfer velocity for a Schmidt number of 600 by about a factor of 3. The total enhancement depends on the value of the constant b. Influence of Surface Films

A film on the water surface creates pressure that works against the contraction of surface elements. This is the point at which the physicochemical structure of the surface influences the structure of the near-surface turbulence as well as the generation of waves. As at a rigid wall, a strong film pressure at the surface maintains a two-dimensional continuity at the interface just as at a rigid wall. Therefore, [11] should be valid for a smooth film-covered water surface and has indeed been verified in wind/ wave tunnel studies as the lower limit for the transfer velocity. As a consequence, both [11] and [13] can only be regarded as limiting cases. A more general approach is required that has not yet been established. One possibility is a generalization of [11] and [13] to kw ¼ uw

1 ScnðsÞ bðsÞ

½14

where both b and n depend on dimensionless parameters describing the surface conditions s. Even films with low film pressure may easily decrease the gas transfer rate to half of its value at clean water surface conditions. But still too few measurements at sea are available to establish the influence of surfactants on gas transfer for oceanic conditions more quantitatively. Influence of Waves

Wind waves cannot be regarded as static roughness elements for the liquid flow because their characteristic particle velocity is of the same order of magnitude as the velocity in the shear layer at the surface.

This fact causes a basic asymmetry between the turbulent processes on the air and on the water sides of the interface. Therefore, the wave effect on the turbulent transfer in the water is much stronger and of quite different character than in the air. This basic asymmetry can be seen if the transfer velocity for CO2 is plotted against the transfer velocity for water vapor (Figure 3(a)). At a smooth water surface the points fall well on the theoretical curve predicted by the theory for a smooth rigid wall. However, as soon as waves occur at the water surface, the transfer velocity of CO2 increases significantly beyond the predictions. Even at high wind speeds, the observed surface increase is well below 20%. When waves are generated by wind, energy is not only transferred via shear stress into the water but a second energy cycle is established. The energy put by the turbulent wind into the wave field is transferred to other wave numbers by nonlinear wave–wave interaction and finally dissipated by wave breaking, viscous dissipation, and turbulence. The turbulent wave dissipation term is the least-known term and of most importance for enhanced near-surface turbulence. Evidence for enhanced turbulence levels below wind waves has been reported from field and laboratory measurements. Experimental results also suggest that the gas transfer rate is better correlated with the ‘mean square slope’ of the waves as an integral measure for the nonlinearity of the wind wave field than with the wind speed. It is not yet clear, however, to what extent ‘microscale wave breaking’ can account for the observed enhanced gas transfer rates. A gravity wave becomes unstable and generates a steep train of capillary waves at its leeward face and has a turbulent wake. This phenomenon can be observed even at low wind speeds, as soon as wind waves are generated. At higher wind speeds, the frequency of microscale wave breaking increases. Influence of Breaking Waves and Bubbles

At high wind speeds, wave breaking with the entrainment of bubbles may enhance gas transfer further. This phenomenon complicates the gas exchange between atmosphere and the oceans considerably. First, bubbles constitute an additional exchange surface. This surface is, however, only effective for gases with low solubility. For gases with high solubility, the gas bubbles quickly come into equilibrium so that a bubble takes place in the exchange only for a fraction of its lifetime. Thus, bubble-mediated gas exchange depends – in contrast to the exchange at the free surface – on the solubility of the gas tracer.

AIR–SEA GAS EXCHANGE

(a)

165

(b)

+ with waves (unlimited fetch) • no waves

0.001 0.80

10−2 Schmidt number exponent n

10−4

0.75 0.70

0.65

0.65

0.60

0.60

0.55

0.55

0.50

0.50

0.45

0.45

0.40 0.001

1

0.3 0.80

0.70

kH2O (cm s−1) 0.1

0.1 Small circular facility Large circular facility

0.75

kCO2 (Sc = 600) (cm s−1)

10−3

0.01

0.01

0.1

0.40 0.3

Mean square surface slope s 2

10

Figure 3 (a) Transfer velocity of CO2 plotted against the transfer velocity of water vapor as measured in a small circular wind wave facility. (b) Schmidt number exponent n as a function of the mean square slope. (a) From Ja¨hne B (1980) Zur Parameterisierung des Gasaustausches mit Hilfe von Laborexperimenten. Dissertation, University of Heidelberg. (b) From Ja¨hne B and HauXecker H (1998) Air–water gas exchange. Annual Review of Fluid Mechanics 30: 443–468.

Second, bubble-mediated gas transfer shifts the equilibrium value to slight supersaturation due to the enhanced pressure in the bubbles by surface tension and hydrostatic pressure. Third, breaking waves also enhance near-surface turbulence during the breaking event and the resurfacing of submerged bubbles. Experimental data are still too sparse for the size and depth distribution of bubbles and the flux of the bubbles through the interface under various sea states for a sufficiently accurate modeling of bubblemediated air–sea gas transfer and thus a reliable estimate of the contribution of bubbles to the total gas transfer rate. Some experiments from wind/wave tunnels and the field suggest that significant enhancements can occur, other experiments could not observe a significant influence of bubbles. Empiric Parametrization

Given the lack of knowledge all theories about the enhancement of gas transfer by waves are rather speculative and are not yet useful for practical application. Thus, it is still state of the art to use semiempiric or empiric parametrizations of the gas exchange rate with the wind speed. Most widely used is the parametrization of Liss and Merlivat. It identifies three physically well-defined regimes (smooth,

wave-influenced, and bubble-influenced) and proposes a piecewise linear relation between the wind speed U and the transfer velocity k: k ¼ 106 8 > 0:472UðSc=600Þ2=3 ; Ur3:6 m s1 > <  7:917ðU  3:39ÞðSc=600Þ1=2 ; U > 3:6 m s1 and Ur13 m s1 > > : 16:39ðU  8:36ÞðSc=600Þ1=2 ; U > 13 m s1

½15 At the transition between the smooth and wavy regime, a sudden artificial jump in the Schmidt number exponent n from 2/3 to 1/2 occurs. This actually causes a discontinuity in the transfer rate for Schmidt number unequal to 600. The empiric parametrization of Wanninkhof simply assumes a quadratic increase of the gas transfer rate with the wind speed: k ¼ 0:861  106 ðs m1 ÞU2 ðSc=600Þ1=2 ½16 Thus, this model has a constant Schmidt number exponent n ¼ 1/2. The two parametrizations differ significantly (see Figure 4). The Wanninkhof parametrization predicts significantly higher values. The discrepancy between the two parametrizations

166

AIR–SEA GAS EXCHANGE

80 70

14

60

C

SF6 − 3He 222

k 600 (cm h−1)

where Fw and hw are the surface area and the effective height Vw/Fw of a well-mixed water body, respectively. The time constant tw ¼ hw/k is in the order of days to weeks. It is evident that the transfer velocities obtained in this way provide only values integrated over a large horizontal length scales and timescales in the order of tw. Thus, a parametrization of the transfer velocity is only possible under steady-state conditions over extended periods. Moreover, the mass balance contains many other sources and sinks besides air–sea gas exchange and thus may cause severe systematic errors in the estimation of the transfer velocity. Consequently, mass balance methods are only poorly suited for the study of the mechanisms of air–water gas transfer.

Wanninkhof relationship Liss−Merlivat relationship

50

Rn

Heat (CFT)

40 30 20 10 0

0

5

10 15 Wind speed u10 (m s−1)

20

Tracer Injection

Figure 4 Summary of gas exchange field data normalized to a Schmidt number of 600 and plotted vs. wind speed together with the empirical relationships of Liss and Merlivat and Wanninkhof. Adapted from Ja¨hne B and HauXecker H (1998) Air–water gas exchange. Annual Review of Fluid Mechanics 30: 443–468.

(up to a factor of 2) mirrors the current uncertainty in estimating the air–sea gas transfer rate.

Experimental Techniques and Results Laboratory Facilities

Laboratory facilities play an important role in the investigation of air–sea gas transfer. Only laboratory studies allow a systematic study of the mechanisms and are thus an indispensible complement to field experiments. Almost all basic knowledge about gas transfer has been gained by laboratory experiments in the past. Among other things this includes the discovery of the influence of waves on air–water gas exchange (Figure 3(a)) and the change in the Schmidt number exponent (Figure 3(b)). Many excellent facilities are available worldwide (Table 3). Some of the early facilities are no longer operational or were demolished. However, some new facilities have also been built recently which offer new experimental opportunities for air–water gas transfer studies. Geochemical Tracer Techniques

The first oceanic gas exchange measurements were performed using geochemical tracer methods such as the 14C, 3He/T, or 222Rn/226Ra methods. The volume and time-average flux density is given by mass balance of the tracer concentration in a volume of water Vw: Vw c_w ¼ Fw j or j ¼ hw c_w

½17

The pioneering lake studies for tracer injection used sulfur hexafluoride (SF6). However, the tracer concentration decreases not only by gas exchange across the interface but also by horizontal dispersion of the tracer. This problem can be overcome by the ‘dual tracer technique’ (Watson and co-workers) simultaneously releasing two tracers with different diffusivities (e.g., SF6 and 3He). When the ratio of the gas transfer velocities of the two tracers is known, the dilution effect by tracer dispersion can be corrected, making it possible to derive gas transfer velocities. But the basic problem of mass balance techniques, that is, their low temporal resolution, remains also with artificial tracer approaches. Eddy Correlation Flux Measurements

Eddy correlation techniques are used on a routine basis in micrometeorology, that is, for tracers controlled by the boundary layer in air (momentum, heat, and water vapor fluxes). Direct measurements of the air–sea fluxes of gas tracers are very attractive because the flux densities are measured directly and have a much better temporal resolution than the mass balance-based techniques. Unfortunately, large experimental difficulties arise when this technique is applied to gas tracers controlled by the aqueous boundary layer. The concentration difference in the air is only a small fraction of the concentration difference across the aqueous mass boundary layer. But after more than 20 years of research has this technique delivered useful results. Some successful measurements under favorable conditions have been reported and it appears that remaining problems can be overcome in the near future. The Controlled Flux Technique

The basic idea of this technique is to determine the concentration difference across the mass boundary

Table 3

Comparison of the features of some major facilities for small-scale air–sea interaction studies (operational facilities are typeset in boldface)

Length (mean perimeter) (m) Width of water channel (m) Outer diameter (m) Inner diameter (m) Total height (m) Max. water depth (m) Water surface area (m2) Water volume (m3) Maximum wind speed (m s  1) Suitable for sea water Wave maker Water current generator (m s  1) Water temperature control (1C) Air temperature control (1C) Air humidity control Gastight air space

HH

M

D

SIO

C

UM

W

SU

HD1

HD2

HD3

HD4

15 1.8

40 2.6

100 8.0

40 2.4

33 0.76

15 1.0

18.3 0.91

2.0 0.8 104 83 15 N Y Y Y Y Y N

3.0 0.8 800 768 15 N Y Y N N N N

2.4 1.5 96 144 12 Y Y N N N N N

0.85 0.25 24.8 8 25 N Y 70.6 Y N N Y

1.0 0.5 15 10 30 Y Y 70.5 Y N N Y

1.22 0.76 16.7 ? 25 N Y 70.5 Y Y N Y

1.57 0.10 0.60 0.40 0.50 0.08 0.16 0.01 11 Y N N 5–35 5–35 Y Y

11.6 0.20 4.0 3.4 0.70 0.25 3.5 0.87 12 Y N N N N N Y

29.2 0.62 9.92 8.68 2.40 1.20 18.0 20.7 15 Y N o 0.6 5–35 5–35 Y Y

3.90 0.37

1.5 0.3 27 8 25 N N Y N N N N

119 2.0 40 36 5.6 3.0 239 716 19 Y N N N N N N

0.33 0.10 1.44 0.14 8 Y N o 0.1 Y N N Y

HH, Bundesanstalt fu¨r Wasserbau, Hamburg; M, IMST, Univ. Marseille, France; D, Delft Hydraulics, Delft, The Netherlands (no longer operational); SIO, Hydraulic Facility, Scripps Institution of Oceanography, La Jolla, USA; C, Canada Center for Inland Waters (CCIW); UM, University of Miami; W, NASA Air–Sea Interaction Research Facility, Wallops; SU, Storm basin, Marine Hydrophysical Institute, Sevastopol, Ukraine (no longer operational), HD1, Small annular wind/wave flume, Univ. Heidelberg (no longer operational), HD2, Large annular wind/wave flume, Univ. Heidelberg (dismantled); HD3, Aeolotron, Univ. Heidelberg (HD3, in operation since June 2000); HD4, Teflon-coated small Heidelberg linear wind/wave flume (N ¼ No, Y ¼ Yes).

AIR–SEA GAS EXCHANGE

14

28

13 12

Wind (20 Hz)

26

Wind (60 s average)

24

11 Gas transfer rates 10

(k600, CFT data)

22 20

9

18

8

16

7

14

6

12

5

10

4

8

3

6

2

4

1

2

0 03:00

03:30

04:00

k600 (cm h−1)

Wind (u10) (m s−1)

168

0 04:30

Time (UTC) (hh:mm) Figure 5 Wind speeds and gas transfer velocities computed with the controlled flux technique (CFT) during the 1995 MBL/CoOP West Coast experiment (JD133) for a period of 90 min. The transfer velocities are normalized to Schmidt number 600 and averaged over 4 min each. From Ja¨hne B and HauXecker H (1998) Air–water gas exchange. Annual Review of Fluid Mechanics 30: 443–468.

layer when the flux density j of the tracer across the interface is known. The local transfer velocity can be determined by simply measuring the concentration difference Dc across the aqueous boundary layer (cold surface skin temperature) according to [4] with a time constant t˜ for the transport across the boundary layer [8]. This technique is known as the ‘controlled flux technique’ (CFT). Heat proves to be an ideal tracer for the CFT. The temperature at the water surface can then be measured with high spatial and temporal resolution using IR thermography. A known and controllable flux density can be applied by using infrared radiation. Infrared radiation is absorbed in the first few 10 mm at the water surface. Thus, a heat source is put right at top of the aqueous viscous boundary layer. Then the CFT directly measures the waterside heat transfer velocity. A disadvantage of the CFT is that the transfer velocity of gases must be extrapolated from the transfer velocity of heat. The large difference in the Schmidt number (7 for heat, 600 for CO2) casts some doubt whether the extrapolation to so much higher Schmidt numbers is valid. Two variants of the technique proved to be successful. Active thermography uses a CO2 laser to heat a spot of several centimeters in diameter on the water surface. The heat transfer rates are estimated

from the temporal decay of the heated spot. Passive thermography uses the naturally occurring heat fluxes caused by latent heat flux jl, sensible heat flux js, and long wave emission of radiation jr . The net heat flux jn ¼ jl þ js þ jr results according to [4] in a temperature difference across the interface of DT ¼ jh/(rcpkh). Because of the turbulent nature of the exchange process any mean temperature difference is associated with surface temperature fluctuations which can be observed in thermal images. With this technique the horizontal structure of the boundary layer turbulence can be observed. Surface renewal is directly observable in the IR image sequences, which show patches of fluid being drawn away from the surface. With some knowledge about the statistics of the temperature fluctuations, the ‘temperature difference’ DT across the interface as well as the time constant t˜ of heat transfer can be computed from the temperature distribution at the surface. Results obtained with this technique are shown in Figure 5 and also in the overview graph (Figure 4). Summary of Field Data

A collection of field data is shown in Figure 4. Although the data show a clear increase of the transfer

AIR–SEA GAS EXCHANGE

velocity with wind speed, there is substantial scatter in the data that can only partly be attributed to uncertainties and systematic errors in the measurements. Thus, in addition, the field measurements reflect the fact that the gas transfer velocity is not simply a function of the wind speed but depends significantly on other parameters influencing nearsurface turbulence, such as the wind-wave field and the viscoelastic properties of the surface film.

Outlook In the past, progress toward a better understanding of the mechanisms of air–water gas exchange was hindered by inadequate measuring technology. However, new techniques have become available and will continue to become available that will give a direct insight into the mechanisms under both laboratory and field conditions. This progress will be achieved by interdisciplinary research integrating different research areas such as oceanography, micrometeorology, hydrodynamics, physical chemistry, applied optics, and image processing. Optical- and image-processing techniques will play a key role because only imaging techniques give direct insight to the processes in the viscous, heat, and mass boundary layers on both sides of the air– water interface. Eventually all key parameters including flow fields, concentration fields, and waves will be captured by imaging techniques with sufficient spatial and temporal resolution. The experimental data gained with such techniques will stimulate new theoretical and modeling approaches.

Nomenclature 2

D (cm s  1) jc (Mol cm  2 s  1) k (cm s  1) Kc (cm2 s  1) R (cm  1 s) Sc ¼ n/D t˜ ¼ z˜/k (s) u (cm s  1) z˜ (cm) a n (cm2 s  1)

molecular diffusion coefficient concentration flux density transfer velocity turbulent diffusion coefficient transfer resistance Schmidt number boundary layer time constant friction velocity boundary layer thickness dimensionless solubility kinematic viscosity

Air–Sea Transfer: N2O, NO, CH4, CO. Breaking Waves and Near-Surface Turbulence. Bubbles. Surface Gravity and Capillary Waves.

Further Reading Borger AV and Wanninkhof R (eds.) (2007) Special Issue: 5th International Symposium on Gas Transfer at Water Surfaces. Journal of Marine Systems 66: 1--308. Businger JA and Kraus EB (1994) Atmosphere–Ocean Interaction. New York: Oxford University Press. Donelan M, Drennan WM, Saltzman ES, and Wanninkhof R (eds.) (2001) Gas Transfer at Water Surfaces. Washington, DC: American Geophysical Union. Duce RA and Liss PS (eds.) (1997) The Sea Surface and Global Change. Cambridge, UK: Cambridge University Press. Garbe C, Handler R, and Ja¨hne B (eds.) (2007) Transport at the Air–Sea Interface, Measurements. Models, and Parameterization. Berlin: Springer. Ja¨hne B (1980) Zur Parameterisierung des Gasaustausches mit Hilfe von Laborexperimenten. Dissertation, University of Heidelberg. Ja¨hne B and HauXecker H (1998) Air–water gas exchange. Annual Review of Fluid Mechanics 30: 443--468. Ja¨hne B, Heinz G, and Dietrich W (1987) Measurement of the diffusion coefficients of sparingly soluble gases in water. Journal of Geophysical Research 92: 10767--10776. Ja¨hne B and Monahan E (eds.) (1995) Air–Water Gas Transfer. Hanau: Aeon. King DB, De Bryun WJ, Zheng M, and Saltzman ES (1995) Uncertainties in the molecular diffusion coefficient of gases in water for use in the estimation of air–sea exchange. In: Ja¨hne B and Monahan E (eds.) Air–Water Gas Transfer, pp. 13--22. Hanau: Aeon. Liss PS and Merlivat L (1986) Air–sea gas exchange rates: Introduction and synthesis. In: Buat-Menard P (ed.) The Role of Air–Sea Exchange in Geochemical Cycles, pp. 113--127. Dordrecht: Reidel. McGilles WR, Asher WE, Wanninkhof R, and Jessup AT (eds.) (2004). Special Issue: Air Sea Exchange. Journal of Geophysical Research 109. Wanninkhof R (1992) Relationship between wind speed and gas exchange over the ocean. Journal of Geophysical Research 97: 7373--7382. Wilhelms SC and Gulliver JS (eds.) (1991) Air–Water Mass Transfer. New York: ASCE.

Relevant Websites See also Air–Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-Methane Hydrocarbons, Organo-Halogens.

169

http://www.ifm.zmaw.de – Institute of Oceanography, Universita¨t Hamburg. http://www.solas-int.org – SOLAS.

AIR–SEA TRANSFER: DIMETHYL SULFIDE, COS, CS2, NH4, NON-METHANE HYDROCARBONS, ORGANO-HALOGENS J. W. Dacey, Woods Hole Oceanographic Institution, Woods Hole, MA, USA H. J. Zemmelink, University of Groningen, Haren The Netherlands Copyright & 2001 Elsevier Ltd.

The oceans, which cover 70% of Earth’s surface to an average depth of 4000 m, have an immense impact on the atmosphere’s dynamics. Exchanges of heat and momentum, water and gases across the sea surface play major roles in global climate and biogeochemical cycling. The ocean can be thought of as a vast biological soup with myriad processes influencing the concentrations of gases dissolved in the surface waters. The quantities of mass flux across the surface interface, though perhaps small on a unit area basis, can be very important because of the extent of the ocean surface and the properties of the gases or their decomposition products in the atmosphere. Gas exchange across the sea–air surface depends, in part, on differences in partial pressures of the gases between the ocean surface and the atmosphere. The partial pressure of a gas in the gas phase can be understood in terms of its contribution to the pressure in the gas mixture. So the partial pressure of O2, for example, at 0.21 atm means that at 1 atmosphere total pressure, O2 is present as 21% of the gas, or mixing, volume. Trace gases are present in the atmosphere at much lower levels, usually expressed as parts per million (106 atm), parts per billion (109 atm) or parts per trillion (1012 atm, pptv). Dimethylsulfide (DMS), when present at 100 pptv, accounts for about 100 molecules per 1012 molecules of mixed gas phase, or about 1010 of the gas volume. In solution, a dissolved trace gas in equilibrium with the atmosphere would have the same partial pressure as the gas in the air. Its absolute concentration in terms of molecules or mass per unit volume of water depends on its solubility. Gas solubility varies over many orders of magnitude depending on the affinity of water for the gas molecules and the volatility of the gas. Gases range widely in their solubility in sea water, from the permanent gases like nitrogen (N2), oxygen (O2), nitrous oxide (N2O) and methane (CH4) that have a low solubility in sea

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water to the moderately soluble carbon dioxide (CO2) and dimethylsulfide (CH3)2S, to highly soluble ammonia (NH3 and its ionized form NHþ 4 ) and sulfur dioxide (SO2). Sulfur dioxide is more than 106 times more soluble than O2 or CH4. Using the example above of an atmospheric DMS concentration of 100 pptv, the equilibrium concentration of DMS in surface water would be about 0.07 nmol l1. Generally the solubility of any individual gas increases at cooler water temperatures, and solubility of gases in sea water is somewhat less than for fresh water because of the so-called ‘salting out’ effect of dissolved species in sea water. At any moment the partial pressure difference between surface water and the atmosphere depends on an array of variables. The gases in this article are biogenic, meaning that their mode of formation is the result of one or more immediate or proximate biological processes. These dissolved gases may also be consumed biologically, or removed by chemical processes in sea water, or they may flux across the sea surface to the atmosphere. The rates at which the source and sink processes occur determines the concentration of the dissolved gas in solution as well as the turnover, or residence time, of each compound. Similarly, there can be several source and sink processes for the gases in the atmosphere. Long-lived compounds in the atmosphere will tend to integrate more global processes, whereas short-lived compounds are concentrated near their source and reflect relatively short-term influences of source and sink. In this sense, carbonyl sulfide is a global gas. At it has a residence time of several years in the atmosphere, its concentration does not vary in the troposphere to any appreciable degree. On the other hand, the concentration of DMS varies on a diel basis and with elevation, with higher concentrations at night when atmospheric oxidants (most notable hydroxyl) are relatively depleted. The extent of disequilibrium between the partial pressures of a gas in the surface water and in the atmosphere determines the thermodynamic gradient which drives gas flux. The kinetics of flux ultimately depend on molecular diffusion and larger-scale mixing processes. Molecular diffusivity is generally captured in a dimensionless parameter, the Schmidt number (ratio of viscosity of water to molecular

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diffusivity of gas in water), and varies widely between gases depending primarily on the molecular cross-section. From moment to moment, the flux of any particular gas is dependent on interfacial turbulence which is generated by shear between the wind and the sea surface whereby higher wind speed causes increasing turbulence and thus stimulating the onset of waves and eventually the production of bubbles and sea spray. There are considerable uncertainties relating gas exchange to wind speed. These arise due to the various sea-state factors (wave height, swell, breaking waves, bubble entrainment, surfactants, and others) whose individual dependencies on actual wind speed and wind history are not well quantified. The fluxes of gases across the air–sea interface are usually calculated using a wind-speed parameterization. These estimates are considered to be accurate to within a factor of 2 or so. This article summarizes the characteristics of several important trace gases – dimethylsulfide, carbonyl sulfide, carbon disulfide, nonmethane hydrocarbons, ammonia and methylhalides – focusing on their production and fate as it is determined by biological and chemical processes.

Dimethylsulfide Natural and anthropogenic sulfur aerosols play a major role in atmospheric chemistry and potentially in modulating global climate. One theory holds that a negative feedback links the emission of volatile organic sulfur (mostly as DMS) from the ocean with the formation of cloud condensation nuclei, thereby

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regulating, in a sense, the albedo and radiation balance of the earth. The direct (backscattering and reflection of solar radiation by sulfate aerosols) and indirect (cloud albedo) effects of sulfate aerosols may reduce the climatic forcing of trace greenhouse gases like CO2, N2O and CH4. The oxidation products of DMS which also contribute to the acidity of rain, particularly in marine areas, result from industrialized and/or well-populated land. Dimethylsulfide (DMS) is the most abundant volatile sulfur compound in sea water and constitutes about half of the global biogenic sulfur flux to the atmosphere. Studies of the concentration of DMS in the ocean have shown that average surface water concentrations may vary by up to a factor of 50 between summer and winter in mid and high latitudes. Furthermore, there are large-scale variations in DMS concentration associated with phytoplankton biomass, although there are generally poor correlations between local oceanic DMS concentrations and the biomass and productivity of phytoplankton (due to differences between plankton species in ability to produce DMS). The nature and rates of the processes involved in the production and consumption of DMS in sea water are important in determining the surface concentrations and the concomitant flux to the atmosphere. The biogeochemical cycle of DMS (Figure 1) begins with its precursor, b-dimethylsulfoniopropionate (DMSP). DMSP is a cellular component in certain species of phytoplankton, notably some prymnesiophytes and dinoflagellates. The function of DMSP is unclear, although there is evidence for an

Photochemistry SO2 Stratosphere COS DMS CS2

SO2 MSA

Troposphere Photochemistry DMS

Ocean

DMSO

Photochemistry 2_ 4

DMSP

SO Organic matter

Figure 1 Fate and production of dimethylsulfide (DMS), carbonylsulfide (COS) and carbon disulfide. DMSO, dimethylsulfoxide; DMSP, dimethylsulfoniopropionate; MSA, methane sulfonic acid.

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osmoregulatory role as its cellular concentrations have been found to vary with salinity. It is generally thought that healthy algal cells do not leak either DMSP or DMS, although mechanical release into the surrounding sea water can lead to DMS production during cell senescence and grazing by zooplankton or as a consequence of viral attack on phytoplankton cells. Oceanic regions dominated by prolific DMSPproducing phytoplankton tend to have high DMS and DMSP concentrations. Breakdown of DMSP, presumably after transfer from the particulate algal (pDMSP) form to a dissolved (dDMSP) form in sea water, can proceed in different ways, mostly depending on microbiological conditions. One major pathway involves cleavage of DMSP to DMS and acrylic acid. Bacterial metabolism of dDMSP may be a major mechanism for DMS production in sea water, with acrylic acid residue acting as a carbon source for heterotrophic growth. Sulfonium compounds are vulnerable to attack by hydroxide ion; the resulting chemical elimination reaction occurs rapidly and quantitatively in strong base but only slowly at the pH of sea water. DMS in sea water has many potential fates. The volatility of DMS and the concentration gradient across the sea–air interface lead to the ocean being the major source of DMS to the atmosphere. Estimates of the annual sulfur release (as DMS) vary from 13–37 Tg S y1 (Kettle and Andreae, 1999). However, whereas the absolute flux of DMS from sea to air may be large on a global scale, sea–air exchange may represent only a minor sink for seawater DMS. It has been estimated that DMS loss to the atmosphere is only a very small percentage of the DMS sink, but this undoubtedly depends on the biogeochemical conditions in the water column at the time. Photochemical oxidation of DMS, either to dimethylsulfoxide (DMSO) or to other products, occurs via photosensitized reactions. The amount of photochemical decomposition depends on the amount of light of appropriate wavelengths and the concentration of colored organic compounds in solution to convert light energy into reactive radicals. Light declines exponentially with depth; the distribution of colored dissolved organic materials exhibits depth and seasonal variability. Microbial consumption of DMS, although extremely variable in both time and space in the ocean, appears to be a significant sink for oceanic DMS. The residence time of DMS is probably of the order of a day or two in most seawater systems. Since the atmospheric residence time of DMS is about a day or two, the atmospheric consequences of DMS flux are mostly confined to the troposphere. In the troposphere, DMS is oxidized primarily by hydroxyl radical. The main atmospheric oxidation

products are methane sulphonic acid, SO2 and DMSO.

Carbonyl Sulfide Carbonyl sulfide (COS, OCS) is the major sulfur gas in the atmosphere, present throughout the troposphere at 500 pptv. COS has a long atmospheric residence time (B4 years). Because of its relative inertness COS diffuses into the stratosphere where it oxidizes to sulfate particles and contributes in reactions involving stratospheric ozone chemistry. Unlike DMS which is photochemically oxidized in the troposphere, the major sink for COS is terrestrial vegetation and soils. COS is taken up by plants by passing through the stomata and subsequently hydrolyzing to CO2 and H2S through the action of carbonic anhydrase inside plant cells. There is no apparent physiological significance to the process; it appears to just occur accidentally to the normal physiology of plants. COS is produced in the ocean by photochemical oxidation of organic sulfur compounds whereby dissolved organic matter acts as a photosensitizer. The aqueous concentration of COS manifests a strong diel cycle, with the highest concentrations in daytime (concentration range on the order of 0.03– 0.1 nmol l1). COS hydrolyzes in water to H2S at rates dependent on water temperature and pH. The flux of oceanic COS to the atmosphere may represent about one-third of the global COS flux.

Carbon Disulfide Concentrations in surface water are around 1011 mol 11. Although a number of studies have indicated that the ocean forms an important source for atmospheric CS2, the underlying biochemical cycles still remain poorly understood. CS2 is formed by photochemical reactions (possibly involving precursors such as DMS, DMSP and isothiocyanates). CS2 formation has been observed to occur in bacteria in anoxic aquatic environments and in cultures of some marine algae species. The residence time of CS2 in the atmosphere is relatively short (about one week). Although CS2 might contribute directly to SO2 in the troposphere, its main significance is in the formation of COS via photochemical oxidation which results in the production of one molecule each of SO2 and COS per molecule of CS2 oxidized. The resulting COS may contribute to the stratospheric aerosol formation. Concentrations around 14 pmol l1 of carbon disulfide in the mid-Atlantic Ocean were first observed

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(1974); higher concentrations have been found in coastal waters. More than a decade later CS2 concentrations in the North Atlantic were found to be comparable to the earlier observations. However, in coastal waters CS2 concentrations were found to be a factor 10 lower, respectively 33 and 300 pmol l1. The global CS2 flux has been estimated on 6.7 Gmol S y1, and it has been concluded that the marine emission of CS2 provides a significant indirect source of COS, but it forms an insignificant source of tropospheric SO2.

Nonmethane Hydrocarbons Nonmethane hydrocarbons (NMHCs) are important reactive gases in the atmosphere since they provide a sink for hydroxyl radicals and play key roles in the production and destruction of ozone in the troposphere. NMHCs generally refer to the C2–C4 series, notably ethane, ethene, acetylene, propane, propene, and n-butane, but also the five-carbon compound isoprene. Of these, ethene is generally the most abundant contributing 40% to the total NMHC pool in sea water. Published data of concentrations of NMHCs in sea water vary widely sometimes exceeding a factor 100. For example, in one extensive study, ethene and propane were found to be the most abundant species in the intertropical South Pacific, with mixing ratios of 2.7 to 58 and 6 to 75 pptv, respectively; whereas in the equatorial Atlantic these species showed mixing ratios of 20 pptv and 10 pptv, respectively. The water-column dynamics of NMHCs are poorly understood. NMHCs have been detected in the surface sea and with maxima in the euphotic zone and tend to be present at concentrations in sea water at around 1010 mol l1. Evidence suggests that photochemical oxidation of dissolved organic matter results in the formation of NMHCs. There can be very little doubt that the physiology of planktonic organisms is also involved in NMHC formation. Ethene and isoprene are freely produced by terrestrial plants where the former is a powerful plant hormone but the function of the latter less well understood. It is likely that similar processes occur in planktonic algae. NMHC production tends to correlate with light intensity, dissolved organic carbon and biological production. A simplified scheme of marine NMHC production is shown in Figure 2. The flux of NMHCs to the atmosphere (with estimates ranging from o10 Mt y1 to 50 Mt y1) is minor on a global scale, but has a potential significance in local atmospheric chemistry. Although oceans are known to act as sources of NMHCs, the

NMHC (R-H) + OH

173

Products (R + H2O) Troposphere

NMHC (R-H) Photosynthetic organisms

Ocean Photolysis and chemical conversion Dissolved organic matter

Zooplankton Bacteria

Figure 2 Simplified scheme of marine nonmethane hydrocarbon (NMHC) production. In the marine troposphere NMHC acts as a sink for hydroxyl (OH) radicals and thereby plays a key role in ozone chemistry.

sources of individual NMHCs in the marine boundary layer are not always clear. Those NMHCs with a life time of more than a week (e.g., ethane, ethyne, propane, cyclopropane) show latitudinal gradients consistent with a continental source, whereas variations of NMHCs with life times shorter than a week (all alkenes and pentane) are more consistent with a marine source.

Ammonia Ammonia is an extremely soluble gas, reacting with water and dissociating into an ammonium ion at ambient pH. At pH 8.2, about one-tenth of dissolved ammonia is present as NH3. Ammonium is also a rapidly cycling biological nutrient; it is taken up by bacteria and phytoplankton as a source of fixed nitrogen, and released by sundry physiological and decompositional processes in the food web. Anthropogenic loading of ammonium (and other nutrients) into the coastal marine environment results in increased phytoplankton growth in a phenomenon called eutrophication. Ammonium is oxidized to nitrate by bacteria in a process known as nitrification (Figure 3). Conversely, in anoxic environments, ammonium can be formed by nitrate-reducing bacteria. Ammonia plays an important role in the acid–base chemistry in the troposphere where the unionized ammonia (NH3) is converted into ionized ammonia (NHþ 4 ) via a reaction that neutralizes atmospheric acids as HNO3 and H2SO4. This leads to the formation of ammonium aerosols such as the stable ammonium sulfate. Eventually the ammonia returns to the surface by dry or wet deposition. Few data exist on the fluxes of NH3 over marine environments. Evidence suggests that most of the ocean surface serves as a source of NH3 to the

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NOx

Oxidation

NH3(g)

Reduction

NH4-aerosol

Dry and wet deposition

Troposphere NH3(g)

Ammonification

NH4+

Water

Nitrification

NO3

Denitrification

Organic nitrogen Nitrogen fixation

N2

Figure 3 Simplified scheme of marine NHx chemistry. In the marine boundary layer NO3 acts as an initiator for the degradation of many organic compounds, in particular dimethylsulfide (DMS).

atmosphere, even in regions of very low nutrients. In the North Sea, an area situated in the middle of densely populated and industrialized countries of Western Europe, air from nearby terrestrial sources may act as a source of NH3 into surface waters. It has been estimated that the annual biogenic emission of ammonia from European seas is around 30 kt N y1, which is comparable to the emissions of smaller North European countries, leading to the conclusion (amongst others) that seas are among the largest sources of imported ammonium for maritime countries. The net emission of ammonia from coastal waters of the north-east Pacific Ocean to the atmosphere has been shown to be in the order of 10 mmol m2 d1.

Organohalogens Halogenated compounds, such as methyl chloride (CH3Cl), methyl bromide (CH3Br) and methyl iodide (CH3I) are a major source of halogens in the atmosphere, and subsequently form sources of reactive species capable of catalytically destroying ozone. Among these CH3I is likely to play an important role in the budget of tropospheric ozone, through production of iodine atoms by photolysis. Due to their higher photochemical stability methyl chloride and

methyl bromide are more important in stratospheric chemistry; it has been suggested that BrO species are responsible for losses of tropospheric ozone in the Arctic (Figure 4). Atmospheric methyl halides, measured over the ocean by several cruise surveys, have been shown to have average atmospheric mixing ratios of: CH3Cl, 550–600 pptv; CH3Br, 10–12 pptv; CH3I, 0.5–1 pptv. Their temporal and spatial variations are not well understood, neither is their production mechanism in the ocean known. Measurements of atmospheric and seawater concentrations of CH3Cl and CH3I have indicated that the oceans form natural sources of these methyl halides. In contrast, CH3Br appears to be undersaturated in the open ocean and exhibits moderate to 100% supersaturation in coastal and upwelling regions, leading to a global atmosphere to ocean flux of 13 Gg y1. Coastal salt marshes, although they constitute a minor area of the global marine environment, may produce roughly 10% of the total fluxes of atmospheric CH3Br and CH3Cl and thus contribute significantly to the global budgets. Macrophytic and phytoplanktonic algae produce a wide range of volatile organohalogens including di- and tri-halomethanes and mixed organohalogens. There is evidence for the involvement of enzymatic synthesis of methyl halides, but the metabolic

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Temporary reservoirs HBr, BrONO2, etc. O3 Hv

Hv

CH3Br

O2

Br Hv

Coupling with CIOx

BrO HO2

O3 HOBr

Stratosphere

O3

Troposphere

NO2 HO2

Ozone loss CH3Br

OH

Photolabile fraction

Br O2 BrO

Hv

RCHO

and different dynamics spatially and with depth in the ocean. As conditions change in an apparently warming world, changes in the dynamics of surface ocean gases can be expected. The behavior of these trace gases or even the dynamics of the planktonic community are not understood sufficiently to allow good quantitative predictions about changes in trace gas flux to be made. Changes in flux of some gases could lead to an acceleration of warming, while changes in others could lead to cooling. It is, thus, important to understand the factors controlling trace gas dynamics in the surface ocean.

_

Br

BrO

175

See also

Soluble fraction (HBr) Dry deposition

Wet deposition

Air–Sea Gas Exchange. Air–Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-Methane Hydrocarbons, Organo-Halogens.

Ocean CH3Br

Photochemistry

Dissolved organic matter

Further Reading

Phytoplankton/Bacteria

Figure 4 Schematic illustration of the circulation of methyl bromide.

production pathways are not well known. In free sea water, photochemical processes, ion substitution and, possibly the alkylation of halide ions (during the oxidation of organic matter by an electron acceptor such as Fe(III)) are also potential formation mechanisms. Sunlight or microbial mediation are not required for these reactions. In the ocean, chemical degradation of CH3Br occurs by nucleophilic substitution by chloride and hydrolysis. Microbial consumption is also a likely sink for halogenated compounds.

Conclusions The biogenic trace gases are influenced by the complete range of biological processes – from the biochemical and physiological to the ecological level of food web dynamics. The gases that are influenced directly by plant physiology (probably the light NMHCs and isoprene, for example) tend to be most closely related to phytoplankton biomass or primary productivity. Other gases produced during grazing and decomposition (e.g., DMS, NH3), or gases formed by photochemical reactions in dissolved organic material show differing temporal dynamics

Andreae MO (1990) Ocean–atmosphere interactions in the global biogeochemical sulfur cycle. Marine Chemistry 10: 1--29. Andreae MO and Crutzen PJ (1997) Atmospheric aerosols: biogeochemical sources and role in atmospheric chemistry. Science 276: 1052--1058. Barrett K (1998) Oceanic ammonia emissions in Europe and their transboundary fluxes. Atmospheric Environment 32(3): 381--391. Chin M and Davis DD (1993) Global sources and sinks of OCS and CS2 and their distributions. Global Biogeochemical Cycles 7: 321--337. Cox RA, Rattigana OV, and Jones RL (1995) Laboratory studies of BrO reactions of interest for the atmospheric ozone balance. In: Bandy RA (ed.) The Chemistry of the Atmosphere; Oxidants and Oxidation in the Earth’s Atmosphere. Cambridge: The Royal Society of Chemistry. Crutzen PJ (1976) The possible importance of COS for the sulfate layer of the stratosphere. Geosphysical Research Letters 3: 73--76. Graedel TE (1995) Tropospheric budget of reactive chlorine. Global Biogeochemical Cycles 9: 47--77. Kettle AJ and Andreae MO (1999) Flux of dimethylsulfide from the oceans: a comparison of updated datasets and flux models. Journal of Geophysical Research 105: 26793--26808. Lovelock JE (1974) CS2 and the natural sulfur cycle. Nature 248: 625--626. Turner SM and Liss PS (1985) Measurements of various sulfur gases in a coastal marine environment. Journal of Atmospheric Chemistry 2(3): 223--232.

AIR–SEA TRANSFER: N2O, NO, CH4, CO C. S. Law, Plymouth Marine Laboratory, The Hoe, Plymouth, UK Copyright & 2001 Elsevier Ltd.

identifies the marine contribution to total atmosphere budgets. There is also a brief examination of the approaches used for determination of marine trace gas fluxes and the variability in current estimates.

Nitrous Oxide (N2O) Introduction The atmospheric composition is maintained by abiotic and biotic processes in the terrestrial and marine ecosystems. The biogenic trace gases nitrous oxide (N2O), nitric oxide (NO), methane (CH4) and carbon monoxide (CO) are present in the surface mixed layer over most of the ocean, at concentrations which exceed those expected from equilibration with the atmosphere. As the oceans occupy 70% of the global surface area, exchange of these trace gases across the air–sea interface represents a source/sink for global atmospheric budgets and oceanic biogeochemical budgets, although marine emissions of NO are poorly characterized. These trace gases contribute to global change directly and indirectly, by influencing the atmospheric oxidation and radiative capacity (the ‘greenhouse effect’) and, together with their reaction products, impact stratospheric ozone chemistry (Table 1). The resultant changes in atmospheric forcing subsequently influence ocean circulation and biogeochemistry via feedback processes on a range of timescales. This article describes the marine sources, sinks, and spatial distribution of each trace gas and

The N2O molecule is effective at retaining long-wave radiation with a relative radiative forcing 280 times that of a CO2 molecule. Despite this the relatively low atmospheric N2O concentration results in a contribution of only 5–6% of the present day ‘greenhouse effect’ with a direct radiative forcing of about 0.1 Wm2. In the stratosphere N2O reacts with oxygen to produce NO radicals, which contribute to ozone depletion. N2O is a reduced gas which is produced in the ocean primarily by microbial nitrification and denitrification. N2O is released during ammonium  (NHþ 4 ) oxidation to nitrite (NO2 ) (Figure 1), although the exact mechanism has yet to be confirmed. N2O may be an intermediate of nitrification, or a byproduct of the decomposition of other intermediates, such as nitrite or hydroxylamine. Nitrification is an aerobic process, and the N2O yield under oxic conditions is low. However, as the nitrification rate decreases under low oxygen, the relative yield of N2O to nitrate production increases and reaches a maximum at 10–20 mmol dm3 oxygen (mmol ¼ 1  106 mol). Conversely, denitrification is an anaerobic process in which soluble oxidized nitrogen

Table 1 The oceanic contribution and atmospheric increase and impact for methane, nitrous oxide, nitric oxide, and carbon monoxidea Trace gas

Atmospheric concentration (ppbv)

Atmospheric lifetime (years)

Major impact in atmosphere

Increase in atmosphere (1980– 90)

Oceanic emission as % of total global emissions

Nitrous oxide (N2O) Nitric oxide (NO)

315

110–180

0.25% (0.8 ppbv y1)

7–34%

0.01

o0.2

Not known

Not known

Methane (CH4)

1760

10

0.8% (0.6 ppbv y1)

1–10%

Carbon monoxide (CO)

120

0.2–0.8

Infrared active Ozone sink/source Ozone sink/source OH sink/oxidation capacity Infrared active OH sink/oxidation capacity Ozone sink/source OH sink/oxidation capacity Ozone sink/source Infrared active

 13 to 0.6%

0.9–9%

a

ppbv, parts per billion by volume. (Adapted from Houghton et al., 1995.)

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177

Nitrification +

NH 4 H2O O2

_

NO2

_

N2O

NO

NO3

N2 Denitrification Figure 1 ‘Leaky Pipe’ flow diagram of nitrification and denitrification indicating the potential exchange and intermediate role of NO and N2O (Reprinted by permission from Nature copyright (1990), Macmillan Magazines Ltd.)

compounds, such as nitrate and nitrite, are converted to volatile reduced compounds (N2O and N2) in the absence of oxygen. Oxygen availability inhibits denitrification at ambient levels, and also determines the products of denitrification. An enzymatic gradient of sensitivity to oxygen results in the accumulation of N2O under sub-oxia (3–10 mmol dm3) due to the inhibition of the enzyme nitrous oxide reductase. At lower oxygen (o3 mmol dm3) the reaction continues through to N2 and so anoxic environments are sinks for N2O. N2O yields from nitrification are 0.2–0.5%, whereas denitrification yields may be as high as 5% at optimal levels of sub-oxia. An inverse correlation between N2O and oxygen, and associated linear relationship between nitrate and N2O, suggest that N2O in the ocean originates primarily from nitrification. This may not be the case for sediments, in which denitrification is the dominant source of N2O under variable oxygen tension, with nitrification only contributing in a narrow suboxic band. Attribution of source is difficult as nitrification and denitrification may occur simultaneously and interact, with exchange of products and intermediates (Figure 1). This is further complicated, as denitrification will be limited to some extent by nitrate supply from nitrification. Isotopic data from the surface ocean in oligotrophic regions imply that N2O originates primarily from nitrification. However, recent evidence from waters overlying oxygen-deficient intermediate layers suggests that the elevated surface mixed-layer N2O arises from coupling between the two processes, as the observed isotope signatures cannot be explained by nitrification or denitrification alone. An additional N2O source from the dissimilatory reduction of nitrate to ammonium is restricted to highly anoxic environments such as sediments. The oceanic N2O distribution is determined primarily by the oxygen and nutrient status of the water

column. Estuaries and coastal waters show elevated supersaturation in response to high carbon and nitrogen loading, and the proximity of sub-oxic zones in sediment and the water column. As a result the total marine N2O source tends to be dominated by the coastal region. The N2O flux from shelf sea sediments is generally an order of magnitude lower than estuarine sediments, although the former have a greater spatial extent. A N2O maximum at the base of the euphotic zone is apparent in shelf seas and the open ocean, and is attributed to production in suboxic microzones within detrital material. Oceanic surface waters generally exhibit low supersaturations (o105%), although N2O supersaturations may exceed 300% in surface waters overlying low oxygen intermediate waters and upwelling regions, such as the Arabian Sea and eastern tropical North Pacific. These ‘natural chimney’ regions dominate the open ocean N2O source, despite their limited surface area (Table 2). The surface N2O in upwelling regions such as the Arabian Sea originates in part from the underlying low-oxygen water column at 100– 1000 m, where favorable conditions result in the accumulation of N2O to supersaturations exceeding 1200%. N2O transfer into the surface mixed layer will be limited by vertical transport processes and a significant proportion of N2O produced at these depths will be further reduced to N2. The oceans account for 1–5 Tg N-N2O per annum (Tg ¼ 1  1012 g) or 6–30% of total global N2O emissions, although there is considerable uncertainty attached to this estimate (Figure 2). A recent estimate with greater representation of coastal sources has resulted in upward revision of the marine N2O source to 7–10.8 Tg N-N2O per annum; although this may represent an upper limit due to some bias from inclusion of estuaries with high N2O supersaturation. However, this estimate is in agreement with a total

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AIR–SEA TRANSFER: N2O, NO, CH4, CO

Table 2 N2O and CH4 regional surface water supersaturations (from Bange et al., 1996; 1998) (supersaturation is >100%, undersaturation is o100% with equilibrium between atmosphere and water at 100%)

Estuaries Coastal/shelf Oligotrophic/transitional ocean Upwelling ocean

Surface % N2O saturation mean (range)

Surface % CH4 saturation mean (range)

607 (101–2500) 109 (102–118) 102.5 (102–104) 176 (108–442)

1230 (146–29 000) 395 (85–42 000) 120 (80–200) 200 (86–440)

N2O sources _1 (Tg N yr ) Oceans 3 (1_ 5) Soil natural 6 (3.3 _ 9.7)

Soil anthropogenic 3.3 (0.6 _14.8)

Biomass burning 0.5 (0.2 _1) Industrial 1.3 (0.7 _1.8)

½2

  NO2 þ hn-NO þ O 3 P

½3

O þ O2 þ M-O3 þ M

½4

At high concentrations (>50 ppbv; ppbv ¼ parts per billion by volume), O3 in the atmospheric boundary layer becomes a toxic pollutant that also has important radiative transfer properties. The production of nitric acid from NO influences atmospheric pH, and contributes to acid rain formation. In addition, the oxidation of NO to the nitrate (NO3) radical at night influences the oxidizing capacity of the lower troposphere. Determination of the magnitude and location of NO sources is critical to modeling boundary layer and free tropospheric chemistry. NO cycling in the ocean has received limited attention, as a result of its thermodynamic instability and high reactivity. Photolysis of nitrite in surface waters occurs via the formation of a nitrite radical with the production of NO:

Livestock feed 2.1 (0.6 _ 3.1)

N2O sinks _1 (Tg N yr ) Atmospheric increase 3.9 (3.1_ 4.7)

Stratospheric photolysis 12.3 (9 _ 16) Figure 2 Atmospheric nitrous oxide sources and sinks (adapted from Houghton et al., 1995). Units: Tg ¼ 1  1012 g.

oceanic production rate of 11 Tg N-N2O per annum calculated from new production and nitrification.

Nitric Oxide (NO) Nitric oxide (NO) plays a central role in atmospheric chemistry, influencing both ozone cycling and the tropospheric oxidation capacity through reactions with hydroperoxy- and organic peroxy-radicals. When the NO concentration exceeds B40 pptv (pptv ¼ parts per trillion by volume) it catalyzes the production of ozone (O3): CO þ OH * þ O2 -HO2* þ CO2

HO2* þ NO-OH * þ NO2

½1



* þ HOH-NO þ OH þ OH NO 2 þ hn-NO2

This reaction may account for 10% of nitrite loss in surface waters of the Central Equatorial Pacific, resulting in a 1000-fold increase in dissolved NO at a steady-state surface concentration of 5 pmol dm3 during light periods (pmol ¼ 1  1012 mol). This photolytic production is balanced by a sink reaction with the superoxide radical (O 2 ) to produce peroxynitrite:  O 2 þ NO- OONO

This reaction will be dependent upon steady-state concentration of the superoxide radical; however, as the reaction has a high rate constant, NO is rapidly turned over with a half-life on the order of 10–100 seconds.

AIR–SEA TRANSFER: N2O, NO, CH4, CO NO sources _1 (Tg N yr )

Soils 15 (10 _ 20) Aircraft emmisions 0.6

Biomass burning 5.5 (2.5 _ 8.5)

Lightning 6 (2 _ 8) Transport from stratosphere 0.5

Fossil fuel combustion 21

Figure 3 Atmospheric nitric oxide sources (from Graedel and Crutzen, 1992). Units: Tg ¼ 1  1012 g.

As with N2O, NO may also be produced as a byproduct or intermediate of denitrification and nitrification (Figure 1). NO production by soils is better characterized than in marine systems, and is significant both in terms of nitrogen loss and the global NO budget (Figure 3). The greater oxygen availability in soils limits reduction of NO via denitrification and so enhances NO efflux. Sediment pore water NO maxima have been attributed to denitrification, although, as this process also represents a sink for NO (Figure 1), this may reflect poising at an optimal redox potential for NO production. Conversely, the NO maximum in low oxygen intermediate waters in the east tropical North Pacific derives from nitrification. Current understanding of the oceanic NO distribution is that it is limited to the surface ocean and intermediate low oxygen water column. There is potential for higher NO concentrations in coastal and estuarine waters from sediment and photolytic sources, and nitrite photolysis to NO may also be significant in upwelling regions. Despite the short half-life of NO in surface waters, the maintenance of steady-state NO concentration suggests that photolytic production may support an, as yet unquantified, source of atmospheric NO. Surface concentrations in the Central Equatorial Pacific suggest that the oceanic NO source would not exceed 0.5 Tg N per annum, which is relatively insignificant when compared with other sources (Figure 3).

Methane (CH4) CH4 is the most abundant organic volatile in the atmosphere and, next to CO2, is responsible for 15% of the current greenhouse radiative forcing, with a direct radiative forcing of 0. 5 Wm2. CH4 reacts with OH and so limits the tropospheric oxidation capacity and influences ozone and other greenhouse

179

gases. The reaction with OH generates a feedback that leads to a reduction in the rate of CH4 removal. CH4 is a reduced gas which, paradoxically, is supersaturated in the oxidized surface waters of the ocean (see Table 2). CH4 is produced biotically and abiotically, although its oceanic distribution is controlled primarily by biological processes. Methanogenesis is classically defined as the formation of CH4 from the fermentation and remineralization of organic carbon under anoxic conditions. Methanogens require a very low reducing potential and are generally obligate anaerobes, although there is evidence that they can tolerate some exposure to oxygen. However, methanogens cannot utilize complex organic molecules and often coexist with aerobic consortia to ensure a supply of simple C1 substrates. Methanogens utilize formate, acetic acid, CO2, and hydrogen in sulfate-rich anoxic environments, although they are generally out-competed by sulfatereducing bacteria which have a greater substrate affinity. However, the methanogens can also utilize other noncompetitive substrates such as methanol, methylamines, and reduced methylated compounds, when out-competed for the C1 compounds. A significant fraction of CH4 is oxidized before exiting the marine system and so the oxidation rate is critical in determining the air–sea flux. This is accomplished by methanotrophs that obtain their carbon and energy requirements from CH4 oxidation under aerobic conditions via the following reactions: CH4 -CH3 OH -HCHO-HCOOC -CO2 methane - methanol - formaldehyde - formate - carbon dioxide

Methanotrophs are found in greater numbers in sediments than in oxic sea water, and consequently the oceanic water column CH4 oxidation is an order of magnitude lower than in sediments. Methanotrophs have a high inorganic nitrogen requirement and so methanotrophy is highest at the oxic–anoxic interface where ammonium is available. Anaerobic CH4 oxidation also occurs but is less well characterized. It is generally restricted to anaerobic marine sediments, utilizing sulfate as the only oxidant available, and is absent from anaerobic freshwater sediments which lack sulfate. A significant proportion of CH4 produced in anaerobic subsurface layers in sediments is oxidized during diffusive transport through the sulfate-CH4 transition zone by anaerobic oxidation and subsequently by aerobic oxidation in the overlying oxic layers. Anaerobic oxidation represents the main sink for CH4 in marine sediments, where it may account for 97% of CH4 production.

180

AIR–SEA TRANSFER: N2O, NO, CH4, CO

CH4 production is characteristic of regions with high input of labile organic carbon such as wetlands and sediments, but is usually restricted to below the zone of sulfate depletion. The oceanic CH4 source is dominated by coastal regions, which exhibit high CH4 fluxes as a result of bubble ebullition from anoxic carbon-rich sediments, and also riverine and estuarine input. Some seasonality may result in temperate regions due to increased methanogenesis at higher temperatures. The predominant water column source in shelf seas and the open ocean is CH4 production at the base of the euphotic zone. This may arise from lateral advection from sedimentary sources, and in situ CH4 production. The latter is accomplished by oxygen-tolerant methanogens that utilize methylamines or methylated sulfur compounds in anoxic microsites within detrital particles and the guts of zooplankton and fish. Lateral advection and in situ production may be greater in upwelling regions, as suggested by the increased CH4 supersaturation in surface waters in these regions. Oceanic CH4 concentration profiles generally exhibit a decrease below 250 m due to oxidation. Methanogenesis is elevated in anoxic water columns, although these are not significant sources of atmospheric CH4 due to limited ventilation and high oxidation rates. Other sources include CH4 seeps in shelf regions from which CH4 is transferred directly to the atmosphere by bubble ebullition, although their contribution is difficult to quantify. Abiotic CH4 originating from high-temperature fluids at hydrothermal vents also elevates CH4 in the deep and intermediate waters in the locality of oceanic ridges. A significant proportion is oxidized and although the contribution to the atmospheric CH4 pool may be significant in localized regions this has yet to be constrained. Hydrates are crystalline solids in which methane gas is trapped within a cage of water molecules. These form at high pressures and low temperatures in seafloor sediments generally at depths below 500 m. Although CH4 release from hydrates is only considered from anthropogenic activities in current budgets, there is evidence of catastrophic releases in the geological past due to temperature-induced hydrate dissociation. Although oceanic hydrate reservoirs contain 14 000 Gt CH4, there is currently no evidence of significant warming of deep waters which would preempt release. Other aquatic systems such as rivers and wetlands are more important sources than the marine environment. Shelf regions are the dominant source of CH4 from the ocean (14(11–18) Tg CH4 per annum), accounting for 75% of the ocean flux (Table 2). The ocean is not a major contributor to the atmospheric

CH4 sources _1 (Tg CH4 yr )

Other natural 35 (20 _ 90) Wetland 115 (55 _ 150) Anthropogenic (other) 275 (200 _ 350)

Oceans 14 (11_ 18) Anthropogenic (fossil fuel) 100 (70 _ 120)

CH4 sinks _1 (Tg CH4 yr )

Tropospheric OH oxidation 445 (360 _ 530)

Loss to stratosphere 40 (32 _ 48) Soils 30 (15 _ 45) Atmospheric increase 37 (35 _ 40)

Figure 4 Atmospheric methane sources and sinks (adapted from Houghton et al. (1995)). Units: Tg ¼ 1  1012 g.

CH4 budget, as confirmed by estimates of the oceanic CH4 source (Figure 4).

Carbon Monoxide (CO) The oxidation of CO provides the major control of hydroxyl radical content in the troposphere and limits the atmospheric oxidation capacity. This results in an increase in the atmospheric lifetime of species such as CH4, N2O, and halocarbons, and enhances their transfer to the stratosphere and the potential for subsequent ozone destruction. It has been suggested that decreasing stratospheric ozone and the resultant increase in incident ultraviolet (UV) radiation may increase marine production and efflux of CO, thereby generating a positive feedback loop. However, this may be compensated by a negative feedback in which increased UV reduces biological production and dissolved organic matter, so reducing the CO source. CO also influences tropospheric ozone by its interaction with NOx, and is a minor greenhouse gas with a radiative forcing of 0.06 Wm2 at current atmospheric concentrations.

AIR–SEA TRANSFER: N2O, NO, CH4, CO

The principal source of dissolved CO is the abiotic photodegradation of dissolved organic matter (DOM) by UV-R, and CO represents one of the major photoproducts of DOM in the ocean. Quantum yields for CO are highest in the UV-B range (280–315 nm) and decrease with increasing wavelengths. However, the UV-A (315–390 nm) and blue portion of the visible spectrum contribute to marine CO production as a greater proportion of radiation at these wavelengths reaches the Earth’s surface. Humics represent approximately half of the DOM and account for the majority of the chromophoric dissolved organic matter (CDOM), the colored portion of dissolved organic matter that absorbs light energy. The CO photoproduction potential of humics is dependent upon the degree of aromaticity. Terrestrial humics are characterized by an increased prevalence of phenolic groups, and addition of precursor compounds containing phenolic moieties to natural samples stimulates CO production. Direct photo-oxidation of humics and compounds containing carbonyl groups, such as aldehydes, ketones, and quinones, occurs via the production of a carbonyl radical during a-cleavage of an adjacent bond:R þ ðCOR0 Þ * -R0 þ CO RCOR0 þ hvðRCOÞ * þR0 -R0 þ CO CO production may also occur indirectly by a photosensitized reaction in which light energy is transferred via an excited oxygen atom to a carbonyl compound. This may occur with ketonic groups via the photosensitized production of an acetyl radical. Whereas light and CDOM are the primary factors controlling CO production there may also be additional influence from secondary factors. For example, organo-metal complexes have increased light absorption coefficients and their photo-decomposition will enhance radical formation and CO production at higher levels of dissolved metals such as iron. There is also minor biotic production of CO by methanogens, but this does not appear to be significant. CO can be oxidized to carbon dioxide by selected microbial groups including ammonia oxidizers and methylotrophs that have a broad substrate specificity and high affinity for CO. However, only the carboxidotrophs obtain energy from this reaction, and these may be unable to assimilate CO efficiently at in situ concentrations. CO turnover times of 4 hours are typical for coastal waters, whereas this varies between 1 and 17 days in the open ocean. The lower oxidation rate in the open ocean may be due to light inhibition of CO oxidation. Extrapolation from laboratory

181

measurements suggests that only 10% of photochemically produced CO is microbially oxidized. Dissolved CO exhibits diurnal variability in the surface ocean in response to its photolytic source, although this is also indicative of a strong sink term. The decline in the surface mixed-layer CO concentration in the dark results from a combination of CO oxidation, vertical mixing, and air–sea exchange. As the equilibration time between atmosphere and oceanic surface mixed layer is on the order of a month, this suggests that the former two processes dominate. Superimposed upon the diurnal cycle of CO in the surface ocean are spatial and seasonal gradients that result from the interaction of photoproduction and the sink processes. Below the euphotic zone CO is uniformly low throughout the intermediate water column. CO production potential is highest in wetland regions, which are characterized by high CDOM and enhanced light attenuation. Photochemical production of CO represents a potential sink for terrestrial dissolved organic carbon (DOC) in estuaries and coastal waters. This pathway may account for some of the discrepancy between the total terrestrial DOC exported and the low proportion of terrestrial DOC observed in the marine pool. Although a strong lateral gradient in CDOM exists between rivers and the open ocean, estuarine CO production may be limited by reduced UV light penetration. CO photoproduction may occur down to 80 m in the open ocean, and 20 m in the coastal zone, but is restricted to the upper 1 m in wetlands and estuaries. In addition, estuarine and coastal CO flux may also be restricted by the higher CO oxidation rates. There is evidence that upwelling regions may support enhanced CO production, in response to upwelled CDOM that is biologically refractory but photolabile. The presence of a CO gradient in the 10 m overlying the surface ocean suggests that the photolytic source of CO may influence the marine boundary layer. The marine source of CO is poorly constrained, with estimates varying from 10 to 220 Tg CO per annum. A flux of 1200 Tg CO per annum was estimated on the assumption that low rates of oceanic CO oxidation would only remove a small proportion of photoproduced CO, and that the residual would be ventilated to the atmosphere. The discrepancy between this and other flux estimates implies that a significant CO sink has been overlooked, although this may reflect shortcomings of different techniques. The oceanic contribution to the global source is between 1 and 20%, although Extrapolation of photochemical production rates from wetlands, estuaries, and coasts suggests that these

182

AIR–SEA TRANSFER: N2O, NO, CH4, CO

Table 3

Atmospheric CO sources and sinksa (adapted from Zuo et al., 1998)

CO sources (Tg CO y1) Industrial/fuel combustion Biomass burning Vegetation and soils Methane oxidation NMHC oxidation Ocean (Coast/Shelf Total sources

CO sinks (Tg CO y1) 400–1000 300–2200 50–200 300–1300 200–1800 10–220 300–400) 1260–6720

Tropospheric hydroxyl oxidation Soils Flux to stratosphere

1400–2600 250–530 80–140

Total sinks

1730–3270

a

Note that a separate estimate of the coastal/shelf CO source is shown for comparison, but does not contribute to the total source. Tg ¼ 1  1012 g.

alone may account for 20% of the total global CO flux. Although the marine source is responsible for o10% of the total global flux (Table 3), it may still dominate atmospheric oxidation conditions in remote regions at distance from land.

Air–Sea Exchange of Trace Gases The flux of these trace gases across the air–sea interface is driven by physical transfer processes and the surface concentration anomaly, which represents the difference between the partial pressure observed in surface water and that expected from equilibrium with the atmosphere. Direct determination of the oceanic emission of a trace gas is difficult under field conditions. Atmospheric gradient measurements above the ocean surface require enhanced analytical resolution, whereas more advanced micrometeorological techniques have yet to be applied to these trace gases. Determination of the accumulation rate in a floating surface flux chamber is a simpler approach, but may generate artefactual results from the damping of wave- and wind-driven exchange, and enhanced transfer on the inner chamber surfaces. Consequently the majority of flux estimates are calculated indirectly rather than measured. The surface anomaly is derived from the difference between the measured surface concentration (Cw), and an equilibrium concentration calculated from the measured atmospheric concentration (Cg) and solubility coefficient (p) at ambient temperature and salinity. This is then converted to a flux by the application of a dynamic term, the gas transfer velocity, k: F ¼ kðCw  aCgÞ The transfer velocity k is the net result of a variety of molecular and turbulent processes that operate at different time and space scales. Wind is the primary driving force for most of these turbulent processes,

and it is also relatively straightforward to obtain accurate measurements of wind speed. Consequently, k is generally parameterized in terms of wind speed, with the favored approaches assuming tri-linear and quadratic relationships between the two. These relationships are defined for CO2 at 201C in fresh water and sea water and referenced to other gases by a Schmidt number (Sc) relationship: k gas ¼ k ref ðSc gas=Sc ref Þn where n is considered to be  1/2 at most wind speeds. This dependency of k is a function of the molecular diffusivity (D) of the gas and the kinematic viscosity of the water (m), and is expressed in terms of the Schmidt number (Sc ¼ m/D). Determination of marine trace gas fluxes using different wind speed–transfer velocity relationships introduces uncertainty, which increases at mediumhigh wind speeds to a factor of two. Furthermore, additional uncertainty is introduced by the extrapolation of surface concentration gradient measurements to long-term climatological wind speeds. Current estimates of oceanic fluxes are also subject to significant spatial and temporal bias resulting from the fact that most studies focus on more productive regions and seasons. This uncertainty is compounded by the extrapolation of observational data sets to unchartered regions. With the exception of N2O, the ocean does not represent a major source for these atmospheric trace gases, although spatial variability in oceanic source strength may result in localized impact, particularly in remote regions. In the near future, advances in micrometeorological techniques, improved transfer velocity parameterizations and the development of algorithms for prediction of surface ocean concentrations by remote sensing should provide further constraint in determination of the oceanic source of N2O, NO, CH4, and CO.

AIR–SEA TRANSFER: N2O, NO, CH4, CO

See also Air–Sea Gas Exchange. Air–Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-Methane Hydrocarbons, Organo-Halogens. Gas Exchange in Estuaries. Surface Films.

Further Reading Bange HW, Bartel UH, et al. (1994) Methane in the Baltic and North Seas and a reassessment of the marine emissions of methane. Global Biogeochemical Cycles 8: 465--480. Bange HW, Rapsomanikis S, and Andreae MO (1996) Nitrous oxide in coastal waters. Global Biogeochemical Cycles 10: 197--207.

183

Carpenter EJ and Capone DG (eds.) (1983) Nitrogen in the Marine Environment. London: Academic Press. Graedel TE and Crutzen PJ (eds.) (1992) Atmospheric Change: An Earth System Perspective. London: W. H. Freeman and Co. Houghton JT, Meira Filho M, Bruce J, et al. (1995) Climate Change 1994. Radiative Forcing of Climate Change and an Evaluation of the IPCC IS92 Emission Scenarios, Intergovernmental Panel on Climate Change. Cambridge: Cambridge University Press. Liss PS and Duce RA (eds.) (1997) The Sea Surface and Global Change. Cambridge: Cambridge University Press. Zuo Y, Guerrero MA, and Jones RD (1998) Reassessment of the ocean to atmosphere flux of carbon monoxide. Chemistry and Ecology 14: 241--257.

GAS EXCHANGE IN ESTUARIES M. I. Scranton, State University of New York, Stony Brook, NY, USA M. A. de Angelis, Humboldt State University, Arcata, CA, USA Copyright & 2001 Elsevier Ltd.

the gas in either the gas or liquid phase.) For gases that make up a large fraction of the atmosphere (O2, N2, Ar), pgas does not vary temporally or spatially. For trace atmospheric gases (carbon dioxide (CO2), methane (CH4), hydrogen (H2), nitrous oxide (N2O), and others), pgas may vary considerably geographically or seasonally, and may be affected by anthropogenic activity or local natural sources.

Introduction Many atmospherically important gases are present in estuarine waters in excess over levels that would be predicted from simple equilibrium between the atmosphere and surface waters. Since estuaries are defined as semi-enclosed coastal bodies of water that have free connections with the open sea and within which sea water is measurably diluted with fresh water derived from land drainage, they tend to be supplied with much larger amounts of organic matter and other compounds than other coastal areas. Thus, production of many gases is enhanced in estuaries relative to the rest of the ocean. The geometry of estuaries, which typically have relatively large surface areas compared to their depths, is such that flux of material (including gases) from the sediments, and fluxes of gases across the air/water interface, can have a much greater impact on the water composition than would be the case in the open ocean. Riverine and tidal currents are often quite marked, which also can greatly affect concentrations of biogenic gases.

Gas Solubility The direction and magnitude of the exchange of gases across an air/water interface are determined by the difference between the surface-water concentration of a given gas and its equilibrium concentration or gas solubility with respect to the atmosphere. The concentration of a specific gas in equilibrium with the atmosphere (Ceq) is given by Henry’s Law: Ceq ¼ pgas =KH

½1

where pgas is the partial pressure of the gas in the atmosphere, and KH is the Henry’s Law constant for the gas. Typically as temperature and salinity increase, gas solubility decreases. (Note that Henry’s Law and the Henry’s Law constant also may be commonly expressed in terms of the mole fraction of

184

Gas Exchange (Flux) Across the Air/ Water Interface The rate of gas exchange across the air/water interface for a specific gas is determined by the degree of disequilibrium between the actual surface concentration of a gas (Csurf) and its equilibrium concentration (Ceq), commonly expressed as R: R ¼ Csurf =Ceq

½2

If R ¼ 1, the dissolved gas is in equilibrium with the atmosphere and no net flux or exchange with the atmosphere occurs. For gases with Ro1, the dissolved gas is undersaturated with respect to the atmosphere and there is a net flux of the gas from the atmosphere to the water. For gases with R41, a net flux of the gas from the water to the atmosphere occurs.

Models of Gas Exchange The magnitude of the flux (F) in units of mass of gas per unit area per unit time across the air/water interface is a function of the magnitude of the difference between the dissolved gas concentration and its equilibrium concentration as given by Fick’s First Law of Diffusion: F ¼ kðCsurf  Ceq Þ

½3

where k is a first order rate constant, which is a function of the specific gas and surface water conditions. The rate constant, k, also known as the transfer coefficient, has units of velocity and is frequently given as k ¼ D=z

½4

where D is the molecular diffusivity (in units of cm2 s1), and z is the thickness of the laminar layer at the

GAS EXCHANGE IN ESTUARIES

air/water interface, which limits the diffusion of gas across the interface. In aquatic systems, Csurf is easily measured by gas chromatographic analysis, and Ceq may be calculated readily if the temperature and salinity of the water are known. In order to determine the flux of gas (F) in or out of the water, the transfer coefficient (k) needs to be determined. The value of k is a function of the surface roughness of the water. In open bodies of water, wind speed is the main determinant of surface roughness. A number of studies have established a relationship between wind speed and either the transfer coefficient, k, or the liquid laminar layer thickness, z (Figure 1). The transfer coefficient is also related to the Schmidt number, Sc, defined as: Sc ¼ n=D

½5

where n is the kinematic viscosity of the water. In calmer waters, corresponding to wind speeds of o5 m s1, k is proportional to Sc2/3. At higher wind speeds, but where breaking waves are rare, k is proportional to S1/2. Therefore, if the transfer coefficient of one gas is known, the k value for any other gas can be determined as: Scn1 =Scn2

k1 =k2 ¼ Smooth surface regime

150

Rough surface regime

_ 2/ 3

Breaking wave (bubble) regime

_ 1/ 2

Kw Sc

Kw Sc

125

½6

185

where n ¼ the exponent. For short-term steady winds, the transfer coefficient for CO2 has been derived as kCO2 ¼ 0:31ðU10 Þ2 ðSc=600Þ0:5

½7

where U10 is the wind speed at a height of 10 m above the water surface. Eqns [6] and [7] can be used to estimate k for gases other than CO2. In most estuarine studies wind speeds are measured closer to the water surface. In such cases, the wind speed measured at 2 cm above the water surface can be approximated as 0.5U10. In restricted estuaries and tidally influenced rivers, wind speed may not be a good predictor of wind speed due to limited fetch or blockage of prevailing winds by shore vegetation. Instead, streambed-generated turbulence is likely to be more important than wind stress in determining water surface roughness. In such circumstances, the large eddy model may be used to approximate k as: k ¼ 1:46ðDul1 Þ1=2

½8

where u is the current velocity, and l is equivalent to the mean depth in shallow turbulent systems. Much of the reported uncertainty (and study to study variability in fluxes) is caused by differences in assumptions related to the transfer coefficient rather than to large changes in concentration of the gas in the estuary.

Less soluble gas (e.g O2)

Transfer velocity cm h

_1

Direct Gas Exchange Measurements 100

75

More soluble gas (e.g. CO2)

50

25

0.25 0

0

5

0.5 10

_1

0.75 U ms

_1

15 U ms

1 20

Figure 1 Idealized plot of transfer coefficient (Kw) as a function of wind speed (u) and friction velocity (u*). (Adapted with permission from Liss PS and Merlivat L (1985) Air–sea gas exchange rates. In: Buat-Menard P (ed.) The Role of Air–sea Exchange in Geochemical Cycling, p. 117. NATO ASI Series C, vol. 185. Dordrecht: Reidel.)

Gas exchange with the atmosphere for gases for which water column consumption and production processes are known can be estimated using a dissolved gas budget. Through time-series measurements of biological and chemical cycling, gas loss or gain across the air/water interface can be determined by difference. For example, in the case of dissolved O2, the total change in dissolved O2 concentration over time can be attributed to air/water exchange and biological processes. The contribution of biological processes to temporal changes in dissolved O2 may be estimated from concurrent measurements of phosphate and an assumed Redfield stoichiometry, and subtracted from the total change to yield an estimate of air/water O2 exchange. Gas fluxes also may be measured directly using a flux chamber that floats on the water surface. The headspace of the chamber is collected and analyzed over several time points to obtain an estimate of the net amount of gas crossing the air/water interface enclosed by the chamber. If the surface area enclosed

186

GAS EXCHANGE IN ESTUARIES

by the chamber is known, a net gas flux can be determined. Some flux chambers are equipped with small fans that simulate ambient wind conditions. However, most chambers do not use fans and so do not take account of the effects of wind-induced turbulence on gas exchange. Despite this limitation, flux chambers are important tools for measuring gas exchange in environments (such as estuaries or streams) where limited fetch or wind breaks produced by shoreline vegetation make wind less important than current-induced turbulence in shallow systems. While flux chambers may alter the surface roughness and, hence, gas/exchange rates via diffusion, flux chambers or other enclosed gas capturing devices also are the best method for determining loss of gases across the air/water interface due to ebullition of gas bubbles from the sediment. Measurement of radon (Rn) deficiencies in the upper water column can be used to determine gas exchange coefficients and laminar layer thickness, which can then be applied to other gases using eqn [3]. In this method, gaseous 222Rn, produced by radioactive decay of 226Ra, is assumed to be in secular equilibrium within the water column. 222Rn is relatively short-lived and has an atmospheric concentration of essentially zero. Therefore, in nearsurface waters, a 222Rn deficiency is observed, due to flux of 222Rn across the air/water interface. The flux of 222Rn across the air/water interface can be determined by the depth-integrated difference in measured 222Rn and that which should occur based on the 226 Ra inventory. From this flux, the liquid laminar layer thickness, z, can be calculated. Other volatile tracers have been used in estuaries to determine gas exchange coefficients. These tracers, such as chlorofluorocarbons (CFCs) or sulfur hexafluoride (SF6), are synthetic compounds with no known natural source. Unlike 222Rn, these gases are stable in solution. These tracers may be added to the aquatic system and the decrease of the gas due to flux across the air/water interface monitored over time. In some estuaries, point sources of these compounds may exist and the decrease of the tracer with distance downstream may be used to determine k or z values for the estuary.

Individual Gases Methane (CH4)

Atmospheric methane plays an important role in the Earth’s radiative budget as a potent greenhouse gas, which is 3.7 times more effective than carbon dioxide in absorbing infrared radiation. Despite being present in trace quantities, atmospheric methane

plays an important role controlling atmospheric chemistry, including serving as a regulator of tropospheric ozone concentrations and a major sink for hydroxyl radicals in the stratosphere. Methane concentrations have been increasing annually at the rate of approximately 1–2% over the last two centuries. While the contribution of estuaries to the global atmospheric methane budget is small because of the relatively small estuarine global surface area, estuaries have been identified as sources of methane to the atmosphere and coastal ocean and contribute a significant fraction of the marine methane emissions to the atmosphere. Surface methane concentrations reported primarily from estuaries in North America and Europe range from 1 to 42000 nM throughout the tidal portion of the estuaries. Methane in estuarine surface waters is generally observed to be supersaturated (100% saturation B2–3 nM CH4) with R-values ranging from 0.7 to 1600 (Table 1). In general, estuarine methane concentrations are highest at the freshwater end of the estuary and decrease with salinity. This trend reflects riverine input as the major source of methane to most estuaries, with reported riverine methane concentrations ranging from 5 to 10 000 nM. Estuaries with large plumes have been observed to cause elevated methane concentrations in adjacent coastal oceans. In addition to riverine input, sources of methane to estuaries include intertidal flats and marshes, ground-water input, runoff

Table 1 Methane saturation values (R) and estimated fluxes to the atmosphere for US and European estuaries Geographical regiona

Rb

Flux CH4 (mmol m2 h1)

North Pacific coast, USA North Pacific coast, USA Columbia River, USA Hudson River, USA Tomales Bay, CA, USA Baltic Sea, Germany European Atlantic coast Atlantic coast, USA Pettaquamscutt Estuary, USA

3–290 1–550 78 18–376 2–37 10.5–1550 0.7–1580 n.a. 81–111

6.2–41.7c 3.6–8.3c 26.0c 4.7–40.4c 17.4–26.3c 9.4–15.6c 5.5c 102–1107c 0.8–14.2c 541–3375d

a

For studies that report values for a single estuary, the major river feeding the estuary is provided. For studies that report values for more than one estuary, the oceanic area being fed by the estuaries is given. b R ¼ degree of saturation ¼ measured concentration/atmospheric equilibrium concentration. n.a. indicates values not available in reference. c Diffusive flux. d Ebullition (gas bubble) flux.

GAS EXCHANGE IN ESTUARIES

from agricultural and pasture land, petroleum pollution, lateral input from exposed bank soils, wastewater discharge and emission from organic-rich anaerobic sediments, either diffusively or via ebullition (transport of gas from sediments as bubbles) and subsequent dissolution within the water column. Anthropogenically impacted estuaries or estuaries supplied from impacted rivers tend to be characterized by higher water-column methane concentrations relative to pristine estuarine systems. Seasonally, methane levels in estuaries are higher in summer compared with winter, primarily due to increased bacterial methane production (methanogenesis) in estuarine and riverine sediments. Methane can be removed from estuarine waters by microbial methane oxidation and emission to the atmosphere. Methane oxidation within estuaries can be quite rapid, with methane turnover times of o2 h to several days. Methane oxidation appears to be most rapid at salinities of less than about 6 (on the practical salinity units scale) and is strongly dependent on temperature, with highest oxidation rates occurring during the summer, when water temperatures are highest. Methane oxidation rates decrease rapidly with higher salinities. Methane diffusive fluxes to the atmosphere reported for estuaries (Table 1) fall within a narrow range of 3.6–41.7 mmol m2 h1 (2–16 mg CH4 m2 day1). Using a global surface area for estuaries of 1.4  106 km2 yields an annual emission of methane to the atmosphere from estuaries of 1–8 Tg y1. Because the higher flux estimates given in Table 1 generally were obtained close to the freshwater endmember of the estuary, the global methane estuarine emission is most likely within the range of 1– 3 Tg y1, corresponding to approximately 10% of the total global oceanic methane flux to the atmosphere, despite the much smaller global surface area of estuaries relative to the open ocean. Methane is also released to the atmosphere directly from anaerobic estuarine sediments via bubble formation and injection into the water column. Although small amounts of methane from bubbles may dissolve within the water column, the relatively shallow nature of the estuarine environment results in the majority of methane in bubbles reaching the atmosphere. The quantitative release of methane via this mechanism is difficult to evaluate due to the irregular and sporadic spatial and temporal extent of ebullition. Where ebullition occurs, the flux of methane to the atmosphere is considerably higher than diffusive flux (Table 1), but the areal extent of bubbling is relatively smaller than that of diffusive flux and, except in organic-rich stagnant areas such as tidal marshes, probably does not contribute

187

significantly to estuarine methane emissions to the atmosphere. Methane emission via ebullition has been observed to be at least partially controlled by tidal changes in hydrostatic pressure, with release of methane occurring at or near low tide when hydrostatic pressure is at a minimum. Nitrous Oxide (N2O)

Nitrous oxide is another important greenhouse gas that is present in elevated concentrations in estuarine environments. At present, N2O is responsible for about 5–6% of the anthropogenic greenhouse effect and is increasing in the atmosphere at a rate of about 0.25% per year. However, the role of estuaries in the global budget of the gas has only been addressed recently. Nitrous oxide is produced primarily as an intermediate during both nitrification (the oxidation of ammonium to nitrate) and denitrification (the reduction of nitrate, via nitrite and N2O, to nitrogen gas), although production by dissimilatory nitrate reduction to ammonium is also possible. In estuaries, nitrification and denitrification are both thought to be important sources. Factors such as the oxygen level in the estuary and the nitrate and ammonium concentrations of the water can influence which pathway is dominant, with denitrification dominating at very low, but non-zero, oxygen concentrations. Nitrous oxide concentrations are typically highest in the portions of the estuary closest to the rivers, and decrease with distance downstream. A number of workers have reported nitrous oxide maxima in estuarine waters at low salinities (o5–10 on the PSU scale), but this is not always the case. The turbidity maximum has been reported to be the site of maximum nitrification (presumably because of increased residence time for bacteria attached to suspended particulate matter, combined with elevated substrate (oxygen and ammonium)). Table 2 presents a summary of the data published for degree of saturation and air–estuary flux of nitrous oxide from a variety of estuaries, all of which are located in Europe and North America. Concentrations are commonly above that predicted from air–sea equilibrium, and estimates of fluxes range from 0.01 mmol m2 h1 to 5 mmol m2 h1. Ebullition is not important for nitrous oxide because it is much more soluble than methane. Researchers have estimated the size of the global estuarine source for N2O based on fluxes from individual estuaries multiplied by the global area occupied by estuaries to range from 0.22 Tg N2O y1 to 5.7 Tg y1 depending on the characteristics of the rivers studied. Independent estimates based on budgets of nitrogen

188

GAS EXCHANGE IN ESTUARIES

Table 2 Nitrous oxide saturation values (R) and estimated fluxes to the atmosphere for US and European estuaries Estuary

R

Flux a N2O (mmol m2 h1)

Europe Gironde River Gironde River Oder River Elbe Scheldt Scheldt

1.1–1.6 E1.0–3.2 0.9–3.1 2.0–16 E1.0–31 E1.2–30

n.a. n.a. 0.014–0.165 n.a. 1.27–4.77 3.56

UK Colne Tamar Humber Tweed Mediterranean Amvraikos Gulf

0.9–13.6 1–3.3 2–40 0.96–1.1

0.9–1.1

1.3 0.41 1.8 E0

Table 3 Fluxes of carbon dioxide from estuaries in Europe and eastern USA Estuary

European rivers Northern Europe Scheldt estuary Portugal UK Clyde estuary East coast USA Hudson River (tidal freshwater) Georgia rivers

R

Fluxa CO2 (mmol m2 h1)

0.7–61.1 0.35–26.2 1.6–15.8 1.1–14.4 E0.7–1.8

1.0–31.7 4.2–50 10–31.7 4.4–10.4 n.a.

1.2–5.4

0.67–1.54

Slight supersaturation to 22.9

1.7–23

a n.a., insufficient data were available to permit calculation of this value.

0.043 7 0.0468

North-west USA Yaquina Bay Alsea River

1.0–4.0 0.9–2.4

0.165–0.699 0.047–0.72

East coast USA Chesapeake Bay Merrimack

0.9–1.4 1.2–4.5

n.a. n.a.

a

All fluxes given are for diffusive flux to the atmosphere. n.a. indicates that insufficient data were given to permit calculation of flux.

input to rivers, assumptions about the fraction of inorganic nitrogen species removed by nitrification or denitrification, and the fractional ‘yield’ of nitrous oxide production during these processes indicate that nitrous oxide fluxes to the atmosphere from estuaries is about 0.06–0.34 Tg N2O y1. Carbon Dioxide (CO2) and Oxygen (O2)

Estuaries are typically heterotrophic systems, which means that the amount of organic matter respired within the estuary exceeds the amount of organic matter fixed by primary producers (phytoplankton and macrophytes). Since production of carbon dioxide then exceeds biological removal of carbon dioxide, it follows that estuaries are likely to be sources of the gas to the atmosphere. At the same time, since oxidation of organic matter to CO2 requires oxygen, the heterotrophic nature of estuaries suggests that they represent sinks for atmospheric oxygen. In many estuaries, primary productivity is severely limited by the amount of light that penetrates into the water due to high particulate loadings in the water. In addition, large amounts of organic matter may be supplied to the estuary by runoff from agricultural and forested land, from ground water, from

sewage effluent, and from organic matter in the river itself. There are many reports of estuarine systems with oxygen saturations below 1 (undersaturated with respect to the atmosphere), but few studies in which oxygen flux to the estuary has been reported. However, estuaries are often dramatically supersaturated with respect to saturation with CO2, especially at low salinities, and a number of workers have reported estimates of carbon dioxide flux from these systems (Table 3).

Dimethylsulfide (DMS)

DMS is an atmospheric trace gas that plays important roles in tropospheric chemistry and climate regulation. In the estuarine environment, DMS is produced primarily from the breakdown of the phytoplankton osmoregulator 3-(dimethylsulfonium)-propionate (DMSP). DMS concentrations reported in estuaries are generally supersaturated, ranging from 0.5 to 22 nM, and increase with increasing salinity. DMS levels in the water column represent a balance between tightly coupled production from DMSP and microbial consumption. Only 10% of DMS produced from DMSP in the estuarine water column is believed to escape to the atmosphere from estuarine surface water, since the biological turnover of DMS (turnover time of 3–7 days) is approximately 10 times faster than DMS exchange across the air/water interface. A large part of the estuarine DMS flux to the atmosphere may occur over short time periods on the order of weeks, corresponding to phytoplankton blooms. The DMS flux for an estuary in Florida, USA, was estimated to be on the order of o1 nmoles m2 h1. Insufficient

GAS EXCHANGE IN ESTUARIES

data are available to determine reliable global DMS air/water exchanges from estuaries. Hydrogen (H2)

Hydrogen is an important intermediate in many microbial catabolic reactions, and the efficiency of hydrogen transfer among microbial organisms within an environment helps determine the pathways of organic matter decomposition. Hydrogen is generally supersaturated in the surface waters of the few estuaries that have been analyzed for dissolved hydrogen with R-values of 1.5–67. Hydrogen flux to the atmosphere from estuaries has been reported to be on the order of 0.06–0.27 nmol m2 h1. The contribution of estuaries to the global atmospheric H2 flux cannot be determined from the few available data. Carbon Monoxide (CO)

Carbon monoxide in surface waters is produced primarily from the photo-oxidation of dissolved organic matter by UV radiation. Since estuarine waters are characterized by high dissolved organic carbon levels, the surface waters of estuaries are highly supersaturated and are a strong source of CO to the atmosphere. Reported R-values for CO range from approximately 10 to 410 000. Because of the highly variable distributions of dissolved CO within surface waters (primarily as the result of the highly variable production of CO), it is impossible to derive a meaningful value for CO emissions to the atmosphere from estuaries. Carbonyl Sulfide (OCS)

Carbonyl sulfide makes up approximately 80% of the total sulfur content of the atmosphere and is the major source of stratospheric aerosols. Carbonyl sulfide is produced within surface waters by photolysis of dissolved organosulfur compounds. Therefore, surface water OCS levels within estuaries exhibit a strong diel trend. Carbonyl sulfide is also added to the water column by diffusion from anoxic sediments, where its production appears to be coupled to microbial sulfate reduction. Diffusion of OCS from the sediment to the water column accounts for B75% of the OCS supplied to the water column and is responsible for the higher OCS concentrations in estuaries relative to the open ocean. While supersaturations of OCS are observed throughout estuarine surface waters, no trends with salinity have been observed. Atmospheric OCS fluxes to the atmosphere from Chesapeake Bay have been reported to range from 10.4 to 56.2 nmol m2 h1. These areal fluxes are over 50 times greater than those determined for the open ocean.

189

Elemental Mercury (Hg0)

Elemental mercury is produced in estuarine environments by biologically mediated processes. Both algae and bacteria are able to convert dissolved inorganic mercury to volatile forms, which include organic species (monomethyl- and dimethyl-mercury) and Hg0. Under suboxic conditions, elemental mercury also may be the thermodynamically stable form of the metal. In the Scheldt River estuary, Hg0 correlated well with phytoplankton pigments, suggesting that phytoplankton were the dominant factors, at least in that system. Factors that may affect elemental mercury concentrations include the type of phytoplankton present, photo-catalytic reduction of ionic Hg in surface waters, the extent of bacterial activity that removes oxygen from the estuary, and removal of mercury by particulate scavenging and sulfide precipitation. Fluxes of elemental mercury to the atmosphere have been estimated for the Pettaquamscutt estuary in Rhode Island, USA, and for the Scheldt, and range from 4.2–29 pmol m2 h1, although the values are strongly dependent on the model used to estimate gas exchange coefficients. Volatile Organic Compounds (VOCs)

In addition to gases produced naturally in the environment, estuaries tend to be enriched in byproducts of industry and other human activity. A few studies have investigated volatile organic pollutants such as chlorinated hydrocarbons (chloroform, tetrachloromethane, 1,1-dichloroethane, 1,2-dichloroethane, 1,1,1-trichloroethane, trichloroethylene and tetrachloroethylene) and monocyclic aromatic hydrocarbons (benzene, toluene, ethylbenzene, oxylene and m- and p-xylene). Concentrations of VOCs are controlled primarily by the location of the sources, dilution of river water with clean marine water within the estuary, gas exchange, and in some cases, adsorption onto suspended or settling solids. In some cases (for example, chloroform) there also may be natural biotic sources of the gas. Volatilization to the atmosphere can be an important ‘cleansing’ mechanism for the estuary system. Since the only estuaries studied to date are heavily impacted by human activity (the Elbe and the Scheldt), it is not possible to make generalizations about the importance of these systems on a global scale.

Conclusions Many estuaries are supersaturated with a variety of gases, making them locally, and occasionally

190

GAS EXCHANGE IN ESTUARIES

regionally, important sources to the atmosphere. However, estuarine systems are also highly variable in the amount of gases they contain. Since most estuaries studied to date are in Europe or the North American continent, more data are needed before global budgets can be reliably prepared. Air–water interface: The boundary between the gaseous phase (the atmosphere) and the liquid phase (the water). Catabolic: Biochemical process resulting in breakdown of organic molecules into smaller molecules yielding energy. Denitrification: Reduction of nitrate via nitrite to gaseous endproducts (nitrous oxide and dinitrogen gas). Ebullition: Gas transport by bubbles, usually from sediments. Estuary: Semi-enclosed coastal body of water with free connection to the open sea and within which sea water is measurably diluted with fresh water derived from land drainage. Gas solubility: The amount of gas that will dissolve in a liquid when the liquid is in equilibrium with the overlying gas phase. Henry’s Law Constant: Proportionality constant relating the vapor pressure of a solute to its mole fraction in solution. Liquid laminar thickness: The thickness of a layer at the air/water interface where transport of a dissolved species is controlled by molecular (rather than turbulent) diffusion. Molecular diffusivity (D): The molecular diffusion coefficient. Nitrification: Oxidation of ammonium to nitrite and nitrate. Practical salinity scale: A dimensionless scale for salinity. Redfield stoichiometry: Redfield and colleagues noted that organisms in the sea consistently removed nutrient elements from the water in a fixed ratio (C : N : P ¼ 106 : 16 : 1). Subsequent workers have found that nutrient concentrations in the sea typically are present in those same ratios. Transfer coefficient: The rate constant which determines the rate of transfer of gas from liquid to gas phase.

See also Air–Sea Gas Exchange. Air–Sea Transfer: Dimethyl Sulfide, COS, CS2, NH4, Non-Methane Hydrocarbons, Organo-Halogens. Air–Sea Transfer: N2O, NO, CH4, CO.

Further Reading Bange HW, Rapsomanikis S, and Andreae MO (1996) Nitrous oxide in coastal waters. Global Biogeochemical Cycles 10: 197--207. Bange HW, Dahlke S, Ramesh R, Meyer-Reil L-A, Rapsomanikis S, and Andreae MO (1998) Seasonal study of methane and nitrous oxide in the coastal waters of the southern Baltic Sea. Estuarine, Coastal and Shelf Science 47: 807--817. Barnes J and Owens NJP (1998) Denitrification and nitrous oxide concentrations in the Humber estuary, UK and adjacent coastal zones. Marine Pollution Bulletin 37: 247--260. Cai W-J and Wang Y (1998) The chemistry, fluxes and sources of carbon dioxide in the estuarine waters of the Satilla and Altamaha Rivers, Georgia. Limnology and Oceanography 43: 657--668. de Angelis MA and Lilley MD (1987) Methane in surface waters of Oregon estuaries and rivers. Limnology and Oceanography 32: 716--722. de Wilde HPJ and de Bie MJM (2000) Nitrous oxide in the Schelde estuary: production by nitrification and emission to the atmosphere. Marine Chemistry 69: 203--216. Elkins JW, Wofsy SC, McElroy MB, Kolb CE, and Kaplan WA (1978) Aquatic sources and sinks for nitrous oxide. Nature 275: 602--606. Frankignoulle M, Abril G, Borges A, et al. (1998) Carbon dioxide emission from European estuaries. Science 282: 434--436. Frost T and Upstill-Goddard RC (1999) Air–sea exchange into the millenium: progress and uncertainties. Oceanography and Marine Biology: An Annual Review 37: 1--45. Law CS, Rees AP, and Owens NJP (1992) Nitrous oxide: estuarine sources and atmospheric flux. Estuarine, Coastal and Shelf Science 35: 301--314. Liss PS and Merlivat L (1986) Air–sea gas exchange rates: introduction and synthesis. In: Baut-Menard P (ed.) The Role of Air–Sea Exchange in Geochemical Cycling, pp. 113--127. Dordrecht: Riedel. Muller FLL, Balls PW, and Tranter M (1995) Processes controlling chemical distributions in the Firth of Clyde (Scotland). Oceanologica Acta 18: 493--509. Raymond PA, Caraco NF, and Cole JJ (1997) Carbon dioxide concentration and atmospheric flux in the Hudson River. Estuaries 20: 381--390. Robinson AD, Nedwell DB, Harrison RM, and Ogilvie BG (1998) Hypernutrified estuaries as sources of N2O emission to the atmosphere: the estuary of the River Colne, Essex, UK. Marine Ecology Progress Series 164: 59--71. Sansone FJ, Rust TM, and Smith SV (1998) Methane distribution and cycling in Tomales Bay, California. Estuaries 21: 66--77. Sansone FJ, Holmes ME, and Popp BN (1999) Methane stable isotopic ratios and concentrations as indicators of

GAS EXCHANGE IN ESTUARIES

methane dynamics in estuaries. Global Biogeochemical Cycles 463--474. Scranton MI, Crill P, de Angelis MA, Donaghay PL, and Sieburth JM (1993) The importance of episodic events in controlling the flux of methane from an anoxic basin. Global Biogeochemical Cycles 7: 491--507.

191

Seitzinger SP and Kroeze C (1998) Global distribution of nitrous oxide production and N input in freshwater and coastal marine ecosystems. Global Biogeochemical Cycles 12: 93--113.

PENETRATING SHORTWAVE RADIATION C. A. Paulson and W. S. Pegau, Oregon State University, Corvallis, OR, USA Copyright & 2001 Elsevier Ltd.

Introduction The penetration of solar radiation into the upper ocean has important consequences for physical, chemical, and biological processes. The principal physical process is the heating of the upper layers by the absorption of solar radiation. To estimate the solar radiative heating rate, the net downward shortwave irradiance entering the ocean and the rate of absorption of this energy as a function of depth must be determined. Shortwave irradiance is the flux of solar energy incident on a plane surface (W m2). Given the downward shortwave radiance field just above the sea surface, the rate of shortwave absorption as a function of depth is governed primarily by sea surface roughness, molecular structure of pure sea water, suspended particles, and dissolved organic compounds. The optical properties of pure sea water are considered a baseline; the addition of particles and dissolved compounds increases absorption and scattering of sunlight. The dissolved organic compounds are referred to as ‘colored dissolved organic matter’ (CDOM) or ‘yellow matter’ because they color the water yellowish-brown. The source of CDOM is decaying plants; concentrations are highest in coastal waters. Suspended particles may be of biological or geological origin. Biological (organic) particles are formed as the result of the growth of bacteria, phytoplankton, and zooplankton. The source of geological (inorganic) particles is primarily weathering of terrestrial soils and rocks that are carried to the ocean by the wind and rivers. Phytoplankton particles are the main determinant of optical properties in much of the ocean and the concentration of chlorophyll associated with these plants is used to quantify the effect of phytoplankton on optical properties. Case 1 waters are defined as waters in which the concentration of phytoplankton is high compared with inorganic particles and dissolved compounds; roughly 98% of the world ocean falls into this category. Case 2 waters are waters in which inorganic particles or CDOM are the dominant influence on optical properties. Case 2 waters are usually coastal, but not all coastal water is case 2.

192

Inherent optical properties (IOPs), such as attenuation of a monochromatic beam of light, depend only on the medium, i.e. IOPs are independent of the ambient light field. It is often assumed that the inherent optical properties of the upper ocean are independent of depth. To the extent that the upper ocean is well-mixed, the assumption of homogeneous optical properties is reasonable. However, in the stratified layers below the mixed layer, the concentration of particles is likely to vary with depth. The consequences of this variation on radiant heating are expected to be small because the magnitude of the downward irradiance decreases rapidly with depth. Apparent optical properties (AOPs) depend both on the medium and on the directional properties of the ambient light field. Some AOPs, such as the ratio of downward irradiance in the ocean to the surface value, are sufficiently independent of directional properties of the light field to be useful for characterizing the optical properties of a water body.

Albedo Albedo, A, is the ratio of upward to downward short-wave irradiance just above the sea surface and is defined by: A

Eu Ed

where Eu and Ed are the upwelling and downward irradiances just above the sea surface, respectively. The upwelling irradiance is composed of two components: emergent irradiance due to back-scattered light from below the sea surface; and irradiance reflected from the sea surface. Emergent irradiance is typically o10% of reflected irradiance. The rate at which net short-wave irradiance penetrates the sea surface is the rate at which the sea absorbs solar energy and is given by: ð1  AÞ Ed ðWm2 Þ R.E. Payne analyzed observations to represent albedo as a function of solar altitude y and atmospheric transmittance G defined by: G ¼ Ed r2 =S siny where S is the solar constant (1370 W2) and r is the ratio of the actual to mean Earth–sun separation. The transmittance is a measure of the effect of the

PENETRATING SHORTWAVE RADIATION

Table 1 Latitude 801N 701N 601N 501N 401N 301N 201N 101N 01 101S 201S 301S 401S 501S 601S

193

Mean albedos for the Atlantic Ocean by month and latitude Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

0.28 0.11 0.10 0.09 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06

0.41 0.12 0.10 0.09 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07

0.33 0.15 0.09 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.08

0.14 0.10 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.08 0.08 0.11

0.10 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.08 0.09 0.10 0.13

0.09 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.09 0.11 0.13

0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.08 0.10 0.11 0.27

0.08 0.09 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.08 0.08 0.07

0.12 0.11 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.08 0.08

0.25 0.10 0.08 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.07 0.07

0.16 0.11 0.10 0.08 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

0.44 0.12 0.11 0.09 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06

(Reproduced with permission from Payne, 1972.)

Earth’s atmosphere, including clouds, on the radiance distribution at the Earth’s surface. If there were no atmosphere, the transmittance would equal one and the radiance would be a direct beam from the sun. For very heavy overcast, the transmittance can be o0.1 and the downward radiance distribution may be approximately independent of direction. Payne’s observations were taken from a fixed platform off the coast of Massachusetts from 25 May to 28 September. Solar altitude ranged up to 721 and the mean wind speed was 3.7 m s1. The transmittance varied from near zero to about 0.75. Payne fitted smooth curves to the albedo as a function of transmittance for observations in intervals of 21 of solar altitude and 0.1 in transmittance. The smoothed albedos ranged from 0.03 to 0.5. Payne extrapolated the curves to values of solar altitude and transmittance for which there were no observations by use of theoretical calculations of reflectance for a sea surface roughened by a wind speed of 3.7 m s1. Albedo was obtained for the limiting case of G ¼ 1 by adding 0.005 to the calculated reflectance to account for the irradiance emerging from beneath the surface. Wind speed affects albedo through its influence on the surface roughness. The clear-sky reflectivity from a flat water surface is a strong function of solar altitude for altitudes o301. As wind speed and roughness increase, reflectivity decreases because of the nonlinear relationship between reflectivity and the incidence angle. Payne investigated the variation of albedo with wind speed for solar elevations from 171 to 251 and found that albedo decreased at the rate of 2% per meter per second increase in wind speed.

Wind speed may also affect albedo by the generation of breaking waves that produce white caps. The albedo of whitecaps is higher, on average, than the whitecap-free sea surface. Hence the qualitative effect of whitecaps is to increase the albedo. Monahan and O’Muircheartaigh estimate an increase of 10% in albedo due to whitecaps for a wind speed of 15 m s1 and of 20% for a wind speed of 20 m s1. Payne estimated monthly climatological values of albedo at 101 latitude intervals for the Atlantic Ocean (Table 1). For the range of latitudes from 401S to 401N, mean albedo varies from a minimum of 0.06 at the equator in all months to a maximum of 0.11 at 401S and 401N for the months containing the winter solstice. The symmetry exhibited by mean albedo values for the winter solstice in the North and South Atlantic suggests that the values in Table 1 may be reasonable estimates for the world ocean.

Spectrum of Downward Irradiance The spectrum of downward short-wave irradiance at various depths in the ocean (Figure 1) illustrates the strong dependence of absorption on wavelength. The shape of the spectrum at the surface is determined primarily by the temperature of the sun (Wien displacement law) and wavelength-dependent absorption by the atmosphere. The area under the spectrum at each depth is proportional to the downward irradiance. The downward irradiance at a depth of 1 m is less than half the surface value because of preferential absorption at wavelengths in excess of 700 nm. The downward irradiance below 10 m depth is in a relatively narrow band centered

194

PENETRATING SHORTWAVE RADIATION

_

_

Downward irradiance (W m 2 nm 1)

1.5

1.0

0.5 10 m Surface 1m 1 cm

100 m

500

1000

1500

2000

2500

Wavelength Figure 1 The spectrum of downward irradiance Ed(z, l) in the sea at different depths. (Adapted with permission from Jerlov, 1976.)

Modeled Irradiance 10

4

Radiative transfer models are useful tools for investigating the characteristics of underwater light fields and their dependence on suspended particles and dissolved organic matter. The Hydrolight model, constructed by Mobley, is used to illustrate the diffuse attenuation coefficient for downward irradiance for three cases with different concentrations of chlorophyll and values of CDOM beam attenuation at 440 nm (Figure 2). The diffuse attenuation coefficient for downward irradiance Kd is defined by

_ Kd (m 1)

103 102 101 100 _ 10 1

200

500

800

1100 1400 1700 2000 2300 Wavelength (nm)

Figure 2 The diffuse attenuation coefficient for downward irradiance in sea water versus wavelength. Data for the wavelength band from 350 to 800 nm are from the Hydrolight radiative transfer model with the following conditions: depth 10 m, solar altitude 601, cloudless sky, wind speed 2 m s1. The three lines (thin, thick, dashed) result from specified concentrations of chlorophyll (0.05, 1, 10 mg m3) and the beam attenuation coefficient at 440 nm for CDOM (0.01, 0.05, 0.1 m1). Data for the wavelength band from 800 to 2200 nm are for pure water. (Tabulations taken from Kuo et al., 1993.)

near 470 nm (blue-green). Pure sea water is most transparent near a wavelength of 450 nm which, by coincidence, is close to the peak in the downward irradiance spectrum at the surface.

Kd ðz;lÞ ¼

d lnEd ðz;lÞ dz

where z is the vertical space coordinate, zero at the surface and positive upward, and l is the wavelength. If Kd is independent of z, monochromatic irradiance decreases exponentially with depth, consistent with Beer’s law. Kd increases five orders of magnitude as wavelength increases from 500 to 2000 nm (Figure 2), consistent with the strong dependence of absorption on wavelength shown in Figure 1. In the wavelength band centered around 500 nm, Kd varies by a factor of 10 between the least absorbent and most absorbent cases. The order of magnitude variation in Kd at 500 nm among the three cases (Figure 2) has a dramatic effect

PENETRATING SHORTWAVE RADIATION

195

0

0

10 60

Depth (m)

Depth (m)

30

90

120

150 _ 10 4

20

30 _3

10

_ 10 2 E / E(0)

_1

10

100

40 0.01

Figure 3 Net irradiance E ¼ (Ed  Eu) versus depth from the Hydrolight model for the conditions specified in the caption for Figure 2.

on net irradiance (Figure 3) because the peak of the solar spectrum is near 500 nm (Figure 1). Net irradiance, E, is the difference between net (wavelength integrated) downward and net upward irradiance. The difference between downward and net irradiance is negligible for most purposes because upward irradiance is typically 3% of downward irradiance within the ocean. At depths where the net downward irradiance is o10% of its surface value, the decay with z is approximately exponential because light at these depths is roughly monochromatic. The three cases with different optical properties (Figures 2 and 3) can be characterized biologically as oligotrophic, mesotrophic, and eutrophic (ranging from least to most absorbent). Oligotrophic water has low biological production and low nutrients. Eutrophic water has high biological production and high nutrients and mesotrophic water is moderate in both respects. The oligotrophic case illustrated in Figures 2 and 3 is typical of open-ocean water. Mesotrophic and eutrophic water are likely to be found near the coast.

0.1 Ed / Ed(0)

1.0

Figure 4 Measurements in the upper 40 m of downward irradiance normalized by downward irradiance just below the surface. The measurements were made in the North Pacific (351N, 1551W) in February. The open circles are the average of five sets of observations with similar irradiance profiles; solar altitude ranged from 301 to 381 and the sky was overcast. The plus signs show one set of observations for which solar altitude was 161 and the sky was clear. The curves are the sum of two exponential terms fitted to the observations (see eqn [1]). (Adapted with permission from Paulson and Simpson, 1977.)

Table 2 Values of parameters determined by fitting the sum of two exponential terms (see eqn [1]) to values of downward irradiance which define Jerlov’s (1976) water types Water type

F1

K1 (m1)

K2 (m1)

C (mg m1)

I IA IB II III

0.32 0.38 0.33 0.23 0.22

0.036 0.049 0.058 0.069 0.13

0.8 1.7 1.0 0.7 0.7

0–0.01 B0.05 B0.1 B0.5 B1.5–2.0

Values of downward irradiance in the upper 100 m were used, except that values were limited to the upper 50 m for type I because of a change in slope below 50 m. F2 is 1  F1. (Adapted from Paulson and Simpson, 1977.) The column labeled C is the approximate chlorophyll concentration for each water type as determined by Morel (1988).

Parameterized Irradiance versus Depth Observations show that downward short-wave irradiance decreases exponentially with depth below a depth of about 10 m (Figure 4) as the result of absorption by the overlying sea water of all irradiance except for blue-green light. This suggests that Ed can be approximated by a sum of n exponential terms: Ed =E0 ¼

n X i¼l

Fi expðKi zÞ

½1

n X

Fi ¼ 1

i¼l

where Fi is the fraction of downward irradiance in a wavelength band i and Ki is the diffuse attenuation coefficient for the same band. The leading term in eqn [1] is defined as the short wavelength band that describes the exponential decay below 10 m (Figure 4). At least one additional term is required. A total of two terms fits the observations in Figure 4

196

PENETRATING SHORTWAVE RADIATION

Depth (m)

0

20

III

II

I

40

0.001

0.01

0.1

1.0

Ed / Ed(0) Figure 5 Normalized downward irradiance versus depth for water types I, II, and III. The data (open circles, squares, and triangles) are from Jerlov (1976) and the curves are a fit to the data with the parameters given in Table 2. (Adapted with permission from Paulson and Simpson, 1977.)

reasonably well, although accuracy in the upper few meters is lacking. Jerlov has proposed a scheme for classifying oceanic waters according to their clarity. He defined five types (I, IA, IB, II, and III) ranging from the clearest open-ocean water (type I) to increasingly turbid water. Parameters for the sum of two exponential terms (eqn [1]) fit to the values of downward irradiance which define the Jerlov water types are given in Table 2 and plots of values and fitted curves are shown in Figure 5. Apart from systematic disagreement in the upper few meters, the fit is good. Differences in the values of K2 in Table 2 are not significant. Water types IA and IB are not shown in Figure 5. However, the fits to the observations shown in Figure 4 yield parameters very similar to those for types IA and IB (open circles and plus signs, respectively, in Figure 4). Most open-ocean water is intermediate between types I and II. The approximate chlorophyll concentration for each of the Jerlov water types is given in Table 2. The Jerlov water types can be compared to the oligotrophic, mesotrophic, and eutrophic cases illustrated in Figures 2 and 3. The oligotrophic case is intermediate between types IA and IB. The mesotrophic case is similar to type III and the eutrophic case is similar to Jerlov’s coastal type 5 water. The sum of two exponential terms (eqn [1]) is adequate for modeling purposes when the required

vertical resolution is a few meters or greater. For a vertical resolution of 1 m or less, additional terms are required. These additional terms can be constructed with knowledge of the surface irradiance spectrum (Figure 1) and the diffuse attenuation coefficient versus wavelength (Figure 2).

See also Heat and Momentum Fluxes at the Sea Surface. Radiative Transfer in the Ocean. Upper Ocean Heat and Freshwater Budgets. Wind- and BuoyancyForced Upper Ocean.

Further Reading Dera J (1992) Marine Physics. Amsterdam: Elsevier. Jerlov NG (1976) Marine Optics. Amsterdam: Elsevier. Kou L, Labrie D, and Chylek P (1993) Refractive indices of water and ice in the 0.65- to 2.5-mm spectral range. Applied Optics 32: 3531--3540. Kraus EB and Businger JA (1994) Atmosphere–Ocean Interaction, 2nd edn. New York: Oxford University Press. Mobley CD (1994) Light and Water. San Diego: Academic Press. Mobley CD and Sundman LK (2000) Hydrolight 4.1 User’s Guide. Redmond, WA: Sequoia Scientific.

PENETRATING SHORTWAVE RADIATION

Monahan EC and O’Muircheartaigh IG (1987) Comments on glitter patterns of a wind-roughened sea surface. Journal of Physical Oceanography 17: 549--550. Morel A (1988) Optical modeling of the upper ocean in relation to its biogenous matter content. Journal of Geophysical Research 93: 10749--10768. Paulson CA and Simpson JJ (1977) Irradiance measurements in the upper ocean. Journal of Physical Oceanography 7: 952--956.

197

Payne RE (1972) Albedo of the sea surface. Journal of Atmospheric Sciences 29: 959--970. Thomas GE and Stamnes K (1999) Radiative Transfer in the Atmosphere and Ocean. Cambridge: Cambridge University Press. Tyler JE and Smith RC (1970) Measurements of Spectral Irradiance Underwater. New York: Gordon and Breach.

RADIATIVE TRANSFER IN THE OCEAN C. D. Mobley, Sequoia Scientific Inc., WA, USA Copyright & 2001 Elsevier Ltd.

Introduction Understanding how light interacts with sea water is a fascinating problem in itself, as well as being fundamental to fields as diverse as biological primary production, mixed-layer thermodynamics, photochemistry, lidar bathymetry, ocean-color remote sensing, and visual searching for submerged objects. For these reasons, optics is one of the fastest growing oceanographic research areas. Radiative transfer theory provides the theoretical framework for understanding light propagation in the ocean, just as hydrodynamics provides the framework for physical oceanography. The article begins with an overview of the definitions and terminology of radiative transfer as used in oceanography. Various ways of quantifying the optical properties of a water body and the light within the water are described. The chapter closes with examples of the absorption and scattering properties of two hypothetical water bodies, which are characteristic of the open ocean and a turbid estuary, and a comparison of their underwater light fields.

Terminology The optical properties of sea water are sometimes grouped into inherent and apparent properties.

•

•

Inherent optical properties (IOPs) are those properties that depend only upon the medium and therefore are independent of the ambient light field. The two fundamental IOPs are the absorption coefficient and the volume scattering function. (These quantities are defined below.) Apparent optical properties (AOPs) are those properties that depend both on the medium (the IOPs) and on the directional structure of the ambient light field, and that display enough regular features and stability to be useful descriptors of a water body. Commonly used AOPs are the irradiance reflectance, the remote-sensing reflectance, and various diffuse attenuation functions.

‘Case 1 waters’ are those in which the contribution by phytoplankton to the total absorption and

198

scattering is high compared to that by other substances. Absorption by chlorophyll and related pigments therefore plays the dominant role in determining the total absorption in such waters, although covarying detritus and dissolved organic matter derived from the phytoplankton also contribute to absorption and scattering in case 1 waters. Case 1 water can range from very clear (oligotrophic) to very productive (eutrophic) water, depending on the phytoplankton concentration. ‘Case 2 waters’ are ‘everything else,’ namely, waters where inorganic particles or dissolved organic matter from land drainage contribute significantly to the IOPs, so that absorption by pigments is relatively less important in determining the total absorption. Roughly 98% of the world’s open ocean and coastal waters fall into the case 1 category, but near-shore and estuarine case 2 waters are disproportionately important to human interests such as recreation, fisheries, and military operations. Table 1 summarizes the terms, units, and symbols for various quantities frequently used in optical oceanography.

Radiometric Quantities Consider an amount DQ of radiant energy incident in a time interval Dt centered on time t, onto a surface of area DA located at position (x,y,z), and arriving through a set of directions contained in a solid angle DO about the direction (y, j) normal to the area DA, as produced by photons in a wavelength interval Dl centered on wavelength l. The geometry of this situation is illustrated in Figure 1. Then an operational definition of the spectral radiance is Lðx; y; z; t; y; j; lÞ 

DQ DtDADODl ½Js1 m2 sr1 nm1 

½1

In the conceptual limit of infinitesimal parameter intervals, the spectral radiance is defined as Lðx; y; z; t; y; j; lÞ 

@4Q @t @A @O @l

½2

Spectral radiance is the fundamental radiometric quantity of interest in optical oceanography: it completely specifies the positional (x,y,z), temporal (t), directional (y, j), and spectral (l) structure of the light field. In many oceanic environments, horizontal

RADIATIVE TRANSFER IN THE OCEAN

Table 1

199

Quantities commonly used in optical oceanography

Quantity

SI units

Symbol

Radiometric quantities Quantity of radiant energy Power Intensity Radiance Downwelling plane irradiance Upwelling plane irradiance Net irradiance Scalar irradiance Downwelling scalar irradiance Upwelling scalar irradiance Photosynthetic available radiation

J nm1 W nm1 W sr1 nm1 W m2 sr1nm1 W m2 nm1 W m2 nm1 W m2 nm1 W m2 nm1 W m2 nm1 W m2 nm1 Photonss1 m2

Q F I L Ed Eu E Eo Eou Eou PAR

Inherent optical properties Absorption coefficient Volume scattering function Scattering phase function Scattering coefficient Backscatter coefficient Beam attenuation coefficient Single-scattering albedo

m1 m1 sr1 sr1 m1 m1 m1 –

a b b˜ b bb c oo

Apparent optical properties Irradiance reflectance (ratio) Remote-sensing reflectance Attenuation coefficients of radiance L(z, y, j) of downwelling irradiance Ed(z) of upwelling irradiance Eu(z) of PAR

– sr1 m1 m1 m1 m1 m1

R Rrs

ΔQ

ΔΩ

ΔA

x  

y

z Figure 1 Geometry used to define radiance.

K(y, j) Kd Ku KPAR

variations (on a scale of tens to thousands of meters) of the IOPs and the radiance are much less than variations with depth, in which case it can be assumed that these quantities vary only with depth z. (An exception would be the light field due to a single light source imbedded in the ocean; such a radiance distribution is inherently three-dimensional.) Moreover, since the timescales for changes in IOPs or in the environment (seconds to seasons) are much greater than the time required for the radiance to reach steady state (microseconds) after a change in IOPs or boundary conditions, time-independent radiative transfer theory is adequate for most oceanographic studies. (An exception is time-of-flight lidar bathymetry.) When the assumptions of horizontal homogeneity and time independence are valid, the spectral radiance can be written as L(z, y, j, l). Although the spectral radiance completely specifies the light field, it is seldom measured in all directions, both because of instrumental difficulties and because such complete information often is not

200

RADIATIVE TRANSFER IN THE OCEAN

needed. The most commonly measured radiometric quantities are various irradiances. Suppose the light detector is equally sensitive to photons of a given wavelength l traveling in any direction (y, j) within a hemisphere of directions. If the detector is located at depth z and is oriented facing upward, so as to collect photons traveling downward, then the detector output is a measure of the spectral downwelling scalar irradiance at depth z, Eod(z, l). Such an instrument is summing radiance over all the directions (elements of solid angle) in the downward hemisphere; thus Eod(z, l) is related to L(z, y, j, l) by Eod ðz; lÞ ¼

Z

Lðz; y; j; lÞdO ½Wm2 nm1  ½3 2pd

Here 2pd denotes the hemisphere of downward directions (i.e., the set of directions (y, j) such that 0ryrp/2 and 0rjo2p, if y is measured from the þ z or nadir direction). The integral over 2pd can be evaluated as a double integral over y and j after a specific coordinate system is chosen. If the same instrument is oriented facing downward, so as to detect photons traveling upward, then the quantity measured is the spectral upwelling scalar irradiance Eou(z, l). The spectral scalar irradiance Eo(z, l) is the sum of the downwelling and upwelling components: EO ðz; lÞ  Eod ðz; lÞ þ Eou ðz; lÞ Z ¼ Lðz; y; j; lÞdO

½4

4p

Eo(z, l) is proportional to the spectral radiant energy density (J m3 nm1) and therefore quantifies how much radiant energy is available for photosynthesis or heating the water. Now consider a detector designed so that its sensitivity is proportional to |cos y|, where y is the angle between the photon direction and the normal to the surface of the detector. This is the ideal response of a ‘flat plate’ collector of area DA, which when viewed at an angle y to its normal appears to have an area of DA|cos y|. If such a detector is located at depth z and is oriented facing upward, so as to detect photons traveling downward, then its output is proportional to the spectral downwelling plane irradiance Ed(z, l). This instrument is summing the downwelling radiance weighted by the cosine of the photon direction, thus Ed ðz; lÞ ¼

R

2pd

Lðz; y; j; lÞjcosyjdO

½Wm2 nm1 

½5

Turning this instrument upside down gives the spectral upwelling plane irradiance Eu(z, l). Ed and Eu are useful because they give the energy flux (power per unit area) across the horizontal surface at depth z owing to downwelling and upwelling photons, respectively. The difference Ed  Eu is called the net (or vector) irradiance. Photosynthesis is a quantum phenomenon, i.e., it is the number of available photons rather than the amount of radiant energy that is relevant to the chemical transformations. This is because a photon of, say, l ¼ 400 nm, if absorbed by a chlorophyll molecule, induces the same chemical change as does a photon of l ¼ 600 nm, even though the 400 nm photon has 50% more energy than the 600 nm photon. Only a part of the photon energy goes into photosynthesis; the excess is converted to heat or is re-radiated. Moreover, chlorophyll is equally able to absorb and utilize a photon regardless of the photon’s direction of travel. Therefore, in studies of phytoplankton biology, the relevant measure of the light field is the photosynthetic available radiation, PAR, defined by Z 700 nm lZ Eo ðz; lÞdl PARðzÞ  350 nm hc ½photonss1 m2 

½6

where h ¼ 6.6255  1034 J s is the Planck constant and c ¼ 3.0  1017 nm s1 is the speed of light. The factor l/hc converts the energy units of Eo to quantum units (photons per second). Bio-optical literature often states PAR values in units of mol photons s1 m2 or einst s1 m2 (where one einstein is one mole of photons).

Inherent Optical Properties Consider a small volume DV of water, of thickness Dr as illuminated by a collimated beam of monochromatic light of wavelength l and spectral radiant power Fi(l) (W nm1), as schematically illustrated in Figure 2. Some part Fa(l) of the incident power Fi(l) is absorbed within the volume of water. Some part Fi(c, l) is scattered out of the beam at an angle c, and the remaining power Ft(l) is transmitted through the volume with no change in direction. Let Fs(l) be the total power that is scattered into all directions. The inherent optical properties usually employed in radiative transfer theory are the absorption and scattering coefficients. In the geometry of Figure 2, the absorption coefficient a(l) is defined as the limit of the fraction of the incident power that is absorbed within the volume, as the thickness becomes small

RADIATIVE TRANSFER IN THE OCEAN

ΔΩ

4

10

Φs ( )

ΔV

Φt

_1 _1

id

Co

Cle

VSF,  (m sr )

Φa

Φi

Tu rb

2

10



201

ha

rbo

r

as

tal

ar

0

10

oc

ea

oc

n

ea

n

_2

10

Δr

Pure sea water

Figure 2 Geometry used to define inherent optical properties.

10

_4

0.1

1.0

10.0

100.0

Scattering angle, (deg)

aðlÞ 

lim Dr-0

1 Fa ðlÞ Fi ðlÞ Dr

½m1 

½7

The scattering coefficient b(l) has a corresponding definition using Fs(l). The beam attenuation coefficient c(l) is defined as c(l) ¼ a(l) þ b(l). Now take into account the angular distribution of the scattered power, with Fs(c, l)/Fi(l) being the fraction of incident power scattered out of the beam through an angle c into a solid angle DO centered on c, as shown in Figure 2. Then the fraction of scattered power per unit distance and unit solid angle, b(c, l), is lim lim Fs ðc; lÞ bðc; lÞ  Dr-0 DO-0 Fi ðlÞDrDO

½m1 sr1  ½8

The spectral power scattered into the given solid angle DO is just the spectral radiant intensity scattered into direction c times the solid angle: Fs(c, l) ¼ Is(c, l) DO. Moreover, if the incident power Fi(l) falls on an area DA, then the corresponding incident irradiance is Ei(l) ¼ Fi(l)/DA. Noting that DV ¼ DrDA is the volume of water that is illuminated by the incident beam gives lim IS ðc; lÞ bðc; lÞ ¼DV-0 Ei ðlÞDV

½9

This form of b(c, l) suggests the name volume scattering function (VSF) and the physical interpretation of scattered intensity per unit incident irradiance per unit volume of water. Figure 3 shows measured VSFs (at 514 nm) from three greatly different water bodies; the VSF of pure water is shown for comparison. VSFs of sea water typically increase by five or six orders of magnitude in going from c ¼ 901 to c ¼ 0.11 for a given water sample, and scattering at a given angle c can vary by two orders of magnitude among water samples.

Figure 3 Volume scattering functions (VSF) measured in three different oceanic waters. The VSF of pure sea water is shown for comparison.

Integrating b(c, l) over all directions (solid angles) gives the total scattered power per unit incident irradiance and unit volume of water, in other words the spectral scattering coefficient: bðlÞ ¼

Z

bðc; lÞdO ¼ 2p

Z

p

bðc; lÞ sin c dc

½10

0

4p

Eqn. [10] follows because scattering in natural waters is azimuthally symmetric about the incident direction (for unpolarized light sources and randomly oriented scatterers). This integration is often divided into forward scattering, 0rcrp/2, and backward scattering, p/2rcrp, parts. Thus the backscatter coefficient is bb ðlÞ  2p

Z

p

bðc; lÞ sin c dc

½11

p=2

The VSFs of Figure 3 have b values ranging from 0.037 to 1.824 m1 and backscatter fractions bb/b of 0.013 to 0.044. The preceding discussion assumed that no inelastic-scattering processes are present. However, inelastic scattering does occur owing to fluorescence by dissolved matter or chlorophyll, and to Raman scattering by the water molecules themselves. Power lost from wavelength l by scattering into wavelength l0 al appears as an increase in the absorption a(l). The gain in power at l0 appears as a source term in the radiative transfer equation. Two more inherent optical properties are commonly used in optical oceanography. The single-scattering albedo is oo(l) ¼ b(l)/c(l). The single-scattering albedo is the probability that a photon will be

202

RADIATIVE TRANSFER IN THE OCEAN

scattered (rather than absorbed) in any given interaction, hence oo(l) is also known as the probability of photon survival. The volume scattering phase func˜ tion, bðc; lÞ is defined by bðc; lÞ 

bðc; lÞ ½sr1  bðlÞ

½12

Apparent Optical Properties

Writing the volume scattering function b(c, l) as the product of the scattering coefficient b(l) and the phase ˜ function bðc; lÞ partitions b(c, l) into a factor giving the strength of the scattering, b(l) with units of m1, and a factor giving the angular distribution of the ˜ scattered photons, bðc; lÞ with units of sr1. A striking feature of the sea water VSFs of Figure 3 is that their phase functions are all similar in shape, with the main differences being in the detailed shape of the functions in the backscatter directions (c4901). The IOPs are additive. This means, for example, that the total absorption coefficient of a water body is the sum of the absorption coefficients of water, phytoplankton, dissolved substances, mineral particles, etc. This additivity allows the development of separate models for the absorption and scattering properties of the various constituents of sea water.

The equation that connects the IOPs and the radiance is called the radiative transfer equation (RTE). Even in the simplest situation of horizontally homogeneous water and time independence, the RTE is a formidable integro-differential equation: dLðz; y; j; lÞ ¼  cðz; lÞLðz; y; j; lÞ dz Z Lðz; y0 ; j0 ; lÞ þ 4p

 bðz; y0; j0 -y; j; lÞdO0 þ Sðz; y; j; lÞ

Apparent optical properties are always a ratio of two radiometric variables. This ratioing removes effects of the magnitude of the incident sky radiance onto the sea surface. For example, if the sun goes behind a cloud, the downwelling and upwelling irradiances within the water can change by an order of magnitude within a few seconds, but their ratio will be almost unchanged. (There will still be some change because the directional structure of the underwater radiance will change when the sun’s direct beam is removed from the radiance incident onto the sea surface.) The ratio just mentioned, Rðz; lÞ 

Eu ðz; lÞ Ed ðz; lÞ

½14

is called the irradiance reflectance (or irradiance ratio). The remote-sensing reflectance Rrs(y, j, l) is defined as

The Radiative Transfer Equation

cos y

radiometric quantities. For example, it is not possible to write down an equation that can be solved directly for the irradiance Ed; one must first solve the RTE for the radiance and then compute Ed by integrating the radiance over direction.

½13

The scattering angle c in the VSF is the angle between the incident direction (y0 , j0 ) and the scattered direction (y, j). The source term S(z, y, j, l) can describe either an internal light source such as bioluminescence, or inelastically scattered light from other wavelengths. The physical environment of a water body – waves on its surface, the character of its bottom, the incident radiance from the sky – enters the theory via the boundary conditions necessary to solve the RTE. Given the IOPs and suitable boundary conditions, the RTE can be solved numerically for the radiance distribution L(z, y, j, l). Unfortunately, there are no shortcuts to computing other

Rrs ðy; j; lÞ 

Lw ðy; j; lÞ Ed ðlÞ

½sr1 

½15

where Lw is the water-leaving radiance, i.e., the total upward radiance minus the sky and solar radiance that was reflected upward by the sea surface. Lw and Ed are evaluated just above the sea surface. Both Rrs(y, j, l) and R(z, l) just beneath the sea surface are of great importance in remote sensing, and both can be regarded as a measure of ‘ocean color.’ R and Rrs are proportional (to a first-order approximation) to bb/(a þ bb), and measurements of Rrs above the surface or of R within the water can be used to estimate water quality parameters such as the chlorophyll concentration. Under typical oceanic conditions, for which the incident lighting is provided by the sun and sky, the radiance and various irradiances all decrease approximately exponentially with depth, at least when far enough below the surface (and far enough above the bottom, in shallow water) to be free of boundary effects. It is therefore convenient to write the depth dependence of, say, Ed(z, l) as Ed ðz; lÞ  Ed ð0; lÞexpleft½

Z

z 0

Kd ðz0 ; lÞdz0 

½16

RADIATIVE TRANSFER IN THE OCEAN

where Kd(z, l) is the spectral diffuse attenuation coefficient for spectral downwelling plane irradiance. Solving for Kd(z, l) gives dlnEd ðz; lÞ dz 1 dEd ðz; lÞ ¼ Ed ðz; lÞ dz

203

be the dominant absorber at the blue end of the spectrum, especially in coastal waters influenced by river runoff. Organic Particles

Kd ðz; lÞ ¼ 

Biogenic particles occur in many forms. 1

½m 

½17

The beam attenuation coefficient c(l) is defined in terms of the radiant power lost from a collimated beam of photons. The diffuse attenuation coefficient Kd(z, l) is defined in terms of the decrease with depth of the ambient downwelling irradiance Ed(z, l), which comprises photons heading in all downward directions (a diffuse, or uncollimated, light field). Kd(z, l) clearly depends on the directional structure of the ambient light field, hence its classification as an apparent optical property. Other diffuse attenuation coefficients, e.g., Ku, Kod, or KPAR, are defined in an analogous manner, using the corresponding radiometric quantities. In most waters, these K functions are strongly correlated with the absorption coefficient a and therefore can serve as convenient, if imperfect, descriptors of a water body. However, AOPs are not additive, which complicates their interpretation in terms of water constituents.

Optical Constituents of Seawater Oceanic waters are a witch’s brew of dissolved and particulate matter whose concentrations and optical properties vary by many orders of magnitude, so that ocean waters vary in color from the deep blue of the open ocean, where sunlight can penetrate to depths of several hundred meters, to yellowish-brown in a turbid estuary, where sunlight may penetrate less than a meter. The most important optical constituents of sea water can be briefly described as follows. Sea Water

Water itself is highly absorbing at wavelengths below 250 nm and above 700 nm, which limits the wavelength range of interest in optical oceanography to the near-ultraviolet to the near infrared. Dissolved Organic Compounds

These compounds are produced during the decay of plant matter. In sufficient concentrations these compounds can color the water yellowish brown; they are therefore generally called yellow matter or colored dissolved organic matter (CDOM). CDOM absorbs very little in the red, but absorption increases rapidly with decreasing wavelength, and CDOM can

Bacteria Living bacteria in the size range 0.2– 1.0 mm can be significant scatterers and absorbers of light, especially at blue wavelengths and in clean oceanic waters, where the larger phytoplankton are relatively scarce. Phytoplankton These ubiquitous microscopic plants occur with incredible diversity of species, size (from less than 1 mm to more than 200 mm), shape, and concentration. Phytoplankton are responsible for determining the optical properties of most oceanic waters. Their chlorophyll and related pigments strongly absorb light in the blue and red and thus, when concentrations are high, determine the spectral absorption of sea water. Phytoplankton are generally much larger than the wavelength of visible light and can scatter light strongly. Detritus Nonliving organic particles of various sizes are produced, for example, when phytoplankton die and their cells break apart, and when zooplankton graze on phytoplankton and leave cell fragments and fecal pellets. Detritus can be rapidly photooxidized and lose the characteristic absorption spectrum of living phytoplankton, leaving significant absorption only at blue wavelengths. However, detritus can contribute significantly to scattering, especially in the open ocean. Inorganic Particles

Particles created by weathering of terrestrial rocks can enter the water as wind-blown dust settles on the sea surface, as rivers carry eroded soil to the sea, or as currents resuspend bottom sediments. Such particles range in size from less than 0.1 mm to tens of micrometers and can dominate water optical properties when present in sufficient concentrations. Particulate matter is usually the major determinant of the absorption and scattering properties of sea water and is responsible for most of the temporal and spatial variability in these optical properties. A central goal of research in optical oceanography is to understand how the absorption and scattering properties of these various constituents relate to the particle type (e.g., microbial species or mineral composition), present conditions (e.g., the physiological

204

RADIATIVE TRANSFER IN THE OCEAN

state of a living microbe, which in turn depends on nutrient supply and ambient lighting), and history (e.g., photo-oxidation of pigments in dead cells). Biogeo-optical models have been developed that attempt (with varying degrees of success) to predict the IOPs in terms of the chlorophyll concentration or other simplified measures of the composition of a water body.

Examples of Underwater Light Fields Solving the radiative transfer equation requires mathematically sophisticated and computationally intensive numerical methods. Hydrolight is a widely used software package for numerical solution of oceanographic radiative transfer problems. The input to Hydrolight consists of the absorption and scattering coefficients of each constituent of the water body (microbial particles, dissolved substances, mineral particles, etc.) as functions of depth and wavelength, the corresponding scattering phase functions, the sea state, the sky radiance incident onto the sea surface, and the reflectance properties of the bottom boundary (if the water is not assumed infinitely deep). Hydrolight solves the one-dimensional, time-independent radiative transfer equation, including inelastic scattering effects, to obtain the radiance distribution L(z, y, j, l). Other quantities of interest such as irradiances or reflectances are then computed using their definitions and the solution radiance distribution. To illustrate the range of behavior of underwater light fields, Hydrolight was run for two greatly different water bodies. The first simulation used a

chlorophyll profile measured in the Atlantic Ocean north of the Azores in winter. The water was well mixed to a depth of over 100 m. The chlorophyll concentration Chl varied between 0.2 and 0.3 mg m3 between the surface and 116 m depth; it then dropped to less than 0.05 mg m3 below 150 m depth. The water was oligotrophic, case 1 water, and commonly used bio-optical models for case 1 water were used to convert the chlorophyll concentration to absorption and scattering coefficients (which were not measured). A scattering phase function similar in shape to those seen in Figure 3 was used for the particles; this phase function had a backscatter fraction of bb/b ¼ 0.018. The second simulation was for an idealized, case 2 coastal water body containing 5 mg m3 of chlorophyll and 2 g m3 of brown-colored mineral particles representing resuspended sediments. Bio-optical models and measured mass-specific absorption and scattering coefficients were used to convert the chlorophyll and mineral concentrations to absorption and scattering coefficients. The large microbial particles of low index of refraction were assumed to have a phase function with bb/b ¼ 0.005, and the small mineral particles of high index of refraction had bb/b ¼ 0.03. The water was assumed to be well mixed and to have a brown mud bottom at a depth of 10 m. Both simulations used a clear sky radiance distribution appropriate for midday in January at the Azores location. The sea surface was covered by capillary waves corresponding to a 5 m s1 wind speed. Figure 4 shows the component and total absorption coefficients just beneath the sea surface for these two hypothetical water bodies, and Figure 5

Case 2 0.8

0.06

0.6

_1

Absorption coefficient, a (m )

Case 1 0.08

Total 0.04

0.4 Total

0.02

Chl

Min

Water

Water

0.2

Chl CDOM

CDOM 0 400

500

600

Wavelength,  (nm)

700

0 400

500

600

700

Wavelength,  (nm)

Figure 4 Absorption coefficients for the case 1 and case 2 water bodies. The contributions by the various components are labeled.

RADIATIVE TRANSFER IN THE OCEAN

Case 2

Case 1

0.30

205

3.0 Total

_1

Scattering coefficient, b (m )

0.25 2.0

0.20 Total

Minerals

0.15

Chl

Chl

0.10

1.0

0.05 Water

Water

0

0 400

500

600

700

400

500

Wavelength,  (nm)

600

700

Wavelength,  (nm)

Figure 5 Scattering coefficients for the case 1 and case 2 water bodies. The contributions by the various components are labeled (CDOM is nonscattering).

log [radiance, L (W m sr nm )]

th 0

_1

_1

log [radiance, L (W m sr nm )]

Dep

_2

_1 _2

_1 _2 _3 _4 0

_ 90

1

90 0 )

0

700 00 6 nm) 500 ,( h t 400 g elen Wav

)

eg

eg

(d

(d

v

v



90

n,

n,

700 600 ) 0 50 (nm th,  400 g n e el Wav

0

io

io

18

ct

ct

80

re

re

di

di

_8 0

g

g

_7

in

_ 90

in

_6

ew

ew

_5

Vi

Vi

_4

_1

0

00 m

th 1

Dep

Figure 6 The case 1 water radiance distribution in the azimuthal plane of the sun at depth 0 (just below the sea surface) and at 100 m.

shows the corresponding scattering coefficients. For the case 1 water, the total absorption is dominated by chlorophyll at blue wavelengths and by the water itself at wavelengths greater than 500 nm. However, the water makes only a small contribution to the total scattering. In the case 2 water, absorption by the mineral particles is comparable to or greater than that by the chlorophyll-bearing particles, and water dominates only in the red. The mineral particles are the primary scatterers.

Figure 6 and 7 show the radiance in the azimuthal plane of the sun as a function of polar viewing direction and wavelength, for selected depths. For the case 1 simulation (Figure 6), the depths shown are zero, just beneath the sea surface, and 100 m; for the case 2 simulation (Figure 7), the depths are zero and 10 m, which is at the bottom. Note that the radiance axis is logarithmic. A viewing direction of yv ¼ 0 corresponds to looking straight down and seeing the upwelling radiance (photons traveling straight up).

RADIATIVE TRANSFER IN THE OCEAN

th 0

_1

m

_3

_2

_1

1

_2

th 10

Dep

_2

_1

_1

log [radiance, L (W m sr nm )]

Dep

log [radiance, L (W m sr nm )]

206

0 _1 _2

g in ew Vi

_3 0 _ 90

_4 _5 _6 0 _ 90

Vi

ew

in

g

re di

180

re

700 600 m) 0 50  (n 400 length, e v a W

0

,

n io ct

90

180

di

ct

90

io

n,



v

0

v

eg

)

)

eg

(d

(d

700 600 m) 0 n 0 ( 5  400 length, e Wav

Figure 7 The case 2 water radiance distribution in the azimuthal plane of the sun at depth 0 (just below the sea surface) and at 10 m.

10

Case 1

2

0

10

Case 2

1

Depth = 0

Depth = 0 10

0

5m

_

_1

Scalar irradance Eo (W m 2 nm )

10

_2

100 m

_4

200 m

10

_1

10

10 m 10

_2

10

_3

_6

10

10

_4

_8

10

10

_5

_ 10

10

400

500

600 700 Wavelength,  (nm)

10

400

500 600 700 Wavelength,  (nm)

Figure 8 The scalar irradiance Eo at selected depths for the case 1 and case 2 waters.

Near the sea surface, the angular dependence of the radiance distribution is complicated because of boundary effects such as internal reflection (the bumps near yv ¼ 901, which is radiance traveling horizontally) and refraction of the sun’s direct beam (the large spike near yv ¼ 1401). As the depth increases, the angular shape of the radiance distribution smooths out as a result of multiple scattering. By 100 m in the case 1 simulation, the shape of the radiance distribution is approaching its asymptotic shape, which is determined only by the IOPs. In the case 2 simulation, the upwelling radiance

(  901ryvr901) at the bottom is isotropic; this is a consequence of having assumed the mud bottom to be a Lambertian reflecting surface. As the depth increases, the color of the radiance becomes blue for the case 1 water and greenish-yellow for the case 2 water. In the case 1 simulation at 100 m, there is a prominent peak in the radiance near 685 nm, even though the solar radiance has been filtered out by the strong absorption by water at red wavelengths. This peak is due to chlorophyll fluorescence, which is transferring energy from blue to red wavelengths, where it is emitted isotropically.

RADIATIVE TRANSFER IN THE OCEAN

207

_1

Remote-sensing reflectance, Rrs (sr )

0.015 Case 2 Case 1 0.010

0.005

0 400

500

600

700

Wavelength,  (nm) Figure 9 The remote-sensing reflectance Rrs for the case 1 and case 2 waters.

As already noted, the extensive information contained in the full radiance distribution is seldom needed. A biologist would probably be interested only in the scalar irradiance Eo, which is shown at selected depths in Figure 8. This irradiance was computed by integrating the radiance over all directions. Although the irradiances near the surface are almost identical, the decay of these irradiances with depth is much different in the case 1 and case 2 waters. The remote-sensing reflectance Rrs, the quantity of interest for ‘ocean color’ remote sensing, is shown in Figure 9 for the two water bodies. The shaded bars at the bottom of the figure show the nominal SeaWiFS sensor bands. The SeaWiFS algorithm for retrieval of the chlorophyll concentration uses a function of the ratio Rrs(490 nm)/Rrs (555 nm). When applied to these Rrs spectra, the SeaWiFS algorithm retrieves a value of Chl ¼ 0.24 mg m3 for the case 1 water, which is close to the average value of the measured profile over the upper few tens of meters of the water column. However, when applied to the case 2 spectrum, the SeaWiFS algorithm gives Chl ¼ 8.88 mg m3, which is almost twice the value of 5.0 mg m3 used in the simulation. This error results from the presence of the mineral particles, which are not accounted for in the SeaWiFS chlorophyll retrieval algorithm. These Hydrolight simulations highlight the fact that it is now possible to compute accurate underwater radiance distributions given the IOPs and boundary conditions. The difficult science lies in learning how to predict the IOPs for the incredible variety of water constituents and environmental

conditions found in the world’s oceans, and in learning how to interpret measurements such as Rrs. The development of bio-geo-optical models for case 2 waters, in particular, is a research topic for the next decades.

Further Reading Bukata RP, Jerome JH, Kondratyev KY, and Pozdnyakov DV (1995) Optical Properties and Remote Sensing of Inland and Coastal Waters. New York: CRC Press. Caimi FM (ed.) (1995) Selected Papers on Underwater Optics. SPIE Milestone Series, vol. MS 118. Bellingham, WA: SPIE Optical Engineering Press. Jerlov NG (1976) Marine Optics. Amsterdam: Elsevier. Kirk JTO (1994) Light and Photosynthesis in Aquatic Ecosystems 2nd. New York: Cambridge University Press. Mobley CD (1994) Light and Water Radiative Transfer in Natural Waters. San Diego: Academic Press. Mobley CD (1995) The optical properties of water. In Bass M (ed.) Handbook of Optics, 2nd edn, vol. I. New York: McGraw Hill. Mobley CD and Sundman LK (2000) Hydrolight 4.1 Users’ Guide. Redmond, WA: Sequoia Scientific. [See also www.sequoiasci.com/hydrolight.html]. Shifrin KS (1988) Physical Optics of Ocean Water. AIP Translation Series. New York: American Institute of Physics. Spinrad RW, Carder KL, and Perry MJ (1994) Ocean Optics. New York: Oxford University Press. Walker RE (1994) Marine Light Field Statistics. New York: Wiley.

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS J. M. Prospero, University of Miami, Miami, FL, USA R. Arimoto, New Mexico State University, Carlsbad, NM, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The atmosphere is the primary pathway for the transport of many geochemically important substances to the oceans. Although the magnitude of these wind-borne transports is not accurately known, there is growing evidence that atmospheric deposition significantly impacts chemical and biological processes in the oceans. It is only over the past several decades that marine scientists have come to appreciate the importance of atmospheric transport. Historically it had been assumed that the fluxes of continental materials to the oceans were dominated by rivers. But over time it was recognized that much of the riverine load was deposited in estuaries or on the continental shelves. In contrast, winds can rapidly span great distances to reach even the most remote ocean regions. The transport and deposition of particulate matter (PM) to the oceans depends on many factors including the distribution of sources, physical and chemical properties of the particles, meteorological conditions, and removal mechanisms. Our interest here focuses on particles between about 0.1 and 10 mm in diameter which, because of their small size, have atmospheric lifetimes ranging from days to several weeks. These are commonly referred to as aerosol particles or aerosols. Larger particles are deposited close to their sources and do not contribute substantially to ocean deposition except in some coastal regions. Smaller aerosols carry little mass and, while they are important for other atmospheric issues, they are not particularly important for air/sea chemical exchange. Winds carry billions of tons of PM to the ocean. Some of the PM is emitted by natural processes and some is produced anthropogenically, that is, as a result of human activities. Much PM is emitted directly as primary particles; this includes mineral (soil) dust, organic particles from plants, and emissions from anthropogenic combustion processes (e.g., from industry, homes, and vehicles) and biomass burning,

208

which can be natural (e.g., started by lightning) or anthropogenic (e.g., in clearing land, burning agricultural waste). An important PM fraction – secondary PM – is that produced from gases, natural and anthropogenic, that react in the atmosphere to form particles. One goal of marine scientists is to characterize atmospheric transport and chemical deposition to the ocean and to assess the impact of the air/sea exchange process. This is a difficult task which can only be achieved when we know the kinds of materials deposited and their temporal and spatial variability. Because of the patchy distribution of sources and the relatively short tropospheric residence times of aerosols, PM concentrations over the oceans vary by orders of magnitude in time and space. Here we review the sources and composition of aerosols and the removal mechanisms relevant to deposition issues. We then present estimates of deposition rates of some PM classes to the oceans.

Aerosol Sources, Composition, and Concentrations The composition of PM over the oceans varies greatly due to the myriad sources and variations in their strengths. Soils emit fine mineral particles. Plants produce a wide range of organic particles, ranging from decayed leaf matter, to plant waxes, and condensed organic compounds. Volcanoes sporadically inject many tons of material into the atmosphere, and much of this is deposited in the oceans; but the total amount of PM deposited over time is relatively small compared with other sources. Smelters, power plants, and incinerators emit PM highly enriched with trace metal pollutants. Combustion sources, both natural (wild fires, biomass burning) and anthropogenic (power plants, vehicles, home heating) emit thousands of organic compounds. Combustion processes and the use of fertilizers contribute to the production of nitrogen-rich particles. Pesticides and other synthetic organic compounds are emitted from industrial and domestic sources. Typically the dominant marine aerosol species by mass are: (a) sea salt, produced by breaking waves and bursting bubbles; (b) sulfate, including that from sea salt aerosol and non-sea-salt sulfate (nss-SO4 2 ), the latter of which is both transported from pollution sources on the continents and produced from

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

gaseous precursors such as dimethyl sulfide (DMS) emitted from the oceans; (c) nitrate, originating from pollution sources on the continents and produced by lightning; (d) ammonium, derived mostly from continental sources but in some areas from ocean sources; (e) mineral dust, from arid lands; (f) organic carbon (OC), largely from anthropogenic and natural sources on the continents; and (g) black carbon (BC), from biomass burning and anthropogenic sources. PM composition is strongly size dependent not only as a result of various production mechanisms but also because size-selective removal occurs during transport. Physical production mechanisms (grinding of rocks, bursting bubbles) normally produce large particles with most of the mass in PM greater than 1-mm diameter (coarse particles). For example, the mass median diameter (MMD: 50% of the mass is greater than the MMD and 50% is less) of dust particles over deserts can be extremely large, many tens or hundreds of micrometers, but over the oceans, it is typically only several micrometers. The MMD of sea salt particles is generally in the range of about 5–10 mm, depending on wind conditions. Other primary particles including soot emitted from smoke stacks, diesel exhaust, and particles shed by plants (e.g., plant waxes, fibers), tending to be in the size range of B0.1–1.0 mm.

Table 1

Lat 1N North Pacific Western Pacific Cheju, 33.5 Korea Central Pacific Midway 28.2 Oahu 21.36 North Atlantic Mace 53.5 Head Bermuda 32.3 Barbados 13.2 South Pacific American  14.3 Samoa

a

Gas-phase reactions produce secondary PM ranging in size range of 0.001–0.1 mm diameter. Examples are sulfate particles produced from SO2 emitted from power plants or from the oxidation of DMS emitted from the oceans. Particles in this very fine particle mode are highly mobile. They can rapidly coagulate to form larger particles (typically 0.1–1 mm) or they can diffuse to the surface of cloud or fog droplets or to larger particles (e.g., sea salt, mineral dust). Table 1 presents concentration data for the aerosols that make up most of the PM mass over the oceans; it also includes data for vanadium, which is included as an example of an element strongly affected by pollution sources. The column on the extreme right shows the total aerosol concentration less that of sea salt, so as to better illustrate the impact of transported PM. These data are the product of longterm measurements obtained from a global ocean network. In general, PM concentrations are much higher in the Northern Hemisphere relative to the Southern Hemisphere. Mineral dust shows an extremely wide range of concentrations over the oceans; the highest are over the tropical North Atlantic and the western North Pacific. These reflect the impact of dust transport from North Africa and China, respectively. Dust concentrations in the southern oceans tend to be extremely low due to the

Annual mean aerosol concentrations measured at remote ocean stations Station locationa

Antarctic Mawson

 67.6

Sea salt (mg m  3)

NO3 (mg m  3)

nss-SO4 (mg m  3)

NH4 (mg m  3)

Dust (mg m  3)

V (mg m  3)

Total b (mg m  3)

Total SSc (mg m  3)

126.5

19.8

4.1

7.2

3.0

15.5

4.1

49.5

29.8

 177.4  157.7

13.8 15.1

0.3 0.4

0.5 0.5

0.8 0.0

0.7 0.7

0.2 0.3

15.4 16.7

1.6 1.6

 9.9

14.1

1.5

2.0

0.9

0.5

0.9

19.0

4.9

 64.9  59.4

13.7 16.5

1.1 0.5

2.2 0.8

0.3 0.1

5.6 14.6

1.3 1.9

22.8 32.5

9.2 16.0

 170.6

16.7

0.1

0.4

0.0

0.1

17.2

0.5

62.5

0.3

0.0

0.1

0.5

0.1

Lon 1E

0.0

Station Location: negative latitudes – Southern Hemisphere: negative longitudes – Western Hemisphere. Total aerosol: the sum of the major aerosol components – sea salt, soil dust, nss-SO4, NO3, and NH4. c Total SS: Total aerosol minus sea salt. b

209

210

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

dearth of strong dust sources combined with the great distances to central ocean regions. The impact of air pollution is evident over much of the Northern Hemisphere. Extremely high NO3  and nss-SO4 2 concentrations are seen in the western Pacific near Asia; these are attributable to continental outflow and exacerbated by limited emission controls. Moderately high pollutant levels are seen over the North Atlantic as well, a result of emissions from North America and Europe. In contrast, the concentrations of NO3  and nss-SO4 2 at American Samoa and the Antarctic stations Mawson and Palmer are extremely low; these represent conditions that one might expect when pollution impacts are minimal. The larger-scale picture of PM distributions is provided by sensors such as the advanced very high resolution radiometer (AVHRR), which measures solar radiation backscattered to space by PM to estimate aerosol optical thickness (AOT) (Figure 1). There are three characteristics of the global distributions of AOT, all consistent with the data in Table 1. First, the highest AOT (i.e., the greatest column loadings of PM) is found close to the continents. This distribution affirms the fact that over most of the ocean PM is largely the result of material transported from the continents. Second, there are large seasonal differences in PM concentrations due to the seasonality of emissions and meteorology. Third, some continents emit more PM than others, illustrating the large-scale differences in production and transport. Especially notable in satellite images is a large plume of AOT over the tropical Atlantic, extending from the coast of Africa to South America (December– February) and to the Caribbean (June–August). This plume is mainly African dust. A large region of high AOT over the Arabian Sea (June–August) is attributed to dust from Africa and the Middle East. In this same season, a large PM plume seen off the west coast of southern Africa is attributed to smoke from intense biomass burning. Substantial aerosol plumes are also seen over the North Atlantic; these are caused by pollutants from North America and Europe. Large regions of high AOT are seen along the coast of Asia; but the Asian plume is most prominent in spring when large quantities of soil dust mix with pollution aerosols. The attribution of these plumes to these dominant aerosol types is supported by evidence from field studies.

Aerosol Removal Mechanisms Estimating Wet and Dry Deposition

PM is deposited to the ocean by two broadly characterized mechanisms: (1) dry deposition, in which a

particle is transferred directly from the atmosphere to the surface; and (2) wet deposition, in which a particle is first incorporated into a cloud or rain droplet that subsequently falls to the surface. The relative efficiency of the removal processes is dependent on a number of factors, especially the particle-size distribution and the hygroscopic properties of the aerosol. In most ocean regions, wet removal is thought to dominate for most aerosol species. Wet deposition is relatively easy to measure using precipitation collectors, such as automatic bucket systems that open only when precipitation falls. Dry deposition, on the other hand, is much more difficult to collect because this process is affected by a variety of factors, all highly variable: the properties of the aerosol and the water surface, atmospheric stability, relative humidity, wind velocity, etc. Furthermore, while there is a vast quantity of data on wet deposition to continental areas, there is very little for ocean environments. There are some long-term records for selected species in precipitation at a few island stations but there are no matching data for dry deposition. Wet Deposition

Long-term studies show that on average the wet deposition rates of many species are related to their concentrations in the atmospheric aerosol and to rainfall rates. This relationship is expressed in terms of a dimensionless scavenging ratio, S: S ¼ Cp rC1 a

½1

where Cp is the concentration of the substance in precipitation (g kg 1), r the density of air (B1.2 kg m 3), and Ca the aerosol concentration of the species of interest (g m 3). Wet deposition rates depend on the vertical distribution of PM and the type of precipitation event (e.g., frontal, cumulus, and stratus). In practice, comprehensive, long-term, aerosol data are only available from surface sites; consequently one must assume that over the longer term the PM concentrations in surface-level air are correlated with their vertical distributions. Therefore, S is appropriately calculated only when data have been obtained over periods of a year or more. That is, the variability in the aerosol and precipitation events must be smoothed by the averaging process. Typically used values for S fall in the range of 200–1000. Scavenging ratios can be applied to regions where no precipitation data exist using the following

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

211

(a) Dec., Jan., Feb.

EAOT 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Husar, Stove and Prospero, 1996

(b) Jun., Jul., Aug.

EAOT 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Husar, Stove and Prospero, 1996

Figure 1 The distribution of aerosols over the oceans inferred from aerosol optical thickness estimated (EAOT) by National Oceanic and Atmospheric Administration (NOAA) AVHRR. Aerosol optical thickness is a measure of the attenuation of direct solar radiation at a specific wavelength due to the scattering and absorption caused by aerosols. Large values of optical thickness suggest high concentrations of aerosols. Distributions are shown for the months (a) Dec.–Feb. and (b) Jun.–Aug. Adapted by permission of American Geophysical Union from Husar RB, Prospero JM, and Stowe LL, Characterization of tropospheric aerosols over the oceans with the NOAA advanced very high resolution radiometer optical thickness operational product, Journal of Geophysical Research, vol. 102(D14), pp. 16889–16909, 1997. Copyright 1997 American Geophysical Union.

expression: Fp ¼ PCp ¼ PSr1 Ca

½2

where Fp is the wet deposition flux (g m  2 yr  1) and P is the precipitation rate (m yr  1), using conversion factors to translate rainfall amounts to the mass of water deposited per unit area. Note

212

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

that the combined terms PSr  1Ca have a unit of velocity. Dry Deposition

There are no widely accepted methods for directly measuring PM dry deposition to water surfaces. In practice, dry deposition is almost always calculated by assuming that the deposition rate is proportional to PM concentration times a ‘deposition velocity’, vd. The dry PM flux, Fd (g m  2s  1), is given by Fd ¼ vd Ca

½3

where vd is the dry deposition velocity (m s  1) and Ca is the mass concentration of the substance in the atmosphere (g m  3). Deposition velocities have been modeled based on physical principles, and they have been empirically derived by concurrently measuring the concentration of PM species in the atmosphere and the amount deposited to a surrogate surface (typically a flat plate). While there have been some determinations of deposition velocities at continental sites, the data for ocean regions are scant. In eqn [3], vd incorporates all the processes of dry deposition, but it is difficult to accurately parametrize vd for the ambient aerosol because the importance of these processes varies with particle size. For PM between 0.1 and 1.0 mm, dry deposition is inefficient, and wet removal is normally the major sink. Gravitational settling and surface impaction control the dry removal of PM larger than about 1 mm while below about 0.1 mm, Brownian diffusion dominates. Each of these mechanisms depends on wind speed, aerosol hygroscopicity, relative humidity near the surface, and other factors which are poorly characterized. Unfortunately, there is no general agreement on how to resolve these uncertainties. Nonetheless, many estimates of dry deposition make the following assumptions about the dependence on particle size and the uncertainties in the resulting estimated deposition rate:

• • •

submicrometer aerosol particles: 0.001 m s  17 a factor of 3, supramicrometer crustal particles not associated with sea salt: 0.01 m s  17 a factor of 3, giant sea salt particles and materials carried by them: 0.03 m s  17 a factor of 2.

Despite the widespread use of these values, it should be emphasized that they are only estimates and that the error range is, if anything, probably optimistic. For example, wind speed has a great influence on deposition velocities: for PM B0.1–1.0 mm in diameter, the deposition velocity ranges from c. 0.005 cm s  1 at 5 m s  1 to c. 0.1 cm s  1 at 15 m s  1.

Deposition of Aerosols to the Oceans In this section, we present estimates of the deposition of a number of PM species that are potentially important for biogeochemical processes in the oceans. Estimates of deposition to specific locations can be made using the above relationships, assuming that the necessary aerosol concentration data are available. Larger-scale estimates of deposition are best obtained with atmospheric chemical transport models as discussed below. These models are subject to large uncertainties because they generally rely on estimates of aerosol properties and concentrations over the oceans and they incorporate highly parametrized removal schemes. For illustrative purposes, we present results for mineral dust, selected trace elements, and a group of nitrogen-containing species. A wide range of natural and anthropogenic organic species are also transported to the oceans and deposited there. Of these, certain persistent organic pollutants are known to have a harmful impact on marine biological systems. There are, however, relatively little data on the large-scale transport of organics that would enable us to address this issue on a global scale. Consequently, we do not include organic species in this report.

Deposition of Mineral Dust and Eolian Iron

In many ocean regions, primary (photosynthetic) biological production is limited by the classical nutrients such as nitrate and phosphate. In nutrient-rich surface waters, biological activity is usually high which results in high chlorophyll concentrations. But in large areas of the world’s oceans, nutrient concentrations are high, yet chlorophyll remains low which suggests low productivity. These are termed high-nutrient, low-chlorophyll (HNLC) waters; prominent examples include the equatorial Pacific and much of the high-latitude southern oceans. In the 1980s, it was found that primary production in some HNLC regions was limited by the availability of iron, an essential micronutrient in certain enzymes involved in photosynthesis. Remote ocean regions are largely dependent on atmospheric dust for the input of iron. The deposition of this eolian iron and its impact on productivity has important implications for the global CO2 budget and, hence, climate. Increased iron fluxes could conceivably fertilize the oceans, thereby increasing productivity and drawing down atmospheric CO2. In addition, certain nitrogen fixers (e.g., Trichodesmium sp.) have a high iron requirement; an increased eolian iron flux could stimulate the growth

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

of nitrogen fixers, thereby increasing nitrate levels and further contributing to the CO2 drawdown. Much effort has gone into estimating the temporal and spatial patterns of dust deposition to the oceans. Some studies have used satellite aerosol measurements coupled with network measurements of dust to calculate wet and dry deposition fluxes using scavenging ratios and deposition velocities. Recently, regional- and global-scale models have been developed to provide estimates of dust emissions, transport, and deposition. Dust is generally included as a passive tracer, and its removal is highly parametrized. Dry deposition is calculated using the model’s dust size distribution and size-dependent deposition (see section ‘Dry deposition’ above). Wet removal is also modeled, but the interaction of dust with clouds is not well constrained, partly because aerosol–cloud interactions in general are not well understood, and also because there are few measurements of cloud microphysical measurements in dusty regions. Mineral dust is not readily soluble in water; so some models assume that mineral aerosols do not interact with clouds directly, but rather are scavenged via subcloud removal – hence, simple scavenging ratios are used (see section ‘Wet deposition’ above). As a consequence of these uncertainties, current models show large differences in dust wet deposition lifetimes, ranging from 10 to 56 days. A typical model estimate of dust deposition rates to the oceans is shown in Figure 2. In the tropical North Atlantic, rates typically range from 2 to 10 g m  2 yr  1; over the Arabian Sea, they are as

0.000

0.2

0.5

1

2

213

high as 20 g m  2 yr  1. Over much of the North Pacific, rates are in the range 0.5–1 g m  2 yr  1, increasing to 1–2 g m  2 yr  1 closer to the coast of Asia. Dust deposition rates in Figure 2 tend to mirror the PM distribution shown in Figure 1, which, as previously stated, is largely linked to the presence of dust and, in some regions, smoke from biomass burning. Table 2 shows estimates of deposition rates to the major ocean basins produced by eight commonly used models. There is considerable agreement for the North Atlantic which is heavily impacted by African dust. In contrast, there is a considerable spread in the estimates for other regions, especially the Indian Ocean and South Pacific. Despite these differences, current models yield a reasonable, albeit broad, match with sediment trap measurements in the oceans. These various studies show that North Africa is clearly the world’s most active dust source followed by the Middle East and Central Asia. In effect, these combine to form a global dust belt that dominates transport to the oceans. These sources account for the much greater deposition rates to the northern oceans compared with southern oceans. Nonetheless, there are some substantial and important dust sources in the Southern Hemisphere in Australia, southern Africa, and southern South America. Within these continental regions, certain specific environments are particularly active dust sources, and they are sensitive to changes in climate, especially rainfall and wind speed. The presence of large, deep, alluvial deposits, usually deposited in the Pleistocene or

5

10

20

50

Figure 2 Model estimates of dust deposition rates (units: g m  2 yr  1) to the continents and the oceans. Reproduced by permission of American Geophysical Union from Mahowald NM, Baker AR, Bergametti G, et al., Atmospheric global dust cycle and iron inputs to the ocean. Global Biogeochemical Cycles, vol. 19, GB4025, 2005. Copyright 2005 American Geophysical Union.

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ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

Table 2

Estimate of mean annual dust deposition to the global ocean and to various ocean basins Annual dust deposition ratea (1012 g yr  1)

Reference

Duce et al. (1991) Prospero (1996) Ginoux et al. (2001) Zender et al. (2003) Luo et al. (2003) Ginoux et al. (2004) Tegen et al. (2004) Kaufman et al. (2005)

GO

NAO

SAO

NPO

SPO

NIO

SIO

910 358 478 314 428 505 422

220 220 184 178 230 161 259 140

24 5 20 29 30 20 35

480 96 92 31 35 117 56

39 8 28 8 20 28 11

100 20 154 36 113 164 61

44 9 12 15

a

GO, Global Oceans; NAO, North Atlantic Ocean; SAO, South Atlantic Ocean; NPO, North Pacific Ocean; SPO, South Pacific Ocean; NIO, North Indian Ocean; SIO, South Indian Ocean. Adapted from Engelstaedter S, Tegen I, and Washington R (2006) North African dust emissions and transport. Earth-Science Reviews 79(1–2): 73–100.

Table 3 Metal

Iron Copper Nickel Zinc Arsenic Cadium Lead

Atmospheric and riverine fluxes of dissolved and particulate trace metals to the oceana Atmospheric transports

Riverine transports

Dissolved transports

AtmDiss

AtmPart

Ratio Diss/Part

RivDiss

RivPart

Ratio Diss/Part

Ratio AtmDiss/RivDiss

3200 30 10 102 4 3 75

28 000 5 16 33 2 1 9

0.1 6.6 0.6 3.1 1.7 4.7 8.3

1100 10 11 6 10 0.3 2

11 000 1 500 1 400 3 900 80 15 1 600

0.100 0.007 0.008 0.002 0.125 0.020 0.001

2.9 3.0 0.9 16.9 0.4 8.7 37.5

Units: 109 g yr  1. Atm, atmospheric; Riv, riverine; Diss, Dissolved; Part, particulate. Adapted from Duce RA, Liss PS, Merrill JT, et al. (1991) The atmospheric input of trace species to the world ocean. Global Biogeochemical Cycles 5: 193–259.

a

Holocene, is a common prerequisite for strong dust sources. Trace Element Deposition

Some oceanographers study the biogeochemical cycling of trace elements and seek to quantify the elements’ oceanic sources and sinks. In regions dominated by mineral dust, the ratios of many trace elements (e.g., Al, Ba, Ca, Cs, Fe, Hf, Mn, Rb, Sc, Ta, Th, and Yb) are similar to those in geological materials such as soils, thus implicating mineral dust as their main source. Several elements (Co, Cr, Eu, Mg, and Na) show slight enrichments over crustal values while others such as As, Cd, Cu, Ni, Pb, Sb, Se, V, and Zn are strongly enriched. Such large enrichments are typically associated with pollution impacts, but emissions from natural sources such as volcanoes can also be responsible.

The impact of trace element deposition on ocean biogeochemistry depends to a great extent on the degree to which the elements are soluble in seawater. Extensive studies of trace element solubilities in various natural aqueous media or in aqueous solutions of similar composition yield a wide range of solubilities; these depend on the types of aerosols used, the exposure times, and other experimental variables, especially pH. Thus it is difficult to convert the estimated air/sea exchange rates into an effective or bioavailable flux of trace elements, and therefore it is difficult to accurately assess the impact of PM deposition on ocean processes. Comparison of Trace Element Transport by Rivers and by the Atmosphere

Rivers carry large quantities of dissolved and suspended materials to the oceans. Table 3 compares the

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

amounts of selected trace elements carried by rivers with that carried by winds and also the relative amounts of particulate and dissolved or soluble phases in the transported material. Riverine transports of trace elements are overwhelmingly in the particulate phases and the ratio of dissolved to particulate phases ranges from about 0.001 to 0.13. In comparison, the corresponding ratio for atmospheric transport is much larger, ranging from 0.1 to 8.3. Table 3 also shows that the dissolved or soluble inputs to the ocean from the atmosphere exceed those from rivers; in almost all cases the ratio is greater than unity, in some cases much larger. The comparison of river versus atmospheric inputs is based on the measured concentrations in rivers before they reach the oceans. However, much of the material carried by rivers is rapidly deposited when the rivers reach the sea; therefore, the impact of air/ sea exchange on ocean systems is in reality much greater than suggested by Table 3. We emphasize, however, that the data in Table 3 are rather old. Recent work shows that the concentrations of some trace elements have changed significantly over time. For example, during the mid-1900s, aerosol lead greatly increased in response to increasing pollution emissions but in later years concentrations decreased as controls were implemented. In recent decades, other trace elements have increased due to growing industrialization in developing nations. Also, recent research suggests that there is considerably more uncertainty in PM trace metal solubility than shown in Table 3. Nonetheless, one would still expect that the impact of atmospheric transport is much greater than river transport, especially for the open ocean. Nitrogen Deposition

Anthropogenic activities have greatly increased the amounts of nitrogenous materials that enter the atmosphere and find their way into the rivers (Table 4). There is interest in the possible impacts of these materials on the marine environment, especially about chemicals such as nitrate that can affect primary production. This issue is of particular importance in regions where nitrogen is the limiting nutrient, for example, the oligotrophic central oceanic gyres where an enhancement in productivity would increase the drawdown of CO2 and hence affect climate. In coastal waters, atmospheric inputs could contribute to eutrophication although one would expect the inputs from rivers to dominate. There are two broad classes of nitrogen compounds of interest: oxidized and reduced. The most important oxidized species are NO and NO2

Table 4

215

Atmospheric emissions of fixed nitrogen, 1993a

Sources

NOx

NH3

Anthropogenic Biomass burning Agricultural activity Fossil fuel combustion Industry Total anthropogenic

6.4 2.6 20.9 6.4 36.3

4.6 39.7 0.1 2.8 47.2

2.9 5.4 0.8 0.6 0.0 6.8

4.6 0.0 0.8 0.0 5.6 11.0

43.1 5.3

58.2 4.3

Natural Soils, vegetation, and animals Lightning Natural fires Stratosphere exchange Ocean exchange Total natural Grand total Ratio: anthropogenic/natural

Units Tg/g(N) yr  1. Adapted from Jickells TD (2006) The role of air–sea exchange in the marine nitrogen cycle. Biogeosciences 3: 271–280. a

(collectively referred to as NOx) and NOy (termed reactive odd nitrogen) which is comprised of NOx plus the compounds produced from its oxidation, including HNO3 and other compounds. NOx is rapidly oxidized in the atmosphere to a wide range of compounds, many of which are ultimately converted to HNO3 and aerosol NO3  . In the marine boundary layer, HNO3 reacts rapidly with sea salt particles and promptly deposits on the sea surface. Indeed, NO3  is the N-containing compound of greatest interest in terms of impact on the oceans, and it is the N compound most commonly measured and modeled. Table 4 lists the major sources of oxidized and reduced N emitted to the atmosphere. The primary natural sources of NOx are biological fixation and lightning, the latter being rather minor. In modern times, fossil-fuel combustion along with industry and biomass burning dominate the oxidized N cycle. The ratio of anthropogenic NOx emissions to that of natural sources is 2.5 and continues to increase. While most research has focused on inorganic N (IN), there is evidence that organic nitrogen (ON) also may play an important role in marine biogeochemical cycling. However, there are relatively few data on ON compounds and most focus only on dissolved ON. The major reduced nitrogen species (NHx) are aerosol NH4 þ and NH3, the latter being the only gas-phase species that significantly titrates atmospheric acidity. The major natural sources of NH3 (Table 4) include soils, vegetation, and excreta from wild animals. However, the emissions of NH3 to the atmosphere are now dominated by fertilizers and the

216

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

excreta from dairy and beef cattle. Indeed, the ratio of anthropogenic and natural NH3 emissions is 9. The oceans can also be a source of NH3 under certain conditions in some regions, but the continental sources clearly dominate. Models provide estimates of the present-day atmospheric N fluxes to the oceans and their spatial distribution. Figure 3 presents the deposition rate of

reactive nitrogen NOy þ NHx for the year 2000. As was the case for dust, emissions and deposition in the Northern Hemisphere are much greater than those of the Southern Hemisphere. Deposition rates are extremely high adjacent to the continental coastlines; thus one would expect that the adjacent water bodies would be most heavily impacted. The total N flux to the ocean (NOy and NHx but not including ON) in mg N m−2 6000 3000

60° N

2000 1000 900

30° N

800

Latitude (deg)

700 600 EQ.

500 400 300 200

30° S

100 75 60° S

50 25 10

180° W 150° W 120° W

90° W

60° W

30° W

0° E

30° E

60° E

90° E

120° E 150° E 180° E

Longitude (deg)

Figure 3 Model estimates of the deposition rate of total reactive nitrogen (NOy þ NHx) (units: mg N m  2 yr  1) in the year 2000. Reprinted with permission from Dentener F, Stevenson D, Ellingsen K, et al. (2006) The global atmospheric environment for the next generation. Environmental Science and Technology 40(11): 3586–3594 (doi:10.1021/es0523845). Copyright (2006) American Chemical Society.

Table 5

NOy and NHx deposition for the year 2000

Deposition region

Ocean Coastal ocean NH SH World Ratio: ocean/ world

NOy þ NHx

NHx

NOy Total (Tg(N) yr  1)

Mean rate (mg(N) m2 yr  1)

Total (Tg(N) yr  1)

Mean rate (mg(N) m2 yr  1)

Total (Tg(N) yr  1)

23 4 38 14 52 0.43

61 192 150 54 102

24 4 48 16 65 0.36

63 206

47 8 87 30 117 0.40

126

Adapted from Dentener F, Drevet J, Lamarque JF, et al. (2006) Nitrogen and sulfur deposition on regional and global scales: A multimodel evaluation. Global Biogeochemical Cycles 20: GB4003 (doi:10.1029/2005GB002672).

ATMOSPHERIC TRANSPORT AND DEPOSITION OF PARTICULATE MATERIAL TO THE OCEANS

2000 was 46 Tg N yr  1 of which 8 Tg N yr  1 is deposited to the coastal ocean (Table 5). The deposition of oxidized forms (NOy) is essentially equal to that of reduced forms (NHx). The ocean N deposition amounts to c. 40% of global emissions. The total reactive N transport by rivers is about 48 Tg N yr  1, essentially the same as the atmospheric source. However, there is evidence that fluvial nitrogen inputs to the oceans are denitrified on the shelf and that the shelf region is a sink rather than a source of nitrogen for the open oceans. Thus it appears that air/sea transfer is the major source of N transported to the open ocean. ON compounds could also be playing a significant role in total N fluxes. Studies from many different environments suggest that ON constitutes about a third of the total atmospheric reactive nitrogen. Thus, ON could add significantly to the total global flux to the oceans, conceivably raising the total to about 69 Tg N yr  1.

Conclusions It is now recognized that atmospheric transport plays a central role in ocean biogeochemical processes. There is increased interest in the chemically coupled ocean/atmosphere system, how this system has changed over time, and how it might respond to global change. Although many models are currently focusing on this question, the development of these models is handicapped by the dearth of measurements over many ocean regions. It remains a formidable challenge to the community to carry out the necessary measurements over such large ocean areas.

Further Reading Arimoto R, Kim YJ, Kim YP, et al. (2006) Characterization of Asian dust during ACE-Asia, global and planetary change. Monitoring and Modelling of Asian Dust Storms 52(1–4): 23--56. Arimoto R, Ray BJ, Lewis NF, Tomza U, and Duce RA (1991) Mass-particle size distributions of atmospheric dust and the dry deposition of dust to the remote ocean. Journal of Geophysical Research – Atmospheres 102(D13): 15867--15874.

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Boyd PW, Watson A, Law CS, et al. (2000) A mesoscale phytoplankton bloom in the polar Southern Ocean stimulated by iron fertilization of waters. Nature 407: 695--702. Dentener F, Drevet J, Lamarque JF, et al. (2006) Nitrogen and sulfur deposition on regional and global scales: A multimodel evaluation. Global Biogeochemical Cycles 20: GB4003 (doi:10.1029/2005GB002672). Dentener F, Stevenson D, Ellingsen K, et al. (2006) The global atmospheric environment for the next generation. Environmental Science and Technology 40(11): 3586--3594 (doi:10.1021/es0523845). Duce RA, Liss PS, Merrill JT, et al. (1991) The atmospheric input of trace species to the world ocean. Global Biogeochemical Cycles 5: 193--259. Engelstaedter S, Tegen I, and Washington R (2006) North African dust emissions and transport. Earth-Science Reviews 79(1–2): 73--100. Harrison SP, Kohfeld KE, Roeland C, and Claquin T (2001) The role of dust in climate today, at the Last Glacial Maximum and in the future. Earth-Science Reviews 54: 43--80. Husar RB, Prospero JM, and Stowe LL (1997) Characterization of tropospheric aerosols over the oceans with the NOAA advanced very high resolution radiometer optical thickness operational product. Journal of Geophysical Research 102(D14): 16889--16909. Jickells TD (2006) The role of air–sea exchange in the marine nitrogen cycle. Biogeosciences 3: 271--280. Jurado E, Jaward F, Lohmann R, et al. (2005) Wet deposition of persistent organic pollutants to the global oceans. Environmental Science and Technology 39(8): 2426--2435 (doi:10.1021/es048599 g). Mahowald NM, Baker AR, Bergametti G, et al. (2005) Atmospheric global dust cycle and iron inputs to the ocean. Global Biogeochemical Cycles 19: GB4025 (doi:10.1029/2004GB002402). Parekh P, Follows MJ, and Boyle EA (2005) Decoupling of iron and phosphate in the global ocean. Global Biogeochemical Cycles 19: GB2020 (doi:10.1029/ 2004GB002280). Prospero JM (1996) The atmospheric transport of particles to the ocean. In: Ittekkot V, Scha¨fer P, Honjo S and Depetris PJ (eds.) Particle Flux in the Ocean. SCOPE Report 57, pp. 19--52. Chichester: Wiley. Wesely ML and Hicks BB (2000) A review of the current status of knowledge on dry deposition. Atmospheric Environment 34(12–14): 2261--2282.

SURFACE FILMS W. Alpers, University of Hamburg, Hamburg, Germany Copyright & 2001 Elsevier Ltd.

Introduction Surface films floating on the sea surface are usually attributed to anthropogenic sources. Such films consist, for example, of crude oil discharged from tankers during cleaning operations or accidents. However, much more frequently surface films that are of natural origin are encountered at the sea surface. These natural surface films consist of surfaceactive compounds that are secreted by marine plants or animals. According to their physico-chemical characteristics the film-forming substances tend to be either enriched at the sea surface (more hydrophobic character, sometimes referred to as ‘dry surfactant’) or they prevail within the upper water layer (more hydrophobic character ‘wet surfactants’). The first type of surface-active substances (‘dry surfactants’) are able to form monomolecular slicks at the airwater interface and damp the short-scale surface waves (short-gravity capillary waves) much more strongly than the second type. This implies that they have a strong effect on the mass, energy, and momentum transfer processes at the air–water interface. They also affect these transfer processes by reducing the turbulence in the subsurface layer which is instrumental in transporting water from below to the surface. Both types of surface films are easily detectable by radars because radars are roughness sensors and surface films strongly reduce the short-scale sea surface roughness.

Orgin of Surface Films Surface films at the sea surface can be either of anthropogenic or natural origin. Anthropogenic surface films consist, e.g., of crude or petroleum oil spilled from ships or oil platforms (‘spills’), or of surfaceactive substances discharged from municipal or industrial plants (‘slicks’). Natural surface films may also consist of crude oil which is leaking from oil seeps on the seafloor, but usually they consist of surface-active substances, which are produced by biogenic processes in the sea (‘biogenic slicks’). In

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general, the biogenic surface slicks consisting of sufficiently hydrophobic substances (‘dry surfactants’) are only one molecular layer thick (approximately 3 nm). This implies that it needs only few liters of surface-active material to cover an area of 1 km2. The prime biological producers of natural surface films in the sea are algae and some bacteria. Also zooplankton and fish produce surface-active materials, but the amount is usually small in comparison with the primary biological production. Primary production depends on the quantity of light energy available to the organism and the availability of inorganic nutrients. In the higher latitudes light energy depends strongly on the season of the year which results in a seasonal variation of the primary biological production in the ocean and thus of the slick coverage by natural surface films. At times when the biological productivity is high, i.e., during plankton blooms, the probability of encountering natural biogenic surface films is strongly enhanced. In other regions of the world’s ocean where the primary production is not mainly determined by the quantity of light energy available to the organisms, but by the nutrient levels, the seasonal variation of the slick coverage is smaller, however, still observable. Surface slicks of biogenic origin are mainly encountered in sea regions where the nutrition factor favors productivity. This is the case on continental shelves, slopes, and in upwelling regions where nutrition-rich cold water is transported to the sea surface. The areal extent, the concentration and the composition of the surface films vary strongly with time. At high wind speeds (typically above 8–10 m s1) breaking waves disperse the films by entrainment into the underlying water such that they disappear from the sea surface. In general, the probability of encountering surface films of biogenic origin decreases with wind speed. Furthermore, after storms, enhanced coverage of the sea surface with biogenic slicks consisting of ‘dry surfactants’ is often observed which is due to the fact that, firstly, the secretion of surface-active material by plankton is being increased during higher wind speed periods, and secondly, the surface-active substances are being transported to the sea surface from below by turbulence and rising air bubbles generated by breaking waves. The composition of the sea slicks varies also with time because constituents of the surface films are selectively removed by dissolution, evaporation, enzymatic degradation and photocatalytic oxidation.

SURFACE FILMS

Modifications of Air–Sea Interaction by Surface Films Numerous processes that take place at the air–sea interface are affected by surface films. Among other things, surface films: (1) attenuate the surface waves; (2) reduce wave breaking; (3) reduce gas transfer; (4) increase the sea-surface temperature; (5) change the reflection of sunlight; and (6) reduce the intensity of the radar backscatter. Attenuation of Surface Waves

Two main factors contribute to the damping of short-scale surface waves by surface films: 1. the enhanced viscous dissipation in a thin water layer below the water surface caused by strong velocity gradients induced by the presence of viscoelastic films at the water surface; and 2. the decrease in energy transfer from the wind to the waves due to the reduction of the aerodynamic roughness of the sea surface. The enhanced viscous dissipation caused by the surface films results from the fact that, due to the different boundary condition imposed by the film at the sea surface, strong vertical velocity gradients are encountered in a thin layer below the water surface. This layer, also called shear layer, has a thickness of the order of 104 m. Here strong viscous dissipation takes place. In the case of mineral oil films floating on the sea surface, the shear layer may lie completely within the oil layer, but more often it extends also into the upper water layer since the thickness of mineral oil films is typically in the range of 103–106 m. In the case of biogenic monomolecular surface films accumulating at a rough sea surface, the surface films are compressed and dilated periodically, causing variations of the concentration of the molecules and thus of the surface tension. In this case not only the well-known gravity-capillary waves (surface waves) are excited, but also the so-called Marangoni waves. The Marangoni waves are predominantly longitudinal waves in the shear layer. They are heavily damped by viscous dissipitation; at a distance of only one wavelength from their source their amplitude has already decreased to less than onetenth of the original value. This is the reason why Marangoni waves escaped detection until 1968. When these gravity-capillary waves and Marangoni waves are in resonance as given by linear wave theory, the surface waves experience maximum damping. Depending on the viscoelastic properties of the surface film, maximum damping of the surface waves usually occurs in the centimeter to decimeter wavelength region.

219

Reduction of Wave Breaking

Since surface films reduce roughness of the sea surface, the stress exerted by the wind on the sea surface is reduced. Furthermore, the steepness of the shortscale waves is decreased which leads to less wave breaking. Reduction of Gas Transfer

Biogenic surface films do not constitute a direct resistance for gas transfer. However, they do have a major effect on the structure of subsurface turbulence and thus on the rate at which surface water is renewed by water from below. Furthermore, surface films reduce the air turbulence above the ocean surface and thus also the surface renewal. As a consequence, the gas transfer rate across the air–sea interface is reduced in the presence of surface films. Change of Sea Surface Temperature

In infrared images the sea surface areas covered with biogenic surface films usually appear slightly warmer than the adjacent slick-free sea areas (typical temperature increase 0.2–0.5 K). This is due to the fact that surface films reduce the mobility of the near-surface water molecules and slow down the conventional overturn of the surface layer by evaporation. Change of Reflection of Sunlight

Slick-covered areas of the sea surface are easily visible by eye when they lie in the sun-glitter area. This is an area where facets on the rough sea surface are encountered that have orientations that reflect the sunlight to the observer. When the surface is covered with a surface film, the sea surface becomes smoother and thus the orientation of the facets is changed such that the amount of light reflected to the observer is increased. Thus surface slicks become detectable in sun-glitter areas as areas of increased brightness. Outside the sun-glitter area they are sometimes also visible, but with a much fainter contrast. In this case they appear as areas of reduced brightness relative to the surrounding. Radar Backscattering

Surface slicks floating on the sea surface also become visible on radar images because they reduce the short-scale sea surface roughness. Since the intensity of the radar backscatter is determined by the amplitude of short-scale surface waves, slick-covered sea surfaces appear on radar images as areas of reduced radar backscattering. Since radars have their own illumination source and transmit electromagnetic

220

SURFACE FILMS

Figure 1 Radar image acquired by the synthetic aperture radar (SAR) aboard the First European Remote Sensing satellite (ERS-1) on 20 May 1994 over the coastal waters east of Taiwan. The imaged area is 70 km  90 km. It shows a ship (the bright spot at the front of the black trail) discharging oil. The oil trail, which is approximately 80 km long, widens towards the rear because the oil disperses with time. Copyright ^ 2000, European Space Agency.

waves with wavelengths in the centimeter to decimeter range, radar images of the sea surface can be obtained independent of the time of the day and independent of cloud conditions. This makes radar an ideal instrument for detecting oil pollution and natural surface films at the sea surface. Consequently, most oil pollution surveillance aircraft which patrol coastal waters for locating illegal discharges of oil from ships are equipped with imaging radars. Unfortunately the reduction in backscattered radar intensity caused by mineral oil films is often of the same order (typically 5–10 decibels) as that of natural surface films. This makes it difficult by using the information contained in the reduction of the backscattered radar intensisty to differentiate whether the black patches visible on radar images of the sea surface originate from one or the other type of film. However, in many cases the shape of the black patches on the radar images can be used for discrimination. A long elongated dark patch is indicative of an oil spill originating from a travelling ship. Examples

Figure 2 Radar image acquired by the SAR aboard the Second European Remote Sensing satellite (ERS-2) on 10 May 1998 over the Western Baltic Sea which includes the Bight of Lu¨beck (Germany). The imaged area is 90 km  100 mm. Visible are the lower left coastal areas of Schleswig-Holstein with the island of Fehmarn (Germany) and in the upper right part of the Danish island of Lolland. The black areas are sea areas covered with biogenic slicks which are particularly abundant in this region during the time of the spring plankton bloom. The slicks follow the motions of the sea surface and thus render oceanic eddies visible on the radar image. Copyright ^ 2000, European Space Agency.

of radar images on which both types of surface films are visible are shown in Figures 1 and 2.

See also Air–Sea Gas Exchange.

Further Reading Alpers W and Hu¨hnerfuss H (1989) The damping of ocean waves by surface films: a new look at an old problem. Journal of Geophysical Research 94: 6251--6265. Levich VG (1962) Physico-Chemical Hydrodynamics. Englewood Cliffs, NJ: Prentice-Hall. Lucassen J (1982) Effect of surface-active material on the damping of gravity waves: a reappraisal. Journal of Colloid Interface Science 85: 52--58. Tsai WT (1996) Impact of surfactant on turbulent shear layer under the air–sea interface. Journal of Geophysical Research 101: 28557--28568.

BUBBLES D. K. Woolf, Southampton Oceanography Centre, Southampton, UK Copyright & 2001 Elsevier Ltd.

Introduction Air–sea interaction does not solely occur directly across the sea surface, but also occurs across the surface of bubbles suspended in the upper ocean, and across the surface of droplets in the lower atmosphere. This article describes the role of bubbles in air–sea interaction. There are three quite different types of bubbles in the oceans that can be distinguished by their sources (atmospheric, benthic, and cavitation). Benthic sources of bubbles include vents and seeps and consist of gases escaping from the seafloor. Common gases from benthic sources include methane and carbondioxide. Cavitation is largely an unintentional byproduct of man’s activities; typically occurring in the wake of ship propellors. It consists of the rapid growth and then collapse of small bubbles composed almost entirely of water vapor. Cavitation may be thought of as localized boiling, where the pressure of the water falls briefly below the local vapor pressure. Cavitation is important in ocean engineering due to the damage inflicted on man-made structures by collapsing bubbles. Both cavitation bubbles and bubbles rising from the seafloor are encountered in the upper ocean, but are peripheral to air–sea interaction. Atmospheric sources of bubbles are a product of air–sea interaction and, once generated, the bubbles are themselves a peculiar feature of air–sea interaction. The major sources of bubbles in the upper ocean are the entrapment of air within the flow associated with breaking waves and with rain impacting on the sea surface. Once air is entrapped at the sea surface, there is a rapid development stage resulting in a cloud of bubbles. Some bubbles will be several millimeters in diameter, but the majority will be o0.1 mm in size. Each bubble is buoyant and will tend to rise towards the sea surface, but the upper ocean is highly turbulent and bubbles may be dispersed to depths of several meters. Small particles and dissolved organic compounds very often collect on the surface of a bubble while it is submerged. Gas will also be slowly exchanged across the surface of bubbles, resulting in

a continual evolution of the size and composition of each bubble. The additional pressure at depth in the ocean will compress bubbles and will tend to force the enclosed gases into solution. Some bubbles will be forced entirely into solution, but generally the majority of the bubbles will eventually surface carrying their coating and altered contents. At the surface, a bubble will burst, generating droplets that form most of the sea salt aerosol suspended in the lower marine atmosphere. The measurement of bubbles in the upper ocean depends largely on their acoustical and optical properties. At the same time, the effect of bubbles on ocean acoustics has long been a major motivation for bubble studies. The generation of noise at bubble inception may be exploited. For example, acoustic measurements of rainfall depend on bubble phenomena. Fully formed bubble clouds attenuate and scatter both sound and light in the upper ocean. Climatologies of the distribution of bubbles in the upper ocean are based on both acoustical and optical measurements of bubbles. The global distribution of bubbles reflects the dominance of wave breaking as a source of bubbles, and the high sensitivity of wave breaking to wind speed. Bubbles are an important component of global geochemical cycling through their transport of material in the upper ocean and surface microlayer (see Surface Films), and especially their role in the air–sea exchange of gases and particles.

Sources of Bubbles As described already, bubbles may originate in a variety of ways, but this section will concentrate on the major natural processes of air bubble formation. The atmosphere is clearly a potential source of air bubbles, and generation involves the ‘pinching off’ of part of the atmosphere, or the ‘condensation’ of gases dissolved from the atmosphere within a body of water. Generation of bubbles within the body of water, when the surface water is sufficiently supersaturated with air, is similar to ‘vapor’ cavitation, but involves the major constituents of the atmosphere (nitrogen, oxygen, etc.) rather than water vapor alone. In the absence of hydrodynamic pressure effects associated with flow, the radial pressure into a cavity, Pb, is the sum of the atmospheric pressure, Pa, the hydrostatic pressure at a depth, z, and a component associated with the surface tension, g, and the curvature of the

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222

BUBBLES

cavity (or radius ‘r’): Pb ¼ Pa þ rgz þ 2g=r For a bubble to grow, the pressure within a bubble (equal to the sum of partial pressures of the gases diffusing into the bubble), must exceed atmospheric pressure by a margin that increases both with water depth and the curvature of the cavity. A sufficiently large initial cavity is necessary for inception. The explosive dynamics of ‘true’ cavitation are associated with the rapid transport of water vapor across the surface of the cavity. However, the conditions for water vapor cavitation can only be achieved at normal temperatures where pressure is very low. A sufficient pressure anomaly may occur in an intense acoustic pulse, or in the wake of a fast moving solid object, but is not a common natural phenomenon. The conditions for growth of a bubble by diffusion of atmospheric gases are possible within the normal range of natural variability. For gases other than water vapor, molecular transport of the dissolved gases near the surface of the bubble is sufficiently slow that a virtual equilibrium between the internal and external pressures on the bubble must exist. We might observe bubble generation at home within a bucket of water, or in a soda bottle, where warming induces supersaturation (the solubility of most gases decreases with increasing temperature) and defects in the container walls provide the initial cavity. Warming, mixing, or bubble injection may occasionally force supersaturations of several percent at sea, in which case growth of bubbles on natural particles and microbubbles may release the excess pressure. Entrapment of air at the sea surface is more common than inception within the body of the water. Most of us are familiar with plumes of bubbles generated by paddling and by boats, but the entrapment of air in the absence of a solid boundary is less intuitive. In general, air is rarely entrapped by enclosure of a large air volume, but is usually drawn into the interior (‘entrained’) where there is intense and convergent flow of water at the sea surface. Sufficiently energetic convergence occurs where precipitation impacts on the sea surface, and where waves break. Bubble formation is associated with all common forms of precipitation (rain, hail, and snow), but the details of bubble formation are highly specific to the details of the precipitation. In particular, bubble formation by rain is known to be sensitive to the size, impact velocity, and incidence angle of the rain drops. Large drops, exceeding 2.2 mm in diameter, entrain most air. In heavy tropical downfalls, the

volume of air entrained can be fairly significant (B 106 m3 m2 s1), although much lower than rates associated with wave breaking in high winds. Bubbles up to 1.8 mm in radius are entrained by large rain drops, but smaller drops (0.8–1.1 mm in diameter) generate bubbles of only 0.2 mm in radius. When waves break at the seashore, the large ‘dominant’ waves dissipate their energy partly in entraining and submerging quite large volumes of air. On the open ocean, some of the largest and longest waves break, but wave breaking also occurs at much smaller scales. Some very small breaking events may be too weak to entrain air; however, small but numerous breaking events entraining small volumes of air occur on steep waves as short as 0.3 m in wavelength. The energy dissipated in wave breaking is derived from wind forcing of surface waves, and the amount of wave breaking and air entrainment is very sensitive to wind speed. The stage of development of the wave field also has some influence on air entrainment – the size of the largest breaking event is limited to the largest wave that has developed. An important feature of bubble generation at the sea surface is that a myriad of very small (o0.1 mm radius) bubbles is produced. Very large cavities several millimeters in diameter are likely to be torn apart by large shear forces at the sea surface, but it is difficult to explain how bubbles of o1 mm might be fragmented. Also, the same processes in fresh water (e.g. a lake or a waterfall) do not produce many small bubbles. The explanation can be found in the influence of dissolved salts on surface forces. In sea water, a surface deformation will tend to grow more and more contorted, so that when a large bubble is fragmented it will often shatter into numerous much smaller bubbles. The same factors will usually prevent the coalescence of bubbles in sea water.

Dispersion and Development Bubbles entrained by a breaking wave may be carried rapidly to a depth of the order of the height of the breaking wave by its energetic turbulent plume. For some wave breaking and other forms of bubble production the initial injection will be much shallower (B1–100 mm). Most of the bubbles are very small, but the majority of the volume of air is comprised of fairly large (B1 mm) bubbles entrained by breaking waves. Most of these larger bubbles will soon rise to the surface (typically in B1 s) in a highly dynamic plume close behind the breaking wave. The less buoyant, smaller bubbles are generally carried to a greater depth and are easily dispersed by mixing processes in the upper ocean.

BUBBLES

Bubbles are mixed into the ocean by small-scale turbulence associated with the ‘wind-driven upper ocean boundary layer’, but also by relatively large and coherent turbulent structures, especially Langmuir circulation (see Langmuir Circulation and Instability). Langmuir circulation comprises sets of paired vortices (cells) aligned to the wind. Bubbles will be drawn to the downwelling portions of the Langmuir cells, producing lines of enhanced bubble concentration, parallel to the wind. Langmuir cells can be up to tens of meters deep and wide, and downwelling speeds may exceed 0.1 m s1. In principle, even quite large bubbles may be forced downwards, but generally bubbles of only B20 mm

radius are most common at depths of Z1 m. Concentrations fall off rapidly with increasing radius, at radii exceeding the modal radius. The development of a bubble cloud does not solely concern the movement of bubbles, but also concerns the development of each and every bubble. Material will be transferred between the bubble and the surrounding water as a result of the flow of water around the bubble (largely induced by the buoyant rise of bubbles relative to their surroundings) and molecular diffusion close to the surface of the bubble. This transport plays a large part in the role of bubbles in geochemical cycling, which is illustrated schematically in Figure 1. The transport of both

Aerosol production Deposition

Jet drops

Film drops Atmosphere

Bursting bubble

Microlayer Ocean

Bubble scavenging

223

Gas exchange

Figure 1 A schematic illustration of the role of bubbles in geochemical cycling.

224

BUBBLES

volatile (i.e. gases) and nonvolatile substances is of interest. Nonvolatile substances will not penetrate the bubble itself, but may be transported between the surface of the bubble and the surrounding water. Many substances are ‘surface-active’, that is, they tend to stick to the surface and alter the dynamic properties of the surface. Some substances will already be adsorbed on the surface at the point of formation at the sea surface. During the lifetime of a bubble, further material (both dissolved and small particles) will accumulate on the surface of a bubble. One consequence of the ‘bubble scavenging’ process is the cycling of surface-active substances. Also, the surface-active material will alter the dynamic properties of the bubble, critically affecting the rise velocity of the bubble and transport across the surface of the bubble. A pure water surface is ‘mobile’, but it may be immobilized by surface-active material. The flow near a mobile (or free) surface and a rigid surface is quite different. Generally, a ‘dirty’ bubble with a contaminated, rigid surface will rise more slowly and will exchange gas at a much slower rate compared with a ‘clean’ bubble of the same size. The surface of small bubbles is immobilized by only a small amount of contamination, and bubbles o100 mm radius are likely to behave as dirty bubbles for most or all of their life. Larger bubbles will also be contaminated, but their dynamic behavior may remain close to that of a ‘clean’ bubble for several seconds (depending on bubble radius and the contamination level of the water). The transfer of gases across the surface of bubbles is important to the evolution of each bubble, and to the atmosphere–ocean transport of gases. Gases will diffuse across the surface of a bubble. The net transport of each gas across the surface of a single bubble depends on its concentration in the two media and the mechanics of transport: bubblewater flux ¼ j4pr2 ½Cw  Spb  As explained in the previous section, the gases within a bubble are compressed so that the pressure of gases in the bubble generally exceeds those in the atmosphere. This excess leads to a tendency for bubbles to force supersaturation of gases in the upper ocean. Many bubbles may be forced entirely into solution (possibly leaving a fragment enclosed in a shell of organics and small particles – a microbubble). The total (integral) effect of bubble clouds on air–sea gas exchange can be described by the following formula (see Air–Sea Gas Exchange): airsea flux ¼ KT ½Cw  Spa ð1 þ DÞ

(per unit area of sea surface) ¼ Kb ½ð1 þ dÞCa =H  Cw  þ Ko ½Ca =H  Cw  The effect of bubbles on air–sea exchange is described by two coefficients: the contribution to the transfer coefficient, Kb, and a ‘saturation anomaly’, D. Both of these coefficients depend greatly on the solubility of the gas and the bubble statistics. For relatively soluble gases, such as carbon dioxide, the saturation anomaly due to bubble injection is generally negligible, but for less soluble gases, including oxygen the anomaly is usually significant, particularly at high wind speeds. The contribution to the transfer coefficient is again greater for less soluble gases, but is likely to be significant for most gases, at least for high wind speeds. We have focused on unstable bubbles that will either surface or dissolve within a few minutes of their creation. When a bubble totally dissolves it may leave a conglomeration of the particles and the organic material it accumulated. Some of the bubbles may not entirely dissolve, but may be stabilized at a radius of a few micrometers by their collapsed coating. (The mechanism of stabilization is rather mysterious, external pressures will be high and the coating can not entirely prevent the diffusion of gas, therefore total collapse must be resisted by the structural integrity of the coating – perhaps like a traditional stone wall.) Stable microbubbles might also be generated by a biological mechanism. Microbubble populations are denser in coastal waters where biological productivity and organic loading are generally higher. Microbubbles influence the acoustic properties of natural waters and are a common nucleus for cavitation.

Surfacing and Bursting Many small bubbles dissolve in the upper ocean, but generally the majority of the bubbles (and almost all the large bubbles) eventually surface. Phenomena that occur when a bubble surfaces are again significant to geochemical cycling (Figure 1). The release of gas from a bubble to the atmosphere completes the process of air–sea gas exchange mediated by the bubble. The approach of a bubble, or more especially a plume of bubbles, can disrupt the surface microlayer, enhancing turbulent transport directly across the sea surface. The bubble carries material to the sea surface accumulated by scavenging within the upper ocean. Most important are the energetic processes that occur when a bubble bursts on the sea surface. Bubble bursting is responsible for ejecting droplets into the atmosphere, creating the sea salt aerosol.

BUBBLES

Droplets can also be torn directly from wave crests, but bubbles generate almost all of the very small droplets that are easily suspended in the lower atmosphere and that will be dispersed over large distances. When a bubble surfaces its upper surface will project beyond the sea surface. This ‘film cap’ will drain and shatter. The shattering of the film cap produces ‘film drops’. In some cases, the film cap can shatter into many remarkably small (o1 mm) droplets, while in other cases a few large B10 mm radius droplets will be produced. The open cavity left after the film cap shatters will collapse inwards, leading to the upward ejection of a ‘Worthington jet’. This jet will pinch off into a few ‘jet drops’. The drop radii will typically be one-tenth of the radius of the parent bubble, producing drops from B2 mm to tenths of a millimeter in radius from a typical bubble population. The droplets generated by bursting bubbles will be enriched by material brought to the sea surface by the bubble and drawn from the sea surface. The sea salt aerosol will include organic material, metals, viruses, and bacteria.

Acoustical and Optical Properties Our knowledge of bubble distributions in the upper ocean is based on acoustical and optical measurements. Bubbles also have a significant impact on the acoustical and optical properties of the upper ocean. The acoustic properties of bubbles have attracted a great deal of attention. The generation of bubbles, both by breaking waves and rain, is an important source of noise in the upper ocean. Bubbles also absorb and scatter sound. The scattering of sound by an individual bubble is frequency-dependent with three primary regimes: close to, above, and below the ‘breathing frequency’ of the bubble. The breathing frequency of a bubble is the natural frequency at which a bubble will oscillate radially (‘breathe’) and is determined by its radius, surface tension, and the external pressure. The breathing frequency is inversely related to bubble radius, and in the upper ocean, bubbles of different radii will respond in resonance to acoustic frequencies from 10 kHz to a few hundred kHz. Scattering cross-sections close to resonance are very high. When the acoustic frequency is much higher than the breathing frequency of the bubble, the scattering by the bubble is related simply to its physical size (‘geometric scattering’). At low acoustic frequencies the acoustic crosssection of an individual bubble is much lower than its geometric cross-section (Rayleigh scattering). The scattering by a bubble is equal in every direction (isotropic) at most practical frequencies, but

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becomes more anisotropic at very low frequencies. At low acoustic frequencies (o10 kHz), scattering is largely a communal response of clouds rather than of individual bubbles. Many measurements of bubbles have taken advantage of the resonant response of bubbles to sound. In particular, measurements at a number of acoustic frequencies can be inverted to calculate the size distribution of bubbles. A pair of transmitting and receiving ‘transducers’ can measure backscatter remotely along a profile. This technique has been used to infer the concentration and size of bubbles as a function of depth. The very high scattering by the concentrated plumes near breaking waves defy remote measurement. Instead bubbles near the surface may be studied by measuring absorption or scattering along a short path length. Other techniques include applying the influence of air void on the conductivity of the water, and optical measurements. Casual observation of the milky water marking a developing bubble cloud is enough to understand that bubbles in the upper ocean can alter the optical properties (e.g. color and brightness) of the sea, but among the numerous and complicated influences on ocean optics, bubbles have received relatively little attention. Bubble populations have been measured photographically, but for the sparse populations a meter or so beneath the sea surface this method is tedious if ultimately effective. Video footage of wave breaking and the early development of bubble plumes can be used to understand the many processes involved.

Summary of Bubble Distribution Measurements of bubbles in the ocean are still fairly sparse, and the relationship of wave breaking and bubble injection to environmental conditions is only partly understood, but we can at least summarize the general relationship of unstable bubble populations to wind forcing. Away from the immediate plume of a breaking wave, the mean concentration of bubbles of radius, r, at a depth z, typically follows a distribution of the form, Npr4 expðz=LÞ for radii as small as 30 mm, but there is a maximum in N typically at 25 mm radius. A typical attenuation depth, L, is 1 m. Some studies have suggested only a weak, approximately linear relationship between the attenuation depth and wind speed, but recent extensive studies imply attenuation depths proportional to the square of wind speed. There are fewer measurements of bubbles in the upper ocean

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within half a meter of the sea surface, but it is clear that concentrations are much higher near a breaking wave, and larger bubbles are far more common. The injection rate of bubbles is expected to increase with the third or fourth power of the wind speed. As vertical dispersion of the bubbles (and the attenuation depth) also increase with wind speed, the concentration of bubbles below the sea surface is extremely sensitive to wind speed. Air–sea gas exchange, scavenging, and other geochemical transport processes associated with bubbles will share this sensitivity to wind speed, suggesting that a large fraction of activity may occur in fairly rare storm conditions.

See also Air–Sea Gas Exchange. Air–Sea Transfer: Dimethyl NH4, Non-Methane Sulfide, COS, CS2, Hydrocarbons, Organo-Halogens. Air–Sea Transfer: N2O, NO, CH4, CO. Breaking Waves and Near-Surface Turbulence. Evaporation and Humidity. Heat and Momentum Fluxes at the Sea

Surface. Langmuir Circulation and Instability. Surface Films. Three-Dimensional (3D) Turbulence. Upper Ocean Mixing Processes. Wave Generation by Wind. Whitecaps and Foam.

Further Reading Blanchard DC (1983) The production, distribution, and bacterial enrichment of the sea-salt aerosol. In: Liss PS and Slinn WGN (eds.) The Air–Sea Exchange of Gases and Particles, pp. 407--454. Dordrecht: Kluwer. Leighton TG (1994) The Acoustic Bubble. San Diego: Academic Press. Medwin H and Clay CS (1998) Fundamentals of Acoustical Oceanography. San Diego: Academic Press. Monahan EC (1986) The ocean as a source for atmospheric particles. In: Buat-Me´nard P (ed.) The Role of Air–Sea Exchange in Geochemical Cycling, pp. 129--163. Dordrecht: Kluwer. Woolf DK (1997) Bubbles and their role in gas exchange. In: Liss PS and Duce RA (eds.) The Sea Surface and Global Change, pp. 173--205. Cambridge: Cambridge University Press.

BOUNDARY LAYERS: THE UPPER OCEAN BOUNDARY LAYER

UPPER OCEAN VERTICAL STRUCTURE J. Sprintall, University of California San Diego, La Jolla, CA, USA M. F. Cronin, NOAA Pacific Marine Environmental Laboratory, Seattle, WA, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The upper ocean connects the surface forcing from winds, heat, and fresh water, with the quiescent deeper ocean where this heat and fresh water are sequestered and released on longer time- and global scales. Classically the surface layer includes both an upper mixed layer that is subject to the direct influence of the atmosphere, and also a highly stratified zone below the mixed layer where vertical property gradients are strong. Although all water within the surface layer has been exposed to the atmosphere at some point in time, water most directly exposed lies within the mixed layer. Thus, the surface layer vertical structure reflects not only immediate changes in response to the surface forcing, but also changes associated with earlier forcing events. These forcing events may have occurred either locally in the region, or remotely at other locations and transferred by ocean currents. This article first defines the major features of the upper ocean vertical structure and discusses what causes and maintains them. We then show numerous examples of the rich variability in the shapes and forms that these vertical structures can assume through variation in the atmospheric forcing.

radiometers. In contrast, in situ sensors generally measure the ‘bulk’ SST over the top few meters of the water column. The cool skin temperature is generally around 0.1–0.5 K cooler than the bulk temperature. As the air–sea fluxes are transported through the molecular layer almost instantaneously, the upper mixed layer can generally be considered to be in direct contact with the atmosphere. For this reason, when defining the depth of the surface layer, the changes in water properties are generally made relative to the bulk SST measurement. The upper mixed layer is the site of active air–sea exchanges. Energy for the mixed layer to change its vertical structure comes from wind mixing or through a surface buoyancy flux. Wind mixing causes vertical turbulence in the upper mixed layer through waves, and by the entrainment of cooler water through the bottom of the mixed layer. Wind forcing also results in advection by upper ocean currents that can change the water properties and thus the vertical structure of the mixed layer. Surface buoyancy forcing is due to heat and fresh water fluxed across the air–sea interface. Cooling and evaporation induce convective mixing and overturning, whereas heating and rainfall cause the mixed layer to restratify in depth and display alternate levels of greater and lesser vertical

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Major Features of the Upper Ocean Vertical Structure The vertical structure of the upper ocean is primarily defined by the temperature and salinity, which together control the water column’s density structure. Within the ocean surface layer, a number of distinct layers can be distinguished that are formed by different processes over different timescales: the upper mixed layer, the seasonal pycnocline, and the permanent pycnocline (Figure 1). Right at the ocean surface in the top few millimeters, a cool ‘skin’ exists with lowered temperature caused by the combined heat losses from long-wave radiation, sensible and latent heat fluxes. The cool skin is only a few millimeters thick, and is the actual sea surface temperature (SST) measured by airborne infrared

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Figure 1 Conceptual diagram of the vertical structure in the surface layer, and the forcing and physics that govern its existence. The depth of the mixed layer, the seasonal pycnocline, and the main pycnocline are indicated.

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property gradients. Thus, if strong enough, the wind and buoyancy fluxes can generate sufficient turbulence so that the upper portion of the surface layer has a thick, homogeneous (low vertical gradient or stratification), well-mixed layer in temperature, salinity, and density. Wind and buoyancy forcing also affect the vertical structure of the velocity or shear (vertical gradient of horizontal velocity) in the upper mixed layer. Upper ocean processes, such as inertial shear, Langmuir circulations, internal gravity waves, and Kelvin–Helmholtz instabilities, that alter the velocity profile in the surface layer are an active area of research, and are more fully discussed in Upper Ocean Mixing Processes. Temporal and spatial variations in the strength and relative contributions of the atmospheric forcing can cause substantial variability in the water properties and thickness of the upper mixed layer. Large temporal variation can occur on daily and seasonal timescales due to changes in the solar radiation. For example, during the daily cycle the sun heats the ocean, causing the upper surface to become increasingly warm and weakly stratified. The ‘classic’ vertically uniform mixed layer, as depicted in Figure 1, may not be present in the upper ocean surface layer. As the sun sets, the surface waters are cooled and sink, generating turbulent convection that causes entrainment of water from below and mixing that produces the vertically well-mixed layer. Similarly, the mixed layer structure can exhibit significant horizontal variations. The large latitudinal differences in solar radiation result in mixed layers that generally increase in depth from the equator to the Poles. Even in the east–west direction, boundary currents and differential surface forcing can result in mixed layers that assume different vertical structures, although generally the annual variations of temperature along any given latitude will be small. Temporal and spatial variability in the vertical structure of the mixed layer, and the physics that govern this variability are covered elsewhere (see Upper Ocean Mean Horizontal Structure, Upper Ocean Space and Time Variability, and Wind- and Buoyancy-Forced Upper Ocean). Separating the upper mixed layer from the deeper ocean is a region typically characterized by substantial vertical gradients in water properties. In temperature, this highly stratified vertical zone is referred to as the thermocline, in salinity it is the halocline, and in density it is the pycnocline. To maintain stability in the water column, lighter (less dense) water must lie above heavier (denser) water. It follows then, that the pycnocline is a region where density increases rapidly with depth. Although the thermocline and the halocline may not always

exactly coincide in their depth range, one or the other property will control the density structure to form the pycnocline. In mid-latitudes during summer, surface heating from the sun can cause a shallow seasonal thermocline (pycnocline) that connects the upper mixed layer to the deeper more permanent thermocline or ‘main pycnocline’ (see Figure 1). Similarly, in the subpolar regions, the seasonal summer inputs of fresh water at the surface through rainfall, rivers, or ice melt can result in a seasonal halocline (pycnocline) separating the fresh surface from the deeper saltier waters. Whereas the seasonal pycnocline disappears every winter, the permanent pycnocline is always present in these areas. The vertical density gradient in the main pycnocline is very strong, and the turbulence within the upper mixed layer induced by the air–sea exchanges of wind and heat cannot overcome the great stability of the main pycnocline to penetrate into the deeper ocean. The stability of the main pycnocline acts as a barrier against turbulent mixing processes, and beneath this depth the water has not had contact with the surface for a very long time. Therefore the main pycnocline marks the depth limit of the upper ocean surface layer. In some polar regions, particularly in the far North and South Atlantic, no permanent thermocline exists. The presence of an isothermal water column suggests that the cold, dense waters are continuously sinking to great depths. No stable permanent pycnocline or thermocline exists as a barrier to the vertical passage of the surface water properties that extend to the bottom. In some cases, such as along the shelf in Antarctica’s Weddell Sea in the South Atlantic, salinity can also play a role in dense water formation. When ice forms from the seawater in this region, it consists primarily of fresh water, and leaves behind a more saline and thus denser surface water that must also sink. The vertical flow of the dense waters in the polar regions is the source of the world’s deep and bottom waters that then slowly mix and spread horizontally via the large-scale thermohaline ocean circulation to fill the deep-ocean basins. In fact, the thermohaline circulation also plays an important role in maintaining the permanent thermocline at a relatively constant depth in the low and middle latitudes. Despite the fact that the pycnocline is extremely stable, it might be assumed that on some long-enough timescale it could be eroded away through mixing of water above and below it. Humboldt recognized early in the nineteenth century that ocean circulation must help maintain the low temperatures of the deeper oceans; the equatorward movement of the cold deep and bottom water masses are continually renewed through

UPPER OCEAN VERTICAL STRUCTURE

sinking (or ‘convection’) in the polar region. However, it was not until the mid-twentieth century that Stommel suggested that there was also a slow but continual upward movement of this cool water to balance the downward diffusion of heat from the surface. It is this balance, that actually occurs over very small space and timescales that sustains the permanent thermocline observed at middle and low latitudes. Thus, the vertical structure of the upper ocean helps us to understand not only the wind- and thermohaline-forced ocean circulation, but also the response between the coupled air–sea system and the deeper ocean on a global scale.

Definitions Surface Layer Depth

There is no generally accepted definition of the surface layer depth. Conceptually the surface layer includes the mixed layer, where active air–sea exchanges are occurring, plus those waters in the seasonal thermocline that connect the mixed layer and to the permanent thermocline. Note the important detail that the surface layer includes the mixed layer, a fact that has often been blurred in the criteria used to determine their respective depth levels. A satisfactory depth criterion for the surface layer should thus include all the major features of the upper ocean surface layer described above and illustrated in Figure 1. Further, the surface layer depth criterion should be applicable to all geographic regimes, and include those waters that have recently been in contact with the atmosphere, at least on timescales of up to a year. Finally, the definition should preferably be based on readily measurable properties such as temperature, salinity, or density. Ideally then, we could specify the surface layer to be the depth where, for instance, the temperature is equal to the previous winter’s minimum SST. However in practice, this surface layer definition would vary temporally, making it difficult to decipher the year-to-year variability. Oceanographers therefore generally prefer a static criterion, and thus modify the definition to be the depth where the temperature is equal to the coldest SST ever observed using any historical data available at a particular geographic location. This definition is analogous to a local ‘ventilation’ depth: the deepest surface to which recent atmospheric influence has been felt at least over the timescale of the available historical data. The definition suggested for the surface layer is also primarily one-dimensional, involving only the temperature and salinity information from a given location. Lateral advective effects have not been

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included. The roles of velocity and shear, and other three-dimensional processes in the surface layer structure (e.g., Langmuir circulations, internal gravity waves, and Kelvin–Helmholtz instabilities), may on occasion be important. However, their roles are harder to quantify and have not, as yet, been adequately incorporated into a working definition for the depth of the surface layer. Mixed Layer Depth

The mixed layer is the upper portion of the surface layer where active air–sea exchanges generate surface turbulence which causes the water to mix and become vertically uniform in temperature and salinity, and thus density. Very small vertical property gradients can still occur within the mixed layer in response to, for example, adiabatic heating or thermocline erosion. Direct measurements of the upper layer turbulence through dissipation rates provide an accurate and instant measurement of the active ‘mixing’ depth. However, while the technology is improving rapidly, turbulence scales are very small and difficult to detect, and their measurement is not widespread at present. Furthermore, the purpose of defining a mixed layer depth is to obtain more of an integrated measurement of the depth to which surface fluxes have penetrated in the recent past (daily and longer timescales). For this reason, as in the surface layer depth criterion, definitions of the mixed layer depth are most commonly based on temperature, salinity, or density. The mixed layer depth must define the depth of the transition from a homogeneous upper layer to the stratified layer of the pycnocline. Several definitions of the mixed layer depth exist in the literature. One commonly used mixed layer depth criterion determines the depth where a critical temperature or density gradient corresponding to the top of the maximum property gradient (i.e., the thermocline or pycnocline) is exceeded. The critical gradient criteria range between 0.02 and 0.05 1C m 1 in temperature, and 0.005 and 0.015 kg m 3 in density. This criterion may be sensitive to the vertical depth interval over which the gradient is calculated. Another mixed layer depth criterion determines a net temperature or density change from the surface isotherm or isopycnal. Common values used for the net change criterion are 0.2–1 1C in temperature from the surface isotherm, or 0.03–0.125 kg m 3 from the surface isopycnal. Because of the different dynamical processes associated with the molecular skin SST, oceanographers generally prefer the readily determined bulk SST estimate as the surface reference temperature. Ranges of the temperature and density values used in

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(b) 9503 6° S, 166° W

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Figure 2 Temperature (black line), salinity (blue line), and density (green line) during March 1995 from expendable conductivity– temperature–depth profiles in the Pacific Ocean at (a) 6.91 N, 173.21 W and (b) 61 S, 1661 W. In temperature, mixed layer depth is calculated using criteria of a net temperature change of 0.51 C (crossed box) and 1 1C (circle) from the sea surface; and a temperature gradient criteria of 0.01 1C m1 (small cross). In density, mixed layer depth is determined using criteria of a net density change of 0.125st units from the surface isopycnal (crossed box), a density gradient of 0.01st units m1 (circle), and the thermal expansion method of eqn [1] (cross). Note the barrier layer defined as the difference between the deeper isothermal layer and the shallow density-defined mixed layer in (b).

both mixed layer depth definitions will distinguish weakly stratified regions from unstratified. Another form of the net change criterion used to define the mixed layer depth (mld) takes advantage of the equivalence of temperature and density changes based upon the thermal expansion coefficient (a0 ¼ dTdr/dT, where dT is the net change in temperature from the surface, e.g., 0.2–1 1C, and dr/dT is calculated from the equation of state for seawater using surface temperature and salinity values). This criterion thus determines the depth at which density is greater than the surface density by an amount equivalent to the dT temperature change. In this way, this definition has the advantage of revealing mixed layers where salinity stratification may be important, such as in barrier layers, which are discussed further below. Criteria based on salinity changes, although inherent in the density criterion, are not evident in the literature as typically heat fluxes are large compared to freshwater fluxes, and the gravitational stability of the water column is often controlled by the temperature stratification. In addition, subsurface salinity observations are not as regularly available as temperature. To illustrate the differences between the mixed layer depth criteria, Figure 2(a) shows the mixed

layer depth from an expendable conductivity– temperature–depth (XCTD: see ) profile, using the net temperature (05 1C)and density (0.125 kg m 3) change criteria, the gradient density criterion (0.01 kg m 3), and a net change criterion based on the thermal expansion coefficient with dT ¼ 0.51C. In this particular case, there is little difference between the mixed layer depth determined from any method or property. However, Figure 2(b) shows an XCTD cast from the western Pacific Ocean, and the strong salinity halocline that defines the bottom of the upper mixed layer is only correctly identified using the density-defined criteria. Finally, to illustrate the distinction between the surface layer and the upper mixed layer, Figure 3(a) shows a temperature section of the upper 300 m from Auckland to Seattle during April 1996. The corresponding temperature stratification (i.e., the vertical temperature gradient) is shown in Figure 3(b). The surface layer, determined as the depth of the climatological minimum SST isotherm, and also the mixed layer depth from a 1 1C net temperature change from the surface (i.e., SST – 1 1C) are indicated on both panels. This cross-equatorial north–south section also serves to illustrate the seasonal differences expected in the mixed layer. In the early fall of the Southern

UPPER OCEAN VERTICAL STRUCTURE

Hemisphere, the net temperature mixed layer depth criterion picks out the top of the remaining seasonal thermocline, as depicted by the increase in temperature stratification in Figure 3(b). The mixed layer depth criterion therefore excludes information about the depth of the prior winter local wind stirring or heat exchange at the air–sea surface that has been successfully captured in the surface layer using the historical minimum SST criterion. In the Northern Hemisphere tropical regions where there is little seasonal cycle, the surface layer and the mixed layer criteria are nearly coincident. The depth of the mixed layer and the surface layer extend down to the main thermocline. Finally, in the early-spring northern latitudes, the mixed layer criterion again mainly picks out the upper layer of increased stratification that was likely caused through early seasonal surface heating. The surface layer definition lies deeper in the water column near the main thermocline, and below a second layer of

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relatively low stratification (Figure 3(b)). The deeper, weakly stratified region indicates the presence of fossil layers, which are defined in the next section.

Variability in Upper Ocean Vertical Structure Fossil Layers

Fossil layers are nearly isothermal layers that separate the upper well-mixed layer from a deeper wellstratified layer (see Figure 3(b), 31–371 N). The fact that these layers are warmer than the local minimum SST defining the surface layer depth, indicates that they have at some time been subject to local surface forcing. The solar heating and reduced wind stirring of spring can cause the upper layer to become thermally restratified. The newly formed upper mixed layer of light, warm water is separated from the

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older, deeper winter mixed layer by a well-stratified thermocline. The fairly stable waters in this seasonal thermocline may isolate the lower isothermal layer and prevent further modification of its properties, so that this layer retains the water characteristics of its winter formation period and becomes ‘fossilized’. Hence, fossil layers tend to form in regions with significant seasonal heating, a large annual range in wind stress, and deep winter mixed layers. These conditions can be found at the poleward edges of the subtropical gyres. In the northeast Pacific Ocean off California and in the southwest Pacific Ocean near New Zealand, particularly deep and thick fossil layers have been associated with the formation of subtropical mode waters. As with the fossil layers, the mode waters are distinguishable by low vertical gradients in temperature and density, and thus a narrow range or ‘mode’ of property characteristics. The isothermal layer or thermostad of winter water trapped in the fossilized layers may be subducted into the permanent thermocline through the action of Ekman pumping, in response to a curl in the wind field. The mode waters are then transported, retaining their characteristic thermostad, with flow in the subtropical gyre. Not all fossil layers are associated with mode water formation regions. Shallow fossil layers have also been observed where there are strong diurnal cycles, such as in the western equatorial Pacific Ocean. Here, the fossil layers are formed through the same alternating processes of heating/cooling and wind mixing as found in the mode water formation regions. Fossil layers have also been observed around areas of abrupt topography, such as along-island chains, where strong currents are found. In this case, the fossil layers are probably formed by the advection of water with properties different from those found in the upper mixed layer. Barrier Layers

In some regions, the freshwater flux can dominate the mixed-layer thermodynamics. This is evident in the Tropics where heavy precipitation can cause a surfacetrapped freshwater pool that forms a shallower mixed layer within a deeper nearly isothermal layer. The region between the shallower density-defined well-mixed layer and the deeper isothermal layer (Figure 2(b)) is referred to as a salinity-stratified barrier layer. Recent evidence suggests that barrier layers can also be formed through advection of fresh surface water, especially in the equatorial region of the western Pacific. In this region, westerly wind bursts can give rise to surface-intensified freshwater jets

that tilt the zonal salinity gradient into the vertical, generating a shallow halocline above the top of the thermocline. Furthermore, the vertical shear within the mixed layer may become enhanced in response to a depth-dependent pressure gradient setup by the salinity gradient and the trapping of the wind-forced momentum above the salinity barrier layer. This increased shear then leads to further surface intensified advection of freshwater and stratification that can prolong the life of the barrier layer. The barrier layer may have important implications on the heat balance within the surface layer because, as the name suggests, it effectively limits interaction between the ocean mixed layer and the deeper permanent thermocline. Even if under light wind conditions water is entrained from below into the mixed layer, it will have the same temperature as the water in this upper layer. Thus, there is no heat flux through the bottom of the mixed layer and other sinks must come into play to balance the solar warming that is confined to the surface, or more likely, the barrier layer is transient in nature. Inversions

Occasionally temperature stratification within the surface layer can be inverted (i.e., cool water lies above warmer water). The temperature inversion can be maintained in a stable water column since it is density-compensated by a corresponding salinity increase with depth throughout the inversion layer. Inversions are a ubiquitous feature in the vertical structure of the surface layer from the equator to subpolar latitudes, although their shape and formation mechanisms may differ. Inversions that form in response to a change in the seasonal heating at the surface are most commonly found in the subpolar regions. They can form when the relatively warmer surface water of summer is trapped by the cooler, fresher conditions that exist during winter. The vertical structure of the surface layer has a well-mixed upper layer in temperature, salinity, and density, lying above the inversion layer that contains the halocline and subsequent pycnocline (Figure 4(a)). Conversely, during summer, the weak subpolar solar heating can trap the very cold surface waters of winter, sandwiching them between the warmer surface and deeper layers. In this case, the vertical structure of the surface layer consists of a temperature minimum layer below the warm stratified surface layer, and above the relatively warmer deeper layer (Figure 4(b)). The density-defined mixed layer occurs above the temperature minimum. With continual but slow summer heating, the cold water found in this inversion layer slowly mixes with the

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Figure 4 Temperature (black line), salinity (blue line), and density (st, green line) from XCTD casts at (a) 58.21 N, 147.31 W in March 1996, (b) 611 S, 63.91 W in January 1997, (c) 11.91 S, 176.11 W in August 1998, and (d) 33.51 N, 134.61 W in May 1995. Note the presence of temperature inversions at the base of the mixed layer in all casts.

warmer water masses above and below, and erodes away. Inversions can also form through horizontal advection of water with different properties known as water-mass interleaving. For example, in the Tropics

where there may be velocity shear between opposing currents, inversions are typically characterized as small abrupt features (often only meters thick) found at the base of a well-mixed upper layer and at the top of the halocline and pycnocline (Figure 4(c)). Just west of San

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Francisco (130–1401 W), the low temperature and salinity properties of the Subantarctic Water Mass found in the California Current transition toward the higher-salinity water masses formed in the evaporative regime of the mid-subtropical gyre. The interleaving of the various water masses results in inversions that are quite different in structure from those observed in the tropical Pacific or the subpolar regions (Figure 4(d)). The surface layer vertical structure may be further complicated by frequent energetic eddies and meanders that perturb the flow and have their own distinctive water properties. In the transition zone, the inversions can be thick, and occur well within the pycnocline and not at the base of the mixed layer (Figure 4(d)). Typically there may be sharp gradients in temperature and salinity, both horizontally and vertically, that are characteristic of water-mass interleaving from the advective penetrations of the currents and eddies.

Other Properties That Define the Upper Ocean Vertical Structure Other water properties, such as dissolved oxygen and nutrients (e.g., phosphates, silica, and nitrates), can also vary in structure in the upper ocean surface layer. These properties are considered to be nonconservative, that is, their distribution in the water column may change as they are produced or consumed by marine organisms. Thus, although they are of great importance to the marine biology, their value in defining the physical structure of the upper ocean surface layer must be viewed with caution. In addition, until recently these properties were not routinely measured on hydrographic cruises. Nonetheless, the dissolved oxygen saturation of the upper ocean has been a particularly useful property for determining the depth of penetration of air–sea exchanges, and also for tracing water masses. For example, in the far North Pacific Ocean, it has been suggested that the degree of saturation of the dissolved oxygen concentration may be a better indicator than temperature or density for determining the surface-layer depth of convective events. During summer, the upper layer may be restratified in temperature and salinity through local warming or freshening at the surface, or through the horizontal advection of less dense waters. However, these surface processes typically do not erode the high-oxygen saturation signature of the deeper winter convection. Thus the deep high-oxygen saturation level provides a clear record of the depth of convective penetration from the air–sea

exchange of the previous winter, and a unique signal for defining the true depth of the surface layer.

Conclusions In its simplest form the vertical structure of the upper surface layer can be characterized as having a nearsurface well-mixed layer, below which there may exist a seasonal thermocline, where temperature changes relatively rapidly, connected to the permanent thermocline or main pycnocline. The vertical structure is primarily defined by stratification in the water properties of temperature, salinity, and density, although in some regions oxygen saturation and nutrient distribution can play an important biochemical role. The vertical structure of the surface layer can be complex and variable. There exist distinct variations in the forms and thickness of the upper-layer structure both in time and in space, through transient variations in the air–sea forcing from winds, heat, and fresh water that cause the turbulent mixing of the upper ocean. Understanding the variation in the upper ocean vertical structure is crucial for understanding the coupled air–sea climate system, and the storage of the heat and fresh water that is ultimately redistributed throughout the world oceans by the general circulation.

See also Air–Sea Gas Exchange. Deep Convection. Heat and Momentum Fluxes at the Sea Surface. Open Ocean Convection. Penetrating Shortwave Radiation. Upper Ocean Heat and Freshwater Budgets. Upper Ocean Mean Horizontal Structure. Upper Ocean Mixing Processes. Upper Ocean Space and Time Variability. Water Types and Water Masses. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading Cronin MF and McPhaden MJ (2002) Barrier layer formation during westerly wind bursts. Journal of Geophysical Research 107 (doi:10.1029/2001JC00 1171). Kraus EB and Businger JA (1994) Oxford Monographs on Geology and Geophysics: Atmosphere–Ocean Interaction, 2nd edn. New York: Oxford University Press. Philips OM (1977) The Dynamics of the Upper Ocean, 2nd edn. London: Cambridge University Press. Reid JL (1982) On the use of dissolved oxygen concentration as an indicator of winter convection. Naval Research Reviews 3: 28--39.

WIND- AND BUOYANCY-FORCED UPPER OCEAN M. F. Cronin, NOAA Pacific Marine Environmental Laboratory, Seattle, WA, USA J. Sprintall, University of California San Diego, La Jolla, CA, USA Published by Elsevier Ltd.

Introduction Forcing from winds, heating and cooling, and rainfall and evaporation has a profound influence on the distribution of mass and momentum in the ocean. Although the effects from this wind and buoyancy forcing are ultimately felt throughout the entire ocean, the most immediate impact is on the surface mixed layer, the site of the active air–sea exchanges. The mixed layer is warmed by sunshine and cooled by radiation emitted from the surface and by latent heat loss due to evaporation (Figure 1). The mixed layer also tends to be cooled by sensible heat loss since the surface air temperature is generally cooler than the ocean surface. Evaporation and precipitation change the mixed layer salinity. These salinity and temperature changes define the ocean’s surface buoyancy. As the surface loses buoyancy, the surface water can become denser than water below it, causing convective overturning and mixing to occur. Wind forcing can also cause near-surface overturning and mixing, as well as localized overturning at the base of the mixed layer through shear-flow instability. This wind- and buoyancy-generated turbulence causes the surface water to be well mixed and vertically uniform in temperature, salinity, and density. Furthermore, the turbulence can entrain deeper water into the surface mixed layer, causing the surface temperature and salinity to change and the layer of well-mixed, vertically uniform water to thicken. Wind forcing also sets up oceanic currents and can cause changes in the mixed layer temperature and salinity through horizontal and vertical advection. Although the ocean is forced by the atmosphere, the atmosphere can also respond to ocean surface conditions, particularly sea surface temperature (SST). Direct thermal circulation, in which moist air rises over warm SSTs and descends over cool SSTs, is prevalent in the Tropics. The resulting atmospheric circulation cells influence the patterns of cloud, rain, and winds that combine to form the wind and buoyancy forcing for the ocean. Thus, the oceans and

atmosphere form a coupled system, where it is sometimes difficult to distinguish forcing from response. Because water has a density and effective heat capacity nearly 3 orders of magnitude greater than air, the ocean has mechanical and thermal inertia relative to the atmosphere. The ocean thus acts as a memory for the coupled ocean–atmosphere system. We begin with a discussion of air–sea interaction through surface heat fluxes, moisture fluxes, and wind forcing. The primary external force driving the ocean–atmosphere system is radiative warming from the Sun. Because of the fundamental importance of solar radiation, the surface wind and buoyancy forcing is illustrated here with two examples of the seasonal cycle. The first case describes the seasonal cycle in the North Pacific, and can be considered a classic example of a one-dimensional (involving only vertical processes) ocean response to wind and buoyancy forcing. In the second example, the seasonal cycle of the eastern tropical Pacific, the atmosphere and the ocean are coupled, so that wind and buoyancy forcing lead to a sequence of events that make cause and effect difficult to determine. The impact of wind and buoyancy forcing on the surface mixed layer and the deeper ocean is summarized in the conclusion.

Air–Sea Interaction Surface Heat Flux

As shown in Figure 1, the net surface heat flux entering the ocean (Q0) includes solar (shortwave) radiation (Qsw), net infrared (long-wave) radiation (Qlw), latent heat flux due to evaporation (Qlat), and sensible heat flux due to air and water having different surface temperatures (Qsen): Q0 ¼ Qsw þ Qlw þ Qlat þ Qsen

½1

The Earth’s seasons are largely defined by the annual cycle in the net surface heat flux associated with the astronomical orientation of the Earth relative to the Sun. The Earth’s tilt causes solar radiation to strike the winter hemisphere more obliquely than the summer hemisphere. As the Earth orbits the Sun, winter shifts to summer and summer shifts to winter, with the Sun directly overhead at the equator twice per year, in March and again in September. Thus, one might expect the seasonal cycle in the Tropics to be semiannual, rather than annual.

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Figure 1 Schematic drawing of wind- and buoyancy-forced upper ocean processes. Courtesy Jayne Doucette, Woods Hole Oceanographic Institution.

WIND- AND BUOYANCY-FORCED UPPER OCEAN

Visible radiation

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However, as discussed later, in some parts of the equatorial oceans, the annual cycle dominates due to coupled ocean–atmosphere–land interactions. Solar radiation entering the Earth’s atmosphere is absorbed, scattered, and reflected by water in both its liquid and vapor forms. Consequently, the amount of solar radiation which crosses the ocean surface, Qsw, also depends on the cloud structures. The amount of solar radiation absorbed by the ocean mixed layer depends on the transmission properties of light in water and can be estimated as the difference between the solar radiation entering the surface and the solar radiation penetrating through the base of the mixed layer. The Earth’s surface also radiates energy at longer wavelengths similar to a black body (i.e., proportional to the fourth power of the surface temperature in units kelvin). Infrared radiation emitted by the atmosphere and clouds can reflect against the ocean surface and become upwelling infrared radiation. Thus net long-wave radiation, Qlw, is the combination of the outgoing and incoming infrared radiation and tends to cool the ocean. The ocean and atmosphere also exchange heat via conduction (‘sensible’ heat flux). When the ocean and atmosphere have different surface temperatures, sensible heat flux will act to reduce this temperature difference. Thus when the ocean is warmer than the air (which is nearly always the case), sensible heat flux will tend to cool the ocean and warm the atmosphere. Likewise, the vapor pressure at the air–sea interface is saturated with water while the air just above the interface typically has relative humidity less than 100%. Thus, moisture tends to evaporate from the ocean and in doing so, the ocean loses heat at a rate of Qlat ¼ Lðrfw EÞ

½2

where Qlat is the latent heat flux, L is the latent heat of evaporation, rfw is the freshwater density, and E is the rate of evaporation. Qlat has units W m–2, and (rfwE) has units kg s  1 m–2. The latent heat flux is nearly always larger than the sensible heat flux due to conduction. When the evaporated moisture condenses in the atmosphere to form clouds, heat is released, affecting the large-scale wind patterns. Air–sea heat and moisture transfer occur through turbulent processes and is amplified by sea spray, bubble production, and wave breaking. Sensible and latent heat loss thus also depend on the speed of the surface wind relative to the ocean surface flow, |ua – us|. Using similarity arguments, the latent (Qlat) and sensible (Qsen) heat fluxes can be expressed in

239

terms of ‘bulk’ properties at and near the ocean surface: Qlat ¼ ra LCE jua  us jðqa  qs Þ

½3

Qsen ¼ ra cpa CH jua  us jðTa  Ts Þ

½4

where ra is the air density, cpa is the specific heat of air, CE and CH are the transfer coefficients of latent and sensible heat flux, qs is the saturated specific humidity at Ts, the SST, and qa and Ts are, respectively, the specific humidity and temperature of the air at a few meters above the air–sea interface. The sign convention used here is that a negative flux tends to cool the ocean surface. The transfer coefficients, CE and CH, depend upon the wind speed and stability properties of the atmospheric boundary layer, making estimations of the heat fluxes quite difficult. Most algorithms estimate the turbulent heat fluxes iteratively, using first estimates of the heat fluxes to compute the transfer coefficients. Further, the dependence of heat flux on wind speed and SST causes the system to be coupled since the heat fluxes can change the wind speed and SST. Figure 2 shows the climatological net surface heat flux, Q0, and SST for the entire globe. Several patterns are evident. (Note that the spatial structure of the climatological latent heat flux can be inferred from the climatological evaporation shown in Figure 3(a).) In general, the Tropics are heated more than the poles, causing warmer SST in the tropics and cooler SST at the poles. Also, there are significant zonal asymmetries in both the net surface heat flux and SST. The largest ocean surface heat losses occur over the mid-latitude western boundary currents. In these regions, latent and sensible heat loss are enhanced due to the strong winds which are cool and dry as they blow off the continent and over the warm water carried poleward by the western boundary currents. In contrast, the ocean’s latent and sensible heat loss are reduced in the eastern boundary region where marine winds blow over the cool water. Consequently, the eastern boundary is a region where the ocean gains heat from the atmosphere. These spatial patterns exemplify the rich variability in the ocean–atmosphere climate system that occurs on a variety of spatial and temporal scales. In particular, seasonal conditions can often be quite different from mean climatology. The seasonal warming and cooling in the north Pacific and eastern equatorial Pacific are discussed later. Thermal and Haline Buoyancy Fluxes

Since the density of seawater depends on temperature and salinity, air–sea heat and moisture fluxes can

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change the surface density, making the water column more or less buoyant. Specifically, the net surface heat flux (Q0), rate of evaporation (E), and precipitation (P) can be expressed as a buoyancy flux (B0): B0 ¼ gaQ0 =ðrcp Þ þ gbðE  PÞS0

½5

where g is gravity, r is ocean density, cp is specific heat of water, S0 is surface salinity, a is the effective thermal expansion coefficient ðr1 @r=@TÞ, and b is the effective haline contraction coefficient ðr1 @r=@SÞ. Q0 has units W m–2 and E and P have units m s–1. B0 has units m2 s–3 and can be interpreted (when multiplied by density and integrated over a volume) as the buoyant production of turbulent kinetic energy (or destruction of available potential energy). A negative (i.e., downward) buoyancy flux, due to either surface warming or precipitation, tends to make the ocean surface more buoyant and stable. Conversely, a positive buoyancy flux, due to either

surface cooling or evaporation, tends to make the ocean surface less buoyant. As the water column loses buoyancy, it can become convectively unstable with heavy water lying over lighter water. Turbulent kinetic energy, generated by the ensuing convective overturning, can then cause deeper, generally cooler water to be entrained and mixed into the surface mixed layer (Figure 1). Thus entrainment mixing typically causes the SST to cool and the mixed layer to deepen. As discussed in the next section, entrainment mixing can also be generated by wind forcing, through wind stirring and shear at the base of the mixed layer. Figure 3 shows the climatological evaporation and precipitation fields. Note that in terms of buoyancy, a 20 W m–2 heat flux is approximately equivalent to a 5 mm day–1 rain rate. Thus, in some regions of the world oceans, the freshwater flux term in eqn [5] dominates the buoyancy flux, and hence is a major factor in the mixed layer thermodynamics. For

WIND- AND BUOYANCY-FORCED UPPER OCEAN

example, in the tropical regions, heavy precipitation can result in a surface-trapped freshwater pool that forms a shallower mixed layer within a deeper, nearly isothermal layer. The difference between the shallower mixed layer of uniform density and the deeper isothermal layer is referred to as a salinitystratified barrier layer. As the name suggests, a barrier layer can effectively limit turbulent mixing of heat between the ocean surface and the deeper thermocline since the barrier layer water has nearly the same temperature as the mixed layer. In subpolar latitudes, freshwater fluxes can also dominate the surface layer buoyancy profile. During the winter season, atmospheric cooling of the ocean, and stronger wind mixing leaves the water-column isothermal to great depths. Then, wintertime ice formation extracts fresh water from the surface layer, leaving a saltier brine that further increases the surface density, decreases the buoyancy, and enhances the deep convection. This process can lead to deepwater formation as the cold and salty dense water sinks and spreads horizontally, forcing the deep, slow thermohaline circulation. Conversely, in summer when the ice shelf and icebergs melt, fresh water is released, and the density in the surface layer is reduced so that the resultant stable halocline (pycnocline) inhibits the sinking of water. Wind Forcing

The influence of the winds on the ocean circulation and mass field cannot be overstated. Wind blowing over the ocean surface causes a tangential stress (‘wind stress’) at the interface which acts as a vertical flux of horizontal momentum. Similar to the air–sea fluxes of heat and moisture, this air–sea flux of horizontal momentum, s0, can be expressed in terms of bulk properties as s0 ¼ ra CD jua  us jðua  us Þ

½6

where ra is the air density, and CD is the drag coefficient. The direction of the stress is determined by the orientation of the surface wind, ua, relative to the ocean surface flow, us. The units of the surface wind stress are N m–2. Wind stress can also be expressed in terms of an oceanic frictional velocity, u (i.e., s0 ¼ ru 2 ). With frictional velocity related to the wind-generated velocity shear through the nondimensional ‘von Ka´rma´n constant’, k, the shear production of turbulent kinetic energy by the wind can be expressed as: rðkzÞ1 u 3 . The mechanisms by which the momentum flux extends below the interface are not well understood. Some of the wind stress goes into generating ocean

241

surface waves. However, most of the wave momentum later becomes available for generating currents through wave breaking, and wave–wave and wave–current interactions. For example, wave– current interactions associated with Langmuir circulation can set up large coherent vortices that carry momentum to near the base of the mixed layer. As with convective overturning, wind stirring can entrain cooler thermocline water into the mixed layer, producing a colder and deeper mixed layer. Likewise, current shear at the base of the mixed layer can cause ‘Kelvin–Helmholtz’ shear instability that further mix properties within and at the base of the mixed layer. Variability in the depth of the well-mixed layer can be understood through consideration of the turbulent kinetic energy (TKE) budget. For example, for a stable buoyancy flux (i.e., ‘forced convection’), the depth, LMO, at which there is just sufficient mechanical energy available from the wind to mix the input of buoyancy uniformly is referred to as the Monin–Obukhov depth scale: LMO ¼ u 3 =ðkB0 Þ

½7

At depths below LMO, buoyant suppression of turbulence exceeds the mechanical production and there tends to be little surface-generated turbulence. Typically, however, other terms in the TKE budget cannot be ignored. In particular, for an unstable buoyancy flux (i.e., ‘free convection’), the production of potential energy through entrainment becomes important. Thus the mixed layer depth is rarely equivalent to the Monin–Obukhov depth scale. Over timescales at and longer than roughly a day, the Earth’s spinning tends to cause a rotation of the vertical flux of momentum. From the noninertial perspective of an observer on the rotating Earth, the tendency to rotate appears as a force, referred to as the Coriolis force. When the wind ceases, inertial motion tends to continue and accounts for a significant fraction of the total kinetic energy in the global ocean. Vertical shear in the currents and inertial oscillations generated by the winds can cause ‘Kelvin– Helmholtz’ shear instability and be a significant source of TKE. For sustained winds beyond the inertial timescale, Coriolis turning causes the wind-forced surface layer transport (‘Ekman transport’) to be perpendicular to the wind stress. Because the projection of the Earth’s axis onto the local vertical axis (direction in which gravity acts) changes sign at the equator, the Ekman transport is to the right of the wind stress in the Northern Hemisphere and to the left of the wind stress in the Southern Hemisphere. Convergence and divergence of this Ekman transport leads to vertical

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motion which can deform the thermocline and thereby generate pressure gradients that set the subsurface waters in motion. In this way, meridional variations in the prevailing zonal wind stress drive the steady, large-scale ocean gyres. The influence of Ekman upwelling on SST can be seen along the eastern boundary of the ocean basins and along the equator (Figure 2(b)). Equatorward winds along the eastern boundaries of the Pacific and Atlantic Oceans cause an offshore-directed Ekman transport. Mass conservation requires that this water be replaced with upwelled water, water that is generally cooler than the surface waters outside the upwelling zone. Likewise, in the tropics, prevailing easterly trade winds cause poleward Ekman transport. At the equator, this poleward flow results in substantial surface divergence and upwelling. As with the eastern boundary, equatorial upwelling results in relatively cold SSTs (Figure 2(b)). Because of the geometry of the continents, the thermal equator favors the Northern Hemisphere and is generally found several degrees of latitude north of the equator. In the tropics, winds tend to flow from cool SSTs to warm SSTs, where deep atmospheric convection can occur. Thus, surface wind convergence in the Intertropical Convergence Zone (ITCZ) is associated with the thermal equator, north of the equator. The relationship between the SST gradient and winds accounts for an important coupling mechanism in the Tropics.

The Seasonal Cycle The North Pacific: A One-Dimensional Ocean Response to Wind and Buoyancy Forcing

From 1949 through 1981, a ship (Ocean Weather Station Papa) was stationed in the North Pacific at 501 N 1451 W with the primary mission of taking routine ocean and atmosphere measurements. The seasonal climatology observed at this site (Figure 4) illustrates a classic near-one-dimensional ocean response to wind and buoyancy forcing. A one-dimensional response implies that only the vertical structure of the ocean is changed by the forcing. During springtime, layers of warmer and lighter water are formed in the upper surface in response to the increasing solar warming. By summer, this heating has built a stable (buoyant), shallow seasonal thermocline that traps the warm surface waters. In fall, storms are more frequent and net cooling sets in. By winter, the surface layer is mixed by wind stirring and convective overturning. The summer thermocline is eroded and the mixed layer deepens to the top of the permanent thermocline.

To first approximation, horizontal advection does not seem to be important in the seasonal heat budget. The progression appears to be consistent with a surface heat budget described by @T=@t ¼ Q0 =ðrcp HÞ

½8

where qT/qt is the local time rate of change of the mixed layer temperature, and H is the mixed layer depth. Since only vertical processes (e.g., turbulent mixing and surface forcing) affect the depth and temperature of the mixed layer, the heat budget can be considered one-dimensional. A similar one-dimensional progression occurs in response to the diurnal cycle of buoyancy forcing associated with daytime heating and nighttime cooling. Mixed layer depths can vary from just a few meters thick during daytime to several tens of meters thick during nighttime. Daytime and nighttime SSTs can sometimes differ by 41 1C. However, not all regions of the ocean have such an idealized mixed layer seasonal cycle. Our second example shows a more complicated seasonal cycle in which the tropical atmosphere and ocean are coupled.

The Eastern Equatorial Pacific: Coupled Ocean–Atmosphere Variability

Because there is no Coriolis turning at the equator, water and air flow are particularly susceptible to horizontal convergence and divergence. Small changes in the wind patterns can cause large variations in oceanic upwelling, resulting in significant changes in SST and consequently in the atmospheric heating patterns. This ocean and atmosphere coupling thus causes initial changes to the system that perpetuate further changes. At the equator, the Sun is overhead twice per year: in March and again in September. Therefore one might expect a semiannual cycle in the mixed layer properties. Although this is indeed found in some parts of the equatorial oceans (e.g., in the western equatorial Pacific), in the eastern equatorial Pacific the annual cycle dominates. During the warm season (February–April), the solar equinox causes a maximum in insolation, equatorial SST is warm, and the meridional SST gradient is weak. Consequently, the ITCZ is near the equator, and often a double ITCZ is observed that is symmetric about the equator. The weak winds associated with the ITCZ cause a reduction in latent heat loss, wind stirring, and upwelling, all of which lead to further warming of the equatorial SSTs. Thus the warm SST and surface heating are mutually reinforcing.

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Figure 4 Seasonal climatologies at the ocean weather station Papa in the north Pacific: (a) Wind speed, (b) net surface heat flux, and (c) upper ocean temperature. The bold line represents the base of the ocean mixed layer defined as the depth where the temperature is 0.5 1C cooler than the surface temperature. Wind speed and net surface heat flux climatologies are from da Silva et al. (1994).

Beginning in about April–May, SSTs begin to cool in the far eastern equatorial Pacific, perhaps in response to southerly winds associated with the continental monsoon. The cooler SSTs on the equator cause an increased meridional SST gradient that intensifies the southerly winds and the SST cooling in the far eastern Pacific. As the meridional SST gradient increases, the ITCZ begins to migrate northward. Likewise, the cool SST anomaly in the far east sets up a zonal SST gradient along the equator that intensifies the zonal trade winds to the west of the cool anomaly. These enhanced trade winds then produce SST cooling (through increased upwelling, wind stirring, and latent heat loss) that spreads westward (Figure 5).

By September, the equatorial cold tongue is fully formed. Stratus clouds, which tend to form over the very cool SSTs in the tropical Pacific, cause a reduction in solar radiation, despite the equinoctial increase. The large meridional gradient in SST associated with the fully formed cold tongue causes the ITCZ to be at its northernmost latitude. After the cold tongue is fully formed, the reduced zonal SST gradient within the cold tongue causes the trade winds to weaken there, leading to reduced SST cooling along the equator. Finally, by February, the increased solar radiation associated with the approaching vernal equinox causes the equatorial SSTs to warm and the cold tongue to disappear, bringing the coupled system back to the warm season conditions.

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Conclusion Because the ocean mixed layer responds so rapidly to surface-generated turbulence through wind- and buoyancy-forced processes, the surface mixed layer can often be modeled successfully using onedimensional (vertical processes only) physics. Surface heating and cooling cause the ocean surface to warm and cool; evaporation and precipitation cause the ocean surface to become saltier and fresher. Stabilizing buoyancy forcing, whether from net surface heating or precipitation, stratifies the surface and isolates it from the deeper waters, whereas wind stirring and destabilizing buoyancy forcing generate surface turbulence that cause the surface properties to mix with deeper water. Eventually, however, one-dimensional models drift away from observations, particularly in regions with strong ocean– atmosphere coupling and oceanic current structures. The effects of horizontal advection are explicitly not included in one-dimensional models. Likewise, vertical advection depends on horizontal convergences and divergences and therefore is not truly a

one-dimensional process. Finally, wind and buoyancy forcing can themselves depend on the horizontal SST patterns, blurring the distinction between forcing and response. Although the mixed layer is the principal region of wind and buoyancy forcing, ultimately the effects are felt throughout the world’s oceans. Both the winddriven motion below the mixed layer and the thermohaline motion in the relatively more quiescent deeper ocean originate through forcing in the surface layer that causes an adjustment in the mass field (i.e., density profile). In addition, buoyancy and wind forcing in the upper ocean define the property characteristics for all the individual major water masses found in the world oceans. On a global scale, there is surprisingly little mixing between water masses once they acquire the characteristic properties at their formation region and are vertically subducted or convected from the active surface layer. As these subducted water masses circulate through the global oceans and later outcrop, they can contain the memory of their origins at the surface through their water mass properties and thus can potentially

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induce decadal and centennial modes of variability in the ocean–atmosphere climate system.

Nomenclature B0 CD CE CH cp E g H L LMO P qa qs Q0 Qlat Qlw Qsen Qsw S0 Ta Ts ua us u* a b k r ra rfw s0

surface buoyancy flux drag coefficient latent heat flux transfer coefficient sensible heat flux transfer coefficient specific heat capacity of water rate of evaporation gravity mixed layer depth latent heat of evaporation Monin–Obkuhov depth scale precipitation specific humidity of the air saturated specific humidity at the sea surface temperature net surface heat flux entering ocean latent heat flux due to evaporation net infrared (long-wave) radiation sensible heat flux net solar (shortwave) radiation surface salinity air temperature surface ocean temperature air velocity surface ocean velocity oceanic frictional velocity thermal expansion coefficient haline contraction coefficient von Ka´rma´n constant ocean density air density density of fresh water wind stress

See also Breaking Waves and Near-Surface Turbulence. Langmuir Circulation and Instability. Penetrating

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Shortwave Radiation. Sea Surface Exchanges of Momentum, Heat, and Fresh Water Determined by Satellite Remote Sensing. Upper Ocean Heat and Freshwater Budgets. Upper Ocean Space and Time Variability. Upper Ocean Vertical Structure. Water Types and Water Masses.

Further Reading da Silva AM, Young CC, and Levitus S (1994) Atlas of Surface Marine Data 1994, Vol. 1: Algorithms and Procedures, NOAA Atlas NESDIS 6. Washington, DC: US Department of Commerce. Fairall CF, Bradley EF, Hare JE, Grachev AA, and Edson JB (2003) Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. Journal of Climate 16: 571--591. Kraus EB and Businger JA (1994) Oxford Monographs on Geology and Geophysics: Atmosphere–Ocean Interaction, 2nd edn. New York: Oxford University Press. Large WG (1996) An observational and numerical investigation of the climatological heat and salt balances at OWS Papa. Journal of Climate 9: 1856--1876. Niiler PP and Kraus EB (1977) One-dimensional models of the upper ocean. In: Kraus EB (ed.) Modelling and Prediction of the Upper Layers of the Ocean, pp. 143--172. New York: Pergamon. Philander SG (1990) El Nin˜o, La Nin˜a, and the Southern Oscillation. San Diego, CA: Academic Press. Price JF, Weller RA, and Pinkel R (1986) Diurnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing. Journal of Geophysical Research 91: 8411--8427. Reynolds RW and Smith TM (1994) Improved global sea surface temperature analysis using optimum interpolation. Journal of Climate 7: 929--948.

UPPER OCEAN SPACE AND TIME VARIABILITY D. L. Rudnick, University of California, San Diego, USA Copyright & 2001 Elsevier Ltd.

Introduction The upper ocean is the region of the ocean in direct contact with the atmosphere. Air–sea fluxes of momentum, heat, and fresh water are the primary external forces acting upon the upper ocean (see Heat and Momentum Fluxes at the Sea Surface; Evaporation and Humidity; Wind- and Buoyancy-Forced Upper Ocean). These fluxes impose the temporal and spatial scales of the overlying atmosphere. The internal dynamics of the ocean cause variability at scales distinct from the forcing. This combination of forcing and dynamics creates the tapestry of oceanic phenomena at timescales ranging from minutes to decades and length scales from centimeters to thousands of kilometers. This article is concerned primarily with the physical processes causing time and space variability in the upper ocean. The physical balances to be considered are the conservation of mass, heat, salt, and momentum. Thus, physical phenomena are discussed with special reference to their effects on the temporal and spatial variability of temperature, salinity, density, and velocity. While many other biological, chemical, and optical properties of the ocean are affected by the phenomena outlined below, their discussion is covered by other articles in this volume. The most striking feature often seen in vertical profiles of the upper ocean is the surface mixed layer, a layer that is vertically uniform in temperature, salinity, and horizontal velocity (see Upper Ocean Vertical Structure and Upper Ocean Mean Horizontal Structure). The turbulence that mixes this layer derives its energy from wind and surface cooling. The region immediately below the mixed layer tends to be stratified, and is often called the seasonal thermocline because its stratification varies with the seasons. The seasonal thermocline extends down a few hundred meters to roughly 1000 m. Beneath the seasonal thermocline is the permanent thermocline whose stratification is constant on timescales of at least decades. Here the discussion is concerned with variability of the mixed layer and seasonal thermocline.

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The processes discussed below are ordered roughly by increasing time and space scales (Figure 1). Most of the processes are covered in greater detail elsewhere in this volume. It is hoped that this section will provide a convenient introduction to the variability of the upper ocean, and that the reader can proceed to the more in-depth articles as needed.

Turbulence and Mixing The upper ocean is distinguished from the interior of the ocean partly because of the very high levels of turbulence present (see Breaking Waves and NearSurface Turbulence and Upper Ocean Mixing Processes). The smallest scale of motion worthy of note in the ocean is the Kolmogoroff scale, on the order of 1 cm, where energy is dissipated by molecular viscosity. At this scale, the ocean can be considered isotropic; that is, properties vary in the same way regardless of the direction in which they are measured. At much larger scales than the Kolmogoroff scale, the vertical stratification of the ocean becomes important. In the seasonal thermocline, a dominant mechanism for mixing is the Kelvin-Helmholtz instability, in which a vertical shear of horizontal velocity causes the overturn of stratified water (see Internal Waves). The resulting ‘billows’ are observed tobe on the order of 1 m thick and to decay on the order of an hour. A great deal of observational and theoretical work in the last 20 years has been devoted to relating the strength of this mixing to larger (in the order of 10 m) and more easily measurable quantities such as shear and stratification. The resulting Henyey-Gregg parameterization is one of the most fundamentally important achievements of modern oceanography.

Langmuir Circulation and Convection Turbulence in the mixed layer is fundamentally different from that in the seasonal thermocline. Because the mixed layer is nearly unstratified, the largest eddies can be as large as the layer is thick, often about 100 m. These large eddies have come to be called Langmuir cells in honor of Irving Langmuir, the Nobel laureate in chemistry who first described them. Langmuir cells are elongated vortices whose axes are horizontal and oriented nearly parallel to the wind. The cells have radii comparable in size to the mixed layer depth, and can be as long as 1–2 km. Langmuir cells often appear in pairs with opposite

UPPER OCEAN SPACE AND TIME VARIABILITY

247

Seasonal cycle

106

Atmospheric storms

~

Tides

107 El Nino climate

105

Horizontal length scale (m)

104 Fronts and eddies Internal waves

103

Solar heating 102

Langmuir circulation convection

101

100

10

_1

Turbulent mixing day

10

month

year

_2 101

102 minute

103

104 hour

105 Timescale (s)

106

107

108

109

Figure 1 A schematic diagram of the distribution in time and space of upper ocean variability. The temporal and spatial limits of the phenomena should be considered approximate.

senses of rotation. The cells thus create alternating regions of surface convergence and divergence. The regions of convergence collect material floating on the surface such as oil and seaweed. Langmuir first became aware of these cells after noticing lines of floating seaweed during a crossing of the Atlantic. Langmuir cells are forced by a combination of wind and surface waves, and are established typically within an hour after the wind starts blowing. Langmuir cells disappear quickly after the wind stops. Recent research indicates that Langmuir cells often vacillate in strength on the timescale of roughly 15 minutes. Convection cells forced by surface cooling also cause the mixed layer to be homogenized and to deepen (see Open Ocean Convection). A typical feature in the mixed layer is the daily cycle of stratification, with daytime heating causing nearsurface stratification and nighttime cooling causing convection that destroys this stratification and

deepens the mixed layer. The vertical extent of convection cells corresponds to the depth of the mixed layer (of order 100 m); the cells have an aspect ratio of one so their horizontal and vertical scales are equal. Because solar heating has a large, essentially global, scale the daily heating and cooling of the upper ocean is coherent and predictable over large scales. Horizontal velocity in the mixed layer also varies strongly at a 24 h period, as the daily cycle of stratification affects the depth to which the wind forces currents. The deepest mixed layers in the oceans, at high latitudes, are convectively mixed. Convection cells are thus more effective at deepening the mixed layer than are Langmuir cells.

Internal Waves Just as there are gravity waves on the surface of the ocean, there are gravity waves in the thermocline.

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These thermocline gravity waves, modified by the Earth’s rotation, are known as internal waves (see Internal Waves). They exist in a range of frequencies bounded at the lower end by the inertial frequency f and at the upper end by the buoyancy frequency N. A parcel of water given an initial velocity will travel in a circle under the influence of the Coriolis force. The inertial frequency f, twice the local vertical component of the Earth’s rotation vector, is the frequency of rotation around such a circle. The resulting horizontal current is known as an inertial oscillation. The inertial period is 12 h at the poles, 24 h at 301 latitude, and infinite at the equator because local vertical is normal to the Earth’s axis of rotation. The buoyancy frequency N, proportional to the square root of the vertical density gradient, is the frequency of oscillation of a water parcel given a displacement in the vertical. The resulting vertical motion has a frequency of less than one to several cycles per hour in typical ocean stratification. Internal waves oscillate in planes tilted from the horizontal as a function of the frequency between f and N. Internal waves have amplitudes on the order of tens of meters. They may be coherent over vertical scales that approach the depth of the ocean, particularly at high frequencies near N. Lower frequency internal waves, approaching f, have shorter vertical wavelengths often of order 100 m or less. The horizontal wavelength of an internal wave is related to its frequency and vertical wavelength through the internal wave dispersion relation. For a given vertical wavelength, a high frequency internal wave will have shorter horizontal wavelength than a low frequency wave. At the low frequency end of the internal wave spectrum, the near-inertial waves are especially important in the upper ocean. Near-inertial waves are quite ubiquitous because they are so readily excited by wind forcing on the ocean’s surface. In measurements of horizontal current, inertial oscillations are often the most obvious variability because horizontal currents ‘ring’ at the resonant inertial frequency. Just as a bell has a distinctive tone when struck, the ocean has inertial currents when hit, for example, by a storm. Strong inertial currents are one of the indications in the ocean of the recent passage of a hurricane. The radius of an inertial current circle is its speed divided by its rotation rate, U/f. If the current speed is 0.1 ms1, then for a midlatitude inertial frequency of 104 s1, the radius is 1 km. In the aftermath of a storm, the inertial currents and radii may be nearly an order of magnitude larger. Nearinertial waves are a dominant mechanism for transporting wind-driven momentum downward from the mixed layer to the seasonal thermocline and into the

interior. Because near-inertial motions have short vertical scales, they dominate the shear spectrum in the ocean. This shear eventually leads to enhanced turbulence and mixing the penetration of inertial shear into the ocean and the geography of shear and mixing are active topics of research. Tides are well known to anyone who has spent at least a day at the beach. The dominant tidal periods are near one day and one-half day. Tides are most obvious to the casual observer of the sea surface, and they are easily seen in records of horizontal current in the open ocean. Internal tides exist as well, for example forced by tidal flow over bumps on the ocean bottom (see Internal Waves). These internal tides, seen as variability in density and velocity at a location, are a form of internal wave and are governed by the same dynamics. Isolated pulses of tidal internal waves, known as ‘solitons,’ are prevalent in certain regions of rough bottom topography, and are a field of current research.

Fronts and Eddies While vertically uniform, the mixed layer can vary in the horizontal on a wide range of scales. We have already discussed Langmuir circulation and convection cells on scales of order 100 m, but there may be horizontal variability on longer scales. Just as there are fronts in the atmosphere, visible for example in the satellite pictures of clouds shown on the evening television news, there are fronts in the ocean. Fronts in the ocean separate regions of warm and cool water, or fresh and salty water. The most obvious fronts in the mixed layer have widths on the order of 10–100 km, and typically persist for weeks. Fronts of this size have currents directed along the front as a result of the geostrophic momentum balance. That is, the Coriolis force balances the pressure gradient due to having water of varying density across the front. The less dense (usually warmer) water is on the right side of the current in the Northern Hemisphere (the sense of the current is the opposite in the Southern Hemisphere). Fronts in the mixed layer are sites of enhanced vertical circulation on the order of tens of meters per day. Strong biological productivity at fronts is attributed to this vertical circulation which brings deeper water rich in nutrients to the surface. Fronts at scales shorter than 10 km also exist in the mixed layer. At these shorter scales, the geostrophic balance may not be expected to hold. Typical fronts at these scales are observed to be warm and salty on one side and cold and fresh on the other such that the density contrast across the front vanishes. Such a

UPPER OCEAN SPACE AND TIME VARIABILITY

front is often said to be compensated, since temperature and salinity gradients compensate in their effect on density. The presence of compensated fronts in the mixed layer is consistent with a horizontal mixing that is an increasing function of the horizontal density gradient. That is, small-scale horizontal density fronts do not persist as long as compensated fronts. Because of their small scale, fronts of order 1 km are poorly observed in the ocean, and are a topic of current research. Observed fronts are usually not observed to be perfectly straight, rather they wiggle. The wiggles, or perturbations, often grow to be large in comparison with the width of the front. When the perturbations grow large enough, the front may turn back on itself and a detached eddy is formed. The eddies often have sizes on the order of 10 km, when they are confined in depth to the mixed layer. This length scale is related to the Rossby radius of deformation; at scales larger than the Rossby radius flows tend to be geostrophic. The Rossby radius for the mixed layer is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gHDr=r f where g is acceleration due to gravity, H is the depth of the mixed layer, r is the density of the water, and Dr is the change in density across the mixed layer base. For a typical mixed layer, H is 100 m and Dr is 0.2 kg m3, g is 9.8 m s2, and r is 1025 kg m3, so the Rossby radius is about 6 km. Eddies that extend deeper have larger radii, as can be inferred from the formula for the Rossby radius. Large eddies can persist for as long as several months, while smaller eddies are shorter lived. The small-scale mixed layer eddies, a prominent feature in satellite photos of the sea surface, are typically observed to rotate in the counterclockwise direction in the Northern Hemisphere, and clockwise south of the equator. Again, because of their small size, they have been inadequately observed and are a topic of current research.

Wind-Forced Currents One of the oldest theories of ocean circulation is due to V.W. Ekman, who in 1905 suggested a balance between the Coriolis force and the stress due to wind blowing over the ocean surface. The prediction of this theory for a steady wind is a current that spirals to the right (in the Northern Hemisphere) and decays with depth. This spiral structure was not clearly observed in the ocean until the 1980s with the advent of moorings with modern current meters. Although

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the details of the stress parameterization used by Ekman were found to be inadequate to describe observations, the general picture of a spiral remains valid to this day. An alternative theoretical construct to explain upper ocean structure is the bulk mixed layer model. Oceanic properties, such as temperature, salinity, and velocity, are assumed to be vertically uniform in the mixed layer, with a region of very strong vertical gradients at the mixed layer base. The mixed layer is then forced by air–sea fluxes of heat, fresh water, and momentum at the surface, and by turbulent fluxes at the base. The bulk mixed layer model has proven remarkably successful at predicting some basic features of the upper ocean, particularly the vertical temperature structure. Interestingly, the disparate conceptual models of the Ekman spiral and the bulk mixed layer can be rationalized. The upper ocean velocity structure is often, but certainly not always, observed to be vertically uniform near the surface with a region of high shear beneath, in accordance with the bulk mixed layer model. On the other hand, long time averages of ocean current tend to have a spiral structure, in qualitative agreement with the Ekman spiral. This is so if the averages are long enough to span many cycles of mixed layer shoaling and deepening, as due to the daily cycle of surface heating. Thus the timeaverage current spiral may be very different from a typical snapshot of a nearly vertically uniform current. The averaged wind-driven spiral extends downward to a depth comparable to, but slightly deeper than, the mixed layer. The shape of the spiral is strongly influenced by higher frequency variability in the stratification, such as the daily cycle in mixed layer depth discussed above. A spiral is observed in response to temporally variable winds, as well as to steady winds. The temporally variable spiral may have a different vertical structure to the steady spiral. In particular, the current spirals to the left with depth in response to a wind that rotates more rapidly than f in a clockwise direction, in contrast to the steady spiral to the right. Regardless of the detailed velocity structure in the upper ocean, the net transport caused by a steady wind is 901 to the right of the wind in the Northern Hemisphere (and to the left in the Southern Hemisphere). This transport (the vertical integral of velocity) is called the Ekman transport. The Ekman transport is proportional to the wind stress and inversely proportional to the inertial frequency. Thus wind of a given strength will cause more transport near the equator than it would closer to the poles.

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The spectrum of wind over the midlatitude ocean peaks at periods of a few to several days. These periods correspond to the time required for a typical storm to pass. The wind-driven current and transport is thus prominent at these periods. Atmospheric storms have typical horizontal sizes of a few to several hundred kilometers, and the direct oceanic response to these storms has similar horizontal scales. The prominent large-scale features of the wind field such as the westerlies in midlatitudes and the trade winds in the tropics directly force currents in the upper ocean. These currents have large horizontal length scales that reflect the winds.

Seasonal Cycles Just as the seasons cause well-known changes in weather, the annual cycle is one of the most robust signals in the ocean. Summer brings greater heat flux from the atmosphere to the ocean, and warmer ocean temperatures. As the ocean warms up at the surface, stratification increases and the mixed layer becomes shallower. The heat flux reverses in many locations during the winter and the ocean cools at the surface. The resulting convection causes the mixed layer to deepen; at some high latitude locations the mixed layer can deepen to several hundred meters in the winter. Winter conditions in high and midlatitude mixed layers are very important to the general circulation of the oceans, as it is these waters that penetrate into the thermocline and set properties that persist for decades. Along with cooler temperatures, winter brings typically stormier weather and more wind and precipitation. Wind-driven currents often peak during the winter in midlatitudes, at the same time that salinity decreases in response to the increased precipitation. Seasonal cycles occur over the whole globe in an extremely coherent fashion, because they are driven primarily by the solar heat flux. However, the seasonal cycle can vary at different oceanic locations. For example, the seasonal cycle at the equator is smaller than that at midlatitudes because solar heat flux varies less over the year. The Arabian Sea has a pronounced semi-annual cycle. Cold northerly winds in winter cool the ocean and deepen the mixed layer as typical for midlatitudes. More unusual is a second period of relatively low ocean temperatures and deep mixed layers during the summer south-west monsoon. Wind-driven mixing causes the cooling during the south-west monsoon as cool water is mixed up to the surface. The Arabian Sea monsoon is the classic

example of a seasonal wind driven by land–sea temperature differences. Monsoons also exist overthe south-west USA and south-east Asia, among others. Additional local seasonal effects may be caused by river outflows and weather patterns influenced by orography.

Climatic Signals The ocean has significant variability at periods longer than 1 year. The most well known recurrent interannual climatic phenomenon is El Nin˜o (see El Nin˜o Southern Oscillation (Enso)). An El Nin˜o occurs when trade winds reverse at the equator causing upwelling to cease off the coast of South America. The most obvious consequence of an El Nin˜o is dramatically elevated ocean temperatures at the equator. These high temperatures progress poleward from the equator along the coast of the Americas, affecting water properties in large regions of the Pacific. El Nin˜o has been hypothesized to start with anomalous winds in the western equatorial Pacific, eventually having an effect on the global ocean and atmosphere. El Nin˜os occur sporadically every roughly 3–7 years, and are becoming more predictable as observations and models of the phenomenon improve. The reverse phase of El Nin˜o, the so-called La Nina, is remarkable for exceptionally low equatorial temperatures and strong trade winds. Oscillations with periods of a decade and longer also exist in the ocean and atmosphere. Such oscillations are apparent in the ocean as basin-scale variations in sea surface temperature, for example. Salinity and velocity are also likely variable on decadal timescales, although the observational database for these is sparse in comparison with that for temperature. Atmospheric decadal oscillations in temperature and precipitation are well established. Scientists are actively researching whether and how the ocean and atmosphere are coupled on decadal timescales. The basic idea is that the ocean absorbs heat from the atmosphere and stores it for many years because of the ocean’s relatively high heat capacity. This heat may penetrate into the ocean interior and be redistributed by advective processes. The heat may resurface a decade or more later to affect the atmosphere through anomalous heat flux. The coupled ocean–atmosphere process just described is controversial, and the observations to support its existence are inadequate. A major challenge for the immediate future is to obtain the measurements needed to resolve such processes of significance to climate.

UPPER OCEAN SPACE AND TIME VARIABILITY

Conclusion The upper ocean varies on a wide range of temporal and spatial scales. Processes range from mixing occurring on scales of centimeters and minutes to decadal climatic oscillations of entire ocean basins. Fundamental to the ocean is the fact that these processes can rarely be studied in isolation. That is, processes occurring on one scale affect processes on other scales. For example, decadal changes in ocean stratification are strongly affected by turbulent mixing at the smallest scales. Turbulent mixing is modulated by the internal wave field, and internal waves are focused and steered by geostrophic fronts and eddies. The interaction among processes of different scales is likely to receive increasing attention from ocean scientists in the coming years.

See also Breaking Waves and Near-Surface Turbulence. Double-Diffusive Convection. El Nin˜o Southern Oscillation (Enso). Evaporation and Humidity. Heat and Momentum Fluxes at the Sea Surface. Internal Waves. Open Ocean Convection. Upper Ocean Mean Horizontal Structure. Upper Ocean Mixing Processes. Upper Ocean Vertical Structure. Windand Buoyancy-Forced Upper Ocean.

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Further Reading Davis RE, de Szoeke R, Halpern D, and Niiler P (1981) Variability in the upper ocean during MILE. Part I: The heat and momentum balances. Deep-Sea Research 28: 1427--1452. Ekman VW (1905) On the influence of the earth’s rotation on ocean currents. Arkiv Matematik, Astronomi och Fysik 2: 1--52. Eriksen CC, Weller RA, Rudnick DL, Pollard RT, and Regier LA (1991) Ocean frontal variability in the Frontal Air–Sea Interaction Experiment. Journal of Geophysical Research 96: 8569--8591. Gill AE (1982) Atmosphere–Ocean Dynamics. New York: Academic Press. Gregg MC (1989) Scaling turbulent dissipation in the thermocline. Journal of Geophysical Research 94: 9686--9698. Langmuir I (1938) Surface motion of water induced by wind. Science 87: 119--123. Lighthill MJ and Pearce RP (eds.) (1981) Monsoon Dynamics. Cambridge: Cambridge University Press. Munk W (1981) Internal waves and small-scale processes. In: Warren BA and Wunsch C (eds.) Evolution of Physical Oceanography, pp. 264--291. Cambridge, USA: MIT Press. Philander SG (1990) El Nin˜o, La Nin˜a, and the Southern Oscillation. San Diego: Academic Press. Roden GI (1984) Mesoscale oceanic fronts of the North Pacific. Annals of Geophysics 2: 399--410.

UPPER OCEAN MEAN HORIZONTAL STRUCTURE M. Tomczak, Flinders University of South Australia, Adelaide, SA, Australia

following an introductory overview of some elementary property fields.

Copyright & 2001 Elsevier Ltd.

Horizontal Property Fields Introduction The upper ocean is the most variable, most accessible, and dynamically most active part of the marine environment. Its structure is of interest to many science disciplines. Historically, most studies of the upper ocean focused on its impact on shipping, fisheries, and recreation, involving physical and biological oceanographers and marine chemists. Increased recognition of the ocean’s role in climate variability and climate change has led to a growing interest in the upper ocean from meteorologists and climatologists. In the context of this article the upper ocean is defined as the ocean region from the surface to a depth of 1 km and excludes the shelf regions. Although the upper ocean is small in volume whencompared to the world ocean as a whole, it is of fundamental importance for life processes in the sea. It determines the framework for marine life through processes that operate on space scales from millimeters to hundreds of kilometers and on timescales from seconds to seasons. On larger space and timescales, its circulation and water mass renewal processes span typically a few thousand kilometers and several decades, which means that the upper ocean plays an important role in decadal variability of the climate system. (In comparison, circulation and water mass renewal timescales in the deeper ocean are of the order of centuries, and the water masses below the upper ocean are elements of climate change rather than climate variability.) The upper ocean can be subdivided into two regions. The upper region is controlled by air–sea interaction processes on timescales of less than a few months. It contains the oceanic mixed layer, the seasonal thermocline and, where it exists, the barrier layer. The lower region, known as the permanent thermocline, represents the transition from the upper ocean to the deeper oceanic layers. It extends to about 1 km depth in the subtropics, is some what shallower near the equator and absent poleward of the Subtropical Front. These elements of the upper ocean will be defined and described in more detail,

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The annual mean sea surface temperature(SST) is determined by the heat exchange between ocean and atmosphere. If local solar heat input would be the only determinant, contours of constant SST would extend zonally around the globe, with highest values at the equator and lowest values at the poles. The actual SST field (Figure 1) comes close to thissimple distribution. Notable departures occur for two reasons. 1. Strong meridional currents transport warm water poleward in the western boundary currents along the east coasts of continents. Examples are theGulf Stream in the North Atlantic Ocean and the Kuroshio in the North Pacific Ocean. In contrast, cold water is transported equatorward along the west coast of continents. 2. In coastal upwelling regions, for example off the coasts of Peru and Chile or Namibia, SST is lowered as cold water is brought to the surface from several hundreds of meters depth. The annual mean sea surface salinity(SSS) is controlled by the exchange of fresh water between ocean and atmosphere and reflects it closely (Figure 2), the only departures being observed as a result of seasonal ice melting in the polar regions. As a result, the subtropics with their high evaporation and low rainfall are characterized by high salinities, while the regions of the westerly wind systems with their frequent rain-bearing storms are associated with low salinities(Figure 3). Persistent rainfall in the intertropical convergence zone produces a regional minimum in the SSS distribution near the equator. Departures from a strict zonal distribution are again observed, for the same reasons listed for the SST distribution. In addition, extreme evaporation rates in the vicinity of large deserts are reflected in high SSS, and large river run-off produced by monsoonal rainfall over south east Asia results in low SSS in the Gulf of Bengal. As a result, the SSS distribution of the north-west Indian Ocean shows a distinct departure from the normal zonal distribution. Seasonal variations of SST and SSS are mainly due to three factors.

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Figure 1 Annual mean sea surface temperature (1C) (the contour interval is 21C). (Reproduced from World Ocean Atlas 1994.)

1. Variations in heat and freshwater exchange between ocean and atmosphere are significant for the SST distribution, which shows a drop of SST in winter and a rise in summer, but much less important for the SSS distribution, since rainfall and evaporation do not vary much over the year inmost ocean regions. 2. Changes in the ocean current system, particularly in monsoonal regions where currents reverse twice a year, cause the water of some regions to be replaced by water of different SST and SSS. 3. Monsoonal variations of freshwater input from major rivers influences SSS regionally. The temperature distribution at 500 m depth (Figure 4) reflects the circulation of the upper ocean. At this depth the temperature shows little horizontal variation around a mean of 8–101C. Departures from this mean temperature are, however, observed. (1) The western basins of the subtropics have the highest temperatures in all oceans. They indicate the centres of the subtropical gyres (see below). (2) Polewardof 351 latitude temperatures fall rapidly as the polar regions are reached, an indication of the absence of the permanent thermocline (see below). The salinity distribution at 500 m depth (Figure 5) shows clear similarities to the temperature distribution

Figure 2 Mean meridional distribution of sea surface salinity and mean meridional freshwater balance (evaporation precipitation).

and a strong correlation between high temperatures and high salinities. The salinity field displays a totalrange nearly as large as the range seen at the surface (Figure 3). The mean salinity varies strongly between ocean basins, with the North Atlantic Ocean having

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Figure 3 Annual mean sea surface salinity. (Reproduced from World Ocean Atlas 1994.)

Figure 4 Annual mean potential temperature (1C) at 500 m depth. (Reproduced from World Ocean Atlas 1994.)

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Figure 5 Annual mean salinity (PSU) at 500 m depth.(Reproduced from World Ocean Atlas 1994.)

the highest salinity at this depth and the North Pacific Ocean the lowest. The horizontal oxygen distribution is chosen to represent conditions for marine life. Nutrient levels are inversely related to oxygen, and although the relationship varies between ocean basins, an oxygen maximum can always be interpreted as a nutrient minimum and an oxygen minimum as a nutrient maximum. At the sea surface the ocean is always saturated with oxygen. A map of sea surface oxygen would therefore only illustrate the dependence of the saturation concentration on temperature (and to a minor degree salinity) and show an oxygen concentration of 8 mll 1 or more at temperatures near freezing point and 4 mll 1 at the high temperatures in the equatorial region. The oxygen distribution at 500 m depth carries a dual signal. It reflects the dependence of the saturation concentration on temperature and salinity in the same way as at the surface but modified by the effect of water mass aging. If water is out of contact with the atmosphere for extended periods of time it experiences an increase in nutrient content from the remineralization of falling detritus; this process consumes oxygen. Water in the permanent thermocline can be a few decades old, which reduces its

oxygen content to 60–80% of the saturation value (Figure 6). The northern Indian Ocean is an exception to this rule; its long ventilation time (see below) produces oxygen values below 20% saturation. In the polar regions oxygen values at 500 m depth are generally closer to saturation as a result of winter convection in the mixed layer (see below).

The Mixed Layer and Seasonal Thermocline Exposed to the action of wind and waves, heating and cooling, and evaporation and rainfall, the ocean surface is a region of vigorous mixing. This produces a layer of uniform properties which extends from the surface down as far as the effect of mixing can reach. The vertical extent or thickness of this mixed layer is thus controlled by the time evolution of the mixing processes. It is smallest during spring and summer when the ocean experiences net heat gain (Figure 7).The heat which accumulates at the surface is mixed downward through the action of wind waves. During this period of warming the depth of the mixed layer is determined by the maximum depth which wave mixing can affect. Because winds

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Figure 6 Annual mean oxygen saturation (%) at 500 m depth(the contour interval is 10%). (Reproduced from World Ocean Atlas 1994.)

Figure 7 Time evolution of the seasonal mixed layer. Left, the warming cycle; right, the cooling cycle. Numbers can be approximately taken as successive months, with the association shown in Table 1.

areoften weaker during midsummer than during spring, wind mixing does not reach quite so deep during the summer months, and the mixed layer may consist of two or more layers of uniform properties (Figure 7, line 4 of the warming cycle). During fall and winter the ocean loses heat. This cooling produces a density increase at the sea

surface. As a result, mixing during the cooling period is no longer controlled by wave mixing but by convection. The convection depth is determined by the depth to which the layer has to be mixed until static stability is reached. The mixed layer therefore increases with time during fall and winter and reaches its greatest vertical extent just before spring. The thin region of rapid temperature change below the mixed layer is known as the seasonal thermocline. It is strongest (i.e., is associated with the largest change in temperature) in summer and disappears in winter. In the tropics (within 201 of the equator) the heat loss during winter is not strong enough to erase the seasonal thermocline altogether, and the seasonal character of the thermocline is then only seen as a variation of the associated vertical temperature gradient. In the subtropics the mixed layer depth varies between 20–50 m during summer and 70–120 m during winter. In subpolar regions the mixed layer depth can grow to hundreds of meters during winter. Three locations of particularly deep winter mixed layers are the North Atlantic Ocean between the Bay of Biscay and Iceland, the eastern South Indian Ocean south of the Great Australian Bight and the region to the west

UPPER OCEAN MEAN HORIZONTAL STRUCTURE

Table 1

257

Association between number in Figure 7 and months Northern Hemisphere

Southern Hemisphere

Number in Figure 7

Warming cycle

Cooling cycle

Warming cycle

Cooling cycle

1 2 3 4 5

February April May June

June August September October January

August October November December

December February March April July

of southern Chile. In these regions mixed layer depths can exceed 500 m during late winter.

The Barrier Layer The mixed layer depth is often equated with the depth of the seasonal thermocline. Historically this view is the result of the paucity of salinity or direct density observations and the resulting need to establish information about the mixed layer from a vertical profile of temperature alone. This approach is acceptable in many situations, particularlyin the temperate and subpolar ocean regions. There are, however, situations where it can be quite misleading. A temperature profile obtained in the equatorial western Pacific Ocean, for example, can show uniform temperatures to depths of 80–100 m. Such deep homogeneity in a region where typical wind speeds

rarely exceed those of a light breeze cannot be produced by wave mixing. The truth is revealed in a vertical profile of salinity which shows a distinct salinity change at a much shallower depth, typically 25–50 m, indicating that wave mixing does not penetrate beyond this level and that active mixing is restricted to the upper 25–50 m. In these situations the upper ocean contains an additional layer known as the barrier layer (Figure 8). The mixed layer extends to the depth where the first density change is observed. This density change is the result of a salinity increase with depth and therefore associated with a halocline (a layer of rapid vertical salinity change). The temperature above and below the halocline is virtually identical. The barrier layer is the layer between the halocline and thethermocline. The barrier layer is of immense significance for the oceanic heat budget. In most ocean regions the mixed

Figure 8 The structure of the upper ocean in the absence (A) and presence (B) of a barrier layer. T: temperature (1C),S: salinity, st: density. Note the uniformity of temperature(T) from the surface to the bottom of the barrier layer in(B). The stations were taken in the central South China Sea during September 1994.

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UPPER OCEAN MEAN HORIZONTAL STRUCTURE

layer experiences a net heat gain at the surface during spring and summer and has to export heatin order to maintain its temperature in a steady state. If (as described in a previous section) the mixed layer extends down to the seasonal thermocline, this is achieved through the entrainment of colder water into the mixed layer from below. The presence of the barrier layer means that the water entrained from the region below the mixed layer is of the same temperature as the water in the mixed layer itself. The entrainment process is still active but does not achieve the necessary heat export. The barrier layer acts as a barrier to the vertical heat flux, and the heat gained by the mixed layer has to be exported through other means, mainly through horizontal advection by ocean currents and, if the mixed layer is sufficiently transparent to the incoming solar radiation, through direct downward heat transfer from the atmosphere to the barrier layer. The existence of the barrier layer has only come to light in the last decade or two when high-quality salinity measurements became available in greater numbers. It has now been documented for all tropical ocean regions. In the Pacific Ocean the regional extent of the barrier layer is closely linked with high local rainfall in the Intertropical and South Pacific Convergence Zones of the atmosphere. This suggests that the Pacific barrier layer is formed by the lowering of the salinity in the shallow mixed layer in response to local rainfall. In contrast, the barrier layer in the Indian Ocean varies seasonally in extent, and the observed lowering of the mixed layer salinity seems to be related to the spreading of fresh water from rivers during the rainy monsoon season. In the Atlantic Ocean the barrier layer is most likely the result of subduction of high salinity water from the subtropics under the shallow tropical mixed layer. There are also observations of seasonal barrier layers in other tropical ocean regions, such as the South China Sea.

The Subtropical Gyres and the Permanent Thermocline The permanent or oceanic the rmocline is the transition from the upper ocean to the deeper oceanic layers. It is characterized by a relatively rapid decrease of temperature with depth, with a total temperature drop of some 151C over its vertical extent, which varies from about 800 m in the subtropics to less than 200 m near the equator. This depth range does not display the relatively strong currents experienced in the upper ocean but still forms part of the general wind driven circulation, so its water moves with the same current systems seen at the sea surface but with lesser speed.

The permanent thermocline is connected with the atmosphere through the Subtropical Convergence, broad region of the upper ocean poleward of the subtropics where the wind-driven surface currents converge, forcing water to submerge (‘subduct’) under the upper ocean layer and enter the permanent thermocline. This convergence is particularly intense in the subtropical front, a region of enhanced horizontal temperature change within the Subtropical Convergence found at about 351Nand 401S. The Subtropical Front is therefore considered the poleward limit of the permanent thermocline (Figure 9). There is also a zonal variation in the vertical extent, with smallest values in the east and largest values in the west. Taken together, the permanent thermocline appears bowl shaped, being deepest in the western parts of the subtropical ocean (25–301N and 30–351S). The shape is the result of geostrophic adjustment in the wind-driven circulation, which produces anticyclonic water movement in the subtropics known as the subtropical gyres. In most ocean regions the permanent thermocline is characterized by a tight temperature–salinity(TS) relationship, lower temperatures being associated with lower salinities. If temperature or salinity is plotted on a constant depth level across the permanent thermocline, the highest temperature and salinity values are found in the western subtropics (Figures 4 and 5). The tight TS relationship indicates the presence of a stable water mass, known as Central Water. This water mass is formed at the surface in the subtropical convergence, particularly at the downstream end of the western boundary currents, where it is subducted and from where it renews (‘ventilates’) the permanent thermocline by circulating in the subtropical gyres, moving equatorward in the east, westward with the equatorial current system and returning to the ventilation region in the west. As a result the age of the Central Water does not increase in a simple meridional direction from the subtropical front towards the equator but is lower in the east and higher in the west. As the Subtropical Front is a feature of both hemispheres, each ocean, with the exception of the Indian Ocean which does not reach far enough north to have a Subtropical Front in the northern hemisphere, has Central Water of northern and southern origins (Figure 9). Fronts between the different varieties of Central Water are a prominent feature of the permanent thermocline. These fronts are characterized by strong horizontal temperature and salinity gradients but relatively small density change because the effect of temperature on density is partly compensated by the effect of salinity. As a result smallscale mixing processes such as double diffusion,

UPPER OCEAN MEAN HORIZONTAL STRUCTURE

259

Figure 9 Regional distribution of the water masses of the permanent thermocline.

filamentation and interleaving are of particular importance in these fronts.

The Equatorial Region The equatorial current system occupies the region 151S–151N and is thus more than 3000 km wide. Most of itis taken up by the North and South Equatorial Currents, the westward flowing equatorial elements of the subtropical gyres discussed above. Between these two currents flows the North Equatorial Countercurrent as a relatively narrow band eastward along 51N in the Atlantic and Pacific Oceans and, during the north-east monsoon season, along 51S in the Indian Ocean. Another eastward current, the Equatorial Undercurrent, flows submerged along the equator, where it occupies the depth range 50–250 m as a narrow band ofonly 200 km width. Currents near the equator are generally strong, and for dynamical reasons transport across the equator is more or less restricted to the upper mixed layer and to a narrow regime of a few hundred kilometers width along the western boundary of the oceans. This restriction andthe narrow eastward

currents embedded in the general westward flow, shape the distribution of properties in the permanent thermocline near the equator. Insituations where subtropical gyres exist (the Atlantic and Pacific Oceans) in both hemispheres they enter the equatorial current system from the north east and from the south east, leaving a more or less stagnant region (‘shadow zone’) between them near the eastern coast. Figure 10 shows the age distribution for the Atlantic Ocean. The presence of particularly old water in the east indicates a stagnant region or ‘shadow zone’ between the subtropical gyres. The strong eastward flowing currents in the equatorial current system modify the age distribution in the permanent thermocline further. In Figure 10 the Equatorial Undercurrent manifests itself as a band of relatively young water, which is carried eastward. The Indian Ocean does not extend far enough to the north to have a subtropical convergence in the Northern Hemisphere. In the absence of a significant source of thermocline water masses north of theequator the water of the Northern Hemisphere can only be ventilated from the south. Figure 11 shows property fields of the permanent thermocline in the

260

UPPER OCEAN MEAN HORIZONTAL STRUCTURE

Figure 10 Pseudo age of Central Water in equatorial region of the Atlantic Ocean at 500 m depth. The quantity pseudo age expresses the time elapsed since the water had last contact with the atmosphere; it is determined by using anarbitrary but realistic oxygen consumption rate for the permanent thermocline. (Reproduced from Poole and Tomczak m (19) Optimum multiparameter analysis of the water mass structure in the Atlantic Ocean thermocline. Deep-Sea Research 46: 1895–1921.)

Indian Ocean and pathways of its water masses. The region between 51S and the equator is dominated by the westward flow of Australasian Mediterranean Water (AAMW), a water mass formed in the Indonesian seas. Its mass transport is relatively modest, and it is mixed into the surrounding waters before it reaches Africa. Indian Central Water(ICW) originates near 301S in large volume; it joins the anticyclonic circulation of the subtropical gyre and can be followed (at the depth level shown in Figure 11 by its temperature of 11.71C and salinity of 35.1) across the equator along the African coast and into the Northern Hemisphere. The flow into the Northern Hemisphere is thus severely restricted, and the ventilation of the northern Indian Ocean thermocline is unusually inefficient. This is reflected in the extremely low oxygen content throughout the northern Indian Ocean.

The Polar Regions Poleward of the subtropical front the upperocean changes character. As polar latitudes are approached the distinction between upper ocean and deeper layers disappears more and more. There is no

Figure 11 Climatological mean temperature (1C) (A),salinity (PSU) (B) and oxygen concentration (ml l 1) (C) in the Indian Ocean for the depth range 300–450 m, with pathways for Indian Central Water (ICW)and Australasian Mediterranean Water (AAMW). (Reproduced from Tomczak and Godfrey, 1994.)

permanent thermocline; temperature, salinity and all other properties are nearly uniform with depth. The surface mixed layer is, of course, still well defined as the layer affected by wave mixing, but its significance for the heat exchange with the atmosphere is greatly reduced because frequent convection events produced by surface cooling penetrate easily into the waters below themixed layer. Because in the polar regions the upper ocean and the deeper layers form a single dynamic unit, the horizontal structure of the upper ocean in these regions is strongly influenced by features of the deeper layers. Figure 12 shows the arrangement of the

UPPER OCEAN MEAN HORIZONTAL STRUCTURE

261

Figure 12 Fronts in the Southern Ocean. (Reproduced from Tomczak and Godfrey, 1994) STF, Subtropical Front; SAF, Subantarctic Front; PF, Polar Front; CWB, Continental Water boundary; AD, Antarctic Divergence.

various fronts in the Southern Ocean. The fronts are associated with the Antarctic Circumpolar Current. They occupy about 20% of its area but carry 75% of its transport. These fronts extend from the surface to the ocean floor and are thus not exclusive features of the upper ocean. At the low temperatures experienced in the polar seas the density is very insensitive to temperature changes and iscontrolled primarily by the salinity. During ice formation salt seeps out and accumulates under the ice, increasing the water density and causing it to sink. Salt from the upper ocean is thus transferred to the deep ocean basins. As a result, a significant amount of fresh water is added to the upper ocean when the ice melts and floats over the oceanic water. The resulting density gradient guarantees stability of the water column even in the presence of temperature inversions. A characteristic feature of the upper ocean in the polar regions is

therefore the widespread existence of shallow temperature maxima. In the Arctic Ocean the water below the upper ocean can be as much as 41C warmer than the mixed layer. Intermediate temperature maxima in the Antarctic Ocean are less pronounced (up to 0.51C) but occur persistently around Antarctica.

See also Heat Transport and Climate. Wind- and BuoyancyForced Upper Ocean.

Further Reading Tomczak M and Godfrey JS (1994) Regional Oceanography: an Introduction. Oxford: Pergamon.

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS L. K. Shay, University of Miami, Miami, FL, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction Ocean temperature structure changes from profiler and remotely sensed data acquired during hurricane passage have been documented in the literature. These oceanic response measurements have emphasized the sea surface temperature (SST) cooling and deepening of the wind-forced ocean mixed layer (OML). The level of SST cooling and OML deepening process are associated with the oceanic current response, which has two major components (Figure 1). First, the momentum response is associated with the OML current divergence in the nearfield with a net transport away from the storm center. This divergent flow causes upwelling of the isotherms and an upward vertical velocity. Over the next half of the cycle, currents and their transport converge toward the track, forcing downwelling of warmer water into the thermocline. This cycle of upwelling and downwelling regimes occurs over distances of an inertial wavelength and is proportional to the product of the storm translation speed and the local inertial period. Over these distances, horizontal pressure gradients couple the wind-forced OML to the thermocline as part of the three-dimensional cold wake. In the Northern Hemisphere, wind-forced currents rotate anticyclonically (clockwise) with time and depth where the period of oscillation is close to the local inertial period (referred to as near-inertial). This near-inertial current vector rotation with depth creates significant vertical current shears across the OML base and the top of the seasonal thermocline that induces vertical mixing and cooling and deepening of the layer. For these two reasons, the upper ocean current transport and vertical current shear are central to understanding the ocean’s thermal response to hurricane forcing. The SST response, and by proxy the OML temperature response, typically decreases by 1–5 1C to the right of the storm track at one to two radii of maximum winds (Rmax) due to surface wind field asymmetries, known as the ‘rightward bias’. Although warm SSTs (Z26 1C) are required to maintain a

262

hurricane, maximum SST decreases and OML depth increases of 20–40 m are primarily due to entrainment mixing of the cooler thermocline water with the warmer OML water. Ocean mixing and cooling are a function of forced current shear (qv/qz ¼ s) that reduce the Richardson number (defined as the ratio of buoyancy frequency (N2) and (s2)) to decrease below criticality. The proportion of these physical processes to the cooling of the OML heat budget are sheardriven entrainment mixing (60–85%), surface heat and moisture fluxes (Qo) (5–15%), and horizontal advection by ocean currents (5–15%) under relatively quiescent initial ocean conditions (no background fronts or eddies). As per Figure 1, vertical motion (upwelling) increases the buoyancy frequency associated with more stratified water that tends to increase the Richardson number above criticality. In strong frontal regimes (e.g., the Loop Current (LC) and warm core rings (WCRs)) with deep OML, cooling induced by these physical processes is considerably less than observed elsewhere. During hurricane Opal’s passage in the Gulf of Mexico (GOM), SST cooling within a WCR was B1 1C compared to B3 1C on its periphery. In these regimes where the 26 1C isotherm is deep (i.e., 100 m), more turbulent mixing induced by vertical current shear is required to cool and deepen an already deep OML compared to the relatively thin OMLs. That the entrainment heat fluxes at the OML base are not significantly contributing to the SST cooling implies there is more heat for the hurricane itself via the heat and moisture surface fluxes. These regimes have less ‘negative feedback’ to atmosphere than typically observed over the cold wake. To accurately forecast hurricane intensity and structure change in coupled models, the ocean needs to be initialized correctly with both warm and cold fronts, rings and eddies observed in the tropical and subtropical global oceans. The objective of this article is to build upon the article by Shay in 2001 to document recent progress in this area of oceanic response to hurricanes with a focus on the western Atlantic Ocean basin. The rationale here is that in the GOM (Figure 1(c)), in situ measurements are more comprehensive under hurricane conditions than perhaps anywhere else on the globe. Second, once a hurricane moves over this basin, it is going to make landfall along the coasts of Mexico, Cuba, and the United States. In the first

263

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

(c)

(a)

32° N

Mississippi Alabama Louisiana

30° N

Florida

Texas

−200 m

28° N 26° N

WCR Envelope

Warm Core Ring Loop Current

24° N −200 m 22° N

Qo

−200 m

Yucatan Straits

Florida Current Cuba

Northwest Caribbean Sea

Mexico

20° N 100° W

(b)

Georgia

96° W

92° W

88° W

84° W

80° W

xu *

′ Transport

h

(z)

Δh

v mixing z

w Thermocline Upwelling −8

−6

−4

−2

0 R max

2

4

6

8

Figure 1 (a) Tropical cyclone image and (b) a cross-section schematic of the physical processes that alter the OML depth (h: light gray line) forced by hurricane winds (u *) such as shear-induced mixing (qv/qz ¼ shear) and OML depth changes (Dh: dark gray line), upwelling (w) due to transport (arrows) by currents away from the storm center relative to the surface depression (Z0 ), and surface heat fluxes (Qo) from the ocean to the atmosphere, all of which may contribute to ocean cooling during TC passage. (c) States and countries surrounding the Gulf of Mexico and northwest Caribbean Sea and identification of the key oceanic features and processes and areas relative to the 200-m isobath. (a, b) Adapted from Shay LK (2001) Upper ocean structure: Response to strong forcing events. In: Weller RA, Thorpe SA, and Steele J (eds.) Encyclopedia of Ocean Sciences, pp. 3100–3114. London: Academic Press.

section following this, progress on understanding the wind forcing and the surface drag coefficient behavior at high winds is discussed within the context of the bulk aerodynamic formula. In the next section, the importance of temperature, current, and shear measurements with respect to model initialization are described. While cold wakes are usually observed in relatively quiescent oceans (i.e., hurricanes Gilbert (1988); Ivan, Frances (2004)), the oceanic response is not nearly as dramatic in warm features. This latter point has important consequences for coupled models to accurately simulate the atmospheric response where the sea–air transfers (e.g., surface fluxes) may not decrease to significant levels as observed over cold wakes. These physical processes for oceanic response are briefly documented here for recent hurricanes such as the LC, WCR, and cold core ring (CCR) interactions and coastal ocean response during hurricanes Lili in 2002, Ivan in 2004, and Katrina and Rita in 2005. Concluding remarks as well as suggested avenues for future research efforts are in the final section.

Atmospheric Forcing Central to the question of storm forcing and the ocean response is the strength of the surface wind stress and the wind stress curl defined at 10 m above the surface. Within the framework of the bulk aerodynamics formula, the wind stress is given by t ¼ ra cd W10 ðu10 i þ v10 jÞ where ra is the air density, cd is the surface drag coefficient, the magnitude of the 10-m wind (W10 ¼ O(u10 2 þ v10 2 ), where u10 and v10 represent the surface winds at 10 m in the east (i) and north (j) directions, respectively). Momentum transfer between the two fluids is characterized by the variations of wind speed with height and a surface drag coefficient that is a function of wind speed and surface roughness. It is difficult to acquire flux measurements for the high wind and wave conditions under the eyewall at 10 m; however, profilers have been deployed from

264

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

aircraft to measure the Lagrangian wind profiles in hurricanes. These profiler data suggest a logarithmic variation of mean wind speed in the lowest 200 m of the boundary layer. Based on this variation, the surface wind stress, roughness length, and neutral stability drag coefficient determined by the profile method indicate a leveling of the surface momentum flux as winds increase above hurricane force with a slight decrease of the drag coefficient with increasing winds. Donelan and colleagues found the characteristic behavior cd since surface conditions change from aerodynamically smooth to aerodynamically rough (cd increasing with wind speed) conditions. In rough flow, the drag coefficient is related to the height of the ‘roughness elements’ per unit distance downwind or the spatial average of the downwind slopes. In a hurricane, rapid changes in wind speed and direction occur over short distances compared to those required to approach full-wave development. The largest waves in the wind-sea move slowly compared to the wind and travel in directions differing from the surface winds. Under such circumstances, longer waves contribute to the roughness of the sea and a ‘saturation’ of the drag coefficient occurs after wind speeds exceed 33 m s  1 (Figure 2). Beyond this threshold, the surface does not become any rougher. These results suggest that there may be a limiting state in the aerodynamic roughness of the sea surface.

× 10−3

5

The oceanic response is usually characterized as a function of storm translation speed (Uh), radius of maximum winds (Rmax), surface wind stress at 10-m level (tmax), OML depth (h), and the strength of the seasonal thermocline either by reduced gravity (g0 ¼ g(r2 – r1)/r2 where r1 is the density of the upper layer of depth h1, and r2 is the density in the lower layer of depth h2 where r24r1) or buoyancy frequency (see Table 1). The latitude of the storm sets the local planetary vorticity through the local Coriolis parameter (f ¼ 2O sin(j), where O is the angular rotation rate of the Earth (7.29  10  5 s  1), and j is the latitude). The inverse of the local Coriolis parameter (f  1) is a fundamental timescale referred to as the inertial period (IP ¼ 2pf  1). The local IPs decrease poleward, for example, at 101 N, IP B70 h, at 241 N IP B30 h, and at 351 N IP B20 h. The relative importance of this parameter cannot be overemphasized in that at low latitudes such as the eastern Pacific Ocean (EPAC) warm pool, the nearinertial current and shear response will require over a day to develop during hurricane passage. By contrast, at the mid-latitudes, near-inertial motions will develop significant shears across the base of the OML much more quickly. Thus, the initial SST cooling and OML deepening will be minimal at lower latitudes compared to the mid-latitudes for the same oceanic stratification and storm structure.

Measured drag coefficients by various methods

Green squares = profile method (Ocampo-Torres et. al., 1994) Blue asterisks = profile method (Donelan et. al., 2004) Red circles = surface slope (Donelan et. al., 2004) Magenta dots = dissipation (Large and Pond, 1981)

4.5 Drag coeff. referred to 10 m

Air–Sea Parameters

4 3.5 3 2.5

∗

2

∗

∗ ∗

∗

1.5

∗ ∗

1 0.5 0

0

5

10 15 20 25 30 35 40 Wind speed extrapolated to height of 10 m, U10 (m s−1)

45

50

Figure 2 Laboratory measurements of the neutral stability drag coefficient (10  3) by profile, eddy correlation (‘Reynolds’), and momentum budget methods. The drag coefficient refers to the wind speed measured at the standard anemometer height of 10 m. The drag coefficient formula of Large and Pond (1981) is also shown along with values from Ocampo-Torres et al. (1994) derived from field measurements. From Donelan MA, Haus BK, Reul N, et al. (2004) On the limiting aerodynamic roughness of the ocean in very strong winds. Geophysical Research Letters 31: L18306, figure 2 (doi: 1029/2004GRL019460).

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

265

Table 1 Air–sea parameters and scales for hurricanes Lili (2002) for both the LC and GOM common water, and Ivan (2004), Katrina (2005), and Rita (2005) over the GOM basin Parameter

Lili (LC)

Lili (GOM)

Ivan

Katrina

Rita

Radius of max. winds Max. wind stress Translational speed Wavelength Mixed layer depth Inertial period Thermocline thickness Barotropic phase speed Barotropic deformation radius Baroclinic phase speed Baroclinic deformation radius

Rmax (km) tmax (N m  2) Uh (m  1 s) L (km) h (m) IP (d) b (m) c0 (m  1 s) a0 (km) c1 (m  1 s) a1 (km)

25 7.1 6.9 770 110 1.3 200 120 2100 1.5 26

18 8.0 7.7 775 35 1.16 200 150 2400 2.8 46

32 6.7 5.5 594 35 1.25 200 72 1002 2.8 40

42 7.6 6.3 608 74 1.12 200 147 2250 2.5 38

19 8.7 4.7 454 70 1.12 200 150 2300 1.9 29

Froude number (Fr)

Uh/c1

2.5

2.8

2.2

2.5

2.5

Note that these parameters are based on where measurements were acquired; for example, Ivan moved over the DeSoto Canyon and over the shelf compared to Lili moving over the eastern side of the Yucatan Shelf, then into the central GOM. Katrina and Rita scales are based on the north-central GOM.

Ocean Structure

An important parameter governing the response is the wave phase speed of the first baroclinic mode due to oceanic density changes between the OML and the thermocline. In a two-layer model, both barotropic and baroclinic modes are permitted. The barotropic (i.e., depth-independent) mode is referred to as the external mode whereas the first baroclinic (depthdependent) mode is the first internal mode associated with vertical changes in the stratification. The phase speed of the first baroclinic mode (c1) is given by c1 2 ¼ g0 h1 h2 =ðh1 þ h2 Þ where the depth of the upper layer is h1, and the depth of the lower layer is h2. In the coastal ocean, phase speeds range from 0.1 to 0.5 m s  1, whereas in the deep ocean, this phase speed ranges between 1 and 3 m s  1 depending on the density contrast between the two layers. The barotropic mode has a phase speed c0 ¼ OgH where H represents the total depth (h1 þ h2), and is typically 100 times larger than the first baroclinic mode phase speed. An important nondimensional number for estimating the expected baroclinic response depends on the Froude number (ratio of the translation speed to the first baroclinic mode phase speed Uhc1 1). If the Froude number is less than unity (i.e., stationary or slowly moving storms), geostrophically balanced currents are generated by the positive wind stress curl causing an upwelling of cooler water induced by upper ocean transport directed away from the storm track (Figure 1). When the hurricane moves faster than the first baroclinic mode

phase speed, the ocean response is predominantly baroclinic associated with upwelling and downwelling of the isotherms and the generation of strong nearinertial motions in a spreading three-dimensional wake. The predominance of a geophysical process also depends upon the deformation radius of the first baroclinic mode (a1 1) defined as the ratio of the first mode phase speed (c1) and f. In the coastal regime, the deformation radius is 5–10 km, but in deeper water, it increases to 20–50 km due to larger phase speeds. For observed scales exceeding the deformation radius, Earth’s rotational effects, through the variations of f, dominate the oceanic dynamics where timescales are equal or greater than IP. Thus the oceanic mixed layer response to hurricanes is characterized as rotating, stratified shear flows forced by winds and waves. Basin-to-Basin Variability

Profiles from the background GOM, LC subtropical water, and the tropical EPAC are used to illustrate differences in the buoyancy frequency profile (Figure 3). In an OML, the vertical density gradients (N) are essentially zero because of the vertical uniformity of temperature and salinity. Maximum buoyancy frequency (Nmax) in the Gulf is 12–14 cycles per hour (cph) located between the mixed layer depth (40 m) and the top of the seasonal thermocline compared to c. 5–6 cph in the LC water mass distributed over the upper part of the water column. In the EPAC, however, Nmax B20 cph due to the sharpness of the thermocline and halocline

266

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

(a)

(b) 0

Depth (m)

100

200

300 EPAC GOM LC

400

500 0

5

10 15 20 Temperature (°C)

25

(c)

30

33

34

35 36 Salinity (psu)

37

(d) 0

Depth (m)

100

200

300

400

500 1020

1022

1024

1026

Density (kg

1028

1030

m−3 )

0

5

10

15

20

25

N (cph)

Figure 3 (a) Temperature (1C), (b) salinity (practical salinity units, psu), (c) density (kg m  3), and (d) buoyancy frequency (N : cycles per hour) profiles from the eastern Pacific Ocean (red) , the GOM common water (green), and the LC water (blue) as measured from airborne expendable ocean profilers. Notice the marked difference between the gradients at the base of the OML between the three profiles.

(pycnocline) located at the base of the OML (i.e., 30 m). Beneath this maximum, buoyancy frequencies (Z3 cph) are concentrated in the seasonal thermocline over an approximate thermocline scale (b) of 200 m and exponentially decay with depth approaching 0.1 cph. In the LC water, Nmax ranges from 4 to 6 cph and remains relatively constant, and below the 20 1C isotherm depth (B250 m), buoyancy frequency decreases exponentially. The Richardson number increases with increases in the buoyancy frequency for a given current shear (s). This implies that a higher shear is needed in a regime like the EPAC to lower the Richardson number to below-critical values for the upper ocean to mix and cool compared to the water mass in the

GOM. Given a large N at lower latitudes (121 N) where the IP is long (B58 h) in the EPAC warm pool, SST cooling and OML deepening will be much less than in the GOM as observed during hurricane Juliette in September 2001 (not shown). Significant SST cooling of more than 5 1C only occurred when Juliette moved northwest where Nmax decreased to B14 cph at higher latitudes. Levels of SST cooling similar to those for the same hurricane in the GOM would be observed in the common water but not in the LC water mass since the 26 1C isotherm depth is 3–4 times deeper. These variations in the stratification represent a paradox for hurricane forecasters and are the rationale underlying the use of satellite radar altimetry in mapping isotherm depths and

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

estimating oceanic heat content (OHC) from surface height anomalies (SHAs) and assimilating them into oceanic models.

the 26 1C isotherm are large. In the LC regime, for example, this isotherm may be deeper than 150 m whereas in the common water the 26 1C isotherm is located at 40 m. The corresponding OHC relative to the 26 1C isotherm is given by

Gulf of Mexico Basin Warm subtropical water is transported poleward by upper-ocean currents from the tropics through the Caribbean Sea and into the GOM (see Figure 1(c)). This subtropical water exits the northwestern Caribbean Sea through the Yucatan Straits where the transport of B24 Sv (1 Sv ¼ 106 m3 s  1) forms the LC core. Given upper ocean currents B1 m s  1 of the LC, horizontal density gradients between this ocean feature and surrounding GOM common water occur over smaller scales due to markedly different temperature and salinity structure (Figure 4). Variations in isotherm depths and OHC values relative to In situ observations

267

OHC ¼ cp

D26 Z

r½TðzÞ  26 dz

0

where cp is specific heat at constant pressure, D26 is the 26 1C isotherm depth, and OHC is zero wherever SST is less than 26 1C. Within the context of a twolayer model approach and a ‘hurricane season’ climatology, the 26 1C isotherm depth and its OHC relative to this depth are monitored using satellite techniques by combining SHA fields from satellite altimeters onboard the NASA Jason-1, US Navy Geosat Follow On, and European Research Satellite-2 Temperature (°C), 19.7° N, 85.0° W

Derived 0

GDEM3 Climatology WOA01 Climatology Pre-Isidore MODAS Pre-Isidore, measured Pre-Isidore HYCOM-OI Pre-Isidore HYCOM-MODAS

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Figure 4 OHC (kJ cm  2) in the northwest Caribbean Sea and southeast GOM from an objective analysis of in situ aircraft measurements, satellite altimetry, HYCOM NRL-CH nowcast, and HYCOM NRL-MODAS nowcast (four left panels). Temperature (right top) and salinity (right bottom) vertical profiles at a location in the northwest Caribbean Sea, where red lines are climatological profiles (GDEM3 dashed, WOA01 solid), solid blue lines are observed profiles, dashed blue lines are MODAS profiles, and black lines are model nowcasts (HYCOM-NRL dashed and HYCOM-MODAS solid). Adapted from Halliwell GR, Jr., Shay LK, Jacob SD, Smedstad OM, and Uhlhorn EW (in press) Improving ocean model initialization for coupled tropical cyclone forecast models using GODAE nowcasts. Monthly Weather Review.

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(and Envisat) missions with observed SSTs. In the 1970s, Leipper coined the phrase ‘hurricane heat potential’, which represents integrated thermal structure relative to 26 1C water. In the LC regime, OHC values relative to this isotherm depth often exceed 100 kJ cm  2. For oceanic response studies, the key science issue is that such deep isotherms (and OHC levels) tend to be resistive to significant storm-induced cooling by forced near-inertial current shears across the base of deep OML. Loop Current Cycle

The LC is highly variable and when it penetrates beyond latitudes of 251 N, WCR shedding events occur at periods of 6–11 months when CCRs are located on their periphery prior to separation. By contrast, south of this latitude, WCR shedding periods increase to more than 17 months based on a series of metrics developed by Leben and colleagues. These WCRs, with diameters of approximately 200 km, then propagate west to southwest at average phase speeds of B5 km day  1, and remain in the GOM for several months. At any given time, two or three WCRs may be embedded within the complex GOM circulation patterns. Theoretical developments suggest that the LC cycle can be explained in terms of the momentum imbalance paradox theory. This theory predicts that when a northward-propagating anomalous density current (i.e., Yucatan Current) flows into an open basin (GOM) with a coast on its right (Cuba), the outflow balloons near its source forming a clockwiserotating bulge (e.g., LC) since the outflow cannot balance the along-shelf momentum flux after turning eastward. The ballooning of the current satisfies the momentum flux balance along the northern Cuban coast. The subsequent WCR separation from the LC is due to the planetary vorticity gradients where most of the inflow forces a downstream current and the remaining inflow forms a warm ring. Subtropical water emerging from the Caribbean Sea may enter the LC bulge prior to shedding events and impact the OHC distribution, and if in phase with the height of hurricane season may spell disaster for residents along the GOM. Model Initialization

Ocean models that assimilate data are an effective method for providing initial and boundary conditions in the oceanic component of coupled prediction models. The thermal energy available to intensify and maintain a hurricane depends on both the temperature and thickness of the upper ocean

warm layer. The ocean model must be initialized so that features associated with relatively large or small OHC are in the correct locations and T–S (and density) profiles, along with the OHC, are realistic. Ocean forecast systems based on the hybrid coordinate ocean model (HYCOM) have been evaluated in the northwest Caribbean Sea and GOM for September 2002 prior to hurricanes Isidore and Lili, and in September 2004 prior to Ivan. An examination of the initial analysis prior to Isidore is from an experimental forecast system in the Atlantic basin (Figure 4). This model assimilates altimeter-derived SHAs and SSTs. Comparison of OHC maps by the model and observations demonstrate that the analysis (labeled NRL-CH) reproduces the LC orientation but underestimates values of the heat content. In the Caribbean Sea, the thermal structure (T(z)) hindcast tends to follow the September ocean climatology but does not reproduce the larger observed OHC values. The model ocean is less saline than both climatology and profiler measurements above 250 m and less saline than those between 250 and 500 m. Evaluations of model products are needed prior to coupling to a hurricane model to insure that ocean features are in the correct locations with realistic structure. Mixing Parametrizations

One of the significant effects on the upper ocean heat budget and the heat flux to the atmosphere is the choice of entrainment mixing parametrizations at the OML base (see Figure 1). Sensitivity tests have been conducted using five schemes: K-profile parametrization (KPP); Goddard Institute for Space Studies level-2 closure (GISS); Mellor–Yamada level2.5 turbulence closure scheme (MY); quasi-slab dynamical instability model (Price–Weller–Pinkel dynamical instability model, PWP); and a turbulent balance model (Kraus–Turner turbulence balance model, KT). Simulated OML temperatures for realistic initial conditions suggest similar response except that the magnitude of the cooling differs as well as its lateral extent of the cooling patterns (Figure 5). Three higher-order turbulent mixing schemes (KPP, MY, and GISS) seem to be in agreement with observed SST cooling patterns with a maximum of 4 1C whereas PWP (KT) over- (under-) estimate SST cooling levels after hurricane Gilbert. This case is an example of ‘negative feedback’ to the atmosphere given these cooling levels due primarily to shear instability at the OML base. Similar to the post-season hurricane forecast verifications, more oceanic temperature, current, and salinity measurements must be acquired to evaluate these schemes to build a larger

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

269

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Figure 5 Simulated mixed layer temperatures during hurricane Gilbert for mixing schemes (a) KPP, (b) PWP, (c) KT, (d) MY, and (e) GISS. Differences between these five cases are visible with PWP being the coolest and KT being the warmest. Black line indicates track of the storm at 06 GMT 16 Sep. 1988.

statistical base for the oceanic response to high-wind conditions in establishing error bars for the models.

Oceanic Response Recently observed interactions of severe hurricanes (category 3 or above) with warm ocean features such

as the LC and WCR (Lili in October 2002, Katrina and Rita in August and September 2005) are contrasted with hurricanes that interact with CCR (Ivan in September 04) and cold wakes (Gilbert in September 1988) in the GOM. The levels of observed upper cooling and OML depth patterns are predicated on the amount of shear-induced mixing in the upper ocean (see Figure 1). The SST response is

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UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

determined from an optimal interpolation scheme using the NASA TRMM microwave imager (TMI) and advanced microwave sensing radiometer (AMSR- E) where the diurnal cycle was removed from the data. LC Interactions

Hurricanes Isidore (21 September 2002) and Lili (02 October 2002) interacted with the LC in nearly the same area spaced about 10 days apart. For negative feedback regimes, one would anticipate that after the first hurricane, there would have been a significant ocean response with little thermal energy available for the second storm as Isidore moved slowly from Cuba to the Yucatan Peninsula. The cyclonic (counterclockwise) rotating surface wind stress (in the Northern Hemisphere) should have upwelled isotherms due to divergent wind-driven transport that may have been balanced by horizontal advection due to strong northward currents through the Yucatan Straits. While observed cooling levels in the straits were less than 1 1C, the upper ocean cooled by

4.5 1C over the Yucatan Shelf. Since upwelling induced by the persistent trade wind regime maintains a seasonal thermocline close to the surface over this shelf, impulsive wind events force upwelling of colder thermocline water quickly due to transport away from the coast. Isidore remained over the Yucatan Peninsula and weakened to a tropical storm that then moved northward creating a cool wake of B28.5 1C SSTs across the central GOM. Lili reached hurricane status on 26 September while passing over the Caribbean Sea along a similar northwest trajectory as Isidore, making a first landfall along the Cuban coast (Figure 6). As Lili moved into the GOM basin, the storm intensified to a category 4 storm along the LC boundary just as Rmax decreased to form a new eyewall (where winds are a maximum). In the common water, the SST cooling was more than 2 1C due to shear-induced mixing compared to less than 1 1C SST cooling in the LC (Figure 6(c)). This suggests that ‘less negative feedback’ (minimal ocean cooling) to hurricane Lili occurred over the LC than over the common water. Afterward, Lili began a weakening cycle to category

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Figure 6 (a) Pre-Lili, (b) post-Lili, (c) pre–post-Lili SST (1C) field from AVHRR data, courtesy of RSMAS Remote Sensing Laboratory, and (d) measurement grid conducted by NOAA research aircraft on 2 Oct. 2002 (open symbols represent nonfunction probes). Panel (c) is relative to the track and intensity of Lili and the position of the LC. Notice the cold wake in the GOM common water compared to essentially no cold wake in the LC. More details of the response in the LC is given in Figure 9. Black box represents the region where in situ measurements from aircraft expendable were acquired during Lili’s passage.

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

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suggestive of ‘less negative feedback’ to the storm as it crossed over the Yucatan Straits. Given the 10-day time interval between Isidore and Lili, pre-Lili SSTs warmed to over 29 1C in the experimental domain. After Lili’s passage, SSTs decreased to 28.5 1C in the LC; however, along the northern extremity of the measurement domain, SSTs cooled to 27 1C in the Gulf of Mexico common water (GCW), which equates to more than 2 1C cooling. The GCW mixes quickly due to current shears across the OML base forcing the layer to deepen. In the LC itself, there was little evidence of cooling and layer deepening. Given the advective timescale (LV  1 where L is cross-stream scale and V is the maximum current of the LC) of about a day, heat transport from the Caribbean Sea occurs rapidly and will offset temperature decreases induced by upwelling of the isotherms and mixing as in the hurricane Isidore case. The observed current shears during the hurricane were 1.5  10  2 s  1 or about a factor of 2–3 less

Y/R max

1 status due to enhanced atmospheric shear, dry-air intrusion along the western edge, and interacting with the shelf water cooled by Isidore. As shown in Figure 6(d), oceanic and atmospheric profilers were deployed in the south-central part of the GOM from research aircraft. The design strategy was to measure upper-ocean response to a propagating and mature hurricane over the LC. Multiple research flights deployed profilers in the same location before, during, and after passage, which captured not only the LC response to Lili but also to Isidore as the hurricane intensified to category 3 status moving across the Yucatan Straits 10 days early. The minimal LC response highlights the importance of this current system for intensity changes. These profiler data were objectively analyzed over a 31  31 domain in latitude and longitude with a vertical penetration to 750-m depth and aligned with the hurricane path (Figure 7). A day after Isidore, SSTs that remained were above 28 1C, which is

271

−8

12

Figure 7 Objective analysis of SSTs (1C) and MLDs (m). Left columns are SSTs and DSSTs and right columns show MLDs and DMLDs for pre-storm, storm, and post-storm (Wake 1) measurements from Lili in the southeastern GOM as per Figure 6(c). Panels are in storm-coordinate system for cross-track (X/Rmax) and along-track (Y/Rmax) based on Rmax and the storm track orientation at 292 1 T North as in Figure 6(c) centered at 23.21 N and 86.11 W. The DSST (1C) and DMLD (m) are estimated by subtracting the prestorm data from the storm and post-storm data and the arrows represent current measurements from airborne expendable current profilers. Blue shaded areas are more than 2 1C consistent with satellite-derived SSTs in Figure 6(c).

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UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

than observed previously in quiescent regimes due to the strength of this background upper ocean flow of the LC. This lack of shear-induced mixing has implications for hurricane intensity as they move over the deep, warm pools of the LC, which represent a reservoir of thermal energy for hurricanes to tap. Cold and Warm Core Ring Interactions

Hurricane Ivan (Sep. 04) entered the GOM as a category 5 storm and then weakened to a category 4 storm due to a combination of lower OHC, vertical shear in the atmosphere associated with an upperlevel trough, and drier air being drawn into its circulation. During its GOM trajectory, Ivan encountered two CCRs and a WCR where the surface pressure decreased by B10 mb during a brief encounter. Shelf water, cooled by hurricane Frances (10 days earlier) along the northern GOM along with increasing atmospheric shear, acted to oppose intensification during an eyewall replacement cycle (defined as the formation of a secondary eyewall that replaces a collapsing inner eyewall). As shown in Figure 8, pre- and post-SSTs to Ivan reveal the location of both WCR and CCR located

along the track of hurricane Ivan and the cold wake due to enhanced current shear instability. The SST difference field, shown in Figure 8(c), indicates that both the WCR and CCR SSTs are eroded away by the strong forcing. The SSTs over the CCRs indicate cooling levels exceeding 4 1C along and to the right of Ivan’s track that were embedded within the cool wake of about 3.5 1C of Ivan. The northern CCR may have been partially responsible for the observed weakening of Ivan as suggested by Walker and colleagues. Notice that just as in the case of Lili, SST cooling of less than 1 1C was observed in the LC in the southern part of the basin. Prior to landfall, Ivan moved over 14 acoustic Doppler current profiler (ADCP) moorings that were deployed as part of the Slope to Shelf Energetics and Exchange Dynamics (SEED) project (Figure 8(d)), as discussed by Teague and colleagues. These profiler measurements provided the evolution of the current (and shear) structure from the deep ocean across the shelf break and over the continental shelf. The current shear response, estimated over 4-m vertical scales, is shown in Figure 9 based on objectively analyzed data from these moorings. Over the shelf, the current shears increased due to hurricane Ivan strong

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Figure 8 Same as Figure 6 except for hurricane Ivan in Sep. 2004 and panel (d) represents ADCP mooring locations during the SEED experiment in the northern GOM in the white box in panel (c).

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

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Figure 9 Spatial evolution of the rotated current shear magnitude normalized by observed shears from the ADCP measurements (white dots) normalized by observed shears in the LC of 1.5  10  2 s  1 (color) during Lili starting at 2100 GMT 15 Sep. every 6 h. Black contours (25-m intervals) represent the depth of the maximum shears based on the current profiles from the moored ADCP. Cross-track (x) and along-track (y) are normalized by the observed Rmax of 32 km. These ADCP data were provided by the Naval Research Laboratory through their SEED project

winds. The normalized shear magnitude over the shelf (depths of 100 m) is larger by a factor of 4 compared to normalized values over the deeper part of the mooring array (500–1000 m). Notice that the current shear rotates anticyclonically (clockwise) in time over 6-h intervals associated with the forced near-inertial response (periods slightly shorter than the local inertial period). In this measurement domain, the local inertial period is close to 24 h which is close to the diurnal tide. By removing the relatively weak tidal currents and digitally filtering the records, the analysis revealed that the predominant response was due to

forced near-inertial motions. These motions have a characteristic timescale for the phase of each mode to separate from the wind-forced OML current response when the wind stress scale (2Rmax) exceeds the deformation radius associated with the first baroclinic mode (B40 km). This timescale increases with the number of baroclinic modes due to decreasing phase speeds. The resultant vertical energy propagation from the OML response is associated with the predominance of the anticyclonic (clockwise) rotating energy with depth and time that is about 4 times larger than the cyclonic (counterclockwise) rotating component.

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In 2005, hurricane Katrina deepened to a category 5 storm over the LC’s western flank with an estimated wind stress of B7 N m  2. The variations of Katrina’s intensity correspond well with the large OHC values in the LC and the lobe-like structure (eventually a WCR) in the northern GOM. Since SSTs exceeding 30 1C were nearly uniformly distributed in this regime, the LC structure was not apparent in the SST signals. This deeper heat reservoir of the LC provided more heat for the hurricane where satellite-inferred OHC values exceeded 120 kJ cm  2 or more than 5 times the threshold suggested by early studies to sustain a hurricane. Within the next 2 weeks, Rita formed and moved through the Florida Straits into the GOM basin (Figure 10(c)). While Rita’s path did not exactly follow Katrina’s trajectory in the south-central Gulf, Rita moved toward the north-northwest over

(a)

the LC and rapidly intensified to similar intensity as Katrina. After Rita interacted with the eastern tip of the WCR, the hurricane began a weakening cycle due to the cooler water associated with a CCR located on the periphery of the WCR similar to Ivan and cooler water on the shelf. Pre- and post-SST analyses include an interval a few days prior and subsequent to hurricane passage to quantify cooling levels in the oceanic response (Figure 11). Prior to Katrina, SSTs exceeded 31 1C in the GOM without any clear evidence of the LC. Subsequent to Katrina, maximum cooling occurred on the right side of the track with SST decreasing to about 28 1C over the outer West Florida Shelf where OML typically lies close to the surface. Observed SSTs decreased by more than 4 1C along the LC’s periphery, mainly due to shear-induced mixing and upwelling over the shelf. As Katrina moved over the LC, the SST response was less than 2 1C as expected

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UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

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Figure 11 Same as Figure 6 except for the hurricane Katrina case where SST were from optimally interpolated TMI data from http:// www.remss.com. Panel (d) represents sampling pattern from aircraft centered on the WCR on 15 September 2005.

over the deeper subtropical water, consistent with the weaker LC response to Lili. These deeper warm pools tend to resist the development of strong shear-induced mixing episodes. Similarly, pre-Rita SSTs ranged from 28.5 to 29 1C over most of the GOM except for the shelf waters cooled by Katrina. However, after Rita’s passage, the dramatic SSTs cooling of 3–4 1C occurred because of the combination of upwelling and cold water advection associated with a CCR that moved between the WCR and the LC. This scenario was analogous to the Ivan case with the CCRs embedded in the cold wake. To illustrate this effect, oceanic profiler measurements were acquired on 15 and 26 September 2005 in a pattern centered on the LC and the lobe-like structure that eventually became the WCR. The earlier research flight was originally conceived as a post-Katrina experiment in an area where it rapidly intensified over the LC and WCR complex to assess altimeter-derived estimates of isotherm depths and OHC variations. Pairs of profilers, deployed in the

center of this WCR structure, confirmed similar depths of the OML of 75 m where the 26 1C isotherm was located at about 120 m. Hurricane Rita’s trajectory clipped the northeastern part of this warm structure as the storm was weakening prior to landfall on the Texas–Louisiana border. While the OHC levels remained relatively the same in this area between pre- and post-Rita (Figure 12), the dramatic cooling between the LC and shed WCR on 26 September was primarily due to the advection of a CCR moving between these ocean features. In addition to upwelling, vertical mixing cooled the ocean as suggested by the vertical sections (Figures 12(c) and 12(d)). Over this period, the WCR propagated westward at a translation speed of 12 km day  1, or nearly double their speeds. Within the WCR, the 26 1C isotherm depth decreased from a maximum depth of 115 m to B88 m. An important research question emerging from the profiler analysis is whether the strong winds associated with Rita forced the WCR to separate prematurely and propagate faster toward the west.

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Figure 12 (a) Pre-Rita and (b) post-Rita analysis of observed (color) and satellite-inferred (contours) of OHC (kJ cm  2) relative to Rita’s intensity (colored circles) and track and corresponding OHC (kJ cm  2: top panels) and vertical thermal structure sections (1C) along 26.51 N transect from (c) pre-Rita and (d) post-Rita.

Summary Progress has been made in understanding the basic oceanic and atmospheric processes that occur during hurricane passage. There is a continuing need to isolate fundamental physical processes involved in these interactions through focused experimental, empirical, theoretical, and numerical approaches. The GOM is one such basin where detailed process studies can focus on the oceanic response to the hurricane forcing as well as the atmospheric response to ocean forcing. Observational evidence is mounting that the warm and cold core features and the LC system are important to the coupled response during hurricane passage. This is not unique to the GOM as this behavior has also been recently observed in other regions such as the western Pacific Ocean and the Bay of Bengal. Thus, it is a global problem that needs to be addressed.

This coupled variability occurs over the storm scales that include fundamental length scales such as the radius of maximum winds and radius to galeforce winds. The fundamental science questions are that how the ocean and atmosphere are coupled, and that what are the appropriate timescales of this interaction? These questions are not easily answered, given especially the lack of coupled measurements spanning the spectrum of hurricane parameters such as strength, radius, and speed. One school of thought is that the only important process with respect to the ocean is under the eyewall where ocean cooling occurs. However, observed cooling under the eyewall is not just due to the surface flux alone (see Figure 1). In this regime, the maximum winds and heat and moisture fluxes occur; however, the broad surface circulation over the ocean also has nonzero fluxes that contribute the thermal

UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

energy buildup toward the eyewall of a hurricane. The importance of stress-induced mixing and current shear instabilities in surface cooling and deepening of the surface mixed layer cannot be overstated. The deeper this layer (and 26 1C isotherm depth), the more is the heat available to the storm through the heat and moisture fluxes. Notwithstanding, it is not just the magnitude of the OHC, since the depth of the warm water is important to sustaining these surface fluxes. Future research needs to focus on these multiple scale aspects associated with the atmospheric response to ocean forcing (minimal negative feedback) and to continue studies of the

oceanic response to hurricanes over a spectrum of oceanic conditions. High-quality ocean measurements are central to addressing these questions and improving coupled models. For the first time, a strong near-inertial current response was observed by newly developed Electromagnetic Autonomous Profiling Explorer (EMAPEX) floats deployed in front of hurricane Frances (2004) by Sanford and colleagues (Figure 13). These profiling floats have provided the evolving near-inertial, internal wave radiation in unprecedented detail that includes not only the temperature and salinity (and thus density), but also the horizontal current 1.5

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Figure 13 Current (U, V in m s  1), salinity (psu), and density or st (kg m  3) response at Rmax during the passage of hurricane Frances (2004) as measured by an EM-APEX float deployed from USAF WC-130 1 day ahead of the storm. Three floats were successfully deployed in the projected cross-track direction as part of the ONR Coupled Boundary Layer Air–Sea Transfer program. Reprinted from Sanford TB, Dunlap JH, Carlson JA, Webb DC, and Girton JB (2005) Autonomous velocity and density profiler: EMAPEX. In: Proceedings of the IEEE/OES 8th Working Conference on Current Measurement Technology, IEEE Cat No. 05CH37650, pp. 152–156 (ISBN: 0-7803-8989-1), @ 2005 IEEE.

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structure. Notice that the phase propagation of the forced near-inertial currents is upward associated with downward energy propagation from the OML as current vectors rotate anticyclonically (clockwise) with time and depth (in the Northern Hemisphere). Velocity shears associated with these near-inertial currents force mixing events as manifested in a large fraction of the observed SST cooling of more than 2 1C (and layer deepening). Given these measurements of the basic state variables, the evolution of the Richardson numbers forced by a hurricane can be determined to evaluate mixing parametrization schemes used in coupled models for forecasting at the national centers. The variability of the surface drag coefficient has received considerable attention over the last 5 years. Several treatments have concluded that there is a leveling off or a saturation value at B3073 m s  1. The ratio of the enthalpy (heat and moisture) coefficient and the drag coefficient is central to air–sea fluxes impacting the hurricane boundary layer. In this context, the relationship between the coupled processes such as wave breaking and the generation of sea spray and how this is linked to localized air–sea

fluxes remains a fertile research area. A key element of this topic is the atmospheric response to the oceanic forcing where there seem to be contrasting viewpoints. One argument is that the air–sea interactions are occurring over surface wave (wind-wave) time and space scales and cause significant intensity changes by more than a category due to very large surface drag coefficients. While, these sub-mesoscale phenomena may affect air–sea fluxes, the first-order balances are primarily between the atmospheric and oceanic mixed layers. The forced surface waves modulate the heat and momentum fluxes. Future Research

A promising avenue of research has focused on the upper ocean’s role on intensity change. Climatologically, for the western Atlantic basin, the expected number of category 5 storms is one approximately every 3 years. Over the last 4 years, there have been a total of six category 5 storms, well above this mean. Based on extensive deliberations by the international tropical cyclone community, intensity and structure changes are primarily due to environmental

32° N 1 Labor Day 35 2 Camille 69 3 Frederic 79 4 Allen 80 5 Gilbert 88 6 Andrew 92 7 Opal 95 8 Bret 99 9 Isidore 02 10 Lili 02 11 Ivan 04 12 Emily 05 13 Katrina 05 14 Rita 05 15 Wilma 05

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UPPER OCEAN STRUCTURE: RESPONSES TO STRONG ATMOSPHERIC FORCING EVENTS

conditions such as atmospheric circulation, internal dynamics, and oceanic circulation processes. Gyrescale ocean circulation redistributes ocean heat throughout the basins primarily through poleward advection and transport along its western boundary. While there is an open scientific question whether the increased frequency of occurrence of severe hurricanes is due to global warming or natural cycle associated with geophysical processes, the severe hurricanes during the 2005 season interacted with the warm Caribbean Current and the LC. As shown here and in recently published papers, the oceanic response over these regimes differs considerably from that observed quiescent regimes. The key issue is the level of observed ocean cooling in these regimes that is considerably less (i.e., ‘less negative feedback’) than compared to other areas where the cooling is more dramatic. Since winds begin to mix the thin ‘skin’ layer of SST well in front of the storm, the surface temperature reflects the temperature of the oceanic mixed layer under high winds. This point is often overlooked in atmospheric models where SSTs are prescribed or weakly coupled to an ocean where the basic state is at rest. As discussed above, intense hurricanes in the GOM may have encountered the LC and WCR during their lifetimes (Figure 14). With the exception of hurricane Allen (1980), which maintained severe status outside the envelope of this oceanic variability, when hurricanes encounter these features, changes in hurricane intensity are often observed even though warm SSTs prevail during summer months over most of the basin. As noted above, during a 7-week period in 2005, Katrina, Rita, and Wilma all rapidly deepened to catagory 5 status in less than 24 h. Lowest central pressures for this unprecedented hurricane trifecta over a 7-week timescale were 896, 892, and 882 mb. Until Wilma, Gilbert in 1988 held the lowest surface pressure record of 888 mb in the basin. With surface winds in excess of 70 m s  1 within 36 h of landfall over the LC and WCR complex, Katrina and Rita had a pronounced impact on the northern GOM coast as well as offshore structures such as oil rigs. If these oceanic conditions had prevailed during the summer of 1969, hurricane Camille, the strongest land-falling hurricane on record in the Atlantic Ocean basin, may have aligned with the axis of this warm current system. Given the natural variability of this deep warm reservoir, such interactions must be explored in more detail for not only the oceanic response, but also the potential feedbacks to the hurricanes where ocean cooling is minimized with respect to the next-generation forecast models.

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Acknowledgments L.K. Shay gratefully acknowledges NSF support and the support of the NOAA Aircraft Operations Center. Mr. Bill Teague provided Ivan current data; and Drs. Mark Donelan, Brian Haus, George Halliwell, S. Daniel Jacob, and Tom Sanford shared material. SSTs were provided by Remote Sensing Systems website (http://www.remss.com), courtesy of Dr. Chelle Gentemann. Benjamin Jaimes, Eric Uhlhorn, and Jodi Brewster also contributed to this article.

See also Upper Ocean Mixing Processes. Upper Ocean Space and Time Variability. Upper Ocean Vertical Structure. Breaking Waves and Near-Surface Turbulence.

Further Reading Chassignet EP, Smith L, Halliwell GR, and Bleck R (2003) North Atlantic simulations with the hybrid coordinate ocean model (HYCOM): Impact of the vertical coordinate choice and resolution, reference density, and thermobaricity. Journal of Physical Oceanography 33: 2504--2526. D’Asaro EA (2003) The ocean boundary layer under hurricane Dennis. Journal of Physical Oceanography 33: 561--579. Donelan MA, Haus BK, Reul N, et al. (2004) On the limiting aerodynamic roughness of the ocean in very strong winds. Geophysical Research Letters 31: L18306 (doi: 1029/2004GRL019460). Gentemann C, Donlon CJ, Stuart-Menteth A, and Wentz F (2003) Diurnal signals in satellite sea surface temperature measurements. Geophysical Research Letters 30(3): 1140--1143. Halliwell GR, Jr., Shay LK, Jacob SD, Smedstad OM, and Uhlhorn EW (in press) Improving ocean model initialization for coupled tropical cyclone forecast models using GODAE nowcasts. Monthly Weather Review. Jacob SD and Shay LK (2003) The role of oceanic mesoscale features on the tropical cyclone-induced mixed layer response. Journal of Physical Oceanography 33: 649--676. Large WG and Pond S (1981) Open ocean momentum flux measurements in moderate to strong wind. Journal of Physical Oceanography 11: 324--336. Leben RR (2005) Altimeter derived Loop Current metrics. In: Sturges W and Lugo-Fernandez A (eds.) Geophysical Monograph, No. 161: Circulation in the Gulf of Mexico: Observations and Models, pp. 181--201. Washington, DC: American Geophysical Union. Lugo-Fernandez A (2007) Is the Loop Current a chaotic oscillator? Journal of Physical Oceanography 37: 1455--1469.

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Nof D (2005) The momentum imbalance paradox revisited. Journal of Physical Oceanography 35: 1928--1939. Ocampo-Torres FJ, Donelan MA, Merzi N, and Jai F (1994) Laboratory measurements of mass transfer of carbon dioxide and water vapour for smooth and rough flow conditions. Tellus 46B: 16--32. Powell MD, Vickery PJ, and Reinhold TA (2003) Reduced drag coefficient for high wind speeds in tropical cyclones. Nature 422: 279--283. Sanford TB, Dunlap JH, Carlson JA, Webb DC, and Girton JB (2005) Autonomous velocity and density profiler: EM-APEX. In: Proceedings of the IEEE/OES 8th Working Conference on Current Measurement Technology, IEEE Cat No. 05CH37650, pp. 152–156 (ISBN: 0-7803-8989-1). Shay LK (2001) Upper ocean structure: Response to strong forcing events. In: Weller RA, Thorpe SA, and Steele J (eds.) Encyclopedia of Ocean Sciences, pp. 3100--3114. London: Academic Press. Shay LK and Uhlhorn EW (2008) Loop Current response to hurricanes Isidore and Lili. Monthly Weather Review 136 (doi: 10.1175/2008MWR2169).

Sturges W and Leben R (2000) Frequency of ring separations from the Loop Current in the Gulf of Mexico: A revised estimate. Journal of Physical Oceanography 30: 1814--1819. Teague WJ, Jarosz E, Carnes MR, Mitchell DA, and Hogan PJ (2006) Low frequency current variability observed at the shelf break in the northern Gulf of Mexico: May–October 2004. Continental Shelf Research 26: 2559--2582 (doi:10.1016/j.csr.2006.08.002). Vukovich FM (2007) Climatology of ocean features in the Gulf of Mexico using satellite remote sensing data. Journal of Physical Oceanography 37: 689--707. Walker N, Leben RR, and Balasubramanian S (2005) Hurricane forced upwelling and chlorophyll a enhancement within cold core cyclones in the Gulf of Mexico. Geophysical Research Letter 32: L18610 (doi: 10. 1029/2005GL023716).

Relevant Website http://www.remss.com – Remote Sensing Systems Home Page.

UPPER OCEAN MIXING PROCESSES J. N. Moum and W. D. Smyth, Oregon State University, Corvallis, OR, USA Copyright & 2001 Elsevier Ltd.

Introduction The ocean’s effect on weather and climate is governed largely by processes occurring in the few tens of meters of water bordering the ocean surface. For example, water warmed at the surface ona sunny afternoon may remain available to warm the atmosphere that evening, or it may be mixed deeper into the ocean not to emerge for many years, depending on near-surface mixing processes. Local mixing of the upper ocean is predominantly forced from the state of the atmosphere directly above it. The daily cycle of heating and cooling, wind, rain, and changes in temperature and humidity associated with mesoscale weather features produce a hierarchy of physical processes that act and interact to stir the upper ocean. Some of these are well understood, whereas others have defied both observational description and theoretical understanding. This article begins with an example of insitu measurements of upper ocean properties. These observations illustrate the tremendous complexity of the physics, and at the same time reveal some intriguing regularities. We then describe a set of idealized model processes that appear relevant to the observations and in which the underlying physics is understood, at least at a rudimentary level. These idealized processes are first summarized, then discussed individually in greater detail. The article closes with a brief survey of methods for representing upper ocean mixing processes in large-scale ocean models. Over the past 20 years it has become possible to make intensive turbulence profiling observations that reveal the structure and evolution of upper ocean mixing. An example is shown in Figure 1, which illustrates mixed-layer1 evolution, temperature

1 Strictly, a mixed layer refers to a layer of fluid which is not stratified (vertical gradients of potential temperature, salinity and potential density, averaged horizontally or in time, are zero. The terminology is most precise in the case of a convectively forced boundary layer. Elsewhere, oceanographers use the term loosely to describe the region of the ocean that responds most directly to

structure and small-scale turbulence. The small white dotsin Figure 1 indicate the depth above which stratification is neutral or unstable and mixing is intense, and below which stratification is stable and mixing is suppressed. This represents a means of determining the vertical extent of the mixed layer directly forced by local atmospheric conditions. (We will call the mixed-layer depth D.) Following the change in sign from negative (surface heating) to positive (surface cooling) of the surface buoyancy flux, Jb0 , the mixed layer deepens. (Jb0 represents the flux of density (mass per unit volume) across the sea surface due to the combination of heating/cooling and evaporation/ precipitation.) The mixed layer shown in Figure 1 deepens each night, butthe rate of deepening and final depth vary. Each day, following the onset of daytime heating, the mixed layer becomes shallower. Significant vertical structure is evident within the nocturnal mixed layer. The maximum potential temperature (y) is found at mid-depth. Above this, y is smaller and decreases toward the surface at the rate of about 2 mK in 10 m. The adiabatic change in temperature, that due to compression of fluid parcels with increasing depth, is 1 mKin 10 m. The region above the temperature maximum is superadiabatic, and hence prone to convective instability. Below this superadiabatic surface layer is a layer of depth 10–30 m in which the temperature change is less than 1 mK. Within this mixed layer, the intensity of turbulence, as quantified by the turbulent kinetic energy dissipation rate, e, is relatively uniform and approximately equal to Jb0 . (e represents the rate at which turbulent motions in a fluid are dissipated to heat. It is an important term in the evolution equation for turbulent kinetic energy, signifying the tendency for turbulence to decay inthe absence of forcing.) Below the mixed layer, e generally (but not always) decreases, whereas above, e increases by 1–2 factors of 10. Below the mixed layer is a region of stable stratification that partially insulates the upper ocean from the ocean interior. Heat, momentum, and chemical species exchanged between the atmosphere and the ocean interior must traverse the centimeters thick cool skin at thevery surface, the surface layer, and the

surface processes. Late in the day, following periods of strong heating, the mixed layer may be quite shallow (a few meters or less), extending to the diurnal thermocline. In winter and following series of storms, the mixed layer may extend vertically to hundreds of meters, marking the depth of the seasonal thermocline at midlatitudes.

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mixed layer to modify the stable layer below. These vertical transports are governed by a combination of processes, including those that affect only the surface itself (rainfall, breaking surface gravity waves), those that communicate directly from the surface throughout the entire mixed layer (convective plumes) or a good portion of it (Langmuir circulations) and also those processes that are forced at the surface but have effects concentrated at themixed-layer base (inertial shear, Kelvin–Helmholtz instability, propagating internal gravity waves). Several of these processes are represented in schematic form in Figure 2. Whereas Figure 1 represents the observed time evolution of the upper ocean at a single location, Figure 2 represents an idealized three-dimensional snapshot of some of the processes that contribute to this time evolution. Heating of the ocean’s surface, primarily by solar (short-wave) radiation, acts to stabilize the water column, thereby reducing upper ocean mixing. Solar radiation, which peaks at noon and is zero at night, penetrates the air–sea interface (limited by absorption and scattering to a few tens of meters), but heat is lost at the surface by long-wave radiation,

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evaporative cooling and conduction throughout both day and night. The ability of the atmosphere to modify the upper ocean is limited by the rate at which heat and momentum can be transported across the air–sea interface. The limiting factor here is theviscous boundary layer at the surface, which permits only molecular diffusion through to the upper ocean. This layer is evidenced by the ocean’s coolskin, a thin thermal boundary layer (a few millimeters thick), across which a temperature difference of typically 0.1 K is maintained. Disruption ofthe cool skin permits direct transport by turbulent processes across theair–sea interface. Once disrupted, the cool skin reforms over a period of some tens of seconds. A clear understanding of processes that disrupt the coolskin is crucial to understanding how the upper ocean is mixed.

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Thermo cline Breakin g interna l waves Figure 2 Diagram showing processes that have been identified by a widerange of observational techniques as important contributors to mixing the upperocean in association with surface cooling and winds. The temperature (y) profiles shown here have the adiabatic temperature (that due to compression of fluid parcels with depth) removed; thisis termed potential temperature. The profile of velocity shear (vertical gradient of horizontal velocity) indicates no shear in the mixed layer and nonzero shear above. The form of the shear in the surface layer is a current area of research. Shear-induced turbulence near the surface may be responsible for temperature ramps observed from highly resolved horizontal measurements. Convective plumes and Langmuir circulations both act to redistribute fluid parcels vertically; during convection, they tend to movecool fluid downward. Wind-driven shear concentrated at the mixed-layer base (thermocline) may be sufficient to allow instabilities to grow, from which internal gravity waves propagate and turbulence is generated. At the surface, breaking waves inject bubbles and highly energetic turbulence beneath the sea surface and disrupt the ocean’s cool skin, clearing a pathway for more rapid heat transfer into the ocean.

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by the local stratification in a process known as convection. Cooling occurs almost every night and also sometimes in daytime in association with weather systems such as cold air outbreaks from continental landmasses. Convection may also be causedby an excess of evaporation over precipitation, which increases salinity, and hence density, at the surface. Winds aid convection by a variety of mechanisms that agitate the sea surface, thereby disrupting the viscous sublayer and permitting rapid transfer of heat through the surface (see below). Convection in the ocean is analogous to that found in the daytime atmospheric boundary layer, which is heated from below, and which has been studied in great detail. Recourse to atmospheric studies of convection has helped in understanding the ocean’s behavior. Surface tension and viscous forces initially prevent dense, surface fluid parcels from sinking. Once the fluid becomes sufficiently dense, however, these forces are overcome and fluid parcels sinkin the form of convective plumes. The relative motion of the plumes helps to generate small-scale turbulence, resulting in a turbulent field encompassing a range of scales from the depth of the mixed layer (typically 100 m) to a few millimeters. A clear feature of convection created bysurface cooling is the temperature profile of the upper ocean (Figure 1). Below the cool skin is an unstable surface layer that is the signature of plume formation. Below that is a well-mixed layer in which density (as well as temperature and salinity) is relatively uniform. The depth of convection is limited bythe local thermocline. Mixing due to penetrative convection into the thermocline represents another source of cooling of the mixed layer above. Within the convecting layer, there is an approximate balance between buoyant production of turbulent kinetic energy and viscous dissipation, as demonstrated by the observation eEJb0 . The means by which the mixed layer is restratified following nighttime convection are not clear. Whereas someone-dimensional models yield realistic time series of sea surface temperature, suggesting that restratification is a one-dimensional process (see below), other studies of this issue have shown onedimensional processes to account for only 60% of the stratification gained during the day. It has been suggested that lateral variations in temperature, due to lateral variations in surface fluxes, or perhaps lateral variations in salinity due to rainfall variability, may be converted by buoyant forces into vertical stratification. These indicate the potential importance of three-dimensional processes to restratification.

Wind Forcing Convection is aided by wind forcing, in part because winds help to disrupt the viscous sublayer at the sea surface, permitting more rapid transport of heat through the surface. In the simplest situation, winds produce a surface stress and a sheared current profile, yielding a classic wall-layer scaling of turbulence and fluxes in the surface layer, similar to the surface layer of the atmosphere. (Theory, supported by experimental observations, predicts a logarithmic velocity profile and constant stress layer in the turbulent layer adjacent to a solid boundary. This is typically found in the atmosphere during neutral stratification and is termed wall-layer scaling.) This simple case, however, seems to berare. The reason for the difference in behaviors of oceanic and atmospheric surface layers is the difference in the boundaries. The lower boundary of the atmosphere is solid (at least over land, where convection is well-understood), but the ocean’s upper boundary is free to support waves, ranging from centimeter-scale capillary waves, through wind waves (10s of meters) to swell (100s of meters). Thesmaller wind waves lose coherence rapidly, and are therefore governed by local forcing conditions. Swell is considerably more persistent, and may therefore reflect conditions at a location remote in space and time from the observation, e.g., a distant storm. Breaking Waves

Large scale breaking of waves is evidenced at the surface by whitecapping and surface foam, allowing visual detection from above. This process, which is not at all well understood, disrupts the ocean’s cool skin, a fact highlighted by acoustic detection of bubbles injected beneath the sea surface by breaking waves. Small-scale breaking, which has no visible signature (and is even less well understood but is thought to be due to instabilities formed in concert with the superposition of smaller-scale waves) also disrupts the ocean’s cool skin. An important challenge for oceanographers is to determine the prevalence of small-scale wave breaking and the statistics of cool skin disruption at the sea surface. The role of wave breaking in mixing is an issue of great interest at present. Turbulence observations in the surface layer under a variety of conditions have indicated that at times (generally lower winds and simpler wave states) the turbulence dissipation rate (and presumably other turbulence quantities including fluxes) behaves in accordance with simple wall-layer scaling and is in this way similar to the atmospheric surface layer. However, under higher winds, and perhaps more complicated wave states, turbulence dissipation rates greatly exceed those

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predicted by wall-layer scaling. This condition has been observed to depths of 30 m, well below a significant wave height from the surface, and constituting a significant fraction of the ocean’s mixed layer. (The significant wave height is defined as the average height of the highest third of surface displacement maxima. A few meters is generally regarded as a large value.) Evidently, an alternative to wall-layer scaling is needed for these cases. This is a problem of great importance in determining both transfer rates across the air–sea interface to the mixed layer below and the evolution of the mixed layer itself. It is at times when turbulence is most intense that most of the air–sea transfers and most of the mixed layer modification occurs. Langmuir Circulation

Langmuir circulations are coherent structures within the mixed layer that produce counter rotating vortices with axes aligned parallel to the wind. Their surface signature is familiar as windrows: lines of bubbles and surface debris aligned with the wind that mark the convergence zones between the vortices. These convergence zones are sites of downwind jets in the surface current. They concentrate bubble clouds produced by breaking waves, or bubbles produced by rain, which are then carried downward, enhancing gas-exchange rates with the atmosphere. Acoustical detection of bubbles provides an important method for examining the structure and evolution of Langmuir circulations. Langmuir circulations appear to be intimately related to the Stokes drift, a small net current parallel to the direction of wave propagation, generated by wave motions. Stokes drift is concentrated at the surface and is thus vertically sheared. Small perturbations in the wind-driven surface current generate vertical vorticity, which is tilted toward the horizontal (downwind) direction by the shear of the Stokes drift. The result of this tilting is a field of counterrotating vortices adjacent to the ocean surface, i.e., Langmuir cells. It is the convergence associated with these vortices that concentrates the wind-driven surface current into jets. Langmuir cells thus grow by a process of positive feedback. Ongoing acceleration of the surface current by the wind, together with convergence of the surface current by the Langmuir cells, provides a continuous source of coherent vertical vorticity (i.e., the jets), which is tilted by the mean shear to reinforce the cells. Downwelling speeds below the surface convergence have been observed to reach more than 0.2 m s1, comparable to peak downwind horizontal flow speeds. By comparison, the vertical velocity scale associated with convection, w ¼ ðJb0 DÞ1=3 is

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closer to 0.01 m s1. Upward velocities representing the return flow to the surface appear to be smaller and spread over greater area. Maximum observed velocities are located well below the sea surface but also well above the mixed-layer base. Langmuir circulations are capable of rapidly moving fluid vertically, thereby enhancing and advecting the turbulence necessary to mix the weak near-surface stratification which forms in response to daytime heating. However, this mechanism does not seem to contribute significantly to mixing the base of the deeper mixed layer, which is influenced more by storms and strong cooling events. In contrast, penetration of the deep mixed layer base during convection (driven by the conversion of potential energy of dense fluid plumes created by surface cooling/evaporation into kinetic energy and turbulence) is believed to be an important means of deepening the mixed layer. So also is inertial shear, as explained next. Wind-Driven Shear

Wind-driven shear erodes the thermocline at the mixedlayer base. Wind-driven currents often veer with depth due to planetary rotation (cf. the Ekman spiral). Fluctuations in wind speed and direction result in persistent oscillations at near-inertial frequencies. Such oscillations are observed almost everywhere in the upper ocean, and dominate the horizontal velocity component of the internal wave field. Because near-inertial waves dominate the vertical shear, they are believed to be especially important sources of mixing at the base of the mixed layer. In the upper ocean, near-inertial waves are generally assumed to be the result of wind forcing. Rapid diffusion of momentum through the mixed layer tends to concentrate shear at the mixed layer base. This concentration increases the probability of small-scale instabilities. The tendency toward instability is quantified by the Richardson number, Ri ¼ N2/S2, where N2 ¼  (g/r)  dr/dz, represents the stability of the water column, and shear, S, represents an energy source for instability. Small values of Ri (o1/4) are associated with Kelvin–Helmholtz instability. Through this instability, the inertial shear is concentrated into discrete vortices (Kelvin–Helmholtz billows) with axes aligned horizontally and perpendicular to the current. Ultimately, the billows overturn and generate small-scale turbulence and mixing. Some of the energy released by the instabilities propagates along the stratified layer as high frequency internal gravity waves. These processes are depicted in Figure 2. The mixing of fluid from below the mixed layer by inertial shear contributes to increasing the density of the mixed-layer and to mixedlayer deepening.

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Temperature Ramps

Another form of coherent structure in the upper ocean has been observed in both stable and unstable conditions. In the upper few meters temperature ramps, aligned with the wind and marked by horizontal temperature changes of 0.1 K in 0.1 m, indicate the upward transport of cool/warm fluid during stable/unstable conditions. This transport is driven by an instability triggered by the wind and perhaps similar to the Kelvin–Helmholtz instability discussed above. It is not yet clearly understood. Because it brings water of different temperature into close contact with the surface, and also because it causes large lateral gradients, this mechanism appears to be a potentially important factor in near-surface mixing.

Effects of Precipitation Rainfall on the sea surface can catalyze several important processes that act to both accentuate and reduce upper oceanmixing. Drops falling on the surface disrupt the viscous boundary layer, andmay carry air into the water by forming bubbles. Rain is commonly said to ‘knock downthe seas.’ The evidence for this is the reduction in breaking wave intensity and whitecapping at the sea surface. Smaller waves (o20 cm wavelength) may be damped by subsurface turbulence as heavy rainfall actsto transport momentum vertically, causing drag on the waves. The reduced roughness of the small-scale waves reduces the probability of the waves exciting flow separation on the crests of the long waves, and hence reduces the tendency of the long waves to break. While storm winds generate intense turbulence near the surface, associated rainfall can confine this turbulence tothe upper few meters, effectively insulating the water below from surface forcing. This is due to the low density of fresh rainwater relative to the saltier ocean water. Turbulence must work against gravity to mix the surface water downward, and turbulent mixing is therefore suppressed. So long asvertical mixing is inhibited, fluid heated during the day will be trapped near the sea surface. Preexisting turbulence below the surface will continue to mixfluid in the absence of direct surface forcing, until it decays due to viscous dissipation plus mixing, typically over the time scale of a buoyancy period, N1. Deposition of pools of fresh water on the sea surface, such as occurs during small-scale squalls, raises some interesting prospects for both lateral spreading and vertical mixing of the fresh water. In the warm pool area of the western equatorial Pacific, intense squalls are common. Fresh light puddles at the surface cause the surface density field to be

heterogeneous. Release of the density gradient may then occur as an internal bore forming on the surface density anomaly, causing a lateral spreading of the fresh puddle. Highly resolved horizontal profiles of temperature, salinity and density reveal sharp frontal interfaces, the features of which depend on the direction of the winds relative to the buoyancy-driven current. These are portrayed in Figure 3. When the wind opposes the buoyancy current, the density anomaly at the surface is reduced, possibly as a result of vertical mixing in the manner suggested in Figure 3B(B). This mechanism results in a rapid vertical redistribution of fresh water fromthe surface pool and a brake on the propagation of the buoyancy front. Similarly, an opposing ambient current results in shear at the base of the fresh layer, which may lead to instability and consequent mixing. The nature of these features has yet to be clearly established, as has the net effect on upper ocean mixing.

Ice on the Upper Ocean At high latitudes, the presence of an icelayer (up to a few meters thick) partially insulates the oceana Wind

No convection Buoyancy-driven current

Dense water No entrainment

Wind drift current

(A) Wind

Buoyancy-driven current

Convection Entrainment

Dense water

Wind drift current

(B)

Figure 3 Two ways in which the frontal interface of a fresh surface poolmay interact with ambient winds and currents. (A) The case in which the buoyancy-driven current, wind and ambient current are all in the same direction. In this case, the buoyancydriven current spreads and thins unabated. In (B), the buoyancydriven current is opposed by wind and ambient current. In this case, the frontal interface of the buoyancy-driven current may plunge below the ambient dense water, so that convection near the surface intensifies mixing at the frontal interface. Simultaneously, shearforced mixing at the base of the fresh puddle may increase entrainment of dense water from below.

UPPER OCEAN MIXING PROCESSES

gainst surface forcing. This attenuates the effects of wind forcing on the upper ocean except at the lowest frequencies. The absence of surface waves prevents turbulence due to wave breaking and Langmuir circulation. However, a turbulence source is provided by the various topographic features found on the underside of the ice layer. These range in size from millimeter-scaledendritic structures to 10 m keels, and can generate significant mixing nearthe surface when the wind moves the ice relative to the water below or currents flow beneath the ice. Latent heat transfer associated with melting and freezing exerts a strong effect on the thermal structure of the upper ocean. Strong convection can occur under ice-free regions, in which the water surface is fully exposed to cooling and evaporative salinity increase. Such regions include leads (formed by diverging ice flow) and polynyas (where wind or currents remove ice as rapidly as it freezes). Convection can also be caused by the rejection of salt by newly formed ice, leaving dense, salty water near the surface.

Parameterizations of Upper Ocean Mixing Large-scale ocean and climate models are incapable of explicitly resolving the complex physics of the upper ocean,and will remain so for the foreseeable future. Since upper ocean processes are crucial in determining atmosphere–ocean fluxes, methods for their representation in large-scale models, i.e., parameterizations, are needed. The development of upper-ocean mixing parameterizations has drawn on extensive experience in the more general problem of turbulence modeling. Some parameterizations emphasize generality, working from first principles as much as possible, whereas others sacrifice generality to focus on properties specific to the upper ocean. An assumption common to all parameterizations presently in use is that the upper ocean is horizontally homogeneous, i.e., the goal is to represent vertical fluxes in terms of vertical variations in ocean structure, leaving horizontal fluxes to be handled by other methods. Such parameterizations are referred to as‘one-dimensional’ or ‘column’ models. Column modeling methods may be classified aslocal or nonlocal. In a local method, turbulent fluxes at a given depth are represented as functions of water column properties at that depth. For example, entrainment at the mixed-layer base may be determined solely by the local shear and stratification. Nonlocal methods allow fluxes to be influenced directly by remote events. For example, during nighttime

287

convection, entrainment at the mixed-layer base may be influenced directly by changes in the surface cooling rate. In this case, the fact that large convection rolls cannot be represented explicitly in a column model necessitates the nonlocal approach. Nonlocal methods include ‘slab’ models, in which currents and water properties do not vary at all across the depth of the mixed layer. Local representations may often be derived systematically from the equations of motion, whereas nonlocal methods tend tobe ad hoc expressions of empirical knowledge. The most successful models combine local and nonlocal approaches. Many processes are now reasonably wellrepresented in upper ocean models. For example, entrainment via shear instability is parameterized using the local gradient Richardson number and/or a nonlocal (bulk) Richardson number pertaining to the whole mixed layer. Other modeling issues are subjects of intensive research. Nonlocal representations of heat fluxes have resulted in improved handling of nighttime convection, but the corresponding momentum fluxes have not yet been represented. Perhaps the most important problem at present is there presentation of surface wave effects. Local methods are able to describe the transmission of turbulent kinetic energy generated at the surface into the ocean interior. However, the dependence of that energy flux on surface forcingis complex and remains poorly understood. Current research into the physics of wave breaking, Langmuir circulation, wave-precipitation interactions, and other surface wave phenomena will lead to improved understanding, and ultimately to useful parameterizations.

See also Breaking Waves and Near-Surface Turbulence. Bubbles. Deep Convection. Heat and Momentum Fluxes at the Sea Surface. Internal Tides. Langmuir Circulation and Instability. Penetrating Shortwave Radiation. Surface Gravity and Capillary Waves. Three-Dimensional (3D) Turbulence. Under-Ice Boundary Layer. Upper Ocean Vertical Structure. Whitecaps and Foam. Wave Generation by Wind.

Further Reading Garrett C (1996) Processes in the surface mixed layer of the ocean. Dynamics of Atmospheres and Oceans 23: 19--34. Thorpe SA (1995) Dynamical processes at the sea surface. Progress in Oceanography 35: 315--352.

LANGMUIR CIRCULATION AND INSTABILITY S. Leibovich, Cornell University, Ithaca, NY, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The surface of a wind-driven sea often is marked by streaks roughly aligned with the wind direction. These streaks, or windrows, are visible manifestations of coherent subsurface motions extending throughout the bulk of the ocean surface mixed layer, extending from the surface down to the seasonal thermocline. These may be regarded as the large scales of the turbulence in the mixed layer. Windrows and their subsurface origins were first systematically studied and described by Irving Langmuir in 1938, and the phenomenon since has become known as Langmuir circulation. The existence of a simple deterministic description making these large scales theoretically accessible distinguishes this problem from coherent structures in other turbulent flows. The theory traces these patterns to a convective instability mechanically driven by the wind waves and currents. Recent advances in instrumentation and computational data analysis have led to field observations of Langmuir circulation of unprecedented detail. Although the body of observational data obtained since Langmuir’s own work is mainly qualitative, ocean experiments now can yield quantitative measurements of velocity fields in the near-surface region. New measurement methods are capable of producing data comprehensive enough to characterize the phenomenon, and its effect on the stirring and maintenance of the mixed layer, although the labor and difficulties involved and the shear complexity of the processes occurring in the surface layer leave much work to be done before this can be said to be accomplished. Nevertheless, the combination of new experimental techniques and a simple and testable theoretical mechanism has stimulated rapid progress in the exploration of the stirring of the ocean surface mixed layer.

mechanical processes through the action of the wind, as Langmuir originally indicated. At the surface, rolls act to sweep surface water from regions of surface divergence overlying upwelling water into convergence zones overlying downwelling water. Floating material is collected into lines of surface convergence visible as windrows. In confined bodies of water, such as lakes and ponds, windrows are very nearly parallel to the wind, and can have a nearly uniform spacing as shown in Figure 2. In the open ocean, evidence indicates windrows tend to be oriented at small angles to the wind (typically to the right in the Northern Hemisphere), spacing is more variable, and individual windrows can be traced only for a modest multiple of the mean spacing. A windrow may either terminate, perhaps due to local absence of surface tracers, coalesce with an adjacent windrow, or split into two daughter windrows. Thus in the ocean, the general surface appearance is of a network of lines, occasionally interecting, yet roughly aligned with the wind. Windrows are visible in nature only when both Langmuir circulation and surface tracers are present. In the ocean, bubbles from breaking waves are the most readily available tracers, and Langmuir

z

Wind 0

〈w ′b ′〉0 < 0

2

 max = ∗ Λ∗u∗0 /|f |

Ekman layer

 max →  zpyc  max =Λ∗u∗0/|f | → RcLp

u∗p

z = zpyc →RcLp

Pycnocline

〈w ′b ′〉p

ρ

(A)

(B)

(C)

Figure 2 Schematic diagram of mixing length distributions in the UBL under conditions of (A) dynamically negligible surface buoyancy flux (neutral stratification in the well mixed layer), (B) upward buoyancy flux from summer melting, with formation of a seasonal pycnocline and a negative density gradient in the ‘well mixed’ layer, and (C) downward buoyancy flux from rapid freezing, with positive density gradient to the pycnocline. u * , Friction velocity; /w0 b0 S, buoyancy flux; k, Ka´rma´n’s constant, 0.4; L * , similarity constant, 0.028; Rc, critical flux Richardson number, 0.2; f, Coriolis parameter; L ¼ u 3 =ðk/w 0 b 0 SÞ, Obukhov length; Z * ¼ (1 þ L * u * / * kRc|f|L))1/2, stability parameter.

energy equation, which is dominated by three terms: production of TKE by shear ðPS ¼ tˆ  qU=qzÞ, production by buoyancy (Pb ¼  /w0 b0 S), and dissipation by molecular forces (e). Relating stress and shear by the mixing-length hypothesis, the balance of TKE production with dissipation is u3 =l  /w0 b0 S ¼ e *

½8

The negative ratio of buoyancy production to shear production is the flux Richardson number: Pb =PS ¼

l/w0 b0 S l ¼ 3 u kL

½9

*

where L ¼ u3 =ðk/w0 b0 SÞ is known as the Obukhov * length. Studies of turbulence in stratified flows have shown that the ratio eqn [9] does not exceed a limiting value (the critical flux Richardson number, Rc) of about 0.2. This establishes a limit for mixing length in stratified flow: lrRckL, and it is assumed that in the pycnocline this limit is approached, where

L is based on pycnocline fluxes of momentum and buoyancy. Estimates of mixing length in a near neutral UBL from the Ice Station Weddell data (Figure 1) are illustrated in Figure 3(A). Points marked lpeak were taken from the inverse of the wavenumber at the peak in the vertical velocity spectra (averaged over all 1-h flow realizations), as described above. Values marked le were obtained using eqn [8] assuming negligible buoyancy flux, with measured values for u * and e (obtained from spectral levels in the inertial subrange). They show clearly that the ‘wall layer’ scaling, l ¼ k|z| does not hold for depths greater than about 4 m. Rapid melting reduces the extent of the surface layer and the maximum mixing length (Figure 2B). The stability factor Z* ¼ ð1 þ L* u* =ðkRc jf jLÞÞ1=2 derives from similarity theory and ensures that the mixing length varies smoothly from the neutral limit ðlmax -L* u*0 =jf jÞ to the stable limit (lmax-kRcL0) for increasing stability. A consequence of reduced scales during melting is formation of a seasonal

332

UNDER-ICE BOUNDARY LAYER _

0

0

1

2

3

4

5

4

m2 s 2 (× 10 ) 0.4 0.8 1.2

m 0

_

1.6

0

m2 s 1 0.02 0.01

Kfit

az

 = oe

⎥ z⎥

0.03

_5

Klocal

_ 10

m



peak

_ 15

Ksim

_ 20

_ 25

_ 30 Turbulent length scale (A)

Turbulent stress magnitude (B)

Eddy viscosity (C)

Figure 3 (A) Mixing length determined from the TKE equation (le) and from the inverse of the wavenumber at the peak in the weighted w spectrum (lpeak). Error bars indicate twice the standard deviation from the spectra calculated from 1-h segments of data. (B) Average Reynolds stress magnitude, with a least-squares fitted exponential decay with depth. Fit coefficients are t0 ¼ 1.44  104 m2 s2 and a ¼ 0.051 m1. (C) Eddy viscosity estimated by three methods as described in the text. (Reproduced from McPhee MG and Martinson DG (1994) Science 263, 218–221.)

pycnocline, above a ‘trapped’ layer with properties indicative of the mixed layer that existed before the freshwater influx. Rapid ice growth produces negative buoyancy via enhanced salinity at the interface, increasing TKE by the buoyancy production term in eqn [8]. The result is that mixing length and eddy viscosity increase in the UBL, sometimes dramatically. During the 1992 Lead Experiment, turbulent flux and dissipation measured from the edge of a freezing lead in a forced convective regime showed that, compared with the neutral UBL, there was a tenfold increase in mixing length (based on w spectral peaks) and in eddy heat and salt diffusivity (based on measured fluxes and gradients). The Obukhov length was  12 m, about 40% of the mixed layer extent, indicating relatively mild convection, yet the turbulence was greatly altered, apparently by the generation of quasi-organized roll structures in the lead, reminiscent of Langmuir circulations (a thin ice cover precluded any surface waves at the time of the measurements). Mixing length inferred from the lead measurements increased away from the surface following Monin– Obukhov similarity (adapted from atmospheric boundary layer studies), reaching a maximum value roughly comparable to the pycnocline depth scaled by von Ka´rma´n’s constant.

The density profiles in Figure 2(B) and (C) are drawn schematically with slight gradients in the socalled mixed layer. This is at odds with conceptual models of the upper ocean which treat the boundary layer as completely mixed, but is consistent with measurements in the UBL. Wherever scalar fluxes of temperature and salinity are measurable, vertical gradients (albeit small) of mean temperature and salinity are found in the fully turbulent UBL, including statically unstable profiles as in Figure 2(C).

Effective Eddy Viscosity and Diffusivity

Figure 3(C) illustrates different methods for estimating bulk eddy viscosity in the UBL. The distribution labeled Ksim is from the similarity model used to construct the stress profile of Figure 1(B) by matching observed stress at 4 m. The vertical distribution labeled Klocal is the product lpeaku * at each level (Figure 3A and B). Its vertical average value is 0.019 m2 s1. Finally, the dashed line labeled Kfit in Figure 3(B) is from the least-squares fitted extinction coefficient ðRefdˆ gÞ for the Ekman stress solution eqn [2]. The last method is sensitive to small stress values at depth: if the bottommost cluster is ignored, Kfit ¼ 0.020 m2 s1.

UNDER-ICE BOUNDARY LAYER

The mixing length hypothesis holds for scalar properties of the UBL as well as momentum, so that it is reasonable to express, e.g., kinematic heat flux as /w0 T 0 S ¼ ; u* lT

qT qT ¼  KH qz qz

½10

In flows where turbulence is fully developed with large eddies and a broad inertial subrange, scalar eddy diffusivity and eddy viscosity are comparable (Reynold’s analogy). In stratified flows with internal wave activity and relatively low turbulence levels, momentum may be transferred by pressure forces that have no analog in scalar conservation equations, hence scalar mixing length may be considerably less than l. By measuring turbulent heat flux and the mean thermal gradient, it is possible to derive an independent estimate of eddy diffusivity in the UBL from eqn [10]. An example of this method is shown in Figure 4, where heat flux measurements averaged over five instrument clusters are compared with the negative thermal gradient. The data are from the same Ice Station Weddell storm as the other turbulence measurements of Figures 1 and 3. The mean thermal diffusivity, KH ¼ 0.018 m2 s1, is similar to the eddy viscosity (Figure 3C). Close correspondence between eddy viscosity and heat diffusivity was also found during the 1989 CEAREX drift north of Fram Strait, and during the 1992 LEADEX project. In the forced convective regime of the latter, salinity flux 270

20

_1

135

10

μK m

Wm

_2

15

5 0

0

_5 87.0

87.5

88.0

88.5

89.0

Day of 1992

Figure 4 Time series of turbulent heat flux, rcp /w 0 T 0 S(W m2, circles) and temperature gradient  qT=qz(mK m1 curve). The overbar indicates a vertical average over five turbulence clusters from 4 to 24 m. Error bars are twice the sample standard deviation. The temperature gradient was calculated by linear regression, after the calibration of each thermometer was adjusted by a constant amount so that the gradient was zero at time 86.95 when heat flux was zero (heavy arrow). (Reproduced from McPhee MG and Martinson DG (1994) Science 263: 218– 221.)

333

was measured for the first time, with comparably large values for eddy salt diffusivity as for eddy viscosity and heat diffusivity (but with low statistical significance for the regression of /w0 S0 S against qS/ qz).

Outstanding Problems Mixing in the Pycnocline

Understanding of turbulent mixing in highly stratified fluid just below the interface between the wellmixed layer and pycnocline is rudimentary. Many conceptual models assume, for example, that fluid ‘entrained’ at the interface immediately assumes the properties of the well-mixed layer (i.e., is mixed completely), so that the interface sharpens during storms as it deepens following the mean density gradient. Instead, measurements during severe storms in the Weddell Sea show upward turbulent diffusion of the denser fluid with a ‘feathering’ of the interface. Depending on how it is defined, the pycnocline depth may thus decrease significantly during extreme mixing events. Where the bulk stability of the mixed layer is low and there is large horizontal variability in pycnocline depth (as in the Weddell Sea), advection of horizontal density gradients may have large impact on mixing, both by changing turbulence scales and by conditioning the water column for equation-of-state related effects like cabbeling and thermobaric instability. Even with the advantage of the stable ice platform, observations in the upper pycnocline are hampered by the small turbulence scales, by the difficulty of separating turbulence from high frequency internal wave velocities, and by rapid migration of the interface in response to internal waves or horizontal advection. Convection in the Presence of Sea Ice

The cold, saline water that fills most of the abyssal world ocean originates from deep convection at high latitudes. Sea ice formation is a (geophysically) very efficient distillation process and may play a critical role in deep convection in areas like the Greenland, Labrador, and Weddell Seas where the bulk stability of the water column is low. By the same token, melting sea ice is a strong surface stabilizing influence that can rapidly shut down surface driven convection as soon as warm water reaches the well mixed layer from below. Understanding the physics of turbulent transfer in highly convective regimes is a difficult problem both from theoretical and observational standpoints, complicated not only by uncertainty about how

334

UNDER-ICE BOUNDARY LAYER

large-scale eddies interact with the stably stratified pycnocline fluid, but also by the possibility of frazil ice, small crystals that form within the water column. Depending on where it nucleates, frazil can represent a distributed internal source of buoyancy and heat in the UBL. Zones of intense freezing tend to be highly heterogeneous, concentrated in lead systems or near the ice margins, and require specialized equipment for studying horizontal structure. Measuring difficulties increase greatly in the presence of frazil ice or supercooled water, because any intrusive instruments present attractive nucleation sites. In addition to questions of UBL turbulence and surface buoyancy flux, factors related to nonlinearities in the equation of state for sea water may have profound influence on deep convection triggered initially by ice growth and UBL convection. Recent studies have shown, for example, that certain regions of the Weddell Sea are susceptible to thermobaric instability, arising from nonlinearity of the thermal expansion coefficient with increasing pressure. The importance of thermobaric instability for an ice-covered ocean is that once triggered, the potential energy released and converted in to turbulence as the water column overturns thermobarically, may be sufficient to override the surface buoyancy flux that would result from rapid melting as warm water reaches the surface.

friction velocity, square root of kinematic stress turbulence scale velocity ut V horizontal velocity vector convective turbulence scale velocity w* /w0 b0 S turbulent buoyancy flux, (g/r)/w0 r0 S /w0 T0 S kinematic turbulent heat flux /w0 S0 S turbulent salinity flux dˆ complex attenuation coefficient e dissipation rate of turbulent kinetic energy stability factor, (1 þ L * u * /(kRc|f|L))  1/2 Z* k von K`rmK`n’s constant (0.4) similarity constant (B 0.03) L* l turbulent mixing length scale turbulent scalar mixing length scale lT v kinematic molecular viscosity, units m2 s1 vT molecular scalar (thermal) diffusivity, units m2 s  1 t Reynolds stress: /u0 w0 S þ i/v0 w0 S F latitude

Nomenclature

Further Reading

f g K KH i L Pb PS q Rc S T u

Coriolis parameter acceleration of gravity eddy viscosity scalar eddy diffusivity imaginary number Obukhov length, u3 =ðk/w0 b0 SÞ * production rate of turbulent kinetic energy by buoyancy,  /w0 b0 S production rate of turbulent kinetic energy by shear, u3* /l turbulent kinetic energy scale velocity critical flux Richardson number (B 0.2) salinity temperature three-dimensional velocity vector (u, v, w components)

u*

See also Deep Convection. Ice-Ocean interaction. Internal Waves. Langmuir Circulation and Instability. Windand Buoyancy-Forced Upper Ocean.

Ekman VW (1905) On the influence of the earth’s rotation on ocean currents. Ark. Mat. Astr. Fys 2: 1--52. Gill AE (1982) Atmosphere–Ocean Dynamics. New York: Academic Press. Johannessen OM, Muench RD, and Overland JE (eds.) (1994) The Polar Oceans and Their Role in Shaping the Global Environment: The Nansen Centennial Volume. Washington DC: American Geophysical Society. McPhee MG (1994) On the turbulent mixing length in the oceanic boundary layer. Journal of Physical Oceanography 24: 2014--2031. Pritchard RS (ed.) (1980) Sea Ice Processes and Models. Seattle, WA: University of Washington Press. Smith WO (ed.) (1990) Polar Oceanography. San Diego, CA: Academic Press. Untersteiner N (ed.) (1986) The Geophysics of Sea Ice. New York: Plenum Press.

ICE–OCEAN INTERACTION J. H. Morison, University of Washington, Seattle, WA, USA M. McPhee, McPhee Research Company, Naches, WA, USA Copyright & 2001 Elsevier Ltd.

Introduction The character of the sea ice cover greatly affects the upper ocean and vice versa. In many ways icecovered seas provide ideal examples of the planetary boundary layer. The under-ice surface may be uniform over large areas relative to the vertical scale of the boundary layer. The absence of surface waves simplifies the boundary layer processes. However, thermodynamic and mechanical characteristics of ice–ocean interaction complicate the picture in unique ways. We discuss a few of those unique characteristics. We deal first with how momentum is transferred to the water and introduce the structure of the boundary layer. This will lead to a discussion of the processes that determine the fluxes of heat and salt. Finally, we discuss some of the unique characteristics imposed on the upper ocean by the larger-scale features of a sea ice cover.

Drag and Characteristic Regions of the Under-ice Boundary Layer

u 1 1 z ¼ ln z þ C ¼ ln u k k z0

To understand the interaction of the ice and water, it is useful to consider three zones of the boundary layer: the molecular sublayer, surface layer, and outer layer (Figure 1).Under a reasonably smooth and uniform ice boundary, these can be described on the basis of the influence of depth on the terms of the equation for a steady, horizontally homogeneous boundary layer (eqn [1]). ifV ¼

    @ @V @ @V n þ K  r1 rh p @z @z @z @z

diffusivity. The term nð@V=@zÞ is the viscous shear stress, where n is the kinematic molecular viscosity. The pressure gradient term, r1rhp is equal to r1 ð@p=@x þ i@p=@yÞ. The stress gradient term due to molecular viscosity is of highest inverse order in z. It varies as z2, and therefore dominates the stress balance in the molecular sublayer (Figure 1) where z is vanishingly small. As a result the viscous stress, nð@V=@zÞ, is effectively constant in the molecular sublayer, and the velocity profile is linear. The next layer away from the boundary is the surface layer. Here the relation between stress and velocity depends on the eddy viscosity, which is proportional to the length scale and velocity scale of turbulent eddies. The length scale of the turbulent eddies is proportional to the distance from the boundary, |z|. Therefore, the turbulent stress term varies as z1 and becomes larger than the viscous term beyond z greater than (1/k)(n/u*0), typically a fraction of a millimeter. The velocity scale in the surface layer is u*0, where ru20 is equal to t0, the average shear stress at the top of the boundary layer. Thus, K is equal to ku*0|z|, where Von Ka¨rma¨n’s constant, k, is equal to 0.4. Because the turbulent stress term dominates the equations of motion, the stress is roughly constant with depth in the surface layer. This and the linear z dependence of the eddy coefficient result in the log-layer solution or ‘law of the wall’ (eqn [2]).

½1

The coordinate system is right-handed with z positive upward and the origin at the ice under-surface. V is the horizontal velocity vector in complex notation (V ¼ u þ iv), r is water density, and p is pressure. An eddy diffusivity representation is used for turbulent shear stress, Kð@V=@zÞ ¼ V 0 w0 , where K is the eddy

½2

C ¼  (ln z0)/k is a constant of integration. Under sea ice the surface layer is commonly 1–3 m thick. The surface layer is where the influence of the boundary roughness is imposed on the planetary boundary layer. In the presence of under-ice roughness, the average stress the ice exerts on the ocean, t0, is composed partly of skin friction due to shear and partly of form drag associated with pressure disturbances around pressure ridge keels and other roughness elements. Observations under very rough ice have shown a decrease in turbulent stress toward the surface, presumably because more of the momentum transfer is taken up by pressure forces on the rough surface. The details of this drag partition are not known. Drag partition is complicated further for cases in which stratification exists at depths shallow compared to the depth of roughness elements.

335

ICE–OCEAN INTERACTION

S

〈w ′T ′〉0 = T dT / dz 〈w ′S ′〉0 = S dS/dz

N NV ≈ 0 Nz Nz

(

Molecular sublayer

U  =  du /dz

z0 = hs /30

(

Ice

0

Z0

N K NV ≈ 0 N z Nz

(

Ice

T

(

336

23°

 = u*20

Depth below ice (m) for u *0 = 0.01 m s

s|

v /u *0

|u| = 13.5u*0 45°

u / u*0

_ 0.1

Vice

10 _ 0.2

15

45° Outer layer

20

ifV ≈ N K NV Nz Nz

(

(

hs

_1

Surface layer

Z s| 5

Stress and velocity vectors in plan view

_ 0.3

25

 = zf /u*0 _5

_ 0.4 0

5

10

15

u / u*0 and v /u *0 Figure 1 Illustration of three regions of the planetary boundary layer under sea ice: molecular sublayer, surface layer, and outer layer. The velocity profiles are from the Rossby similarity solution (eqns [8], [9], [10] and [11]) for u*0 ¼ 0.01 m s1, z0 ¼ 0.06 m, Z* ¼ 1. The stress and velocity vector comparisons are from the same solution.

Then it also becomes possible to transfer momentum by internal wave generation. However, for many purposes t0 is taken as the turbulent stress evaluated at z0. Laboratory studies of turbulent flow over rough surfaces suggest that z0 may be taken equal to hs/30, where hs is the characteristic height of the roughness elements. In rare situations the ice surface may be so smooth that bottom roughness and form drag are not factors in the drag partition. In such a hydrodynamically smooth situation, the turbulence is generated by shear induced instability in the flow. The surface length scale, z0, is determined by the level of turbulent stress and is proportional to the molecular sublayer thickness according to the empirically derived relation z0 ¼ 0.13(n/u*0). In the outer layer farthest from the boundary, the Coriolis and pressure gradient terms in eqn [1], which have no explicit z dependence, are comparable to the turbulent stress terms. The presence of the Coriolis term gives rise to a length scale, h, for the outer boundary layer equal to u*0/f under neutral stratification. This region is far enough from the boundary so that the turbulent length scale becomes

independent of depth and in neutral conditions has been found empirically to be l ¼ xnu*0 /f, where xn is 0.05. For neutral stratification, u* and h are the independent parameters that define the velocity profile over most of the boundary layer. The ratio of the outer length scale to the surface region length scale, z0, is the surface friction Rossby number, R0 ¼ u*0/(z0f). Solutions for the velocity in the outer layer can be derived for a wide range of conditions if we nondimensionalize the equations with these Rossby similarity parameters, u*0 /f and u*0. However, the growth and melt of the ice produce buoyancy flux that strongly affects mixing. Melting produces a stabilizing buoyancy flux that inhibits turbulence and contracts the boundary layer. Freezing causes a destabilizing buoyancy flux that enhances turbulence and thickens the boundary layer. We can account for the buoyancy flux effect by adjusting the Rossby parameters dealing with length scale. We define the scale of the outer boundary layer as hm ¼ u0 Z =f . If the mixing length of the turbulence in the outer layer is lm ¼ xn u0 Z2 =f , it interpolates in a reasonable way between known values of lm for neutral stratification

ICE–OCEAN INTERACTION

(xnu*0/f) and stable stratification (RcL) if Z* is given as eqn [3]. Z ¼

  x u 1 1=2 1þ n f Rc L

½3

Rc is the critical Richardson number; the Obukhov length, L, is the ratio of shear and buoyant production of turbulent energy, ru30 =kg/r0 w0 S; and  /r0 w0 Sg/r is the turbulent buoyancy flux. With this Rossby similarity normalization of the equations of motion, we can derive analytical expressions for the under-ice boundary layer profile that are applicable to a range of stratification. For large |z|, V will approach the free stream geostrophic velocity, V¯ g ¼ Ug þ iVg ¼ f 1 r1 rh p. Here we will assume this is zero. However, surface stress-driven absolute velocity solutions can be superimposed on any geostrophic current. We also ignore the time variation and viscous terms and define a normalized stress equal to S ¼ ðK@V=@zÞ=u20 . The velocity is nondimensionalized by the friction velocity and the boundary layer thickness, U ¼ Vfhm =u20 , and depth is nondimensionalized by the boundary layer thickness scale, z ¼ z/hm. With these changes eqn [1] becomes eqn [4]. iU ¼ @S=@z

337

Eqn [8] attenuates and rotates (to the right in the Northern Hemisphere) with depth. It duplicates the salient features found in data and sophisticated numerical models. In the outer layer, eqns [5] and [8] are satisfied for nondimensional velocity given by eqn [10]. ˆ U ¼ idˆ ed z

for zrz0

½10

Thus the velocity is proportional to stress but rotated 451 to the right. As we see in the derivation of the law of the wall [2], the surface layer variation of the eddy viscosity with depth is critical to the strong shear present there. Thus eqn [10] will not give a realistic profile in the surface layer. We define the nondimensional surface layer thickness, zsl, as the depth where the surface layer mixing length, |z|, becomes equal to the outer layer mixing length, lm ¼ xn u0 Z2 =f . We find zsl is equal to  Z*xn and applying the definition [6] gives K* as K*sl ¼  kz/Z* in the surface layer. If we approximate the stress profile [8] by a Taylor series, we can integrate [5] with K*sl substituted for K* to obtain the velocity profile in the surface layer.  

Z zsl ˆ ln þ d ðzsl  zÞ UðzÞ  Uðzsl Þ ¼ k z0

for z  z0

½4

½11

In terms of nondimensional variables the constitutive law is given by eqn [5].

Eqn [11] is analogous to [2] except for the introduction of the dˆðzsl  zÞ term. This is the direct result of accounting for the stress gradient in the surface layer. This term is small compared to the logarithmic gradient. Figure 1 illustrates the stress and velocity vectors at various points in the boundary layer as modeled by eqns [8] through [11]. For neutral conditions the nondimensional boundary layer thickness is typically 0.4 (dimensional thickness is 0.4u*/f). Through the outer layer, the velocity vector is 451 to the right of the stress vector as a consequence of the idˆ peið451Þ multiplier in [10]. As the ice surface is approached through the surface layer, the stress vector rotates 10–201 to the left to reach the surface direction. However, in the surface layer the velocity shear in the direction of the surface stress is great because of the logarithmic profile. Thus, as the surface is approached, the velocity veers to the left twice as much as stress. At the surface the velocity is about 231 to the right of the surface stress. It is commonly useful to relate the stress on underice surface to the relative velocity between ice and water a neutral-stratification drag coefficient,

S ¼ K @U=@z

½5

The nondiemensional eddy coefficient is given by eqn [6]. K ¼ ku0 lm =fh2m ¼ kxn

½6

Eqn [6] is the Rossby similarity relation that is the key to providing similarity solutions for stable and neutral conditions. It even provides workable results for slightly unstable conditions. Eqns [4] and [5] can be combined in an equation for nondimensionalized stress (eqn [7]). ði=K ÞS ¼ dSd=z

½7

This has the solution eqns [8]. ˆ

S ¼ ed z

½8

dˆ ¼ ði=K Þ1=2

½9

338

ICE–OCEAN INTERACTION

2 ru20 ¼ rCz VðzÞ where Cz is the drag coefficient for depth z. If z is in the log-layer, eqn [2] can be used to derive the relation between ice roughness and the drag coefficient. We find that Cz ¼ k2[ln(z/z0)]2. Clearly values of the drag coefficient can vary widely depending on the under-ice roughness. Typical values of z0 range from 1 to 10 cm under pack ice. A commonly referenced value for the Arctic is 6 cm, which produces a drag coefficient at the outer edge of the log layer of 9.4  103 (Figure 1). If the reference depth is outside the log layer, the drag coefficient formulation is poorly posed because of the turning in the boundary layer. For neutral conditions, eqns [10] and [11] can be used to obtain a Rossby similarity drag law that yields the nondimensional surface drift relative to the geostrophic current for unit nondimensional surface stress (eqn [12]).

U0 ¼

V0 1 ¼ ð½lnðR0 Þ  A  iBÞ u0 k

½12

Here sffiffiffiffiffiffiffiffi rffiffiffiffiffiffi! k xn þ A ¼ 1  ln xn  D2:2 2k 2xn sffiffiffiffiffiffiffiffi rffiffiffiffiffiffi k xn D2:3 þ B ¼ 2k 2xn

represents isostatic adjustment to runoff of surface melt and percolation of water through the ice cover. In an infinitesimal control volume following the ice–ocean interface, conservation of heat and salt may be expressed in kinematic form as eqns [14] and [15]. q_ ¼ /w0 T 0 S0  w0 QL ðwith units K m s1 Þ

ðw0 þ wi ÞðS0  Sice Þ ¼ /w0 S0 S0 ðwith units psu m s1 Þ ½15 where q_ ¼ Hice =ðrcp Þ is flux (Hice) conducted away from the interface in the ice; r is water density; cp is specific heat of seawater; /w0 T0 S0 is the kinematic turbulent heat flux from the ocean; QL is the latent heat of fusion (adjusted for brine volume) divided by cp; S0 is salinity in the control volume, Sice is ice salinity, and /w0 S0 S0 is turbulent salinity flux. Fluid in the control volume is assumed to be at its freezing temperature, approximated by the freezing line (eqn [16]). T0 ¼ mS0

½13

This Rossby similarity drag law for outside the surface layer results in a surface stress that is proportional to V1.8 rather than V2, a result that is supported by observational evidence, and can be significant at high velocities.

Heat and Mass Balance at the Ice–Ocean Interface: Wintertime Convection The energy balance at the ice–ocean interface not only exerts major influence over the ice mass balance but also dictates the seasonal evolution of upper ocean salinity and temperature structure. At low temperature, water density is controlled mainly by salinity. Salt is rejected during freezing, so that buoyancy flux from basal growth (or ablation), combined with turbulent mixing during storms, determines the depth of the well-mixed layer. Vertical motion of the ice–ocean interface depends on isostatic adjustment as the ice melts or freezes. The interface velocity is w0 þ wi where w0 ¼ ðrice =rÞh_ b ; h_ b is the basal growth rate, and wi

½14

½16

By standard closure, turbulent fluxes are expressed in terms of mean flow properties (eqns [17] and [18]). /w0 T 0 S0 ¼ ch u0 dT

½17

hw0 S0 i0 ¼ cS u0 dS

½18

u*0 is the square root of kinematic turbulent stress at the interface (friction velocity); dT ¼ T  T0 and dS ¼ S  S0 are differences between far-field and interface temperature and salinity; and ch and cS are turbulent exchange coefficients termed Stanton numbers. The isostatic basal melt rate, w0 is the key factor in interface thermodynamics, and in combination with wi it determines the salinity flux. A first-order approach to calculating w0 that is often sufficiently accurate (relative to uncertainties in forcing parameters) when melting or freezing is slow, is to assume that S0 ¼ S, the far-field salinity, and that ch is constant. Combining [14], [16], and [17] gives eqn [19]. w0 ¼

ch u0 ðT þ mSÞ  q_ QL

½19

Salinity flux is determined from [15]. Note the cS is not used, and that this technique fixes (unrealistically)

ICE–OCEAN INTERACTION

the temperature at the interface to be the mixed layer freezing temperature. A more sophisticated approach is required when melting or freezing is intense. Manipulation of [14] through [18] produces a quadratic equation for w0 (eqn [20]). SL 2 w þ ðST þ SL cS  Sice Þw0 þ ðu0 cS þ wi ÞST u0 0 þ u0 cS S  wi Sice ¼ 0  ST ¼

 q_  T =m ch u0

and

SL ¼ QL =ðmch Þ

½20

½21

Here ch and cS (turbulent Stanton numbers for heat and salt) are both important and not necessarily the same. Melting or freezing will decrease or increase S0 relative to far-field salinity, with corresponding changes in T0. The Marginal Ice Zone Experiments (MIZEX) in the 1980s showed that existing ice–ocean turbulent transfer models overestimated melt rates by a wide factor. It became clear that the rates of heat and mass transfer were less than momentum transfer (by an order of magnitude or more), and were being controlled by molecular effects in thin sublayers adjacent to the interface. If it is assumed that the extent of the sublayers is proportional to the bottom roughness scale, z0, then dimensional analysis suggests that the Stanton numbers (nondimensional heat and salinity flux) should depend mainly on two other dimensionless groups, the turbulent Reynolds number, Re* ¼ u*0z0/n, where n is molecular viscosity, and the Prandtl (Schmidt) numbers, n/nT(S), where nT and nS are molecular diffusivities for heat and salt. Laboratory studies of heat and mass transfer over hydraulically rough surfaces suggested approximate expressions for the Stanton numbers of the form shown in eqn [22]. chðSÞ ¼

  /w0 TðSÞ0 S0 n 2=3 pðRe Þ1=2 u0 dTðSÞ nTðSÞ

½22

The Stanton number, ch, has been determined in several turbulent heat flux studies since the original MIZEX experiment, under differing ice types with z0 values ranging from less than a millimeter (eastern Weddell Sea) to several centimeters (Greenland Sea MIZ). According to [22], ch should vary by almost a factor of 10. Instead, it is surprisingly constant, ranging from about 0.005 to 0.006, implying that the Reynolds number dependence from laboratory results cannot be extrapolated directly to sea ice.

339

If the Prandtl number dependence of [14] holds, the ratio ch /cS ¼ (nh/nS)2/3 is approximately 30. Under conditions of rapid freezing, the solution of [20] with this ratio leads to significant supercooling of the water column, because heat extraction far outpaces salt injection in what is called double diffusion. This result has caused some concern. Because the amount of heat represented by this supercooling is substantial, it has been hypothesized that ice may spontaneously form in the supercooled layer and drift upward in the form of frazil ice crystals. This explanation has not been supported by ice core sampling, which shows no evidence of widespread frazil ice formation beyond that at the surface of open water. The physics of the freezing process suggest that the seeming paradox of the supercooled boundary layer may be realistic without spontaneous frazil formation. When a parcel of water starts to solidify into an ice crystal, energy is released in proportion to the volume of the parcel. At large scales this manifests itself as the latent heat of fusion. However, as the parcel solidifies, energy is also required to form the surface of the solid. This surface energy penalty is proportional to the surface area of the parcel and depends on other factors including the physical character of any nucleating material. In any event, if the parcel is very small the ratio of parcel volume to surface area will be so small that the energy released as the volume solidifies is less than the energy needed to create the new solid surface. For this reason, ice crystals cannot form even in supercooled water without a nucleating site of sufficient size and suitable character. In the clean waters of the polar regions, the nearest suitable site may only be at the underside of the ice cover where the new ice can form with no nucleation barrier. Therefore, it is possible to maintain supercooled conditions in the boundary layer without frazil ice formation. Furthermore, recent results suggest that supercooling in the uppermost part of the boundary layer may be intrinsic to the ice formation process. Sea ice is a porous mixture of pure ice and high-salinity liquid water (i.e., brine). The bottom surface of a growing ice floe consists of vertically oriented pure ice platelets separated by vertical layers of concentrated brine. This platelet–brine sandwich (on edge) structure is on the scale of a fraction of a millimeter, and its formation is controlled by molecular diffusion of heat and salt. The low solid solubility of the salt in the ice lattice results in an increase of the salinity of water in the layer above the advancing freezing interface. Because heat diffuses more rapidly than salt at these scales, the cold brine tends to

340

ICE–OCEAN INTERACTION

supercool the water below the ice–water interface. With this local supercooling, any disturbance of the ice bottom will tend to grow spontaneously. The conditions of sea ice growth are such that this instability is always present. Continued growth results in additional rejection of salt, some fraction of which is trapped in the brine layers, and consequently the interfacial region of the ice sheet continues to experience constitutional supercooling. Also, anisotropy in the molecular attachment efficiency intrinsic to the crystal structure of the ice platelets creates an additional supercooling in the interfacial region. The net result is that heat is extracted from the top of the water column at the rate needed to maintain its temperature near but slightly below the equilibrium freezing temperature as salt is added. This and the convective processes in the growing ice may imply that ch/cS ¼ 1 during freezing. The situation with melting may be quite different, since the physical properties of the interface change dramatically. Observations to date suggest that ch remains relatively unchanged with variable ice type and mixed layer temperature elevation above freezing. A value of 5.5  103 is representative. cS is not so well known, since direct measurements of /w0 S0 S are relatively rare. The dependence of the exchange coefficients on Prandtl and Schmidt numbers is not clear, and will only be resolved with more research.

Effects of Horizontal Inhomogeneity: Wintertime Buoyancy Flux Although the under-ice surface may be homogeneous over ice floes hundreds of meters in extent, the key fluxes of heat and salt are characteristically nonuniform. As ice drifts under the action of wind stress, the ice cover is deformed. Some areas are forced together, producing ridging and thick ice, and some areas open in long, thin cracks called leads. In special circumstances the ice may form large, unit-aspectratio openings called polynyas. In winter the openings in the ice expose the sea water directly to cold air without an intervening layer of insulating sea ice. This results in rapid freezing. As the ice forms, it rejects salt and results in unstable stratification of the boundary layer beneath open water or thin ice. These effects are so important that, even though such areas may account for less than 10% of the ice cover, they may account for over half the total ice growth and salt flux to the ocean. Thus the dominant buoyancy flux is not homogeneous but is concentrated in narrow bands or patches. Similarly, in the summer solar radiation is reflected from the ice but is nearly

completely absorbed by open water. Fresh water from summertime surface melt tends to drain into leads, making them sources of fresh water flux as well. The effect of wintertime convection in leads is illustrated in Figure 2. It shows two extremes in the upper ocean response. Figure 2A shows what we might expect in the case of a stationary lead. As the surface freezes, salt is rejected and forms more dense water that sinks under the lead. This sets up a circulation with fresh water flowing in from the sides near the surface and dense water flowing away from the lead at the base of the mixed layer. Figure 2B illustrates the case in which the lead is embedded in ice moving at a velocity great enough to produce a well-developed turbulent boundary layer (e.g. 0.2 m s1). If the mixed layer is fully turbulent, the cellular convection pattern may not occur; rather, the salt rejected at the surface may simply mix into the surface boundary layer. The impact of nonhomogeneous surface buoyancy flux on the boundary layer can also be characterized by the equations of motion. The viscous terms in eqn [1] can be neglected at the scales we discuss here, but the possibility of vertical motion associated with large-scale convection requires that we include the vertical component of velocity. For steady state we have eqn [23].   @ @ V¯ ¯ ¯ ¯ ¯ ½23 K  r1 rp VdrV þ f  V ¼ @z @ V¯ is the velocity vector including the mean vertical velocity w; f¯ is the Coriolis parameter times the vertical unit vector. The advective acceleration term, ¯ ¯ and pressure gradient term are necessary to Vdr V, account for the horizontal inhomogeneity that is caused by the salinity flux at the lead surface. The condition that separates the free convection regime of Figure 2A and the forced convection regime of Figure 2B is expressed by the relative magnitude of the pressure gradient, r1rhp, and turbulent stress, @=@zðK@V=@zÞ, terms in [23]. This ratio can be derived with addition of mass conservation and salt conservation equations, and if we assume the vertical equation is hydrostatic, @p=@z ¼ gr ¼ gMS, where M is the sensitivity of density to salinity. If we nondimensionalize the equations by the ice velocity Ui, mixed-layer depth, d, average salt flux at the lead surface, FS, and friction velocity, u*0, the ratio of the pressure gradient term to the turbulent stress term scales as eqn [24]. L0 ¼

gMFS d r0 Ui u20

½24

ICE–OCEAN INTERACTION

Free convection

Vice ~ 0

z

Forced convection

Lead

y

341

Lead

LL

Vice u*

x

FS

du (x)

u (z )

Thick sea ice

dd (x)

Increased S

dml u (z )

Internal boundary layers, depth = d (x )

 (z )

(A)

(B)

Figure 2 Modes of lead convection. (A) The free convection pattern that results when freezing and salt flux are strong, and the relative velocity of the ice is low. Cellular patterns of convective overturning are driven by pressure gradients that arise from the salinity distrurbance due to ice formation. (B) The forced convection regime that exists when ice motion is strong. The salinity flux and change in surface stress in the lead cause a change in the character of the boundary layer that grows deeper downstream. The balance of forces is primarily Coriolis and turbulent diffusion of momentum. (From Morison JH, MCPhee MG, Curtin T and Paulson CA (1992) The oceanography of winter leads. Journal of Geophysical Research 97: 11199–11218.)

autonomous underwater vehicle. Using the vehicle vertical motion as a proxy for vertical water velocity, it is also possible to estimate the salt flux w0 S0 . The lead was moving at 0.04 m s1, and estimates of salt flux put L0 between 4 and 11 (free convection in Figure 3). Salinity increased in the downstream direction across the lead and reached a sharp maximum Water temperature _ air temperature (˚C)

If this lead number is small because the ice is moving rapidly or the salt flux is small, the pressure gradient term is not significant in [23]. In this forced convection case, illustrated in Figure 2B, the boundary layer behaves as in the horizontally homogeneous case except that salt is advected and diffused away from the lead in the turbulent boundary layer. If the lead number is large because the ice is moving slowly or the salt flux is large, the pressure gradient term is significant. In this free convention case the salinity disturbance is not advected away, but builds up under the lead. This creates pressure imbalances that can drive the type of cellular motion shown in Figure 2A. Figure 3 shows conditions for which the lead number is unity for a range of ice thickness. Here the salt flux has been parametrized in terms of the air– sea temperature difference, and stress has been parametrized in terms of Ui. The figure shows the locus of points where L0 is equal to unity. For typical winter and spring conditions, L0 is close to 1, indicating that a mix of free and forced convection is common. Conditions where lead convection features have been observed are also shown in Figure 3. Most of these are in the free convection regime, probably because they are more obvious during quiet conditions. There have been several dedicated efforts to study the effects of wintertime lead convection. The most recent example was the 1992 Lead Experiment (LeadEx) in the Beaufort Sea. Figure 4 illustrates the average salinity profile at 9 m under a nearly stationary lead. The data was gathered with an

30 20 15 10 hi = 5 cm

′71

25

A3

20

A4

′92 Lead 3

′92 Lead 4

h i = 0 cm A

15 Free convection

10

Forced convection

5

L0 = 1 L0 t = 1

0 0

0.02

0.04 0.06 0.08

0.10 0.12

0.14 0.16

_

Ice velocity (m s 1) Figure 3 Air–water temperature difference versus Ui for L0 equal to 1 for various ice thicknesses, hi. Also shown are the temperature difference and ice velocity values for several observations of lead convection features such as underice plumes. Most of these are in the free convection regime: 0 71 denotes the AIDJEX pilot study; A3 denotes the 1974 AIDJEX Lead Experiment – lead 3 (ALEX3); A4 denotes ALEX4; A denotes the 1976 Arctic Mixed Layer Experiment; and 0 92 Lead 4 denotes the 1992 LeadEx lead 4. LeadEx lead 3 (0 92 Lead 3) was close to L0 ¼ 1. (From Morison JH, McPhee MG, Curtin T and Paulson CA (1992) The oceanography of winter leads. Journal of Geophysical Research 97: 11199–11218.)

342

ICE–OCEAN INTERACTION

ICE

ICE

LEAD

2

0

Relative current

_

Salt flux down (10 5 kg m 2 s 1)

_3

Salinity fluctuation (10 PSU)

4

_

_

~0.02−0.04 m s

_1

Lead average

5

= 6.0 × 10

_6

0

_ 250

_ 200

_ 150

_ 100

_ 50

0

50

100

150

200

Cross-lead distance (m) Figure 4 Composites of S 0 and w 0 S 0 at 9 m depth measured with an autonomous underwater vehicle during four runs under lead 4 at the 1992 Lead Experiment. The horizontal profile data have been collected in 1 m bins. (From Morison JH, McPhee MG (1998) Lead convection measured with an autonomous underwater vechicle, Journal of Geophysical Research 103: 3257–3281.)

at the downstream edge. The salt flux was highest near the lead edges, but particularly at the downstream edge. With even a slight current, the downstream edge plume is enhanced by several factors. The vorticity in the boundary layer reinforces the horizontal density gradient at the downstream edge and counters the gradient at the upstream edge. The salt excess is greatest at the downstream edge by virtue of the salt that is advected from the upstream lead surface. The downstream edge plume is also enhanced by the vertical motion of water at the surface due to water the horizontal flow being forced downward under the ice edge. Figure 5 shows the salt flux beneath a 1000 m wide lead moving at 0.14 m s1 with L0 equal to about 1 (Figure 3). Here the salt flux is more evenly spread under the lead surface. The salt flux derived from the direct w0 S0 correlation method does show some enhancement at the lead edge. This may be partly due to the influence of pressure gradient forces and the reasons cited for the free convection case described above. The other factor that influences the convective pattern is the lead width. In the case of the 100 m lead in even a weak current, the convection may not be fully developed until the downstream edge is reached. For the 1000 m lead of the second case, the convection under the downstream portion of the

lead was a fully developed unstable boundary layer. The energy-containing eddies filled the mixed layer and their dominant horizontal wavelength was equal to about twice the mixed layer depth.

Effects of Horizontal Inhomogeneity: Summertime Buoyancy Flux The behavior of the boundary layer under summer leads is relatively unknown compared to the winter lead process. Because of the important climate consequences, it is a subject of increasing interest. Summertime leads are thought to exhibit a critical climate-related feature of air–sea–ice interaction, icealbedo feedback. This is because leads are windows that allow solar radiation to enter the ocean. The proportion of radiation that is reflected (albedo) from sea ice and snow is high (0.6–0.9) while that from open water is low (0.1). The fate of the heat that enters summer leads is important. If it penetrates below the draft of the ice, it warms the boundary layer and is available to melt the bottom of the ice over a large area. If most of the heat is trapped in the lead above the draft of the ice, it will be available to melt small pieces of ice and the ice

ICE–OCEAN INTERACTION

30

343

LEAD

_6

_2

_1

Salt flux (10 kg m s )

25 20 Salt flux from Ebp-IDM

15 10

Lead average flux _6 7.762 × 10

Downstream average _ flux = 3.362 × 10 6

5 0 w ′S ′ salt flux, 56-m bin composite

_5

−6

average = 9.41× 10 , lead average = 1.19 ×10

−5

_ 10 _ 300

_ 200

_ 100

0

100

200

300

400

Distance upstream from the lead edge (m) Figure 5 Composite average for autonomous underwater vehicle runs 1 to 5 at lead 3 of the 1992 Lead Experiment. The salinity is band-passed at 1 rad m1 and is indicative of the turbulence level and is used to estimate the salt flux by the Ebp  IDM method of Morison and McPee (1998). The salt flux is elevated in the lead and decreases beyond about 72 m downstream of the lead edge. The composite average of w 0 S 0 in 56 m bins for the same runs is also shown in the center panel. The average flux and the decrease downstream are about the same as given by the Ebp  IDM method, but w 0 S 0 suggests elevated fluxes near the lead edge. (From Morison JH, McPhee MG (1998) Lead convection measured with an autonomous underwater vechicle, Journal of Geophysical Research 103: 3257–3281.)

floe edges. In the latter case the area of ice will be reduced and the area of open water increased. This allows even more solar radiation to enter the upper ocean, resulting in a positive feedback. This process may greatly affect the energy balance of an ice-covered sea. The critical unknown is the partition of heating between lateral melt of the floe edges and bottom melt. There are fundamental similarities between the summertime and wintertime lead problems. The equations of motion ([15]–[24]) are virtually identical. Only the sign of the buoyancy flux is opposite. The heat flux is important to summer leads and tends to decrease the density of the surface waters. However, as with winter leads, the buoyancy flux is controlled mainly by salt. As the top surface of the ice melts, much of the water that does not collect in melt ponds on the ice surface instead runs into the leads. If the ambient ice velocity is low, ice melt from the bottom surface will tend to flow upward and collect in the leads as well. Thus leads are the site of a concentrated flux of fresh water accumulated over large areas of ice. If this flux, FS, into the lead is negative enough relative to the momentum flux represented by u*0, the lead number, L0, will be a large negative number and shear production of turbulent energy will not be able to overcome the stabilizing buoyant production. This means turbulent mixing will be weak beneath the lead surface and a layer of fresh water will accumulate near the surface of the lead. The stratification

at the bottom of this fresh water layer may be strong enough to prevent mixing until a storm produces a substantial stress. This will be made even more difficult than in the winter situation because of the effect of stabilizing buoyancy flux on the boundary layer generally. The only way the fresh water will be mixed downward is by forced convection; there is no analogue to the wintertime free convection regime. When there is sufficient stress to mix out a summertime lead, the pattern must resemble that of the forced convection regime in Figure 2A. At the upstream edge of the lead, fresh warm water will be mixed downward in an internal boundary layer that increases in thickness downstream until it reaches the steady-state boundary layer thickness appropriate for that buoyancy flux or the ambient mixed layer depth. The rate of growth should scale with the local value of u*0 (or perhaps u*0Z*). At the downstream edge, another boundary layer conforming to the under-ice buoyancy flux and surface stress will begin to grow at a rate roughly scaling with the local u*0. In spite of the generally stabilizing buoyancy flux, this process has the effect of placing colder, more saline water from under the ice on top of fresher and warmer (consequently lighter) water drawn from the lead. Thus, even embedded in the stable summer boundary layer, the horizontal inhomogeneity due to leads may create pockets of instability and more rapid mixing than might be expected on the basis of average conditions.

344

ICE–OCEAN INTERACTION

Recent studies of summertime lead convection at the 1997–98 Surface Heat Budget of the Arctic experiment saw the salinity decrease in the upper 1 m of leads to near zero and temperatures increase to more than 01C. Only when ice velocities were driven by the wind to speeds of nearly 0.2 m s1 were these layers broken down and the fresh, warm water mixed into the upper ocean. At these times the heat flux measured at 5 m depth reached values over 100 W m2. The criteria for the onset of mixing are being studied along with the net effect of the growing internal boundary layers. Even with an understanding of the mixing process, it will be a challenge to apply this information to larger-scale models, because the mixing is nonlinearly dependent on the history of calm periods and strong radiation.

Internal Waves and Their Interaction with the Ice Cover One of the first studies of internal waves originated with observations made by Nansen during his 1883 expedition. It did not actually involve interaction with the ice cover, but with his ship the Fram. He found that while cruising areas of the Siberian shelf covered with a thin layer of brackish water, the Fram had great difficulty making any headway. It was hypothesized by V. Bjerknes and proved by Ekman that this ‘dead water’ phenomenon was caused by the drag of the internal wave wake produced by the ship’s hull as it passed through the shallow surface layer. This suggests that internal wave generation by deep keels may cause drag on moving ice. Evidence of internal wave generation by keels has been observed by several authors, but estimates of the amount of drag vary widely. This is due mainly to wide differences in the separation of the stratified pycnocline and the keels. The drag produced by under-ice roughness of amplitude h0 with horizontal wavenumber b moving at velocity Vi (magnitude vi) over a pycnocline with stratification given by Brunt–Vaisala frequency, N, a depth d below the ice–ocean interface, can be expressed as an effective internal wave stress (eqn [25]), where Cwd (eqn [26]) accounts for the drag that would exist if there were no mixed layer between the ice and the pycnocline. Siw ¼ GCwd Vi

½25

above which the waves are evanescent (bc ¼ N/vi). G is an attenuation factor that accounts for the separation of the pycnocline from the ice by the mixed layer of depth d (eqn [27]). 8"
an , provide a complete, orthogonal basis for the internal vertical displacements. The corresponding equations for horizontal dependence yield expressions for the horizontal wave number k, phase velocity cp ¼ o=k, and group velocity cg ¼ do=dk in terms of an :   k2 ¼ a2n o2  f 2  

c2p ¼ o2 = a2n o2  f 2

Seafloor

 1 cg ¼ cp a2n

Figure 1 Baroclinic displacement modes 1, 3, and 5, computed for a buoyancy frequency profile from the deep ocean.

If N is taken constant, then Gn can be found analytically: Gn ðzÞ ¼ asinðnpz=DÞ for an ocean depth D. If N is taken more representative of deep-ocean conditions, with a peak at the pycnocline, then the oscillations in Gn are shifted upward (Figure 1). Notice that the displacements are small or zero at the top and bottom of the water column, and that each Gn has n  1 crossings of the origin. Horizontal velocity modes are given by dGn =dz, and they have n crossings and hence nonzero shear for all modes. Most observations of internal tides (except those very near the generation point) are adequately described by a superposition of a few low order modes. In the deep ocean typical phase speeds cp are of order 3 m s1 for n ¼ 1. Corresponding wavelengths l ¼ 2p=k are between 100 and 200 km. Higher order modes have speeds and wavelengths given roughly by cp =n and l=n, respectively. On continental shelves both speeds and wavelengths may be an order of magnitude smaller. These values are for semidiurnal tides; wavelengths of diurnal tides are approximately twice as large. From the above expressions for k and cp it is apparent that internal tides cannot freely propagate unless o > f . They are ‘evanescent’ (exponentially

damped)’ polewards of the critical latitudes where o ¼ f . Freely propagating waves for diurnal tides are therefore confined to the region between latitudes 7301. (In fact, unambiguous observations of diurnal tides are fairly rare, but this is partly due to relatively weak barotropic forcing and higher background noise levels.) Beams

A complementary approach to modal analyses stems from the equation for the two-dimensional stream function, which is hyperbolic in spatial coordinates and may therefore be solved by the method of characteristics. The resulting solution consists of narrow beams of intense motion embedded in an otherwise resting ocean. The group velocity, and hence the energy propagation, follow the characteristics, which are along lines of slope:  2 1=2 o  f2 c ¼ tan y ¼ 7 N 2  o2 From a given internal tide generation point, energy thus propagates along beams at the angle y relative to

INTERNAL TIDES

horizontal, the angle depending (for a given f and N) only on the tidal frequency. Well defined beams comprise a large number of modes, with modal cancellations occurring outside the allowed beam. A numerical example of beam-like propagation from a shelf break is shown in Figure 2. Generation of internal tides is apparently especially efficient when the seafloor slopes at precisely the critical value c. Barotropic flow is then coincident with the motion plane for free internal waves, resulting in near-resonant conditions in which even quite small surface tides can generate internal tides. With nominal values of NB50 cpd, oB2 cpd, f B0:6 cpd, then y is 21. Continental slopes commonly exceed this, so c would be attained near the shelf break, as depicted in Figure 2. When an internal wave is reflected from the ocean bottom or ocean surface, energy propagation is still confined to the angle y, which makes for a curious variation on the usual laws of reflection. If the wave 0 4 80 4 80 4 80 4 80 4

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is incident upon bathymetry that is steeper than y (supercritical case), energy is reflected backwards into deeper water. If the bathymetry is less steep (subcritical case), energy is reflected forward toward shallower water (see Figure 3). Ocean observations of this behavior are not easy to obtain, since mooring instruments must be precisely placed (depending on the ambient N); yet measurements of internal tides in the Bay of Biscay have not only observed the downward energy propagation from the generation point, but also the subsequent reflection from the ocean bottom. In the Bay of Biscay, as in most places, N diminishes with depth, so y grows larger and the beams become steeper as they approach the bottom (Figure 2 and Figure 3 are drawn for constant N.) With such reflection properties, internal waves incident on a subcritical sloping bottom will be focused into the shallows (as in Figure 3A), with energy density correspondingly intensified. The same mechanism tends to trap internal wave energy within steep 80 4 8 0 4 8 0 4 8 0 4 8 0 4 8 0 4

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downward edge the tide has generated packets of internal solitons. These solitary waves, starting from an initial 7 m depth, have extraordinarily nonlinear isotherm displacements of 25 m. Although usually less dramatic, this phenomenon is not uncommon in high tide regions on a continental shelf. The internal tide, generated at the shelf break, propagates shorewards and becomes progressively more nonlinear until the bore disintegrates into a group of solitons, the leading one usually of largest amplitide. Moored Current Meters

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Figure 3 Successive reflections of an internal wave along (A) a subcritical seafloor and (B) a supercritical seafloor. Arrows denote direction of energy propagation.

(supercritical) canyons, where the canyon sides reflect energy ever deeper, focusing it toward the canyon floor. If the floor is subcritical, then energy is further focused toward the canyon head. Intense internal tide currents and large kinetic energy densities have indeed been observed in canyons, and especially near canyon heads. In the presence of internal viscosity or other dissipative mechanisms, internal tidal beams widen. Because group velocities are smaller and decay scales shorter for higher order modes, beams tend to disintegrate rapidly into the few low order modes that are most commonly observed.

Observations Internal tides have been observed with a great multitude of instruments and technique, both in situ and remote. Four distinctly different examples are given here which serve to highlight a number of characteristic features of internal tides. Except for the first example, emphasis is given to deep-sea tides. Vertical Profilers

Vertical profilers, ranging from echo sounders to repeated hydrographic casts to special yo-yo instruments, provide some of the clearest pictures of internal tides. An especially dramatic example from the continental shelf off Oregon is shown in Figure 4. It shows a clear semidiurnal signal in the isotherm displacements, somewhat distorted into a bore-like shape (akin to the nonlinear distortion seen in shoaling surface tides in very shallow water). Along its

Because of their widespread deployments, current meters provide perhaps the most common for observing, or at least detecting, internal tides, especially in the open ocean. Sufficient vertical sampling is required for decoupling the internal modes from the surface tide (and unfortunately sufficient sampling is not common). Figure 5 is an example of marginally adequate vertical sampling; it shows tidal current estimates extracted from moored meters near 1101W on the Pacific equator. Estimates are given for each of 10 months, at 10 depths throughout the water column. The current ellipses are fairly uniform below 1000 m; these depths are dominated by the stable, depth-independent currents of the surface tide. In contrast, large temporal variation, and occasionally much larger amplitudes, are evident in the shallower estimates; in these depths, where the buoyancy frequency (and its change) is maximum, the tidal signal is dominated by the internal tide. Modal analysis reveals that the internal tide is essentially random, isotropic, and without a dominant mode for these 10 months. Such observations are characteristic of in situ observations of internal tides; but in a few locations in the deep ocean, a component of the internal tide has been observed that is not so variable and that maintains phase lock with the astronomical tide. The famous MODE experiment in the western Atlantic found that approximately 50% of the internal tide variance was temporally coherent with the astronomical tide. Such observations imply a nearly constant ocean stratification, at least to the extent that it determines generation and propagation properties. Satellite Altimetry

Recently satellite altimetry has been shown capable of providing a near-global view of the coherent component of internal tides. It does this by detecting the very small surface displacements associated with internal tides. These are given roughly by the tide’s internal displacements scaled by Dr=r, the fractional difference in water density, typically of order 0.2%,

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Figure 4 Color contour time series of temperature profiles from the surface to 35 m depth, obtained by repeated 80 s raising and free-falling of the loose-tethered microstructure profiler, deployed offshore Tillamook, Oregon in October 1995. Top: the semidiurnal internal tide displacement (most clearly seen along the yellow 13.81C isotherm) for a 24 h period. Bottom: a zoom view of a 1.7 h period showing the start of the first soliton displacements. The solitons are separated by roughly 10 min. (Reproduced with permission from Stanton TP and Ostrovsky LA (1998) Observations of highly nonlinear internal solitons over the continental shelf. Geophysical Research Letters 25: 1695–1698.)

thus implying surface displacements of a few centimeters for internal displacement of tens of meters. Altimetry detects such small waves as modulations (with wavelengths 100–200 km for internal mode 1)

of the surface tide as estimated along satellite tracks. Because tides can be estimated from altimeter data only by gathering multi-year time series of elevations at a particular site, only the coherent component of

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Figure 5 M2 tidal current ellipses by month at each of 10 depths obtained from moored current meters near 01, 1101W. Scale bar for velocity is at upper right. Each ellipse indicates how the direction and magnitude of the horizontal current velocity evolves over one tidal cycle. All ellipses are polarized clockwise except those marked with a plus sign. (Reproduced with permission from Weisberg RH, Halpern D, Tang T, Hwang SM (1987) M2 tidal currents in the eastern equatorial Pacific Ocean. Journal of Geophysical Research 92: 3821–3826.)

the internal tide which maintains phase lock with the surface tide is capable of being detected. Figure 6 gives an example of the first detection of such waves, near the Hawaiian Ridge. The waves are roughly 5 cm amplitude near the ridge and decay slowly with distance, but are still detectable 1000 km away. Phase estimates (not shown) reveal clearly that the waves are propagating away from the ridge. Evidently they are created by the barotropic tide striking the ridge (at nearly right angle from the north) and generating an internal tide that propagates both northwards and southwards. The picture reveals three important aspects of deep-ocean internal tides: (1) that in some locations they maintain temporal coherence over several years, thus allowing altimetry to measure them, (2) that they maintain spatial coherence over a wide area, and (3) that they are capable of propagating hundreds to thousands of kilometers before being dissipated. All three aspects contrast sharply with the usual picture of incoherence obtained from in situ observations.

Waves similar to those in Figure 6 have been detected in many regions throughout the global ocean. However, altimetry is incapable of detecting internal tides in a region where they are temporally incoherent. Such is apparently the case, for example, off the northwest European shelf, a region known for some of the largest internal tides in the world, but where the coherent signals in altimeter data are extremely weak. Internal tide studies with satellite altimetry are relatively new, and further work should reveal new facets from a global perspective. Acoustic Methods

A second example of a powerful, but unconventional, technique for studying coherent internal tides is acoustic tomography. Differences in two-way acoustic travel times between reciprocal transceivers are sensitive to barotropic tidal currents within the acoustic path. Similarly, since vertical isotherm displacements perturb the sound speed

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Figure 6 Mean elevations at the sea surface of internal tides near Hawaii, deduced from altimeter measurements of the Topex/ Poseidon satellite. Positive values (north of the trackline) indicate that the internal tide’s surface elevation is in phase with the barotropic tide’s elevation. Scale bar for elevations at upper left. Background shading corresponds to bathymetry, with darker shading denoting shallower depths and the main axis of the Hawaiian Ridge. Only internal tides that are coherent with the surface tide over the entire measurement period (here 3.5 years) can be detected in this manner.

within the path, the sums (or averages) of the travel times are sensitive to the internal tide. From a sufficiently long time series the mean tidal characteristics along a given path can be determined. The seemingly coarse spatial resolution is actually an advantage, because it suppresses short-scale internal waves and other noise that typically plague current meter measurements. And, in fact, an array of acoustic transceivers can act as a very sensitive directional antenna for spatially coherent internal tides. Such an array in the central Pacific, consisting of acoustic paths roughly 1000 km long and located just north of the area shown in Figure 6, has measured the same coherent internal tide field seen in the altimetry and indicates that the primary source is the Hawaiian Ridge, even at that great distance.

Implications for Energetics and Mixing Internal tides are an important energy source for vertical mixing, especially in coastal waters where they help maintain nutrient fluxes from deep water to euphotic zones on the shelf. A good example is the

Scotian shelf off Nova Scotia where internal tides are responsible for a strip of enhanced concentrations of nutrients and biomass along the shelf break. During each tidal cycle one or two strong (50 m) internal solitons (compare Figure 4) are generated near the shelf edge, moving shoreward but dissipating rapidly, possibly within 10 km. Estimated energy fluxes of 500 W m1 appear more than adequate to maintain observed nutrient supply to the mixed layer. Similar mixing mechanisms have been observed in the Celtic Sea and elsewhere. In the open ocean it seems reasonable that internal tides dissipate by transferring energy into the internal wave continuum or by directly generating pelagic turbulence, but the associated energy fluxes, and even the dominant mechanisms, are unclear. Nonlinearity is a common feature of internal tides (e.g., occurrences of higher harmonics), so ‘diffusion’ into the continuum is conceivable via nonlinear (resonant triad) interactions, but the evidence for this is so far more anecdotal than convincing. Bottom scattering of low mode tides into higher modes may play a role, as well as wave reflections off sloping bottoms, which tend to intensify kinetic energy densities and may lead to shear instabilities and wave breaking.

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The traditional view is that both the internal wave continuum and the pelagic turbulence and mixing are maintained by nontidal mechanisms such as wind generation; whether internal tides play a major or minor role in this is not resolved. At a minimum, improved quantitative estimates are needed for the global internal tide energy budget. The internal tide energy budget also has a bearing on a longstanding geophysical problem: finding the energy sink for the global surface tide. If the generation/dissipation rate for internal tides is sufficiently large, then internal tide generation conceivably supplements the traditional sink of botton friction in shallow seas. Dissipation rates for the surface tide are well determined by space geodesy (e.g., lunar laser ranging) at 3.7 TW, with 2.5 TW for the principal tide M2. How much of this is accounted for by conversion into internal tides is not well determined; published estimates range from o100 GW (0.1 TW) to >1 TW. There is fairly wide agreement that generation of internal tides at continental slopes provides a fairly small energy sink. Both models and measurements suggest that typical energy fluxes at shelf breaks are of order 100 W m1, leading to a global total of order 15 GW. This is perhaps an underestimate, because it may not fully account for shelf canyons and other three-dimensional features, but the order of magnitude seems reliable. Internal tide generation by deep-ocean topography, however, may be far more important. Recent research based on global tide models as well as on empirical estimates of tidal dissipation deduced from satellite altimetry suggests that generation of internal tides by deep-sea ridges and seamounts could account for 1 TW of tidal power. Refining such estimates, and

understanding the role that internal tides play in generation of the background internal wave continuum, in vertical mixing, and in maintenance of the abyssal stratification, are some of the outstanding issues of current research.

See also Internal Tidal Mixing. Internal Waves. Tides.

Further Reading Baines PG (1986) Internal tides, internal waves, and nearinertial motions. In: Mooers C (ed.) Baroclinic Processes on Continental Shelves. Washington: American Geophysical Union. Dushaw BD, Cornuelle BD, Worcester PF, Howe BM, and Luther DS (1995) Barotropic and baroclinic tides in the central North Pacific Ocean determined from longrange reciprocal acoustic transmissions. Journal of Physical Oceanography 25: 631--647. Hendershott MC (1981) Long waves and ocean tides. In: Warren BA and Wunsch C (eds.) Evolution of Physical Oceanography. Cambridge: MIT Press. Huthnance JM (1989) Internal tides and waves near the continental shelf edge. Geophysical and Astrophysical Fluid Dynamics 48: 81--106. Mun WH (1997) Once again: once again – tidal friction. Progress in Oceanography 40: 7--35. Ray RD and Mitchum GT (1997) Surface manifestation of internal tides in the deep ocean: observations from altimetry and island gauges. Progress in Oceanography 40: 135--162. Wunsch C (1975) Internal tides in the ocean. Reviews of Geophysics and Space Physics 13: 167--182.

PROCESSES OF DIAPYCNAL MIXING

THREE-DIMENSIONAL (3D) TURBULENCE

The Mechanics of Turbulence Figure 2 illustrates the main physical mechanisms that drive turbulence at the smallest scales. The description is presented in terms of strain and vorticity, quantities that represent the tendency of the flow at any point to deform and to rotate fluid parcels, respectively. A major and recent insight is that vorticity and strain are not distributed randomly in turbulent flow, but rather are concentrated into coherent regions, each of which is dominated by one type of motion or the other. The first mechanism we consider is vortex rollup due toshear instability. This process

n sio ffu ce Di len bu ur +T

This article describes fluid turbulence withapplication to the Earth’s oceans. We begin with the simple, classical picture of stationary, homogeneous, isotropic turbulence. We then discuss departures from this idealized state that occur in small-scale geophysical flows. The discussion closes with a tour of some of the many physical regimes in which ocean turbulence has been observed. Turbulent flow has been a source of fascination for centuries. The term ‘turbulence’ appears to have been used first in reference to fluid flows by da Vinci, who studied the phenomenon extensively. Today, turbulence is frequently characterized as the last great unsolved problem of classical physics. It plays a central role in both engineering and geophysical fluid flows. Its study led to the discovery of the first strange attractor by Lorenz in 1963, and thus to the modern science of chaotic dynamics. In the past few decades, tremendous insight into the physics of turbulence has been gained through theoretical and laboratory study, geophysical observations, improved experimental techniques, and computer simulations. Turbulence results from the nonlinear nature of advection, which enables interaction between motions on different spatial scales. Consequently, an initial disturbance with a given characteristic length scale tends to spread to progressively larger and smaller scales. This expansion of the spectral range is limited at large scales by boundaries and/or body forces, and at small scales by viscosity. If the range of scales becomes sufficiently large, the flow takes a highly complex form whose details defy prediction. The roles played by turbulence in the atmosphere and oceans can be classified into two categories: momentum transport and scalar mixing. In transporting momentum, turbulent motions behave in a manner roughly analogous to molecular viscosity, reducing differences in velocity between different regions of a flow. For example, winds transfer momentum to the Earth via strong turbulence in the planetary boundary layer (a kilometer-thick layer adjacent to the ground) and are thus decelerated.

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Introduction

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Copyright & 2001 Elsevier Ltd.

Scalar mixing refers to the homogenization of fluid properties such as temperature by random molecular motions. Molecular mixing rates are proportional to spatial gradients, which are greatly amplified due to the stretching and kneading (i.e. stirring) of fluid parcels by turbulence. This process is illustrated in Figure 1, which shows the evolution of an initially circular region of dyed fluid in a numerical simulation. Under the action of molecular mixing (or diffusion) alone, an annular region of intermediate shade gradually expands as the dyed fluid mixes with the surrounding fluid. If the flow is turbulent, the result is dramatically different. The circle is distended into a highly complex shape, and the region of mixed fluid expands rapidly.

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W. D. Smyth and J. N. Moum Oregon State University, Corvallis, OR, USA

Figure 1 A comparison of mixing enhanced by turbulence with mixing due to molecular processes alone, as revealed by a numerical solution of the equations of motion. The initial state includes a circular region of dyed fluid in a white background. Two possible evolutions are shown: one in which the fluid is motionless (save for random molecular motions), and one in which the fluid is in a state of fully developed, two-dimensional turbulence. The mixed region (yellow–blue) expands much more rapidly in the turbulent case.

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Figure 2 Schematic illustration of line vortices and strained regions in turbulent flow. Fluid parcels in the vortex interiors rotate with only weak deformation. In contrast, fluid parcels moving between the vortices are rapidly elongated in the direction of the purple arrows and compressed in the direction of the green arrows.

results in a vorticity concentration of dimension close to unity, i.e. a line vortex. Line vortices are reinforced by the process of vortex stretching. When a vortex is stretched by the surrounding flow, its rotation rate increases to conserve angular momentum. Opposing these processes is molecular viscosity, which both dissipates vorticity and fluxes it away from strongly rotational regions. Turbulence may thus be visualized as a loosely tangled ‘spaghetti’ of line vortices, which continuously advect each other in complex ways (Figure 3). At any given time, some vortices are being created via rollup, some are growing due to vortex stretching, and some are decaying due to viscosity. Many, however, are in a state of approximate equilibrium among these processes, so that they appear as long-lived, coherent features of the flow. Mixing is not accomplished within the vortices themselves; in fact, these regions are relatively stable, like the eye of a hurricane. Instead, mixing occurs mainly in regions of intense strain that exist between any two nearby vortices that rotate in the same sense (Figure 2). It is in these regions that fluid parcels are deformed to produce amplified gradients and consequent rapid mixing.

Stationary, Homogeneous, Isotropic Turbulence Although the essential structures of turbulence are not complex (Figure 2), they combine in a bewildering range of sizes and orientations that defies analysis (Figure 3). Because of this, turbulence is most usefully understood in statistical terms. Although the statistical approach precludes detailed

Figure 3 Computer simulation of turbulence as it is believed to occur in the ocean thermocline. The colored meshes indicate surfaces of constant vorticity.

prediction of flow evolution, it does give access to the rates of mixing and property transport, which are of primary importance in most applications. Statistical analyses focus on the various moments of the flowfield, defined with respect to some averaging operation. The average may betaken over space and/or time, or it may be an ensemble average taken overmany flows begun with similar initial conditions. Analyses are often simplified using three standard assumptions. The flow statistics are assumed to be

• • •

stationary (invariant with respect to translations intime), homogeneous (invariant with respect to translations inspace), and/or isotropic (invariant with respect to rotations).

Much of our present understanding pertains to this highly idealized case. Our description will focus on the power spectra that describe spatial variability of kinetic energy and scalar variance. The spectra provide insight into the physical processes that govern motion and mixing at different spatial scales. Velocity Fields

Big whorls have little whorls That feed on their velocity And little whorls have lesser whorls And so on to viscosity L.F. Richardson (1922) Suppose that turbulence is generated by a steady, homogeneous, isotropic stirring force whose spatial

THREE-DIMENSIONAL (3D) TURBULENCE

variability is described by the Fourier wavenumber kF . Suppose further that the turbulence is allowed to evolve until equilibrium is reached between forcing and viscous dissipation, i.e., the turbulence is statistically stationary. Figure 4 shows typical wavenumber spectra of kinetic energy, EðkÞ, and kinetic energy dissipation, DðkÞ, for such a flow. EðkÞdk is the kinetic energy contained in motions whose wavenumber magnitudes lie in an interval of width dk surrounding k. DðkÞdk ¼ nk2 EðkÞdk is the rate at which that kinetic energy is dissipated by molecular viscosity (n) in that wavenumber band. R NThe net rate of energy dissipation is given by e ¼ 0 dk, and is equal (in the equilibrium state) to the rate at which energy is supplied by the stirring force. Nonlinear interactions induce a spectral flux, or cascade, of energy. The energy cascade is directed primarily (though not entirely) toward smaller scales, i.e., large-scale motions interact to create smaller-scale motions. The resulting small eddies involve sharp velocity gradients, and are therefore susceptible to viscous dissipation. Thus, although kinetic energy resides mostly in large-scale motions, it is dissipated primarily by small-scale motions. (Note that the logarithmic axes used in Figure 4 tend to de-emphasize the peaks in the energy and dissipation rate spectra.) Turbulence can be envisioned as a ‘pipeline’ conducting kinetic energy through wavenumber space: in at the large scales, down the spectrum, and out again at the small scales, all at a rate e. The cascade concept was first suggested early

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Figure 4 Theoretical wavenumber spectra of kinetic energy and kinetic energy dissipation for stationary, homogeneous, isotropic turbulence forced at wavenumber kF . Approximate locations of the energy containing, inertial, and dissipation subranges are indicated, along with the Kolmogorov wavenumber kk . Axes are logarithmic. Numerical values depend on Re and are omitted here for clarity.

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in the twentieth century by L.F. Richardson, who immortalized his idea in the verse quoted at the beginning of this section. The energy spectrum is often divided conceptually into three sections. The energy-containing subrange encompasses the largest scales of motion, whereas the dissipation subrange includes the smallest scales. If the range of scales is large enough, there may exist an intermediate range in which the form of the spectrum is independent of both large-scale forcing and small-scale viscous effects. This intermediate range is called the inertial subrange. The existence of the inertial subrange depends on the value of the Reynolds number: Re ¼ ul=n, where u and l are scales of velocity and length characterizing the energy-containing range. The spectral distance between the energy-containing subrange and the dissipation 3=4 subrange, kF =kK , is proportional to Re . A true inertial subrange exists only in the limit of large Re . In the 1940s, the Russian statistician A.N. Kolmogorov hypothesized that, in the limit Re -N, the distribution of eddy sizes in the inertial and dissipation ranges should depend on only two parameters (besides wavenumber): the dissipation rate e and the viscosity n, i.e., E ¼ Eðk; e; nÞ. Dimensional reasoning then implies that E ¼ e1=4 n5=4 f ðk=kK Þ, where kK ¼ ðe=n3 Þ1=4 is the Kolmogorov wavenumber and f is some universal function. Thus, with the assumptions of stationarity, homogeneity, isotropy, and infinite Reynolds number, all types of turbulence, from flow over a wing to convection in the interior of the sun, appear as manifestations of a single process whose form depends only on the viscosity of the fluid and the rate at which energy is transferred through the ‘pipeline’. This tremendous simplification is generally regarded as the beginning of the modern era of turbulence theory. Kolmogorov went on to suggest that the spectrum in the inertial range should be simpler still by virtue of being independent of viscosity. In that case E ¼ Eðk; eÞ, and the function can be predicted from dimensional reasoning alone up to the universal constant CK , namely, E ¼ CK e2=3 k5=3 . This powerlaw spectral form indicates that motions in the inertial subrange are self-similar, i.e., their geometry is invariant under coordinate dilations. Early efforts to identify the inertial subrange in laboratory flows were inconclusive because the Reynolds number could not be made large enough. (In a typical, laboratory-scale water channel, uB0:1ms1 , lB0:1m, and nB106 m2 s1 , giving Re B104 . In a typical wind tunnel, uB1ms1 , B1m, and nB106 m2 s1 , so that Re B105 .) The inertial subrange spectrum was first verified in 1962 using measurements in a strongly turbulent tidal channel

THREE-DIMENSIONAL (3D) TURBULENCE

near Vancouver Island, where typical turbulent velocity scales uB1ms1 and length scales lB100 m combine with the kinematic viscosity of seawater nB106 m2 s1 to produce a Reynolds number Re B108 . From this experiment and others like it, the value of CK has been determined to be near 1.6. Passive Scalars and Mixing

Now let us suppose that the fluid possesses some scalar property y, such as temperature or the concentration of some chemical species, and that the scalar is dynamically passive, i.e., its presence does not affect the flow. (In the case of temperature, this is true only for sufficiently small-scale fluctuations; see Buoyancy Effects later in this article for details.) Suppose also that there is a source of large-scale variations in y, e.g., an ambient temperature gradient in the ocean. Isosurfaces of y will be folded and kneaded by the turbulence so that their surface area tends to increase. As a result, typical gradients of y will also increase, and will become susceptible to erosion by molecular diffusion. Scalar variance is destroyed at a rate w, which is equal (in equilibrium) to the rate at which variance is produced by the large eddies. Thus, the turbulent mixing of the scalar proceeds in a manner similar to the energy cascade discussed above. However, there is an important difference in the two phenomena. Unlike energy, scalar variance is driven to small scales by a combination of two processes. First, scalar gradients are compressed by the strain fields between the turbulent eddies. Second, the eddies themselves are continually redistributed toward smaller scales. (The latter process is just the energy cascade described in the previous section.) Figure 5 shows the equilibrium scalar variance spectrum for the case of heat mixing in water. Most of the variance is contained in the large scales, which are separated from the small scales by an inertialconvective subrange (so-called because temperature variance is convected by motions in the inertial subrange of the energy spectrum). Here, the spectrum depends only on e and w; its form is Ey ¼ bwe1=3 k5=3 , where b is a universal constant. The shape of the spectrum at small scales is very different from that of the energy spectrum, owing to the fact that, in sea water, the molecular diffusivity, k, of heat is smaller than the kinematic viscosity. The ratio of viscosity to thermal diffusivity is termed the Prandtl number (i.e. Pr ¼ n=k) and has a value near 7 for sea water. In the viscous-convective subrange, the downscale cascade of temperature variance is slowed because the eddies driving the cascade are weakened by viscosity. In other words, the first of the two

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Figure 5 Theoretical wavenumber spectra of scalar variance and dissipation for stationary, homogeneous, isotropic turbulence forced at wavenumber kF . Approximate locations of the variancecontaining, inertial-convective, viscous-convective, and viscousdiffusive subranges are indicated, along with the Kolmogorov wavenumber kK and the Batchelor wavenumber kB . Axes are logarithmic. Numerical values depend on Re and are omitted here for clarity.

processes listed above as driving the scalar variance cascade is no longer active. There is no corresponding weakening of temperature gradients, because molecular diffusivity is not active on these scales. As a result, there is a tendency for variance to ‘accumulate’ in this region of the spectrum and the spectral slope is reduced from 5=3 to 1. However, the variance in this range is ultimately driven into the viscous-diffusive subrange, where it is finally dissipated by molecular diffusion. A measure of the wavenumber at which scalar variance is dissipated is the Batchelor wavenumber, kB ¼ ðe=nk2 Þ1=4 . When Pr > 1, as for sea water, the Batchelor wavenumber is larger than the Kolmogorov wavenumber, i.e., temperature fluctuations can exist at smaller scales than velocity fluctuations. In summary, the energy and temperature spectra exhibit many similarities. Energy (temperature variance) is input at large scales, cascaded down the spectrum by inertial (convective) processes, and finally dissipated by molecular viscosity (diffusion). The main difference between the two spectra is the viscous-convective range of the temperature spectrum, in which molecular smoothing acts on the velocity field but not on the temperature field. This difference is even more pronounced if the scalar field represents salinity rather than temperature, for salinity is diffused even more weakly than heat. The ratio of the molecular diffusivities of heat and salt is of order 102, so that the smallest scales of salinity fluctuation in sea water are ten times smaller than those of temperature fluctuations.

THREE-DIMENSIONAL (3D) TURBULENCE

Turbulence in Geophysical Flows The assumptions of homogeneity, stationarityand isotropy as employed by Kolmogorov have permitted tremendous advances inour understanding of turbulence. In addition, approximations based on theseassumptions are used routinely in all areas of turbulence research. However, wemust ultimately confront the fact that physical flows rarely conform to our simplifying assumptions. In geophysical turbulence, symmetries are upset by acomplex interplay of effects. Here, we focus on three important classes of phenomena that modify small-scale turbulence in the ocean: shear, stratification, and boundary proximity. Shear Effects

Geophysical turbulence often occurs in the presence of a current which varies on scales much larger than the energy-containing scales of the turbulence, and evolves much more slowly than the turbulence. Examples include atmospheric jet streams and largescale ocean currents such as the Gulf Stream and the Equatorial Undercurrent. In such cases, it makes sense to think of the background current as an entity separate from the turbulent component of the flow. Shear upsets homogeneity and isotropy by deforming turbulent eddies. By virtue of the resulting anisotropy, turbulent eddies exchange energy with the background shear through the mechanism of Reynolds stresses. Reynolds stresses represent correlations between velocity components parallel to and perpendicular to the background flow, correlations that would vanish if the turbulence were isotropic. Physically, they represent transport of momentum by the turbulence. If the transport is directed counter to the shear, kinetic energy is transferred from the background flow to the turbulence. This energy transfer is one of the most common generation mechanisms for geophysical turbulence. In sheared turbulence, the background shear acts primarily on the largest eddies. Motions on p scales ffiffiffiffiffiffiffiffiffi much smaller than the Corrsin scale, LC ¼ e=S3 (where S ¼ dU=dz, the vertical gradient of the ambient horizontal current) are largely unaffected. Buoyancy Effects

Most geophysical flows are affected to some degree by buoyancy forces, which arise due to spatial variations in density. Buoyancy breaks the symmetry of the flow by favoring the direction in which the gravitational force acts. Buoyancy effects can either force or damp turbulence. Forcing occurs in the case of unstable density stratification, i.e., when heavy

371

fluid overlies light fluid. This happens in the atmosphere on warm days, when the air is heated from below. The resulting turbulence is often made visible by cumulus clouds. In the ocean, surface cooling (at night) has a similar effect. Unstable stratification in the ocean can also result from evaporation, which increases surface salinity and hence surface density. In each of these cases, unstable stratification results in convective turbulence, which can be extremely vigorous. Convective turbulence usually restores the fluid to a stable state soon after the destabilizing flux ceases (e.g., when the sun rises over the ocean). Buoyancy effects tend to damp turbulence in the case of stable stratification, i.e., when light fluid overlies heavier fluid. In stable stratification, a fluid parcel displaced from equilibrium oscillates pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vertically with frequency N ¼ gr1 dr=dz, the buoyancy or Brunt–Vaisala frequency (g represents acceleration due to gravity and rðzÞ is the ambient mass density). A result of stable stratification that can dramatically alter the physics of turbulence is the presence of internal gravity waves (IGW). These are similar to the more familiar interfacial waves that occur at the surfaces of oceans and lakes, but continuous density variation adds the possibility of vertical propagation. Visible manifestations of IGW include banded clouds in the atmosphere and slicks on the ocean surface. IGW carry momentum, but no scalar flux and no vorticity. In strongly stable stratification, motions may be visualized approximately as two-dimensional turbulence (Figure 1) flowing on nearly horizontal surfaces that undulate with the passage of IGW. The quasitwo-dimensional mode of motion carries all of the vorticity of the flow (since IGW carry none), and is therefore called the vortical mode. In moderately stable stratification, three-dimensional turbulence is possible, but its structure is modified by the buoyancy force, particularly at large scales. Besides producing anisotropy, the suppression of vertical motion damps the transfer of energy from any background shear, thus reducing the intensity of turbulence. On scales smaller than the Ozmipffiffiffiffiffiffiffiffiffiffimuch ffi dov scale, L0 ¼ e=N 3 , buoyancy has only a minor effect. (In Passive scalars and mixing above, we used temperature as an example of a dynamically passive quantity. This approximation is valid only on scales smaller than the Ozmidov scale.) The relative importance of stratification and shear depends on the magnitudes of S and N. If SbN, shear dominates and turbulence is amplified. On the other hand, if S5N, the buoyancy forces dominate and turbulence is suppressed. The relationship between IGW and turbulence in stratified flow is exceedingly complex. At scales in

372

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excess of a few meters (Figure 6), ocean current fluctuations behave like IGW, displaying the characteristic spectral slope k1 . At scales smaller than the Ozmidov scale (typically a few tens of centimeters), fluctuations differ little from the classical picture of homogeneous, isotropic turbulence. The intermediate regime is a murky mix of nonlinear IGW and anisotropic turbulence that is not well understood at present. The breaking of IGW is thought to be the major source of turbulence in the ocean interior. Breaking occurs when a superposition of IGW generates locally strong shear and/or weak stratification. IGW propagating obliquely in a background shear may break on encountering a critical level, a depth at which the background flow speed equals the horizontal component of the wave’s phase velocity. (Many dramatic phenomena occur where wave speed matches flow speed; other examples include the hydraulic jump and the sonic boom.) Just as waves may generate turbulence, turbulent motions in stratified flow may radiate energy in the form of waves. In stably stratified turbulence, the distinction between stirring and mixing of scalar properties becomes crucial. Stirring refers to the advection and deformation of fluid parcels by turbulent motion, whereas mixing involves actual changes in the scalar properties of fluid parcels. Mixing can only be accomplished by molecular diffusion, though it is accelerated greatly in turbulent flow due to stirring (cf. Figure 1 and the accompanying discussion). In stable

~10 m

k

~1 m

~10 cm

_2

Energy spectrum

IGW

k Nonlinear IGW + anisotropic turbulence

_ 5/3

Isotropic turbulence

Vertical wavenumber

Figure 6 Energy spectrum (cf. Figure 4) extended to larger scales to include internal gravity waves (IGW) plus anisotropic stratified turbulence. Labels represent approximate length scales from ocean observations.

stratification, changes in the density field due to stirring are reversible, i.e., they can be undone by gravity. In contrast, mixing is irreversible, and thus leads to a permanent change in the properties of the fluid. For example, consider a blob of water that has been warmed at the ocean surface, then carried downward by turbulent motions. If the blob is mixed with the surrounding water, its heat will remain in the ocean interior, whereas if the blob is only stirred, it will eventually bob back up to the surface and return its heat to the atmosphere. Boundary Effects

It is becoming increasingly clear that most turbulent mixing in the ocean takes place near boundaries, either the solid boundary at the ocean bottom, or the moving boundary at the surface. All boundaries tend to suppress motions perpendicular to themselves, thus upsetting both the homogeneity and the isotropy of the turbulence. Solid boundaries also suppress motion in the tangential directions. Therefore, since the velocity must change from zero at the boundary to some nonzero value in the interior, a shear is set up, leading to the formation of a turbulent boundary layer. Turbulent boundary layers are analogous to viscous boundary layers, and are sites of intense, shear-driven mixing (Figure 7). In turbulent boundary layers, the characteristic size of the largest eddies is proportional to the distance from the boundary. Near the ocean surface, the flexible nature of the boundaries leads to a multitude of interesting phenomena, notably surface gravity waves and Langmuir cells. These phenomena contribute significantly to upper-ocean mixing and thus to air–sea fluxes of momentum, heat and various chemical species. Boundaries also include obstacles to the flow, such as islands and seamounts, which create turbulence. If flow over an obstacle is stably stratified, buoyancyaccelerated bottom flow and a downstream hydraulic jump may drive turbulence (Figure 7). Ocean turbulence is often influenced by combinations of shear, stratification, and boundary effects. In the example shown in Figure 7, all three effects combine to create an intensely turbulent flow that diverges dramatically from the classical picture of stationary, homogeneous, isotropic turbulence.

Length Scales of Ocean Turbulence Examples of turbulent flow regimes that havebeen observed in the ocean can be considered in terms of typical values of e and N that pertain to each (Figure 8). This provides the information to estimate

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373

Depth

20 m

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Tidal channel

 (m– 2 s – 3 )

both largest and smallest scales present in the flow. The largest scale is approximated by the Ozmidov scale, which varies from a few centimeters in the ocean’s thermocline to several hundred meters in weakly stratified and/or highly energetic flows. The smallest scale, the Kolmogorov scale LK ¼ k1 K , is typically 1 cm or less. Turbulence in the upper ocean mixed layer may be driven by wind and/or by convection due to surface cooling. In the convectively mixed layer, N is effectively zero within the turbulent region, and the maximum length scale is determined by the depth of the mixed layer. In both cases the free surface limits length scale growth. Turbulence in the upper equatorial thermocline is enhanced by the presence of shear associated with the strong equatorial zonal current system. Stratification tends to be considerably stronger in the upper thermocline than in the main thermocline. Despite weak stratification, turbulence in the main thermocline tends to be relatively weak due to isolation from strong forcing. Turbulence in this region is generated primarily by IGW interactions. Tidal channels are sites of extremely intense turbulence, forced by interactions between strong tidal currents and three-dimensional topography. Length scales are limited by the geometry of the channel. Turbulent length scales in the bottom boundary layer are limited below by the solid boundary and above by stratification. Intense turbulence is also found in hydraulically controlled flows, such as have been

m

Figure 7 Flow over Stonewall bank, on the continental shelf off the Oregon coast. Colors show the kinetic energy dissipation rate, with red indicating strong turbulence. White contours are isopycnals, showing the effect of density variations in driving the downslope flow. Three distinct turbulence regimes are visible: (1) turbulence driven by shear at the top of the rapidly moving lower layer, (2) a turbulent bottom boundary layer and (3) a hydraulic jump.

Internal hydraulic flow on the Continental Shelf

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Wind-mixed layers Convectively Upper equatorial mixed layers thermocline Bottom boundary layer Main thermocline

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Figure 8 Regimes of ocean turbulence located with respect to stratification and energy dissipation. Dotted lines indicate Ozmidov and Kolmogorov length scales.

found in the Strait of Gibraltar, and also over topography on the continental shelf (cf.Figure 7). In these flows the stratification represents a potential energy supply that drives strongly sheared downslope currents, the kinetic energy of which is in turn converted into turbulence and mixing. All of these turbulence regimes are subjects of ongoing observational and theoretical research, aimed at generalizing Kolmogorov’s view of turbulence to encompass the complexity of real geophysical flows.

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See also

Further Reading

Breaking Waves and Near-Surface Turbulence. Heat and Momentum Fluxes at the Sea Surface. Heat Transport and Climate. Internal Waves. Langmuir Circulation and Instability. Open Ocean Convection. Turbulence in the Benthic Boundary Layer. Upper Ocean Mixing Processes. Vortical Modes.

Frisch U (1995) Turbulence. Cambridge: Cambridge University Press. Hunt JCR, Phillips OM, and Williams D (1991) Turbulence and Stochastic Processes; Kolmogoroff ’s Ideas 50 Years On. London: The Royal Society. Kundu PK (1990) Fluid Mechanics. London: Academic Press.

LABORATORY STUDIES OF TURBULENT MIXING J. A. Whitehead, Woods Hole Oceanographic Institution, Woods Hole, MA, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction As described elsewhere in this encyclopedia, turbulence and diffusion act both vertically in mixing up the density field of the ocean and laterally in mixing adjacent density and tracer fields. Vertical mixing of density in a stratified fluid dissipates turbulent kinetic energy by raising the potential energy of the density field, and the ratio of this dissipation to viscous dissipation of turbulence is a fundamental quantity needed to understand the energy balance of the ocean. Estimates of these rates by theory are not yet readily available in a form useful for the ocean. Numerical approaches and computational fluid dynamics are under development, but they are not capable of investigating a wide range of parameter space in present form, nor can results be recovered that are verified by experimental benchmarks. Direct ocean measurements rely on theoretical assumptions about the form of turbulence, which must ultimately be verified in the laboratory. Therefore, laboratory measurements continue to be essential to the quest of determining the rates of turbulent dissipation within flows of stratified fluid. Both the production of turbulence through instability and its dissipation are markedly different from the case of instability and dissipation within a homogeneous fluid. Ignoring internal wave radiation from turbulent regions, dissipation is partitioned between viscous dissipation and the work that increases potential energy. This is shown below by the two integrals for energy of a simple system with no internal body forces and closed bottom and top boundaries in a field of gravity. In this example, the flow must start with an initial value of kinetic energy, and the decrease in kinetic energy of a volume of incompressible fluid is equal to the rate of buoyancy work plus viscous dissipation: D E 1 dhv˜  v˜ i ¼ ghrwi  v ðrv˜ Þ2 2 dt The change in potential energy is the rate of buoyancy work plus buoyancy flux multiplied by elevation: Z h dhrzi z½rw dz ¼ ghrwi þ g g dt 0

where angle brackets are averages over all three spatial dimensions, and the square brackets are over the two lateral dimensions. Here, v˜ is the threedimensional velocity vector, n is viscous diffusivity, the variables z, w are the vertical direction and velocity (the direction of gravity g), and density is r. Clearly, if the following three conditions are met, then the only two terms that can dissipate the kinetic energy are viscous dissipation and buoyancy (heat) flux: first, the potential energy is not changing in time; second, dense (considering it to be cold) fluid enters the bottom with lighter (warm) fluid leaving the top (it would require a downward heat flow into the volume to allow this); and third, the volume is given an initial value of kinetic energy that is allowed to run down. It is the purpose of laboratory experiments to allow measurements of density and velocity fields and to obtain the partition between viscous dissipation and buoyancy flux. No instrument exists for precisely measuring every term within any of the above brackets, so simplified approaches have been necessitated.

Experiments Two types of experiments generate the turbulence, either a shear-flow instability is set up or eddies are directly generated. In addition, there are two groups of density distribution, one with sharp interfaces and the other having continuous stratification. Since both of the dissipation terms shown above are negative, there are no cases with both of the equations in their steady form. Experiments incorporate either transient setups that run down with time, or utilize flowing tanks (mostly with salt-stratified water but a few wind tunnels with thermal stratification are used too). In the latter case, the flows are transient following a fluid parcel. The techniques to produce eddies are numerous. Figure 1 shows some of the laboratory configurations. Some generate turbulence behind grids in tunnels (Figure 1(a)) and others are driven by buoyancy (Figure 1(b)). In some flumes and wind tunnels, narrowing the sides enhances the shear in a test region. Other experiments have a moving grid or rod stirrer (Figure 1(c)), and still others have a moving lid (Figure 1(d)). Not sketched are special studies of pumped jets directed toward an interface, and experiments with double-diffusion driven flows, both being directed toward explicit mechanisms of mixing. Experiments have been motivated by numerous phenomena in addition to oceanographic ones; some

375

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LABORATORY STUDIES OF TURBULENT MIXING

(a)



z

(b)



(c)



z

z



(d)

z

Figure 1 Configurations of various laboratory experiments to measure stratified mixing.

examples are fire prevention, ventilation, snow avalanches, ecosystem studies, solar ponds, mixing of industrial chemicals, and turbidity currents. Flows have been produced in pumped tunnels, as sketched in Figure 1(a), with the stratification produced by either temperature (using heat-transfer devices) or salinity (using multiple sources). A variation is a closed-circuit tunnel with a special device for turbulence-free propulsion of the water. In some cases the vertical salinity distribution is initially set and the density profile runs down with time. Another variant has layers pumped in at multiple elevations with the same amount withdrawn at each corresponding level downstream and returned to cisterns. Also, there are currents driven by buoyant flow in tanks with sloping tops and bottoms, as shown in Figure 1(b), or with exchange flows in passages between reservoirs containing waters of differing salinity. Then, there are experiments in closed containers with oscillating grids or moving rods as sketched in Figure 1(c). In some cases the experiments are in annular chambers and some are rotating on a turntable (Figure 1(d)). Salt-stratified experiments are the most numerous. They possess a vertical salinity distribution that evolves with time, although

there are also thermal experiments using air or water motivated by engineering applications. The dynamics are all characterized by velocity scale of the turbulence u (which may be the same size as velocity difference in a sheared laminar flow whose instability generates the turbulence) and density difference Dr. The force of gravity makes the density difference equivalent to a buoyancy difference g0 ¼ gDr/r0. Geometrically, there is the separation distance d between regions of different velocity and density. Finally, there are two additional fluid properties, the viscosity n and density diffusivity D. Velocity, reduced gravity, length, viscosity, and diffusivity can be reduced to many combinations of three dimensionless numbers. For stratified mixing, they are picked sequentially to represent the important balances in the flow in rank order. The primary dimensionless number is the bulk Richardson number Ri ¼ g0 d=u2. It is a measure of the ratio of buoyancy to fluid inertia. The second number is the Reynolds number Re ¼ u d=v, which is the ratio of inertial to viscous force. The third is the Schmidt number Sc ¼ n/D. It is the ratio of viscosity to density diffusivity. Experiments generally have the objective to determine a buoyancy work rate (frequently called buoyancy flux) as a function of these three numbers. By the early 1990s it was clear that the buoyancy flux obeyed a range of power-law relations with Ri, and that these are sensitive to details of actual experiments in most cases. The relative roles of Re and Sc are less well documented. Virtually any source of turbulence can be used to mix stratified fluid, and one of the challenges of experiments is to separate the influence of the spatial distribution of the turbulent source from the actual processes within the fluid. Turbulence that is shed from a vertical rod that moves laterally and sheds a turbulent wake in fresh water above a salt layer (Figure 2(a)) produces striations that are strongest near the interface and weaker at higher and lower levels. This produces a divergence in buoyancy flux so the interface gets progressively thicker with time. However, continuous stratification (Figure 2(b)) can spontaneously break down to internal layers. Therefore, in both cases, the flux varies locally. To complicate matters, the variation of the stratification is usually about the same size as the scale of the turbulence, so in almost all experiments it is not obvious that statistical turbulence theory applies. Probably the simplest configuration has salt water under fresh water with grid stirring confined to one layer, for example, the top layer. This easily attains high Reynolds numbers using a horizontal grid moving up and down with vigorous oscillatory motion. To determine u at the level of the interface,

LABORATORY STUDIES OF TURBULENT MIXING

(a)

377

(b) 10−1 E ~Ri −1/2

ue /u∗

10−2

10−3 E ~Ri −1 10−4

10−5 Figure 2 Shadowgraphs from a parallel beam of light falling onto a screen. They show the effect of turbulence produced by many excursions of a moving rod (a) for a layered fluid as in Figure 1(c) right (salt water under fresh, the rod has recently reversed direction near the tank wall); and (b) for a stratified fluid that breaks down into layers as in Figure 1(c) left (stratified salt water, the rod is moving toward the left into a placid, previously mixed fluid).

grid velocity should be multiplied by a suitable constant to account for spatial variation of the turbulence between grid and interface. As time progresses, for Ri41 the turbulence in the upper layer causes the interface to remain sharp and it mixes salt water up into the top layer. This ‘entrains’ salt water into the top layer, which increases both the volume and salinity of the top layer and decreases the volume of the bottom one but leaves its salinity unchanged. The interface moves downward with entrainment velocity ue, and the speed quantifies the mixing rate. As density difference between the layers decreases, Ri decreases. The entrainment velocity increases steeply with decreasing Richardson number as shown by solid circles in Figure 3. For Rio1, the interface deflection is as large as d and the subsequent mixing rate is rapid and soon the two layers mix completely. Many other experiments have produced entrainment velocity measurements; a collection of some is shown in the lower cluster of Figure 3. In some of them, the mixing is supplied by shear instability driven by a rotating screen in an annulus with stratified fluid (Figure 1(d)). A mixed layer with a sharp density jump at the bottom penetrates into the stratified fluid. Others involve gravity currents (Figure 1(b)), with buoyant outflows and with counter-flows. All of these configurations successfully give useful data for large stratification (Rio1). The scatter in the points shown in Figure 3 is typical and is due to the statistical nature of the data rather

E ~Ri −3/2 10−1

100

101

102

Ri Figure 3 Entrainment velocity as a function of Richardson number for assorted experiments. Solid circles: Grid experiments where the grid velocity is to define u * . Open circles: Buoyant outflows. Triangles: Density currents. Crosses: Counterflows. Data and slopes taken from Fernando HJS (1991) Turbulent mixing in stratified fluids. Annual Review of Fluid Mechanics 23: 455–493, figure 15, with permission from author, and Turner JS (1973) Buoyant convection from isolated sources. Buoyancy Effects in Fluids, pp. 165–206, Figure 9.3. Cambridge, UK: Cambridge University Press, with permission from author.

than instrumental error. In such experiments, the horizontally averaged density typically breaks up into patches and layers so that local regions have different local values of Richardson number. In spite of such scatter, the results of such experiments are overwhelmingly consistent with each other with respect to the general trends of the data shown in Figure 3. Primarily, the Richardson number is the most important variable governing mixing rate if it is of order 1 or more. The Reynolds number does exhibit some role especially if less than c. 500. Experiments to date range up to almost Reo105 and generally speaking the mixing is sensitive to Reynolds number for the entire range. It is thought that mixing will finally become insensitive for very large Re but data up to such a possible limit are not yet available. Three power laws are sketched as straight lines in this figure. To the left the open circle data have a smaller slope. To the right, the data are inversely proportional to a higher power of Richardson number. The constants of proportionality are functions of the actual configuration and the manner of defining velocity and density. As an example, the

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LABORATORY STUDIES OF TURBULENT MIXING

relation found by Kato and Phillips for shear-driven experiments is

10−1

ue ¼ 2:5 Ri1 u This can be rearranged to gDrue d ¼ 2:5r0 u3 , which states that the rate of change of potential energy is proportional to the rate of delivery of kinetic energy for mixing. The value of power-law dependence has been extensively studied and discussed. There are some circumstances in which there is no dependence because stratification effects are smaller than viscous, diffusive, or turbulent effects. In other cases there are over 40 proposed relations with Ri, but there is still no clear consensus about the range of validity for each of these in Ri, Re, and Sc space. This lack of agreement seems to arise for a number of reasons. First, no tight cluster about one line is found because of the scatter mentioned above. Second, it has always been found that the results tend to be specific for each experimental configuration. Third, the experiments are in water or air, so only a few values of Sc are investigated. One proposed relation between the three dimensionless numbers has the dimensionless entrainment at successively increasing Ri proportional to Ri  3/2, ScRi  1, and Sc  1/2 Re  1/4. It shows that there is an increasingly important role for molecular and turbulence effects as stratification is increased. However, in listing 30 such relations, Fernando found that the  1.5 power law generally tended to be for higher values of Ri rather than lower values. A gravity current down a slope is of particular interest to oceanography because of its relevance to deep overflows in polar regions and salt plumes in aridpregions. In such studies the Froude number Fr ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u = g0 d cos yðERi1=2 Þ (y is angle of the slope) is often used to measure the intensity of mixing compared to stratification. Results for a wide range of gravity-driven currents are shown in Figure 4. As in Figure 3 it is clear that large stratification suppresses entrainment velocity, and that ueEu with smaller stratification. In addition, it is clear that mixing is enhanced at larger Reynolds numbers.

Continuous Stratification The layers of mixed fluid in mechanical stirring experiments are separated by very sharp interfaces, so for large values of Ri the layers remain well defined. Therefore, the results are relatively precise and easy to interpret. Of course, experiments with continuous stratification are more similar to the ocean. Continuously stratified fluid exposed to turbulence tends

10−2 ue u∗ 10−3

10−4 100

101

Fr = u∗ /(g ′dcos())1/2 Figure 4 Entrainment velocity in laboratory gravity currents down a slope compared to estimates of ocean overflows. The solid triangle is for Lake Ogawara, the solid square is for Mediterranean outflow into the Atlantic, the small star is for the Denmark Strait overflow, and the solid diamond is for the Faroe Bank channel overflow. For rotating density currents, the open squares, open triangles, and large stars are experiments with Reo100 and the open diamonds are with ReZ100. The shaded area and open circles are found for large Reynolds number nonrotating density currents. Supplied by C. Cenedese.

to break up into layers for large Ri. Therefore, N varies locally and hence the dimensionless number varies locally. As a result, the mixing becomes concentrated in local regions. In addition, as time progresses the layers evolve and slowly change flux. The dynamics that determine the size of the layers remain controversial. In some cases, the layer depth scales with the scale u/N, and in other cases the scales are linked to vortical modes shed from the stirrers, or even to the stirrer size itself. In continuously stratified experiments, the overall Richardson number can be defined as Ric ¼ (Nd/u )2 or, if shear du/dz is imposed, as Ris ¼ (N/(du/dz))2. The invention of the bathythermograph led to the discovery of extensive layering within the ocean. Although this layering can be explained as a consequence of localized wave breaking, its universal character suggests that there is a more fundamental cause. Laboratory experiments with stirring near the sidewall of a stratified fluid, and later experiments with continuously stratified fluid stirred with a rod, all exhibited the spontaneous growth of layers for Ricc1. Figure 5 shows this growth in salt-stratified water with grid-generated turbulence (with d set to the mesh size) for Ric ¼ 10.7. In some elevations the local stratification (as measured by N) increases, and

0

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Figure 5 Density profiles at successive times showing the spontaneous emergence of layers in a continuously stratified fluid for large Ric. Adapted from Rehmann CR and Koseff JR (2004) Mean potential energy change in stratified grid turbulence. Dynamics of Atmospheres and Ocean 37: 271–294, with permission from Elsevier.

at other elevations it decreases toward zero. This has a profound influence on propagation of both internal and acoustic waves through the water. As time progresses beyond the stage shown in the figure, the layers begin to interact with each other and some will be eliminated, so finally two layers remain with only one interface in between. Ultimately, the density difference between these two layers decreases to zero, and the fluid becomes fully mixed. In contrast, such an experiment with values of Ric approximately 1 or lower does not produce layers (Figure 6). Instead, a mixed layer forms at the bottom and top of the fluid. Simultaneously, the interior stratification gradually decreases. The result is a fluid with values of N decreasing everywhere. Figures 5 and 6 were produced with data from experiments with the configurations shown in Figure 1(c). Accurate resolution of the vertical density field by a conductivity microprobe (developed for stratified turbulent flume measurements) allows precise measurements of the change of potential energy with time. This change is quantified by a flux Richardson number Rfc, defined as rate of change in potential energy divided by power (rate of energy) exerted by the stirrer (which is estimated for the grid using known drag laws). The resulting data are shown in Figure 7. Starting from Ric ¼ 0, experiments with increasing values have increasing values of Rfc, which level off at a value RfcC0.067 at RicC1. Almost all layered and continuously stratified experiments can be interpreted as having a flux Richardson number that reaches its maximum value

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 (kg m−3) Figure 6 The evolution of a density field with stirring source throughout the entire fluid and with Rir1. Adapted from Rehmann CR and Koseff JR (2004) Mean potential energy change in stratified grid turbulence. Dynamics of Atmospheres and Ocean 37: 271–294, with permission from Elsevier.

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LABORATORY STUDIES OF TURBULENT MIXING

10−2

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Figure 7 Mixing efficiency in grid-stirring experiments with continuous stratification and small Ri. Adapted from Rehmann CR and Koseff JR (2004) Mean potential energy change in stratified grid turbulence. Dynamics of Atmospheres and Ocean 37: 271–294, with permission from Elsevier.

of c. 0.1 at a Richardson number of order 1, with values decreasing toward 0 for larger and smaller values. The exact value of the maximum has been widely discussed, with some estimates approaching a maximum value of 0.2 and others only reaching a maximum value of 0.05 or so. Naturally, the exact definition depends on the choice of a length and velocity scale, which is not only a matter of choice, but also subjected to the details of each apparatus and analysis technique. In addition, since most experiments are either transient or possessing a variation in space, the measurement location

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LABORATORY STUDIES OF TURBULENT MIXING

salinity field is known to influence deep wintertime convection.

10−1

Rfc

Summary 10−2

10−3 103

104 Re

Figure 8 Mixing efficiency as a function of Reynolds number in mixing grid experiments of a uniformly stratified fluid. Stratification was from salinity in some experiments, temperature in others, and from both in one case. Adapted from Rehmann CR and Koseff JR (2004) Mean potential energy change in stratified grid turbulence. Dynamics of Atmospheres and Ocean 37: 271–294, with permission from Elsevier.

and time can influence a value. Thus, a maximum of RfC0.1 with a range of roughly 750% has remained unchanged over the last 20 years. However, so far most of the experiments have been conducted with Reynolds numbers of a few thousand or less, and with salt rather than temperature. Therefore the coverage in Re and Sc space is still limited. Finally there is evidence that a large Reynolds number might produce smaller maximum values (Figure 8), but comprehensive results are not yet available. A few studies have been conducted with two ingredients such as salt and temperature contributing density. If the value of Rf for each component differs, the phenomenon called ‘differential mixing’ is said to exist. At present, laboratory experiments do provide a small amount of evidence for differential mixing. Differential mixing is expected to be most important in oceanic regions where both temperature and salinity variations influence density, for instance, in polar regions. Therefore, for climate studies it would be important to incorporate differential mixing accurately in numerical models, especially since the

In summary, laboratory experiments to measure the rate of mixing of stratified fluids by turbulence have been conducted using a wide range of devices. Both qualitative and quantitative data provide information to oceanographers and numerical modelers. To estimate the mixing rates and their consequences in many ocean regions from the top mixed layer to the abyss, information about flux Richardson number from controlled laboratory measurements is vital. The buoyancy flux ratio rises from 0 at small Richardson number to values of about 0.1 at Richardson number equal to 1, and then falls off for greater values. At Richardson number greater than 1, a wide range of power laws spanning the range from  0.5 to  1.5 are found to fit data for different experiments. The layering in this range causes flux clustering and may contribute to the relatively large scatter in the data and to the wide range of proposed power laws.

Further Reading Breidenthal RE (1992) Entrainment at thin stratified interfaces: The effects of Schmidt, Richardson and Reynolds numbers. Physics of Fluids A 10: 2141--2144. Cenedese C, Whitehead JA, Ascarelli TA, and Ohiwa M (2004) A dense current flowing down a sloping bottom in a rotating fluid. Journal of Physical Oceanography 34: 188--203. Fernando HJS (1991) Turbulent mixing in stratified fluids. Annual Review of Fluid Mechanics 23: 455--493. Park YG, Whitehead JA, and Gnanadesikan A (1994) Turbulent mixing in stratified fluids: Layer formation and energetics. Journal of Fluid Mechanics 279: 279--312. Rehmann CR and Koseff JR (2004) Mean potential energy change in stratified grid turbulence. Dynamics of Atmospheres and Ocean 37: 271--294. Turner JS (1973) Buoyant convection from isolated sources. In: Buoyancy Effects in Fluids, pp. 165--206. Cambridge, UK: Cambridge University Press. Turner JS (1986) Turbulent entrainment: The development of the entrainment assumption and its application to geophysical flows. Journal of Fluid Mechanics 170: 431--471.

INTERNAL TIDAL MIXING W. Munk, University of California San Diego, La Jolla, CA, USA Copyright & 2001 Elsevier Ltd.

Introduction Any meaningful attempt towards understanding how the ocean works has to include an allowance for mixing processes. A great deal is known about the transport of heat and solutes by molecular processes in laboratory-scale experiments. Quite naturally, at the dawn of ocean science, these concepts were borrowed to speculate about the oceans. On that basis it was suggested by Zoeppritz that the temperature structure T(z) at 1000 m depth could be interpreted in terms of the time history T(t) at the surface some 10 million years earlier. It would indeed be nice if past climate could be inferred in this simple manner. Once it was realized that molecular diffusion failed miserably to account for the transports of heat and salt, generations of oceanographers attempted to patch up the situation by replacing the molecular coefficients of conductivity and diffusivity by enormously larger eddy coefficients, but leaving the governing laws (equations) unchanged. By an arbitrary choice of the magnitude of the coefficients it was generally possible to achieve a satisfactory (to the author) agreement between theory and observation, a result aided by the uncertainty of the measurements. But the clear danger signal was there; each experiment, each process required a different set of values. The present generation of oceanographers has come to terms with the need for understanding the mixing processes, just as they had come to terms some years ago with the need for understanding how ocean currents, ocean waves etc. are generated. Wind mixing and tidal mixing are very different processes. And the widespread use of parametrization of mixing processes will not succeed unless there is an underlying understanding of what is being parametrized. Here we have come a long way and have a long way to go.

Stirring and Mixing In a Newtonian fluid the down-gradient flux of a quantity y is given by Fy ¼ kdy=dx

½1

It is the very smallness of the molecular diffusivity k which requires large gradients dy/dx in order to attain significant fluxes Fy . The fundamental distinction between stirring and mixing was first made in 1948. Stirring produces the gradients whereas molecular mixing reduces the gradients. For the purpose of this article stirring and mixing are both included in the discussion of the contribution of internal tides to mixing processes. For an ocean in steady-state there needs to be an overall balance between the generation and dissipation of mean-square gradients. Nearly all of ocean dynamics deals with processes that generate gradients. This takes place over a wide variety of scales, all the way up to the scales of ocean basins. Dissipation takes place on the ‘microscale,’ i.e., millimeters to centimeters. This is the scale which includes the dominant contributions to the gradient spectrum. A further increase in the spatial resolution of the measurements does not lead to a significant increase in the measured mean-square gradients.

The Battle for Spatial Resolution It is very difficult to attain a quantitative measure of mixing in the turbulent ocean interior. The problem is the need for very high spatial resolution. An eddy coefficient can be estimated as follows: kmolecular  rmsðdy=dxÞ ¼ keddy  meanðdy=dxÞ

½2

(The subscript ‘molecular’ is introduced here to emphasize the distinction.) The ratio: mean square gradient/square mean gradient (the ‘Cox number’) has been used to estimate the ratio of the eddy coefficient to the molecular coefficient. A typical value away from boundaries is keddy ¼ 105 m2s1, two orders of magnitude in excess of the molecular coefficient. Achieving the required resolution has been a major accomplishment; but there are many problems with the measurements, and even more problems with the interpretation of the measurements along the lines of eqn[2]. It was only with the confirmation by a tracer release experiment that the community has come to accept the value kpelagic ¼ 105 m2s1 for the eddy diffusivity in the interior ocean, away from rough topography. There is of course considerable variability from place to place, but the surprising finding is not how large this variability is but how small it is.

381

382

INTERNAL TIDAL MIXING

Maintaining the Stratification

Tidal Dissipation: The Astronomic Evidence A quite different estimate of eddy diffusivity associated with pelagic mixing can be made from the following considerations. Bottom water is formed in the winter by convective overturning in just a few places: the Greenland Sea, the Labrador Sea and along the Antarctic continent. The formation is estimated at Q ¼ 25 Sverdrups (25  106 m3 s1). This would fill the oceans with cold water in a few thousand years. The reason this does not happen is that turbulent diffusion downward from the warm surface balances the upwelling of cold water. With reasonable assumptions this leads to an estimate of eddy diffusivity kstratification ¼ 104 m2s1, ten times the measured pelagic value. Measurements near topography do indeed give high diffusivities, orders of magnitude above the pelagic value. One simple interpretation is that there are concentrated areas of mixing (just as there are concentrated areas of bottom water formation) from which the water masses (but not the turbulence) are exported into the interior ocean. We can ask the question whether the global stratification can be maintained by vertical mixing in 10% (say) of the ocean volume with an average diffusivity 100 times the pelagic value? The work done against buoyancy by turbulent mixing in a stratified fluid can be written

eb ¼ kðg=rð dr=dzÞÞ ¼ kN2 W kg1

½3

where N is the buoyancy frequency. Only a fraction g (called the ‘mixing efficiency’) of the work goes into increasing potential energy (the rest goes into joule heat). A typical value is g ¼ 0.2. The total work per unit area is etotal ¼ eb/g. For the world ocean of area A, the total work done is

D¼A

Z

retotal dz ¼ gg  1kADr W

½4

where Dr ¼ 1 kg m3 is taken as the difference between surface and bottom density. Then for A ¼ 3.6  1014 m2 and k ¼ kstratification ¼ 104 m2s1, one has D ¼ 2 TW (1 terawatt ¼ 1012 W) for the power required to maintain the global stratification in the face of 25 Sverdrups of bottom water formation. To maintain the pelagic turbulence requires only 0.2 TW.

It is interesting to compare these numbers with the dissipation of tidal energy. We know this number with remarkable accuracy to be 2.570.1 TW for the principal lunar tide (M2); it is obtained from the measured rate of 3.8270.07 cm y1 at which the Moon is moving away from the Earth. For all solar and lunar tides the dissipation is 3.7 TW, but with lesser certainty. We note that the tidal dissipation is of the same magnitude as the 2 TW required for maintaining the ocean stratification. Is this an accident? The astronomic evidence tells us nothing about how and where the dissipation takes place. Allowing for dissipation in the solid Earth and atmosphere leaves 3.4 TW to be dissipated somewhere somehow in the ocean. Ever since it was estimated in 1919 that the dissipation in the Irish Sea is at 0.060 TW, the traditional sink has been in the turbulent bottom boundary layers of marginal seas, about 60 Irish Seas for the world. And before the astronomic estimates settled down to their present value, the ocean estimates kept rising and falling with the astronomic estimates.

Boundary Layer Dissipation Versus Scatter When we speak of tides we usually refer to surface (barotropic) tides which have a nearly uniform current velocity from top to bottom, and a maximum vertical displacement at the surface. However, there is also a class of internal (baroclinic) tides with velocities that vary with depth and with maximum displacements in the interior. A surface (barotropic) tide has essentially no shear in the interior ocean, There is shear near the bottom boundary, but the barotropic tidal velocities are so low in the deep ocean that the dissipation is negligible. In shallow seas the barotropic tidal currents are amplified, and the dissipation (proportional to the current cubed) is greatly amplified. This is the basis on which the global tidal dissipation has been attributed to the marginal seas. Internal tides are part of a larger class of internal waves with frequencies other than tidal frequencies. A possible mechanism of tidal dissipation is the scattering of surface tides into internal tides, with subsequent transfer of energy into the broad spectrum of internal waves, and finally into turbulent dissipation: surface tides-internal tides-internal waves-turbulence. What is required at the second stage is some nonlinear frequency splitting which

INTERNAL TIDAL MIXING

converts the low-frequency tidal line spectrum into a closely packed high-frequency continuum that resembles the observed internal wave spectrum. The final step is associated with the fact that the internal wave spectrum is at or near instability in the Richardson sense: the means-square shear is roughly 4 N2 (N is the buoyancy or Brunt-Va¨isa¨la¨ frequency). Scattering of surface tides into internal tides can take place along wavy bottoms. A second possibility is scattering along submarine ridges. An acoustic tomography experiment north at Hawaii detected internal waves of tidal frequency radiating northward.

Satellite Altimetry to The Rescue A subsequent analysis of satellite altimetry clearly showed internal tides emanating from the Hawaii submarine ridge. This is shown in Figure 1. The radiated power was estimated at 0.015 TW. So 14

383

Hawaiian chains will radiate 0.2 TW, enough to power the pelagic mixing associated with kpelagic ¼ 105 m2s1. The discovery of internal tide signatures by means of satellite altimetry was an altogether unexpected dividend from a technology that has revolutionized tidal analysis. The global sampling of surface elevation has introduced a new element into a subject that had gone to bed (in the opinion of some) with the work of Victorian mathematicians. With the Laplace tide equation as a guide for the tidal response of a nondissipative ocean, the assimilation of TOPEX/POSEIDON altimetry can lead to estimates of where one needs to introduce dissipation for agreement with the satellite data. The most recent result allocates roughly 1 TW to the open ocean, mostly over rough terrain. The tentative conclusion is that tidal dissipation is a significant factor in open ocean turbulent mixing.

Figure 1 Surface manifestation of internal M2 tides emanating from the Hawaii’an Island Chain. (Reproduced from Ray and Mitchum 1997.) The wiggly curves show the amplitudes along the ascending orbits of TOPEX/POSEIDON, with positive elevations on the north side. The dashed lines are the inferred crests of the mode 1 component of internal tides. Background shading corresponds to bathymetry, with darker areas denoting shallower water. The triangle to the north east shows the position of the tomographic array. Adapted from Ray & Mitchum (1997).

384

INTERNAL TIDAL MIXING

Supporting evidence comes from the measurements of tracer dispersion and microstructure in the Brazil Basin. Diffusivities of k ¼ 2  104  4  104m2 s1 at an elevation of 500 m above the abyssal hills of the Mid-Atlantic Ridge, increasing to 10  104 m2 s1 near the bottom have been obtained. Perhaps the most important result is that over a period of a month the diffusivities vary by a factor of two, with the large values occurring at spring tide and the small values at neap tide.

Discussion There is more than enough tidal dissipation to feed the measured pelagic turbulence associated with kpelagic ¼ 105 m2s1. With regard to the larger value kstratification ¼ 104 m2s1, the present best estimates would suggest that tidal dissipation could power half

the turbulence needed to account for the observed ocean stratification. Figure 2 attempts an allocation of tidal energy flux, but there are many uncertainties, some by factors of two or more. The assumed onedimensional balance between upward advection and downward diffusion as a measure of kstratification is itself somewhat uncertain. However, the present conclusion is that tidal dissipation is a significant, possibly dominant factor driving mixing in the ocean interior. The combination of detailed in situ measurements of turbulent mixing subject to a global lid on available tidal energy has led to giant strides towards a meaningful parametrization of ocean mixing, whether or not tidally induced. The conversion of wind energy to turbulent mixing plays a major role, particularly in the upper oceans. In Figure 2, equal weight has arbitrarily been assigned to tides and winds. We shall have to await the outcome of this competition.

See also 2

Dispersion and Diffusion in the Deep Ocean. Internal Tides. Internal Waves. Tides. Turbulence in the Benthic Boundary Layer.

Further Reading

3

Figure 2 Sketch of proposed flux of tidal energy (modified from Munk and Wunsch, 1997). The traditional sink is in the turbulent boundary layer of marginal seas. Scattering into internal tides over ocean ridges (by the equivalent of 14 Hawaii’s) and subsequent degradation into the internal wave continuum feeds the pelagic turbulence at a level consistent with kpelagic ¼ 105 m2s1. Most of the ocean mixing is associated with a few concentrated areas of surface to internal mode convergence over regions of extreme bottom roughness and with severe wind events. Light lines represent speculation with no observational support.

Cartwright DE (1999) Tides; a Scientific History. Cambridge: Cambridge University Press. Dushaw BD, Cornuelle BD, Worcester PF, Howe BM, and Luther DS (1995) Barotropic and baroclinic tides in the central North Pacific Ocean determined from longrange reciprocal acoustic transmission. Journal of Physical Oceanography 25: 631--647. Eckart C (1948) An analysis of the stirring and mixing processes in incompressible fluids. Journal of Marine Research 7: 265--275. Gregg MC (1988) Diapycnal mixing in the thermocline; a review. Journal of Geophysical Research 92: 5249--5286. Ledwell JR, Watson AJ, and Law CS (1993) Evidence for slow mixing across the pycnocline from an open-ocean tracer release experiment. Nature 364: 701--703. Munk W (1997) Once again: once again-tidal friction. Progress in Oceanography 40: 7--36. Munk W and Wunsch C (1997) The Moon, of course. Oceanography 10: 132--134. Munk W and Wunsch C (1998) Abyssal recipes II:: energetics of tidal and wind mixing. Deep-Sea Research I 45: 1977--2010. Ray RD and Mitchum GT (1997) Surface manifestation of internal tides in the deep ocean: observations from altimetry and island gauges. Progress in Oceanography 40: 135--162. Taylor GI (1919) Tidal friction in the Irish Sea. Philosophical Transactions of the Royal Society A 220: 1--93.

ESTIMATES OF MIXING J. M. Klymak, University of Victoria, Victoria, BC, Canada J. D. Nash, Oregon State University, Corvallis, Oregon, OR, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction Mixing in the ocean redistributes tracers, driving physical and biogeochemical dynamics. The mixing of the ‘active’ tracers, temperature and salinity, changes the density of seawater, creating pressure gradients that can drive mean currents. For example, in overturning circulations that range in scale from small estuaries to the global ocean, the mixing of buoyant fluid through the interior sets the conversion rate of potential energy and directly controls the strength of overturning. Momentum is also diffused by turbulent mixing, which transmits forces from the ocean surface and boundaries into the interior. The mixing of ‘passive’ scalars, such as nutrients, carbon dioxide, and oxygen, is important to understanding biological cycles in the ocean. Phytoplankton rely on vertical mixing to transport recycled nutrients into the sunlit near-surface waters. The mixing of carbon dioxide ultimately affects its storage in the ocean and removal from the atmosphere.

0.54

Most of the mixing in the interior of the ocean is thought to take place when internal waves break due to convective or shear instabilities. A numerical simulation serves to illustrate a typical ocean mixing event (Figure 1). A Kelvin–Helmholtz billow is triggered on an interface between warm and cold water when the warm water moves to the right faster than the cold water. A wave-like instability grows and two vortices form (Figure 1(a)). The vortices pair to create a breaking vortex (Figure 1(b)). Further instabilities ensue, creating a fully turbulent and three-dimensional flow field (Figure 1(c)) that decays as small-scale shear and temperature variability if mixed away. Breaking events like this are believed to dominate mixing in the ocean. They dominate because molecular diffusivity acting on ‘large-scale’ gradients (tens of meters) is very ineffective at mixing. Mixing is ultimately accomplished by molecular processes via Fickian diffusion, that is, the irreversible flux of property C is proportional to its three-dimensional gradient and the molecular diffusion coefficient kC: f C ¼ kC rC

½1

For temperature, a thermodynamic tracer, kTE10  7 m2 s  1 and for salinity and other tracers kSE10  9 m2 s  1. At large scales, representing the nonturbulent flow, gradients are small and the molecular flux is slow. In the absence of turbulence, a spike of

(a)

(b)

(c)

(d)

Z (m)

0.27 0.00 −0.27 −0.54 0.5 0.54

0.00

0.0

/o

Z (m)

0.27

−0.27 −0.54

−0.5

Figure 1 A numerical simulation of turbulent mixing (Smyth et al., 2001). The event is triggered by a shear instability between an upper layer of warm water (red) moving to the right and a lower layer of cold water (blue) moving to the left. The initial pair of vortices (a) combine to create a single large breaking event (b). This becomes fully turbulent and three dimensional (c) at which point there is large irreversible diffusion of heat. Diffusion continues until a large volume of mixed fluid results (d).

385

386

ESTIMATES OF MIXING

high temperature introduced into an otherwise isothermal fluid would take over 4 months to spread just 1 m through molecular diffusion alone. However, the stirring driven by the breaking of ‘fine-scale’ (order 1–10 m) waves maintains gradients at the ‘microscale’ (order 1 mm to 1 cm). The microscale gradients can be very large, Figures 1(b) and 1(c), driving a larger molecular flux that mixes large-scale gradients more efficiently. Typical ocean turbulence diffusivities are at least 100 times greater than molecular diffusivities, and can diffuse the above temperature spike over 1 m in O(1 day). It is useful to parametrize the turbulent stirring that drives the turbulent flux in terms of gradients of the mean fields, C. We do this by defining a ‘turbulent diffusivity’ KC so that FC E  KC rC

½2

Note that this parametrization has the same form as eqn [1], so we are drawing a direct analogy between the random walk that accomplishes mixing on the microscale and the stirring that takes place on the finescale of a breaking wave. The power of this concept is that the random walk of the stirring is on ‘large’ scales, and does not change for different tracers in the water. It is generally assumed that all variance created via stirring at large scales is ultimately transformed to small enough scales where it diffuses via molecular processes. Thus, an estimate of the turbulent diffusivity for one tracer may be applied to other tracers experiencing the same turbulent flow. Therefore we drop the subscript and discuss the turbulent diffusivity, K, as a dynamic property of the flow. The methods discussed below find that the turbulent diffusivity in the open ocean is KE100–1000 kT, and much more in shallow water and near topography. The accumulated effect of this mixing becomes an important term in understanding the circulation of the oceans.

Approaches to Quantifying Mixing The Advection–Diffusion Balance

The sequence of events shown in Figure 1 illustrates the different methods of how we quantify mixing in the ocean. Suppose we are interested in the mixing of temperature T in a fluid. It is often assumed that one can separate the scales of turbulent motions from those of the mean flow, allowing one to write: T ¼ T þ T0

½3

u ¼ u þ u0

½4

where the primes represent the ‘turbulent’ part of the flow and the overbars the mean quantities. In Figure 1, the horizontally averaged velocity and temperature represent the mean, and deviations from these the turbulence. This so-called ‘Reynolds decomposition’ allows us to transform the advection– diffusion equation qT/qt þ u  rT ¼ kTr2T into an evolution equation for the mean temperature T: DT ¼ kT r2 T þ r  hu0 T 0 i Dt

½5

¼ r  ðf T þ FT Þ

½6

Here, the material derivative D=Dt ¼ ð@=@t þ u  rÞ is the change in time following a parcel in the mean flow, and the angle brackets denote an average in time and space over a turbulent event. Equation [5] shows that T depends not only on T and u, but also on the correlation /u0 T 0 S which we term the turbulent heat flux, often approximated using the Fickian analogy [2] as: FT ¼ hu0 T 0 iE  KrT

½7

In most places in the ocean, the turbulent flux acting on the mean gradients is much greater than the molecular one, |FT|c|fT|, implying that we can drop the first term on the right-hand side of eqn [5]. Considering Figure 1 again, suppose we integrate eqn [5] over a volume, v, defined by the lower half of the domain, with z ¼ 0 the top of the volume. There are no fluxes out the sides or bottom of the volume, and there is no mean flux out the top ðwðz ¼ 0Þ ¼ 0Þ. The only flux of temperature is the turbulent one through z ¼ 0, so that the change of temperature in the volume can be calculated by @ @t

Z

T dV ¼  V

I

hw0 T 0 i dA

½8

A

where A is the surface at z ¼ 0. In Figure 1, there is an increase in the mean temperature of the volume, so the left-hand side of eqn [8] is greater than zero. The tendrils that drop below z ¼ 0 are warmer than the mean, so T 0 40, and they are moving down, so w0 o0, therefore w0 T 0 o0. If the tendrils are completely diffused away by molecular mixing in the volume, then warm water will have been left behind and the temperature in the volume will increase. The real situation is more complicated. Tendrils are further strained and stretched, and some of the warm water rises again out of the volume. However, on average some is always exchanged so that /w0 T 0 So0 and net warming takes place in the lower volume. Note that there is an equal amount of cooling in the upper volume.

ESTIMATES OF MIXING

The Gradient-Variance Balance

In order for stirring to be irreversible, the gradients produced must be diffused away by molecular processes. For temperature, this is described formally through the evolution equation for turbulent temperature gradient variance |rT 0 |2. Temperaturegradient variance is a nonintuitive quantity to consider, but it is the best measure of ‘stirring’, and is related to the thermodynamic quantity of entropy. For steady state, homogeneous turbulence it can be shown that: D E 2 ½9 hu0 T 0 i  rTEkT jrT 0 j This states that the net production of gradient variance by turbulent velocities is balanced by its destruction by molecular diffusion (there are transport terms that have been dropped, hence the approximation). The averages represented by the angle brackets must be collected over long enough time that the irreversible part of the ‘turbulent’ flux is measured. The rate of destruction of the turbulent gradients is fundamental to quantifying mixing in the ocean and is written as D E 2 ½10 w  2kT jrT 0 j

Large-Scale Estimates Large-scale estimates are made on quantities measured on vertical scales greater than a meter. They are based on estimating the mixing terms in the advection–diffusion equation of the tracer C: @C þ u  rC ¼ r  ðKrCÞ @t

½11

where again, K is the turbulent diffusivity. Often the right-hand side is replaced by @=@zðK@C=@zÞ because mean vertical gradients exceed horizontal ones, further reducing to K@ 2 C=@z2 for spatially uniform K. Purposeful Tracer Releases

The first and conceptually simplest method to measure mixing is to release a man-made tracer and measure its vertical spread over time. Briefly, if we follow the parcel of water and assume a constant vertical diffusivity K, then the spread of the tracer, C, is governed by the diffusion equation @C @2C ¼K 2 @t @z

½12

387

If the tracer is injected as a spatial delta function at t ¼ 0, then the solution is an ever-widening vertical cloud described by a Gaussian. The larger the K, the faster the cloud spreads. Direct tracer releases are elegant and definitive in their results. There are no confounding sources or sinks of the dye in the ocean, so tracking the vertical spread is unambiguous. (Please note that care should be taken interpreting the terms ‘vertical’ and ‘horizontal’ or ‘lateral’. By these terms we really mean perpendicular and parallel to surfaces of constant density respectively, or ‘diapycnal’ and ‘isopycnal’. The distinction is conceptually important, but notationally less convenient, so we use the shorthand here.) However the technique is difficult to perform, limiting its routine use. Specialized equipment and analysis methods are needed to release the dye and then analyze the water samples to find minute quantities of tracer in the water. Furthermore, horizontal stirring and advection of a dye patch can spread it horizontally to such an extent that it is very difficult to find all the dye using finite ship resources. Tracer Budgets (Inverse Methods)

The rate of mixing can be estimated from an integrated version of the mean tracer equation (eqn [11]) if we constrain a volume of water and assume its contents are in steady state. A concrete example of the budget method is from data collected in the Brazil Basin in the Southwest Atlantic. Here, water colder than 1 1C produced in the Antarctic flows north into the basin through the Vema Channel (Figure 2). No water that cold is observed to leave the basin, therefore the incoming water must be warmed by mixing before it leaves. Quantitatively, the mixing estimate is made by integrating the remaining terms in eqn [11] over the volume: Adv: top

in Vema

Turb: top

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflffl I ffl}|fflfflfflfflfflfflfflffl{ zfflfflfflfflffl I ffl}|fflfflfflfflfflffl{ I uT dA1  wT s dAS ¼ FT dAs 1

S

½13

S

We also know that there is just as much water entering the volume as leaving through the upper surface, so QS ¼ Q1, where QS ¼

I

w dAS

½14

S

is the volume flux through the upper surface, and Q1 is similarly defined and is the flux through Vema Channel. Only the advective transport of heat into

388

ESTIMATES OF MIXING

s

FT dAs

Ts uT dA1

Ts

w dAs s

Figure 2 Sketch of heat fluxes in and out of a volume bounded in the vertical by an isotherm Ts, and at a strait by the dashed line.

the basin needs to be measured (the left-hand side of [13]) to determine the turbulent heat flux through the upper bounding surface. Using a combination of moorings and shipboard cruises, Hogg et al. estimated these transports by assuming that the deviations from the mean temperature and velocity values entering Vema Channel are uncorrelated: I

  uT dA1 EQ T 1

½15

1

They then defined the upper surface as the TS ¼ 1 1C isotherm and assumed a constant vertical temperature gradient along that surface, allowing the mean turbulent diffusivity at that surface to be determined as

difficult amount of change to detect in open ocean basins with any confidence. Furthermore, the velocities and tracers measured at the boundaries of the volume must be well constrained and shown to be in steady state. This is very difficult as velocities and tracers are estimated from a few individual ship tracks that each take a month or more to complete, often months or years apart. Not surprisingly, the most convincing inverse estimates have come from well-constrained topographies like the Brazil Basin where the flow and temperatures into the basin can be monitored with a few long-term moorings in the channel, and the bounding isotherm can be mapped with a high degree of confidence.

Fine- and Microscale Estimates 

K¼

  @T Q Ts  T1 As @z

1 ½16

6 3 1 For the Brazil   Basin, Q ¼ 3.7  10 m s , AS ¼ 5  12 2 10 m , T 1 ¼ 0:35 1C, and the mean temperature gradient at the 11 C surface is @T/@zE  2  10  3 1C m  1. The mean turbulent diffusivity across this interface is therefore KE2:5  104 m2 s1 . Inverse estimates of mixing are routinely made from hydrographic sections in the ocean, where the same concepts are used to find the flux of heat and density at different depths in a series of volumes. Often many vertical layers and sections are used. If well constrained, this method of estimating the heat flux (and thus the diapycnal mixing) would be unambiguous in providing basin-average estimates. A one-dimensional advection–diffusion balance can be used to estimate the turbulent diffusivity in the deep ocean from large-scale tracer profiles. This simple inverse estimate finds average turbulent diffusivities of K ¼ 10  4 m2 s  1 in the North Pacific. The principal difficulty with the inverse method is the assumption that the system is in steady state. Inverse estimates in the open ocean indicate vertical velocities of a couple of meters per year. This is a

Fine- and microscale measurements estimate turbulent stirring or mixing by directly observing the turbulence. These methods have the advantage over budget-based estimates in that they also elucidate what causes the turbulence. However, these methods require specialized instrumentation as the sensors used to measure microscale quantities must be small, respond quickly, and be capable of recording a very large dynamic range. In addition, the vehicles they are mounted on must suppress vibrations as much as possible to prevent contamination of the small-scale signals (Figure 3). Vibration and spatial resolution concerns limit the speed with which these profilers can be dropped or towed as well. In the following, we describe a series of methods that (1) directly measure the turbulent stirring of a fluid using the eddy correlation technique, (2) directly measure the molecular destruction of temperature gradients, (3) estimate the mixing by relating the buoyancy flux to the energetics of the turbulence. Finally, we describe two techniques that enable mixing to be estimated from large-scale measurements of (1) statically unstable fluid (Thorpe scales) and (2) energy in the internal wave field (the Gregg– Henyey method).

ESTIMATES OF MIXING

389

Figure 3 Microstructure platforms that the authors have worked with. In all cases the sensors are on the nose of the vehicles. The upper three are profilers. Left to right: Absolute velocity profiler (University of Washington), advanced microstructure profiler (University of Washington) and Chameleon (Oregon State University). The lower instrument, Marlin (Oregon State University), is towed.

Direct Eddy Correlation

With careful and specialized measurements it is possible to estimate mixing by quantifying the stirring of the fluid. As described above, this means quantifying the stirring of cold water upward into warm water by measuring vertical velocity fluctuations, w0, and temperature fluctuations, T 0. One of the few attempts to apply this method in the open ocean demonstrates its difficulty (Figure 4). Temperature and velocity were acquired along a horizontal path using a towed instrument outfitted with thermistors and shear probes. While the raw signals are large and very active (Figure 4(a)), the product w0 T 0 is not one-sided. Instead, it has instantaneous values that are large and can be of either positive and negative sign, such that the fluctuations are far greater than the mean correlation /w0 T 0 S. This is a general problem since turbulence is sporadic, and stirring is both downgradient (i.e., transports heat from regions of warm fluid) and upgradient (i.e., transports heat to regions of warm fluid), the latter representing restratification of partially mixed fluid. Since much of w0 T 0 is reversible (i.e., just stirring fluid

that is not immediately mixed), the eddy correlation technique must be made over long times to produce stable estimates of the irreversible flux. In this case, the background vertical gradient dT=dz is positive, so /w0 T 0 So0 represents a downgradient flux, as appears most frequently in Figure 4(b). This method does not enjoy wide use. Determining what is ‘mean’ and what is ‘turbulent’ is very difficult from the limited measurements possible with a vertical or horizontal profiler. The data presented here were simply bandpassed, with large scale motions considered to be nonturbulent. However choosing what is ‘large scale’ requires some art. A second difficulty is estimating the vertical velocities in the ocean. In this instance, the vertical velocities were w0 E0.03 m s  1, large enough that the method was deemed possible. The final limitation is gathering enough statistics of the turbulence to make estimates of mean fluxes. Microscalars (Osborn–Cox method)

In contrast to measuring the fine-scale stirring of the tracer (i.e., /w0 T 0 S above), the Osborn–Cox method

390

ESTIMATES OF MIXING

(a)

Tope S7 Blocks: 2197-2244 0.08

0.03 0.02

0 −0.04

0.01

−0.08

w (m s−1)

 (°C)

0.04

0

−0.12

−0.01

−0.16 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

0

5

10

15

20

25

30

35 40 x (m)

45

50

55

60

65

70

75

w (°C m s−1) × 104

(b)

1 0

1

Figure 4 (a) Spatial series of high-passed temperature and vertical velocity from a towed vehicle in the upper ocean thermocline (Fleury and Lueck, 1994). The high-passing procedure is meant to emphasize turbulent fluctuations. (b) The correlation between these observations.

quantifies the rate of molecular diffusion of scalar variance at the microscale. By considering the evolution equation for microscale scalar variance, the turbulent diffusivity K is related to the rate of destruction of scalar variance w (eqn [10]). Because this method measures the rate of irreversible molecular mixing, it is one of the most direct measures of quantifying K. The most commonly measured scalar is temperature, as it is relatively easy to measure and as it diffuses at larger scales (i.e., 1 mm to 1 cm) than chemical constituents like salt. To measure such scales, a sensor must be small and respond very rapidly. Microbead thermistors – coated with a thin film of glass to electrically insulate them from seawater (Figure 5) – are generally used for this purpose. This sensor yields high-resolution temperature gradients such as those shown in Figure 6(b). Only one dimension of the gradient is measured, so we estimate w as *  + @T 0 2 ½17 w ¼ 6kT @z where we have assumed that the turbulence is isotropic (i.e., the variance of the gradients is the same

in all directions). The turbulent diffusivity simply relates the intensity of small-scale gradients to the large-scale temperature gradient as D E ð@T 0 =@zÞ2 w ½18 K ¼ 3kT  2 ¼  2 @T=@z 2 @T=@z Like other methods, there are a number of limitations. Temperature probes require a finite amount of time to diffuse heat through their insulation and the thermal boundary that develops in the surrounding seawater. This smooths the signals and coarsens the measurement. If probes can be lowered slowly enough to allow heat to diffuse through the coating, but fast enough to capture a synoptic snapshot of the turbulent event, then all of the gradient variance could be resolved and w measured. However, the required 10–20 cm s  1 profiling speed reduces the number of realizations that may be captured, so there is a trade-off between resolution and statistics. In practice, most sensors are deployed too rapidly and not all of the variance is measured. Corrections can be applied by fitting data to a universal spectrum (i.e., the ‘Batchelor’ or ‘Kraichnan’ spectrum) which allows extrapolation of resolved

ESTIMATES OF MIXING

0.25 mm

2

6.4-mm Stainless Hard diameter steel tube epoxy

0.15mm

3

1

4

391

Rubber tip

5

Bimorph Heat shrink beam tubing

Electrical leads

Figure 5 Three common sensors used on microstructure instruments. Left: A glass-bead thermistor (Gregg, 1999); upper scale in mm. Center: a four-electrode microconductivity probe (Nash and Moum, 1999). 1–4 are electrodes and 5 is an insulating glass. Right: Schematic of a piezoelectric shear probe (Gregg, 1999).

(a) T (°C)

5.12 5.11

(b)

0.5

T′

5.1

0

* v /* x (s1)

(c)

0.5 0.1 0 0.1

* w /* x (s1)

(d)

0.1 0 0.1 100

105

110

115

120

125 Time (s)

130

135

140

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150

Figure 6 Microscale data collected using the towed vehicle Marlin in the open ocean (Moum et al., 2002). This data was collected with the instrument moving at 1 m s  1, so time is equivalent to distance in m. (a) Temperature, (b) temperature gradient, (c) and (d) velocity gradients (shear). Note that where there is high velocity variance there is usually high temperature variance.

measurements to higher wavenumbers (Figure 7). Unfortunately, the spectral levels and wavenumber extent of the spectrum both depend on the turbulent kinetic energy dissipation rate e, so an independent measure of microscale shear variance (see below) is required to accurately apply such corrections. It is also possible to use microconductivity probes to measure temperature on spatial scales of 10  3 m (Figure 5, center panel). Conductivity is a rapid measurement, so speed through the water does not limit probe response, only the physical configuration of the probes. The difficulty with this measurement is

that conductivity depends on both the temperature and salinity of seawater. Thus, microconductivity works best for determining the mixing rate of temperature in water that has small salinity variations. The last difficulty, shared with the other microscale methods, is that turbulence is intermittent, so a large number of samples need to be made in order to characterize the turbulence level of a given locale. However, unlike the estimate /w0 T 0 S, w is a direct measure of irreversible mixing, so does not need multiple realizations of the same turbulent event to converge.

392

ESTIMATES OF MIXING

dT/dz ((K m−1)cpm−1)

10−3

10−3 Weak turbulence

Strong turbulence Response 10−4

10−4

10−5

10−5 raw

fit 10−6 10−1

100

10−6 10−1

101

100

101

kx (cpm) Figure 7 Fit of universal turbulence spectra (red curves) to data spectra collected near Hawaii (Klymak and Moum, 2007). The raw signals (gray) have been corrected (black) for the temporal response of the thermistor. Note that noise at high wavenumbers may contaminate the spectra of weak turbulence.

Estimates from Energy Considerations (Osborn Method)

The most common method of estimating ocean mixing rates is based on quantifying the energetics of the turbulence, not the mixing itself. It is widely employed because the energetics can be measured from rapidly profiling sensors, and because energy measurements are useful in their own right. This method was originally proposed by Osborn in 1980, and is based on the observation that temperature-gradient variance occurs in accord with velocity gradient (shear) variance (i.e., compare Figures 6(b) and 6(c)). The argument is based on the local balance of turbulent kinetic energy. A turbulent event loses energy by viscous dissipation and by changing the background potential energy of the flow due to irreversible mixing. In a steady state, or time-averaged sense, this is expressed as P ¼ e þ Jb

½19

where P is the rate of production of turbulence by the mean flow due to some wave-breaking process, e is the rate of turbulent energy dissipation by viscosity, and Jb is the irreversible buoyancy flux due to mixing. The buoyancy flux is directly related to the turbulent mass flux Jb ¼ ghr0 w0 i=r where g is the gravitational acceleration, and r ¼ r þ r0 is the density. The method assumes that the turbulent buoyancy flux is a fixed ratio G of the dissipation:

unstratified water, the buoyancy flux must be zero by definition, but e can be substantial. However, observations indicate that for much of the stratified ocean GE0.270.05. (Note that G can be related to the ‘mixing efficiency’ Rf ¼ Jb/P ¼ G/(G þ 1) via eqn [19]). Measurements of the dissipation rate of turbulent kinetic energy e are made from microstructure profilers (Figure 3) in much the same way that measurements of w are made. A small shear probe (Figure 5, right panel) measures velocity shear (Figures 6(c) and 6(d)). The shear spectrum is calculated and integrated to get an estimate of the dissipation rate of turbulent kinetic energy e, again using universal spectra as a guide. Fortunately, both the spectral amplitude and wavenumber extent of shear spectra scale with e, so that universal spectra may be fit to a limited range of wavenumber range of the shear spectrum, avoiding poorly resolved wavenumbers. But unlike the measurement of w using thermistors, measurement of turbulent shear variance is easily contaminated by the slightest vibration of the measurement platform. This necessitates the use of specialized profilers that minimize coherent eddy shedding and decouple ship motion from the sensor (Figure 3). Just like temperature, the flux of density can be parametrized by a turbulent diffusivity so that Jb ¼ 

g @r K ¼ KN 2 r0 @z

for stratification Jb EGe

½20

This is a somewhat bold assumption as obvious counterexamples can be found. For instance, in

g N ¼ r0 2

!

dr dz

!

½21

ESTIMATES OF MIXING

Combining, we use measurements of e to estimate e K¼G 2 N

½22

This method has greatly increased the number of estimates of mixing in the ocean. Microstructure profilers have been deployed in a wide array of environments: in the open ocean, over rough and abrupt topography, and in coastal waters, giving us a large variety of environments in which ocean mixing has been estimated. As tenuous as it is, the assumptions used in this method are mitigated by the fact that we know that dissipation rates vary by orders of magnitude throughout the ocean so that the distribution of dissipation roughly mimics the distribution of mixing, except in well-mixed regions. Thorpe scales The dissipation rate can also be estimated from less-specialized instruments. Breaking internal waves, like that shown in Figure 1, lift

p (MPa)

(a)

393

dense water above light water. The dissipation rate of the water that goes into turbulence from this uplift can be estimated from the size of the overturn LT : eE0:64L2T N 3

½23

This has been shown to give unbiased estimates of the dissipation rate if enough profiles are collected. Note that, at open ocean dissipation rates and stratifications, LT is quite small and the small density differences make detecting overturns subject to noise constraints. A coastal example is shown in Figure 8, where braids indicative of shear-instabilities drive density overturns with LTE5 m. The overturns coincide with strong turbulence. There is also a strong correspondence of e and w in these data. This study compared the two estimates of K from these separate microstructure estimates and found similar results.

100 300 500 700 900 1100 1300 1500 Distance (m) + + + + + + + + + ++ + + + + + + + + ++ + + + + + + + + + + + AMP # 51 52 53 54 55 56 57 58 −72 0.2 23.5 VSS (dB) 0.4 23.75 −77 24.0 0.6 24.25

−82

0.8 (b) 0.2 p (MPa)

23.5 0.4

log  (K2 s−1)

0.6

24.0 24.25

23.75

p (MPa)

(c) 0.8 0.2 0.4

−10

23.5 0

10

0.6

24.0

23.75

Lt (m)

24.25

(d) 0.8 0.2 p (MPa)

23.5 0.4

0.8

23.75

24.0

0.6

log  (W kg−1)

24.25 0

0.2

0.4 Time (h)

0.6

0.8

Figure 8 An example of a deterministic turbulent event measured four ways. This is a shear instability, similar in dynamics to the instability in Figure 1, observed in Admiralty Inlet, Washington (Seim and Gregg, 1994). (a) Acoustic backscatter from turbulence microstructure. This visualizes the braids between 0.2 and 0.6 h. Vertical white lines are microstructure profiler drops, horizontal white lines are contours of r. (b) w estimated from profiler. (c) Density overturns measured with the profiler. Note how there are large overturns associated with the braids. (d) The turbulence dissipation rate, e estimated from shear probes. For all the panels, 0.1 MPa ¼ 10 dbar ¼ 10-m depth.

394

ESTIMATES OF MIXING

−3

log(K ) m2 s−2

1000 −4

6000

0

2000

Hawaiian Rise

5000

−5

Sitito Iozima Ridge

4000

Japan

3000

Caroline Ridge

Admiralty Islands Ridge

z (m)

2000

4000

6000

GM IW

−6

8000

10 000

12 000

14 000 Moonless Smts

Isu Trench

Nankei Trench

Papua New Guinea

r (km)

Figure 9 Indirect estimate of mixing from the western Pacific Ocean using large-scale oceanic data (Kunze et al., 2006). Internal wave energy levels were used to estimate e, and hence K.

Gregg–Henyey method The Osborn method may be extended further using the observation that away from boundaries and strongly sheared currents the dissipation rate is directly related to the energy in the internal wave field. Models have been developed that estimate the rate at which energy cascades through a steady-state internal wave field. It is easier to estimate the energy of the wave field with finescale sensors than it is to directly measure the microscale. For instance, these methods have allowed the estimate of mixing using routine hydrographic data of the world oceans (Figure 9). The internal-wave energy method has limited application in regions where the internal wave field is not in equilibrium with external forcing, in particular near topography, or where turbulence is generated by noninternal wave process at boundaries. Somewhat frustratingly, this is where the dissipation rates are the strongest.

microscale methods do not agree as well. Inverse methods indicate turbulent diffusivities on the order of K ¼ 10  4 m2 s  1, while microstructure measurements are challenged to find average turbulent diffusivities this high. Recent attention has been directed toward finding enhanced mixing near boundaries. This zeroth-order problem will continue to require much effort and ingenuity to solve. Higher-order testing of the assumptions that go into these measurements are ongoing, aided by innovations in measurements and numerical methods.

Summary

e

Substantial effort has gone into estimating the rate of mixing in the ocean. The problem is hard to tackle directly, so great ingenuity has been used to devise indirect methods of making the observations. Mixing measurements have been made in many environments in the open and coastal ocean, lakes, and estuaries. In the Brazil Basin, where the source of deep water is well constrained, estimates of K from the basin-scale estimates agree quite well with microstructure estimates. In the open ocean, however, large-scale and

Nomenclature fC FC Jb K N G

kC w

molecular flux of scalar C turbulent flux of scalar C turbulent buoyancy flux turbulent diffusivity buoyancy frequency ratio of buoyancy flux to viscous dissipation rate of turbulent kinetic energy dissipation molecular diffusivity for scalar C rate of temperature variance dissipation

See also Energetics of Ocean Mixing. Internal Tides. ThreeDimensional (3D) Turbulence. Upper Ocean Mixing Processes. Wind- and Buoyancy-Forced Upper Ocean.

ESTIMATES OF MIXING

Further Reading Dillon TM (1982) Vertical overturns: A comparison of Thorpe and Ozmidov length scales. Journal of Geophysical Research 87: 9601--9613. Eckart C (1948) An analysis of the stirring and mixing processes in incompressible fluids. Journal of Marine Research 7: 265--275. Fleury M and Lueck R (1994) Direct heat flux estimates using a towed vehicle. Journal of Physical Oceanography 24: 810--818. Ganachaud A and Wunsch C (2000) Improved estimates of global ocean circulation, heat transport and mixing from hydrographic data. Nature 408: 453--457. Gregg MC (1989) Scaling turbulent dissipation in the thermocline. Journal of Geophysical Research 94: 9686--9698. Gregg MC (1999) Uncertainties in measuring e and wt. Journal of Atmospheric and Oceanic Technology 16: 1483--1490. Henyey FS, Wright J, and Flatte´ SM (1986) Energy and action flow through the internal wave field. Journal of Geophysical Research 91: 8487--8495. Hogg N, Biscaye P, Gardner W, and Schmitz WJ, Jr. (1982) On the transport and modification of Antarctic bottom water in the Vema Channel. Journal of Marine Research 40: 231--263. Johnson HL and Garrett C (2004) Effects of noise on Thorpe scales and run lengths. Journal of Physical Oceanography 34: 2359--2373. Klymak JM and Moum JN (2007) Interpreting spectra of horizontal temperature gradients in the ocean. Part II: Turbulence. Journal of Physical Oceanography 37: 1232--1245. Klymak JM, Moum JN, Nash JD, et al. (2006) An estimate of tidal energy lost to turbulence at the Hawaiian Ridge. Journal of Physical Oceanography 36: 1148--1164. Klymak JM, Pinkel R, and Rainville L (2008). Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. Journal of Physical Oceanography 38: 380–399.

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Kunze E, Firing E, Hummon JM, Chereskin TK, and Thurnherr AM (2006) Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. Journal of Physical Oceanography 36: 1553--1576. Lumpkin R and Speer K (2003) Large-scale vertical and horizontal circulation in the North Atlantic Ocean. Journal of Physical Oceanography 33: 1902--1920. Moum JN (1996) Energy-containing scales of turbulence in the ocean thermocline. Journal of Geophysical Research 101: 14095--14109. Moum JN, Caldwell DR, Nash JD, and Gunderson GD (2002) Observations of boundary mixing over the continental slope. Journal of Physical Oceanography 32: 2113--2130. Nash J, Alford M, Kunze E, Martini K, and Kelley S (2007) Hotspots of deep ocean mixing on the Oregon continental slope. Geophysical Research Letters 34: L01605 (doi:10.1029/2006GL028170). Nash JD and Moum JN (1999) Estimating salinity variance dissipation rate from conductivity measurements. Journal of Atmospheric and Oceanic Technology 16: 263--274. Osborn TR (1980) Estimates of the local rate of vertical diffusion from dissipation measurements. Journal of Physical Oceanography 10: 83--89. Seim HE and Gregg MC (1994) Detailed observations of a naturally occurring shear instability. Journal of Geophysical Research 99: 10049--10073. Smyth WD, Moum JN, and Caldwell DR (2001) The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations. Journal of Physical Oceanography 31: 1969--1992. St. Laurent LC, Toole JM, and Schmitt RW (2001) Buoyancy forcing by turbulence above rough topography in the abyssal Brazil Basin. Journal of Physical Oceanography 31: 3476--3495. Wunsch C and Ferrari R (2004) Vertical mixing, energy, and the general circulation of the oceans. Annual Review of Fluid Mechanics 36: 281--314.

ENERGETICS OF OCEAN MIXING A. C. Naveira Garabato, University of Southampton, Southampton, UK & 2009 Elsevier Ltd. All rights reserved.

Introduction One of the defining features of the ocean’s physical environment is its nearly ubiquitous stable stratification (Figure 1). Aside from a relatively thin and homogeneous layer that is widely found near the surface (the so-called upper ocean mixed layer), the density of the ocean increases monotonically with depth in a perceptible manner, the rate of this increase generally declining toward the ocean floor. Current views on the origin of the ocean stratification began to

take form in the early twentieth century, as the oceanographers Georg Wu¨st and Albert Defant and other pioneers obtained the first clear picture of the temperature, salinity, and density distributions of the deep ocean. These unprecedented observations brought about the revelation that much of the ocean is occupied by a few relatively cold and dense water masses that are formed and sink within two specific high-latitude regions: the northern North Atlantic and the Southern Ocean. Critically, as each of these water masses flows away from its formation region and pervades large areas of the globe, its initially distinct properties are eroded by mixing with surrounding waters that are, on average, lighter. As a result, the density of water masses originating at high latitudes often decreases along their path, to a point where the waters become light enough to return to the surface.

ACC 0

27.5 1000

Depth (m)

2000

28.0

3000

4000

5000

Southern Ocean 6000 − 80 − 60

North Atlantic −40

−20

0

20

40

60

Latitude (° N) Figure 1 Vertical distribution of neutral density in the Atlantic Ocean along 301 W. Contours denote isopycnal surfaces with values between 23 and 28.4 kg m  3 at intervals of 0.1 kg m  3. Colors indicate three density classes (separated by the 27.5 and 28.0 kg m  3 isopycnals) that are subject to different mixing regimes. The stratification in the abyssal ocean (purple) arises primarily from the balance between upwelling of dense water produced in the high-latitude Southern Ocean and turbulent diapycnal mixing elsewhere. In contrast, the stratification within and above the permanent pycnocline (orange) is mainly shaped by the wind- and eddy-driven subduction of near-surface water masses along isopycnals. Finally, the thick layer at the base of the permanent pycnocline, which outcrops into the upper ocean mixed layer within the Antarctic Circumpolar Current (ACC), represents a transition between the pycnocline and abyssal mixing regimes and is subject to a combination of both. The dashed arrows indicate the broad sense of the global ocean overturning driven by this set of mixing processes.

396

ENERGETICS OF OCEAN MIXING

This simple conceptual model lies at the heart of present views of the climatically key overturning circulation of the ocean. Implicit in the model is a competition between the removal of buoyancy from the deep ocean by dense water formation at high latitudes, and the addition of buoyancy to the deep ocean by downward mixing of light upper ocean waters, which are warmed directly by the sun. Come the second half of the century, the realization that the existence of the ocean’s overturning circulation entails the antagonistic interaction between buoyancy forcing at the sea surface and mixing in the ocean interior excited an animated debate around the driving forces of the circulation that has persisted to this day. The focus of the discussion has been on understanding the circulation’s energetics, as it is the way in which energy enters, flows through, and exits the ocean that strictly defines how the overturning circulation is driven. It is on the basis of energy considerations that early notions of surface buoyancy forcing as the governing rate-limiting process of the overturning have been challenged most convincingly in favor of ocean mixing. A central ingredient of this line of reasoning is a result put forward by Johan Sandstro¨m in 1908, stating that a fluid’s motion cannot be sustained by heating and cooling at the fluid’s surface if the source of heating lies at the same level as or above the source of cooling, such as is the case with buoyancy forcing at the sea surface. In other words, heating and cooling at a fluid’s surface do only minimal work on (i.e., input very little energy to) the fluid. Although subtle differences between the ocean and the idealized fluid that Sandstro¨m described have been argued to limit his result’s applicability to the oceanic context, it is now widely believed that mixing processes constitute the primary driving force of the overturning circulation. In this prevalent view, buoyancy forcing at the sea surface exerts an important influence on the structure of the ocean’s overturning and stratification that is particularly pronounced in transient oceanic states associated with large-scale climatic change, but cannot by itself sustain the circulation indefinitely. Thus, the problem of understanding how the overturning circulation is driven can be reduced to that of determining the energetics of ocean mixing.

The Global Ocean’s Energy Budget In order to fully appreciate the intimate link between the overturning circulation and mixing in the ocean interior, it is helpful to consider the global ocean’s energy budget. This can be formally synthesized in the global budgets of kinetic, potential, and internal

397

energy, which can respectively be written as: Z Z Z @=@t

rK dV ¼  

Z Z Z Z

rKðu  us Þ  n dA ½pu þ mrK  n dA

 CK2P þ CI2K  CK-I ½1 Z Z Z @=@t

rP dV ¼ 

Z Z

rPðu  us Þ  n dA Z Z Z þ r@Ptide =@t dV þ CK2P

½2

and Z Z Z @=@t

rI dV ¼  

Z Z Z Z

rIðu  us Þ  n dA ½Frad  rcp kT rT

 @H=@S rkS rS  n dA  CI2K þ CK-I

½3

In these expressions, K, P, and I are the kinetic, potential, and internal energies per unit mass. The terms on the left-hand side of each equation denote the rate of change with time (t) of K, P, and I scaled by the water’s potential density (r) and integrated over the global ocean volume. These terms equal zero in the steady-state limit that is relevant to our discussion. In turn, the first terms on the equations’ right-hand sides describe the advection of the various forms of energy through the ocean surface, with u indicating the oceanic velocity, us the velocity of the free ocean surface, n a unit vector normal to that surface, and the integral being taken over the global ocean surface area. These advective terms are thought to constitute a significant source of energy to the ocean, but much of it is expended in small-scale turbulence within the upper ocean mixed layer and does not penetrate into our domain of interest, the stratified ocean interior. The second terms on the equations’ right-hand sides represent the three remaining candidate sources of the ocean interior’s energy. The term in the kinetic energy equation stands for the work done on the ocean by differential pressure (p) and viscous stresses (m is the kinematic viscosity of seawater) associated with the wind blowing on the sea surface. The differential pressure contribution is the dominant one. The term in the potential energy equation denotes the transfer of energy (expressed as a time-varying potential energy per unit mass, Ptide) from the Earth–

398

ENERGETICS OF OCEAN MIXING

Moon–Sun system to the ocean by the continuous tidal displacement of the oceanic mass by gravitational forces. Finally, the term in the internal energy equation embodies surface and geothermal buoyancy forcing and amalgamates three different contributions: the radiative flux of internal energy between the near-surface ocean and overlying atmosphere/ice (Frad), and the diffusive fluxes of internal energy brought about by molecular-scale mixing of temperature (T) and salinity (S) with diffusivities kT and kS (cp and H are the specific heat capacity of seawater at constant pressure and the enthalpy of water, respectively). As advanced by Sandstro¨m’s result and reiterated by most (though not all) available recent estimates, the net buoyancy work done on the ocean by exchanges with the atmosphere is likely to be minimal. Since this has also been shown to be the case for the geothermal heating contribution, the second term on the right-hand side of the internal energy equation can be neglected, and our discussion of energy supply to the ocean will hereby focus on the two outstanding sources: the winds and the tides. The remaining terms on the right-hand sides of the three expressions above indicate the processes by which energy can be converted R R R between its various ru  rP dV, characterforms. CK2P, defined as izes the transformation of kinetic energy into potential energy (or vice versa) associated with the raising or lowering of the ocean’s center of mass by advection. Although this term is often important in regional energy budgets, it averages out to a negligible R R R value in the global budget. CI2K is defined as p r  u dV and represents a bidirectional transfer between the internal and kinetic energy pools due to the compressibility of seawater, which causes density to vary with pressure. This term has been estimated to be small away from the upper ocean mixed layer. energy conversion term CK-I is the only irreversible RRR and is defined as re dV, where e is the rate at which internal energy (heat) is produced by the viscous dissipation of turbulent kinetic energy per unit mass. This process represents the only significant sink of kinetic (and, indirectly, potential) energy in the ocean, and must therefore balance the energy input by winds and tides. Thus, the dominant global energy budget for the ocean interior can be synthesized as Z Z

ru  n dA Z Z Z þ r@Ptitde =@t dVECK-I 

½4

The validity of this balance in the characterization of

the ocean’s kinetic and potential energy sources and sinks is widely accepted by oceanographers. Nonetheless, establishing the physical controls and sensitivities of the overturning circulation demands that the flow of energy through the ocean be understood as well. It is the physical means of this energy flow that has been the focus of the ocean mixing debate in recent decades. In the following, we review the two most salient views of the subject to date, and provide an outlook on the major avenues of future development.

The Traditional Paradigm of Ocean Mixing: The Abyssal Ocean The longest-standing and most influential paradigm of ocean mixing and its driving of the overturning circulation was first formulated by Walter Munk in 1966. The paradigm describes how the ocean stratification below a nominal depth of 1000 m (i.e., below the ocean’s permanent pycnocline) may be explained by a simple one-dimensional balance between the upwelling (at a rate of c. 1  10  7 m s  1 or 3 m yr  1) of dense abyssal waters formed at high latitudes, and the downward turbulent mixing (at a rate defined by a turbulent diffusivity kr of c. 1  10  4 m2 s  1) of lighter overlying waters. In energetic terms, the balance is established between a decrease in the ocean’s potential energy associated with high-latitude production of dense waters, which lowers the ocean’s center of mass, and a compensating potential energy increase brought about by the lightening of those waters by turbulent diapycnal (i.e., across density surfaces) mixing as they upwell, which restores the ocean’s center of mass to its original level. A key fact that is made evident in this view is that, when oceanic turbulence ensues, not all the turbulent kinetic energy is dissipated into internal energy, but a fraction of it is expended in mixing water masses of different densities and thus leads to a vertical buoyancy flux. The relationship between the turbulent diapycnal diffusivity kr and the rate of turbulent kinetic energy dissipation e may then be expressed as kp ¼ GeN 2

½5

where the buoyancy frequency N ¼ (gr  1 @r/@z)1/2 is a measure of the stratification, g is the acceleration due to gravity, and G is the so-called mixing efficiency, commonly (and somewhat controversially) thought to be about 0.2. Using [5], it has been shown that driving the global overturning circulation across the observed ocean stratification requires that 2–3 TW is dissipated

ENERGETICS OF OCEAN MIXING

by turbulence in the ocean interior, and that c. 0.5 TW is consumed by turbulent mixing in raising the ocean’s center of mass. The plausibility of this ocean mixing paradigm is suggested by the broad correspondence between the power required to support it (2–3 TW) and estimates of the rate at which work is done on the ocean by winds and tides. The wind contribution is thought to be very large, perhaps on the order of 10 TW, but an overwhelming fraction of this is likely dissipated within the upper ocean mixed layer or radiated as

399

surface waves toward the coastal boundaries where the waves’ energy is dissipated. The principal pathway for wind energy to enter the interior ocean is, in all likelihood, the wind work on the surface geostrophic flow (i.e., on the oceanic general circulation), which has been shown to occur at a rate of c. 0.8 TW and to be focused on the Antarctic Circumpolar Current (ACC), the broad, eastwardflowing current system that circumnavigates the Southern Ocean (Figure 2). Approximately 80% of the global wind work on the general circulation is

(a) 60° N

0° N

60° S 0° E

60° E

120° E

−24 −21 −18 −15 −12

−9

180° E

−6

−3

0

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thought to enter the ocean in the ACC. The manner in which this energy flows in the ocean interior and its ultimate fate represent one of the most significant unknowns in the ocean mixing problem. It is generally accepted that the bulk of the wind energy input to the general circulation is transferred to mesoscale eddies via the action of baroclinic instability (a further 0.2 TW may be transferred from the wind to the mesoscale eddy field directly, as eddies are generated by variable wind forcing). However, the energy’s subsequent pathway toward dissipation is uncertain. The traditional paradigm of ocean mixing proposes that a large fraction of the energy in the large-scale circulation and the mesoscale eddies may be eventually passed onto the ocean’s ubiquitous field of internal waves, through one or several poorly understood energy transfer processes. These include the generation of internal waves by geostrophic flow over small-scale topography; the spontaneous emission of internal waves by loss of geostrophic balance in mesoscale motions; and the nonlinear coupling between mesoscale eddies and internal waves propagating through them. Once in the internal wave field, energy is rapidly cascaded to increasingly smaller scales, to a point where wave breaking and turbulence ensue and a large fraction (1 – G) of the energy is dissipated through turbulence to heat. Aside from internal wave processes, it is also thought that a potentially large proportion of the wind work on the general circulation may be dissipated in turbulence generated by flows over sills within spatially confined abyssal passages, fracture zones, and midocean ridge canyons, although estimates of this contribution vary widely. A second significant mechanism via which the wind supplies energy to the ocean interior is the wind work on upper ocean inertial motions. As the wind blows on the sea surface, it generates upper ocean mixed layer currents that rotate at the local inertial frequency and can force downward- and equatorward-propagating near-inertial internal waves. The magnitude of the wind work on upper ocean inertial motions has been estimated as 0.5 TW, although energy losses to turbulence at the base of the mixed layer mean that this figure is likely to be an overestimate of the rate at which near-inertial internal waves transport energy into the ocean interior. Much of the wind work on upper ocean inertial motions occurs at mid-latitudes and exhibits a marked seasonal cycle (Figure 3), being primarily forced by winter storms. Together with the wind work on the general circulation, tides represent the primary source of the energy required to sustain ocean mixing. The rate at which the sun and the moon work on the ocean via

tidal forces has been estimated to be as large as 3.5 TW, but a substantial fraction of this energy (c. 2.6 TW) is dissipated on shallow continental shelves and does not access the ocean interior. The remaining c. 0.9 TW enters the deep ocean as a barotropic tide that forces flow over rough and steep topography and, in doing so, generates internal waves of tidal periodicity (internal tides) and boundary layer turbulence. The spatial distribution of this generation process is patchy, with enhanced barotropic tidal dissipation rates found over midocean ridges, continental slopes, and other elongated features such as island arcs (Figure 4). Although the bulk of tidally induced mixing occurs in the close vicinity of the generating topography, there is observational evidence of low-mode internal tides being able to transmit their energies over long distances and support turbulent mixing many hundreds of kilometres away from their generation site. Current estimates suggest that this process accounts for c. 0.2 TW, a small yet significant fraction of the total tidal energy input to the ocean interior. Recent observations suggest that a further noteworthy contribution to tidal energy dissipation may be associated with sill overflow turbulence within mid-ocean ridge canyons and other canyon-like topographic features. The final potentially significant source of energy to the ocean interior is also the most surprising and uncertain: the kinetic energy input by the marine biosphere. Net primary production in the euphotic zone produces roughly 60 TW of energy bound in carbohydrates, most of which is used in chemical form by organisms in the biosphere. However, it has been suggested that an amount of the order of 1 TW may be ultimately converted to biomechanical work done by animals swimming in the aphotic ocean. This estimate is subject to many uncertainties and remains exploratory. We conclude, therefore, that the traditional paradigm of ocean mixing, applicable below the permanent pycnocline, may be synthesized as a one-dimensional balance between the upward buoyancy flux associated with the upwelling of dense abyssal waters, and the downward buoyancy flux driven by internal wave breaking and nearboundary turbulence, whose primary energy sources are tides and the wind work on the general circulation. In the last two decades, the validity of this conceptual model has been disputed somewhat imprecisely on the basis of a growing body of measurements indicating that kr is often an order of magnitude smaller than the paradigm’s canonical value of 1  10  4 m2 s  1 within and above the permanent pycnocline, that is, outside the

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paradigm’s depth range of applicability. Despite its partially misguided motivation, this challenge has nonetheless stimulated the emergence of an alternative paradigm of ocean mixing and its energetics

that is consistent with observations of weak diapycnal mixing in the permanent pycnocline. The essential elements of this model are outlined in the following section.

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An Alternative Paradigm of Ocean Mixing: The Permanent Pycnocline The role of air–sea interaction in setting the stratification of the permanent pycnocline was first highlighted by Columbus Iselin in the first half of the twentieth century, when he noticed that the temperature–salinity relationships imprinted horizontally in the wintertime upper ocean mixed layer of the North Atlantic are reflected in the pycnocline’s vertical structure. This observation inspired the development of a family of conceptual models describing the renewal of water masses in the permanent pycnocline, and showing that a realistic stratification may be obtained in the upper kilometer of the ocean interior without any diapycnal mixing. A common ingredient of these models is the appeal to a three-way interaction between wind-forced (Ekman) vertical motion, isopycnal (i.e., along-density surfaces) stirring of water masses by mesoscale eddies, and atmospheric buoyancy forcing at the sea surface to explain how the subduction of relatively unmodified upper ocean waters into the ocean interior comes about. In energetic terms, the flow defined by the models is primarily driven by the wind work on the general circulation, which is then transferred to the mesoscale eddy field by the action of baroclinic instability. The models do not address the issues of how the subducted waters return to the surface and how the wind work is ultimately dissipated, that is, the mass and

energy budgets of the modeled circulation are not closed. The Southern Ocean arguably represents the most notorious manifestation of the above mechanism at work, and is thus at the heart of this alternative paradigm of ocean mixing. There, a range of density surfaces found at great depth over much of the global ocean, including some of the waters implicated in the traditional paradigm, are seen to outcrop into the upper ocean mixed layer of the ACC (Figure 1). This suggests that a considerable volume of water in the global ocean interior (roughly the layer between depths of 1000 and 2000 m) may not necessarily undergo turbulent mixing with lighter overlying water masses in order to return to the upper ocean, but that it may upwell along the steeply sloping isopycnals of the Southern Ocean instead. This notion finds support in recent studies of the Southern Ocean circulation and the dynamics of the ACC. The first indicate that upwelling of deep water (with original sources in the North Atlantic) to the surface does indeed occur over a substantial fraction of the ACC water column. Much of the upwelled water is returned northward as a wind-forced Ekman flow in the upper ocean mixed layer, where its properties are modified by air–sea interaction, and is subsequently subducted back into the interior at the ACC’s northern edge, from where it spreads northward and pervades vast areas of the global ocean’s permanent pycnocline. Studies of the dynamical balances of the ACC suggest, in turn, that

ENERGETICS OF OCEAN MIXING

the upwelling of deep water may be largely sustained by the current’s vigorous mesoscale eddy field, in which nonlinear eddies act to drive a rectified southward flow across the time-mean geostrophic ACC streamlines. It thus becomes apparent that the Southern Ocean eddy field is pivotal to the two paradigms of ocean mixing presented here. On the one hand, it channels a large fraction (c. 0.65 TW, around 25–30%) of the net oceanic energy input toward dissipation scales, thereby contributing to sustain turbulent mixing across isopycnals in the traditional paradigm. On the other, it drives isopycnal upwelling of deep water masses that lie at the base of the permanent pycnocline in the mid- and low-latitude oceans. The extent to which these seemingly conflicting roles may be reconciled is unclear, but some light can be shed on the issue by considering the energetics of the alternative ocean mixing paradigm.

Lunisolar tides

The key assumption that this paradigm makes in proposing that mesoscale eddies may sustain isopycnal upwelling across the ACC is that their energy must be largely dissipated in viscous boundary layers at the ocean surface and floor, with minimal turbulent mixing anywhere in the ocean interior. The notion of bottom drag as a significant factor in the dissipation of the eddy field does find some support in the theory of geostrophic turbulence, which predicts that the evolution of newly generated mesoscale eddies involves a gradual vertical stretching that fluxes kinetic energy downward. Nonetheless, recent observations indicate that a significant fraction of the wind work on the ACC may instead contribute to sustain intense internal wave generation in areas of complex topography and ultimately lead to strong turbulent dissipation and mixing in the interior, much as described by the traditional paradigm.

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Figure 5 Schematic energy budget of the global ocean circulation, with uncertainties of at least factors of 2 and possibly as large as 10. The top row of boxes represents possible energy sources, with pa denoting atmospheric pressure loading. Shaded boxes are the principal energy reservoirs in the ocean, with energy values given in exajoules (EJ, 1018 J) and yottajoules (YJ, 1024 J). Fluxes into and out of the reservoirs are in terawatts (TW). The tidal input of 3.5 TW is the only accurate number here. The essential energetics consists of the conversion of c. 1.6 TW of wind work and c. 0.9 TW of tidal work into oceanic potential and kinetic energy through the generation of the large-scale circulation, and the ultimate viscous dissipation of that work into internal energy via internal wave breaking and near-boundary turbulence. The ellipse indicates the likely but uncertain importance of a loss of balance in the geostrophic mesoscale and other related processes in transferring eddy energy to the internal wave field. Dashed-dot lines indicate energy returned to the general circulation by turbulent mixing, and are first multiplied by the mixing efficiency G. Open ocean mixing by internal waves includes the upper ocean. Reproduced from Wunsch C and Ferrari R (2004) Vertical mixing, energy, and the general circulation of the oceans. Annual Review of Fluid Mechanics 36: 281–314, & Annual Reviews.

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Although the evidence available to date points to topographic internal wave generation in the ACC as a key agent in the transfer of eddy energy to the internal wave field, the other mechanisms introduced in the previous section are likely to enhance this transfer, and be more widely spread across the global ocean. On the whole, this points to the fascinating possibility that the two paradigms of ocean mixing presented here may be physically coupled, and suggests that it may no longer be appropriate to consider diapycnal and isopycnal water mass pathways in isolation. The emerging picture of ocean mixing is thus best described by the combination of two spatially and physically intertwined ‘pycnocline isopycnal’ and ‘abyssal diapycnal’ regimes.

Conclusion We conclude that the oceanic stratification and overturning circulation owe their existence to mechanical (rather than buoyancy) forcing. This principle is reflected in a state-of-the-art synthesis of the energy budget of the global ocean shown in Figure 5. The essential energetics consists of the conversion of 2–3 TW of wind and tidal work into oceanic potential and kinetic energy through the generation of the large-scale circulation, and the ultimate viscous dissipation of that work into internal energy via internal wave breaking and near-boundary turbulence. There are many uncertainties regarding the energy flow between the large-scale circulation and the small dissipation scales, but available evidence points to the existence of two physically coupled, spatially overlapping ocean mixing regimes. In the abyssal ocean, at depths in excess of c. 1000 m, the circulation is driven by turbulent mixing, which allows dense waters to upwell across the stable stratification and acts to counteract the decrease of the ocean’s potential energy brought about by high-latitude dense-water production. In contrast, the waters above and in the vicinity of the ocean’s permanent pycnocline, that is, roughly in the upper 2000 m, tend to flow along isopycnals primarily in response to the release by baroclinic instability of the potential and kinetic energy imparted by the wind on the general circulation. The likely subsequent transfer of some of this energy to the internal wave field couples the pycnocline and abyssal mixing regimes physically, and so may introduce important subtleties in the way the ocean responds to climatic changes in forcing. Significant open questions remain regarding all aspects of how energy enters the ocean, cascades to small scales, and dissipates. These are summarized in several major avenues of future development, of which the most prominent are: (1) quantitative

assessment of the energy budget of the upper ocean mixed layer, and of the mechanisms regulating the flow of wind energy across its base; (2) quantification of the energy sources to the internal wave field, and of the processes regulating its rate of dissipation; (3) determination of the mechanisms responsible for coupling internal waves and mesoscale eddies and for dissipating the latter; (4) evaluation of the global significance of sill overflow turbulence within confined passages and mid-ocean ridge canyons; (5) assessment of the global importance of double and differential diffusion, nonlinearities in the equation of state, and biomechanical mixing. In the light of the preceding discussion, it is probable that our attempts to understand the ocean’s state in past and future climates will be dangerously misguided until these issues are resolved.

Nomenclature A cp CI2K

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area specific heat capacity of seawater at constant pressure global rate of transfer between internal and kinetic energies due to the compressibility of seawater global rate of transfer of kinetic to internal energy due to turbulent dissipation global rate of transfer between kinetic and potential energies due to advection radiative flux of internal energy between the near-surface ocean and overlying atmosphere/ice enthalpy of water internal energy per unit mass turbulent diapycnal diffusivity kinetic energy per unit mass unit vector normal to the ocean surface buoyancy frequency pressure potential energy per unit mass tidal potential energy per units mass salinity time temperature three-dimensional velocity vector three-dimensional velocity vector of the free ocean surface volume eastward coordinate northward coordinate vertical coordinate

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G e kS kT m r r

mixing efficiency rate of turbulent kinetic energy dissipation per unit mass molecular diffusivity of salinity molecular diffusivity of temperature kinematic viscosity of seawater potential density three-dimensional gradient operator (@/@x, @/@y, @/@z)

See also Breaking Waves and Near-Surface Turbulence. Dispersion and Diffusion in the Deep Ocean. Double-Diffusive Convection. Internal Tidal Mixing. Internal Tides. Internal Waves. Ocean Circulation. Three-Dimensional (3D) Turbulence. Tidal Energy. Turbulence in the Benthic Boundary Layer. Upper Ocean Mixing Processes. Vortical Modes. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading Alford MH (2003) Improved global maps and 54-year history of wind work on ocean inertial motions. Geophysical Research Letters 30(8): 1424. Bryden HL and Nurser AJG (2003) Effects of strait mixing on ocean stratification. Geophysical Research Letters 33: 1870--1872. Egbert GD and Ray RD (2003) Semi-diurnal and diurnal tidal dissipation from Topex/Poseidon altimetry. Geophysical Research Letters 30(17): 1907. Gnanadesikan A (1999) A simple predictive model for the structure of the oceanic pycnocline. Science 283: 2077--2079. Hughes CW (2002) Oceanography: An extra dimension to mixing. Nature 416: 136--139.

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Hughes GO and Griffiths RW (2006) A simple convection model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Modelling 12: 46--79. Kunze E, Firing E, Hummon JM, Chereskin TK, and Thurnherr AM (2006) Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles. Journal of Physical Oceanography 36: 1553--1576. Marshall J and Radko T (2006) A model of the upper branch of the meridional overturning circulation of the Southern Ocean. Progress in Oceanography 70: 331--345. Munk WH and Wunsch C (1998) Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Research I 45: 1977--2010. Naveira Garabato AC, Stevens DP, Watson AJ, and Roether W (2007) Short-circuiting of the overturning circulation in the Antarctic Circumpolar Current. Nature 447: 194--197. Polzin KL, Toole JM, Ledwell JR, and Schmitt RW (1997) Spatial variability of turbulent mixing in the abyssal ocean. Science 276: 93--96. Rudnick DL, Boyd TJ, Brainard RE, et al. (2003) From tides to mixing along the Hawaiian Ridge. Science 301: 355--357. Samelson RM (2004) Simple mechanistic models of middepth meridional overturning. Journal of Physical Oceanography 34: 2096--2103. St. Laurent L and Simmons H (2006) Estimates of power consumed by mixing in the ocean interior. Journal of Climate 19: 4877--4889. Toggweiler JR and Samuels B (1998) On the ocean’s large-scale circulation near the limit of no vertical mixing. Journal of Physical Oceanography 28: 1832--1852. Webb DJ and Suginohara N (2001) Oceanography: Vertical mixing in the ocean. Nature 409: 37. Wunsch C (1998) The work done by the wind on the oceanic general circulation. Journal of Physical Oceanography 28(11): 2332--2340. Wunsch C and Ferrari R (2004) Vertical mixing, energy, and the general circulation of the oceans. Annual Review of Fluid Mechanics 36: 281--314.

FOSSIL TURBULENCE C. H. Gibson, University of California, San Diego, La Jolla, CA, USA Copyright & 2001 Elsevier Ltd.

Introduction Fossil turbulence processes are central to turbulence, turbulent mixing, and turbulent diffusion in the ocean and atmosphere, in astrophysics and cosmology, and in other natural flows. However, because turbulence is often imprecisely defined, the distinct and crucial role of fossil turbulence may be overlooked. Turbulence occurs when inertial-vortex - forces n  o dominate all other forces for a range of length and timescales to produce rotational, eddylike motions, where n is the velocity and o is the vorticity r  n. Turbulence originates at length scales with the smallest overturn times; that is, at the viscous Kolmogorov length scale LK  ðn3 =eÞ1=4 and timescale TK  ðn=eÞ1=2 , where n is the kinematic viscosity of the fluid and e is the viscous dissipation rate. In stratified fluids with gravity, turbulence appears in bursts on tilted density layers to produce patches (see Figure 3). Turbulence then cascades, - driven by n  o forces, to larger scales where buoyancy forces cause fossilization of vertical motions at the Ozmidov scale LR Eðe=N 3 Þ1=2 , where N is the ambient stratification frequency. Coriolis forces may also cause fossilization, usually of horizontal motions, at the Hopfinger scale LH ¼ ðe=O3 Þ1=2 , where O is the angular velocity. In the ocean and atmosphere this may occur at large horizontal scales when O is the vertical component of the planetary angular velocity, or at any scale where LH is smaller than the eddy size; for example, in the spin up of Kelvin– Helmholtz billows. Turbulence is defined as an eddylike state of fluid motion where the inertial vortex forces of the eddies are larger than any other forces that tend to damp the eddies out. Fossil turbulence is defined as any fluctuation in a hydrophysical field such as temperature, salinity, or vorticity that was produced by turbulence and persists after the fluid is no longer turbulent at the scale of the fluctuation. Fossil turbulence patches persist much longer than the turbulence events that produced them because they must mix away, but with smaller dissipation rates, the same velocity and scalar variances that

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existed in their original turbulence patches. Fossil turbulence preserves information about the original turbulence and completes the mixing and diffusion processes initiated by stratified turbulence bursts. The first printed reference to the concept of fossil turbulence was apparently when George Gamov suggested in 1954 that galaxies might be fossils of primordial turbulence produced by the Big Bang. Although it now appears that the primordial fluid at the time of galaxy formation was too viscous and stratified to be strongly turbulent, the Gamov concept that information about irreversible hydrodynamic states and processes might be preserved by parameters of the structures formed was quite correct and is concisely captured by the term ‘fossil turbulence’. Persistent refractive index patches caused by mountain wakes in the stratified atmosphere and detected by radar were recognized as fossils of turbulence, causing organizers and participants to form a Fossil Turbulence working group for the 1969 Stockholm Colloquium on Spectra of Meteorological Variables. Woods showed that billows made visible by introducing dye in the interior of the stratified ocean formed highly persistent remnants of these turbulent events, as demonstrated by Thorpe in the laboratory using a tilt tube. The first use of the expression ‘fossil turbulence’ was attributed to Woods. Patches of strong oceanic temperature microstructure measured without velocity microstructure from a submarine were termed ‘footprints of turbulence’ by Stewart. However, two important assumptions were that no universal fossil turbulence description is possible and that all vertical velocity fluctuations vanish in fossil turbulence. Both are incorrect. A universal similarity theory of stratified fossil turbulence was presented by Gibson in 1980. Universal constants and spectral forms of stratified fossil turbulence were estimated by the theory (see Figure 1), and hydrodynamic phase diagrams were introduced as a method for classifying temperature and salinity microstructure patches in the ocean interior according to their hydrodynamic states (turbulent, active-fossil turbulence, completely fossil) and for extracting fossilized information about the previous turbulence and mixing. Furthermore, it was shown that turbulent kinetic energy is trapped and preserved in fossil turbulence as saturated internal wave motions termed fossil-vorticity-turbulence. The 1980 theory has been confirmed and extended to describe fossilized turbulence produced by magnetic forces, self-gravitational forces, and space-time inflation in a new theory of gravitational structure formation.

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Fossil turbulence patches have been misinterpreted as turbulence patches in oceanic turbulence sampling experiments that fail to account for the turbulent fossilization process. Such mistakes have led to large underestimates of the true average vertical turbulence flux rates in many ocean layers, and are the source of the so-called by Tom Dillon ‘dark mixing’ paradox of the deep ocean interior. Dark mixing is to the ocean as dark matter is to galaxies. Dark matter is unseen matter that must exist to prevent galaxies from flying apart by centrifugal forces. Dark mixing is mixing by turbulence events that must exist to explain why some layers in the ocean interior are well mixed, but have strong turbulent patches that are undetected except for their fossil turbulence remnants. Both the dark mixing and dark matter paradoxes appear to be manifestations of the same problems; that is, extreme intermittency of nonlinear cascade processes over a wide range of values leading to extreme undersampling errors, and basic misunderstandings about the underlying irreversible fluid mechanics. Observations of fossil turbulence patches of rare, powerful, but undetected turbulence events in the deep ocean support the statistical evidence (Figure 2) that bulk flow estimates of the vertical diffusivity in the deep main thermocline are correct, rather than interpretations of sparse temperature dissipation rate w measurements that claim large discrepancies but do not take into account either the extreme intermittency of w in deep ocean layers or the fossil turbulence evidence.

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Fossil turbulence signatures in hydrophysical fields preserve information about previous turbulence. Temperature fluctuations produced by turbulence are termed fossil-temperature-turbulence for fluctuations at length scales where the turbulence has been damped by buoyancy. Skywriting rapidly becomes fossil-smoke-turbulence above the inversion layer. The larger the vertical fossil temperature turbulence patch size LP , the larger the viscous and temperature dissipation rates e and w must have been in the

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original patch of turbulence. Fossil turbulence remnants are the footprints and scars of previous turbulent events, and redundantly preserve information about their origins in a wide variety of oceanic fields such as temperature, salinity, bubbles, and vorticity. The process of extracting information about previous turbulence and mixing from fossil turbulence is termed hydropaleontology. Stratified and rotating fossils of turbulence are more persistent than their progenitor turbulence events because they possess almost the same velocity variance (kinetic energy) and scalar variance (potential entropy of mixing) as the original turbulent field, but have smaller viscous and scalar dissipation rates of these quantities than they had while they were fully turbulent just prior to the beginning of fossilization. The most powerful turbulence events produce the most persistent fossils, with persistence times proportional to the normalized Reynolds number e0 =eF and inversely proportional to the ambient stratification frequency N, where e0 E0:4L2T N 3 is the estimated dissipation rate at beginning fossilization and eF ¼ 30nN 2 is at complete fossilization, where LT ELP is the maximum Thorpe overturning scale of the patch. Oceanic fossil turbulence processes are more complex and important than in nonstratified nonrotating flows where the only mechanism of fossilization is the viscous damping of turbulence before mixing is complete and all scalar fields become singly connected. Laboratory viscous fossil turbulence without stratification or rotation is thus mentioned in textbooks only as a curiosity of flow visualization, where eddy patterns of dye or smoke that appear to be turbulent are not because the turbulent fluid motions have been damped by viscosity. Buoyancy-fossils and rotation-fossils are difficult to study in the laboratory or in computer simulation because of the wide range of relevant length and timescales. Most of the ocean’s kinetic energy exists as fossilvorticity-turbulence because its motions, driven by thermohaline oceanic circulation, air–sea interaction of atmospheric motions and tidal forces of the sun and moon, are converted into turbulence energy at the top and bottom ocean surfaces by turbulence formation and its cascade to larger scales, and are not immediately or locally dissipated. Instead, oceanic turbulent kinetic energy and its induced scalar-potential-entropy is fossilized by buoyancy and Coriolis forces and distributed oceanwide by advection, leaving fossil-scalar-turbulence and fossil-vorticity-turbulence remnants in a variety of hydrodynamic states. These turbulence fossils move and interact with their environment and each other in the ocean interior by mechanisms that are poorly understood and hardly

recognized. For example, a necessary stage of average double-diffusive vertical fluxes in the ocean may be fluxes driven by double-diffusive convection in the final stages of fossil-temperature-salinity-turbulence decay within fossil-temperature-salinityturbulence patches. The turbulence event scrambles the pre-existing temperature and salinity fields to produce a stirred field in which the full range of possible double-diffusive instabilities occur. These drive motions in the late stages of the turbulent fossil decay, leaving characteristic layered structures of salt fingering. Convective instabilities at low Rayleigh number, insufficient to drive turbulent motion, may also occur. Powerful turbulence events produce fossils that radiate wave energy and trigger secondary turbulence events at their boundaries; this turbulence also fossilizes, and these fossils produce more turbulence (Figure 3). Turbulent mixing and diffusion is initiated by turbulence, but the final stages of the mixing and diffusion are completed only after the flow has become partially or completely fossilized. In practice, complete fossilization rarely occurs in the ocean because propagating internal waves produce strong shears at the strong density gradients of fossil density turbulence boundaries through baroclinic torques rr  rp=r2 , where r is density and p is pressure. Active turbulence that arises in this way from patches of fossil turbulence is known as zombie turbulence.

History of Fossil Turbulence The distinctive, eddy-like-patterns of turbulent motions are beautiful and easily recognized. Many ancient civilizations have woven them into their arts, religions, and sciences. The first attempts at hydropaleontology were applications of Kolmogorovian universal similarity theories of turbulence to cosmology by members of the Soviet school of turbulence. Recent evidence from space telescopes suggests that primordial turbulence and density structure from the Big Bang were fossilized at 1035 s by inflation of space beyond the length scales of causal connection ct of the fluctuations, where c is the speed of light and t is the time. This fossilized microstructure seeded the formation of all subsequent structures in a zombieturbulence fossil-turbulence cascade, similar to that in the ocean, that preserves evidence of the hydrophysical states of each stage of the process in various hydrodynamic fossils. The 1969 working group on fossil turbulence examined patches of persistent refractive index fluctuations produced by turbulence in the stratified

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ω

z

v0=0

Fr(z)=u(z)/zN(z) ≤ Frcrit≈2

v2

v4 v×ω

5LK ε ≥ εο

v1 v

v×ω

∇ρ

p ω = ∇ρ×∇ ρ2

409

Turbulence forms ∇p Tilted shear layer

Tilted density layer (A)

(C)

(B)

Billows form Fr(z)≥ Frcrit

2

εF = 30 νN

3

εo ≈ 0.4 L Tmax N ; LTmax ≈ LP

v5 v×ω

ε = εο 0.6LR

Secondary turbulence events

v6

LP

2

εo ≥ ε ≥ εF

Fossi vorticity turbulence

ε ≥ εο

v×ω εo ≥ εF ≥ ε

Turbulence grows Fossilization begins

Fossilization ends (D)

(E)

(F)

Figure 3 Figure 3 Fossil turbulence formation on a suddenly tilted density layer in a stagnant stratified fluid. Baroclinic torques cause a build up of vorticity on the density surface (A) with laminar boundary layers n1 tol n3 (B). These become turbulent at 5–10 times the Kolmogorov scale LK , steepening the density gradient and permitting storage of potential turbulent kinetic energy (C). Turbulent billows form and absorb the stored kinetic energy. The strong density interface is broken to form the first billow at a critical turbulent boundary layer scale when Fr ðzÞ  Frcrit E2ðDÞ. Turbulence cascades to larger vertical scales limited by 0.6 times the Ozmidov scale LR where fossilization begins (E). leaving a fossil-vorticity-turbulenceremnant. This decays by viscous dissipation and vertical radiation of internal waves that may form secondary turbulence events (F). Microstructure patches are actively turbulent if e  eo , partially fossilized for eo  e  eF , and fossil for eF  e.

atmosphere and fluctuating temperature and dye patches produced by turbulence in the ocean interior which could not possibly be turbulent at the time of their detection. Scuba divers observed dye patch fossils of breaking internal waves from their turbulent beginning until they became motionless. Temperature microstructure patches were reported from a submarine with and without measurable velocity fluctuations, indicating that those without must be fossilized. Fossil turbulence patches produced in the stratified atmosphere by wind over mountains and wakes of other aircraft are dangerous to aircraft because of their long persistence times and powerful fossil-vorticity-turbulence. For example, a fossil turbulence patch sent passengers flying into the ceiling of a Boeing 737 at 24 000 feet on the afternoon of 3 September 1999, bound from Los Angeles to San Francisco, injuring 15 of the 107 passengers and five crew aboard. Detectable refractive index fluctuations

from billow turbulence events behind mountains are observed many kilometers downstream by radar scattering measurements, long after all turbulence and fossil-vorticity-turbulence have been damped by buoyancy forces as shown by the layered anisotropy that develops in the radar returns. Fossils of clear air turbulence (CAT) (referred to as ‘angels’) have long been well known to radar operators. Equivalent dominant turbulent patches of the deep ocean have not yet been detected in their actively turbulent states. The quantitative universal similarity theory of stratified fossil turbulence published in 1980 was based on towed-body small-scale temperature measurements made by Schedvi in 1974 in the upper ocean of the Flinders Current off Australia in a US–Soviet intercomparison cruise on the Dmitri Mendeleev. The data showed clear evidence of buoyancy effects causing departures from spectral forms of the Kolmogorov and Batchelor universal similarity

410

FOSSIL TURBULENCE

theories of turbulence and turbulent mixing. These departures were attributed to stratified turbulence fossilization. Key ideas of the theory are illustrated in Figure 1 by the time evolution of velocity spectra fu and temperature spectra fT at five stages in a turbulent wake as the turbulence is fossilized by buoyancy forces. The integral of fu over wavenumber k is the velocity variance and the integral of fT is the temperature variance, where the Prandtl number n=D is about 10 (corresponding to that of sea water) with D the thermal diffusivity. Spectra are multiplied by k2 so their integrals represent velocity and temperature gradient variances, respectively proportional to the dissipation rates e and w. Stage 1 of the turbulence patch is fully turbulent with turbulent activity coefficient AT > 1ðe > eo , where eo is the value at beginning of fossilization). Velocity dissipation spectra k2 fu for stages 1–5 along the wake are shown on the left of Figure 1. Each has integral e=3n. Inertial subranges with slope þ 1/3 reflect Kolmogorov’s second hypothesis for wavenumbers kE2p=L between the energy (or Obukhov) scale LO (small k) and the Kolmogorov scale LK (large k). The turbulence continues its cascade to larger LO scales by entrainment of external nonturbulent fluid until the increasing LO is matched by the decreasing Ozmidov scale LR ¼ ðe=N 3 Þ1=2 at the beginning of fossilization, where e ¼ eo , with spectral forms 2. The temperature dissipation spectrum k2 fT increases in amplitude from stage 1 to stage 2 as vertical temperature differences are entrained over larger vertical scales (even though the velocity spectrum k2 fu decreases) with 2=3 ¯ , where w is the increasing area w=6D ¼ CðqT=qzÞ diffusive dissipation rate of temperature variance and Cox number C is the mean square over square mean temperature gradient ratio. The dramatic differences in spectral shapes between the velocity dissipation spectra and the temperature dissipation spectra in Figure 1 reflect the theoretical result that without radiation, the kinetic energy of powerful fossilized turbulence events persist as fossil-vorticity-turbulence for longer periods than the temperature variance persists as fossil-temperatureturbulence. This is the basis of the AT  ðe=eo Þ1=2 versus C hydrodynamic phase diagram shown at the bottom right of Figure 1. Because w and C are large in the temperature fossil, the velocity dissipation rate e at the beginning of fossilization eo can be estimated from C using eo E0:4 > L2 T N 3  13DCN 2 . The turbulent activity coefficient AT  ðe=eo Þ1=2 is greater than 1 for stratified turbulence patches before fossilization, and less than 1 after

fossilization begins. A particular patch decays along a straight line trajectory in the hydrodynamic phase diagram until e ¼ eF  30nN 2 at complete fossilization. It can be shown that C averaged over a large horizontal layer in the stratified ocean for a long time period is a good measure of the turbulent heat flux divided by the molecular heat flux, with the vertical turbulent diffusivity K ¼ DC. The motivation for most oceanic microstructure measurements is to estimate K through measurements of the average C for various oceanic layers. The problem is that C, e, and w in the ocean, atmosphere, and in all other such natural flows with wide spacetime cascade ranges, tend to be extremely intermittent.

Intermittency of Oceanic Turbulence and Mixing Turbulence with the enormous range of length scales possible in oceanic layers is very intermittent in space and time. Sampling turbulence and turbulent mixing without recognizing this intermittency and without recognizing that most oceanic microstructure is fossilized or partially fossilized has led to misinterpretations and errors in estimates of turbulent diffusion and mixing rates, especially in the deep ocean interior and in strong equatorial thermocline layers where intermittencies of e and w are maximum. Dissipation rates e and w are random variables produced by nonlinear cascades over a wide range of scales, resulting in lognormal probability density distributions and mean values larger than the mode values by factors in the range 102–105. Since the mode of a distribution is the most probable measured value, sparse microstructure studies will significantly underestimate mean e and w values, as well as any vertical exchange coefficients and flux estimates of heat, mass, and momentum that are derived from these quantities. For lognormal random variables, GX ¼ exp½3s2lnX =2, where the variance s2lnX is the intermittency factor of a lognormal random variable X. Intermittency factors s2lnw and s2lne in the ocean have been measured, and range from typical values of 5 at midlatitudes near the surface, to 6 or 7 in the deep ocean and at equatorial latitudes. Thus, probable undersampling errors for these quantities range from Gw;e ¼ 1800  36 000 in the ocean. For comparison, the intermittency factor s2ln$ of the superrich (upper 3%) in US personal income, which is close to lognormal, has been measured to be 4.3, giving a Gurvich number G$ of 600. Figure 2 shows a lognormality plot of independent deep ocean samples of X ¼ C averaged over 150 m in the

FOSSIL TURBULENCE

vertical. The axes are stretched so lognormal random variables fit a straight line. Figure 2 Shows the effects of undersampling errors due to intermittency in attempts to estimate the Cox number C of the stratified layers of the deep ocean, illustrating the ‘deep dark mixing paradox.’ In 1966 Walter Munk estimated that the vertical turbulent diffusivity of temperature in the deep Pacific Ocean below a kilometer depth should be K ¼ DC ¼ 1  2 cm2 s1 with corresponding Cox number of 500–1100. Dropsonde temperature microstructure measurements find CE30, more than an order of magnitude less. However, the deep C averages are clearly lognormal, with maximum likelihood estimator Cmle, values in better agreement with the previous range than with the microstructure range, as shown in Figure 2. This matter is still controversial in the oceanographic literature, partly because the tests have either been carried out in shallow high latitude layers where there is no disagreement, or in deep layers where adequate microstructure sampling is impossible. In the shallow main thermocline at 0.3 km, C values are less intermittent and K values are less by a factor of 30 by all methods, including tracer release studies. From the Munk and Gibson value for C, the vertical heat flux is constant at about 6 W m2 at depths in the thermocline between 0.3 and 2 km, consistent with computer models of planetary heat transfer. Remarkably, the galactic dark matter paradox arises from errors very similar to those leading to the oceanic dark-mixing paradox. Gas emerging from the Big Bang plasma condensed to form a primordial fog of widely separated objects, each with the mass of a small planet. These PFPs entered into nonlinear gravitational accretional cascades to form stars a million times more massive, and their number density n became lognormal with Gurvich numbers Gn near 106. Only about 1 in 30 finds its way into a star. The rest are now dark and frozen, thirty million per star in a galaxy, dominating the interstellar and inner-halo galactic dark matter. Star microlensing surveys have failed to detect these objects, and have excluded their existence assuming a uniform pdf rather than the expected intermittent lognormal probability density function for n that explains this questionable interpretation.

Turbulence and Fossil Turbulence Definitions Turbulence is defined as a rotational, eddy-like state of fluid motion where the inertial-vortex forces of the eddies are larger than any of the other forces which

411

tend to damp the eddies out. The inertial-vortex - force F¯ I ¼ n  o produces turbulence, and appears in the Newtonian momentum conservation equations, -

qu ! ! ! - ¼ rB þ v  o þ Fv þ FC þ FB þ y; qt p u2 B ¼ þ þ gz r 2 -

-

½1

-

where n is the velocity field, o ¼ r  n is the vorticity, B is the Bernoulli group of mechanical energy terms, p is pressure, r is density, g is gravity, z is 2v is the viscous force, n is the kinematic up, Fn ¼ nr viscosity, FC ¼ 2v  O is the Coriolis force, and FB ¼ N 2 L is the buoyancy force when the buoyancy frequency N is averaged over the largest vertical scale L of the turbulence event (other forces are neglected). The growth of turbulence is driven by FI forces at all scales of the turbulent fluid. Irrotational flows (those with o ¼ 0) are nonturbulent by definition, but supply the kinetic energy of turbulence because the turbulent fluid induces a nonturbulent cascade of the irrotational fluid from large to small scales by sucking irrotational fluid into the interstices between the growing turbulence domains. turbulent flows, - In - viscous and inertial-vortex n  o forces are equal at a universal critical Reynolds number vx=nE100 for separation distances xE10LK , where LK is the Kolmogorov length scale 3 1=4 n LK  e

½2

and e is the viscous dissipation rate per unit mass. Fossil turbulence is defined as a fluctuation in any hydrophysical field produced by turbulence that persists after the fluid is no longer turbulent at the scale of the fluctuation. Examples of fossil turbulence are jet contrails, skywriting, remnants of cold milk poured rapidly into hot coffee, and patches of ocean temperature microstructure observed with little or no velocity microstructure existing within the patches. The best-known fossil turbulence parameter in the ocean is the mixed-layer depth, which persists long after it was produced and the turbulence has been damped. Buoyancy forces match inertial-vortex forces in a turbulent flow at the Ozmidov scale LR 

h e i1=2 N3

½3

where the intrinsic frequency N of a stratified fluid is

gqr 1=2 N rqz

½4

412

FOSSIL TURBULENCE

for the ambient stably stratified fluid affecting the turbulence. Coriolis forces match inertial-vortex forces at the Hopfinger scale

e LH  O3

1=4 ½5

where O is the angular velocity of the rotating coordinate system. Taking the curl of eqn[1] for a stratified fluid gives the vorticity conservation equation -

qo - - - 2 rr  rp þ vr2 o þ u  ro ¼ o  e þ qt r2

½6

where the vorticity of fluid particles (on the left side) tends to increase from vortex line stretching by the rate of strain tensor 2 e (the first term on the right), baroclinic torques on strongly tilted strong density gradient surfaces (the second term), and decreases by viscous diffusion (the third term). From eqn [6] we see that turbulence events in stably stratified natural fluids are most likely to occur where density gradients are large and tilted for long time periods; for example on fronts, because this is where most of the vorticity is produced.

Formation and Detection of Stratified Fossil Turbulence Figure 3 shows how turbulence bursts and fossil turbulence patches may form in the interior of the stratified ocean by vorticity forming on a tilted density surface, starting from a state of rest. The longer the density surface is tilted the more kinetic energy is stored in the resulting boundary layers. The boundary layers formed on both sides of the tilted surface become turbulent when the critical Reynolds number is reached. This occurs when the boundary layer thickness is 5–10 LK based on the viscous dissipation rate of the laminar boundary layer (Figure 3C). The turbulence sharpens the density gradient, keeping the local Froude number FðzÞ ¼ uðzÞ=zNðzÞ less than Frcrit E2 so that the kinetic energy is stored. Billows form when FrðzÞ  2 (Figure 3D), and the turbulent burst occurs, absorbing all the kinetic energy stored on the tilted density layer. Fossilization begins when buoyancy forces match the inertial vortex forces of the turbulence, at a vertical size of about 0.6 LR (Figure 3E) with viscous dissipation rate eo . The dissipation rate e decreases with time during the process so LR decreases as the vertical patch size LP ELTmax increases. The fossil turbulence patch does not collapse, even though the interior turbulent motions decrease in

their vertical extent. The patch size LP preserves information about the Ozmidov scale LRo at the beginning of fossilization when the viscous dissipation rate e is eo . Thus, from eqn[3] and LRo ¼ 0:6LR we have the expression eo ¼ 0:4L2P N 3

½7

from which we can estimate eo from measurements of LP LTmax and N long after the turbulence event. Because the saturated internal waves of fossil vorticity turbulence have frequency N, they propagate vertically, and produce secondary turbulence events above and below the fossil turbulence patch (Figure 3F). Secondary turbulent events also form at the top and bottom of the fossil because these strong gradient surfaces are also likely to be tilted. Motions of the ocean are inhibited in the vertical direction by gravitational forces, so that the turbulence and fossil vorticity turbulence kinetic energy is mostly in the horizontal direction. In eqn [5], O is the vertical component of the Earth’s angular velocity and approaches zero at the equator since O ¼ Oo sin y, where y is the latitude. Large-scale winds and currents develop at equatorial latitudes because they are unchecked by Coriolis forces. These break up into horizontal turbulence which can also cascade to large scales before fossilization by Coriolis forces at LH scales. Ozmidov scales LR and Hopfinger scales LH for the dominant turbulent events of particular layers, times, and regions of the ocean cover a wide range, with typical maximum values LR ¼ 3  30 m and LH ¼ 30  500 km occurring where e is large.

Quantitative Methods A patch of temperature, salinity, or density microstructure is classified according to its hydrodynamic state by means of hydrodynamic phase diagrams as shown in Figure 1 (insert), which compare parameters of the patch to critical values. For the patch to be fully turbulent, both the Froude number Fr ¼ U=NL and the Reynolds number Re ¼ UL=n must be larger that critical values from our definition of turbulence. If both are subcritical the patch is classified as completely fossilized. Most oceanic microstructure patches are found in an intermediate state, termed partially fossilized, where Fr is subcritical and Re is supercritical. This means that the largest turbulent eddies have been converted into saturated internal waves, but smaller-scale eddies exist that are still overturning and fully turbulent. A variety of hydrodynamic phase diagrams have been constructed as fossil turbulence theory has evolved, but all have active, active-fossil, and completely fossil quadrants. Figure 4 shows a hydrodynamic phase

FOSSIL TURBULENCE

diagram applied to turbulence-fossil–turbulencephytoplankton growth interaction. Growth rates of various phytoplankon species are extremely sensitive to both turbulence and the duration of the turbulence. Laboratory experiments reveal that red tide dinoflagellates have two thresholds for growth inhibition; dissipation rate e  eGI for turbulence, and T  TGI for the duration T of turbulence with e  eGI (apparently to detect oceanic fossil turbulence from its greater persistence). If the dissipation rate e exceeds about eGI ¼ 0:3 cm2 s3 for periods T more than TGI ¼ 15 min a day for such microscopic swimmers they stop reproducing and the population dies in a few days. Shorter-duration turbulence events are ignored, no matter how powerful. Diatom growth in the laboratory and field reacts positively to turbulence events with more than several minutes persistence. The hypothesis matching this behavior is that both classes of species have evolved methods of hydrodynamic pattern recognition so that they can maximize their chances of survival with respect to their swimming abilities. Dinoflagellate red tides occur when nutrient-rich upper layers of the sea experience several days of sun with weak winds and waves so that they become strongly stratified. The diatoms settle out of the light zone so that the dinoflagellates can bloom. However, when waves appear with sufficient strength to break and mix the surface layer, this may be detected by the phytoplankton from the long persistence time T > TGI of the fossil-vorticity-turbulence patches produced by breaking waves. It is supposed that when such fossil-vorticity-turbulence patches are detected, both phytoplankton species adjust their growth rates in anticipation of an upcoming sea-state change from

413

strongly stratified to well mixed in order to maximize their survival rates according to their swimming abilities. The expression derived by the author for the persistence time of e  eF in a fossil turbulence patch before it becomes completely fossilized is T ¼ N 1 eo =eF , where the Reynolds number ratio Reo =ReF ¼ eo =eF ¼ eo =30nN 2 (top of Figure 4). The shaded gray zone of the hydrodynamic phase diagram in Figure 4 shows estimates of N and eo that would inhibit growth of a particular red tide dinoflagellate species with eGI and TGI values known from laboratory measurements.

See also Dispersion and Diffusion in the Deep Ocean.

Further Reading http://xxx.lanl.gov http://www-acs.uscd.edu/Bir118. Gibson CH (1991) Kolmogorov similarity hypotheses for scalar fields: sampling intermittent turbulent mixing in the ocean and galaxy. In: Turbulence and Stochastic Processes: Kolmogorov’s Ideas 50 Years On, Proceedings of the Royal Society London, Ser. A, V 434 (N 1890) 149–164. Gibson CH (1996) Turbulence in the ocean, atmosphere, galaxy, and universe. Applied Mechanics Reviews 49(5): 299--315. Gibson CH (1999) Fossil turbulence revisited. Journal of Marine Systems 21: 147--167. Thomas WH, Tynan CT, and Gibson CH (1997) Turbulence– phytoplankton interrelationships. In: Progress in Phycological Research Ch. 5, Vol. 12, Chapman DJ and Round FE (eds) Biopress Ltd.

OPEN OCEAN CONVECTION A. Soloviev, Nova Southeastern University, Oceanographic Center, Florida, USA B. Klinger, Center for Ocean-Land-Atmosphere Studies (COLA), Calverton, MD, USA Copyright & 2001 Elsevier Ltd.

Introduction Free convection is fluid motion due to buoyancy forces. Free convection, also referred to as simply convection, is driven by the static instability that results when relatively dense fluid lies above relatively light fluid. In the ocean, greater density is associated with colder or saltier water, and it is possible to have thermal convection due to the vertical temperature gradient, haline convection due to the vertical salinity gradient, or thermohaline convection due to the combination. Since sea water is about 1000 times denser than air, the air–sea interface from the waterside can be considered a free surface. So-called thermocapillary convection can develop near this surface owing to the dependence of the surface tension coefficient on temperature. There are experimental indications that in the upper ocean layer more than 2 cm deep, buoyant convection dominates. Surfactants, however, may affect in the surface renewal process. This article will mainly consider convection without these capillary effects. Over most of the ocean, the near-surface region is considered to be a mixed layer in which turbulent mixing is stronger than at greater depth. The strong mixing causes the mixed layer to have very small vertical variations in density, temperature, and other properties compared to the pycnocline region below. Convection is one of the key processes driving mixed layer turbulence, though mechanical stirring driven by wind stress and other processes is also important. Therefore, understanding convection is crucial to understanding the mixed layer as well as property fluxes between the ocean and the atmosphere. Thermal convection is associated with the cooling of the ocean surface due to sensible (QT ), latent (QL ), and effective long-wave radiation (QE ) heat fluxes. QT may have either sign; its magnitude is, however, much less than that of QE or QL (except perhaps in some extreme situations). The top of the water column becomes colder and denser than the

414

water below, and convection begins. In this way, cooling is associated with the homogenization of the water column and the deepening of the mixed layer. Warming due to solar radiation occurs in the surface layer of the ocean and is associated with restratification and reductions in mixed layer depth. The most prominent examples of this mixing/restratification process are the diurnal cycle (nighttime cooling and daytime warming) and the seasonal cycle (winter cooling and summer warming). There are also important geographical variations in convection, with net cooling of relatively warm water occurring more at higher latitudes and a net warming of water occurring closer to the Equator. For this reason, mixed layer depth generally increases towards the poles, though at very high latitudes ice-melt can lower the surface salinity enough to inhibit convection. Over most of the ocean, annual average mixed layer depths are in the range of 30–100 m, though very dramatic convection in such places as the Labrador Sea, Greenland Sea, and western Mediterranean Sea can deepen the mixed layer to thousands of meters. This article discusses convection reaching no deeper than a few hundred meters. Dynamically, the convection discussed here differs from deep convection in being more strongly affected by surface wind stress and much less affected by the rotation of the Earth. Convection directly affects several aspects of the near-surface ocean. Most obviously, the velocity patterns of the turbulent flow are influenced by the presence of convection, as is the velocity scale. The convective velocity field then controls the vertical transport of heat (or more correctly, internal energy), salinity, momentum, dissolved gases, and other properties, and the vertical gradients of these properties within the mixed layer. Convection helps to determine property exchanges between the atmosphere and ocean and the upper ocean and the deep ocean. The importance of convection for heat and gas exchange has implications for climate studies, while convective influence on the biologically productive euphotic zone has biological implications as well.

Phenomenology The classical problem of free convection is to determine the motion in a layer of fluid in which the top surface is kept colder than the bottom surface. This is an idealization of such geophysical examples as an ocean being cooled from above or the atmosphere

OPEN OCEAN CONVECTION

being heated from below. The classical problem ignores such complications as wind stress on the surface, waves, topographic irregularities, and the presence of a stably stratified region below the convection region. The study of convection started in the early twentieth century with the experiments of Benard and the theoretical analysis of Rayleigh. One might expect that heavier fluid would necessarily exchange places with lighter fluid below as a result of buoyancy forces. This happens by means of convective cells or localized plumes of sinking dense fluid and rising light fluid. However, such cells or plumes are retarded by viscous forces and are also dissipated by thermal diffusion as they sink into an environment with a different density. When the buoyancy force is not strong enough to overcome the inhibitory effects, the heavy-over-light configuration is stable and no convection forms. The relative strengths of these conflicting forces is measured by the Rayleigh number, a nondimensional number given by eqn [1]. Ra ¼

gaDTh3 ðkT vÞ

½1

Here g is the acceleration of gravity, a is the thermal expansion coefficient of sea water (a ¼ 2:6 104 1C1 at T ¼ 201C and S ¼ 35 PSU), DT is the temperature difference between the top and bottom surfaces, h is the convective layer thickness, and m and kT are the molecular coefficients of viscosity and thermal diffusivity, respectively (n ¼ 1:1 106 m2 s1 and kT ¼ 1:3  107 m2 s1 at T ¼ 201C and S ¼ 35PSU). The term DT ¼ Dr=r represents the fractional density difference between top and bottom. Convection occurs only if Ra is greater than a critical value, Racr , which depends somewhat on geometrical and other details of the fluid. For the classical problem of water bounded above and below by solid plates, Racr ¼ 657. For sea water under typical conditions, even a temperature difference of 0.11C makes Ra > Racr as long as h is greater than a centimeter. For Ra > Racr, the Rayleigh number still serves a useful purpose as a guide to the nature of the convective activity (though the problem also depends on the Prandtl number, Pr ¼ n=kT ). For a fixed Pr and for Ra only slightly larger than Racr , motion occurs in regular, steady cells. As Ra is increased, the motion becomes time-dependent. Regular oscillations occur, and these increase in number and frequency for higher Ra. At sufficiently high Ra, the flow is turbulent and intermittent. The value of Ra in the ocean is very large (typically greater than 1014 for a

415

temperature difference of 0.11C over 10 m), so convection is usually turbulent. Turbulent convection is usually characterized by the formation of descending parcels of cold water. In laboratory experiments, it has been found that water from the cooled surface layer collects along lines, producing thickened regions that become unstable and plunge in vertical sheets (Figure 1). In analogy to atmospheric convection, we will here call these parcels thermals, although — in contrast to the atmosphere — in the ocean they are colder than the surrounding fluid. In 1966, Howard formulated a phenomenological theory that represented turbulent convection as the following cyclic process. The thermal boundary layer forms by diffusion, grows until it is thick enough to start convecting, and is destroyed by convection, which in turn dies down once the boundary layer is destroyed. Then the cycle begins again. This phenomenological theory has implications for the development of parametrizations for the air–sea heat and gas exchange under low wind speed conditions (see later). The descending parcels of water have a mushroomlike appearance. In the process of descending to deeper layers, the descending parcels developing as a result of the local convective instability of the thermal molecular sublayer join and form larger mushroomlike structures. The latter descend faster and eventually form bigger structures. This cascade process produces a hierarchy of convective scales, which

Figure 1 Orthogonal views of convective streamers in warm water that is cooling from the surface. The constantly changing patterns appear as intertwining streamers in the side view. (From Spangenberg WG and Rowland WR (1961)) Convective circulation in water induced by evaporative cooling. Physics of Fluids 4: 743–750. ^ 1961 American Institute of Physics.

416

OPEN OCEAN CONVECTION

is illustrated in Figure 2 on the example of the haline convection.

Penetrative Convection The unstable stratification of the mixed layer is usually bounded below by a stratified pycnocline. One can imagine the mixed layer growing in depth with thermals confined to the statically unstable depth range. Suppose the density at the top of the pycnocline is r1 (Figure 3A). As surface buoyancy loss and convection increase the average density of the mixed layer, the mixed layer density increases to r2 , which is slightly denser than r1 (Figure 3B). The static instability now allows convection to act on the pycnocline down to density r2 (Figure 3C), so that the mixed layer grows at the expense of the pycnocline. This is known as nonpenetrative convection. In reality, the largest thermals acquire enough kinetic energy, as they fall through the mixed layer, that they can overshoot the base of the mixed layer, working against gravity. This is penetrative convection. The penetrative convection produces a

countergradient flux that is not properly accounted for if we model convective mixing as merely a very strong vertical diffusion. Unlike the smooth density profile at the base of a mixed layer that is growing by nonpenetrative convection (Figure 3C), penetrative convection is characterized by a density jump at the base of the mixed layer (Figure 3D). The cooling of the ocean from its surface is compensated by the absorption of solar radiation. The latter is a volume source for the upper meters of the ocean. The thermals from the ocean surface, as they descend deeper into the mixed layer, produce heat flux that is compensated by the volume absorption of solar radiation. This is another type of the penetrative convection in the upper ocean, which will be considered in more detail in a later section.

Relative Contributions of Convection and Shear Stress to Turbulence For the limiting case in which the only motion in the mixed layer is due to convection, there are simple estimates of average speed and temperature fluctuations associated with the plumes. When the Rayleigh number is high enough that the flow is fully turbulent, the plume characteristics should be largely independent of the viscosity and diffusivity throughout most of the mixed layer. In that case, ignoring the Earth’s rotation and influences from the pycnocline, the governing parameters of the system are simply the mixed layer depth h and the surface buoyancy flux B0 . B0 is based on the surface heat fluxes according to eqn [2], where r is the water density, cp is the specific heat capacity of water (E4  103 Jkg1 K1 ), L is the latent heat released by evaporation (E2:5  106 Jkg1 ), S is the surface salinity, and b is the coefficient of salinity expansion (b ¼ 7:4  104 PSU1 at T ¼ 201C and S ¼ 35PSU). h i 1 B0 ¼ gr1 ac1 ð Q þ Q þ Q Þ þ bQ L S E L L S p

½2

The first term in the square bracket in the right side of [2] relates to the buoyancy flux due to surface cooling; the second term relates to the buoyancy flux due to the surface salinity increase because of evaporation. Given all the above restrictions, the velocity scale, o * , is then given by the Priestly formula ([3]). Figure 2 Shadowgraph picture of the development of secondary haline convection. From Foster TD (1974) The hierarchy of convection. Colloques Internationaux du CNRS N215. Processus de Formation Des Eaux Oceaniques Profondes, pp. 235–241. ^ 1974 Centre National de la Recherche Scientifique.

w * ¼ ðB0 hÞ1=3

½3

This is the only combination of B0 and h that will give the proper units for velocity. Similarly, if we define the buoyancy to be b ¼ gDr=r, the buoyancy

OPEN OCEAN CONVECTION

(A) Original density

(B) After cooling

Depth

Mixed layer

Depth

417

Pycnocline

1

Statically unstable

1

2

Density

Density (D) Penetrative

Depth

Depth

(C) Nonpenetrative

1

2

1

2

Density

Density Figure 3 Schematic diagram of nonpenetrative and penetrative convection.

scale, b * , is given in [4]  1=3 b * ¼ B2 =h

½4

Laboratory experiments have shown that these scales are in good agreement with actual fluctuations during convection. For typical oceanic parameters (for instance, heat flux of Q0 ¼ 100Wm2 and h ¼ 100 m), o * is a few centimeters per second and b * is equivalent to temperature fluctuations of about 0.011C.

Two major sources of turbulent kinetic energy in the upper ocean are the wind stress and buoyant convection. Upper ocean convection is usually accompanied by near-surface currents induced by wind and wind waves. The near-surface shear is then an additional source of near-surface turbulent mixing. In the 1950s, Oboukhov proposed the buoyancy length scale, LO ¼ ku3* =B0 , where k is the Von Karman constant (k ¼ 0:4), B0 is the surface buoyancy flux (e.g., defined by [2]), and u * is the boundary layer velocity scale (friction velocity)

418

OPEN OCEAN CONVECTION

defined as u * ¼ ðt=rÞ1=2 , where t represents the bottom stress in the atmospheric case and the windstress in the oceanic case (r is the density of air or water, respectively). Later, Monin and Oboukhov suggested the stability parameter, x ¼ z=LO (where z is the height in the atmosphere or the depth in the ocean), to characterize the relative importance of shear and buoyant convection in the planetary boundary layer. Experimental studies conducted in the atmospheric boundary layer show that at xo  0:1 the flow is primarily driven by buoyant convection. From the analogy between the atmospheric and oceanic turbulent boundary layers, the Monin– Oboukhov theory is often applicable to the analysis of the oceanic processes as well. In particular, it provides us a theoretical basis to separate the layers of free and forced convection in the upper ocean turbulent boundary layer. For a 5 m s1 wind speed and Q0 ¼ 100Wm2 , the Oboukhov scale is LO B15 m. This means that the shear-driven turbulent flow is confined within a 1.5 m thick near-surface layer of the ocean. In a 50 m deep mixed layer, 97% of its depth will be driven by the buoyant convection during nighttime, with the rate of dissipation of turbulent kinetic energy there about equal to the surface buoyancy flux, B0 , as shown by Shay and Gregg.

Convection and Molecular Sublayers Convection is driven by the horizontal–mean vertical density gradient. At high Ra, typical vertical velocities are much lower near the top and bottom boundaries than they are in the bulk of the water column. Since the vertical density gradient is reduced by the convective motion, the velocity distribution causes most of the vertical density gradient to occur near the boundaries. Indeed, under low-wind, lowwave conditions in which convection dominates, the mixed layer temperature gradient is largely confined to a region only about 1 mm deep. Because the vertical heat flux at the base of the convection region is typically much smaller than at the surface, the large temperature gradient only occurs at the surface, where this thermal sublayer is often referred to as the cool skin. The temperature jump across the cool skin can be related to the vertical flux of heat at the air–sea interface and constants of molecular viscosity and heat diffusion in water using convection laws. The vertical heat flux, Q0 , can be written in nondimensional form as the Nusselt number ([5]).   Nu ¼ Q0 =cr r =ðkT DT=hÞ

½5

In [5] the heat flux is normalized by the heat flux due to vertical diffusion in the absence of convection. This quantity must be a function of the given nondimensional parameters of the system, which, for thermal convection in the absence of other driving mechanisms, are just Ra and the Prandtl number Pr (here we ignore the Earth’s rotation and entrainment from the pycnocline). A further simplifying assumption is that for high Ra (greater than 107), typical of the mixed layer, the convection is fully turbulent and does not depend on the mixed layer thickness, h, which implies [6], where AðPrÞ is a dimensionless coefficient depending on Prandtl-number (according to laboratory measurements, AE0:16  0:25). Nu ¼ AðPrÞRa1=3

½6

Given the definitions of Ra and Nu, this relation can be rearranged to yield the temperature difference across the cool skin, DT, as a function of the surface heat flux, Q0 ¼ QL þ QE þ QT ([7]).  1=4  3=4 Q0 =cp r DT ¼ A3=4 agk2T =v

½7

DT is 0.2–0.41C under typical oceanic conditions but can be as much as 11C in regions of very high heat loss to the atmosphere (e.g., Gulf Stream at high latitudes). While the term ‘sea surface temperature’ (SST) is often used to represent the temperature of the mixed layer as a whole, the existence of a cool skin means that the temperature of the literal surface of the ocean can be somewhat lower than the rest of the mixed layer. Satellite measurements of SST are based on infrared emissions from a thin layer of several micrometers, so that these measurements can be somewhat different from ship-based ‘surface’ measurements, which are generally based on sampling within several upper meters of the ocean. Indeed, while the first experimental evidence of the cool skin was obtained in the 1920s, the phenomenon was not widely recognized by the oceanic community until sophisticated methods, including remote sensing by infrared techniques, had been gradually helping to incorporate the cool skin into modern oceanography. The accuracy of current satellite remote sensing techniques is, nevertheless, still below that level at which the cool skin becomes of crucial importance. The effect of the cool skin on the heat exchange between ocean and atmosphere is also basically below the resolution of widely used bulk flux algorithms. However, one interesting practical application of the cool skin phenomenon emerged in the 1990s. Similar laws govern the thermal sublayer of the ocean (the cool skin) and diffusive sublayers

OPEN OCEAN CONVECTION

associated with air–sea gas exchange. Such gas exchange is a key biogeochemical variable, and for greenhouse gases such as CO2 is of climatological importance as well. The rate at which gases cross the air–sea interface is measured by the piston velocity, K. Boundary layer laws relate K to DT in the eqn [8]. A0 Q0 ðm=vÞ1=2 K¼ cp rDT

½8

A0 is a dimensionless constant (E1.85) and m is the molecular gas diffusion coefficient in water (m ¼ 1:6  109 m2 s1 for CO2 at T ¼ 201C and S ¼ 35PSU). The more readily available cool skin data can then be used for an adjustment of the gas transfer parametrization. The convective parametrizations for the cool skin and air–sea gas exchange are valid within the range of wind speed from 0 to 3–4 m s1. Under higher wind speed conditions, the cool skin and the interfacial air–sea gas exchange are controlled by the wind stress and surface waves. The transition is observed when the surface Richardson number,

Rf0i ¼ agQ0 =ðcp ru4 Þ, reaches a value of approxi* mately  1.5  105 (here r is the water density).

Diurnal and Seasonal Cycles of Convection For much of the year, much of the ocean experiences a cycle of daytime heating and nighttime cooling that leads to a strong diurnal cycle in convection and mixed layer depth. Such behavior is illustrated in Figure 4. At night, when there is cooling, the convective plumes reach the base of the mixed layer, which deepens as the mixed layer grows colder and denser. During the day, convection is inhibited within the bulk of the mixed layer but may still occur near the surface of the mixed layer, even if the mixed layer experiences a net heat gain. This is because the vertical distribution of cooling and heating are somewhat different. Heat loss is dominated by latent heat flux associated with evaporation and hence this forcing occurs at the top surface. Heat gain is dominated by solar radiation that is absorbed by the water over a range of depths that can extend tens of meters

2

4

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bold =

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Figure 4 Diurnal cycles in the outer reaches of the California Current (341N, 1271W). Each day the ocean lost heat and buoyancy starting several hours before sunset and continuing until a few hours after sunrise. These losses are shown by the shaded portions of the surface heat and buoyancy fluxes in the top panel. In response, the surface turbulent boundary layer slowly deepened (lower panel). The solid line marks D, the depth of the surface turbulent boundary layer, and the lightest shading shows 108 W kg1oso107 W kg1 where s is the dissipation rate of the turbulent kinetic energy. The shading increases by decades, so that the 0=mn ¼ B0 , and darkest shade is s >105 W kg1. Note that 1 MPa in pressure p corresponds to approximately 100 m in depth, Jb Jq0 ¼ ðQ0 þ QR Þ, where QR is the solar radiation flux penetrating ocean surface. (From Lombardo CP and Gregg MC (1989) Similarity scaling during nighttime convection. Journal of Geophysical Research 94: 6273–6274.) ^ 1989 American Geophysical Union.

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in many parts of ocean. For example, one can have surface heat loss of 100 W m2 occurring at the surface and net radiative heat gain of 500 W m2 distributed over the top 30 m of the ocean. Calculating the rate of change of heat due to the forcing between the surface and a depth z, we find that there is actually heat loss for small z down to a depth known as the thermal compensation depth. Below this depth, the mixed layer restratifies and convection occurs only through the mechanism of penetrative convection. For most of the World Ocean, the thermal compensation depth is less than 1 m between sunrise and sunset. Usually, the rate of turbulent kinetic energy production in the mixed layer is dominated by the convective term at night, but by the wind stress term during most of the day. Because the thermal compensation depth is generally quite small, turbulent kinetic energy generated by convection makes no contribution to turbulent entrainment of water through the bottom of the mixed layer, which lies much deeper. Under low wind speed conditions and strong solar insolation, the thickness of the surface convective layer of the ocean may reduce to only several centimeters. In that case, convection in the upper ocean may be of a laminar or transitional nature. Stable stratification inhibits turbulent mixing below the relatively thin near-surface convection layer. Vertical mixing of momentum is confined to the shallow daytime mixed layer so that, during the day, flow driven directly by the wind stress is confined to a similarly thin current known as the diurnal jet. In the evening, when convection is no longer confined by the solar radiation effect, convective plumes penetrate deeper into the stratified part of the mixed layer, increasing the turbulent mixing of momentum at the bottom of the diurnal jet. The diurnal jet then releases its kinetic energy during a relatively short time. This process is so intensive that the releasing kinetic energy cannot be dissipated locally. As a result, a Kelvin–Helmholtz type instability is formed, which generates billows — a kind of organized structure. The billows intensify the deepening of the diurnal mixed layer. Although the energy of convective elements is relatively small, it serves as a catalyst for the release of the kinetic energy by the mean flow. In the equatorial ocean, the shear in the upper ocean is intensified by the Equatorial Undercurrent; the evening deepening of the diurnal jet is therefore sometimes so intense that it resembles a shock, which radiates very intense high-frequency internal waves in the underlying thermocline. The diurnal cycle is often omitted from numerical ocean models for reasons of computational cost.

However, the mixed layer response to daily-averaged surface fluxes is not necessarily the same as the average response to the diurnal cycle. Neglecting the diurnal cycle replaces periodic nightly convective pulses with chronic mixing that does not reach as deep. Open ocean convection is a mechanism effectively controlling the seasonal cycle in the ocean as well. Resolution of diurnal changes is usually uneconomical when the seasonal cycle is considered. Because of nonlinear response of the upper ocean to the atmospheric forcing, simply averaged heat fluxes cannot be used to estimate the contribution of the convection on the seasonal scale. The sharp transition between the nocturnal period, when convection dominates mixing in the surface layer, and the daytime period, when the sun severely limits the depth of convection, leaving the wind stress to control mixing, may simplify the design of models for the seasonal cycle of the upper ocean. Incorporation of convection adjustment schemes into the oceanic component of the global circulation models leads to an appreciable change of troposphere temperature in high latitudes, which affects the global ocean and atmosphere circulation. Parametrization of the convection on the seasonal and global scales is therefore an important task for the prediction of climate and its changes.

Conclusions Observation of the open ocean convection is a difficult experimental task. Though convective processes have been observed in several oceanic turbulence studies, most of our knowledge of this phenomenon in the ocean is based on the analogy between atmospheric and oceanic boundary layers and on laboratory studies. Many intriguing questions regarding the convection in the open ocean remain, however. Some of them, like the role of penetrative convection in mixed layer dynamics, are of crucial importance for improvement of the global ocean circulation modeling. Others, like the role of surfactants in the surface renewal process, are of substantial interest for studying the air–sea exchange and global balance of greenhouse gases like CO2.

See also Air–Sea Gas Exchange. Breaking Waves and NearSurface Turbulence. Deep Convection. ThreeDimensional (3D) Turbulence. Upper Ocean Mixing Processes.

OPEN OCEAN CONVECTION

Further Reading Busse FH and Whitehead JA (1974) Oscillatory and collective instabilities in large Prandtl number convection. Journal of Fluid Mechanics 66: 67--79. Foster TD (1971) Intermittent convection. Geophysical Fluid Dynamics 2: 201--217. Fru NM (1997) The role of organic films in air–sea gas exchange. In: Liss PS and Duce RA (eds.) The Sea Surface and Global Change, pp. 121--172. Cambridge: Cambridge University Press. Gregg MC, Peters H, Wesson JC, Oakey NS, and Shay TJ (1984) Intense measurements of turbulence and shear in the equatorial undercurrent. Nature 318: 140--144. Holland WR (1977) The role of the upper ocean as a boundary layer in models of the oceanic general circulation. In: Kraus EB (ed.) Modelling and Prediction of the Upper Layers of the Ocean. Oxford: Pergamon Press. Katsaros KB (1980) The aqueous thermal boundary layer. Boundary-Layer Meteorology 18: 107--127. Kraus EB and Rooth CGH (1961) Temperature and steady state vertical heat flux in the ocean surface layers. Tellus 13: 231--238.

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Caldwell DR, Lien R-C, Moum JN, and Gregg MC (1997) Turbulence decay and restratification in the equatorial ocean surface layer following nighttime convection. Journal of Physical Oceanography 27: 1120--1132. Shay TJ and Gregg MC (1986) Convectively driven turbulent mixing in the upper ocean. Journal of Physical Oceanography 16: 1777--1791. Soloviev AV and Schluessel P (1994) Parameterization of the cool skin of the ocean and of the air-ocean gas transfer on the basis of modeling surface renewal. Journal of Physical Oceanography 24: 1339--1346. Thorpe SA (1988) The dynamics of the boundary layers of the deep ocean. Science Progress (Oxford) 72: 189--206. Turner JS (1973) Buoyancy Effects in Fluids. Cambridge: Cambridge University Press. Woods JD (1980) Diurnal and seasonal variation of convection in the wind-mixed layer of the ocean. Quarterly Journal of the Royal Meteorological Society 106: 379--394.

DEEP CONVECTION J. R. N. Lazier, Bedford Institute of Oceanography, Nova Scotia, Canada Copyright & 2001 Elsevier Ltd.

Introduction Density of ocean water generally increases with depth except at the surface where stirring by waves and convection creates a well-mixed homogeneous layer. Breaking waves alone can mix the upper 5–10 m, but convection, forced by an increase in density at the surface via heat loss or evaporation, can greatly increase the mixed layer depth. During winter, heat loss from the surface of the ocean is high and convectively mixed surface layers are the norm in the extratropical oceans. The deepest (4 1500 m) are found in the Labrador Sea, the Greenland Sea, and the Golfe du Lion in the Mediterranean Sea, because of two special features. First, they are near land where cold continental air flows over the water to create the necessary high heat loss. Second, the circulation in each is weakly cyclonic which helps to maintain the convecting water where the high heat loss occurs. The combination of these features provides the persistent heat loss from the same body of water that is needed to force convection to reach great depths. The example of a deepening convection layer in Figure 1 shows profiles of s1.5 (s1.5 þ 1000 ¼ potential density in kg m  3 referenced to 1500 decibars; 1 decibar corresponds to about 1 m) versus depth, on February 25 and March 8, 1997, in the Labrador Sea. The convecting layer is the approximately homogenous layer next to the surface about 750 m deep with a s1.5 of 34.65 kg m  3 on February 25 (Station 1) and 1150 m deep and 34.67 kg m  3 11 days later (Station 2). The buoyancy that was removed between the two profiles is proportional to the area between them, i.e. buoyancy ¼ ðg=r0 Þ

Z Drdz

where g (10 m s  2) is the acceleration due to gravity, r0(E1034 kg m  3) is the reference density, z is the depth and Dr is the difference in density between Stations 1 and 2. For the two stations illustrated this calculation yields a buoyancy loss of 0.17 m2 s  2. By ignoring the small effect of evaporation, precipitation,

422

and any advection, this buoyancy loss can be assumed to be due solely to heat loss from the surface. The loss is converted to joules by dividing by ga/r0c where g and r0 are as before and a (about 10  4 1C  1) is the thermal expansion of water and c (4.2 kJ kg  1 1C  1) is the specific heat of sea water at constant pressure. The conversion suggests that a heat loss of about 0.68  109 J m  2 was required to remove the buoyancy between the two dates. Over the 11 days between the observations this heat loss is equivalent to an average rate of heat loss of 715 W m  2. By a similar calculation, the heat loss required to increase the depth of convection to 2000 m would have been about 1.2  109 J m  2 or 460 W m  2 over a month. If the profiles in Figure 1 had been obtained during an era of mild winters rather than during one of abnormally severe winters they probably would have exhibited a markedly lower density in the upper layers. This might occur because of abnormally large freshwater flows into the surface layers due to increased outflows from the Arctic, warmer summers or a multiyear period of restratification following a vigorous period of convection. As the lower density represents ‘extra’ buoyancy to be removed before convection can proceed to greater depths, the ultimate depth of the convecting layer will be less in this situation, for a given heat loss, than in the illustrated one. Thus the ultimate depth of the convecting layer during a winter depends on the total amount of buoyancy lost from the sea surface and the distribution of that buoyancy with depth. The distribution of the newly convected water in two dimensions is illustrated in the contour plot of salinity across the Labrador Sea in Figure 2. These data were obtained in July following the exceptionally cold winter of 1992–93. The water mass resulting from convection is the large volume of nearly homogeneous water lying between 360 and 800 km on the horizontal scale and between 500 and 2300 m in the vertical. Because of its large volume, unique properties, and the fact that it spreads beyond its region of formation, this water is known as Labrador Sea Water. The upper layer (0–500 m) in the central part of the section is clearly not as well mixed as the layer between 500 and 2300 m. This is because the observations were obtained in July about 3 months after deep convection ceased at the end of the cooling season, about April 1. Since that time the surface layer has been flooded with fresh water derived from melting ice and river runoff. Also, the layer below this low salinity surface layer, to about 500 m, has been

DEEP CONVECTION

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Figure 1 Vertical distribution of s1.5 obtained from R/V Knorr at 56.81N, 54.21W in the Labrador Sea on February 25 (Station 1) and March 8, 1997 (Station 2).

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Distance (km) Figure 2 Salinity distribution across the Labrador Sea between 53.01N, 55.51W and 60.61N, 49.31W obtained between June 19 and 23, 1993. The water between 500 m and 2200 m in the central part of the section is unusually homogeneous because of deep convective mixing during the severe winter of 1992–93. The CTD station positions are indicated by numbered triangles along the surface.

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invaded by higher salinity water from the right (northeast) and to a lesser extent the left (south-west). The newly formed Labrador Sea Water is a mixture of all the water down to 2300 m including the water in contact with the atmosphere at the surface. In these uppermost layers the concentration of gases such as oxygen, carbon dioxide, tritium, and chlorofluorocarbons (CFCs) are at or near equilibrium with the atmosphere. By transporting these gases down from the upper layers of the ocean to intermediate depths, convection provides a mechanism to ventilate the deeper layers, which is one of the most important consequences of deep convection. Dissolved oxygen, for example, is slowly used up in the deep ocean by biological processes and would eventually vanish without the renewal via convection. Also most of the carbon dioxide ever put in the atmosphere by volcanoes since the formation of the Earth became dissolved in the ocean and is now contained in sediments in the bottom of the ocean. As combustion of fossil fuels over the earth raises the carbon dioxide content of the atmosphere it is important to understand the rate at which this gas is entering the deeper layers of the ocean through processes such as deep convection. Subsequent to formation, Labrador Sea Water spreads to other regions of the ocean at intermediate depths. Knowledge of the speed of this flow and its influence increased during the 1990s due to the widespread high quality observations of temperature, salinity, and CFCs across the North Atlantic obtained under the international World Ocean Circulation Experiment. The newly ventilated water formed in the Labrador Sea during the severe winters of the early 1990s moved across the North Atlantic at about 2 cm s  1. This speed is about three to four times greater than the previous estimate, leading to the conclusion that the intermediate flows are much faster than previously thought. A comparison of six decades of data from the Labrador Sea and from the subtropical waters near Bermuda suggest that the products of deep convection in the Labrador Sea impact the waters off Bermuda after about 6 years.

Plumes – the Mixing Agent Convection begins to increase the depth of the mixed layer in the Labrador Sea near the end of September when the surface net buoyancy flux from the surface turns from positive to negative. Deepening continues until about the end of March when the buoyancy flux again becomes positive. When convection is active, water at the surface becomes denser than the underlying water and descends in plumes. This water is replaced by slightly lighter water rising toward the

surface. The physical features of the convecting water including the plumes and the water between have been the subject of a number of investigations; most notably by the group of scientists at Kiel working in the Golfe du Lion in the Mediterranean Sea with moored acoustic Doppler current profilers (ADCP) and current meters. The cartoon in Figure 3 summarizes some of the main features of plumes and the mixing layer. At the surface is the thermal boundary layer where the water is losing heat/buoyancy to the atmosphere. Water in this layer is, on average, slightly denser than in the mixed layer beneath and descends into the mixing layer within plumes which have a horizontal dimension of about 1 km, approximately equal to that vertical extent, i.e. an aspect ratio of E 1. The average rate of descent within the plumes is about 0.02 m s  1 while the maximum is E0.13 m s  1. Rotation of the plumes, due to the horizontal component of the Coriolis force, is expected because water must converge into the plume at its top and presumably diverge out of it near the bottom. However, this effect has not yet been conclusively observed in the field although it has been observed in laboratory experiments and in numerical simulations. Another effect that has not been observed is an increasing horizontal dimension with depth which is expected if water is entrained into the plumes as they descend, or if they decelerate as they go deeper. Observations also indicate that there is no net vertical mass flux within a convecting region or patch. This appears to have solved the long-standing puzzle of whether the descending water was replaced by rising water between the plumes or by converging flow in the upper layer and diverging flow in the deep layer. Finally, on average, the plumes are not penetrative, i.e. the plumes do not have enough energy to descend into water that is denser than the water within the plume. One consequence of this is illustrated in Figure 1 by the fact that the bottom of the mixing layer at Station 2 lies on the s1.5 versus depth curve observed earlier on February 25. If convection was penetrative the bottom of the mixed layer on March 8 would lie below this curve and the s1.5 versus depth gradient at the bottom of the mixing layer would be greater than when it was observed earlier. The plumes in the cartoon suggest that there should be a high correlation between fluctuations in temperature and vertical velocity. However, recent measurements from drifting floats indicate this correlation to be weak, thus making the cartoon a rather simplified view of the true situation. In reality the plumes are probably not vertical over the full depth of the convecting layer but contorted by the largerscale flows.

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Figure 3 A schematic diagram of a E 1000 m convecting layer indicating approximate values for features of individual plumes including the horizontal scale, vertical downward velocity, rotation, and entrainment. Across the patch of convecting water there is no net vertical mass flux and no significant penetration into the layers of denser water beneath the convecting layer. Buoyancy (B0) lost from the surface creates the thermal boundary layer in the upper E 100 m where the denser water is formed which sinks within the plumes. The wiggly up arrow indicates the (slow) upward flow that replaces the (relatively fast) downward flow within the plumes. Note that the horizontal scale in the figure is E 50 times the vertical scale.

A direct view of the motion within convecting plumes has recently been obtained from freely drifting floats. When one of these is launched it immediately sinks to a predetermined depth below the convecting layer where it remains for typically 7 days while its buoyancy adjusts. At the end of this period its buoyancy is decreased slightly and the float rises into the convecting layer. A large attached drogue then causes the float to moved up and down with the vertical motions of convection. In the Labrador Sea over 25 days in February and March 1997 the maximum vertical velocity observed by a set of these floats was downward at 0.2 m s  1 with a rms value for all the observations of 0.02 m s  1. This is equivalent to a round trip of the convecting layer of 1 day for the average water parcel. On a number of occasions the floats were seen to penetrate below the average bottom of the mixing layer. Contrary to the

conclusions mentioned above that the convection is not penetrative, this suggests that a certain amount of plume penetration into denser layers does occur.

Temperature and Salinity Variability Year-long time-series records of temperature and salinity obtained in the middle of the Labrador Sea indicate that there is a marked increase in temperature and salinity variability during and following convection. This is evident in the temperature record in Figure 4 obtained at 510 m in 1994–95. Between June and the middle of February the temperature sensor is below the mixed layer and the temperature slowly increases by about 0.21C. In mid-February the temperature drops by about 0.41C as the deepening convecting layer reaches the depth of the sensor. At this time the magnitude of the variations in

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Months, 1994 – 95 Figure 4 A 1 year record of temperature at 510 m in the central Labrador Sea illustrating the increase in variance during and after the mixed layer reaches the depth of the instrument in early February 1995 plus the sudden upward shift in temperature near April 1 associated with the end of convection.

temperature suddenly increase and continue at a high level for a number of weeks. Spectra calculated from 85-day pieces of this record before and after the arrival of the convection layer show a broadband fourfold increase in the spectral energy of the variability after the sensor is immersed in the mixed layer. Time-series of the energy show a peak in February shortly after the mixed layer arrives followed by a decline to 1/30th of the peak value by the end of the record in June. Similar fluctuations occur in salinity and are largely in phase with those in temperature. Figure 5 shows temperature versus salinity plots at four depths during 21 days in March 1995 when convection was proceeding to about E 2000 m. During this time period density at each of these levels was relatively constant in time. In the figure the hourly observations at each depth show the extent of the fluctuations in temperature and salinity and the fact that they are largely parallel to the constant density surfaces; the T–S fluctuations tend to be parallel to the isopycnals. The most probable explanation for the fluctuations is that they are horizontal variations in temperature and salinity being swept past the mooring by the current. One suggestion is that these variations reflect horizontal variability in the depth of convection. However, recent work in the mixed layer of the tropical Pacific demonstrates that compensating horizontal variations in temperature and

salinity may be ubiquitous features of the mixed layer whether it is convecting or not. These results lead to the alternative suggestion that the compensating temperature and salinity variations exist in the windmixed surface layer before deep convection begins and are propagated downward by the convection. When convection stops, the vertical density stratification is reestablished in the surface layers and the horizontal variations that appeared during convection are no longer renewed. Those variations existing when convection ends are then slowly mixed away by turbulent eddies leading to the decay in the amplitude of the fluctuations as observed in the timeseries. Assuming a horizontal scale L of 100 km for the region of convection with its small-scale horizontal variations, the timescale of eddy mixing will be about L2/KH where KH, the horizontal eddy diffusivity, is about 103 m2 s  1. This gives a timescale for the horizontal mixing of about 4 months, which is about the decay time observed in the records.

Restratification At the end of the cooling season, vertical mixing due to convection ceases and its dominant influence on mid-depth water properties ends. This also marks the beginning of the restratification process during which the vertical stratification existing prior to the homogenization begins to be reestablished. Two

DEEP CONVECTION

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Figure 5 Temperature versus salinity diagrams based on data obtained at 260, 510, 1010, and 1510 m in the central Labrador Sea, between March 12 and April 2, 1995, during the final stage of convection when the density at these depths remained relatively constant.

timescales seem to be involved. At the end of convection a rapid restratification occurs which is indicated by sudden shifts of variables such as temperature. One example is indicated in Figure 4 by the increase of 0.21C near April 1 following roughly 6 weeks of convective activity at this depth. It is not clear if this increase indicates an end to convection over a large area, or the advection of a stratified nonconvecting water column to the observation site. The first option, however, seems more likely as the end of convection appears in other records as a rapid increase in stratification especially in the upper layers. For example, a tomographic array in the Golfe du Lion observed a roughly 40 day restratification period following convection. This seems to be the only observation of restratification over a large area. Another example is the record from a PALACE (profiling autonomous Lagrangian circulation explorer) in the Labrador Sea which shows, during 2

consecutive years, a sudden transition between the low stratification associated with convection and a stratified water column. This record is admittedly like a mooring, from a single point, but it does give a consistent picture in the 2 years. A recent numerical model of the restratification process may describe this rapid phase. It has a homogeneous cylinder of water floating in an ocean of constant stratification. The density gradient between the cylinder and the surrounding ocean gives rise to a narrow cyclonic current which breaks up via baroclinic instability into baroclinic eddies. These mix the homogeneous water horizontally with the stratified waters and so dissipate the homogeneous cylinder in timescale t. For the special case where the stratification in the water surrounding the homogeneous cylinder is concentrated in the upper layer h: tE56 r=ðhDbÞ1=2

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where r is the radius of the homogeneous cylinder, h is the depth of the upper stratified water bounding the homogeneous cylinder and Db is the difference in density between the homogeneous water and the surrounding water in buoyancy units (i.e. Db ¼ gDr/r0). For the Labrador Sea where h E 500 m, Db E 2  10  3 m s  2 and r E 100 km; t E 65 days. This result, being of the same order as the observed rapid changes in stratification, suggests that the model may be appropriate to explain the observations. The long timescale of restratification has now been observed in the Labrador Sea. Observations have been made continuously over one summer and intermittently over a few successive years. Changes over the summer of 1996 are illustrated in Figure 6 by the depths of four of the isopycnals within the upper 1000 m. At stations between 400 and 600 km the isopycnal depths increase significantly between May and October while the 27.72 and 27.74 kg m  3, surfaces between 650 and 790 km, show a decrease in depth. These changes suggest that lighter water, from beyond the region of deepest convection (400– 600 km), moves into the upper water column in the region of deepest convection while denser water in the region of deepest convection moves outward toward the boundaries at mid-depth. Restratification over a number of years is illustrated in the time series of s1.5 through the 1990s in Figure 7. In the early years of the decade the water produced by the convection, indicated by the layer of

low vertical gradient, lies within the 34.64 and 34.70 kg m  3 surfaces. Between 1990 and 1994 the volume of this waterremains roughly constant but the value of s1.5 at the core increases from 34.67 to 34.69 kg m  3 as the winters became more severe and the convecting layer continued to deepen into the stratified layer below. In the years following 1995, convection was limited to 1000–1500 m. The deep reservoir of ‘homogeneous’ water was thereby isolated below the convecting layer and slowly decreased in volume with each passing year as it drained away. This decrease in volume was balanced by an increase in the volume of lighter more stratified water in the upper layers from the boundaries. The large interannual variation in the volume of the Labrador Sea Water illustrated in this figure is a well known property of the water mass; however, its effects on the large-scale ocean currents and processes are not yet well understood.

Discussion While the focus of this article has been on the Labrador Sea, numerous aspects of the convective processes discussed here also apply to the other two locations of open-ocean convection in the North Atlantic: the Greenland Sea and the Mediterranean Sea. But as there are common threads to the overturning in these seas there are also significant regional differences. The Greenland Sea is unique in

Distance (km) 200

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0

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that ice plays a significant role in the preconditioning phase. The deepest convective overturning occurs in late winter just after the ice-free ‘Nord Bukta’ region opens up. In the Mediterranean the winds are more localized than in the other two regions. This clearly influences the convection, as the region of deepest mixed layers generally lies in the path of the Mistral winds. Furthermore, these winds are rarely coldenough to cause convection during daylight hours, so the Mediterranean has a daily cycle of overturning that is not present in the other two seas. Also, the impact of the basin-scale NorthAtlantic Oscillation wind pattern influences the Labrador and Greenland Seas to a much greater extent than the Mediterranean. Finally, the dimensions of the convection zones and convected water masses differ greatly. In the Mediterranean, the convecting patch is of order 50 km wide; in the Greenland Sea it is of order 100 km wide, and in the Labrador Sea the zone of convection approaches 500 km in width, and includes the boundary currents. Convection at each of these three Atlantic sites contributes to the global meridional overturn circulation, although the quantitative measures are not yet known. In the Greenland Sea particularly, the deep convection into the cyclonic gyre seems rather isolated from the processes that produce the dense overflows. Mediterranean Water and Labrador Sea Water both make an obvious contribution to the Upper North Atlantic Deep Water, respectively, as

high and low salinity endpoints. Distant identification of Labrador Sea Water is through its low salinity; potential vorticity; low nutrient concentration; high dissolved oxygen, tritium, and CFCs. Along the western boundary velocity and CFC maxima associated with Labrador Sea Water have been observed near Abaco, nearly 5000 km south. For the era ending in 1977 a dilution of the tritium maxima of the deep western boundary currents by factors of order 10 was observed, from the subpolar gyre to the Blake-Bahama Outer Ridge. This suggested dilution and delay (recirculation) mechanisms en route. Model studies suggest that when convection is initiated or increased at the high-latitude source, a pressure wave propagates south along the western boundary, as a topographic Rossby wave, well before the arrival of tracer-tainted, identifiable water mass. Such model studies point out that sloping topography acts as a wave guide, and rather gently leads dense water masses equatorward from high latitude, as they slowly sink. Thus, ‘sinking’ is minimal in the near-field of the convection, but occurs downstream. Production of kinetic energy of theoverturning circulation by potential energy created by buoyancy forcing requires that dense water sink and less dense water rise, but the sites of sinking and rising are, at least in modestudies, often distant from the convection. In a diapycnal/epipycnal coordinate system, however, convection is more locally associated with

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time-averaged diapycnal transport (‘water-mass conversion’). Such analyses are beginning to be carried out with models and are an insightful way to approach the link between convection, sinking, and global meridional overturning. Thus we are still seeking to quantify production rates of the constituent water masses of the global meridional overturning. Outward transport of Labrador Sea Water is even difficult todefine, because of extensive recirculation within the subpolar gyre, and entrainment once the water mass has left the subpolar gyre. Estimates have ranged from o1 Sv to 410 Sv. Much alsremains unknown about the detailed geography of deep convection and circulation. In the Labrador Sea, both interior and boundary currents are known to participate in the deep convection (estimates of 1–2 Sv of boundary current production). However, the boundary current, is shielded from deep convection by low salinity shelf waters at some sites. Where the circulation crosses from Greenland to Labrador, the boundary currents broaden and slow down, andare generally exposed to some of the most intense air–sea heat flux in the sea; there and over the wide continental slope near Labrador, convection may be particularly deep. Direct velocity and transport measurements are needed to augment water mass observations. Unfortunately the Lagrangian movement of water masses is difficult to observe even with modern ‘quasi-Lagrangian’ floats and drifters. A recent description of the Labrador/Irminger Sea circulation

from PALACE floats notes that ‘no floats travelled southward to the subtropical gyre in the deep western boundary current, the putative main pathway of dense water in the meridional overturning circulation’. If the boundary current is concentrated to a narrow width, for example at the Flemish Cap, then these profiling floats may have difficulty staying within it; tracer observations assure us that the transport does in fact take place.

Further Reading Lazier JR, Pickart RS, and Rhines PB (2001) Deep convection. In: Ocean Circulation and Climate – Observing and Modelling the Global Ocean. London: Academic Press. Lilly J, Rhines P, Visbeck M, et al. (1999) Observing deep convection in the Labrador Sea during winter 1994– 1995. Journal of Physical Oceanography 29: 2065--2098. Marshall J and Schott F (1999) Open-ocean convection observations, theory and models. Reviews of Geophysics 37: 1--64. Schott R, Visbeck M, and Send U (1994) Open ocean deep convection, Mediterranean and Greenland Seas. In: Malanotte-Rizzoli P and Robinson AR (eds.) Ocean Processes on Climate Dynamics: Global and Mediterranean Examples, pp. 203--225. Dordrecht: Kluwer Academic Publishers. Lab Sea Group (1998) The Labrador Sea Deep Convection Experiment. Bulletin of the American Meteorological Society 79: 2033--2058.

DOUBLE-DIFFUSIVE CONVECTION R. W. Schmitt, Woods Hole Oceanographic Institution, Woods Hole, MA, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction The density of seawater is determined by both its temperature and its salt content or salinity. Whereas added heat makes water lighter, added salt makes it denser, so both must be considered when evaluating the gravitational stability of the water column. That is, a given column of water will ‘convect’ or overturn if dense waters overlie lighter waters. In many parts of the world ocean, the distributions of temperature and salinity are opposed in their effects on density. This arises because of the tendency of warm water to easily evaporate in low latitudes, the predominance of rainfall in cold, high-latitude regions, and the deep circulation patterns that bring the cold waters to lower latitudes. The opposing effects of temperature and salinity on density, and the fact that the molecular conductivity of heat is about 100 times as large as the diffusivity of salt in water, makes possible a variety of novel convective motions that have come to be known as double-diffusive convection. In the following, the oceanic double-diffusive mixing phenomena such as ‘salt fingers’, ‘diffusive convection’, and ‘intrusions’ are discussed in turn. Observational evidence suggests their importance in all the oceans, and models indicate a substantial impact on water mass structure and the thermohaline circulation.

Salt Fingers In much of the subtropical ocean, warm, salty water near the surface overlies cooler, fresher water from higher latitudes. If the temperature contrast could be removed there would be a large-scale overturning of the water column, releasing the very substantial energy available in the salt distribution. However, this does not happen except on a small scale, where the greater diffusivity of heat can establish thermal equilibrium in adjacent water parcels that still have strong salt contrasts. A bit of warm, salty water displaced into the cold freshwater beneath loses heat, but not much salt to the surrounding water, leaving a cool, salty water parcel that continues to sink. Similarly, a cold fresh parcel displaced upward gains

heat but not salt, becoming warm and fresh and therefore buoyant. This ‘salt finger’ instability, discovered by M. Stern in 1960, appears as a closepacked array of up and down flowing convection cells which exchange heat laterally but diffuse little salt. The result is an advective transport of salt and, to a lesser extent, heat in the vertical. Typical cell widths in the ocean are 2–3 cm, the scale for effective heat conduction. The salt finger instability is ‘direct’, in the sense that initial displacements are accelerated, and can be modeled accurately with an exponential growth rate. When most intense, the fingers tend to exist on high-gradient interfaces separating wellmixed layers in the adjacent fluid. The significant role of salt fingers in oceanic mixing is now becoming apparent, as there are clear indications that it is the dominant mixing process in certain regions and a contributing process within the main thermocline of the subtropical gyres. As could be expected, the propensity toward salt fingering is a strong function of the intensity of the vertical salinity gradient. The instability can grow at extremely weak values of the salinity gradient, because the diffusivity of salt is 2 orders of magnitude less than the thermal conductivity. When expressed in terms of the effects on density, all that is required is a top-heavy density gradient due to salt that is only about one-hundredth of the gradient due to temperature. That is, the density ratio, Rr, must be less than the diffusivity ratio: 1oRr  aTZ =bSZ okT =kS E100

½1

where a, b are the thermal expansion and haline contraction coefficients, TZ, SZ are the vertical gradients of temperature and salinity, and kS, kT are the molecular diffusivity for salt and the thermal conductivity. This criterion is met over vast regions of the tropical and subtropical thermocline, since the ratio of diffusivities is about 100. However, while the required salt gradient is very small, the growth rate of salt fingers does not become ‘large’ until Rr approaches 1 (at which the vertical density gradient vanishes). Indeed, we find that the primary fine-scale evidence for salt fingers, the ‘thermohaline staircase’, occurs only when the density ratio becomes low. Fingers transport more salt than heat in the vertical and have a net counter-gradient buoyancy flux. Since the growth rate and fluxes increase with the strength of the stratification, high-gradient regions will harbor greater fluxes than adjacent weak-gradient intervals.

431

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This leads to a buoyancy flux convergence that can cause the weaker-gradient region to overturn and mix. The resulting structure has thin interfaces separating thicker, well-mixed layers. The layers are continuously mixed by the downward salt flux, and the convective turbulence of the layers serves to keep the interface thin and limits the length of the fingers. Observations of the ‘thermohaline staircase’ have been reported from several sites with strong salinity gradients. A necessary condition for an organized salt finger staircase seems to be that the density ratio is less than 1.7 (Figure 1). Such conditions are found occasionally near the surface, where evaporation produces the unstable salinity gradient, but more often at depth where the presence of isopycnal

gradients of temperature and salinity can lead to a minimum in Rr, provided there is a component of differential advection (shear) acting on the isopycnal gradients of T and S. Examples of staircases are found beneath the Mediterranean water in the eastern Atlantic, within the Mediterranean and Tyrrhenian Seas, and beneath the subtropical underwater (salinity maximum) of the western tropical Atlantic. Detailed examinations of one particular staircase system in the western tropical Atlantic were made in 1985 (Caribbean Sheets and Layers Transects – C-SALT) and in 2001, when a ‘Salt Finger Tracer Release Experiment’ (SFTRE) was performed. Over a large area in the western tropical North Atlantic (B1 million km2), a sequence of B10–15 mixed

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Figure 1 The occurrence of thermohaline staircases as a function of density ratio. The strong staircases have Rro1.7. Irregular steppiness characterizes the central waters of the subtropical gyres, where RrB2.

DOUBLE-DIFFUSIVE CONVECTION

layers, 5–40-m thick, can be observed. Data from the 1960s to the 2000s indicate that the layers are a permanent feature of the region, despite layer splitting and merging, and a moderately strong eddy field. One of the most remarkable characteristics of this staircase is the observed change in layer properties across the region. Layers get colder, fresher, and lighter from north to south (Figure 2), the inferred flow direction for the upper layers, which appear to be losing salt to the layers below. These unique water mass transformations in temperature, salinity, and density provide strong evidence for salt fingers. That is, such changes can only be due to a flux convergence by salt fingers, which transport more salt than heat; turbulence transports the two components equally and isopycnal mixing, by definition, transports them in density-compensating amounts. Towed microstructure measurements taken in the staircase revealed limited-amplitude, narrow-band temperature structure within the interfaces. The dominant horizontal wavelength was B5 cm, in excellent agreement with the theoretical finger scale. The shape of towed microstructure spectra for this and many other observations is also found to be distinctly different from that of turbulence, leading to useful discrimination tests for towed data.

Vertically profiling instruments which measure the dissipation rates of both thermal variance and turbulent kinetic energy reveal a strong correlation of thermal variance with the interfacial gradients that provide the strongest finger growth rate (Figure 3). Such data also allow the discrimination of salt fingers from turbulence by their relative efficiencies in converting energy sources into changes in the stratification. That is, salt fingers are rather efficient in converting energy from the salt field to the thermal field (B70%), with the result that there is relatively little viscous dissipation for the amount of mixing achieved. In contrast, turbulence is rather inefficient, with only B20% of dissipated kinetic energy converted into an increase in potential energy. Also, salt fingers lead to a net decrease in potential energy, exactly the opposite to turbulence (this is what allows it to maintain staircase-type profiles, whereas turbulence should ultimately smooth the overall profiles). A good way to appreciate this difference in mixing mechanisms is to compare the formulas for estimating the vertical diffusivities from microstructure measurements of the dissipation rates of turbulent kinetic energy (e) and thermal variance (w). These formulas are contrasted below:

•

for turbulence (with flux Richardson number, Rf ¼ 0.1770.03, after Osborn (1980) and Osborn and Cox (1972)):

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Figure 2 Potential temperature–salinity values from a depth cycling CTD (an acronym for ‘conductivity, temperature, and depth’) on a mooring in the center of the tropical Atlantic thermohaline staircase during SFTRE. The color intensity represents the number of observations of any one T–S value; thus the mixed layers appear as distinct high-density lines in this 4.5-month time series. The evolution of layer properties across the region is such that layers become warmer, saltier, and denser from southeast to northwest, and the advection of the layers past the mooring provides the range of T–S values observed. The layer properties cross isopycnals (the 26.8, 27.0, 27.2, and 27.4 potential density surfaces are shown) with an apparent heat/salt density flux convergence ratio near 0.85.

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•

Rf e e e ¼ Gt 2 E0:2 2 1  Rf N2 N N

wy 2yZ

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for salt fingers (with Rr ¼ 1.6, and flux ratio g ¼ 0.7, after St. Laurent and Schmitt (1999)): Rr  1 e e E2 2 1  g N2 N Rr w y KS ¼ E2:3Ky g 2yZ   w g g Rr  1 e Ky ¼ y ¼ KS ¼ 2yZ Rr Rr 1  g N 2 e e ¼ Gf 2 E0:8 2 N N

KS ¼

½3

Note that the ‘mixing efficiencies’ Gt, Gf are distinctly different for turbulence and salt fingers, with the fingers being more efficient and dissipating less energy for a given amount of mixing. A broad-scale microstructure survey capable of addressing these issues was done with the North Atlantic Tracer Release Experiment (NATRE;

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Figure 3 Profiles of potential temperature and salinity (left panel), the dissipation rate of thermal variance (middle panel), and the theoretical salt finger growth rate (right panel) for a high resolution profiler cast during SFTRE. The tracer was injected into the layer with potential temperature near 10 1C (B350-m depth). The salt finger growth rate calculated from the fine-scale temperature and salinity gradients is a good predictor of the microscale dissipation rate of thermal variance.

see Tracer Release Experiments). This region of the eastern North Atlantic thermocline is susceptible to salt fingers, and optical microstrucuture imagery revealed that they were the most frequently observed microstructure in the thermocline. In addition, it was obvious in the sensor data that there were many occurrences of the ‘high-chi, low-epsilon’ signature of salt fingers, that contrasted with the high-epsilon signatures of turbulence. A parametric sorting of the mixing events allows classification of the stronger microstructure patches by the value of the local Richardson number and density ratio. Statistically significant variations in the value of the ‘mixing efficiency’ were observed in this parameter space (Figure 4). When translated into a flux ratio for salt fingers, the oceanic microstructure data are in excellent agreement with laboratory experiments on salt fingers. This parametric approach to the microstructure allows a classification of the mixing events as either turbulent (with low efficiency) or salt fingering (with high efficiency). With each occurring at different

frequencies in the water column, this translates into differences for the net vertical eddy diffusivities for heat and salt. At the depth range of the tracer injection in NATRE, the diffusivity estimated taking salt fingers into account agrees well with the value derived from tracer dispersion. Analysis using the conventional turbulence formula yields a diffusivity that is 50% low and a diapycnal velocity of the wrong sign. The magnitude of the fluxes in NATRE can be contrasted with those in the C-SALT/SFTRE staircase. From the substantial rate of dissipation of thermal variance, we can estimate an eddy diffusivity for salinity of 0.9  10  4 m2 s  1 within the staircase. This is in excellent agreement with the observed dispersion of tracer within the staircase, showing that diffusivities are elevated by an order of magnitude by the formation of steps. Since the staircase occupies about one-fourth of the area of the Atlantic between 101 and 151 N, the vertical salt flux in this high gradient area is predicted to be 3–4 times as large as the flux in the remaining area of this latitude band. This is because the rest of the area is expected to have a

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Figure 4 The ‘mixing efficiency’, G, as a function of density ratio and Richardson number, for the microstructure sampled in the main thermocline of the eastern North Atlantic Ocean (right panel). The left panel shows data from non-double-diffusive regions, where results are consistent with turbulence. The low-Rr, high-Ri regime of the right panel provides clear indications of high-efficiency salt fingers playing a significant role in the mixing of the North Atlantic thermocline.

diffusivity 10 times smaller (like NATRE) as well as a weaker salinity gradient. Thus, the staircase areas appear to be very significant sites of enhanced diapycnal exchange and water mass transformation.

Diffusive Convection The ‘diffusive’ form of double-diffusive convection is realized when the stratification is the opposite of the salt finger situation. That is, cold fresh water overlies warm salty water, with the salt providing the overall stabilization of the water column. However, there is energy to be released in the ‘warm on the bottom’ temperature distribution, and the different rates of heat and salt diffusion allow convection to occur. The essential physics is distinct from salt fingers, as the faster diffusion of heat is releasing energy in its own distribution rather than that of the slowerdiffusing salt. Again considering movement of small parcels of water, we see that the elevation of warm salty water into the cold fresh one will cause it to become cold salty water, and thus heavier than when it started upward. Instead of accelerating upward as in a salt finger, it is actually driven back down with greater force than it took to initially displace it. This is termed an ‘overstability’ and leads to a growing

oscillation. However, the oscillatory behavior is hard to observe except in careful laboratory experiments, as it quickly reaches an amplitude where transition to a layered series of convective cells is realized. This is another form of thermohaline staircase, with temperature and salinity both increasing with depth. A laboratory experiment that involves heating a stable salt gradient from below easily develops a thermohaline staircase in the diffusive sense. The mixed layers are maintained by convective motions driven by the heat flux from below; the thin, stable, gradient regions are sharp interfaces that conduct heat vertically, but transport little salt. In the ocean such ‘diffusive’ staircases are mostly found in highlatitude oceans, where surface cooling and freshening can set up the necessary gradients. Often, diffusive staircases are found under sea ice (Figure 5). It seems that the ice helps to isolate the water column from wind forcing, leading to exceptionally weak internal waves, so that the relatively slow diffusive process can dominate the vertical mixing. The fluxes for the diffusive staircase are generally less than fluxes for salt fingers. This is because fingers advectively carry heat and salt vertically across the interfaces, whereas a diffusive interface must rely largely on vertical conduction across the horizontal interface. The surface area for heat diffusion is much

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Temperature (°C) 1.0 0

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340 0.01 °C 0.1 C Figure 5 Temperature staircase in the Arctic halocline beneath the ice. Five- to ten-meter mixed layers are seen separated by thin ‘diffusive’ interfaces across which heat conduction provides a buoyancy flux to stir the adjacent layers. Such staircase layers are widespread beneath the ice and the upward heat flux they supply may contribute to its melting.

greater in the convoluted structure of a salt finger interface. In many ways, the fully developed diffusive interface is simply a modified form of Rayleigh– Bernard convection, with fluid boundary conditions and the diffusion of salt acting as a weak drag on the intensity. The relative effectiveness of heat and salt diffusion across the interface sets the salt to heat flux ratio. Except for density ratios very close to 1, the flux ratio is rather low. This can be understood by considering an interface made sharp by convection. The heat and salt would diffuse into boundary layers on either side, with different thicknesses depending on the square root of their diffusivities. After a certain time, the thermal boundary layer would be thick enough to be unstable and convection would occur, carrying the heat and salt anomalies away from the interface. The relative amounts of salt and heat transported should depend on the square root of the diffusivity ratio (B0.1), a number in reasonable agreement with laboratory measurements, so long as the density ratio is not too close to 1. At density ratios closer to 1, the interface is increasingly disrupted by turbulent plumes from the mixed layers, and a more direct transport of both heat and salt occurs, resulting in a higher salt-to-heat buoyancy flux ratio. Of course, this ratio is limited by the energetics to be less than 1.

Strong but localized diffusive staircases are found at the hot, salty brines in topographic deep areas found at oceanic spreading centers. There the separation of heat and salt could be contributing to pooling of the brines in topographic depressions and possibly ore formation as well. The diffusive process also plays a role in intrusions and on the upper side of warm, salty water masses such as the Mediterranean water in the Atlantic. It may be a factor in the evolution of fresh mixed layers laid down by rain, river inputs, or ice melt. These can create ‘barrier layers’ whereby the strong salt stratification prevents mixing with underlying water. If the fresh surface layers cool, the conditions for diffusive convection arise at the base of the mixed layer. Since diffusive convection transports heat upward, but not much salt, there is a tendency to maintain the barrier layer stratification. Thus, such barrier layers should persist longer than they would without double diffusion. Barrier layers appear to be important in modifying air–sea interactions in both the Tropics and highlatitude areas of deep convection such as the Labrador Sea. However, the most extensive regions of diffusive convection are found beneath the surface layers in the polar and subpolar oceans. Steps under the Arctic ice were first reported in the 1970s, and have been observed to cover much of the Arctic in recent data. A ‘diffusive’ thermohaline staircase between about 200- and 400-m depth appears to be a ubiquitous feature under most of the Arctic ice field away from the boundaries. It supports a heat flux from the intruding warm, salty Atlantic water to the cooler, fresher Arctic surface waters above. The extensiveness of the staircase may be due to the especially weak internal wave field under the ice. This is likely due to the rigid ice lid, but also possibly due to an enhanced wave decay within the convectively mixed staircase itself. Areas near topography with stronger internal waves and more frequent turbulent mixing events are less likely to harbor a staircase. The downgradient buoyancy flux from the turbulence (an upgradient buoyancy flux is necessary to maintain a staircase) and the destruction of the small-scale property gradients by isotropic turbulence are competing factors to the double diffusion. The waters around Antarctica also display prominent diffusive staircases. The layers in the Weddell Sea are much thicker (10–100 m) than those found in the Arctic and may support an upward heat flux of 15 W m  2 in open waters, if the diffusive interfaces are thin enough, an issue not easily resolved with ordinary instruments. This flux is sufficient to be important in upper ocean heat budgets and may help to maintain ice-free conditions in the summer. Lower

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fluxes are estimated for the Arctic steps. However, recent observations of a much stronger incursion of Atlantic water into the Arctic are characterized by very extensive double-diffusive intrusions. This suggests that the lateral processes (next section) may be dominant over vertical in the halocline of the Arctic Ocean, and that the significant climatic changes currently underway there may be mediated in part by small-scale double-diffusive processes. The importance of diffusive convection in polar regions lies in its ability to produce a cold, salty, and dense water mass without air–sea interaction. That is, heat can be extracted from a subsurface water mass without much change in salinity. The resulting water may be dense enough to become a bottom water mass. This idea has been applied to the formation of Antarctic Bottom Water and Greenland Sea Bottom Water. The T–S characteristics are in agreement with the model predictions but quantification of the rates of mixing remains uncertain. This is now viewed as a critical issue since the upward heat flux may be contributing to the current rapid decay of Arctic sea ice.

Intrusions In the presence of horizontal variations in temperature and salinity along density surfaces, as is common at oceanic fronts, the small-scale double-diffusive processes can drive horizontal motions on 10–100-m vertical scales. These intrusive instabilities arise because of the buoyancy flux convergences due to the mixing by salt fingers and diffusive interfaces. In the presence of horizontal T and S gradients, these vertical flux convergences generate lateral pressure gradients which drive a slow movement of water across the front. This is often manifested as a complex interleaving of warm/salty and cold/fresh water masses (Figure 6). The relative motion of each water type relative to the other is an effective means for keeping the double diffusion most intense, as it has a tendency to drive the density ratio toward 1. Thus, it is a selfreinforcing (direct) instability or a sort of ‘horizontal salt finger’. The sense of the heat and salt flux convergences is such that a warm salty intrusion is expected to lose more salt than heat due to fingering across its lower boundary, and thus should become lighter and rise across density surfaces. Similarly, a cold fresh intrusion should gain more salt than heat and become denser and sink across density surfaces. Such behavior was predicted theoretically by Stern, confirmed in the laboratory by Turner and observed in numerous observational programs. In situations where the temperature increases with depth, diffusive

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convection may dominate the mixing, leading to sinking of a warm salty intrusion. Microstructure observations confirm that enhanced dissipation consistent with double diffusion occurs at the interfacial boundaries of such intrusive fine structure. For more detail on double-diffusive intrusions, see Intrusions.

Global Importance It is fair to ask whether the different heat/salt transport rates achieved in small-scale doublediffusive mixing have any influence on the large-scale circulation. In general, ocean models assume that the small scales are available to consume any necessary variance, and in particular that there is no difference in heat and salt diffusivities. The presence of double diffusion in the ocean means that this assumption is invalid, and that a variety of effects whereby the differential transport rates feedback on the largerscale structure manifest themselves. For the intrusive instabilities described above, one effect is the tendency to destroy small-scale anomalies in temperature and salinity. The role of the relative horizontal motion between the vertically arrayed layers (shear) in forcing the density ratio toward 1 is important here, as this keeps the doublediffusive convection most intense. The strong mixing continues driving the anomalous fluid across density surfaces until it reaches a level with matching properties. Thus, intrusions are a powerful mechanism for removing water-mass anomalies and maintaining the tightness of the mean temperature– salinity relationship. The process can occur anywhere, and there is good evidence that it is a major lateral mixing agent at both polar and equatorial latitudes. Another effect on the T–S relation is due to the strong dependence of the vertical mixing rate on density ratio. The well-documented increase in fingering intensity as the density ratio approaches 1 leads to a number of interesting effects. When vertical variations in density ratio arise this dependence leads to fine-scale flux convergences which act to remove the anomaly in density ratio. Since densitycompensated T–S anomalies are prominent in the mixed layer, such a differential mixing mechanism is needed to explain the tightness and shape of the T–S relation of the subducted waters in the thermocline. Also, for both salt fingers and diffusive convection, there is a forcing of the density ratio away from unity, unless compensated by vertical fluxes or differential lateral advection. In addition, salt fingers are often the dominant mixing process operating on fine-scale intrusions at fronts. Double-diffusive

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600 Figure 6 Intrusive fine structure in the front associated with the North Atlantic Current east of Newfoundland. Warm, salty waters from the south interleave with cold, fresh waters from the north to create strong salinity-compensated temperature inversions with an overall stable density profile (s). An optical microstructure instrument revealed intense double-diffusive mixing at the boundaries of the intruding water masses.

intrusions may be a primary mechanism for accomplishing lateral mixing of water masses at the fine scale, acting efficiently on the horizontal gradients produced by meso-scale stirring by eddies. Since double diffusion acts preferentially on high-gradient regions, it may be responsible for a large fraction of the global dissipation of thermal and haline variance, despite modest eddy diffusivities. This is reinforced by the recent discovery that enhanced open ocean turbulence is found mainly in the weakly stratified abyss, where the contribution to dissipation of scalar variance is necessarily small, even though the eddy diffusivities may be large. In addition to being of regional importance as an enhanced flux site in the thermocline of the tropical

North Atlantic, salt fingers may be important in all of the other oceans and many marginal seas. In the Atlantic at 241 N, fully 95% of the upper kilometer of the ocean is salt finger favorable. Indeed, conditions are favorable for fingering in all of the central waters of the subtropical gyres. Since it is well established that the strength of the thermohaline circulation is very sensitive to the magnitude of the vertical (diapycnal) mixing coefficient, we must be concerned with the large-scale effects of widespread salt fingering. Model studies show that major features of the steady-state solutions are very sensitive to the ratio of the vertical eddy diffusivities for salinity and temperature. A 22% decrease in the strength of the thermohaline circulation was realized

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in one model when salt fingers were added to the vertical mixing scheme. Studies in global models with realistic topography and forcing found that double diffusion helps to bring deep temperature and salinity fields into closer agreement with observations. Models also suggest that double diffusion lowers the net interior density diffusivity sufficiently to make the thermohaline circulation more susceptible to collapse. Since collapse of the thermohaline circulation has occurred rapidly in the past, has dramatic impacts on climate, and is predicted to be a possible outcome of future greenhouse warming, it behooves us to seek a better understanding of the double-diffusive mixing processes in the ocean.

See also Deep Convection. Intrusions. Open Ocean Convection.

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Further Reading Osborn T (1980) Estimates of the local rate of vertical diffusion from dissipation measurements. Journal of Physical Oceanography 10: 83--89. Osborn T and Cox C (1972) Oceanic fine structure. Geophysical Fluid Dynamics 3: 321--345. Schmitt RW (1994) Double diffusion in oceanography. Annual Reviews of Fluid Mechanics 26: 255--285. Schmitt RW, Ledwell JR, Montgomery ET, Polzin KL, and Toole JM (2005) Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science 308(5722): 685--688. Stern ME (1975) Ocean Circulation Physics. New York: Academic Press. St. Laurent L and Schmitt RW (1999) The contribution of salt fingers to vertical mixing in the North Atlantic Tracer Release Experiment. Journal of Physical Oceanography 29(7): 1404--1424. Turner JS (1973) Buoyancy Effects in Fluids. Cambridge, UK: Cambridge University Press.

DIFFERENTIAL DIFFUSION A. E. Gargett, Old Dominion University, Norfolk, VA, USA & 2009 Elsevier Ltd. All rights reserved.

Introduction Because three-dimensional turbulence normally occurs on time and space scales much smaller than those resolved by numerical ocean models, its effects must generally be parametrized in these models. A particularly important quantity is r0 w0 , the averaged (overbar) vertical flux of density associated with small-scale turbulence, in which r0 and w0 are turbulent fluctuations of density and vertical velocity. By analogy with molecular diffusive fluxes, this turbulent flux is routinely parametrized as r0 w0 ¼ Kr r¯ z , that is, as proportional to r¯ z , the vertical gradient of mean density with a proportionality constant, the turbulent eddy diffusivity Kr, that is assumed to be much larger than the molecular diffusivity of density Dr (see Three-Dimensional (3D) Turbulence). In addition, present ocean general circulation models (OGCMs) normally assume that the same diffusivity can be used to parametrize the vertical turbulent flux of all other scalars in terms of their mean gradients. This assumption underlies use of the same eddy diffusivity in separate equations for temperature (T) and salinity (S), the two properties that (with pressure) determine the density of seawater. Finally, frequent usage of constant Kr in OGCMs implicitly assumes that Kr will not change if mean ocean conditions such as stratification change over time. Details of parametrizations of small-scale turbulence would be a matter of relatively minor import were OGCM predictions insensitive to them, but this is not the case. Different constant values of Kr are known to produce large changes in important features of predicted steady-state circulations: for example, heat flux carried by the oceanic meridional overturning circulation, a climatically important variable, is particularly sensitive to the value chosen for Kr. OGCM sensitivity to Kr results from linkages among the vertical turbulent flux of density, the mean vertical density structure, and horizontal circulation. Since density in the ocean is a highly nonlinear function of the two scalar variables T and S, differences in their vertical fluxes will lead to changes in these modeled linkages, hence to potential changes in modeled circulation.

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Differential vertical transfers of T and S have long been recognized in double diffusive processes (see Double-Diffusive Convection), which can occur in regions of the ocean where the mean vertical gradient of either T or S is destabilizing. However, even where mean gradients of T and S are both stabilizing, ordinary turbulent processes may be associated with what has come to be called differential diffusion, the preferential vertical turbulent transfer of T relative to S.

What Is Differential Diffusion? Differential diffusion results from the larger molecular diffusivity of T relative to that of S, by a mechanism whose cartoon is shown in Figure 1. Figure 1(a) shows a box filled with a light layer of warm (W) and fresh (F) water atop a denser layer of cold (C) and salty (S) water: both T and S components contribute to static stability of the system. Figure 1(b) depicts a blob of lower layer fluid that has been displaced into the upper layer: the blob becomes warmer relatively quickly as heat is rapidly communicated by molecular diffusion from its warmer surroundings. However, because the molecular diffusivity of salt (DS) is 100 times smaller than that of temperature (DT), very little salt escapes from the blob during the time it takes for its temperature to equilibrate. This leaves (Figure 1(c)) a blob that is still denser than its surroundings due to the excess salt, hence falls back to an equilibrium position (Figure 1(d)) between the two original layers. In this final state, temperature has been transferred vertically, but very little salt has accompanied it. In the case of actual turbulence, a simplified conceptual picture is that of a locally overturning turbulent eddy that stirs an embedded scalar field. The decay timescale te of the eddy is influenced by the turbulent Reynolds number Re  uc/n (where u and c are characteristic turbulent velocity and length scales

(a)

(b) WF WF

CS T

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Figure 1 Cartoon of the process of differential diffusion of T and S, which has its roots in the much larger molecular diffusivity of temperature than salt in seawater, where DT C 100 DS.

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Laboratory Evidence for Differential Diffusion Differential diffusion was first demonstrated in the laboratory, where J.S. Turner used T and S in turn to produce a two-layer density stratification that was subsequently mixed away by grid-generated turbulence. Figure 2 shows (normalized) entrainment velocity ue, which quantifies the vertical transport of density, as a function of a Richardson number Ri0  g(Dr/r)c/u2 defined in terms of turbulent velocity (u) and length (c) scales and the density difference Dr between the two layers. At high values of Ri0, the entrainment velocity is consistently higher when Dr is produced by T than when the same density difference was produced by S. Effects similar to those observed in Turner’s singly stratified experiments were subsequently demonstrated in laboratory experiments where T and S made simultaneous and equal contributions to the stability of either a density step or linear density stratification. In all of these laboratory settings, the oceanically important scalars T and S, characterized by DTC100 DScDS, exhibit differential diffusion in the sense of enhanced vertical flux of T relative to

Limit at zero Ri0 1.0 5×10−1

2×10−1 10−1 ue /u

and n is fluid kinematic viscosity), which determines the range of scales present in the flow, hence the time required to transfer turbulent kinetic energy to the small spatial scales at which velocity variance is dissipated. Turbulent stirring also transfers variance of an embedded scalar to a scalar dissipation scale that varies as D1/2, where D is the scalar molecular diffusivity (see Three-Dimensional (3D) Turbulence). Important oceanic scalars such as T, S, dissolved oxygen, and chemical nutrients are all characterized by scalar Schmidt number Sc  n/D41, hence by scalar variance dissipation scales that are smaller than velocity variance dissipation scales (see ThreeDimensional (3D) Turbulence). Straining scalar variance to a smaller scalar dissipation scale requires additional time, and this added time increases as Sc increases. As a result, the amount of scalar variance that is dissipated during te depends upon Sc. For ScB1, scalar variance can be entirely erased within the decay timescale of the velocity field, while effectively nondiffusive scalars (Scc1) will experience almost no transfer of variance to their much smaller diffusive scales within the same period of time. Since irreversible scalar mixing is rooted in the elimination of scalar variance by intermingling at the molecular level, these differences can result in differential vertical diffusion of scalars which have different molecular diffusivities.

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Figure 2 Comparison between directly measured entrainment velocity (equivalent to vertical flux) resulting from grid-stirring on one side of a density interface produced respectively by T (filled circles) and S (open circles). Mean stratification increases with Richardson number Ri0 , here calculated with turbulent length c and velocity u scales characteristic of the grid turbulence. Reproduced from Turner JS (1973) Buoyancy Effects in Fluids. Cambridge, UK: Cambridge University Press, with permission from Cambridge University Press.

that of S, that is, greater vertical diffusion of the scalar with the larger molecular diffusivity. In the doubly stratified experiments, where vertical gradients of T and S simultaneously make equal contributions to stability, the differential fluxes can be directly interpreted as KT4KS, that is, a larger turbulent diffusivity for T than for S.

Numerical Simulation of Differential Diffusion Much of what is known about the mechanism of differential diffusion in double stably stratified systems has come from direct numerical simulation (DNS), in which dependence of the process on molecular properties of scalars is provided by explicit resolution of the necessary dissipative scales. Even with modern computers, however, computational limitations do not presently allow three-dimensional resolution of the very small spatial scales associated with dissipation of true salinity variance. Thus to date, simulations have

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been run instead with a variable ‘S’ that has a diffusivity only 10 times that of salt, that is, DT ¼ 10 D‘S’, rather than the appropriate value of DT ¼ 100 DS. While this computational pseudo-salt ‘S’ variable will continue to be referred to as S, it should be kept in mind that numerical results using ‘S’ underestimate the magnitude of differential diffusion of true salt relative to temperature. One of the major contributions of DNS to the investigation of differential diffusion has been documentation of the essential role played by smallscale restratification (counter-gradient flux) associated with the scalar of smaller molecular diffusivity. Figure 3 shows results from a DNS in which turbulence generated impulsively at t ¼ 0 stirs an initially linear density gradient made up equally by T and S. In this visualization, the initial contortions of T and S

T

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isosurfaces situated originally at mid-depth in the box are identical. However, as time goes on, the effect of differences in molecular diffusivity can be seen in the faster disappearance of fluctuations in T, through its more effective molecular diffusion. The slower molecular diffusion of S leaves fragments of excess S that are progressively unsupported as turbulent kinetic energy dies away (see particularly panels 6 and 7). These heavy S fragments fall back toward a stable position (the restratification process), producing a counter-gradient flux and partially reversing the downgradient flux of S that occurred in the initial turbulent overturning process. Thus the net vertical flux of S is smaller than that of T. The essential role of stable stratification in producing the small-scale counter-gradient fluxes that lead to differential diffusion is emphasized in the

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Figure 3 Increasing numbers indicate time evolution of T and S isosurfaces that are initially horizontal and located at the vertical midpoint of a computational box. The computational fluid is linearly stably stratified with equal contributions to density from T and S. Turbulence imposed impulsively at t ¼ 0 stirs both T and S fields as it decays. Molecular diffusion of T is 10 times larger than that of S, so T fluctuations disappear more rapidly, leaving unsupported fragments of anomalous S (panels 6 and 7) which cause countergradient flux (restratification) of S during part of the decay process . Reprinted from Gargett AE, Merryfield WJ, and Holloway G (2002) Direct numerical simulation of differential scalar diffusion in three-dimensional turbulence. Journal of Physical Oceanography 33: 1758–1782.

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Figure 4 (a) Schematic of differential mixing by unstratified and stratified turbulence follows evolution of a fluid element whose T content is represented by the shaded circle and S content by the unshaded region within. In the unstratified case, Lagrangian displacement increases monotonically, and net vertical transport of S exceeds that of the more diffusive (leakier) component T, leading to KS4KT. In the stratified case, a restoring buoyancy force starts to act on displaced particles once some T has preferentially diffused out of the fluid element. Lagrangian displacement attains a maximum and then decreases as the fluid partially restratifies, leading to KSoKT. (b) Vertical plane projection of trajectories of 11 Lagrangian particles tracked in DNS of turbulence in stratified and unstratified systems, showing the behaviors suggested in (a). Reproduced from Merryfield WJ (2005) Dependence of differential mixing on N and Rr. Journal of Physical Oceanography 35: 991–1003, with permission of the American Meteorological Society.

cartoon of Figure 4(a). The impact of the faster molecular diffusivity of T on a particle containing anomalies of both T and S depends upon whether the ambient fluid is unstratified (left) or stably stratified (right). In both cases the faster diffusing T leaks out into the surroundings during Lagrangian displacement of the particle. In the unstratified case, Lagrangian displacement is monotonic, that is, on average the particle continues to move in its initial direction, taken here as upward. In the position shown, the particle has lost much of its T anomaly and retained more of its S anomaly, resulting in greater vertical flux of S than T, hence KS4KT

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in this unstratified case. In the stratified (oceanic) case, however, leakage of T enhances a restoring buoyancy force that eventually causes the Lagrangian displacement of the increasingly salt-heavy particle to reverse direction. Averaged over a turbulent event, the result is greater vertical flux of T, hence KSoKT. Figure 4(b) illustrates the fundamental differences in average Lagrangian displacements proposed in the cartoon, using Lagrangian particles tracked in DNS of stratified and unstratified turbulence. At an initial time, all the particles shown were situated near the x-axis and had upward initial velocities, as assumed in the cartoon of Figure 4(a). Net upward displacements are smaller in the stratified case, as a result of the restratification tendency associated with differential diffusion of T and S. Similar results are found in a DNS of differential diffusion of T and S ¼ ‘S’ associated with Kelvin– Helmholtz shear instability, thought to be the most common source of episodic turbulent mixing in the stratified interior of the ocean. Figure 5 is a visualization of the two scalar distributions, in terms of their (initially equal) contributions to density, during roll-up of the instability. Effects of the different dissipation scales of the two scalars are immediately obvious in the much finer spatial structure seen in rS relative to rT. While the fact that the distribution of rS contains much smaller scales does not itself necessarily imply differential diffusion, these simulations do exhibit differential diffusion: values of diffusivity ratio d  KS/KT lie between 0.5 and 0.85. The density ratio Rr  aT¯ z =ðbS¯ z Þ quantifies the relative contributions of T and S stratification to density stratification in a doubly stable system: extremes of Rr ¼ 0 and Rr ¼ N are respectively cases where either T or S is a passive scalar, that is, has no effect on density. Although initial DNS of differential diffusion used Rr ¼ 1, subsequent computations have investigated cases where T and S make unequal contributions to the basic density stratification. The impact of Rr on d proves to be systematic (d is smaller when Rr ¼ 0 than when Rr ¼ N) but secondary in importance to that of the strength of the turbulence: in all cases, d approaches 1 as the strength of turbulence increases.

Oceanic Values of Diffusivity Ratio The magnitude of flux differences that might actually be realized in the ocean is thus dependent upon the strength of typical ocean turbulence relative to the strength of turbulence in the DNS which are the main source of quantitative predictions. Unfortunately it is difficult or impossible to make

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0

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Figure 5 DNS results showing partial densities (a) rS and (b) rT during roll-up of a Kelvin–Helmholtz instability on a transitional layer between two water masses that are equally stably stratified by T and S. Values colored range from  0.4D (red) to 0.4D (dark blue), where 2D is the density difference across the transitional layer. Values outside this range are transparent. Reprinted from Smyth WD, Nash JD, and Moum JN (2005) Differential diffusion in breaking Kelvin–Helmholtz billows. Journal of Physical Oceanography 35: 1004–1022.

observational determinations of parameters such as turbulence Reynolds and Froude numbers that are typically used to characterize turbulence strength in DNS computations. Various determinations of diffusivity ratio d from laboratory, computational, and observational studies have instead been compiled as a function of Reb ¼ e/nN2, (where e is the rate of turbulent dissipation per unit mass  kinetic energy 1=2 is the buoyancy frequency r and N ¼ gr1 o ¯z determined by the mean density stratification), a form of turbulence Reynolds number that is accessible to both computational and observational determination. Characterization in terms of Reb is doubly useful because Reb is also a commonly used metric of the strength of ocean turbulence: turbulent flows with Rebo200 are considered weak in the sense that a crucial characteristic of turbulence (isotropy) fails even at dissipation scales. Figure 6 shows a range of presently available data: most come from laboratory (thick curve, pluses) and DNS studies with Rr ¼ 1 (gray and solid circles, gray boxes), Rr ¼ 0 (white boxes), and Rr ¼ N (black boxes), but a single observational study has estimated d from microscale measurements of T and S,

albeit with large uncertainty represented by the large light gray circle. Recalling that the computational results (using ‘‘S’’ instead of S) underestimate the magnitude of differential diffusion for true salt, it is clear from Figure 6 that differential diffusion may become significant below values of order RebB1000. Since turbulence in the stratified ocean interior is most frequently characterized by Rebo1000, differential diffusion should act routinely to enhance the vertical diffusion of oceanic T relative to S.

Other Observational Evidence for Differential Diffusion? While the single set of observational determinations of d shown in Figure 6 results does suggest do1, it cannot be regarded as conclusive because d ¼ 1 is within observational uncertainty associated with the extreme difficulty of determining the S dissipation spectrum. However, other fragmentary pieces of observational evidence also point to the possibility that differential diffusion is active in the ocean interior.

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Reb Figure 6 Diffusivity ratio d ¼ KS/KT as a function of buoyancy Reynolds number Reb ¼ e/nN 2. Estimates are from laboratory studies (thick curve: two-layer density stratification; pluses: linear density stratification), computational studies (gray bullets: impulsively generated turbulence; black bullets: turbulence generated by Kelvin–Helmholtz instability; squares ([black, gray, white] correspond to Rr ¼ [0,1,N]): impulsively generated turbulence), and ocean observations of T and S dissipation spectra (large light-gray circle).

One such piece of evidence may be found in the distinctive pattern of T/S fine structure found in overturning patches within doubly stable water columns. DNS results document the appearance of significant differences in turbulent scalar fluxes over timescales of order (0.1  0.2)TN, where TN ¼ 2p/N. Since the buoyancy period TN is the typical timescale for Kelvin–Helmholtz shear instability, the effects of differential diffusion should thus be evident within the lifetime of an overturning patch. Differential diffusion with do1 will result in preferential rotation of an initial T/S line toward horizontal, as weak turbulence preferentially mixes away fluctuations in the component with the larger molecular diffusivity. This sense of T/S rotation is seen in Figure 7 within an overturning event observed in a region of the Ross Sea, Antarctica, where mean T and S gradients are both stabilizing, enabling differential diffusion, and where T is effectively a passive scalar (i.e., RrB0), further enhancing the magnitude of differential diffusion of T relative to S. Preferential rotation in the sense seen in Figure 7 is a common feature of overturns observed in the Ross Sea. While both previous pieces of observational evidence for the action of differential diffusion in the ocean may be considered inconclusive, it is harder to dismiss observations of density-compensating T/S intrusions in a region that is doubly stable in T and S, that is, in a region where double diffusion cannot occur. Figure 8 shows a striking example, a set of

Figure 7 T/S plot through an overturn (red points) observed in a vertical profile from a region in the Ross Sea, Antarctica, where mean T and S gradients (blue line) are both stabilizing. Rotation of the T/S relationship toward horizontal, as observed in the T/S fine structure within the overturn, is a signature that would be expected if differential diffusion were significant in the sense expected from the molecular properties of T and S, i.e., with d ¼ KS /KTo1.

intrusive features in upper polar deep water, a water mass characterized by stable T and S stratification that is found between about 700 and 1000 m in the Arctic Ocean. Making necessary modifications to existing theory for intrusions driven by double diffusion, linear stability analysis determines values of KT and KS consistent with the observed intrusion thicknesses of 40–60 m. The required diffusivity ratio lies in the range 0.6pdp0.7, a not unreasonable value given the compilation seen in Figure 6. Moreover, required values of 1  10  6 m2 s  1p KTp 3  10  6 m2 s  1 are roughly an order of magnitude smaller than values determined from temperature measurements in the interior of other oceans, consistent with recognition that the Arctic interior is an environment of unusually low turbulence intensity; hence it is possibly a particularly favorable site in which to observe consequences of differential diffusion. Finally, with the predicted values of d and KT, growth rates for the intrusions are consistent with the appearance of measurable interleaving structures on timescales shorter than the estimated residence time of the upper polar deep water in the Arctic Ocean.

Does Differential Diffusion Matter? Although the molecular-scale mechanism of differential diffusion is similar to that of double diffusion, the energy supply for mixing is quite different in the two cases. While double-diffusive processes are

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T (°C) Figure 8 Within the Arctic Ocean, potential temperature generally increases with depth (pressure), while salinity exhibits a subsurface maximum, here seen B250 dbar. Horizontal lines bracket density-compensating intrusive features between about 700 and 1100 m, in doubly stable upper polar deep water. Differential diffusion with KT 4KS is a possible explanation for the existence of intrusions in a part of the water column that is stable to double diffusion. Reprinted from Merryfield WJ (2002) Intrusions in doublediffusively stable Arctic waters: Evidence for differential mixing? Journal of Physical Oceanography 32: 1452–1459.

associated with release of potential energy from the gravitationally unstable component of the mean T and S fields, differential diffusion is driven by the energy of ordinary small-scale three-dimensional turbulence. Thus while double-diffusive processes occur only where mean T and S gradients are suitable, differential diffusion will be active regardless of the signs of these gradients, provided only that the turbulent Reynolds number is sufficiently low. The compilation of d values shown in Figure 6 suggests that Rebo1000 may be sufficiently low. Since the majority of mixing events observed in the stratified interior of the ocean fall within this range, it appears that differential diffusion is potentially important for understanding net vertical density fluxes associated with three-dimensional turbulence in the ocean. Because changes in vertical density fluxes affect vertical density gradients (which in turn affect major ocean processes such as deep and bottom water formation, supply of nutrients to the bioactive surface layer, and ocean uptake of atmospheric carbon dioxide), and because such changes must be expected to evolve if density structure evolves, differential diffusion at microscales may have unexpected effects on much larger scales of ocean

variability. Investigation of the effects of parametrizations for double and differential diffusion found only small differences from results with equal constant diffusivities for T and S in a steady-state global ocean model. However, changes in relative vertical transports of T and S may allow previously unsuspected nonlinear feedback processes in ‘timedependent’ models, a possibility suggested by study of unequal diffusivities in a box model of the North Atlantic thermohaline circulation. With equal T and S diffusivities and fixed surface fluxes, the model exhibits the familiar result of convection in either the polar or subtropical set of boxes. However, with unequal diffusivities, there exists a range of atmospheric forcing under which the polar boxes instead oscillate between unstable convection and stable stratification, with a period that decreases with the value of the diffusivity ratio d ¼ KS/KTo1 in the polar boxes. While box models are certainly not the ocean, such results raise questions about unsuspected and unexplored feedback effects that may enter more realistic time-dependent numerical models, were they to properly incorporate the effects of differential (and double) diffusion. Given the importance of modelbased predictions of the evolution of the coupled atmosphere/ocean system under various climate forcing

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scenarios, differential diffusion may yet prove to be more than a laboratory curiosity.

See also Double-Diffusive Convection. Estimates of Mixing. Three-Dimensional (3D) Turbulence.

Further Reading Gargett AE and Ferron B (1996) The effects of differential vertical diffusion of T and S in a box model of thermohaline circulation. Journal of Marine Research 54: 827--866. Gargett AE, Merryfield WJ, and Holloway G (2002) Direct numerical simulation of differential scalar diffusion in three-dimensional turbulence. Journal of Physical Oceanography 33: 1758--1782.

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Jackson PR and Rehmann CR (2003) Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow. Journal of Physical Oceanography 33: 1592--1603. Merryfield WJ (2002) Intrusions in double-diffusively stable Arctic waters: Evidence for differential mixing? Journal of Physical Oceanography 32: 1452--1459. Merryfield WJ (2005) Dependence of differential mixing on N and Rr. Journal of Physical Oceanography 35: 991--1003. Merryfield WJ, Holloway G, and Gargett AE (1999) A global ocean model with double-diffusive mixing. Journal of Physical Oceanography 29: 1124--1142. Nash JD and Moum JN (2002) Microstructure estimates of turbulent salinity flux and the dissipation spectrum of salinity. Journal of Physical Oceanography 32: 2312--2333. Smyth WD, Nash JD, and Moum JN (2005) Differential diffusion in breaking Kelvin–Helmholtz billows. Journal of Physical Oceanography 35: 1004--1022. Turner JS (1973) Buoyancy Effects in Fluids. Cambridge, UK: Cambridge University Press.

DISPERSION AND DIFFUSION IN THE DEEP OCEAN R. W. Schmitt and J. R. Ledwell, Woods Hole Oceanographic Institution, Woods Hole, MA, USA Copyright & 2001 Elsevier Ltd.

Introduction One of the very consistent results of oceanic microstructure measurements over the past two decades is that the rate of turbulent mixing is quite weak below the well-stirred surface layer. With few exceptions, the interior ocean seemed remarkably nonturbulent, indeed, almost laminar. A major puzzle arose, as the rate of renewal of deep waters seemed well in excess of what could be absorbed by turbulent vertical mixing in most of the oceanic gyres. Without mixing to warm the deep water, the large-scale meridional overturning circulation would cease, as this mixing is, in a very fundamental way, its driving mechanism. New observations in the abyssal ocean have shown that there are, in fact, sites of greatly enhanced turbulence, which appear to be sufficiently strong and extensive to provide the necessary mixing. The new observations, the basic physics involved, and the apparent energy sources for this turbulence are reviewed below.

The Thermohaline Circulation A major feature of the ocean circulation is the sinking of cold, dense water at high latitudes, its spread to low latitudes, and eventual upwelling. During this upwelling the water must be warmed and made less dense; this is accomplished by interior vertical mixing. This is a key step in the process, indeed it can be considered its driving agent, if the meridional temperature gradient is accepted as a given. The interior mixing can be thought of as a ‘suction’ which pulls cold water into the various basins; without this suction, the deep circulation would stagnate. Numerous modeling studies have now shown that it is the mixing rate that sets the strength of the thermohaline circulation. One of the long-standing puzzles in oceanography has been the apparent lack of sufficient mixing to accommodate the estimated production of cold bottom waters. Given the strength of bottom-water sources

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and the area and stratification of the ocean basins, the required eddy diffusivity of turbulent vertical diffusion can be readily estimated; this turns out to be about 1  104 m2 s1. However, numerous measurements by sensitive temperature and velocity probes of the microstructure of the ocean yield a typical midgyre value of diffusivity that is at least an order of magnitude smaller. This body of work has been confirmed by tracer release studies in the upper thermocline of the North Atlantic. Thus, the apparent lack of mixing cannot be ascribed to statistical sampling issues or the models used for the interpretation of microstructure data. Rather, it is now apparent that the main problem was one of lack of observations, since the majority of turbulence measurements were confined to the upper few hundred meters. This shows the vital importance of exploratory observations in the vast and undersampled oceans.

Deep-sea Observations of Mixing The clear importance of mixing to the thermohaline circulation has inspired recent attempts to sample the turbulent mixing rate in the deep sea. The variables of interest for estimating mixing rates are the dissipation rates of temperature variance and turbulent kinetic energy. These require measurements of centimeter-scale gradients of temperature and velocity with very sensitive instruments. There were a number of technical difficulties to overcome; sensors capable of detecting the subtle signatures of oceanic turbulence in the weakly stratified abyss can have difficulties withstanding the immense pressure at the bottom of the sea. Also, untethered instruments must be used, in order to reach 5–6 km depth and to avoid the vibrations introduced by trailing cables. This leads to a requirement for significant redundancy in tracking and weight-release mechanisms, to minimize the risk of losing a sophisticated instrument. The ‘High Resolution Profiler’ was developed at the Woods Hole Oceanographic Institution for studies of deep-ocean mixing. Profiles of dissipation below 3000 m depth began to be acquired in the early 1990s. The first were from the area around a submerged seamount in the eastern North Pacific. These showed substantial elevation of turbulence levels in the deep ocean near the seamount but there was no detectable enhancement of

DISPERSION AND DIFFUSION IN THE DEEP OCEAN

dissipation beyond 10 km away from the base of the seamount. Similar patterns were found in the Atlantic in a few deep dissipation measurements made in association with the North Atlantic Tracer Release Experiment. Profiles showed a rather uniform value of the vertical eddy diffusivity with depth except very near the bottom. These dissipation profiles were also the first to show that the models used to interpret turbulence measurements yielded mixing rates in good agreement with tracer dispersion in the upper thermocline (see Double-Diffusive Convection). These hints of enhanced mixing near topography helped to motivate a study of mixing in an abyssal fracture zone. Fracture zones are valleys that cut across midocean ridges, thus providing a passage for flow of cold bottom water from one ocean basin to another. The Romanche Fracture Zone on the Atlantic equator was suspected of harboring strong mixing because of the rapid change in water temperatures along the valley. That is, cold water entering the valley was observed to warm significantly as it flowed through. Turbulence observations confirmed that this was a site of intense mixing. The flow over the rough topography was strongly turbulent, though the mixing rates dropped dramatically above the layer of fast moving water. Though these observations provided the first solid evidence for strong mixing in the deep ocean, they were obtained in a rather specialized site, so that generalizations of mixing rates for an entire basin were not feasible. Thus, an effort was mounted to examine mixing over a more typical region of deep ocean bathymetry. The region next examined was in the Brazil Basin of the western South Atlantic. The Mid-Atlantic Ridge (MAR) poses a barrier to eastward movement of the coldest, densest bottom waters in the basin. Confined between the continental rise to the west and the MAR to the east, the Antarctic Bottom Water (AABW) enters the Brazil Basin through passages in the south and exits through the Romanche fracture zone and across the equator to the North Atlantic. Just as in the Romanche, the bottom waters are seen to warm as they progress through the basin. An estimate of the rate of input of cold water, and knowledge of the surface area of isotherms and vertical temperature gradients within the basin allow the required mixing intensity to be calculated. A number of investigators have made such estimates for the Brazil Basin, since the southern source flows and northern outflows were reasonably well measured. Similar to the earlier global estimates of deep water-formation rates, the required mixing coefficient is B1–4  104 m2 s1.

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Measurements of the turbulent dissipation rate made using the High Resolution Profiler revealed that the mixing was weak and insignificant in the western part of the basin. However, to the east approaching the MAR, there was a dramatic increase in turbulence. The bottom topography also changed character from smooth to rough from west to east, as can be seen in Figure 1. This basic picture of the localization of mixing over rough topography had been suggested by previous work but never before demonstrated. The pattern of turbulent dissipation suggested that a tracer release experiment near the MAR would be of greater interest than elsewhere in the basin. Accordingly, 110 kg of the tracer sulfur hexafluoride (SF6) were released at 4000 m depth at a point about 500 m above the ridge tops of the fracture zones that trend westward from the ridge crest. When sampled 14 and 26 months after the release, a dramatic spread of the tracer was observed both laterally (Figure 2) and vertically (Figure 3). The lateral dispersion is characterized by two features: the bulk of the tracer which drifts and spreads to the southwest, and a secondary maximum which propagates eastward toward the MAR. These two tracer lobes are especially apparent after 26 months. The rate of lateral spread is rather modest compared to similar experiments in the upper ocean, since the mesoscale eddies are weak at these depths. The east–west trending fracture-zone valleys, extending perpendicularly from the main north–south axis of the Ridge, are prominent features of Figure 2. These valleys play an important role in establishing the bimodal structure of the tracer plume. That is, the valleys seem to serve as conducts for the tracer that is carried to the east, a feature easily seen in the sections of Figure 3. It appears that the tracer mixed downward from the injection plume has been carried eastward toward the ridge, and tracer-free water has been advected in below the core of the plume. Propagation both along and across density surfaces is apparent. These sections were obtained in one of the fracture zones, where the tops of the confining ridges are at a depth of 4500 m in the vicinity of the main tracer plume. The diapycnal mixing coefficient estimates from the tracer dispersion is B3  104 m2 s1 at the level of the injection and increases greatly toward the bottom, a pattern also displayed by the turbulence measurements. The separation of the plume into two main clouds by 26 months after injection suggests two mixing regimes are active at this site. The upper, interior, cloud seems to be spreading vertically, advecting southwestward, and sinking deeper across density surfaces. The downward motion is expected for a mixing rate

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that increases toward the bottom, though most oceanic models assume that upward flow is the rule. This finding means that the tracer experiment has added unique new knowledge to our understanding of the deep ocean. The flow of the lower tracer plume to the east is also exciting, as it shows how the deepest water is warmed in the course of eastward flow in the canyon. That is, the downward trend of the isopycnals requires a flow to the east in response to the pressure gradient; in steady state, the density gradient must be maintained by mixing (a steady state is a reasonable assumption, as the same density structure appears in surveys over 3 years). Thus, the fracture zones radiating from the ridge axis act as conduits which draw the bottom water toward the ridge, warming it and effectively upwelling it to lighter density strata. This is a secondary flow driven by the enhancement of mixing within the fracture zone valleys and toward the ridge. Such secondary flows just affect the stratification and circulation in the vicinity of rough bottom topography. These insights into the patterns of deep-ocean mixing are complemented by new information on the mechanisms causing the mixing. The overall pattern of spatial variation on the basin scale showed an

enhancement of mixing over rough topography. There was also an enhancement in the vertical shear in horizontal velocity on scales of B20–200 m in patterns that are indicative of internal waves. The variations in mixing can be further defined by examining the dissipation profiles above a variety of topographic types in the rough area. A simple classification of bottom types into crest, slope or valley profiles, and averaging of the data in a ‘height above bottom’ coordinate system, shows distinct differences between the average mixing rates (Figure 4). The ‘slope’ profiles show the greatest vertical extent of strong mixing above the bottom. This is to be expected if the bottom is a source of low-frequency internal waves or serves to reflect and amplify ambient internal waves. Internal waves propagate horizontally as well as vertically, with the lower frequency waves having a more horizontal propagation direction. Thus, the waves at a ‘slope’ station are likely to have come laterally from up-slope topographic features. The overall pattern of mixing variation has important consequences for estimating the net rate of mixing in the region. For topography to be a source of internal waves, there must be flow over the rough bottom.

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This could be a mean flow, such as in the Romanche Fracture Zone, or a time-varying flow, due to eddies or the tides. Since the deep eddy field was not found to be particularly strong in this region, we suspect the tides are the main source of energy for the enhanced internal waves. A simple comparison of the 3-day averaged vertically integrated dissipation rate (in mW m2) with the tidal velocities estimated from a global model is very suggestive of a dynamical link with the tides (Figure 5).

The record reflects the conditions at the geographical position of the dissipation profiles during the survey, and the tidal speed shows the amplitude of the estimated semidiurnal (12.42 hour period) tidal velocity over the spring–neap cycle during the cruise. The net dissipation is well correlated with the tides and appears to lag the forcing by about a day or two, a reasonable time scale for the vertical propagation of internal waves into the water column above.

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If one accounts for the variation in mixing rate due to the observed differences in slope and time of sampling relative to the tides, it is possible to estimate the net amount of mixing over the surveyed area, and make reasonable extrapolations for the rest of the Brazil Basin. When this is done, the mixing induced by tides and topography appears to account for the warming of Antarctic Bottom Water that is observed in the Brazil Basin. This apparent balance is

an important advance in our understanding of the deep flows in the ocean and the maintenance of the thermohaline circulation. However, it is too early to say whether tidally induced internal waves arising over rough topography generate enough mixing in the global ocean to absorb all the deep water production. Only a tiny fraction of the deep ocean has been surveyed with microstructure instruments. Models of how the tides

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interact with topography are only now being developed. Detailed bathymetric data and improved tidal models will be necessary to develop an estimate of the global mixing rate due to the tides. Consideration of the energetics of mixing for the thermohaline circulation suggests that the tides may cause about half the required mixing globally, making available about 3 mW m2 on average. This is the minimum value of the column integrated dissipation rates observed in the rough regions of the Brazil Basin during the spring–neap cycle, though much lower values are obtained in smooth-bottom regions. The other potential source of enhanced turbulence in the deep ocean may be the flow of the Antarctic Circumpolar current over rough topography. This current receives a great deal of energy from its constant driving ‘tailwind’ and is a flow that extends to the bottom. The main dissipation mechanism is likely to be internal lee wave generation as the currents interact with complex topographic features. These waves can deliver energy to small vertical scales well up into the water column. Preliminary

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data suggest that internal waves are indeed enhanced in rough areas of the Antarctic Circumpolar current, with an energy distribution suggestive of more uniform mixing in the vertical. This would provide a more conventional upwelling pattern than the complex circulations found in and around the valleys of the MAR. Estimates of the energetics involved suggest that it may be as important globally as the tides. However, no deep ocean turbulence measurements have been made in these remote areas, so the rates of mixing remain speculative. We must await the results of future scientific expeditions to explore this mechanism. Indeed, much also remains to be done on the tidal mixing issue, since the currently available data amount to little more than a glimpse into a complex problem.

Summary Whereas a decade ago, the mechanisms for mixing the ocean seemed quite mysterious, and the ‘missing mixing’ seemed to undermine our theories of the thermohaline circulation, we now have some leading candidates for how this mixing occurs. Bottom topography is key to these mixing mechanisms. They are as follows: 1. Flows through passages: fracture zones and other deep-sea passages serve to connect topographic deeps. Dense bottom waters can spill through these valleys at high velocity and experience strong mixing as they cascade over sills and rough topography. The change in deep-water properties has long been noted but now we know that the turbulence levels are indeed enhanced in such areas. 2. Tidal flows over rough topography: though midocean tidal velocities are weak, and thus generate negligible turbulence on their own, they readily interact with bottom topography to produce internal waves. These waves propagate into the water column above and produce enhanced turbulence to 1000–2000 m above the bottom. This mechanism predicts that interior mixing will be concentrated above rough bathymetry in areas with the strongest tides. It is not ‘boundary mixing’ in the traditional sense, since rough topography tends to be in association with midocean ridges and more heavily sedimented continental margins tend to have smooth bathymetry. The bottom source of energy for the waves leads to a decay of turbulence rates with height which leads to a general cross isopycnal flow ‘downward’. A diapycnal flow ‘upward’ is found in canyons, which serves to provide the requisite mixing and

upwelling for the bottom waters. This mechanism may be the leading mixing process serving to convert bottom waters into lighter density classes above midocean ridges throughout the world ocean. Antarctic Bottom Water in particular may be warmed largely by this process. 3. Mean flows over rough topography: this is a candidate mechanism about which little has been documented, but seems likely to play a role in key regions. The leading area of importance is the Southern Ocean, where the deep reaching Antarctic Circumpolar Current flows over significant bottom topography. Preliminary indications are that the vertical distribution of internal wave energy is more uniform, suggesting less variability in turbulent mixing rate, and thus a more uniform upwelling profile. This mixing may be key for converting the deep and intermediate waters into thermocline waters. The North Atlantic Deep Water is the primary candidate for warming in the region of the Antarctic Circumpolar Current.

These mechanisms all involve bottom topography, and thus point to the importance of improved knowledge of bathymetry for progress in understanding the deep and intermediate circulation. The spatial variations in mixing rates must lead to greater complexity in deep flows than has been anticipated in the present generation of models. It should also be noted that these mechanisms are not ‘boundary mixing’ in the usual sense. That is, the mixing occurs well away from the thin benthic boundary layer and is more likely to be concentrated over a midocean ridge than near a lateral boundary. Indeed, no enhancement of mixing has been observed in western boundary currents where the bottom is smooth. It is also important to note that numerous modeling studies have shown that the magnitude of the vertical mixing is limiting to the amplitude of the overall thermohaline circulation itself. Indeed, the role of interior mixing in the thermohaline circulation can be compared to the role of the wind stress in the wind-driven circulation; that is, turbulence provides the essential interior balance of vertical upwelling with downward mixing of heat, just as the wind stress pattern at the surface imparts circulation to the ocean’s horizontal gyres. The high latitude sinking regions are then analogs of the western boundary currents that close the wind-driven flows. This view more clearly shows that it is the interior mixing acting on available density gradients, rather than the surface formation of dense water, that acts as the driving agent for the thermohaline circulation. Indeed, without mixing, the

DISPERSION AND DIFFUSION IN THE DEEP OCEAN

deep circulation would become cold and stagnant and oceanic warmth would be confined to a thin surface boundary layer. This issue is of major concern, since the substantial circulation of warm water poleward is responsible for much of the heat flux carried by the ocean. There is evidence that the North Atlantic limb of the thermohaline circulation was cut off at various times in the past, and some suggest that global warming could shut it off in future, due to surface water freshening by an enhanced hydrologic cycle. Recent modeling work shows that there is a delicate balance between the fresh water forcing and the rate of interior mixing that determines the stability of the thermohaline circulation. A better understanding of oceanic mixing is thus essential for prediction of the future evolution of the Earth’s climate system.

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See also Double-Diffusive Convection. Internal Tidal Mixing. Upper Ocean Mixing Processes.

Further Reading Ledwell JL, et al. (2000) Evidence for enhanced mixing over rough topography in the abyssal ocean. Nature 403: 179--182. Munk W and Wunsch C (1998) Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Research 45: 1977--2010. Polzin KL, Toole JM, Ledwell JR, and Schmitt RW (1997) Spatial variability of turbulent mixing in the abyssal ocean. Science 276: 93--96.

HORIZONTAL DISPERSION, TRANSPORT, AND OCEAN PROPERTIES

VORTICAL MODES E. L. Kunze, University of Washington, Seattle, WA, USA Copyright & 2001 Elsevier Ltd.

Potential Vorticity In a rotating buoyancy-stratified fluid, potential vorticity is: P  q ¼ ð2O þ r  vÞ  rhðr; pÞ

Introduction In the late 1970s, moored measurements in the ocean found that gradient quantities such as shear and strain exhibited frequency behavior inconsistent with linear internal gravity waves. While this behavior could arise from Doppler shifting or other nonlinearities within the internal wave spectrum, it was also realized that geostrophic, or nonlinear potential vorticity-carrying, motions might be contributing to fine-scale variance. In their simplest form, these could take the form of thin layers of varying stratification with large horizontal extent and little shear associated with them, so-called ‘passive fine-structure’. These perturbations would be subinertial in a water-following frame, and spread tracers much more efficiently along isopycnals (density surfaces) through stirring and shear dispersion than internal waves. The term ‘vortical mode’ was originally coined to refer to the zero-frequency eigenmode of the linear stratified f -plane equations associated with potential vorticity-carrying perturbations, that is, geostrophy, regardless of scale. However, the vortical mode has come to denote both linear and nonlinear subinertial (intrinsic frequencies o{f ) ocean finestructure with vertical wavelengths lz o100 m which cannot be described as internal gravity waves. The term vortical mode will be used in this sense here. In the sections below, potential vorticity is defined, its role on basin scales and mesoscales briefly described, then evidence for potential vorticity-carrying fine-structure in the ocean interior is discussed. Such evidence is indirect and inferential. Fine-scale vortical modes are expected to arise from (i) the potential enstrophy cascade of geostrophic turbulence, (ii) mixing in turbulent patches, (iii) bottom friction and eddy-shedding of flow past topography, and (iv) double diffusion. Interpretation of fine-scale observations is challenging because of the presence of finescale internal waves and nonlinear advection by large-scale internal waves. As a result, the spatial and spectral distributions of vortical mode shear and strain variance are still unknown.

½1

where 2O ¼ ð0; f cotðlatitudeÞ; f Þ is the planetary vorticity vector associated with Earth’s O, rotation f ¼ 2jOjsinðlatitudeÞ the Coriolis frequency, the relative r  v vorticity, and hðr; pÞ any well-behaved function of density and pressure. It is convenient to use the buoyancy b ¼ gr0 =r0 for hðr; pÞ where r0 ¼ rðx; y; zÞ  r0 . The vertical gradient of buoyancy @b=@z ¼ N 2 is the stratification, or buoyancy frequency squared. The potential vorticity can be thought of as the dot product of the absolute vorticity vector and the stratification vector, that is, a multiplication of the stratification vector rb with that component of the absolute vorticity vector 2O þ r  v parallel to it (Figure 1). As shown by Hans Ertel in 1942, potential vorticity is conserved following a fluid parcel in the absence of irreversible processes.1 That is, potential vorticity is invariant without forcing by wind stresses, radiation, and evaporation/precipitation at the sea surface, molecular dissipation by microscale turbulence and double diffusion in the ocean interior, or stresses and geothermal heating at the bottom. Potential vorticity perturbations arestable if 2O  P is everywhere of the same sign. Relative to a background potential vorticity ¯ 2 , unstable in a rotating stratified fluid P ¼ f N conditions can arise from (i) fine-scale strati¯ 2 Þ (equivalent of strain fication perturbations BOðN @x=@zBOð1Þ), (ii) relative vorticity xBOðf Þ (equivalent to vorticity Rossby numbers Rz ¼ z=f BOð1Þ), or 1 In a rotating buoyancy-stratified fluid, the potential vorticity of a water parcel is conserved in the absence of dissipative processes. The potential vorticity is

P  q ¼ ð2O þ r  vÞ  rhðr; pÞ where 2O ¼ ð0; f cotðlatitudeÞ; f Þ, f is the planetary vorticity vector associated with Earth’s rotation f ¼ 2jOjsinðlatitudeÞ the Coriolis frequency, r  v the relative vorticity, and hðr; pÞ any well-behaved function of density and pressure. The potential vorticity can be thought of as the dot product of the absolute vorticity vector and the stratification vector, that is, a multiplication of the stratification vector rb with that component of the absolute vorticity vector 2O þ r  v parallel to it. As a conserved quantity, it is a useful dynamical tracer.

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and nonzero vorticity Rossby numbers (thin dashed and dotted), as well as linear internal gravity waves (thick dotted). At low aspect ratios (large horizontal compared to vertical scales), potential exceeds kinetic energy. At highaspect ratios, kinetic energy dominates. At vorticity Rossby numbers of  1, there is little potential energy because the Coriolis and centripetal acceleration terms balance. For vorticities exceeding f of either sign (@Rz @441, there is excess potential energy. These relations are independent of scale.

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Figure 1 Schematic of potential vorticity, which is the buoyancy gradient vector Ob times that component of the absolute vorticity 2O þ r  v vector parallel with it. When isopycnals are flat, the absolute vorticity is just the Coriolis frequency f plus the vertical relative vorticity. When isopycnals are sloped as shown, horizontal vorticities (vertical shears) also become important.

(iii) vertical shears @v=@zBOð1Þ (equivalentto gradient Richardson numbers Ri ¼ N 2 =ð@v=@zÞ2 BOð1Þ. In Figure 2, the relationship between the ratio of available potential to horizontal kinetic energy PE/KE and scaled aspect ratio ðNH=fLÞ2 ¼ ðNkH =fkz Þ2 is shown for geostrophic (thick dashed)

Basin scale potential vorticity structure does not fit into our definition of the vortical mode. However, the dynamics ofpotential vorticity anomalies should be independent of scale and much of our intuition comes from studies on these larger scales. Moreover, it is not yet known whether a spectral gap exists separating large- and small-scale potential vorticity perturbations. On basin scales O (1000 km), baroclinic potential vorticity anomalies are linear and quasigeostrophic, associated with the large-scale gyres. Potential energy greatly exceeds kinetic energy for these very low aspect ratio motions (Figure 2). Potential vorticity

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(NkH)2/(fkz)2 Figure 2 Dynamic diagram describing the dependence of ratio of available potential to horizontal kinetic energy RE ¼ PE=KE (energy Burger number) on scaled aspect ratio RL ¼ ðNH=fLÞ2 (length scale Burger number), where HBkz1 is the vertical scale and LBkH1 the horizontal scale. The thick dashed diagonal corresponds to geostrophy (linear vortical modes). Rz ¼ z/f is the vorticity Rossby number. The thin diagonals correspond to nonzero negative (dotted) and positive (dashed) vorticity Rossby number vortical modes. Nonzero Rossby number vortical modes in the domain above the Ric ¼ 1 curve have vertical shears exceeding the buoyancy frequency N. The thick dotted curve is the relation for linear internal gravity waves for N=f ¼ 40. (Reproduced with permission from Kunze and Sanford, 1993.)

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can be simplified to f ðyÞN 2 ðx; y; zÞ. This ‘stretching vorticity’ is a powerful dynamicaltracer, facilitating diagnosis of the gyre-scale circulation. Ventilatedwind-driven waters can be tracked from their winter outcrop. Under theassumption that, once a water parcel enters the pycnocline, its behavior can beexplained by inviscid quasigeostrophic dynamics on a b-plane, potential vorticity conservation determines the stratification along particlepaths, a powerful constraint in ideal thermocline theory. Unventilated watersin shadow zones (backwaters isolated from direct atmospheric forcing)become homogenized over time. Also on these scales, long planetary Rossby waveshave potential vorticity as their restoring forces.

Mesoscale Similarly, mesoscale O(10–100 km) potential vorticity-carrying structures do not fit into our definition of the vortical mode, but are better understood. They have lower aspect ratios than basin scale anomalies. These include western boundary currents like the Gulf Stream and Kuroshio, rings, eddies, fronts, short Rossby waves, Meddies, and other submesoscale thermocline vortices. Vertical relative vorticity is often important on these scales, PCð f þ r  vÞN 2 . Baroclinic and barotropic instability are means of transferringpotential vorticity toward smaller scales as part of the potential enstrophy cascade of 2-D geostrophic turbulence which tends to coalesce potential vorticity into coherent vortices resembling Meddies. This 2-D upscale energy cascade will be arrested by planetary Rossby wave radiation ifamplitudes are too weak to overcome the b-effect ðb ¼ @f =@yÞ. Rossby waveradiation is unlikely to be important for vortical mode because groupvelocities are very small for small vertical scales.

Fine-scale On the fine-scale O(100–1000 m), the vortical mode has been invoked on vertical scales of 1–10 m to account for (i) fine-structure contamination of internal waves in mooring measurements and (ii) scaledependent isopycnal diffusivities from tracer release experiments that are too large to be explained by internal wave shear dispersion. Sampling designed to minimize instrument motion contamination of internal wave measurements in the Internal Wave Experiment [e.g. (IWEX)] reveals that Eulerian frequency spectra offine-scale fluctuations such as shear and strain are not consistent withlinear

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internal gravity waves. Because of strong advective nonlinearity onthese scales, particularly from internal wave heaving, subinertial geostrophic finestructure would be Doppler shifted into the internal wave frequency band, f oooN. However, Doppler shifting will also smear fine-scale internal waves across all frequencies, so it isunclear whether ‘finestructure’ contamination is not justdue to fine-scale internal gravity waves of different intrinsicfrequencies becoming confused. Efforts to reduce Doppler shifting by examining time-series on isopycnal surfaces or with a water-following float have found signals much more compatible with linear internal wave dynamics. Lagrangian time-series have not yet been of sufficient duration to characterize subinertial variances. Isopycnal diffusivities increase with scaleso that a fine-scale patch diffuses more and more rapidly as it spreads with time. On 0.1–1.0 km scales, 0.0770.04 m2s1 diffusivities were inferred from a North Atlantic Tracer Release Experiment (NATRE). These may be explicable from internal wave shear dispersion in which vertical turbulent diffusion is spread horizontally by vertically varying horizontal displacements. However, 1–30 km diffusivities of 1–3 m2 s1 cannot be accounted for by either shear dispersion due to internal waves or persistent largescale (100 m) vertical shears. The vortical mode has been invoked to explain the O(10 km) diffusivities. Arguing that excess fine-scale strain is associated with the vortical mode, and assuming that dominant aspect ratios for the internal wave and vortical modefields are the same and independent of vertical wavenumber, yields quantitatively plausible horizontal diffusivities.

Generation Mechanisms Potential vorticity can only be modified by irreversible processes, and even then remains conserved within a volume containing all the dissipation. Moreover, it cannot flux across isopycnals. This puts severe restrictions on sources for the vortical mode. Away from atmospheric forcing, potential vorticity can only be altered through molecular dissipation. Fine-scale vortical modes are expected to arise from: 1. The potential enstrophy cascade of geostrophic turbulence, including baroclinic instability, although atmospherically forced mesoscale property anomalies appear to be smoothed out in only a few months. 2. Mixing and dissipation in micro-scale turbulence patches. 3. Bottom friction and eddy-shedding of flow past topography.

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4. Double-diffusive layering and interleaving. Whether any of these mechanisms is sufficient to maintain a widespread or universal vortical mode field is unknown. The third and fourth mechanisms in particular are expected to be highly localized.

Observational Challenge The fine-scale poses considerable observational challenges because of the presence of energetic finescale internal waves, and nonlinear advection by large-scale internal waves.Measurements in the wake of a seamount found potential vorticity structure on vertical wavelengths of 50–200 m and horizontal scales of O(1 km), which was attributed to shedding of bottom boundary layers or flow separation (Figure 3). Strong coherent vortices with horizontal scales O(1 km) have been found in a number of ocean pycnoclines, generated by either bottomhugging flows encountering abrupt changes in topography or deep convection. However, whether a universal vortical mode spectrum exists throughout the ocean, analogous to the canonical internal wave

spectrum, has not been established since it has yet to be isolated from the omnipresent internal wave field. Four methods have been attempted to identifypotential vorticity-carrying fine-structure.

1. Intrinsic frequency should be subinertial ðo{f Þ for the vortical mode and superinertial for gravity waves away from boundaries where Kelvin and other bottom-trapped topographic waves, while not vortical modes, can have subinertial frequencies. For large-scale flows that experience little Doppler shifting ðv  rÞc, where c represents ðu; v; w; bÞ, this isunambiguous from fixed Eulerian measurements. However, fine-structure with vertical wavelengths 1–10 m is strongly advected both vertically and horizontally by largerscale internal wave flows, so that Eulerian frequency measurements such as moorings are no longer unambiguous. Lagrangian time-series are necessary to identify the intrinsic frequency. Water-following measurements of shear and strain have yet to be made forsufficient duration to reliably identify the vortical mode.

Figure 3 Energy ratios versus scaled aspect ratios (as inFigure 2) in the wake of a seamount.Gray bars emanating from the left axis correspond to horizontal wavelengthsexceeding 8.5 km (survey averages) with vertical wavelengths marked.These intersect the geostrophic curve (thick dashed diagonal) forvertical wavelengths lz ¼ 50 and 200 m, lie near theinternal wave curve (thick dotted curve) for lz ¼ 100, 130 and 400 m, and between the curves(corresponding to kinetic energy being dominated by near-inertialwaves and potential energy by geostrophic motions) otherwise. Black dots( ) correspond to scales resolved by the survey. At lower aspectratios, these mostly cluster near the internal wave curve. At higher aspectratios, they fall slightly below the internal wave curve, suggesting excesshorizontal kinetic energy. Gray bars emanating from the right axis denotehorizontal wavelengths 0.3 km (incoherent scales) at various verticalwavelengths. Two of these intersect the internal wave curve. The remainder,with scaled aspect ratios of O(10), lie below it, again suggestingexcess horizontal kinetic energy, possibly from the vortical mode. Very littleenergy is associated with higher aspect ratio estimates, so these may bealiased. (Reproduced with permission from Kunze and Sanford,1993.)

VORTICAL MODES

2. Potential vorticity anomalies should be associated with vortical modes but not internal gravity waves (except possibly advection of background gradients – which should be small given the short timescales of internal waves). From the definition of potential vorticity in [1], this requires resolving fine-scale gradients on both the vertical and horizontal, which is difficult in itself. Moreover, since gradient quantities such as relative vorticity and buoyancygradients rb have blue horizontal wavenumber spectra, i.e. more variance at smaller than larger scales, sampling must be designed to filter out variance at scales smaller than those of interest. 3. The ratio of relative vorticity to horizontal divergence. Forlinear (geostrophic) vortical mode, vorticity greatly exceeds horizontal divergence (which vanishes in the steady geostrophic limit). For internal waves, the horizontal divergence is greater or equalto the relative vorticity. This approach has the same problems of spatial resolution as the potential vorticity method. 4. Ratio of horizontal kinetic to available potential energy (shear/strain ratio) HKE/APE as a function of dynamic lengthscale ratio ð fL=NHÞ2 . These differ for linear internal waves and geostrophic flow (Figure 2). This approach also suffers potential contamination by aliasing, in this case, by larger scales.

Conclusions Observational evidence for vortical mode finestructure in the ocean is sparse, largely indirect, and

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inferential. As a result, the spatial and spectral distributions of vorticalmode variances are unknown. Given their potentially important role in submeso scale isopycnal stirring, the oceanic vortical mode warrants further study.

See also Dispersion and Diffusion in the Deep Ocean. Dispersion in Shallow Seas. Three-Dimensional (3D) Turbulence. Internal Tidal Mixing. Internal Tides. Internal Waves. Upper Ocean Mixing Processes.

Further Reading Schro¨der W (ed.) (1991) Geophysical Hydrodynamics and Ertel’s Potential Vorticity (Selected Papers of Hans Ertel ). Bremen-Ro¨nnebeck, Germany: Interdivisional Commission of History of IAGA. Huang RX (1991) The three-dimensional structure of wind-driven gyres: Ventilation and subduction. Reviews of Geophysics 29: 590--609. Kunze E and Sanford TB (1993) Submesoscale dynamics near a seamount. Journal of Physical Oceanography 23: 2567--2601. Ledwell JR, Watson AJ, and Law CS (1998) Mixing of a tracer in the pycnocline. Journal of Geophysical Research 103: 21499--21529. Mu¨ller P, Olbers DJ, and Willebrand J (1978) The IWEX spectrum. Journal of Geophysical Research 83: 479--500. Mu¨ller P (1995) Ertel’s potential vorticity theorem in physical oceanography. Reviews of Geophysics 33: 67--97.

INTRUSIONS

Copyright & 2001 Elsevier Ltd.

Introduction In most frontal regions, where waters of different salinities and temperatures meet laterally, an interleaving of the different waters is observed. These features are commonly referred to as intrusions. Sometimes a single layer of water from one region is advected into the other region, such as the Mediterranean salt tongue in the North Atlantic, by either a mean flow or eddy motion. Multiple layers of the two different water masses are also seen quite often. The driving mechanism for these multiple intrusions is related to horizontal gradients in salinity and temperature and the small-scale (e.g., smaller than the thickness of the intrusions) mixing occurring between the interleaving layers. In this article, only intrusions produced by this latter process are discussed. Both observational and theoretical studies are presented. Frontal regions are locations where waters of different temperature and salinity meet and interact. They are usually characterized by relatively large horizontal gradients in these two properties. Fronts have been found in the coastal ocean, at the shelfbreak and at the boundaries of major currents, such as the Gulf Stream and Antarctic Circumpolar Current. An example of a front (Figure 1) is shown by the azimuthally averaged salinity structure of a Mediterranean eddy. A Mediterranean eddy (Meddy) is a coherent eddy of Mediterranean Sea water found in the eastern North Atlantic Ocean. The front with its larger horizontal gradients in salinity is located at a depth range of 700–1300 m and with a radius of 15–30 km. The temperature field has a similar structure to the salinity structure shown in Figure 1. With the horizontal change in salinity, it would be expected that there would be a horizontal change in the density of the sea water. However, the effect of the horizontal change of temperature on the density nearly completely compensates the density change due to the salinity change across the front. Thus, the density surfaces are nearly horizontal. However, there is a slight upward (downward) tilt of density surfaces in the lower (upper) half of the Meddy. The resulting pressure gradient balances the geostrophic

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flow of the eddy. Along-front geostrophic flows are found at most fronts. A closer look at the structure of temperature and salinity in the frontal region shows an interleaving of water with the characteristics of the temperature and salinity on the two sides of the front. The temperature and salinity of the water in these interleaving layers show evidence that mixing of the two water types has also occurred. Figure 2 shows a section of closely spaced (e.g., 1–2 km) vertical profiles of salinity starting from the center of the Meddy (Figure 1) and moving towards the outside edge. In each profile, there are wiggles in the salinity field, typically of 1–2 km vertical scale, which represents the water moving horizontally from the center of the Meddy to the edge or vice versa. These wiggles are referred to as intrusions. Intrusions like these are found in most frontal regions such as those associated with the Gulf Stream and Antarctic Circumpolar Current. The observed interleaving of temperature and salinity is thought to develop as an instability of the thermohaline front. These fluctuations lead to regions of enhanced double-diffusive mixing. Two types of double-diffusive mixing can occur: salt-fingering under the warm, salty layers and diffusiveconvection under layers that are relatively cold and

Salinity (PSU) 0 36.5 36.0

500 35.5 36.0

Depth (m)

D. L. Hebert, University of Rhode Island, Rhode Island, USA

1000

35.5

Meddy

35.5

35.5

1500

2000 0

10

20

30

40

50

Radial distance (km)

Figure 1 Azimuthally averaged cross-section of the salinity of a Mediterranean eddy (Meddy) embedded in eastern North Atlantic water. This survey, the second one of this Meddy, was made in June 1985, PSU, practical salinity units.

INTRUSIONS

465

500

Depth (m)

750

1000

1250

1500 35.5

36

36.5

37

37.5

38

38.5

39

Salinity (PSU) Figure 2 A set of closely spaced vertical profiles of salinity taken from the center of the Meddy towards its edge during June 1985. Profiles have been offset by 0.25 PSU (practical salinity units).

Density flux

Warm and salty

Cold and fresh

Salt-fingering interface Diffusive-convection interface

Frontal region Figure 3 A schematic of the interleaving layers representing the intrusions. The open arrows indicate the cross-frontal motion driven by the depth-varying density flux (solid arrows). In this diagram, salt-fingering is the dominant form of double-diffusion; thus, the warm, salty water rises as it crosses the front.

fresh (Figure 3). Both forms of double-diffusive mixing generate a downward density flux, that is, a release of potential energy. The convergence or divergence of this density flux makes the intrusion either heavier or lighter, respectively. These density changes produce pressure gradients which drive the interleaving motions across the front. If the density

flux of salt-fingering exceeds that of diffusive-convection, waters in the warm, salty layers become less dense and, therefore, rise as they cross the front. The cold, salty layers become more dense and sink as they cross the front. It is believed that this case applies for the intrusions found for the lower half of the Meddy. If diffusive-convection dominates (which is believed

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INTRUSIONS

to be the case for the intrusions occurring in the upper half of the Meddy), water in the cold, fresh layers should rise across the front and the warm, salty layers sink.

500

750

There have been many observations of intrusions in vertical profiles of salinity and temperature taken in frontal regions. Other than demonstrating the presence of intrusions and indicating their vertical scale, it is difficult to make any other conclusions about the dynamics of the intrusions. Closely spaced profiles (e.g., Figure 2) show that the horizontal structure of intrusions is complex. Although it is possible to track an intrusion across several kilometers (and several profiles), the structure of the individual intrusion changes significantly. In addition, some intrusions appear to start and end abruptly. One of the problems of interpreting this type of data is that the frontal region usually has a horizontal velocity field associated with it. Although the water in the intrusions is moving across the front, it is also being advected along the front by the geostrophic current of the front. Therefore, some of the observed crossfrontal variability could be due to differential advection of the intrusions along the front. Most of the observations of intrusions have been single surveys of the front; the evolution and the dynamics of the intrusions could not be determined. Even if multiple surveys are undertaken, temporal changes in the intrusions cannot be separated from possible alongfrontal variations of the intrusions. However, there has been one study where some of the dynamics of the intrusions could be investigated. This was an experiment to determine the evolution of a Mediterranean eddy. The front (Figure 1) between these two water masses can be thought of as a circular front. Thus, the problem of differential along-front advection of the intrusions is removed since the front loops back on itself. It would be expected that the individual cross-frontal transects could be typical for all radial sections and that alongfrontal variations are small. Thus, the cross-frontal transects could be used to determine the intrusion dynamics. This Meddy was surveyed four times over a two-year period as it decayed. The vertical structure of the intrusions evolves as the cross-frontal temperature and salinity gradients change (Figure 4). For the first year of the study, the Meddy had a core region unaffected by intrusive mixing. During this time, the intrusions appeared to have a similar wiggly vertical structure at all locations for both surveys with vertical scale of about

Depth (m)

Observational Studies

1000

1250

Oct. 86 Oct. 85

1500 35.0

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Oct. 84 June 85

36.0 Salinity (PSU)

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Figure 4 Vertical profiles of salinity made through the intrusive region of the Meddy during four surveys: October 1984 (solid line), June 1985 (dashed line), October 1985 (dotted line) and October 1986 (bold line). The profiles have been offset by 0.2 PSU (practical salinity units) from each other.

20 m. The wiggles are rather smooth (i.e., sinusoidal) and have approximately the same vertical scale. By the time of the third survey, the intrusions just reached the center of the Meddy. We can imagine that there was a constant cross-frontal gradient (driving the intrusions) for the first year of observation and that the intrusions had passed their initial (exponential) growth stage. By the time of the third survey, some of the intrusions appeared to have a step-like structure. A year later, the intrusions had a more pronounced step-like structure with a larger vertical scale, about 50 m (Figure 4). This structure is probably representative of decaying intrusions. Double-diffusive processes were still active vertically but the horizontal advection mean gradients were less important; there was not a supply of new water unaffected by the mixing. One major question that remains concerns the cross-frontal fluxes of heat and salt by the intrusions. In order to address this question, it is necessary to measure the very weak velocities in the intrusive layers or observe the large-scale changes in the

INTRUSIONS

properties of the frontal region. For the first year of study of the Meddy, it had a core region with very little horizontal variability (Figure 1). Using the rate at which the intrusions moved into this central core region, extremely small cross-frontal velocities, u0 , on the order of 1 mm s1 were found. Using the salinity anomalies associated with the intrusions, S0 , and this order of magnitude estimate for the crossfrontal velocity, the average cross-frontal flux of salt, FS ¼  /u0 S0 S, was calculated. Parameterizing this flux in terms of horizontal diffusion,  FS ¼ KH

1qS r qr

 ½1

where KH is the horizontal eddy diffusivity and qS/qr is the mean horizontal (radial) salinity gradient across the front; an eddy diffusivity coefficient of 0.4 m2 s1 was found. The dominant mechanism responsible for the decay and eventual demise of the Meddy was thermohaline interleaving, presumably driven by double-diffusive buoyancy fluxes. Over the 2-year observation period, intrusions at the edge of the Meddy core eroded the warm and salty central region from an initial diameter of 60 km until the core was no longer detectable. Using the rate at which the salinity and temperature of the Meddy at a specific radius changed, an eddy diffusivity could be estimated. qS q2 S 1qS þ ¼ KH qt qr2 rqr

! ½2

Likewise, by integrating eqn [2] from the center to a specified radius, changes in the salt and heat content of the Meddy can be used to estimate an eddy diffusivity. It was found that an eddy diffusivity of 1–5 m2 s1 could be used to parameterize the crossfrontal fluxes of the intrusions. To date, this Meddy study has been the only one where estimates of the fluxes by intrusions could be made. Estimates of the horizontal eddy diffusivity ranged from 0.5 to 5 m2 s1. An attempt was made to understand the dynamics of the intrusions with these surveys but the temporal sampling (six months) was too infrequent to be of use in investigating the evolution of the intrusions.

Theoretical Studies The driving mechanism for the cross-frontal velocity of intrusions (i.e., the horizontal pressure gradients) is due to divergences in the vertical density fluxes, generally assumed to be due to double-diffusive mixing (Figure 3). Most of the theoretical studies to

467

date have looked at the initial growth of the intrusions using linear stability analysis with parameterizations for the vertical flux by salt-fingers. In these linear stability calculations, the background frontal structure is assumed to have linear gradients, both horizontally and vertically, of salinity and temperature. The horizontal gradients are chosen such that there is no horizontal gradient in density and thus, no along-front velocity. Vertical gradients are chosen such that the background structure is unstable to double-diffusion, usually saltfingering. Double-diffusive mixing is parameterized as a constant eddy diffusivity for salt (or heat) and a constant ratio of the heat to salt flux. The perturbations are assumed to be small, so there are no inversions in temperature and salinity. The linear stability analysis predicts the vertical scale, crossfrontal and along-frontal slopes of the fastest-growing unstable mode given the salinity and temperature gradients. These properties have been compared to observations and have shown general agreement. For typical horizontal gradients of salinity and temperature found in frontal regions, the growth rate of the fastest mode has an e-folding timescale on the order of 10 days. Inclusion of a background velocity shear due to the sloping isopycnals across the front can produce faster-growing intrusions with an e-folding timescale on the order of several days. Linear stability studies predict properties of the initial growth stage, in which fluxes grow exponentially, but say nothing about the finite amplitude ‘steady’ state properties. When the intrusions reach finite amplitude, the fluxes of heat and salt by the interleaving should reach a constant value. Since fronts in the ocean exist much longer than the time for the intrusions to grow, intrusions spend most of their lives in the finite-amplitude state. Therefore, the usefulness of extrapolating intrusion properties and fluxes from linear theory is questionable. For growing intrusions to reach an equilibrium, a three-way balance between salt-finger, diffusiveconvection and (cross-frontal) advective fluxes is necessary. The initial instability may set the vertical scale of the finite-amplitude intrusions, but the crossfrontal fluxes may depend critically on the form of the equilibrium that the growing intrusions eventually reach. A numerical model verified that small amplitude intrusions, predicted by linear stability analysis, evolved into large amplitude, equilibrium, intrusions. When the amplitude of the intrusion becomes large enough that temperature and salinity inversions occur, the growth of the intrusion slows and reaches an equilibrium state. This equilibrium state is characterized by interleaving layers with saltfingering and diffusive-convection occurring at the

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INTRUSIONS

interfaces separating statically unstable ‘convecting’ layers. As expected, the three-way flux balance is achieved. As well as obtaining a balance in the advection and mixing of the salinity and temperature, there must be a momentum (energy) balance. The double-diffusion mixing lowers the potential energy of the system. This potential energy is converted into kinetic energy within the convecting layers. In addition, the convecting layers allow a large flux of momentum from the salt-fingering interface to the double-diffusive interface. The friction between the interleaving layers balances the pressure gradient produced by the density flux divergence.

numerical and observational, will use the three-dimensional structure of intrusions to evaluate the two-dimensional studies done to date.

See also Double-Diffusive Convection. Upper Ocean Mean Horizontal Structure. Upper Ocean Mixing Processes. Water Types and Water Masses.

Further Reading Summary The presence of intrusions in frontal regions has led oceanographers to believe that they must be important in the cross-frontal fluxes of heat and salt. However, at present, these fluxes are almost impossible to observe in the ocean. Thus, we must rely on theoretical and numerical studies to address this important question. In order to be useful, these studies must predict properties of intrusions which can be compared to observations. To date, comparisons have been limited to the vertical length scale of intrusions from single vertical profiles of temperature and salinity. Predictions of the slope of the intrusions (relative to density surfaces) in the crossfrontal and along-frontal directions have been compared to the few cross-frontal sections made. With improvements in navigation with global positioning satellites and the advent of undulating towed bodies, rapid three-dimensional high-resolution mapping of intrusions can be undertaken. Future work, both

Hebert D, Oakey N, and Ruddick B (1990) Evolution of a Mediterranean salt lens: scalar properties. Journal of Physical Oceanography 20: 1468--1483. May BD and Kelley DE (1997) Effect of baroclinicity on double-diffusive interleaving. Journal of Physical Oceanography 27: 1997--2008. McDougall TJ (1985) Double-diffusive interleaving. Part II: Finite amplitude, steady state interleaving. Journal of Physical Oceanography 15: 1542--1556. Ruddick B (1992) Intrusive mixing in a Mediterranean salt lens – intrusion slopes and dynamical mechanisms. Journal of Physical Oceanography 22: 1274--1285. Ruddick BR and Hebert D (1988) The mixing of Meddy ‘Sharon’. In: Nihoul JCJ and Jamart BM (eds.) Small-Scale Mixing in the Ocean. Elsevier Oceanography Series, vol. 46. Amsterdam: Elsevier. Toole JM and Georgi DT (1981) On the dynamics and effects of double-diffusively driven intrusions. Progress in Oceanography 10: 123--145. Walsh D and Ruddick B (1998) Nonlinear equilibration of thermohaline intrusions, Journal of Physical Oceanography 28: 1043--1070.

DISPERSION IN SHALLOW SEAS J. T. Holt and R. Proctor, Proudman Oceanographic Laboratory, Birkenhead, Merseyside, UK

Instantaneous values have been divided into mean and fluctuating components as in eqn[2].

Copyright & 2001 Elsevier Ltd.

u ¼ u¯ þ u0 ;

Introduction The study of marine dispersion is particularly, but not exclusively, concerned with understanding and predicting the fate and impact of pollutants, both from acute spills from shipping and coastal facilities and from longer-term (chronic) discharges. Marine pollutants are primarily of anthropogenic origin and so most significantly affect the regions of the marine environment closest to centres of human activity, namely, estuaries, bays, and the shallow seas of the continental shelves. Some pollutants may be regarded as natural, such as toxic algal blooms; these also primarily impact human activity, such as recreation and fisheries, in shallow seas. The environmental impact of pollutants depends on extremely complicated biogeochemistry and ecotoxicology; however, in almost every instance the effect depends in some way on the amount of contaminant present. Hence an appreciation of how pollutants are transported, dispersed, and diluted in shallow seas is crucial to our understanding of their impact. This draws on all aspects of shallow sea physical oceanography and our aim here is to assess the processes that determine the horizontal dispersion of a patch of contaminant. As with many problems in oceanography, it is difficult to define a completely general set of significant processes, as they will always depend on the particular situation at hand, so we will illustrate the general principles with a number of applications to specific cases.

Fundamentals – The Fluid Mechanics of Dispersion For most practical purposes in shallow sea studies the evolution of the ensemble mean concentration, ¯ Cðx; y; z; tÞ, of some constituent of the water in the velocity field, u ¼ ðu; n; wÞ, can be represented by the advection-diffusion equation [1], ¯ @C ¯ þS þ u¯  rC¯ ¼ r  ðKrCÞ @t where S represents any sources/sinks present.

½1

C ¼ C¯ þ C0

½2

Turbulent fluxes are taken to be proportional to spatial gradients of mean quantities, for example, as in eqn [3].

u0 C0 ¼ Kx

@ C¯ @x

½3

K ¼ ðKx ; Ky ; Kz Þ is the turbulent diffusivity; to appreciate the significance of this quantity it is necessary to consider the processes it represents in some detail. The only way to change the constituent properties of an infinitesimal water parcel is through molecular diffusion. The typical scale of this process is nB106 m2 s1, so diffusion across a 100 km wide shelf sea would take O(108) years if it were the only active process. However, both turbulent and mean flows can reduce this timescale to the extent that they are the only significant processes. Currents will transport water masses and their constituents around shelf seas; however, it is through the process of straining (stretching) the surfaces that separate bodies of water with different constituents that dispersion in shallow seas occurs; an example of such a surface might be the interface between water containing a particular pollutant and water in which that pollutant is absent. Horizontal variations in velocity (shears) can be expressed as the sum of a solid body rotation and a pure strain. As long as the shear persists, this strain field will tend to lengthen material contours, for example at the interfaces described above. This increases the surface area, and reduces the interface width, with a consequent increase in the diffusive flux between the water masses. This process is particularly evident in turbulent flows, where the randomly oriented strain field leads to a continual stretching and thinning of the surfaces between water masses; these quickly become very sinuous, folded, and complicated, and the diffusivity effective on a macro scale, K, is increased by many orders of magnitude above the molecular value, n. There are a number of processes in the shelf sea environment that act to generate these shear and strain fields, some of which are now described.

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DISPERSION IN SHALLOW SEAS

Turbulence and Eddies

The dominant process that generates the strain fields described above is, directly or indirectly, turbulence. This can be characterized by a length scale, or wavenumber k, according to the theories of Kolmogorov. The statistical characteristics of the smallest scales of motion (the viscous range) are determined by the rate of dissipation of kinetic energy (e) and the kinematic viscosity (n), which define the typical dimensions (Lv ), life times (Tv ) and velocities (Uv ) associated with the smallest turbulent vortices:  3 1=4 n Lv ¼ e

Tv ¼

n1=2 e

Uv ¼ ðenÞ1=4

½4

Similarity theory requires the wavenumber spectrum of turbulent kinetic energy to have the form EðkÞ ¼ ðenÞ1=4 FðLv kÞ

½5

where FðLLn kÞ is a universal function. In the range of scales where viscosity is negligible but 1/k is still much smaller than the bulk flow macro scale, L (i.e., in the inertial subrange: Lv 51=k5L), the energy spectrum must be independent of viscosity and depend only on the transfer of energy from larger to smaller scales. This leads to the  5/3 power law of Kolmogorov and Obukhov (eqn [6]). EðKÞBe2=3 k5=3

½6

If it is assumed that dispersion of a patch (of, say, a contaminant) of size l is mainly due to eddies also of size l, then the eddy diffusivity for this patch must be as given by [7] as long as l is within the inertial range. kBe1=3 l4=3

½7

This similarity argument is only valid for fully developed three-dimensional turbulence, so L is limited either by the water depth or by a stratification scale, since eddies larger than this will inevitably be quasitwo-dimensional. Two-dimensional flows have the property, arising from the conservation of vorticity, that the energy cascade is reversed and small scales cascade to large (again obeying a k5/3 law). This results in the comparatively stable and long-lived meso-scale eddies prevalent in the open ocean (the oceanic analogue of atmospheric weather systems). The frictional nature of shelf seas, however, generally limits the occurrence of these large-scale two-dimensional eddies to deeper or stably stratified regions (see later). The dispersion of tracer patches in the North Sea, for example, does not reflect the

behavior expected of a field of horizontal eddies alone, but rather a complex interaction between vertical and horizontal shear and mixing processes as described next. Tidal Currents and Shear Dispersion

At first sight the oscillatory nature of tidal currents makes them an unlikely candidate for a dispersive process. However, a conundrum found in tidal waters is the disparity between estimates of horizontal diffusion coefficients based on tracer releases (KxB100–1000 m2 s1) and those calculated from the typical horizontal eddy velocities and length scales in these waters (KxB0.1–1m2 s1): tidal waters can be strongly dispersive but eddies alone are insufficient to explain this. While the answer to this discrepancy depends on the particular flow field and geometry in question, there are a number of processes that contribute to the enhanced lateral dispersion in tidal waters. One is an interaction between horizontal shear and the horizontal dispersion generated by vertical shear. This is essentially the shearstraining mechanism for enhancing diffusivities described above. The tidal flow over the seabed generates a vertical shear, which in turn generates turbulence, the diffusion coefficient of which may be written as Kz CDUH (where CDB0.005 is a drag coefficient; U is a tidal velocity scale; H is a vertical length scale, say, the water depth), although there are many other forms for this. This turbulence is strained by the vertical shear to give an enhanced horizontal diffusion; the upper limit of this (for the case of a mixing timescale much smaller than the tidal period) is given by eqn [8]. Kx E

U2 H 2 B170Kz 240Kz

½8

This in itself is insufficient to account for the values of Kx quoted above; however, if this dispersion is in turn strained by a lateral shear, values of the required magnitude can be reached. Of course the nature of the horizontal shear depends on the particular situation, but as an example a unidirectional residual current, U0, with a sinusoidal across-stream profile (length scale D) gives an enhanced horizontal diffusivity as in [9]. Kxx ¼

U02 D2 16p2 Kx

½9

For values U0 ¼ 0:1 m s1, D ¼ 5 km, and Kx ¼ 10 m2 s1 this gives Kxx ¼ 160 m2 s1. Two important points to note here are, first, that a steady horizontal shear is much more efficient at dispersion

DISPERSION IN SHALLOW SEAS

than an oscillatory one; and, second, that when a uniform oscillatory current and a sheared residual current are combined, underlying turbulence is not necessarily required for dispersion (as it is in the above argument). Scales of flow can be chosen that lead to two particles that are initially close together in position to diverge rapidly even in the absence of turbulence – an example of deterministic Lagrangian chaotic behavior.

Wind-driven Currents

Away from regions of strong density variation, wind stress drives the dominant residual currents in shelf seas, and since these have large horizontal scales and can be considered quasi-stationary as far as the above analysis is concerned, their horizontal variations contribute strongly to shear-dispersion. Persistent basin-wide jetlike circulations such as the Dooley current from the northern North Sea into the Skagerrak have widths of O(100 km), and hence have very high effective diffusivities at their periphery and disperse material over large distances. Wind stress also contributes to vertical shears and mixing. In combination with the Coriolis force it leads to currents that decrease from the surface value, oriented at 45o right (left) of the wind direction, and turn to the right (left) in the Northern (Southern) Hemispheres: the Ekman spiral. The turbulence resulting from this shear can create a mixed layer in thermally stratified conditions. This mixed layer is bounded at the bottom by a thermocline that isolates the surface from deeper waters and prevents vertical mixing of contaminants. The reduction in vertical mixing by stratification can enhance the transport and dispersion of material in the surface layer as this material is confined to the region where the wind-driven currents are strongest. In regions that stratify seasonally, this suggests that a given wind speed can result in greater dispersion of a surface contaminant in summer than in winter. However, this effect is likely to be more than compensated for by winds generally being stronger in winter than in summer. While much of the focus of wind effects on dispersion is on the generation of large-scale currents, small-scale turbulence, and surface waves, the interaction of wind stress and surface waves can also result in rows of vortices of alternating sign known as Langmuir circulation. These are in the vertical plane and have axes aligned with the wind direction. These structures manifest themselves, through convergent surface currents, as rows of floating material oriented

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in the wind direction, called windrows. While it is known that they are associated with enhanced currents in the wind direction and very strong downward currents underneath the wind rows, their role in surface mixing and horizontal transport is still something of a mystery.

Baroclinic Processes

Density-driven currents are common in shelf seas; they arise as a result of horizontal variations in pressure due to variations in the density field. Examples include jetlike features at tidal mixing fronts where well-mixed water meets thermally stratified water, and coastal currents due to strong salinity variations such as in river plumes. The characteristic horizontal scale of these features is the Rossby radius of deformation, R ¼ Nh/f (where f is the Coriolis parameter, N the buoyancy frequency and h a vertical length scale). These currents can be unstable and lead to the formation of (quasi) two-dimensional meso-scale eddies (for example, in the Norwegian Coastal Current); however, the role of these eddies in dispersion across shelf seas is not well established. Since material circulating in closed orbits does not disperse, a persistent gyre circulation at a fixed location (such as seen around the bottom cold water dome in the western Irish Sea) will tend to restrict the dispersion of material. In contrast a meso-scale eddy detached from a coastal current will transport material large distances and disperse it as the eddy dissipates. The relation between the motion around closed orbits (u0) and the internal diffusion velocity scale (Kx/L) within an eddy is the Peclet number: Pe ¼ u0 L=Kx . With the typical values: u0 ¼ 0:5 m s1, LBRB5 km and Kx ¼ 10 m2 s1, we find PeB250, which means that material makes many closed orbits before diffusing. Because of their longevity and their ability to transport material as a coherent structure, two-dimensional meso-scale eddies are not well represented by diffusion coefficients, but to give an example of the magnitude of their effect, we can estimate Kxx Bu0 L Pe1=2 with the values given above, this gives Kxx B160 m2 s1 (cf. the horizontal shear-dispersion value calculated above). As well as generating meso-scale eddies, baroclinic currents, such as those seen around cold domes, will contribute to the horizontal shear-dispersion described in the last section, particularly because they are often very localized, for example, to a narrow frontal region, and as such are associated with large horizontal gradients (they are often referred to as frontal jet) and so can result in small regions of very high horizontal diffusivity.

472

DISPERSION IN SHALLOW SEAS

Surface Waves

Away from the near shore zone, wind-generated surface waves are generally non-dispersive. Their only mechanism for net transport is by Stokes drift. This results from the orbital motion of particles in the wave’s velocity field being helical (rather than elliptical) because the velocity decreases with depth and the forward velocity of a particle near the surface is greater than its reverse velocity at the bottom of the orbit. This leads to a net flow US in the direction of wave propagation given by eqn [10] for deep water waves of amplitude A. Us ¼ A2 g1=2 k3=2 expð2kzÞ

½10

This can be significant to the transport and dispersion of small-scale patches such as oil slicks (see below).

Dispersion Phenomena There have been numerous tracer, dye release, and drifter experiments, along with theoretical analyses and numerical experiments to examine the many phenomena in which these dispersion processes play a significant role. Passive Dispersion

Passive tracers provide the ideal tool for studying dispersion processes. Dyes and radioactive tracers have proved most useful in marine studies to map advection and diffusion in shelf seas over timescales ranging from minutes to many years. Most notable has been the release of 137Cs (with its half-life of 30.1 years) from the Sellafield (Windscale) nuclear reprocessing facility on the west coast of England during the 1970s and 1980s. This tracer, well mixed in the water column, has been observed to travel around northern Scotland, taking 2 years to reach into the North Sea. This has enabled marine scientists to estimate the contributions of advection and horizontal diffusion to the changing spatial distribution. Model studies that take into account advective and dispersive effects have produced quantitatively good agreement with the observed distributions over 15-year simulations with horizontal dispersion coefficients proportional to aR2 , where a is a constant value and R is the local tidal current amplitude.

variety of degradation and dispersion processes, including advection, turbulent diffusion, spreading, evaporation, emulsification, dissolution, photochemical oxidation, aerosol formation, sedimentation, and biodegradation. The composition and position of release of the oil greatly influence the relative importance of these processes. Certain refined products like petroleum and light kerosenes may be subject to almost complete evaporation; heavier oils do not undergo significant weathering and are more likely remain in the water and therefore be subject to the advection–diffusion processes arising from tides, wind, and wave action. If a slick is introduced into a uniform current then as it diffuses and increases in size it will be advected by the current without any distortion of the patch slick shape. However, if the current is spatially nonuniform (horizontally and vertically) then the slick will be distorted and elongated and display apparent enhanced diffusion in the direction of the dominant current (Figure 1). The elongation of the oil slick in the direction of the wind and waves suggests the presence of a shear-diffusion process. Vertical shears in the surface water are produced by the action of wind and waves and these processes can also mix oil droplets down into the water column, where they are diffused by the shear. The interaction between the turbulent diffusion of oil droplets and the vertical shear beneath the slick determines the growth dimensions of the slick, with shear diffusion elongating the slick in the direction of the wind/ waves and Fickian diffusion affecting the slick width. Tides will act to move the slick backward and forward but will contribute little (except in very shallow water with strong tidal currents) to the diffusion process. Typically it is considered that the wind-driven surface current moves at about 2–3.5% of the wind speed and the surface wave drift (Stokes drift)

Oil Slicks

When crude oil or petroleum products are released into the sea they are immediately subjected to a

Figure 1 Dye diffusion elongation of dye patch.

experiment,

showing

horizontal

DISPERSION IN SHALLOW SEAS

has a velocity of about 1–2% of the wind speed. However, for typical wave conditions the component of the shear diffusion will only be about 10% of that due to the wind-induced flow.

k

k

_ 5/ 3

Plankton Patchiness

The vast majority of marine organisms are planktonic and are thus largely at the mercy of the motions in the sea. It has been known for many years that plankton are neither randomly nor uniformly distributed in the sea; rather they exist in patches, or exhibit patchiness. It is still not clear in which situations the spatial distribution of plankton is controlled by physical processes (dispersion/ concentration) and in which it is controlled by biological ones (growth/reproduction/behavior). The spectra of chlorophyll patches (of size L) in the sea have been shown to follow the k5=3 shape described above (here kB1=L). While spectra for higher trophic levels (e.g., zooplankton) seem to be less steep than k2 at low wavenumbers, in general many biological quantities have spatial distributions that appear to be determined solely by the physics. It has, however, been shown that biological interaction can modify the spectral distribution shape. In particular, spectral shape is modified by interaction between species. If there is no interaction in the population then, at length scales smaller than the ‘kiss length’ (the minimum size patch that can maintain itself in the presence of diffusion), the distribution follows the k5=3 distribution (i.e., solely controlled by the physical turbulence), whereas at larger scales the spectrum is proportional to k1 (i.e., flatter (‘whiter’) than the physical turbulence). Thus, small-scale plankton structures appear totally controlled by turbulence, whereas large-scale structures show less variance than the underlying physical structures. When species interact (e.g., predator–prey interaction) in the inertial sub-range the interaction produces a k3 spectral shape; that is, there is increased variance at the larger length scales. However, if the underlying turbulence is fully two-dimensional (enstrophy-conserving) then the converse is true: the population interaction leads to less (k1) intense patchiness at larger scales (Figure 2). Thus biological interaction can either redden (increase) or whiten (decrease) the spectral variability of patchiness due to physical diffusion alone, depending on the turbulent length scale. A similar result is seen in terrestrial environments. In simple terms, at low turbulence levels (too low to disperse patches) predator–prey encounter is increased; at intermediate levels patches are dispersed, making predator–prey encounter difficult; at high levels the prey

ln (C f)

k

473

_3

Inertial sub-range − spectrum 'redder'

_1

Two-dimensional turbulence − spectrum 'whiter'

ln (k ) Figure 2 Schematic of power spectrum of concentration fluctuation (Cf) against wavenumber (k) for two interacting species in different turbulence regimes. The line k 5=3 represents the spectrum of physical turbulence.

distribution is homogenized, resulting in increased predator–prey encounter rates. It has, however, been suggested that the diffusion approach to plankton patchiness is too readily applied and that it neglects the fact that true diffusion within the surface layers of the sea is insignificant at larger than centimeter scales. It has been shown that stirring by a turbulent flow causes variability to be transferred from large to small scales and that under the influence of turbulent advection a patch of tracer (e.g., phytoplankton) develops fine tendrils and filaments (as described above). In simple models of coupled phytoplankton and zooplankton, fine structure in the zooplankton distributions can be generated by the transfer of variability from larger scales by stirring over the lifetime of zooplankton. The timescale at mid-latitudes for the transfer of variance from the large (100 km) scale motions to the small (1 km) scale is typically 10 days, which is less than the lifetime of larger zooplankton such as copepods (typically 25 days to reach maturity). Thus any largescale variation in juvenile copepod distribution will be stirred down to kilometer length scales. Spectral shapes resulting from these studies show the zooplankton to have a flatter spectra than the phytoplankton with exponents representative of fully twodimensional turbulent flow and the experiments indicate that zooplankton lifetime is an important determinant in their spatial pattern. Suspended Particulate Matter Dispersion

The transport of the organic and inorganic suspended material typically found in sea water

474

DISPERSION IN SHALLOW SEAS

represents a somewhat different class of dispersion phenomena important for a number of reasons: its role in biogeochemical cycling and the optical properties of sea water important for biological production; the transport of pollutants that preferentially adhere to particles (such as many heavy metals); and in the long term because of its implications to bed forms and coastal morphology. This material, collectively referred to as suspended particulate matter (SPM), will tend to settle out of the water column and deposit on the seabed. Its horizontal transport is crucially dependent on the rate of settling (a factor of the particle size), and the ability of currents (due to wind, waves and tides) to scour the seabed and erode/resuspend the material back into the water column. Coarse material (grain sizes larger than B0.1 mm) generally only moves as bed load or during strong storm events and, because of its short residence time in the water column, is primarily involved only in benthic processes, bedforms, and coastal morphology, rather than pollutant transport, or optical properties. In contrast fine material, with settling velocities up to about 10 mm s1 (grain size of B0.1 mm) can be treated in a similar fashion to dissolved tracers with the addition of settling, resuspension, and deposition terms to the advection– diffusion equation. The former is simply vertical advection at the settling velocity, while empirical forms for the other terms must be found. These often relate the rates of erosion and deposition to critical bed stresses ðtero ; tdep Þ, for example, as in eqn [11], where B is the amount of deposited material (m2), w is the settling velocity, and M is an empirical erosion rate. @B ¼ Mðt=tero  1Þ; @t

t > tero

@B ¼ wCjz¼H ðt=tdep  1Þ; @t

½11 totdep

This can have a marked effect on the dispersion and transport, since under conditions of weak bed stress (for example, neap tides) the material remains on the seabed and is not transported (unlike dissolved material in the water column). This can lead

to the transport being dependent on correlations between the tidal cycles, and variations in the wind and wave climates. There is a tendency for material to settle out in the summer month and be resuspended during winter storms. This seasonality can have a marked effect on the transport (compared with dissolved material) if winter and summer currents differ significantly, for example, owing to density driven currents and variation in the wind forcing.

See also Langmuir Circulation and Instability. Gravity and Capillary Waves.

Surface

Further Reading Abraham ER (1998) The generation of plankton patchiness by turbulent stirring. Nature 391: 577--580. Beckers JM (ed.) (1999) Marine Turbulence Revisited. Journal of Marine Systems 21, special volume. Bowden KF (1983) Physical Oceanography of Coastal Waters. Chichester: Ellis Horwood. Charnock H, Dyer KR, Huthnance JM, Liss PS, et al. (1994) Understanding the North Sea System. London: Royal Society of London. Geyer WR and Signell RP (1992) A reassessment of the role of tidal dispersion in estuaries and bays. Estuaries 15(2): 97--108. Monin AS and Ozmidov RV (1985) Turbulence in the Ocean. D. Reid: Dordrecht, 247 pp. Morales R, Elliott AJ, and Lunel T (1997) The influence of tidal currents and wind on mixing in the surface layers of the sea. Marine Pollution Bulletin 34(1): 15--25. Prandle DP and Beechey J (1991) Marine dispersion of caesium 137 released from Sellafield and Chernobyl. Geophysical Research Letters 18(9): 1723--1726. Simecek-Beatty D, Lehr WR, Lae R and Overstreet R (2001) Special Issue on Langmuir circulation and oil spill modelling. Spill Science and Technology Bulletin 6(3/4). Zimmerman JTF (1986) The tidal whirlpool: a review of horizontal dispersion by tidal and residual currents. Netherlands Journal of Sea Research 20: 133--154.

DISPERSION FROM HYDROTHERMAL VENTS K. R. Helfrich, Woods Hole Oceanographic Institution, Woods Hole, MA, USA Copyright & 2001 Elsevier Ltd.

Introduction Among the most significant scientific events of the last century is the discovery of hydrothermal vent fields and their unusual ecological communities along the crests of the mid-ocean ridges. The venting consists of localized sources of very hot (B3501C) water that rises 100–300 m above the vent before it spreads laterally, similar to the plume from a smokestack. Venting also occurs as less intense and relatively cool diffuse flow (B101C above the ambient ocean temperature) spread out over a much broader area than the focused high-temperature vents. Diffuse flow rises only a few meters above the seafloor before it is mixed with the ambient sea water. While the diffuse flow carries about half of the total hydrothermal heat flux, its effect on the overlying water column is much less dramatic than the high-temperature vents. The hydrothermal venting from diffuse and localized high-temperature venting is essentially continuous over periods of years to decades. On longer timescales the individual vent sites will dissipate and new sites will emerge at other locations along the ridge crest. This nearly continuous venting is also punctuated by intense short duration venting events. These intense events are produced by magma eruptions on the seafloor or tectonic activity that rapidly exposes large quantities of sea water to hot rock or releases large quantities of very hot water from the crust. The result is the creation of huge ‘megaplumes’ that can rise 500–1000 m above the ridge crest to form mesoscale eddies with diameters of O(20 km) and thickness of O(500 m). Because of its large buoyancy and the dynamical control exerted by the Earth’s rotation, vent fluid is not simply advected away by background flow. The venting is capable of forcing circulation on a variety of temporal and spatial scales and this may have important consequences on how the vent fluid is ultimately dispersed. This article focuses on the flow produced by high-temperature venting and megaplumes, since they are most relevant for long-range

dispersal of vent fluids as a consequence of their large vertical penetration into the water column. The fate of the heat, chemicals, and biological material released by the vent is of interest for many reasons. To geophysicists, the hydrothermal heat flux represents a substantial fraction of the total heat flux (conductive plus convective) from mid-ocean ridges. For chemists, the vent fluid is laden with chemicals and minerals leeched from the subseafloor rock that over geologic time may contribute to the geochemical state of the oceans. The unique biological communities that accompany venting depend upon the chemical and thermal energy delivered by the venting. Since most of these unusual animals can survive only at vent sites, the dispersal of vent fluid is the primary mechanism of larvae dispersal and the colonization of remote new vent sites.

The Rising Plume The cascade of scales initiated by a high-temperature vent begins with the fast O(1 h) rise of the buoyant fluid from the vent to the spreading level O(100 m) above the source. Fluid emerging from an isolated hot vent rises as a turbulent plume, entraining and mixing with the ambient sea water as it rises (see Figure 1). Because the entrained ambient water is denser than the fluid in the turbulent plume, the plume buoyancy decreases continually with height above the source. If the ambient environment had uniform density the plume fluid would remain less dense than the environment and it would rise indefinitely. However, even in the deep ocean the ambient water is stratified. Eventually the plume density increases until it equals the background density. After a short overshoot of this neutral density level due to the momentum of the rising fluid, the plume spreads horizontally as an intrusive density current, or it may be swept downstream by ambient currents. Figure 2 shows a transect through a hydrothermal plume on the Juan de Fuca Ridge in the North Pacific. The figure is typical of many such observations made worldwide over the last two decades. In the figure temperature and light attenuation anomalies (defined relative to the background values along isolines of density) are contoured as functions of depth and horizontal distance along the centerline of the axial valley. High values of light attenuation anomaly are due to particulates introduced into the water column by venting. Indeed, in many cases light attenuation is a more useful indicator of

475

476

DISPERSION FROM HYDROTHERMAL VENTS

Outflow

ZS ZM

Entrainment

Figure 1 Sketch of a plume from a localized high-temperature hydrothermal vent. The plume rises to a maximum height ZM above the source. The rising stem of the plume continually entrains ambient fluid so that the density of the plume buoyancy decreases with height above the bottom. Eventually the plume density equals the ambient density and it spreads laterally at some height ZS above the source.

hydrothermal activity than temperature anomaly. As discussed below, the temperature anomaly may be very small, or even negative. The main features of the buoyant rise, entrainment, and spreading processes can be determined with a theoretical plume model that conserves momentum, mass, and buoyancy integrated on a horizontal slice across the plume. The key assumption is that the entrainment velocity, or the rate at which ambient fluid is drawn into the plume, is linearly proportional to the vertical velocity within the plume. (Details of the basic plume models and the justification of the assumptions are discussed in Morton et al. (1956), see Further Reading section.) Modeling, laboratory experiments, and observations show that the maximum rise height the plume above the source ZM is given by:

where

 1=4 ZM ¼ 3:8 F0 N 3

½1

  r0  rs F0 ¼ Qg r0

½2

and N2 ¼

gdra r0 dz

½3

Here F0 is the buoyancy flux from the vent and N is the buoyancy frequency of the ambient water, which over the rise height of the plume is assumed to be constant. Q is the source volume flux and rs (r0 ) is the source (ambient) fluid at the level of the vent. The background density is ra ðzÞ; z is the height above the source, and g is the acceleration due to gravity. While the maximum plume rise is ZM , the radial outflow is centered at a slightly lower height ZS E 0:8ZM . The thickness of the spreading layer over the source is E 0:2ZM . Typical values of F0 ¼ 102 m4 s3 and N ¼ 103 s1 give ZM E210 m. The time taken for a parcel of fluid to ascend from the vent to the spreading level BN 1 . Doubling the vent buoyancy flux leads to only a very minor change in ZM . This weak quarter power dependency on F0 is significant because observation of ZM and N are often used to estimate the heat flux from a vent H ¼ rs cp QðTs  T0 Þ ¼ rs cp F0 =ga, where cp is the specific heat, a is the coefficient of thermal expansion and Ts and T0 are the temperatures of the source and ambient fluids, respectively. From eqn [1], HpZ4M N 3 . Estimates of H are very sensitive to small errors in either ZM or N. The model and eqn [1] were derived under ideal conditions. Others effects will affect plume behavior. For example, in an ambient flow with velocity U, ZM pðF0 U1 N 2 Þ1=3 . Increasing U leads to decreasing rise heights. Mid-ocean ridge crests are locations of rough, variable topography and this may affect plume behavior. For example, the slow spreading Mid-Atlantic Ridge is characterized by axial valleys that are typically deeper than the plume rise spreading level, ZS , while the fast spreading Pacific ridges have axial valleys shallower than ZS . Deep-valley topography will constrain the plume outflow and direct it along the ridge axis, limiting off-ridge dispersal of vent fluids. Despite limitations the basic plume model provides useful insight into the dispersal of vent fluids. The entrainment of ambient water into the plume causes a substantial dilution of a parcel of vent fluid. The volume flux into the spreading level QM ¼ 1:3ðF03 N 5 Þ1=4 . For F0 ¼ 102 m4 s3 and N ¼ 103 s1 , QM ¼ Oð102 m3 s1 Þ. This gives a dilution of Oð104 Þ for a typical source flux Q ¼ Oð102 m3 s1 Þ. Entertainment occurs at all levels, but the largest velocities of background fluid into the plume occur in the lower quarter of the rise height. Larvae of bottom dwelling vent organisms can easily be swept into the plume and rapidly transported up to the spreading level. They then have a greater likelihood of dispersal over the distances typical of individual vent spacing (O(10 km)), Furthermore, these larvae are in water that is chemically

DISPERSION FROM HYDROTHERMAL VENTS

477

SLU 2, SSS Temperature anomaly (˚C) 1800

_

Light attenuation anomaly (m 1 )

1800 1900

1900 0.015 0.020

0.005 0.010

27.690 27.695

2000

0.025 0.030 0.035

2100

H 0.040

27.700

H

27.685

0.005 0.010 0.015

27.685

Depth (m)

Depth (m)

1 km

2000

27.690 27.695 0.050 0.030

27.700

0.025

2100

0.025

0.010 L

0.035

L

2200

2200 (A)

(B)

X Y

4.3 7.0

3.8 6.0

3.5 5.0

3.2 4.0

Figure 2 Transect of temperature (A) and light attenuation (B) anomalies through a hydrothermal plume on the Juan de Fuca Ridge in the North Pacific. The transect was taken along the axis of the axial valley. The maxima of temperature and light attenuation are located directly over the vent. (Reproduced with permission from Baker and Massoth, 1987).

distinct from the ambient environment and this may enhance survival during the dispersal process. The temperature and salinity anomalies at the spreading level (where the density anomaly is zero) are dependent on the ambient temperature and salinity gradients and can be counter-intuitive. In the deep Pacific salinity decreases with height above the bottom, as does temperature. These background gradients results in relatively warm and salty spreading plume water. An example of the temperature and salinity vertical profiles through the effluent layer of a plume on the Juan de Fuca Ridge is shown in Figure 3. The spreading plume is easily distinguished as a layer of nearly uniform temperature and salinity in Figure 3A. In Figure 3B and C the potential temperature y and salinity are plotted against potential density, s2 , and clearly show the relatively warm and salty effluent layer. In comparison, in the deep Atlantic where the salinity increases with height above the bottom the spreading plume is relatively cold and fresh. The temperature of the plume at the neutral level is colder than the ambient water despite the enormous temperature of the source fluid. Thus temperature alone may not always be an obvious indicator of hydrothermal activity. In either case, temperature anomalies at the spreading level are Oð101 1CÞ despite source temperature anomalies of B3501C The rise characteristics of event megaplumes are similar to the continuous venting, except that the source duration is limited and the buoyancy flux, F0 , is typically one to two orders of magnitude larger. For comparison, the heat flux from a typical high temperature vent is 1–100 MW, while megaplume sources are estimated to be >1000 MW. If the source

duration is small compared with the parcel rise time, N1, then the plume model must be replaced by a model for an isolated thermal. In this case ZM ¼ 2:7ðB0 N 2 Þ1=4 , where B0 ¼ V 0 gðr0  rs =r0 Þ is the buoyancy and V0 the volume of the pulse of hot fluid forming the release. Entrainment into and dilution of a thermal are comparable to the continuous release.

Mesoscale Flow and Vortices A high temperature vent continuously delivers plume fluid to the spreading level. Ambient currents can simply advect this fluid away from the vent location, but if the currents are weak, or oscillatory with small mean, then plume fluid accumulates over the vent and a radial outflow must develop. On a timescale f 1 this radial flow will be retarded by rotation. Here f ¼ 2OsinðfÞ is the Coriolis parameter, O is the rotation rate of Earth and f the latitude. At 451N f ¼ 104 s1 . The outward-flowing fluid parcels turn to the right (looking from above in the northern hemisphere) and an anticyclonic circulation will develop. With time a slowly growing lens of plume fluid will be formed. Below the spreading level, entrainment into the rising limb of the plume causes a radial inflow of ambient water. The Coriolis acceleration again results in fluid parcels turning to the right as they move inward and cyclonic circulation is established. The result is a baroclinic vortex pair: an anticyclonic lens of plume fluid at the spreading level and cyclonic circulation of ambient fluid around the rising buoyant plume. This circulation is sketched in Figure 4. The dynamical balance is geostrophic

478

DISPERSION FROM HYDROTHERMAL VENTS

1700

36.80

Density 2 36.84 36.88

36.92

34.54

Salinity (PSU) 34.58 34.62

34.66

1.8

Temperature (˚C) 2.2 2.0

2.4

Depth (m)

TT-175 STA.6

2000

2400 (A)

Effluent layer

S

T

r2

2.0

 (˚C)

1.9

1.8

1.7

1.6 (B)

Salinit y (PSU)

34.62

34.60

34.58

34.56 36.84 (C)

2

36.88

36.92

Figure 3 (A) Vertical profiles of temperature, salinity and density through a hydrothermal effluent layer on the Juan de Fuca Ridge. The relatively warm and salty effluent layer is clear in plots of potential temperature, y, versus potential density, s2 (B) and salinity versus density (C). (Reproduced with permission from Lupton et al., 1985.)

wherein the radial pressure gradients are balanced by the Coriolis acceleration. Figure 5 shows results from a laboratory experiment that illustrates the effects of rotation on plume structure. In the photographs dense fluid, dyed for visualization, is released from a small source into a tank of water that has been stratified with salt to give a constant density gradient (constant N). The tank is on a table rotating about the vertical axis to simulate the Coriolis effect. These photographs are taken looking in from the side a short time after the source has been turned on. The experiments were done with dense fluid which falls, rather than light fluid that rises. This is inconsequential for the physics and the hydrothermal vent situation can be envisioned simply by turning the figures upside down. As the rotation increases, as measured by decreasing values of the ratio N=f , lateral spreading of the plume is retarded and the anticyclonic lens of plume fluid becomes thicker. Dynamical scaling arguments and experiments show that the aspect ratio of the resulting eddy, h=L E 0:75f =N. Here h is the central thickness of the anticyclonic eddy (dyed fluid in the figure) and L is the radius. These arguments also give the eddy azimuthal, or swirling, velocity vBðF0 f Þ1=4. For the typical values of F0 and f ¼ 104 s1 , vB0:03 ms1 . This is comparable to observed background flows over ridges and suggests that plume vortex flow can persist in the presence of a background flow. The anticyclonic plume eddy will continue to grow until it reaches a critical radius LEZM N=f at which it becomes unstable and breaks up. An example of plume break up is shown in Figure 6, which contains a sequence of photographs looking down on the experiment. The plume vortex was initially circular (not shown), but eventually the eddy elongates (Figure 6A). It then splits into two separate vortex pairs (Figure 6B) which propagate away from the source (Figure 6C). The process of formation and instability process then begin again. A steady source results in the unsteady production of vent vortices as depicted in Figure 4. The timescale for this production process is tB B102 Nf 2 . For typical midlatitude values of N and f and ZM , LE2 km and tB B2 months. Note that f decreases as the equator is approached, resulting in larger diameter eddies which take longer to grow if other factors remain constant. The small size and long production time contribute to difficulty in directly observing these eddies, although observations of the water column properties do show indications of eddy-like features with the expected scales. Futhermore, ambient flows can

DISPERSION FROM HYDROTHERMAL VENTS

479

Anticyclonic circulation

Cyclonic circulation

Figure 4 Sketch of the effect of the Earth’s rotation on a hydrothermal plume. Rotation causes an anticyclonic horizontal circulation in the spreading fluid and cyclonic circulation below. These flows are indicated by the arrows. Lateral spreading of plume fluid is retarded by rotation and eventually the plume may become unstable, producing isolated vortices of plume fluid which have a radius LEZM N=f , which is about 2 km at mid-latitudes. A continuous vent could result in the production of numerous eddies which propagate away from the vent site.

be expected to influence this idealized scenario. But even with ambient flows that would tend to carry plume fluid from a vent, the tell-tale anticyclonic circulation at the spreading level and cyclonic flow below is expected. Indeed, there is observational evidence for this vorticity signature in time mean measurements of flow in the vicinity of a vent. However, the most compelling evidence for this dynamical scenario comes from megaplume observations. Figure 7 shows temperature and light attenuation (a measure of particulate concentration indicative of hydrothermal source fluid) anomaly sections across a megaplume observed near the Juan de Fuca Ridge in the North Pacific. Note the much larger rise height and lateral scales of this plume compared with the example in Figure 2. The structure of the megaplume is indicative of anticyclonic circulation within the core and this has been confirmed by detailed analysis. The production of eddies from either continuous high temperature venting or episodic megaplume

events is important for the dispersal of the vent fluid. While dispersal by simple advection and stirring by prevailing flows may be the dominant dispersal mechanism, even occasional eddy formation is significant. Coherent anticyclonic vortices are known to have closed streamlines and can retain their anomalous properties over long distances and large time periods. These eddies provide a mechanism for the long-range dispersal of vent organisms which are entrained into the rising plumes and then trapped in the eddies. Within the eddies larvae are suspended in water with anomalous properties that may enhance survival.

Large-scale flow Small-scale localized convection over the ridge crest can result in a large-scale circulation extending O(1000 km) from the ridge. From the point of view of the large-scale mid-depth (2000–3000 m) circulation, venting at numerous locations along a ridge

480

DISPERSION FROM HYDROTHERMAL VENTS

(A)

(B)

Zs Zmax

(C) Figure 5 Side-view photographs showing the effects of rotation on convective plumes. In (A) the rotation is zero. The classic turbulent plume and spreading layer are evident. Panel (B) has weak rotation, N=f ¼ 5:02. The lateral spreading is inhibited and the falling plume is partially obscured by the cyclonic circulation which has developed around the plume. In (C) the rotation is stronger, N=f ¼ 1:42, and the anticyclonic lens of dyed fluid is thicker and has a smaller radius. See the text for a description of the experiment. (Reproduced with permission from Helfrich and Battisti, 1991.)

crest segment produces an average net upwelling localized over the ridge crest characterized by divergent isopycnals over the ridge crest. This can set up a mean circulation similar to the circulation from an individual vent plume, anticyclonic flow at the spreading level and cyclonic below, but now extending along the length of the ridge crest segment. Fluid entrained into the plumes and upwelled to the spreading level must be replaced. This requires a broad downwelling flow to close the mass balance. However, on these scales of 100–1000 km the variation of the Corriolis parameter due to the spherical shape of the earth, the beffect, causes the two circulation cells to extend to the west of the ridge

(regardless of hemisphere) to form what has been termed a beta;-plume. The ideal b-plume described here will be affected by the ridge crest topography and any background mid-depth flow. However, there is some observational evidence suggestive of this model of long-range dispersal of plume fluid. Observations near 151S in the eastern Pacific (Figure 8) show a plume of anomalously high values of 3He (a distinctive signature of hydrothermal origin water) centered in the water column just above the depth of the ridge crest. The plume extends over 2000 km west of the ridge. As predicted by the b-plume dynamics the westward extension of the plume is greatest closer to the equator. There are no similar

DISPERSION FROM HYDROTHERMAL VENTS

481

observation in the Atlantic; this is perhaps explained by the deep axial topography.

Discussion

(A)

Localized high-temperature hydrothermal venting along ridge crest is capable of forcing circulations on scales many orders of magnitude larger than the vent field size. This is a consequence of the combination of the large buoyancy flux of hydrothermal vents and the dynamical effects of the Earth’s rotation. Rotating flows are very sensitive to vertical motions such that small vertical flows are amplified into large horizontal circulations. The immense buoyancy flux of the high-temperature vents and megaplumes gives

1200

0.08 0.12

1400

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Megaplume l Temperature anomaly

Profile

0.16

0.20 0.24

27.58

27.60 27.62

0.24

27.64

0.20 0.16 0.12 0.08 0.04 0.04

27.66

0.04 H

0.04

2200

0.04 H

2400 0

2

4

(B)

6

8 10 12 14 16 18 20 22 Distance (km) Megaplume l Attenuation

Profile

1200 1400

0.43 0.45 0.47

27.58

0.49 0.51

Depth (m)

1600 1800

27.60

0.51

27.62 0.49 0.47 0.45 0.43

27.64

2000

0.41 27.66

0.49 0.47 0.45

2200

0.43

0.41

H 0.41

2400 0

(C) Figure 6 Photographs of a laboratory experiment showing the formation and break up of a plume vortex. The view is from above and time increases from panel (A) to (C). A single continuous source produces one plume vortex which eventually becomes unstable, (A), and forms two smaller baroclinic vortex pairs which move away from the source (B), after which the process of plume vortex formation begins again (C). (Reproduced with permission from Helfrich and Battisti, 1991.)

2

4

6

8 10 12 14 16 18 20 22 Distance (km)

Figure 7 Observations of the temperature and light attenuation anomalies of a megaplume found near the Juan de Fuca Ridge. The figure shows a slice in depth and horizontal distance through the center of a nearly circular (in plan view) plume. The eddy aspect ratio b=LBf =N as predicted by the scaling theory and laboratory experiments. The lower level high in light attenuation may be the result of a separate, less intense hydrothermal source. The horizontal dashed lines are density isolines (sy contours) and the saw-tooth lines indicate the trajectory of the measurement package. (Reproduced with permission from Baker et al., 1989.)

482

DISPERSION FROM HYDROTHERMAL VENTS 3

STN.

7

 ( He)% 4

5

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25 Depth (km)

2

3

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East Pacific Rise 0

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Figure 8 Transect along 151S in the Pacific showing a plume of 3He anomaly, dð3 HeÞ, extending over 2000 km west of the East Pacific Rise. (Reproduced with permission from Lupton, 1995.)

rise to rapid vertical ascent and just as importantly large entrainment of background fluid into the rising plumes. These combine to force a localized net upwelling many times larger than the mass flux of the individual vent. The stacked nature of the resulting horizontal flow, anticyclonic circulation at one level and cyclonic below, is typically unstable and produces eddies which have scales comparable to the local Rossby radius of deformation, Ld ¼ ZM N=f , based on the plume rise height. In reality the ultimate dispersion of high-temperature vent fluid probably occurs through a combination of simple advection and stirring by background flow and the formation of long-lived coherent vortices and b-plumes. The reader might wonder whether these rotationally influenced convective processes are at work in the atmosphere where smokestacks and fires routinely cause localized plumes. There is one important difference between the atmosphere and the ocean in this regard. The scale at which rotation influences the flow and would produce eddies, the deformation radius Ld , is very much larger in the atmosphere than the ocean due to the greater static stability of the atmosphere (larger N). So these features are not likely to occur as a consequence of smokestacks and fires, which are simply too small to be affected by rotation. However, hurricanes are an example of the interaction of convection and rotation which produces intense vortices. Also, it would be possible for large volcanic eruptions which rise into the stratosphere to produce the atmospheric equivalent

of oceanic megaplumes. Finally, oceanic deep convection produced by surface cooling and sinking induces some of the same circulation characteristics discussed here, but over typically much larger horizontal scales than isolated high-temperature vents.

Nomenclature ZM Zs F0 N g Q r0 rs ra z y F H cp Ts T0 a s2 B0 V0 f O

maximum plume rise height. plume spreading level height. source buoyancy flux. background buoyancy frequency. gravitational acceleration. source volume flux. density of the ambient fluid at vent level. source fluid density. ambient density. depth above the source. latitude, potential temperature. latitude. heat flux. specific heat at constant pressure. source temperature. ambient temperature at vent level. coefficient of thermal expansion. measure of density. initial buoyancy of a thermal. initial volume of a thermal. Coriolis parameter. rotation rate of the Earth.

DISPERSION FROM HYDROTHERMAL VENTS

h L v tB Ld

vertical thickness of the anticyclonic plume eddy. radius of the anticyclonic plume eddy. azimuthal velocity within the plume eddy. timescale for plume break up. Rossby radius of deformation.

See also Double-Diffusive Convection. Internal Tidal Mixing. Upper Ocean Mixing Processes.

Further Reading Baker ET and Massoth GJ (1987) Characteristics of hydrothermal plumes from two vent fields on the Juan de Fuca Ridge, northeast Pacific Ocean. Earth and Planetary Science Letters 85: 59--73. Baker ET, Lavelle JW, and Feely RA (1989) Episodic venting of hydrothermal fluids from Juan de Fuca Ridge. Journal of Geophysical Research 94(B7): 9237--9250.

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Helfrich KR and Battisti T (1991) Experiments on baroclinic vortex shedding from hydrothermal plumes. Journal of Geophysical Research 96: 12511--12518. Humphries SE, Zierenberg RA, Mullineaux LS, and Thomson RE (eds.) (1995) Seafloor Hydrothermal Systems, Physical, Chemical, Biological and Geological Interactions, Geophysical Monograph 91. Washington, DC: American Geophysical Union. Lupton JE (1995) Hydrothermal plumes: near and far field. In: Humphries SE, Zierenberg RA, Mullineaux LS, Thomson RE Seafloor Hydrothermal Systems, Physical, Chemical, Biological and Geological Interactions, pp. 317–346. Geophysical Monograph 91. Washington, DC: American Geophysical Union. Lupton JE, Delaney JR, Johnson HP, and Tivey MK (1985) Entrainment and vertical transport of deep-ocean water by buoyant hydrothermal plumes. Nature 316: 621--623. Morton BR, Taylor GI, and Turner JS (1956) Turbulent gravitational convection from maintained and instantaneous sources. Proceedings of the Royal Society of London, Series A 234: 1--13. Parsons LM, Walker CL, and Dixon DR (1995) Hydrothermal Vents and Processes. London: Geological Society.

NEPHELOID LAYERS I. N. McCave, University of Cambridge, Cambridge, UK & 2009 Elsevier Ltd. All rights reserved.

Introduction A remarkable feature of the lower water column in most deep parts of the World Ocean is a large increase in light scattering and attenuation conferred by the presence of increased amounts of particulate material. This part of the water column is termed the bottom nepheloid layer (BNL). Another class of nepheloid layers found especially at continental margins are intermediate nepheloid layers (INLs) (Figures 1 and 2). These occur frequently at high levels off the upper continental slope and at the depth of the shelf edge. From here, they spread out across the continental margin. These INLs are similar to the inversions observed in the BNL on some profiles (Figure 1). (The surface nepheloid layer (SNL), not treated here, is simply the upper ocean layer in which particles are produced by biological activity, and which may have material from river plumes close to shore.) The increase in light scattering is perceived relative to minimum values found at mid-water depths of 2000–4000 m (shallower on continental margins). The increased scattering is due to fine particles. This has been determined by particle-size measurements and filtration of seawater with determination of concentration by weight and volume. Most data on the distribution and character of nepheloid layers have been acquired by optical techniques, principally by the Lamont photographic nephelometer and the SeaTech transmissometer, and more recently the WetLabs transmissometer and light scattering sensor (LSS). The optical work has revealed that the BNL is up to 2000-m thick (can be more in trenches) and generally has a basal uniform region, the bottom mixed nepheloid layer (BMNL), corresponding quite closely to the bottom mixed layer defined by uniform potential temperature (Figure 1). Above the BMNL there is a more or less exponential fall-off in intensity of light scattering up to the clear-water minimum marking the top of the BNL. Both bottom and INLs are principally produced by resuspension of bottom sediments. Their distribution indicates the dispersal of resuspended sediment in the ocean basins and is thus a signature of both material and water transport away from boundaries (some of

484

which may be internal such as ridges and seamounts). Most concentrated nepheloid layers occur on the continental shelf, upper slope, or deep continental margin. They indicate the locus of active resuspension and redeposition by strong bottom currents and internal waves.

Optics of Nephelometers: What They ‘See’ Detection of deep-ocean nepheloid layers has been mainly through measurement of light scattering. The Lamont nephelometer has made the largest number of profiles in all oceans but is no longer in use. It used an incandescent bulb as the source and photographic film as the detector of the light scattered from angles between y ¼ 81 and 241 from the forward axis of the light beam. The film was continuously wound on as the instrument was lowered, resulting in an averaging of the received signal over about 25-m depth. The short-lived Geochemical Ocean Section Study (GEOSECS) nephelometer used a red (l ¼ 633 nm) laser source and a photoelectric cell to detect light scattered from y ¼ 3–151 off the axis of the beam. The SeaTech (now WetLabs, Inc.) LSS measures infrared light (880 nm) backscattered (1801) from particles in the sample volume using a solar-blind silicon detector. Because of multiple scattering, nephelometers do not yield precise optical parameters. Optical transmission with a narrow beam can yield the attenuation coefficient (c). The most commonly used SeaTech and WetLabs transmissometers have a red light source (l ¼ 660 or 670 nm) and usually a 0.20– 0.25-m path length. Most of the contribution to the total scattering b comes from near-forward angles (low values of y). Jerlov (1976) shows that, for surface waters, 47% of b occurs between y ¼ 01 and 31, 79% between 01 and 151, and 90% between 01 and 301. The GEOSECS instrument records about 32% of b and the Lamont nephelometer about 16%. The total scattering is given by Mie theory (assumed spherical particles) as a function of particle size d and relative refractive index (relative RI) n, and wavelength of light l. The relative indices of refraction of suspended material are dominated by components with n ¼ 1.05 and 1.15, values probably characteristic of organic and mineral matter, respectively (e.g., RI of seawater 1.34, quartz 1.55, ratio n ¼ 1.15). Particles from clear ocean waters and weak nepheloid layers (concentration Co40 mg m3) tend

NEPHELOID LAYERS

485

c 1.0

0.8

0.6

0.4

 1.80

1.85

41 24

4700

300

88 16

109

4800

200

179

Pressure (dbar)

215

Meters above bottom

157

234

252

4900

100

227

BMNL 5000 2

4

8 μm

16

32

0

Figure 1 Data from the SeaTech transmissometer in the Atlantic showing the BMNL and a nepheloid layer comprising multiple steps in temperature and turbidity. Turbidity is given as c the attenuation coefficient, and y is potential temperature in 1C. Also shown on the right are particle-size spectra determined by Coulter counter. Reproduced from McCave IN (1983) Particulate size spectra, behavior and origin of nepheloid layers over the Nova Scotian Continental Rise. Journal of Geophysical Research 88: 7647–7666.

to have particle-size distributions by volume which are flat, equivalent to k ¼ 3 in a particle number distribution of Junge type, N ¼ Kd  k where N is the cumulative number of particles larger than diameter d and K and k are constants. However, this distribution does not appear to be maintained at sizes finer than about 2 mm where k decreases toward 1.5. In concentrated nepheloid layers, this distribution does not occur at all and a peaked distribution with a peak between 3 and 10 mm is encountered.

Morel has calculated scattering according to Mie theory for suspensions with Junge distributions and several indices of refraction. Recalculation into cumulative curves of percentage scattering in Figure 3 illustrates the fact that most recorded scattering is produced by fine particles. The cases shown are for values of k of 2.1, 3.2, and 4.0 and a two-component (peaked) distribution with k ¼ 2.1 up to a ¼ pdn/ l ¼ 32 and k ¼ 4.0 for larger sizes (a ¼ 32 is equivalent to d ¼ 5.6 mm for l ¼ 633 nm). In each case, three

486

NEPHELOID LAYERS

(a)

0 Knorr 51 sta. 698 Rockall Trough 54° 23.1′ N, 15° 18.7′ W INL.

Depth (m)

1000

2000 BNL 900m

{

INV 2 INV 1

Nephels, arbitrary scale 3000  (b)

3.0

2.8

2.6

3.2

2200 Nephel inversion 2

Higher-salinity core 

2300 NEADW

Depth (m)

2400

2500

Nephel inversion 1 Knorr 51 Station 698 Rockall Trough 54° 28.1′ N 15° 18.7′ W 24 Aug. 1975

2600

AABW Source _ High silica_

2700 40

50

60

70

80

90

Ne

15

20

25

30

35

Silica (μg at/l)

Figure 2 (a) Full-depth profile taken in the Rockall Trough with the GEOSECS nephelometer. An INL and two inversions are apparent. (b) Detail of the lower 500 m of the profile in (a) showing the relationship between the nephel (turbidity) inversions and hydrography. Reproduced from McCave IN (1986) Local and global aspects of the bottom nepheloid layers in the world ocean. Netherlands Journal of Sea Research 20: 167–181.

forward-scattering angles are given. It is clear that scattering close to the beam is more sensitive to large particles than that at 201. (At yo0.51 we are essentially dealing with a transmissometer.) In the case of k ¼ 2.1 only about 22% of the scattering is from

smaller sizes (ao32) at y ¼ 20 1. Thus the curves for y ¼ 10 1 and 20 1 are generally representative, and the distribution for both k ¼ 3.2 and the composite case show that 95% of the scattering is by particles o5 mm for l ¼ 633 nm.

NEPHELOID LAYERS

16

32

 64

128 256

4

8

16

487

32

100 k = 3.2

Cumulative percent scattering

k = 2.1

32

32 32

20°

20° 10°

k = 4.0

2°

20° 10°

2°

50

2° 2°

32

k = 2.1 for  < 32

20° 10°

k = 4.0 for  < 32

10°

0 1

2

4

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16

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64

128 256

0.5

1

2

4

8

1

2

4

8

16

32

 = Πdnw /  Figure 3 Cumulative percentage of scattering calculated from the data of Morel (1973) by McCave (1986). The right-hand case is for a peaked distribution with the peak at a ¼ 32, equivalent to 5.6 mm for l ¼ 633 nm and n ¼ 1.15. Note that material larger than the peak contributes virtually nothing to the scattering in this case. Reproduced from McCave IN (1986) Local and global aspects of the bottom nepheloid layers in the world ocean. Netherlands Journal of Sea Research 20: 167–181.

So it is dominantly the fine fraction of particles in nepheloid layers that is seen and recorded by nephelometers. These particles have very low settling velocities, less than B5  106 m s1. Although larger particles are present, they are rare and their properties and behavior cannot be invoked to explain features of distribution shown by nephelometers. The SeaTech 0.25 m path length transmissometer has been used for most of the modern work on the structure and behavior of nepheloid layers. The transmission T is related to beam attenuation coefficient c over path length l as T ¼ecl. The major control of c is due to particles. Attenuation is due to absorption a and scattering b, thus c ¼ a þ b. The value of c for pure seawater is about 0.36 m1 for this instrument operating at l ¼ 660 nm, and any excess is due to particulate effects. Because scattering very close to the beam is more sensitive to large particles, the transmissometer is more sensitive to larger particles and the nephelometer to smaller ones (Figure 4).

Nepheloid Layer Features The principal features of nepheloid layers that must be accounted for are the facts that the concentration is generally highest close to the bed, decreasing upward, but also that this is not universally the case, because: inversions, upward increases of concentration, are

also found; steeper gradients in concentration as well as inversions are often found at the boundaries of distinct density (temperature and/or salinity) changes, but their frequency decreases upward. The thickness of the BNL is generally in the region of 500–1500 m, and exceptionally up to 2000 m. This is clearly greater than the thickness of the bottom mixed layer (Figures 2 and 4), a fact which rules out the possibility of simple mixing by boundary turbulence being a sufficient mechanism for BNL generation. In several cases, the nepheloid layer is seen to transcend water masses. That is to say, the nepheloid layer shows a general decline in turbidity upward through interleaved water masses of differing sources and temperature/salinity characteristics though there may be a steeper turbidity gradient at the boundary between water masses. The highest suspended sediment concentrations occur in the BMNL in regions of strong bottom currents where they are typically 100–500 mg m3 (this is the same as mg l1). In general, deep western boundary currents and regions of recirculation carry high particulate loads. However, high turbidity is also found beneath regions of high surface eddy kinetic energy (variance of current speed) when located over strong thermohaline bottom currents. High surface eddy kinetic energy is connected with high bottom eddy kinetic energy; thus, intermittent variability, when added to a strong steady

488

NEPHELOID LAYERS

mixing (which occurs mainly in the BMNL) up to a kilometer above the bed. It is not possible to mix sediment across sharp density steps without breaking them down. However, these features are explicable if the layers are recently separated from the bottom. Some layers marked by steps in potential temperature contain excess radon-222 (originating from bottom sediments) with a 3.8 day half-life, suggesting detachment of bottom layers within 2–3 weeks before sampling. It is anticipated that with time these layers become thinner by mixing at their boundaries and by lateral spreading to yield, eventually, a uniform stratification. In this, the upper part of the BNL, sheared-out mixed layers that have, on average, come further from the sloping sides of the basins and from regions with less frequent resuspension, have lower concentrations. The basal layers are on average more recently resuspended and also gain material by fallout from above; thus, there is an overall decreasing particulate concentration, and increasing age upward.

−1

PMC (μg l ) 0

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100

150

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250

Surface nepheloid layer Attenuation excess Coarser/organicdominated particles

200

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Depth (m)

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1000

t −1

Clear water (~10 μg l ) Good agreement between instruments

1200

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Deep intermediate nepheloid layer 1800

Bottom nepheloid layer Scattering excess Finer/inorganic dominated particles

2000 26.00

26.25

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Decay of Concentration: Aging of Particulate Populations 28.00

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t (kg m )

Figure 4 Profiles of particulate matter concentration calculated from beam attenuation (solid line), and light scattering (dotted line) against depth, together with the density structure (st) of the water column (dashed line). Reproduced from Hall IR, Schmidt S, McCave IN, and Reyss JL (2000) Particulate matter distribution and Th-234/U-238 disequilibrium along the Northern Iberian Margin: Implications for particulate organic carbon export. DeepSea Research I 47: 557–582.

component, may be responsible for very high current speeds which produce intense sediment resuspension.

Separated Mixed-Layer Model Nepheloid layer structure is consistent with a quasivertical transport mechanism involving turbulent mixing in bottom layers of B10–50-m thickness (see Turbulence in the Benthic Boundary Layer), followed by their detachment and lateral advection along isopycnal (equal density) surfaces. The detachment occurs in areas of steep topography as well as in areas of lower gradient at benthic fronts where sloping isopycnals intersect the bottom. In many nepheloid layers, there are sharp upward increases in sediment content associated with steps in other properties such as temperature and salinity (Figure 1). Both the step structure and inversions in particulate matter concentration are incompatible with vertical turbulent

The particles composing the nepheloid layer may be modified due to aggregation with similar-sized particles and scavenging by larger rapidly settling ones. Aggregation may also be caused through biological activity, although little is known about such processes at great depths. Particles tend to settle and to be deposited onto the bed, from the bottom mixed layer. The larger particles should be deposited in a few weeks to months, 10–20 mm particles taking 50– 20 days to settle from a 60-m-thick layer. This will not affect the layer perceived by nephelometers so quickly because the timescale of fine particle removal initially involves Brownian aggregation with a ‘halflife’ of several months to years. The direct rate of deposition of very fine particles (0.5–1 mm) from a layer which remained in contact with the bed would be very slow. Concentration would halve in about 8 years. Thus the rate of decrease in concentration of 0.5–1-mm particles is due more to their being moved to another part of the size spectrum by aggregation (and then deposited) than to their being deposited directly. The fine material in dilute nepheloid layers has a mean residence time measured in years, demonstrated by the residence time of particle-reactive short half-life radionuclides such as 210Pb (t1/2 ¼ 22.3 years). In more concentrated nepheloid layers, a large proportion of this material will be removed in under a year, and in the BMNL residence times are tens to a hundred or so days estimated via 234Th (t1/2 ¼ 24.1 days). The

NEPHELOID LAYERS

dilute nepheloid layers in tranquil parts of the oceans could thus contain material that was resuspended very far away. The contribution of this material to the net sedimentation rate of these tranquil regions may not be negligible. The rate of deposition in the central South Pacific of only 0.5–2 mm ky1 could include up to 1 mm ky1 of fine material from the nepheloid layer. With aging, the individual detached layers comprising the nepheloid layer lose material by aggregation and settling and lose their identity by being thinned through shearing. An originally discontinuous vertical profile of concentration with inversions is converted to one of relatively smooth upward decline

60°

40° E

in concentration. Present understanding of particle aggregation and sinking rates suggests that this takes a few years to achieve.

Chemical Scavenging by Particles in Nepheloid Layers Many chemical species are particle-reactive and rapidly become adsorbed onto surfaces. This is why a number of elements are present in only trace quantities in seawater as outlined by Robert Anderson in 2004. The phenomenon of ‘boundary scavenging’, preferential removal of particle active species at

100°

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120° Mid-ocean and aseismic ridges and plateaus

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Figure 5 Distribution of excess turbidity for the Indian Ocean expressed as log(E/Ec) where E is the maximum light scattering near the bed and Ec is the value at the clear water minimum. A value of 1 thus represents a factor of 10 increase from the clear-water value. Reproduced from McCave IN (1986) Local and global aspects of the bottom nepheloid layers in the world ocean. Netherlands Journal of Sea Research 20: 167–181; based on Lamont nephelometer data presented by Kolla V, Sullivan L, Streeter SS, and Langseth MG (1976) Spreading of Antarctic bottom water and its effects on the floor of the Indian Ocean inferred from bottom water potential temperature, turbidity and sea-floor photography. Marine Geology 21: 171–189; and Kolla V, Henderson L, Sullivan L, and Biscaye PE (1978) Recent sedimentation in the southeast Indian Ocean with special reference to the effects of Antarctic bottom Water circulation. Marine Geology 27: 1–17.

490

NEPHELOID LAYERS

60

30

0

30

60

60

45

45

0 200 00 5 100

30

30

50 15

15

0

0 >2000 μg cm−2

15

500_ 2000 100 _ 500

15

50 _ 100 0, which means that 901 þ yoWo1801 – arctan(g)E1501, and then sxy>0. Equations [12] give sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   tay þ gtax grhf 2 grhf  þ v¼ CN 2CN 2CN

½13

558

SEA ICE DYNAMICS

where CN ¼ rwCw(cos yw  gsin yw). The southward flow is analyzed in a similar way. The velocity solution is independent of the exact form of the rheology and even of the absolute magnitude of the stresses as long as the proportionality |sxy| ¼ g|sxx| holds. The general solution is illustrated in Figure 8 for the Northern Hemisphere case, and the Southern Hemisphere case is symmetric. The resulting ice compactness increases to almost 1 in a very narrow ice edge zone, and further in the ice, thickness increases due to ridge formation.

Numerical Modeling In mesoscale and large-scale sea ice dynamics, all workable numerical models are based on the continuum theory. A full model consists of four basic elements: (1) ice state J, (2) rheology, (3) equation of motion, and (4) conservation of ice. The elements (1) and (2) constitute the heart of the model and are up to the choice of the modeler: one speaks of a threelevel (dim( J) ¼ 3) viscous-plastic sea ice model, etc. The unknowns are ice state, ice velocity, and ice stress, and the number of independent variables is dim( J) þ 2 þ 3. Any proper ice state has at least two levels. The model parameters can be grouped into those for (a) atmospheric and oceanic drag, (b) rheology, (c) ice redistribution, and (d) numerical design. The primary geophysical parameters are the drag coefficients and compressive strength of ice. The drag coefficients together with the Ekman angles tune the free drift velocity, while the compressive strength tunes the length scale in the presence of internal friction. The secondary geophysical parameters come from the rheology (other than the compressive strength) and the ice state redistribution scheme. The redistribution parameters would be probably very important but the distribution physics lacks good data. The numerical design parameters include the choice of the grid; also since the system is highly nonlinear, the stability of the solution may require smoothing techniques. Since the continuum particle size D is fairly large, the grid size can be taken as DxBD. Because the inertial timescale of sea ice is quite small, the initial ice velocity can be taken as zero. At solid boundary, the no-slip condition is employed, while in open boundary the normal stress is zero (a practical way is to define open water as ice with zero thickness and avoid an explicit open boundary). In short-term modeling, the timescale is 1 h–10 days. The objectives are basic research of the dynamics of drift ice and coupled ice–ocean system, ice forecasting, and applications for marine technology.

In particular, the basic research has involved rheology and thickness redistribution. Leads up to 20 km wide may open and close and heavy-pressure ridges may build up in a 1-day timescale, which has a strong influence on shipping, oil drilling, oil spills, and other marine operations. Also, changes in ice conditions, such as the location of the ice edge, are important for weather forecasting over a few days. In long-term modeling, the timescale is 1 month– 100 years. The objectives are basic research, ice climatology, and global climate. The role of ice dynamics is to transport ice with latent heat and freshwater and consequently modify the ice boundary and air–sea interaction. Differential ice drift opens and closes leads which means major changes to the air–sea heat fluxes. Mechanical accumulation of ice blocks, like ridging, adds large amount to the total volume of ice. An example of long-term simulations in the Weddell Sea is shown in Figure 9. The drag ratio Na and Ekman angle were 2.4% and 101, both a bit low, for the Antarctica, but it can be explained that the upper layer water current was the reference in the water stress and not the geostrophic flow. The compressive strength constant was taken as P ¼ 20 kPa. The grid size was about 165 km, and the model was calibrated with drift buoy data for the ice velocity and upwardlooking sonar data for the ice thickness. There is a strong convergence region in the southwest part of the basin, and advection of the ice shows up in larger ice thickness northward along the Antarctic peninsula. The width of the compressive region east of the peninsula is around 500 km. The key areas of modeling research are now ice thickness distribution and its evolution, and use of satellite synthetic aperture radars (SARs) for ice kinematics. The scaling problem and, in particular, the downscaling of the stress from geophysical to local (engineering) scale is examined for combining scientific and engineering knowledge and developing ice load calculation and forecasting methods. The physics of drift ice is quite well represented in shortterm ice forecasting models, in the sense that other questions are more critical for their further development, and the user interface is still not very good. Data assimilation methods are coming into sea ice models which give promises for the improvement of both the theoretical understanding and applications.

Concluding Words The sea ice dynamics problem contains interesting basic research questions in geophysical fluid dynamics. But perhaps, the principal science motivation is connected to the role of sea ice as a dynamic

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air–ocean interface. The transport of ice takes sea ice (with latent heat and fresh water) to regions, where it would not be formed by thermodynamic processes, and due to differential drift leads open and close and hummocks and ridges form. Sea ice has an important role in environmental research. Impurities are captured into the ice sheet from the seawater, sea bottom, and atmospheric fallout, and they are transported with the ice and later released into the water column. The location of the ice edge is a fundamental boundary condition for the marine biology in polar seas. A recent research line for sea ice dynamics is in paleoclimatology and paleoceanography. Data archive of drift ice and icebergs exists in marine sediments, and via its influence on ocean circulation, the drift ice has been an active agent in the global climate history. In the practical world, three major questions are connected with sea ice dynamics. Sea ice models have been applied for tactical navigation to provide shortterm forecasts of the ice conditions. Ice forcing

on ships and fixed structures are affected by the dynamical behavior of the ice. Sea ice information service is an operational routine system to support shipping and other marine operations such as oil drilling in ice-covered seas. In risk assessment for oil spills and oil combating, proper oil transport and dispersion models for ice-covered seas are needed.

See also Ice-Ocean interaction. Sea Ice. Sea Ice: Overview.

Further Reading Coon MD, Knoke GS, Echert DC, and Pritchard RS (1998) The architecture of an anisotropic elastic-plastic sea ice mechanics constitutive law. Journal of Geophysical Research 103(C10): 21915--21925. Dempsey JP and Shen HH (eds.) (2001) IUTAM Symposium on Scaling Laws in Ice Mechanics, 484pp. Dordrecht: Kluwer.

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Doronin YuP and Kheysin DYe (1975) Morskoi Led (Sea Ice), (English trans. (1977), 323pp. New Delhi: Amerind). Leningrad: Gidrometeoizdat. Hibler WD, III (1979) A dynamic-thermodynamic sea ice model. Journal of Physical Oceanography 9: 815--846. Hibler WD, III (1980) Sea ice growth, drift and decay. In: Colbeck S (ed.) Dynamics of Snow and Ice Masses, pp. 141–209. New York: Acamemic Press. Hibler WD, III (2004) Modelling sea ice dynamics. In: Bamber JL and Payne AJ (eds.) Mass Balance of the Cryosphere: Observations and Modelling of Contemporary and Future Changes, 662pp. Cambridge, UK: Cambridge University Press. Leppa¨ranta M (1981) An ice drift model for the Baltic Sea. Telbus 33(6): 583–596. Leppa¨ranta M (ed.) (1998) Physics of Ice-Covered Seas, vols. 1 and 2, 823pp. Helsinki: Helsinki University Press. Leppa¨ranta M (2005) The Drift of Sea Ice, 266pp. Heidelberg: Springer-Praxis. Pritchard RS (ed.) (1980) Proceedings of the ICSI/AIDJEX Symposium on Sea Ice Processes and Models, 474pp. Seattle, WA: University of Washington Press. Richter-Menge JA and Elder BC (1998) Characteristics of ice stress in the Alaskan Beaufort Sea. Journal of Geophysical Research 103(C10): 21817--21829. Rothrock DA (1975) The mechanical behavior of pack ice. Annual Review of Earth and Planetary Sciences 3: 317--342. Timmermann R, Beckmann A, and Hellmer HH (2000) Simulation of ice–ocean dynamics in the Weddell Sea I: Model configuration and validation. Journal of Geophysical Research 107(C3): 10.

Timokhov LA and Kheysin DYe (1987) Dynamika Morskikh L’dov, 272pp. Leningrad: Gidrometeoizdat. Untersteiner N (ed.) (1986) Geophysics of Sea Ice, 1196pp. New York: Plenum. Wadhams P (2000) Ice in the Ocean, 351pp. Amsterdam: Gordon & Breach Science Publishers.

Relevant Websites http://psc.apl.washington.edu – AIDJEX Electronic Library, Polar Science Center (PSC). http://www.aari.nw.ru – Arctic and Antarctic Research Institute (AARI). http://ice-glaces.ec.gc.ca – Canadian Ice Service. http://www.fimr.fi – Finnish Ice Service, Finnish Institute of Marine Research. http://www.hokudai.ac.jp – Ice Chart Off the Okhotsk Sea Coast of Hokkaido, Sea Ice Research Laboratory, Hokkaido University. http://IABP.apl.washington.edu – Index of Animations, International Arctic Buoy Programme (IABP). http://nsidc.org – National Snow and Ice Data Center (NSIDC). http://www.awi.de – Sea Ice Physics, The Alfred Wegener Institute for Polar and Marine Research (AWI).

SEA ICE P. Wadhams, University of Cambridge, Cambridge, UK & 2009 Elsevier Ltd. All rights reserved.

Introduction This article considers the seasonal and interannual variability of sea ice extent and thickness in the Arctic and Antarctic, and the downward trends which have recently been shown to exist in Arctic thickness and extent. There is no evidence at present for thinning or retreat of the Antarctic sea ice cover.

Sea Ice Extent Arctic

The seasonal cycle The best way of surveying sea ice extent and its variability is by the use of satellite imagery, and the most useful imagery on the large scale is passive microwave, which identifies types of surface through their natural microwave emissions, a function of surface temperature and emissivity. Figure 1 shows ice extent and concentration maps for the Arctic for each month, averaged over the period 1979–87, derived from the multifrequency scanning multichannel microwave radiometer (SMMR) sensor aboard the Nimbus-7 satellite. This instrument gives ice concentration and, through comparison of emissions at different frequencies, that percentage of the ice cover which is multiyear ice (i.e., ice which has survived at least one summer of melt). The ice concentrations are estimated to be accurate to 77%. This is an excellent basis for considering the seasonal cycle, although ice extent, particularly in summer, is now significantly less than these figures show. At the time of maximum advance, in February and March (Figure 1(a)), the ice cover fills the Arctic Basin. The Siberian shelf seas are also ice-covered to the coast, although the warm inflow from the Norwegian Atlantic Current keeps the western part of the Barents Sea open. There is also a bight of open water to the west of Svalbard, kept open by the warm West Spitsbergen Current and formerly known as Whalers’ Bay because it allowed sailing whalers to reach high latitudes. It is here that the open sea is found closest to the Pole in winter – beyond 811 in some years. The east coast of Greenland has a sea ice

cover along its entire length (although in mild winters the ice fails to reach Cape Farewell); this is transported out of Fram Strait by the Transpolar Drift Stream and advected southward in the East Greenland Current, the strongest part of the current (and so the fastest ice drift) being concentrated at the shelf break. At 72–751 N these averaged maps show a distinct bulge in the ice edge, visible from January until April with an ice concentration of 20–50%. During any particular year, this bulge will often appear as a tongue, called Odden, composed mainly of locally formed pancake ice, which covers the region influenced by the Jan Mayen Current (a cold eastward offshoot of the East Greenland Current). Moving round Cape Farewell there is a thin band of ice off West Greenland (called the ‘Storis’), the limit of ice transported out of the Arctic Basin, which often merges with the dense locally formed ice cover of Baffin Bay and Davis Strait. The whole of the Canadian Arctic Archipelago, Hudson Bay, and Hudson Strait are ice-covered, and on the western side of Davis Strait the ice stream of the Labrador Current carries ice out of Baffin Bay southward toward Newfoundland. The southernmost ice limit of this drift stream is usually the north coast of Newfoundland, where the ice is separated by the bulk of the island from an independently formed ice cover filling the Gulf of St. Lawrence, with the ice-filled St. Lawrence River and Great Lakes behind. Further to the west, a complete ice cover extends across the Arctic coasts of NW Canada and Alaska and fills the Bering Sea, at somewhat lower concentration, as far as the shelf break. Sea ice also fills the Sea of Okhotsk and the northern end of the Sea of Japan, with the north coast of Hokkaido experiencing the lowest-latitude sea ice (441) in the Northern Hemisphere. In April, the ice begins to retreat from its low-latitude extremes. By May, the Gulf of St. Lawrence is clear, as is most of the Sea of Okhotsk and some of the Bering Sea. The Odden ice tongue has disappeared and the ice edge is retreating up the east coast of Greenland. By June, the Pacific south of Bering Strait is ice-free, with the ice concentration reducing in Hudson Bay and several Arctic coastal locations. August and September (Figure 1(b)) are the months of greatest retreat, constituting the brief Arctic summer. During these months the Barents and Kara Seas are ice-free as far as the shelf break, with the Arctic pack retreating to, or beyond, northern Svalbard and Franz Josef Land. The Laptev and East

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