Preface
The book is devoted to the rotating shallow water model (RSW). This model, in spite of its simplicity and physi...

Author:
Vladimir Zeitlin

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Preface

The book is devoted to the rotating shallow water model (RSW). This model, in spite of its simplicity and physical transparency, contains essential ingredients of large-scale atmospheric and oceanic dynamics, which explains its frequent use in applications. Besides, nonlinear dynamics of the model is of general physical interest, as it embraces vortex dynamics, wave–vortex interaction, and nonlinear dynamics of both dispersive and almost dispersionless waves. For several years the contributors of the present volume were actively working in this area, and the book is mostly based on the progress achieved by their, often joint, work. This explains the choice of accents in the book. We apologize in advance for not giving an exhaustive list of references on RSW studies, which is an almost impossible task. Recent advances in analytical, numerical, and experimental treatment of the nonlinear RSW dynamics will be presented. The volume is organized as follows. In Chapter 1 an introduction to geophysical fluid dynamics and the derivation of the model are given in order to make the whole volume self-contained. Although a large part of material is standard and is presented for convenience of a non-specialist reader, some of it is not widely known, such as Lighthill radiation or ageostrophic baroclinic instability. In Chapter 2 the analytic approach based on multi-scale asymptotic expansions is explained and recent results on the geostrophic adjustment and decoupling of wave and vortex motions are presented. Chapter 3 contains a description of the method of Poincaré normal forms and its ramifications with applications to the RSW dynamics. Chapter 4 gives an introduction to the finite-volume numerical methods and presents new efficient numerical schemes for RSW. Chapter 5 deals with essentially nonlinear RSW dynamics. Some exact results on shock formation and existence of the adjusted state are given. Exact solutions for nonlinear travelling waves are also presented. These analytic advances are combined with direct numerical simulations using the new methods of Chapter 4. Finally, Chapter 6 gives a critical review of the experimental methods for studying the RSW dynamics and the results of laboratory experiments. V. Zeitlin

v

Contents

Preface

v

Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective

1

V. Zeitlin 1. 2. 3. 4. 5. 6. 7.

Introduction Derivation of the model General properties of the RSW model An overview of the two-layer RSW model Wave–vortex interactions RSW model on the equatorial β-plane Concluding remarks Acknowledgements References

Asymptotic Methods with Applications to the Fast–Slow Splitting and the Geostrophic Adjustment

2 3 8 22 28 33 41 43 43

47

G.M. Reznik and V. Zeitlin 1. 2. 3. 4. 5. 6.

Introduction Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW Nonlinear geostrophic adjustment in the presence of a lateral boundary Nonlinear geostrophic adjustment in the equatorial region Summary and discussion Acknowledgements References

The Method of Normal Forms and Fast–Slow Splitting

48 55 73 80 106 116 118 118 121

S.B. Medvedev 1. 2. 3. 4. 5.

Introduction Slow manifold in the spatial variables on the f -plane Slow manifold in the spectral variables on the f -plane Poincaré normal forms Skew-gradient normal forms for gradient systems vii

122 124 132 137 156

Contents

viii

6. Hamiltonian normal forms 7. Normal form of the Poisson bracket for one-dimensional fluid 8. Conclusion References

163 175 183 184

Efficient Numerical Finite Volume Schemes for Shallow Water Models 189 François Bouchut 1. A few notions on hyperbolic systems 191 2. Finite volume schemes for conservative systems 197 3. Finite volumes for systems with source terms 214 4. Second-order well-balanced schemes 231 5. Two-dimensional finite volumes on a rectangular grid 240 6. Numerical tests 246 References 254 Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment V. Zeitlin 1. 2. 3. 4.

Introduction Nonlinear wave phenomena and geostrophic adjustment in 1dRSW Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide 5. Summary and conclusions Acknowledgements References

257 258 259 281 297 317 319 319

Experimental Reality of Geostrophic Adjustment A. Stegner

323

1. 2. 3. 4.

323 339 352 375 376 377

The Holy Graal of rotating shallow-water flows Potential vorticity measurements: a new challenge Simple case studies of geostrophic adjustment What do we learn from laboratory experiments? Acknowledgements References

List of Contributors Author Index Subject Index

381 383 389

Chapter 1

Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction 2. Derivation of the model 2.1. Primitive equations model of GFD 2.2. Vertical averaging of the primitive equations and multi-layer RSW models

3. General properties of the RSW model 3.1. The conservation laws 3.2. Hamiltonian formulation of the RSW model

2 3 3 5 8 8 10

3.3. The spectrum of linear excitations in the RSW model on the f -plane and the first

11

notion of the geostrophic adjustment 3.4. Characteristic parameters of the RSW model on the f -plane. Geostrophic balance and slow motions 3.5. Slow motions on the β-plane. Rossby waves 3.6. The fast-slow splitting and geostrophic adjustment 3.7. Lateral boundaries and Kelvin waves 3.8. Topographic effects in RSW

4. An overview of the two-layer RSW model 4.1. Conservation laws. Linearization. Fast and slow motions 4.2. Characteristic parameters and scaling 4.3. Slow motion: the QG regime 4.4. Slow motion: the FG regimes 4.5. Baroclinic instabilities in the two-layer RSW

5. Wave–vortex interactions Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02001-6 1

13 16 19 19 20 22 22 23 24 24 25 28 © 2007 Elsevier B.V. All rights reserved

2

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model 5.1. Wave emission by vortex motions (Lighthill radiation) in the shallow water model 5.2. Brief comments on the wave–mean flow interactions

6. RSW model on the equatorial β-plane 6.1. General properties of the equatorial RSW 6.2. A reminder on equatorial waves and their dispersion properties in the RSW model

7. Concluding remarks Acknowledgements References

28 32 33 33 34 41 43 43

1. Introduction The oceans and the atmosphere are thin, with respect to the Earth radius, stratified fluid layers on the rotating sphere with variable bottom topography under the influence of gravity. The effects of sphericity (the small Earth’s non-sphericity is usually neglected) depend on the horizontal scale of the motions of interest. They are essential for the planetary-scale motions, like tides, but for description of the so-called synoptic-scale motions with characteristic scales of the order of thousand kilometers in the atmosphere, and hundreds kilometers in the ocean, the tangent plane approximation is largely sufficient (Gill, 1982; Holton, 1979; Pedlosky, 1982). The effects of sphericity may be then introduced via the variable Coriolis parameter. Although the vertical motions may be of considerable intensity in the oceans and the atmosphere, like e.g. deep convection, on the synoptic scale average the typical horizontal velocities are at least two orders of magnitude larger than the vertical ones. Moreover, on average at these scales the vertical accelerations are negligible, and the motion within a good accuracy is in the hydrostatic balance (cf. e.g. Holton, 1979). These observations explain why the rotating shallow water model (RSW in what follows) is widely used in geophysical fluid dynamics (GFD). Taken literally, the model describes the motion of quasi two-dimensional incompressible thin fluid layer of constant density with a free surface in hydrostatic balance on the rotating plane with constant or variable Coriolis parameter, and with the centrifugal acceleration absorbed into (effective) gravity. The key ingredients of the geophysical fluid dynamics at synoptic scales, such as rotation, quasi-bidimensionality, and hydrostatic balance, are, thus, incorporated into the model. The stratification is rudimentary: the model describes a single isopycnal surface, the free surface of the fluid layer. Stratification may be taken into account in a more realistic manner by superimposing several shallowwater layers of different densities. Bottom topography is easily introduced. Thus, the model is plausible already at such purely heuristic level. We will demonstrate below how it may be obtained from the full, so called primitive GFD equations, and remind its basic properties.

2. Derivation of the model

3

The literature on RSW and its GFD applications is vast. Needless to say that beyond GFD the shallow water model has numerous and practically important applications in hydraulics. The goal of the present chapter is not to review all of the shallow-water literature (which is an impossible task already in the GFD context), but to introduce the concepts and notions necessary for understanding of the subsequent chapters and, thus, to make the whole volume self-contained. The accents in what follows are correspondingly made on the problems addressed in the following chapters, as for example the problem of splitting of balanced and non-balanced motions, which is one of the main topics of the book. Part of material of this chapter is standard and may be found in the cited textbooks, it is given below for completeness.

2. Derivation of the model The standard derivation of the (non-rotating) shallow-water equations from the Euler equations for incompressible constant-density fluid with a free surface (Whitham, 1974) is based on the expansion in vertical to horizontal scale ratio. Rotation may be easily incorporated and we will not repeat this derivation here. Instead, we will show how the RSW and multi-layer RSW equations can be derived from the full equations of the stratified rotating fluid (“primitive equations”). 2.1. Primitive equations model of GFD The hydrostatic primitive equations can be written in standard notation as follows: ∂t vh + v · ∇vh + f zˆ ∧ vh +

1 ∇h p = 0, ρ

∂z p + ρg ∗ = 0, ∂t ρ + v · ∇ρ = 0,

(2.1) (2.2)

∇ · v = 0.

(2.3)

Here ρ is density, p is pressure, the subscript “h” denotes the horizontal part of the fields, or operators which depend on r = (x, y, z) and t. The velocity field is three-dimensional: v = (vh , w) = (u, v, w), as well as the operator nabla ∇ = (∇h , ∂z ) = (∂x , ∂y , ∂z ). These equations are written on the tangent plane to a sphere rotating with angular velocity Ω. The coordinate in the meridional direction is y (northward in the geophysical context), the coordinate in the zonal direction (eastward) is x, and z is the vertical coordinate in the direction normal to the tangent plane, i.e. opposite to the effective gravity g ∗ , which includes a (small) correction due to the centrifugal acceleration (see e.g. Holton, 1979; the asterisk will be omitted from now on). zˆ is the normal unit vector in z-direction. The Coriolis parameter f entering the horizontal momentum equation is equal to f0 = 2Ω sin φ = const, if the f -plane approximation is used, and to f = f0 +βy,

4

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

if the β-plane approximation is used. Here φ is the contact point latitude which is considered as fixed or variable in the first and second approximations, respectively. Equations (2.1)–(2.3) are non-dissipative. They express momentum and mass conservation of the fluid. Molecular (or turbulent) viscosity and diffusivity may be easily introduced in the r.h.s. of the momentum and mass equations. It should be noted, however that the Reynolds numbers for free (i.e. far from boundary layers) atmospheric and oceanic motions of synoptic scale are extremely high (Pedlosky, 1982; Holton, 1979). This is why, being interested in what follows mostly in the intrinsic dynamics of the system, and not in the forced-dissipative one, we will almost never use the dissipative form of equations. We will work for simplicity with a rigid lid and flat bottom boundary conditions for equations (2.1)–(2.3): w|z=H = w|z=0 = 0,

(2.4)

which are of frequent use in GFD. Other upper boundary conditions, like free surface boundary, or prescribed form of the pressure (geopotential) function at a given vertical level (cf. e.g. Holton, 1979) may be used as well. In (2.3) the continuity equation ∂t ρ + ∇ · (ρv) = 0

(2.5)

is replaced by the incompressibility condition and mass conservation by each fluid parcel. Such (Boussinesq) approximation is totally justified in the oceanic context taking into account the negligible compressibility of water. Further simplification may be made in this case, using the fact that in the ocean ρ = ρ0 + ρs (z) + ρ (x, y, z; t),

ρ0 ρs ρ .

(2.6) ρ

Here ρ0 is a constant, ρs is a background oceanic stratification, and represents the spatio-temporal density variations. Then, the variable part of the density may be neglected in the horizontal momentum equations, and we get ∂t vh + v · ∇vh + f zˆ ∧ vh + ∇Φ = 0, ρ ∂z Φ + g = 0, ρ0 ∂t ρ + v · ∇ρ = 0, ∇ · v = 0,

(2.7)

is introduced. Remarkably, the same equations, where the geopotential Φ = up to the change of sign in the hydrostatic equation, arise in the atmospheric context if the so-called pseudo-height Z, which was first introduced by Hoskins and Bretherton (1972), is used as the vertical coordinate: (γ −1)/γ p . Z = Za 1 − (2.8) Pr 1 ρ0 p

2. Derivation of the model

5

Pseudo-height is a modified pressure coordinate (see Holton, 1979, for pressure coordinates in the atmosphere). The constants are given by Za = γ γ−1 ρPrrg , and c

γ = cpv . In these definitions, Pr and ρr are (constant) surface pressure and density, cv and cp are the specific heats of air at constant volume and constant pressure respectively. The variable ρ has a different physical meaning in the atmosphere: it becomes up to a sign the so-called potential temperature θ which is directly related to the entropy density of air parcels (cf. Holton, 1979). The pseudo-height Z and the physical height z are related as follows: θ dZ = θr dz. 2.2. Vertical averaging of the primitive equations and multi-layer RSW models The derivation of the RSW model by vertical averaging was proposed in the classical works of Jeffreys (1925) in the linear approximation, and Obukhov (1949) in the nonlinear case. We follow below the method of Obukhov by generalizing it to the multi-layer case. To lose no generality we will consider a general case of equations (2.1), together with the continuity equation (2.5), and the hydrostatic balance equation (2.2). Equations (2.1), (2.5) may be rewritten as evolution equations of the horizontal momentum density: ∂t (ρu) + ∂x ρu2 + ∂y (ρvu) + ∂z (ρwu) − fρv = −∂x p, (2.9) 2 ∂t (ρv) + ∂x (ρuv) + ∂y ρv + ∂z (ρwv) + fρu = −∂y p, (2.10) As a preliminary step we integrate these equations in the vertical direction between a pair of material surfaces z1,2 . By definition w|zi =

dzi = ∂t zi + u∂x zi + v∂y zi , dt

i = 1, 2.

(2.11)

By using (2.11) and the formula z2

z2 dz Fx = ∂x

z1

dz F − ∂x z2 F |z2 + ∂x z1 F |z1

(2.12)

z1

which is valid for any function F , we get from (2.9) z2

z2 dz ρu + ∂x

∂t z1

z2 dz ρu + ∂y

z1

z2 dz ρuv − f

2

z1

dz ρv z1

z2 = −∂x

dz p − ∂x z1 p|z1 + ∂x z2 p|z2 . z1

(2.13)

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

6

Analogously, from (2.10) we get z2

z2 dz ρv + ∂x

∂t z1

z2 dz ρuv + ∂y

z1

z2 dz ρv + f 2

z1

dz ρu z1

z2 dz p − ∂y z1 p|z1 + ∂y z2 p|z2 .

= −∂y

(2.14)

z1

From (2.5) and (2.11) we obtain z2

z2 dz ρ + ∂x

∂t z1

z2 dz ρu + ∂y

z1

dz ρv = 0.

(2.15)

z1

We introduce the integrated layer density: z2 dz ρ = −

μ=

1 p|z2 − p|z1 , g

(2.16)

z1

and define for any quantity F its density-weighted vertical average over the layer: 1 F = μ

z2 dz ρF.

(2.17)

z1

We thus obtain the averaged horizontal momentum and continuity equations from (2.13)–(2.15): ∂t μu + ∂x μ u2 + ∂y μuv − f μv z2 = −∂x

dz p − ∂x z1 p|z1 + ∂x z2 p|z2 ,

(2.18)

z1

∂t μv + ∂x μuv + ∂y μ v 2 + f μu z2 = −∂y

dz p − ∂y z1 p|z1 + ∂y z2 p|z2 ,

(2.19)

z1

∂t μ + ∂x μu + ∂y μv = 0.

(2.20)

With the help of hydrostatic relation (2.2), the pressure at any point inside the layer (z1 , z2 ) is expressed in terms of the pressure at the lower material surface

2. Derivation of the model

7

and the position of this latter as follows: z p(x, y, z, t) = −g

dz ρ(x, y, z , t) + p|z1 .

(2.21)

z1

Up to now no approximation was made whatsoever, and no advantage gained with respect to original equations. In order to advance, as usual, some closure hypothesis should be made. Supposing that all variations of all the variables are weak in the vertical direction (the quasi-bidimensionality hypothesis), the correlations may be decoupled: 2

2

v ≈ vv. uv ≈ uv, u ≈ uu, (2.22) By introducing the mean density ρ: ¯ 1 ρ¯ = (z2 − z1 )

z2 dz ρ,

μ = ρ(z ¯ 2 − z1 ),

(2.23)

z1

the integral in (2.21) may be expressed in terms of ρ: ¯ p(x, y, z, t) ≈ −g ρ(z ¯ − z1 ) + p|z1 .

(2.24)

An interesting possibility is to consider ρ¯ as a given function of horizontal coordinates and/or time. We will adopt however a conventional approach and consider only the case ρ¯ = const below. Note, that what is just described is the mean field approximation. The neglected “Reynolds stresses” may be added, if needed, in the r.h.s. of the equations via e.g. the turbulent-viscosity like closures. Omitting the angle brackets in the mean fields from now on we get from (2.18), (2.19), (2.22), (2.24), with the help of (2.20), (2.23): ρ(z ¯ 2 − z1 )(∂t vh + vh · ∇vh + f zˆ ∧ vh ) (z2 − z1 )2 + (z2 − z1 )p|z1 − ∇h z1 p|z1 + ∇h z2 p|z2 . = −∇h −g ρ¯ 2 (2.25) All dependent variables in this equation are functions of horizontal coordinates and time only. We apply now the general equations (2.25) to the fluid contained between a flat bottom and a rigid lid, with boundary conditions (2.4). We choose a material surface z = z2 (x, y, t) ≡ h(x, y, t) inside the fluid, and consider two layers, one above and one below this surface. The generalization to the N -layer case is straightforward. The vertical boundaries are material surfaces. Hence, for the lower layer its lower boundary is constant: z1 = 0, while for the upper layer its upper boundary is constant: z3 = H . The generalization to the nontrivial bottom

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

8

topography is provided by taking z1 = b(x, y), a given topography function. Hence, μ1 = ρ1 (z2 − z1 ), μ2 = ρ2 (H − z2 ). From (2.20) the equations for the evolution of the interface between the layers follow, while the equations for the evolution of the horizontal momentum of the layers arise from (2.25). The two-layer rotating shallow water model thus results: ∂t vi + vi · ∇vi + f zˆ ∧ vi +

1 ∇πi = 0, ρ¯i

i = 1, 2,

(2.26)

∂t h + ∇ · (v1 h) = 0, ∂t (H − h) + ∇ · v2 (H − h) = 0,

(2.27)

π1 = (ρ¯1 − ρ¯2 )gh + π2 ,

(2.29)

(2.28)

where the subscripts 1 (2) denote the lower (the upper) layer, respectively, π2 is the pressure value at the upper boundary, the bottom topography is not introduced, for simplicity, and the subscript h is omitted in velocities and nabla. The one-layer RSW model arises in the limit ρ¯2 → 0: ∂t v + v · ∇v + f zˆ ∧ v + g∇h = 0,

(2.30)

∂t h + ∇ · (vh) = 0.

(2.31)

In the presence of nontrivial bottom topography h should be replaced by h − b in the second equation. Equations (2.30), (2.31) are barotropic, in the sense that the fluid layer moves horizontally as a whole, while equations (2.26)–(2.29) are baroclinic, as vertical shears of the horizontal velocity are possible across the whole fluid layer (0, H ). Below in this chapter, and throughout this volume, the study is centered most of the time at the one-layer RSW model, which will be simply called RSW. However, on several occasions we will illustrate the role of the baroclinic effects by using the two-layer RSW model. Note also that equations (2.30), (2.31) are equivalent to the two-dimensional barotropic gas dynamics in the presence of the Coriolis force, where h plays the role of the gas density ρ, and the equation of state is: P ∝ ρ 2 . This analogy is one of the advantages of the RSW model as it allows, for example, to apply the powerful computation methods developed in gas dynamics (see below, and Chapter 4 of the volume).

3. General properties of the RSW model 3.1. The conservation laws By construction equations (2.30), (2.31) express the (local) conservation of the horizontal momentum and mass. It is easy to see that the energy density e=h

v2 h2 +g 2 2

(3.1)

3. General properties of the RSW model

9

obeys the local conservation equation 2 v + gh = 0, ∂t e + ∇ · vh (3.2) 2 and hence the total energy, E = dx dy e, is constant for the isolated system. A peculiar Lagrangian conservation law is present in the RSW model. Its existence is, perhaps, the most specific feature of the rotating stratified systems which alone is sufficient to understand many of their dynamical properties. The Lagrangian invariant is potential vorticity q (PV in what follows) which is constructed from the vorticity ζ = vx − uy , the Coriolis parameter f , and the height h as follows: q=

ζ +f . h

(3.3)

The Lagrangian conservation of this quantity: (∂t + v · ∇)q = 0,

(3.4)

follows by combining the equation for vorticity: (∂t + v · ∇)(ζ + f ) + (ζ + f )∇ · v = 0,

(3.5)

and the continuity equation (2.31). Note that in the two- (or more) layer RSW, PV is conserved layer-wise. In Eulerian terms the PV conservation is expressed as time-independence of any integral of the form dx dy hF (q), (3.6) over the domain of the flow, if proper boundary conditions are used. Here F is an arbitrary function. The PV conservation is a consequence of the Kelvin circulation theorem (Allen and Holm, 1996). ζ is called relative vorticity in GFD, and ζ +f is called absolute vorticity because, as it is easy to see (cf. e.g. Holton, 1979), f is the planetary vorticity, i.e. vorticity due to the Earth rotation. It is constant in the f -plane approximation, and varies linearly with y in the β-plane approximation. A slight inconvenience of the PV is that it is non-zero even in the absence of motion. That is why the PV-anomaly defined by subtracting from the full PV the background PV equal to Hf00 , where H0 is the rest height, is frequently used. Thus the intuitive picture of the dynamics in the RSW model is that of fluid columns of variable height moving horizontally in such a way that each change of height is compensated by the change of vorticity, in order to preserve the PV.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

10

3.2. Hamiltonian formulation of the RSW model The RSW model possesses the Hamiltonian structure. The Hamiltonian is given by the following expression following from the Lagrangian description of the model (see Allen and Holm, 1996 for details, and also Chapter 3 of the present volume; for Hamiltonian formulation of the two-layer RSW see Bokhove, 2005): v2 h2 ¯ ·v+h +g H = dx dy (m − m) (3.7) . 2 2 Here ¯ = Rh + hv, m

(3.8)

is the Eulerian momentum density corrected due to rotation entering via the vector R defined as ∇ ∧ R = f zˆ .

(3.9)

The Hamiltonian (3.7) is constrained as δH m − hR (3.10) = . δm h The Hamiltonian equation of motion are (summation over repeated indices is understood): δH ∂t h = {h, H} = −∂i h (3.11) , i = 1, 2, δmi δH δH ∂t mi = {h, H} = −(∂j mi + mj ∂i ) (3.12) − h∂i , δmj δh v=

where the Lie–Poisson bracket for RSW is defined as (Allen and Holm, 1996): δA δB δB {A, B}(m, h) = − dx dy + h∂i (∂j mi + mj ∂i ) δmi δmj δh δA δB + (3.13) ∂i h . δh δmi Thus, equation (3.11) gives ∂t h = −∇ · (hv), and equation (3.12) together with (3.10) gives v2 ∂t v + (f zˆ + ∇ ∧ v) ∧ v + ∇ gh + = 0, 2

(3.14)

(3.15)

3. General properties of the RSW model

11

which, using the vector analysis identity: (∇ ∧ A) ∧ B + ∇(A · B) = (B · ∇)A + ai ∇bi ,

(3.16)

is equivalent to the horizontal momentum equation (2.30) in RSW. The Lagrangian conservation of PV follows from the symmetry of the Hamiltonian (or of the action) with respect to relabelling of Lagrangian particles (Salmon, 1998). The Hamiltonian formulation of the RSW model allows to apply the standard nonlinear stability analysis (Holm, Marsden, Ratiu and Weinstein, 1985) to the stationary RSW flows, as the stationary solutions realize the extrema of the Hamiltonian. 3.3. The spectrum of linear excitations in the RSW model on the f -plane and the first notion of the geostrophic adjustment The first idea of the dynamics of the RSW model may be obtained, as usual, by considering small perturbations about the rest state v = 0, h = H0 = const. The linearized equations (2.30), (2.31) in the f -plane approximation f = f0 = const are: ∂t u − f v + g∂x η = 0, ∂t v + f u + g∂y η = 0, ∂t η + H0 (∂x u + ∂y v) = 0,

(3.17)

where u, v are two components of the velocity perturbation, and η is the free surface perturbation. By looking for solutions in the form of harmonic waves with phases ωt −k·x, where ω and k are wave frequency and wave vector, respectively, we obtain the solvability condition of (3.17) in the form: ω ω2 − gH0 k2 − f 2 = 0. (3.18) The three roots of this equation correspond to the stationary solutions ω = 0 and propagative waves with the dispersion relation ω2 − gH0 k2 − f 2 = 0.

(3.19)

The polarisation relation for these waves, which are called inertia–gravity waves (IGW in what follows) are easy to obtain from (3.17), (3.19). A remarkable feature of the RSW model becomes evident from (3.18), namely that two different kinds of motion exist—the slow one and the fast one, which are separated by the spectral gap. Indeed, the minimal frequency of the fast waves is f , and the frequency of the slow motions is zero (they are, thus, infinitely slow in this approximation). Obviously the zero root of (3.18) is related to the PV conservation, because the PV conservation law (3.4) is reduced in the linear approximation to time-independence of the PV linearized about the rest state, as

12

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

it is easy to check. By using the polarisation relations for IGW, it may be seen that the waves carry no PV anomaly. Hence, the slow motions are PV-bearing, vortex ones, while the fast ones are the PV-less IGW. Moreover, nonlinear wave motions in the RSW model can be defined in an invariant way as motions with zero PV-anomaly (Falkovich and Medvedev, 1992). Examples of nonlinear waves verifying this property will be given in Chapter 5 of this volume. In the short-wave limit the IGW become dispersionless, as their phase velocity √ tends to a constant: c = gH0 . In the acoustic analogy perspective, these are just sound waves. Rotation introduces dispersion in the long-wave limit. Indeed, for the waves with frequencies close to the inertial frequency f (inertial oscillations), the group velocity is almost zero. This means that the packets of such waves, once created, will tend to stay at the same location for a long time. The solutions of the dispersion equation (3.18) are presented in Figure 1. The linear analysis above suggests that the solution of the initial-value problem with localized initial data in RSW in linear approximation will consist, in general, of IGW going out of the initial perturbation and leaving the steady vortex component at the location of the initial perturbation. We will see in the next subsection that the vortex component is in geostrophic balance in the leading order. Thus, the just described process represents adjustment to the geostrophic equilibrium, or simply geostrophic adjustment. This process, which was first introduced

√ Figure 1. Dispersion curves for inertia–gravity waves. c = gH0 and f are taken to be one, k is the absolute value of the wavenumber, and only the upper branch of the dispersion curve is presented. The ω = 0 solution is also shown in order to illustrate the spectral gap.

3. General properties of the RSW model

13

in the pioneering paper by Rossby (1938) in the RSW context is universal and of fundamental importance in GFD. 3.4. Characteristic parameters of the RSW model on the f -plane. Geostrophic balance and slow motions A peculiar property of the RSW model is the existence of the internal length-scale which is called the Rossby deformation radius: √ gH0 Rd = (3.20) . f For motions with a single typical velocity scale U and a single typical length scale L, the following non-dimensional parameters may be introduced: Ro =

U , f0 L

Bu =

Rd2 , L2

(3.21)

which are called Rossby and Burger numbers, respectively, together with the nonlinearity parameter λ = H /H0 , where H is the characteristic free-surface displacement. The characteristic time scale for vortex-type motions is T = L/U (∼ vortex turnover time) and, hence, for slow motions the Rossby number, which is the ratio of the vortex turnover frequency to the double frequency of the Earth rotation, is small. The non-dimensional RSW equations for velocity and freesurface displacement η for the motions with the typical scales L, T , U are λ Bu Ro(∂t v + v · ∇v) + zˆ ∧ v = ∇η, Ro λ∂t η + ∇ · v(1 + λη) = 0.

(3.22) (3.23)

Hence, for small Rossby numbers the Coriolis acceleration dominates in the l.h.s. of (3.22). In order to balance this term by the acceleration due to the pressure force the following relation among parameters should hold: λ Bu = O(Ro).

(3.24)

In this case the geostrophic balance f zˆ ∧ v + g∇η = 0

(3.25)

arises in the leading order in the Rossby number. In this sense the slow motions are balanced in the RSW. As is well known, the synoptic-scale motions in the atmosphere and oceans verify the geostrophic balance within a good accuracy (Holton, 1979; Pedlosky, 1982). The physical mechanism leading to balance is the geostrophic adjustment through the IGW emission. This process will be studied in detail in the next chapter.

14

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

The relation (3.25) is diagnostic as it predicts no time-evolution allowing only to express the velocity of the balanced motions via the pressure anomaly (the geostrophic velocity, or “geostrophic wind” relation). In order to get prognostic relations equations (3.22), (3.23) should be considered in the next-to-leading order in Rossby number. For this purpose the freedom in parameters ratios remaining after imposing (3.24) should be removed. This opens a possibility to have different dynamical regimes depending on the ratio of nonlinearity to Rossby number (Charney and Flierl, 1981; Romanova and Zeitlin, 1984; Williams and Yamagata, 1984). Two main regimes are possible: the standard (cf. Holton, 1979; Pedlosky, 1982) quasi-geostrophic one (QG) corresponding to the case λ ∼ Ro 1 and, hence, to L ∼ Rd . The frontal dynamics (FD), or frontal geostrophic (FG) regime (Romanova and Zeitlin, 1984; Williams and Yamagata, 1984; Cushman-Roisin, 1986) is characterized by strong displacements of the free surface λ = O(1) and, consequently, by large scales L Rd , according to (3.24). The intermediate regime called nonlinear geostrophic (NLQG) in Stegner and Zeitlin (1995), also exists and corresponds to Ro λ 1 and to large but smaller than in FD scales. The experimental confirmation of the existence of these regimes was given by Stegner and Zeitlin (1998). We start from the conventional QG regime. By calculating the first-order correction to the geostrophic velocity: va = zˆ ∧ (∂t + vg · ∇)vg ,

(3.26)

and its divergence ∇ · va = −(∂t + vg · ∇)∇ 2 η,

(3.27)

where vg is given by (3.25), and plugging this result in (3.23) we obtain the following slow-motion equation: ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η = 0. (3.28) Here and below all parameters of the order one are taken to be equal to unity for simplicity of notation and J (A, B) denotes the Jacobian of two functions A, B. This balanced equation expresses the PV conservation with PV being calculated in the leading order in the Rossby number. Indeed, the leading-order potential vorticity anomaly in non-dimensional variables is: qqg = ∇ 2 η − η,

(3.29)

and the physical meaning of equation (3.28) is that fluid particles moving with the QG-velocity (3.25) transport their QG potential vorticity (3.29). This “QG-Lagrangian” PV conservation is expressed in Eulerian terms as t-independence of any function of qqg integrated over the domain of the flow—cf. (3.6).

3. General properties of the RSW model

15

It is instructive to compare equations (3.28), (3.29) with the vorticity equation, and vorticity–streamfunction relation for the two-dimensional incompressible Euler equations. In the latter case we have ∂t ζ + J (η, ζ ) = 0,

ζ = ∇ 2 η,

(3.30)

where ζ denotes vorticity and we deliberately used the same notation and the same conventions for the streamfunction as in (3.25). The additional with respect to (3.30) terms in (3.28), (3.29) arise due to the presence of the finite deformation radius. Indeed, by restoring dimensions a factor 12 appears in front of the terms Rd

containing ∂t η in (3.28) and η in (3.29), respectively. Thus, slow-motion dynamics in the QG regime of the RSW model is vortex dynamics with modified vorticity– streamfunction relation, which tends to the standard one as Rd → ∞. While for the two-dimensional Euler equations in the unbounded domain the Green function, i.e. the streamfunction corresponding to a point vortex situated at x = x1 , is given by: 1 log|x − x1 |; ∇ 2 G(x − x1 ) = −δ(x − x1 ), 2π the related expressions for the QG equations are: 1 |x − x1 | K0 ; G(x − x1 ) = 2π Rd 1 ∇ 2 − 2 G(x − x1 ) = −δ(x − x1 ), Rd G(x − x1 ) = −

(3.31)

(3.32)

where K0 (x) is the modified Bessel function of the second kind. It has logarithmic asymptotics at small values of the argument, and exponentially decreasing asymptotics at large values of the argument (Abramowitz and Stegun, 1964): 2 −x K0 (x)|x→0 = − log x, (3.33) K0 (x)|x→∞ = e . x Hence, there is a screening of QG-vortices at distances larger than the deformation radius, while for distances smaller than Rd they behave as usual ones. This difference is however not important for the application of the powerful numerical methods based on the Lagrangian point-wise conservation of vorticity to the QG dynamics, like for example the contour advection method (Dritschel, 1997). It is worth noting that, as well as the parent RSW, or multi-layer RSW model, the QG equations admit the variational principle and Hamiltonian formulation (Holm and Zeitlin, 1998). It should not be forgotten, however, that the QG equation, as well as the FG one below, is not exact and that corrections to these equations should be taken into account e.g. for times much longer than T . Such corrections, as well as conditions

16

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

of applicability of the QG and FG models were analysed in Reznik, Zeitlin and Ben Jelloul (2001) (see also Chapter 2 of the present volume). Another remark is that IGW were completely filtered out by the choice of the time-scale T above. The consistency of this procedure will be discussed in Chapter 2. It should be emphasized that the problem of initialization, i.e. of projection of arbitrary initial data onto the slow manifold arises if equation (3.28) alone is to be used to describe the evolution of the system. It was shown by Reznik, Zeitlin and Ben Jelloul (2001) (see also Chapter 2 of the present volume) how this may be done order by order in (small) Rossby number. It should be remembered also that the derivation of the balanced QG equations above (and FG equations below) is based on the smallness of the Rossby number. At large Rossby numbers new physical mechanisms arise preventing the fast–slow splitting, such as Lighthill radiation (see Section 5 below). In the FG regime the nonlinearity is of the order one and it makes no sense to distinguish between the full height h of the free surface and its perturbation η. The formulas for the ageostrophic velocity are the same as (3.26), (3.27), with the replacement η → h. It is immediately seen from equation (3.23) that the timescale T is not appropriate for the FG regime, as ∂t h = 0 and, hence a slower time τ = Ro t should be used. The slow-motion equation in the FG regime follows: 2 h (∇h)2 ∂τ h − J (3.34) , ∇ 2 h − J h, = 0. 2 2 The physical meaning of this equation is, again, the PV conservation. Indeed, at λ ∼ 1, the leading-order contribution to PV is 1/ h in non-dimensional terms, and equation (3.34) may be rewritten in the form of advection equation for h (or, as a consequence, of any function of h): (∇h)2 ,h . ∂τ h = −J h∇ 2 h + (3.35) 2 As in the QG case, this “FG-Lagrangian” conservation of h gives rise to the infinity of Eulerian conservation laws: any function of h integrated over the domain of the flow is τ -independent. The intermediate NLQG regime may be treated along the same lines, see Reznik, Zeitlin and Ben Jelloul (2001) for details. 3.5. Slow motions on the β-plane. Rossby waves On the β-plane f = f0 + βy in (2.30), and the fourth parameter, the nondimensional β, appears in the model. If it is of the same order as Ro (which is roughly the case for synoptic-scale atmospheric and oceanic motions (Pedlosky, 1982)), the geostrophic balance still arises in the leading order, and the β-terms

3. General properties of the RSW model

17

enter only in the expressions for ageostrophic velocity: ∇ · va = −(∂t + vg · ∇)∇ 2 η − yvg .

(3.36)

(As before the order one ratios of the non-dimensional parameters are absorbed into dynamical variables.) Correspondingly, the slow-motion equation also changes: ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η − ∂x η = 0. (3.37) It still expresses the advection of the geostrophic PV anomaly, but takes into account the variable part of the planetary vorticity, y in non-dimensional terms. This new effect, with respect to the f -plane dynamics, leads to a qualitatively new phenomenon, the appearance of Rossby waves. Indeed, a formal linearization of (3.37) gives the equation ∂t η − ∂t ∇ 2 η − ∂x η = 0,

(3.38)

which has propagating wave solutions with the dispersion relation: ω = −β

kx kx2

+ ky2 +

1 Rd2

,

(3.39)

where ω is the wave frequency, and k = (kx , ky ) the wave vector, and the dimensions were restored. The dispersion curve for Rossby waves is shown in Figure 2.

Figure 2.

Dispersion curve for Rossby waves. Rd and β are taken to be one, k is the zonal component of the wave vector, and the meridional component is taken to be 0.5.

18

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Note that this curve is well separated from the dispersion curve of the IGW, if both are plotted in dimensionful units and smallness of β is taken into account. A peculiar property of Rossby waves is that their zonal phase velocity is always negative. In contradistinction with inertia–gravity waves, these waves are slow and “vortical”, as they are waves of vorticity whose propagation is possible due to the presence of the underlying gradient of the planetary vorticity (see e.g. Pedlosky (1982), for explanation of the physical mechanism of the Rossby wave propagation). As seen from (3.39) Rossby waves are strongly dispersive and strongly anisotropic. It is worth emphasizing that plane-parallel harmonic Rossby waves of arbitrary amplitude are exact solutions of (3.37). Remarkably, equation (3.37) which is called Charney–Obukhov equation coincides with the so-called Hasegawa–Mima equation which arises in the physics of drift waves in plasma (cf. e.g. Nezlin and Snezhkin, 1993). Another remarkable property of this equation is existence of localized steady-moving exact dipolar solutions, the modons which may be obtained in the way similar to the Lamb dipole in two-dimensional hydrodynamics by matching a dipolar solution and a decaying solution across a circle on the β-plane (Larichev and Reznik, 1976). Another way to introduce the Rossby waves is to consider the limit of small nonlinearities λ → 0, and large Rossby and Burger numbers in (3.22), (3.23) considered on the β-plane. In this limit the continuity equation (3.23) is reduced to the incompressibility condition for velocity, and hence the latter may be expressed in terms of a streamfunction ψ. The r.h.s. of equations (3.22) becomes negligible, and they become equivalent to the vorticity equation (3.5), i.e. the system is reduced to incompressible two-dimensional Euler equations in the presence of (non-uniform) rotation. After linearization, the equation ∂t ∇ 2 ψ + ∂x ψ = 0

(3.40)

results. This equation is equivalent to (3.38) in the limit Rd → ∞, which is consistent with the hypothesis of large Burger numbers. Finally, the corrections to the FG regime on the β-plane lead to the following equation: 2 h (∇h)2 ∂τ h − J (3.41) , ∇ 2 h − J h, − h∂x h = 0. 2 2 A peculiar combination of the so-called scalar nonlinearity hhx and the advective (Jacobian) one in this equation is to be noted. A formal linearization of this equation about the rest state h = 1 leads to the dispersionless wave equation: ∂τ η − ∂x η = 0.

(3.42)

This equation is nothing else than the long Rossby waves equation, because, as seen from (3.39), the dispersion of these latter is given by ω ∝ −kx . This result is consistent with the hypothesis of small Bu underlying (3.41).

3. General properties of the RSW model

19

3.6. The fast-slow splitting and geostrophic adjustment Thus, our semi-heuristic analysis (see the next chapter for the rigorous one) shows that the slow vortex and the fast inertia–gravity wave motions in the RSW model are dynamically split, at least for small Rossby numbers. The refined, with respect to the simple linear version discussed above, notion of the geostrophic adjustment then arises, see Blumen (1972) for a review. Any initial perturbation in RSW (and in the atmosphere or oceans, in general) will produce an inertia–gravity waves response which will rapidly propagate out of the (localized) perturbation leaving the geostrophically adjusted, or balanced part to slowly evolve. This process, at least for small Ro, is universal and accompanies the relaxation of any perturbation. Up to now we considered the RSW model in the infinite domain and in the absence of topography. We will now briefly sketch the new phenomena appearing in the presence of these new elements, which we will still treat in the dissipationless limit. 3.7. Lateral boundaries and Kelvin waves The principal modification introduced by lateral boundaries concerns the sector of the fast motions of the model, as a new type of waves, the Kelvin waves, appear in this case (see e.g. Gill, 1982). Let us consider the RSW model in a semi-infinite domain with a straight boundary along the y-axis. The boundary condition v · nˆ = 0

(3.43)

should be imposed, where nˆ is the normal vector to the boundary. Hence u|x=0 = 0.

(3.44)

By considering the linearized RSW equations (3.17) and looking for the planewave solutions with wave-frequency ω, and wavevector l in the y-direction, while keeping the x-dependence arbitrary, we reduce the system to a single equation, e.g. for the variable η: 2 ω − f 2 − c02 l 2 η (x) + (3.45) η(x) = 0, c02 where prime denotes x-differentiation. The boundary condition (3.44) takes the form:

lf η − ωη

(3.46) = 0. ω2 − f 2 x=0 Hence, two situations are possible.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

20

• If ω2 − f 2 − c02 l 2 c02

= k 2 > 0,

(3.47)

then the standard IGW solution results, with reflection at the boundary, and the dispersion relation (3.19). • If ω2 − f 2 − c02 l 2 c02

= −k 2 < 0,

(3.48)

then we get an exponentially decaying off the boundary solution η ∝ e−kx , with ω2 = c02 l 2 ,

k=

1 , Rd

(3.49)

and u ≡ 0.

(3.50)

The latter solution is the boundary Kelvin wave. It is dispersionless and propagating along the boundary. Note that its presence destroys the spectral gap, because it is spectrally well separated from the IGW only for small wavenumbers, cf. Figure 3. It follows from (3.50), (3.17) that the velocity field and the free surface elevation are in geostrophic balance in the Kelvin wave and, hence, this wave belongs to the sector of balanced motions. The dispersion curves for RSW with a straight lateral boundary are presented in Figure 3, showing the IGW and the Kelvin-wave branches. 3.8. Topographic effects in RSW The RSW equations in the presence of the bottom topography b(x, y) are written as follows: ∂t v + v · ∇v + f zˆ ∧ v + g∇h = 0, ∂t (h − b) + ∇ · v(h − b) = 0.

(3.51) (3.52)

The PV in this case is expressed as ζ +f . (3.53) h−b It is obvious from these equations that topography will influence both wave and vortex components of the flow. Thus, linearization of equations (3.51), (3.52) about the rest state h = H0 gives a linear system with variable coefficients, q=

3. General properties of the RSW model

21

√ Figure 3. Dispersion curves for inertia–gravity waves and boundary Kelvin waves. c = gH0 , and f are taken to be one, l is the meridional wavenumber, and only the upper branch of the dispersion curves for positive l is presented.

which may be solved in a number of simple cases, like idealized sea-shelves, canyons, beaches, sea-mountains, etc., see LeBlond and Mysak (1978) and references therein, leading to a variety of topographically trapped waves. Reviewing this subject, which is well represented in the book of LeBlond and Mysak (1978), is out of the scope of our presentation. Let us only mention a simplification frequently used in the oceanic context (Gill, 1982) for motions/topographies with a typical scale corresponding to a large Burger number. In this case the variations of the free surface in equation (3.52) may be neglected in the first approximation, and this equation becomes: ∇ · (H v) = 0,

(3.54)

where H (x, y) is the full variable depth of the water layer. Hence a streamfunction ψ may be introduced: H u = −∂y ψ,

H v = ∂x ψ,

(3.55)

and injected into vorticity equation (3.30) to give an equation for ψ. For instance, for one-dimensional topography H = H (x) on the f -plane this equation reads: H 1 ∂x ψ + ∂yy ψ + f 2 ∂y ψ = 0, ∂t ∂x (3.56) H H H and may be solved by the method of separation of variables, see LeBlond and Mysak (1978) for examples.

22

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

If the topography is weak, i.e. the typical variations of the bottom topography are of the order of the Rossby number, supposed to be small, then the topographic effects in the balanced motion may be obtained by expanding the PV in Rossby number. Thus, in the QG regime the PV modified by topography is: qqg = ∇ 2 η − η + b(x, y). And the QG equation is ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η − J (η, b) = 0.

(3.57)

(3.58)

Thus, topography introduces a “topographic beta-effect” (Pedlosky, 1982), whence the existence, for example, of the topographic Rossby waves on the linear bottom slope becomes evident.

4. An overview of the two-layer RSW model 4.1. Conservation laws. Linearization. Fast and slow motions As before, we will consider the two-layer RSW with a rigid lid and a flat bottom on the f -plane, for simplicity. We rewrite equations (2.26)–(2.29) in a more symmetric form by introducing the deviation of the interface η between the layers, with unperturbed depths H1 and H2 (upper and lower layer, respectively: H1 + H2 = H ). The equations of motion are the horizontal momentum equations: 1 ∂t vi + vi · ∇vi + f zˆ ∧ vi + ∇πi = 0, i = 1, 2; (4.1) ρi (no summation over i; the combination i + 1 is understood modulo 2 everywhere below), the mass conservation equations in each layer: ∂t Hi − (−1)i+1 η + ∇ · vi Hi − (−1)i+1 η = 0, i = 1, 2; (4.2) and the equation of continuity of pressure at the interface: (ρ2 − ρ1 )gη = π2 − π1 .

(4.3)

The PV conservation equations in each layer readily follow: ζi + f , Hi − (−1)i+1 η where ζi = zˆ · ∇ ∧ vi is the relative vorticity in each layer. The conservation of energy: v2 η2 E = dx dy ρi Hi − (−1)i+1 η i + g 2 2 (∂t + vi · ∇)qi = 0,

i=1,2

may be easily established.

qi =

(4.4)

(4.5)

4. An overview of the two-layer RSW model

23

The straightforward linearization of the equations of motion with the harmonic wave ansatz for the solution ∝ ei(ωt−k·x) leads to the dispersion relation ω2 ω2 − gHe k2 − f 2 = 0 (4.6) between the wave frequency and wavenumber, where He =

H1 H2 ρ H1 + ρ1 H2

is called

2

the equivalent height. The two zero roots of this equation correspond to the linearized PVs (4.4). The remaining opposite roots give the dispersion relation of the internal inertia–gravity waves propagating along the interface between the layers. Their phase speed is less than the phase speed of the inertia–gravity waves propagating on the surface of the shallow water layer of the depth H1 + H2 . This is a general property of the baroclinic waves which are always slower than their barotropic counterparts. Otherwise, the properties of these waves are the same as for the surface waves in the one-layer RSW. 4.2. Characteristic parameters and scaling By introducing a typical horizontal velocity scale U and a typical horizontal scale of the motion L, the Rossby number is defined as usual. The nonlinearity parameter λ measures typical amplitudes of η. The natural characteristic scale of the vertical displacements is He , and that of the pressures Pi = ρi U Lf . Two new parameters, with respect to the one-layer RSW, appear in the model: N =2

ρ2 − ρ1 , ρ2 + ρ1

d=

H1 , H2

(4.7)

the stratification parameter, and the layers depth ratio, respectively. Correspondingly, the baroclinic deformation radius arises: Rd = g He /f, (4.8) where g = gN is the reduced gravity. It gives an internal length-scale for the baroclinic motions. Obviously, the baroclinic deformation radius is smaller than the barotropic one constructed from the full gravity and the total depth of the fluid. The baroclinic deformation radius appears naturally while non-dimensionalizing equation (4.3). Note, that in the oceanic context the two densities ρ1,2 are usually taken to be close. The stratification parameter then tends to zero and is retained only in the reduced gravity arising in (4.3), cf. e.g. Zeitlin, Reznik and Ben Jelloul (2003). R2

The Burger number, Bu = Ld2 , hence, should be defined with the help of the baroclinic deformation radius. Similarly to the one-layer case, if the (vortex turnover) time-scale T = L/U is chosen, the Rossby number arises in front of the acceleration terms in non-dimensionalized horizontal momentum equations (4.1),

24

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

and the geostrophic balance in each layer follows at small Ro, provided relation (3.24) is verified. Thus, as in the one-layer case, the spectral gap guarantees the fast-slow motions splitting at small Rossby numbers. 4.3. Slow motion: the QG regime Following the same lines as in the one-layer case the QG equations are straightforwardly derived: Di 2 ∇ πi + (−1)i+1 h¯ i+1 η = 0, Dt

i = 1, 2.

(4.9)

Di (. . .) := ∂t (. . .) + J (πi , . . .), Dt

i = 1, 2,

(4.10)

Here

i h¯ i = H1H+H , i = 1, 2, for simplicity it is supposed that d = O(1), and the 2 hypothesis of small stratification parameter is made, such that

π2 − π1 ≈ η

(4.11)

in non-dimensional terms. As in the one-layer case, the physical meaning of the QG equations is the geostrophic PV advection by the geostrophic velocity in each layer. As follows from (4.9) the dynamics of the upper and lower layers is coupled. This coupling leads to the appearance of the crucially important phenomenon of baroclinic instability which manifests itself already in this simple two-layer model and is of primary importance in oceanic and atmospheric dynamics (Holton, 1979; Pedlosky, 1982)—see below. 4.4. Slow motion: the FG regimes The FG regime corresponds to the interface displacements of the order one. The FG scaling, thus, differs from the QG one used before. It is as follows. The interface displacement η is scaled as H1 . Choosing the characteristic length-scale L we rescale the pressure perturbations πi by ρi f Ui L, where Vi , i = 1, 2, are the velocity scales in each layer. The velocity scales U1 and U2 are of the same order when the parameter d is of the order one, which corresponds to the so-called FGH (homogeneous FG) sub-regime (cf. Benilov and Reznik, 1996) or are chosen to be U2 ∼ Ro U1 for the FGI (inhomogeneous FG) sub-regime where the parameter d is small, d ∼ ε2 (cf. Cushman-Roisin, Sutyrin and Tang, 1992). The consistency of these scalings with the dynamical boundary condition at the interface (4.3) requires that in order to have order one interface displacements the Burger number Bu = ( RLd )2 should be small Bu = O(ε). Here the Rossby deformation radius is

4. An overview of the two-layer RSW model

defined with the help of H1 : Rd =

25

g H1 /f . In terms of the new variable

η2 (4.12) , 2 and η the following slow-motion equations result in the respective sub-regimes (Benilov and Reznik, 1996; Zeitlin, Reznik and Ben Jelloul, 2003): P = π1 + d −1 π2 +

• FGH sub-regime 1 J (η, P ), (4.13) 1 + d −1 1 ∂t ∇ 2 P + J P , ∇ 2 P + ∇ · (1 − η) d −1 + η J η, ∇η = 0. −1 1+d (4.14) • FGI sub-regime (∇η)2 ∂τ η + J (P , η) + J η, (1 − η)∇ 2 η − (4.15) = 0, 2 (∇η)2 = 0, ∂τ ∇ 2 P + J P , ∇ 2 P − J η, (1 − η)∇ 2 η + (4.16) 2 ∂t η =

where, as in the FG regime in the one-layer case, the slower time τ is necessary to introduce. 4.5. Baroclinic instabilities in the two-layer RSW The most drastic consequence of the baroclinicity of the two-layer RSW is the appearance of specific baroclinic instabilities. The most famous one, playing a crucial role in the atmosphere and ocean dynamics (Gill, 1980; Pedlosky, 1982) is the classical quasigeostrophic baroclinic instability. 4.5.1. The standard baroclinic instability The standard configuration to illustrate the baroclinic instability is the geostrophically balanced flow with a vertical shear of horizontal velocity between the layers: vi = (Ui , 0), Ui = ∂y π¯ i : π¯ i = −Ui y + φi (x, y, t) (the Phillips, 1954, model). The two-layer QG equations on the f -plane (4.9) are linearized around this state. Imposing periodic boundary conditions in the direction of the mean flow and periodic or channel boundary conditions in the transverse direction and looking for solutions in the form φi = Ai ei(k·x−ωt) , the condition of solvability of the algebraic equations for Ai gives for c = kωx (Pedlosky, 1982): U1 (k2 + 2h¯ 1 ) + U2 (k2 + 2h¯ 2 ) ± (U2 − U1 )2 (k4 − 4h¯ 1 h¯ 2 ) c= . (4.17) 2(k2 + h¯ 1 + h¯ 2 )

26

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Thus, the flow is unstable for perturbations with long enough wavelengths. This is the instability of balanced flow with respect to balanced slow perturbations, as all calculations are done within the framework of the balanced QG model. 4.5.2. Ageostrophic instabilities of the geostrophic flow The stability of the same initial configuration may be analyzed within the full 2-layer RSW equations. The corresponding stability analysis in the channel configuration with free-slip lateral boundaries was performed in the pioneering work by Sakai (1989) who reproduced the standard baroclinic instability for small Rossby numbers and small-k perturbations, whereas a new instability with respect to unbalanced perturbations arises at larger Rossby numbers and larger wavenumbers. It was called Rossby–Kelvin (RK) by Sakai. The classical Kelvin– Helmholtz (KH) instability of stratified shear flows arises at even larger Ro and smaller k. In Figures 4 and 5 the instability zones in the wavenumber – Froude number plane, and typical growth rates are shown. The results were obtained following Sakai in the configuration of equal layers depths by the collocation method.

Figure 4. The instability zones in the wavenumber – Froude number plane for the two-layer RSW flow in the channel with constant and different velocities in the upper and lower layer. The isolines of the growth rates are shown. The Froude number is defined as a ratio of Rossby and Burger numbers. The calculations are made following Sakai (1989) with greater resolution using the improved version of the spectral collocation method (Poulin and Flierl, 2003).

4. An overview of the two-layer RSW model

Figure 5.

27

The growth rates along the section 3 Fr = k on the previous figure. Note that the growth rate of the RK instability may exceed that of the standard baroclinic one.

There are different kinds of linear waves in the system: Rossby waves propagating due to the equivalent beta-effect produced by the inclined interface between the layers, Kelvin waves appearing due to the presence of lateral boundaries (see the preceding section), and IGW. The dynamical origin of the displayed instabilities is resonance between these waves propagating in each layer (Sakai, 1989): respectively, Rossby–Rossby for the standard baroclinic instability, Rossby–Kelvin for the RK instability, and Kelvin–Kelvin, or IGW–Kelvin, or IGW–IGW for KH instability. While the RK instability is produced by the presence of boundaries, which in any case destroy the spectral gap in the RSW (see Section 3.8), other ageostrophic instabilities persist even in the absence of boundaries. Apart from the classical KH one, characterized by high Rossby numbers, the “Rossby–Gravity” instability seen as the descending from the KH region branch in Figure 4 and corresponding to the resonance between the Rossby and the IG wave is presumably not sensitive to the presence of lateral boundaries. (Its typical growth rates are of the same order as for RK, not shown.) As in the one-layer RSW, the formal validity of the QG and FG equations and, hence, of the dynamical splitting of fast and slow variables in the two-layer RSW may be established by using the asymptotic expansions in Rossby number, see the next chapter. The corrections to these equations for longer times t ∼ Ro12 f

28

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

may be derived as well, although the fast component (waves) still does not influence the slow one at this time-scale. The just displayed ageostrophic instabilities of geostrophic flows indicate that there are limits to dynamical splitting of fast and slow variables at large Rossby numbers. Another obstacles for this splitting at large Rossby numbers are provided by wave–vortex interactions becoming effective in this limit, and in particular the Lighthill radiation.

5. Wave–vortex interactions As we have seen previously, at small Rossby numbers the vortex motions are decoupled from the wave motions. This is not longer the case at large Rossby numbers. Even in the absence of the baroclinic effects discussed in the previous section, waves and vortices do interact in this limit. We will consider below a specific wave emission by vortices, which is less familiar than the classical wavemean flow interactions due to the wave scattering on vortex structures, which we will sketch after. 5.1. Wave emission by vortex motions (Lighthill radiation) in the shallow water model The derivation of the QG and FD equations in the preceding sections indicates that, at least for small Rossby numbers, vortex and wave motions are dynamically split and non-interacting. It is, however, intuitively clear that vortex motions which stir the fluid at frequencies larger than the inertia–gravity waves threshold frequency f cannot avoid generating waves. We will show below that it is indeed the case. To introduce the phenomenon of wave radiation by vortex motions (Lighthill, 1952) in the shallow-water context we begin with the non-rotating shallow water model which is, as already said, equivalent to the two-dimensional acoustics. If the free surface is “frozen”, the motion obeys the two-dimensional Euler equations, which are equivalent to the vorticity equation (3.30). The elementary solutions of this equation are point vortices with the streamfunction given by (3.31) times the intensity κ of the vortex. Dynamics of arbitrary vorticity distributions can be modelled by representing them as superpositions of point vortices. The dynamics of a system of point vortices is essentially Lagrangian, because vorticity is a Lagrangian invariant of the two-dimensional Euler equations, and hence, any point vortex preserves its own intensity and is advected by the velocity field obtained by superposition of velocities due to the other vortices (cf. e.g. Lamb, 1932). A vortex pair is the simplest exact solution of the vorticity equation. If vortex intensities are not equal and opposite (the case of a steady moving vortex

5. Wave–vortex interactions

29

dipole), the vortices rotate around the center of vorticity with a constant angular speed: κ1 + κ2 . Ω= (5.1) 2πa 2 This vortex system has, thus, a single proper frequency and, following Gryanik (1983), we will use it to construct an archetype example of Lighthill radiation. The singular character of point vortices is not important in this context. The results are very similar if the distributed vorticity configuration is chosen, like an elliptic Kircchoff vortex (Zeitlin, 1991). We present below the technically simpler vortex pair example. For spatially two-dimensional problems, like shallow-water ones, it is convenient to introduce the complex coordinates on the (x, y)-plane: z = x + iy, z∗ = x −iy. The complex velocity field V = u−iv of a point vortex of intensity κ situated at point z0 may be described with the help of the complex potential φ(z): κ log(z − z0 ). V = ∂z φ, φ1 = i (5.2) 2π The complex potential of a vortex pair is obtained by superposition: κ2 κ1 log z − a1 eiΩt + i log z − a2 eiΩt , φ2 = i (5.3) 2π 2π where a1,2 are the algebraic distances of the vortices 1 and 2, respectively, from the origin chosen at the center of vorticity, i.e. a1 κ1 + a2 κ2 = 0,

(5.4)

and it is taken into account that the pair rotates uniformly with angular velocity (5.1). Introducing a, the distance between the vortices, and using the polar representation of z: z = reiθ , we obtain the following multipolar expansion of the complex potential (5.3) at distances much larger than the distance a between the vortices: 2 i κ1 κ2 κ1 + κ2 a φ2 |r→∞ = i e2i(Ωt−θ) + · · · . (5.5) log reiθ − 2π 4π κ1 + κ2 r Note that the dipolar term ∝ 1r is absent due to (5.4), and hence the first timedependent term in this expansion is quadrupolar. The vortex-pair solution is relevant for the full shallow water equations, provided the distance a between the vortices is much smaller than the typical wavelength of the gravity waves. In the context of the gravity-wave emission by the pair, the frequency of √ the wave is, obviously, Ω. Therefore, the wavelength is λ= Ω , where c = gH0 is the phase velocity of the gravity waves. Hence the 0 c0 characteristic Froude (Mach) number of the pair is M=

a aΩ , =√ λ gH0

(5.6)

30

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

and for small M the vortex-pair solution may be used at small distances from the origin. The linear gravity waves are solutions of equations (3.17) at f = 0. These equations can be easily reduced to a single equation for η in this case: ∂tt η − gH0 ∇ 2 η = 0.

(5.7)

Thus, in the absence of f the waves are strictly dispersionless (they are sound waves) and have constant phase velocity c0 . Note that there is no spectral gap anymore in the absence of rotation, although ω = 0 is still a solution of the dispersion relation corresponding to Lagrangian conservation of the (rotationless) PV. In what follows the flux of energy carried by the gravity waves is important. It may be obtained from the local form of the energy conservation law (3.2) in the lowest order in wave amplitudes: v2 η2 ∂t H0 + g (5.8) = −H0 ∇ · (vgη). 2 2 The total flux of energy due to waves over any closed contour on the plane can be obtained by integrating the r.h.s. of (5.8) with the help of the Stokes theorem. The physical picture of the presumable wave emission by the vortex pair is as follows. Close to the origin, where the pair is situated (inner region), the vortex pair solution is a good approximation for small M, cf. (5.6). Further off (the intermediate region) the vortex solution should be matched with the solution of the SW equation with non-frozen free surface. As the time-dependent part of the vortex solution rapidly decays with the distance to the origin, cf. (5.5), it is reasonable to take this full solution at small amplitude, i.e. in the form of gravity waves. In the outer region far from the vortex, the (non-stationary) solution is the outgoing gravity wave field carrying the energy towards infinity. If matching respecting these conditions is possible, the wave emission by the vortex is proved. Let us consider the wave solution in the outer region. In the polar coordinates which are convenient for description of the wave field produced by the localized source, η can be sought in the form: η(r, θ, t) = (5.9) ηn (r)ein(Ωt−θ) , n

and in terms of the new variable ρ = Bessel equation, as usual: 1 n2 ηn + ηn + 1 − 2 ηn = 0. ρ ρ

nΩ c0 r,

equation (5.7) takes the form of the

(5.10) (2)

The Hankel function of the second kind Hn is a solution corresponding to the outgoing waves with the energy flux (5.8) directed outward. This function has the

5. Wave–vortex interactions

following asymptotics (Abramowitz and Stegun, 1964): −n

i ρ , Hn(2) (ρ) ρ→0 → n! π 2

2 −i(ρ−n π − π ) (2) 2 4 . e Hn (ρ) ρ→∞ → πρ

31

(5.11) (5.12)

According to the general idea exposed above, the near wave field should be matched to the far vortex field. The continuity of pressure should be used for the purpose. (We do not pay attention to the monopolar time-independent part of the vortex field, which is the zero-mode solution of the wave equation (5.10), as well.) The pressure is the time derivative of the complex potential for the wave field. Hence the matching condition is

(vortex)

gη(waves) r→0 = −∂t φ2 (5.13) . r→∞ Comparing (5.5) with (5.11), we see that only the n = 2 harmonic should be present in the wave field, and the following choice of its amplitude ensures the matching (5.13): i κ1 κ2 (κ1 + κ2 )2 (2) 2Ω (waves) = H2 r e2i(Ωt−θ) . gη (5.14) c0 8π 3 c0 a4 The far-field asymptotics of the wave field can be easily calculated with the help of (5.12) giving the total energy flux at infinity ∝ M 4 (Gryanik, 1983). As was shown by Gryanik (1983) such energy loss results in adiabatic change of a (the backreaction of the wave radiation), the only variable parameter in the vortex system as the conservation of vortex intensities is guaranteed by the Kelvin circulation theorem, and gives rise to a slow separation of vortices of the same sign, and to their closing and collapse in finite time for vortices of opposite signs. A similar analysis was performed by Zeitlin (1991) for the distributed Kircchoff vortex. The behavior close to that of the vortex pair of the same sign vortices was demonstrated, with an important new phenomenon of stability loss by the vortex due to the wave radiation. The phenomenon of destabilization of vortices by wave radiation at further stages was investigated by Ford (1994). Thus, the process of wave-emission by vortices, although weak, exists and its backreaction may lead to qualitative changes in the vortex system at long enough times. What changes in the presence of rotation? In the rotating case at large Burger numbers the vortex dynamics is still described by equation (3.30), cf. (3.33) and the discussion of the Rossby waves in Section 3. Hence, if the same reasoning as above is to be applied to the rotating case, the vortex part may be considered the same if its characteristic scale is much less than the deformation radius. The ratio of the double vortex pair turnover

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

32

frequency to f is the Rossby number of the system: Ro2 =

2Ω . f

(5.15)

In what concerns the wave part, equations (3.17) are equivalent to a single equation for η: −

1 ∂ 2η − 2 η + ∇ 2 η = 0, ∂t 2 Rd

(5.16)

where the deformation radius is purposedly reintroduced in explicit form. If solutions of this system are sought in the form (5.9), the same equation (5.10) results with a modified variable ρ for the nth angular mode: nΩ nΩ ρ= (5.17) r Ro2n − 1; Ron = , c0 f where the Rossby number Ron corresponding to a given mode is introduced. An immediate consequence of this formula is that the propagating waves exist only for Ron > 1, which is consistent with the existence of the spectral gap (see discussion in Section 3). A second conclusion is that the same procedure as in the non-rotating case above (up to the constant arising in (5.17)) will allow to match the near wave field with the far field of the vortex at r such that a r Rd , and thus obtain the wave-radiation from the vortex. Therefore, the necessary condition of the wave radiation is Ro > 1, while Bu 1 is a sufficient condition. The technical condition M 1 appearing in the irrotational argument becomes Bu Ro22 , as it is easy to see. This simple example illustrates the mechanism of Lighthill radiation in the RSW model. A more elaborated study, with similar results, was performed by Ford, McIntyre and Norton (2000, 2002). The role of the spectral gap in suppression of this process at small Ro was emphasized by Saujani and Shepherd (2002). 5.2. Brief comments on the wave–mean flow interactions The Lighthill radiation is an example of how vortices engender waves at high Rossby numbers. Note that this mechanism is purely dissipationless. The opposite, i.e. vorticity generation by waves is also possible at large Rossby numbers, although it happens through dissipative effects. Indeed, as was explained at the beginning of this chapter, waves do not carry the PV anomaly. Hence, an explicit source in the PV conservation equation (3.4) is needed to produce it. Such source can be provided by enhanced dissipation due to the wave breaking. As was already stressed, the RSW model is equivalent to the (rotating) gas dynamics. Hence, shock formation can be expected for waves of large enough amplitude (and, hence, necessarily of high enough Rossby numbers), or for strong enough

6. RSW model on the equatorial β-plane

33

initial gradients. Of course, formally speaking the RSW model, e.g. as it was derived earlier in this chapter from the primitive equations, reaches the limits of its validity at the threshold of shock formation. Nevertheless, extending, as usual, the model to include weak solutions allows to treat the shocks. Those may be considered just as zones of enhanced dissipation and mixing (cf. Whitham, 1974). It is known (e.g. Lighthill, 1978) that shock fronts of variable strength and/or curvature lead to local production (and global redistribution) of vorticity. This process in the RSW context was analysed by Bühler (2000). It will be studied in detail below in Chapter 5, where, in particular, it will be proved that rotation does not prevent shock formation and specific wave-breaking patterns of equatorial waves will be presented. The second wave-breaking mechanism is wave encounter with critical layers, i.e. the locations where the phase-speed of the wave is equal to the local speed of the mean flow (e.g. Lighthill, 1978). Indeed, linearizing the RSW equations about nontrivial coordinate-dependent mean (vortex) flow shows that there are non-removable singularities at these locations, and dissipative effects are necessary to treat them. The momentum exchange between the wave-field and the mean flow due to the presence of critical layers is at the origin of fundamental atmospheric and oceanic processes (e.g. Holton, 1979). They are, however, out of the scope of the present book. (For a comprehensive introduction into the subject see Bühler, 2005.) It should be noted that dissipationless wave–vortex interactions, leading to changes in vortex flow due to the wave propagation, are also possible within the shallow-water dynamics, as was shown by Bühler and McIntyre (2003). This mechanism was not studied in the presence of rotation but, presumably, the vortex screening due to finite deformation radius will make it effective only at high enough Burger and Rossby numbers, as it was the case with Lighthill radiation.

6. RSW model on the equatorial β-plane 6.1. General properties of the equatorial RSW The tangent plane approximation to the atmospheric and oceanic motions on the sphere, which is at the base of the primitive and, as a consequence, of the RSW equations has particular properties on the equator. Indeed, the axis of rotation lying in the equatorial tangent plane, the Coriolis parameters has no constant part and its expansion in coordinates starts from βy. Thus, the equatorial β-plane RSW equations are: ∂t v + v · ∇v + βy zˆ ∧ v + g∇h = 0,

(6.1)

∂t h + ∇ · (vh) = 0.

(6.2)

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

34

The absence of f0 necessitates a redefinition of the deformation radius, which becomes Re =

(gH0 )1/4 , √ β

(6.3)

which is a natural scale in the meridional (y-) direction: Ly ∼ Re . Due to the x–y anisotropy of equations (6.1), (6.2), an important parameter is the aspect ratio of the equatorial motions: δ=

Ly . Lx

(6.4)

The nonlinearity parameter may be introduced, as usual, via the ratio of the typical perturbation of the free surface to its unperturbed position: λ=

H . H0

(6.5)

Natural time- and velocity scales are related to Re : T =

1 ; βRe

U=

gH . βRe2

(6.6)

In terms of this velocity scale, the nonlinearity parameter has also a meaning of the Rossby or Froude number: λ=

U U = Ro = √ = Fr. 2 βRe gH0

(6.7)

Another consequence of the absence of f0 at the equator is that the natural geostrophically balanced states are zonal flows with v = 0, as the inspection of Coriolis and pressure terms in (6.1), (6.2) shows. Hence the notion of slow vortex dynamics, as it was used on the midlatitude tangent plane, is not quite adapted to the equatorial region. On the contrary, the wave dynamics is very rich. The equatorial dynamics is basically the wave dynamics. As will be shown below and in Chapter 5, the fast-slow separation takes place at the equator between the fast and the slow waves, which is distinctively seen in particular in the long-wave sector, i.e. for δ 1. 6.2. A reminder on equatorial waves and their dispersion properties in the RSW model We remind below the standard linear theory of the equatorial waves (Gill, 1980; Majda, 2003) with special attention paid to initialization of different types of waves (Le Sommer, Reznik and Zeitlin, 2004), which will be used later in Chapter 5 of the present volume.

6. RSW model on the equatorial β-plane

35

The non-dimensional linearized RSW equations are: ut − yv + hx = 0,

(6.8)

vt + yu + hy = 0,

(6.9)

ht + ux + vy = 0.

(6.10)

(We use the subscript notation for partial derivatives hereafter.) Wave solutions are sought in the equatorial waveguide, by imposing exponential decay boundary condition in the meridional y-direction. The complete base of parabolic cylinder functions φn (y) (Abramowitz and Stegun, 1964): φn (y) + 2n + 1 − y 2 φn (y) = 0, (6.11) Hn (y)e−y /2 φn (y) = √ , 2n n! π 2

(6.12)

where Hn are Hermite polynomials, is used while separating independent variables x, t and y. 6.2.1. Kelvin waves By a change of variables (Gill, 1982): 1 1 (u + h); g = (u − h) 2 2 equations (6.8)–(6.10) simplify: f =

(6.13)

1 ft + fx + (vy − yv) = 0, 2 1 gt − gx − (vy + yv) = 0, 2 vt + y(f + g) + (f − g)y = 0.

(6.14) (6.15) (6.16)

They admit a particular solution with v = 0 and f = F (x − t, y);

g = G(x + t, y).

(6.17)

For solutions bounded in the meridional direction G = 0 and the equatorial Kelvin wave solution is given by u = F0 (x − t)e−y

2 /2

;

h = F0 (x − t)e−y

2 /2

;

v = 0.

(6.18)

In the context of initial-value problems (geostrophic adjustment) the function F0 is determined by expanding the initialcondition fI = 12 (uI + hI ) in a series in parabolic cylinder functions: fI = ∞ n=0 fIn φn (y). Hence, the function F0 is

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

36

Figure 6.

The Kelvin mode.

given by 1 F0 (x) = fI0 (x) = 2

+∞ (uI + hI )φ0 (y) dy.

(6.19)

−∞

The initial conditions for the Kelvin-wave part of arbitrary initial perturbation uI , vI , hI are (K)

uI

= fI0 φ0 ,

(K)

hI

= fI0 φ0 ,

(K)

vI

= 0.

(6.20)

The velocity and height pattern corresponding to the Kelvin wave is presented in Figure 6. 6.2.2. Yanai waves The Yanai-wave solution corresponds to g = 0, f = 0, v = 0. In this case from (6.8)–(6.10) it follows that 1 ft + fx + (vy − yv) = 0, 2 vy + yv = 0,

(6.21) (6.22)

6. RSW model on the equatorial β-plane

37

vt + yf + fy = 0,

(6.23)

v = v0 (x, t)φ0 (y),

(6.24)

and by expanding the function f in series in φn we get f = F1 (x, t)φ1 (y).

(6.25)

Hence 1 F1t + F1x − √ v0 = 0, 2

v0t +

√ 2F1 = 0.

(6.26)

The initial conditions for the Yanai-wave part of arbitrary initial perturbation are 1 (uI + hI1 )φ1 , 2 1

(Y)

=

(Y)

= vI0 φ0 ,

uI vI

(Y)

hI

=

1 (uI + hI1 )φ1 , 2 1 (6.27)

where, as above, we suppose that initial conditions are expanded in φn . In terms of the variable v the Yanai wave is a solution of the following problem, cf. (6.24)–(6.26), (6.27): (Y) (Y) vtt(Y) + vxt + v (Y) = 0, vI = vI0 φ0 , 1 (Y) vt t=0 = − √ (uI1 + hI1 )φ0 . 2

(6.28)

The solution has the form (6.24) with v0 given by v0 =

√ √ |σ2 |v˜I0 − i 2f˜I1 i(kx−σ1 t) σ1 v˜I0 + i 2f˜I1 i(kx+|σ2 |t) e e dk + dk , √ √ k2 + 4 k2 + 4 (6.29)

where k σ1 = + 2

k2 + 1, 4

k |σ2 | = − + 2

k2 + 1, 4

and we defined the Fourier-transforms of the initial fields: 1 ikx vI0 (x) = dk v˜I0 (k)e , (uI + hI1 ) = dk f˜I1 (k)eikx . 2 1

(6.30)

(6.31)

The velocity and height pattern corresponding to a Yanai wave is presented in Figure 7.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

38

Figure 7.

The lowest westward-propagating Yanai mode.

6.2.3. Rossby and inertia–gravity waves The use of the full equation for the v-field which follows from (6.8)–(6.10) by eliminating u and h is necessary for determining this part of the spectrum: ∂t v − y 2 v − ∂tt v + ∂x v = 0.

(6.32)

The three initial conditions for this equation are: (RG)

v|t=0 = vI

, (RG)

vt |t=0 = − yuI (RG)

vtt |t=0 = vIyy

(RG)

+ hIy

(RG)

− y 2 vI

,

(RG) (RG) , + ∂x uIy + yhI

(6.33)

where (RG)

uI

(K)

= uI − uI

(Y)

− uI

∞

=

1 1 uIn (x)φn (y), (uI0 − hI0 )φ0 + (uI1 − hI1 )φ1 + 2 2 n=2

(6.34)

6. RSW model on the equatorial β-plane (RG)

hI

(K)

= hI − hI

39

(Y)

− hI

∞

1 1 = − (uI0 − hI0 )φ0 − (uI1 − hI1 )φ1 + hIn (x)φn (y), 2 2

(6.35)

n=2

(RG)

vI

(Y)

= vI − vI

=

∞

(6.36)

vIn (x)φn (y).

n=1

By expanding v in a series in φn , v = n vn (x, t)φn (y) we get for vn 2 ∂t ∂xx vn − (2n + 1)vn − ∂tt2 vn + ∂x vn = 0, while for its Fourier-transform v˜n (k, t) = dx e−ikx vn (x, t) we get 3 ∂ttt v˜n + k 2 + 2n + 1 ∂t v˜n − ik v˜n = 0.

(6.37)

(6.38)

Hence v˜n = vn1 (k)e−iσn1 t + vn2 (k)e−iσn2 t + vn3 (k)e−iσn3 t where σnα , α = 1, 2, 3, are the three roots of the dispersion equation σn3α − k 2 + 2n + 1 σnα − k = 0.

(6.39)

(6.40)

By introducing the projections vIn , vtIn , vttIn of the initial conditions vI , vtI , vttI , respectively, onto the corresponding φn , and their Fourier-transforms, we get the following system of algebraic equations for vn(α) 3

vnα = v˜nI ,

α=1

3

σnα vnα = iv˜tnI ,

α=1

3

σn2α vnα = −v˜ttnI .

(6.41)

α=1

The lowest eigenvalue σ1 defines the Rossby mode, the other two—the inertia– gravity wave modes. For the Rossby mode we get (1)

v1 =

1 2 k v ˜ . + iσ v ˜ − σ v ˜ n t n tt n n n I 1 1 I I 2σn31

(6.42)

Therefore, the part of the initial conditions corresponding to this wave may be determined, ∞ φn (y) dk eikx vn1 (k) vI = (6.43) n=1

and similarly for the gravity-wave modes. The Rossby wave and inertia–gravity velocity and height distributions are presented in Figures 8–10. Note that dispersion relations for Kelvin and Yanai modes follow from (6.40) at n = −1 and n = 0, respectively, while the spectrum of Rossby and inertia–gravity waves starts at n = 1.

40

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Figure 8.

The lowest Rossby mode.

Figure 9. The lowest eastward-propagating inertia–gravity mode.

7. Concluding remarks

Figure 10.

41

The lowest westward-propagating inertia–gravity mode.

6.2.4. Dispersion and spectral separation of equatorial waves The dispersion curves for lowest meridional eigenmodes of the equatorial waves are displayed in Figure 11 with special emphasis on the long waves, whose dynamics will be studied in more detail in Chapters 2, 5 of the present volume. The spectral gap between the fast and the slow long waves is indicated. The curves show the spectral separation between the long Rossby and Kelvin waves, which represent the slow motion in the equatorial waveguide, and the long Yanai and inertia–gravity waves, representing the fast motion.

7. Concluding remarks We hope that it is clear from the preceding material that the RSW model is a useful conceptual tool for understanding fundamental processes on GFD. It should be emphasized that the model is far from being academic and some crucial advances in atmospheric and ocean dynamics were achieved with its help. Without reviewing them, which is an impossible task in the limited volume, let us mention as an example a classical paper by Gill (1980), treating tropical convection with the help of the RSW model. Another example of utmost importance is the use of RSW model for developing the initialization algorithms for weather prediction (Baer and Tribbia, 1977; Machenhauer, 1977, see also Chapter 3 of this volume).

42

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

√ Figure 11. Dispersion curves for equatorial waves. c = gH0 , k is the zonal wavenumber. The domain of long waves and the spectral gap between fast and slow waves is indicated. Only two first Rossby modes and two first inertia–gravity modes are shown.

Finally, let us mention for completeness that models beyond the standard RSW may be derived by vertical averaging of the full three-dimensional equations. Thus, starting from the three-dimensional non-hydrostatic Euler equations the socalled Green–Naghdi (GN) equations may be obtained in the leading order in small deviations from hydrostaticity by using the variational principle and vertical averaging (Miles and Salmon, 1985); these equations were rediscovered many times, see Camassa, Holm and Levermore (1996) for the history of the question. The GN equations take into account the dispersive effects for short gravity waves which are absent in RSW. The small-amplitude limit of the GN equations gives the so-called “great lake” equations (Camassa, Holm and Levermore, 1996). Although the GN equations do give useful insights in some circumstances (see e.g. Nadiga, Margolin and Smolarkevich, 1996), the SW model remains a universal leading-order approximate model, which is, in addition, “computation-friendly” (see Chapter 4 of this volume). In the subsequent chapters some recent theoretical, computational, and experimental advances in understanding the RSW dynamics will be presented.

References

43

Acknowledgements The author thanks J. Gula and J. Le Sommer for help with calculations and preparation of illustrations. Part of the presentation stems from the collaboration with G. Reznik which is gratefully acknowledged.

References Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions. Dover, New York. Allen, J.S., Holm, D.D., 1996. Extended geostrophic Hamiltonian models for rotating shallow water motion. Physica D 98, 229–248. Baer, F., Tribbia, J.J., 1977. On complete filtering of gravity modes through nonlinear initialization. Monthly Weather Rev. 105, 1536–1539. Benilov, E.S., Reznik, G.M., 1996. The complete classification of large-amplitude geostrophic flows in a two-layer fluid. Geophys. Astrophys. Fluid Dynam. 82, 1–22. Blumen, W., 1972. Geostrophic adjustment. Rev. Geophys. Space Phys. 10, 485–528. Bokhove, O., 2005. Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations. Appl. Math. Lett. 18, 1418–1425. Bühler, O., 2000. On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263. Bühler, O., 2005. Wave-mean interaction theory. In: Grimshaw, R. (Ed.), Nonlinear Waves in Fluids: Recent Advances and Modern Applications. Springer-Verlag, Vienna/New York. Bühler, O., McIntyre, M.E., 2003. Remote recoil: a new wave–mean interaction effect. J. Fluid Mech. 492, 207–230. Camassa, A., Holm, D.D., Levermore, C.D., 1996. Long-time effects of bottom topography in shallow water. Physica D 98, 258–286. Charney, J.G., Flierl, G.R., 1981. Oceanic analogs of large-scale atmospheric motions. In: Warren, B.A., Wunsch, C. (Eds.), Evolution of Physical Oceanography. MIT Press, Cambridge, MA, pp. 504–548. Cushman-Roisin, B., 1986. Frontal geostrophic dynamics. J. Phys. Oceanogr. 16, 132–143. Cushman-Roisin, B., Sutyrin, G.G., Tang, B., 1992. Two-layer geostrophic dynamics. Part I: Governing equations. J. Phys. Oceanogr. 22, 117–127. Dritschel, D.G., 1997. Introduction to “contour dynamics for the Euler equations in two dimensions”. J. Comp. Phys. 135, 217–219. Falkovich, G.E., Medvedev, S.B., 1992. Kolmogorov-like spectrum for turbulence of inertia–gravity waves. Europhys. Lett. 19, 279–284. Ford, R., 1994. The response of the rotating ellipse of uniform potential vorticity to gravity wave radiation. Phys. Fluids 6, 3694–3704. Ford, R., McIntyre, M.E., Norton, W.A., 2000. Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57, 1236–1254. Ford, R., McIntyre, M.E., Norton, W.A., 2002. Reply to Saujani and Shepherd. J. Atmos. Sci. 59, 2878–2882. Gill, A.E., 1980. Some simple solutions for heat-induced tropical circulation. Quart. J. R. Meteorolog. Soc. 106, 447–462. Gill, A.E., 1982. Atmosphere–Ocean Dynamics. Academic Press, New York. Gryanik, V., 1983. Emission of sound by linear vortex filaments. Izvestija—Atmosphere Ocean Phys. 19, 150–153.

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Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A., 1985. Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116. Holm, D.D., Zeitlin, V., 1998. Hamilton’s principle for quasi-geostrophic dynamics. Phys. Fluids 10, 800–806. Holton, J.R., 1979. An Introduction to Dynamic Meteorology. Academic Press, New York. Hoskins, B.J., Bretherton, F.P., 1972. Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29, 11–37. Jeffreys, H., 1925. On the dynamics of geostrophic winds. Quart. J. R. Meteorolog. Soc. 52, 85–104. Lamb, H.H., 1932. Hydrodynamics. Dover, New York. Larichev, V.D., Reznik, G.M., 1976. 2-dimensional solitary Rossby waves. Sov. Phys. Dokl. 231, 1077–1079. LeBlond, P.H., Mysak, L.A., 1978. Waves in the Ocean. Elsevier, New York. Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170. Lighthill, J., 1952. On sound generated aerodynamically, I. General theory. Proc. R. Soc. London A 211, 564–571. Lighthill, J., 1978. Waves in Fluids. Cambridge Univ. Press, Cambridge. Machenhauer, B., 1977. On the dynamics of gravity oscillations in a shallow water model, with application to normal-mode initialization. Contrib. Atmos. Phys. 50, 253–271. Majda, A.J., 2003. Introduction to PDEs and Waves for the Atmosphere and Ocean. Amer. Math. Soc., New York. Miles, J., Salmon, R., 1985. Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531. Nadiga, B.T., Margolin, L.G., Smolarkevich, P.K., 1996. Different approximations of shallow fluid flow over an obstacle. Phys. Fluids 8, 2066–2077. Nezlin, M.V., Snezhkin, E.N., 1993. Rossby Vortices, Spiral Structures, Solitons. Astrophysics and Plasma Physics in Shallow Water Experiments. Springer-Verlag, Berlin. Obukhov, A.M., 1949. On the problem of geostrophic wind. Izvestija—Geography and Geophysics 13, 281–306 (in Russian). Pedlosky, J., 1982. Geophysical Fluid Dynamics. Springer-Verlag, New York. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286. Poulin, F.J., Flierl, G.R., 2003. The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173–2192. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part I. Rotating shallow-water model. J. Fluid Mech. 445, 93–120. Romanova, N.N., Zeitlin, V., 1984. On quasigeostrophic motions in barotropic and baroclinic fluids. Atmos. Ocean Physics—Izvestija 20, 85–91. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res. 1, 239–263. Sakai, S., 1989. Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176. Salmon, R., 1998. Geophysical Fluid Dynamics. Oxford Univ. Press, New York. Saujani, S., Shepherd, T.G., 2002. Comments on “Balance and the slow quasimanifold. Some explicit results”, by Ford, McIntyre and Norton. J. Atmos. Sci. 59, 2874–2877. Stegner, A., Zeitlin, V., 1995. What can asymptotic expansions tell us about large-scale quasigeostrophic anticyclonic vortices? Nonlin. Proc. Geophys. 2, 186–193. Stegner, A., Zeitlin, V., 1998. From quasi-geostrophic to strongly nonlinear monopolar vortices in paraboloidal shallow-water-layer experiment. J. Fluid Mech. 356, 1–24. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Williams, G.P., Yamagata, T., 1984. Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci. 41, 453–478.

References

45

Zeitlin, V., 1991. On the back-reaction of acoustic radiation for distributed vortex structures. Phys. Fluids A 3, 1677–1680. Zeitlin, V., Reznik, G.M., Ben Jelloul, M., 2003. Nonlinear theory of the geostrophic adjustment. Part II: Two-layer and continuously stratified models. J. Fluid. Mech. 491, 207–228.

Chapter 2

Asymptotic Methods with Applications to the Fast–Slow Splitting and the Geostrophic Adjustment G.M. Reznik P.P. Shirshov Institute of Oceanology, Moscow, Russia

V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction

48 48 54

1.1. Basic notions and general context 1.2. A synopsis of the method

2. Nonlinear geostrophic adjustment in the unbounded domain. Onelayer RSW 2.1. The QG regime 2.2. The FG regime 2.3. Comments on the influence of the β-effect

3. Nonlinear geostrophic adjustment in the unbounded domain. Twolayer RSW 3.1. Two-layer RSW model, a reminder 3.2. The QG regime 3.3. The FG regime

55 56 65 70 73 74 75 77

3.4. Discussion of the geostrophic adjustment in the two-layer RSW and comparison with the one-layer case

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary 4.1. Preliminaries 4.2. The lowest-order solution (the linear adjustment) 4.3. Dynamics of the lowest-order slow motion 4.4. The first-order solution Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02002-8 47

79 80 80 83 88 91

© 2007 Elsevier B.V. All rights reserved

48

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting 4.5. Improved QG equation in the presence of the lateral boundary 4.6. The problem of mass conservation in the localized case 4.7. The summary of the results on the adjustment in the laterally bounded domain

5. Nonlinear geostrophic adjustment in the equatorial region 5.1. The long-wave scaling and classification of possible dynamical regimes 5.2. The resumé of possible weakly nonlinear dynamical regimes in the long-wave approximation 5.3. The long-wave adjustment at Ro = O(δ 2 )

6. Summary and discussion Acknowledgements References

97 99 104 106 106 110 110 116 118 118

1. Introduction 1.1. Basic notions and general context The one- and multi-layer rotating shallow water models, as well as their parent primitive equation model, possess a gap in the spectrum of linear excitations which separate the fast inertia–gravity wave (IGW) motions and slow vortex motions (see Chapter 1 of this volume). The vortex motions at small Rossby numbers are in geostrophic balance, i.e. close to the equilibrium between the Coriolis force and the pressure force. As discussed in Chapter 1, already these observations allow to anticipate the ubiquitous character of the geostrophic adjustment process i.e. the process of relaxation of arbitrary initial perturbations to a geostrophic equilibrium state via emission of inertia–gravity waves. The oceanic and atmospheric data provide the confirmation of this fact (Blumen, 1972). At the same time, these properties of the spectrum are at the origin of the notion of dynamical splitting of the fast and slow variables. Under hypothesis of splitting, i.e. (almost) noninteraction of fast and slow motions, useful balanced models may be derived from the full dynamical equations by using exclusively slow time-scales. The QG and FG models discussed in Chapter 1 arise in this way. Their advantage is that dynamics is reduced to that of the balanced component alone, which allows for conceptual and computational simplifications, as compared to the full original (“primitive”) equations. Especially the QG model is of wide and frequent use in applications ranging from the analysis of particular oceanic and atmospheric processes to studies of global climate dynamics. It is worth mentioning that the first successful weather forecasts were accomplished with the help of the QG model. However, the QG model is not exact, as ageostrophic velocity component (cf. Chapter 1) gives rise to corrections at higher orders in Rossby number. Going beyond the standard QG, but preserving the concept of balance (possibly modified with respect to the simplest geostrophic balance) is the philosophy of the entire

1. Introduction

49

branch of research in GFD dedicated to balance and balanced models. Reviewing this activity is beyond the scope of the present chapter. We will, however, briefly expose the main ideas in order to put in this context the results of the direct asymptotic approach, which will be developed further in this chapter. We illustrate the balanced models approach using the simplest one-layer RSW model. We remind that the equations of the model are: ∂t v + v · ∇v + f zˆ ∧ u + g∇H = 0,

(1.1)

∂t H + ∇ · (vH ) = 0.

(1.2)

The Lagrangian conservation of potential vorticity (PV), or its anomaly, q=

ζ +f f , − H H0

(1.3)

is expressed by ∂t q + v · ∇q = 0.

(1.4)

The PV anomaly proves to be often more useful than the full PV (we will make no distinction between the two when it does not lead to confusion). Here ζ = ∂x v − ∂y u is the relative vorticity, H = H0 + h(x, y, t) and H0 are the full fluid depth and its unperturbed value, respectively (no topographic effects will be considered in this chapter). Another useful quantity is the linearized potential vorticity (LPV) which is equal to Ω=ζ−

f h H0

(1.5)

and satisfies in the f -plane approximation the conservation equation ∂t Ω + ∇ · (vΩ) = 0.

(1.6)

As was shown in Chapter 1, the spectrum of linear perturbations in RSW consists of zero-frequency vortex mode and the IGW spectrum: ω = ± gH0 k2 + f 2 . (1.7) The solution of the linearized RSW equations can be represented as a sum (v, h) = (vg , hg ) + (va , ha ),

(1.8)

where (vg , hg ) correspond to the vortex mode and satisfy the geostrophic balance relation: f zˆ ∧ vg + g∇hg = 0.

(1.9)

50

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

The ageostrophic part represents the fast IGW and is characterized by zero PV-anomaly and zero LPV: qa = Ωa = 0.

(1.10)

If at the initial moment PV is zero then the slow geostrophic part of the solution is absent and, vice versa, if the initial conditions are geostrophically balanced then the fast IGW do not appear in the linearized model. As to the full nonlinear system (1.1), (1.2), the condition of zero PV completely filters out the vortex component, just as in the linear case. Under this condition equations of conservation of PV and LPV are satisfied identically and the resulting equations describe uniquely the fast ageostrophic motion (Falkovich, 1992; Falkovich and Medvedev, 1992). On the contrary, the complete filtering of the wave motions from the non-linear system (1.1), (1.2) is generally impossible (excluding, of course, the case of exact stationary vortex solutions). The simplest way to see this is to consider equations for the velocity divergence D and the so called geostrophic departure ξ , sometimes called ageostrophic vorticity (cf. e.g. Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999; Mohebalhojeh and Dritschel, 2001): D = ∇ · v,

ξ = f ζ − g∇ 2 h.

(1.11)

Both are equal to zero for the linear vortex mode and non-zero for linear IGW. Equations for D, ξ show that even if both quantities are zero at the initial moment, the nonlinear self-interactions of the slow vortex mode generate these fields at subsequent times. However, in the practically important case when the typical Rossby number is small: Ro =

U = ε 1, f0 L

(1.12)

where the typical horizontal velocity scale is U , and the typical horizontal length scale is L, thus induced IGW contributions are small. The smallness of the Rossby number means smallness of the ratio of the typical wave period to the vortex advective time-scale (see Chapter 1). The QG model derived along the standard lines for small Rossby numbers in Chapter 1 may be introduced in an alternative way. Assuming that all fields change at the slow time-scale L/U f −1 , equations (1.1), (1.5) give, up to small corrections: f zˆ ∧ v + g∇h = 0, h=H

−1

Ω,

(1.13) (1.14)

equation (1.6) remaining unchanged. Here H is the Helmholtz operator ∇ 2 −

1 Rd2

and H−1 is its inverse (proper boundary conditions are understood), Rd =

1. Introduction √

51

gH0 f

is the Rossby deformation radius. The first of these equations represents the balance condition, the second gives the diagnostic relation between the height field and the prognostic variable Ω. The evolution equation for Ω is provided by the conservation law (1.6). The QG equations rewritten in the form (1.6), (1.13), (1.14) give an archetype example of the balanced model consisting, generally, of two components (Warn, Bokhove, Shepherd and Vallis, 1995): • Diagnostic balance relations filtering out the fast oscillations (in the present case this is the geostrophic relation (1.13)). • Prognostic equations determining the slow time evolution (in the present case this is equation (1.6) for LPV). Vast literature exists on balance models and their properties (see e.g. Hoskins, 1975; Gent and McWilliams, 1983; Pedlosky, 1984; Allen, 1993; Warn, Bokhove, Shepherd and Vallis, 1995; Mohebalhojeh and Dritschel, 2001, and references therein). Balance conditions and diagnostic relations vary in different models. Reviewing this literature is beyond the scope of the present chapter. Let us only mention in this context that the QG system (1.6), (1.13), (1.14) contains no IGW contributions at all, since the functions D, ξ in (1.11) are identically zero there. However, in the higher-order QG models (e.g. Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999) the fields D, ξ are, generally, non-zero and in this sense the contribution of IGW in the balanced dynamics is also non-zero. To comment on this important point we note that “slowness” of a function generally does not mean the absence of harmonics with arbitrarily high frequency in its spectrum. For example,√the Fourier transformation of the function exp(−ε 2 t 2 ) is equal to exp(−ω2 /ε 2 )/ 2ε, i.e. all harmonics are present in the spectrum of this function. For small ε 1, however, the contribution of harmonics with ω = O(1) is very small, of the order of exp(−1/ε 2 ). Similarly, the slow QG dynamics, generally, contains the contributions from the fast IGW but these contributions are exponentially small and do not appear in the straightforward asymptotic expansions in the Rossby number (see in this context also Saujani and Shepherd, 2002, and references therein). Note that the notion of slowness of the motion we used in the previous argument is well defined only for small Rossby numbers. In the jargon of balanced models the slow “prognostic” variable is called slaving variable (Warn, Bokhove, Shepherd and Vallis, 1995) or master variable (Vautard and Legras, 1986), and the variables determined from the geostrophic relation—the slaved or subordinate variables. The choice of the slaved and slaving variables is not unique. In principle any variable can be taken as a slaving one (Warn, Bokhove, Shepherd and Vallis, 1995). The general idea (applicable to general dynamical systems) is that the slaving variables are described by

52

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

non-stationary equations, and that slaving and slaved variables are related by diagnostic balance relations. In symbolic form these relations can be written in the form (Warn, Bokhove, Shepherd and Vallis, 1995): F (x, y) = 0,

(1.15)

where the slaved y and the slaving x variables constitute a complete set of the dependent variables. Equation (1.15) is called superbalance condition (Lorenz, 1980) and determines some invariant manifold in (generally, infinite-dimensional) phase space x, y. Note that in general the superbalance condition (1.15) does not require smallness of any parameter of the considered problem, i.e. some “fast” motions can belong to the balanced manifold. However, if the slaving variables are slow then the invariant manifold determines a set of slow solutions of the systems. In the RSW case one would choose x = Ω and try to find the balance conditions in the form: y = Y(Ω),

(1.16)

where y corresponds to the horizontal velocity v, or to the pair D, ξ , and Y(Ω) is some functional. In the limit ε → 0, (1.16) together with (1.6), guarantees the slowness of motion. If the slow manifold could be found, the problem of initialization, i.e. that of constructing the initial conditions for the full non-balanced equations, in such a way that the contribution of the fast IGW is minimized, would be solved. Needless to say that such initialization is of extreme importance in the meteorological applications (see the pioneering works of Baer and Tribbia, 1977 and Machenhauer, 1977 which used the RSW model for constructing the initialization algorithms). The problem with this tempting approach is that the superbalance conditions (1.15) can be written and analyzed in explicit form only for a finite dimensional systems of ordinary differential equations (cf. Vautard and Legras, 1986; Bokhove and Shepherd, 1996). Such systems may be obtained, for example, in the case of a doubly-periodic motion when each variable is expanded in a double Fourier series which is truncated then in a proper way (see e.g. Lorenz, 1986). In the general case of a system with infinite number of degrees of freedom the existence of explicit conditions of the type (1.15), and therefore the existence of the invariant slow manifold, is not proved. One can, however, construct approximate balance equations for small Rossby numbers. To do this the special asymptotic technique is used when the slaved variables are expanded in the series in ε, and the slaving variables are not (see Chapter 3 of this volume and Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999). The condition (1.16) is then represented in the form: y = Y0 (Ω) + εY1 (Ω) + · · · ,

(1.17)

1. Introduction

53

where Yi (Ω), i = 0, 1, . . . , are some integro-differential operators. Convergence of this series would mean that an invariant slow manifold exists. Unfortunately, nothing is known about such convergence. However, using the expansion (1.17) one can construct a hierarchy of balanced models where the influence of the IGW is minimized. In this case the n-term expansion (1.17) provides the absence of IGW contributions within accuracy O(ε n+1 ) at least on times of the order of (f ε)−1 . We, thus, see that the problem of existence of the balanced slow manifold, which is, hypothetically, the endpoint of the geostrophic adjustment process, remains open. Some exact results may be obtained in this context within the simplified models using strictly rectilinear frontal configurations, see Chapter 5 of the present volume. Even if the slow manifold does exist, a question of its attainability in course of the geostrophic adjustment process arises (McWilliams, 1988). This is a question to be answered by studying the joint evolution of balanced and unbalanced components of motion in the relaxation (i.e. adjustment) process. The question of possible influence of the unbalanced (fast) component upon the balanced (slow) one, i.e. the question of eventual fast component drag, if the nonlinear effects are taken into account, is of primary importance in the context of such joint evolution. We will show below how the straightforward approach based on multiple time-scale expansions in Rossby number and on timeaveraging allows to answer these questions. The slow manifold will be described order by order in Rossby number with initialization of the slow fields well defined at each order. Up to the fourth order in Rossby number the problem of nonlinear geostrophic adjustment of single-scale (vortex-like) perturbations will be fully resolved. The advantage of such approach is that balance nor diagnostic relations are never imposed ad hoc. They arise from the time-averaging and removal of resonances in the equations of the fast motion. The whole procedure is thus completely algorithmic. The key element of the approach is the time-scale separation, which is well defined for small Rossby numbers because, as already explained, the Rossby number gives the ratio of fast to slow time-scales. There is no need to separate the flow in vortex and wave parts, which can not be done unambiguously in the presence of nonlinear interactions—only the well-defined separation in slow and fast components should be applied. To have a unique Rossby number characterizing the motion an assumption of a single spatial scale will be made. We will consider below the nonlinear geostrophic adjustment in unbounded domains, and then show what are the modifications introduced by the presence of lateral boundaries (as shown in Chapter 1, lateral boundaries destroy the spectral gap). A particular case of the geostrophic adjustment arises in the equatorial region with its specific spectrum of linear excitations (see Chapter 1). This case will be also treated below.

54

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

1.2. A synopsis of the method The geostrophic adjustment problem will be considered as initial value problem for arbitrary initial disturbance and treated by multi time-scale perturbation theory in Rossby number Ro. The main features of the method are summarized as follows: • technical tools are asymptotic expansions in Ro and fast-time averaging with subsequent separation of slow and fast parts of the flow; 1 ,...; • multiple time scales are used: T0 ∼ f10 , T1 ∼ Rof 0 • arbitrary surface elevations may be treated; • typical spatial scales are of the order of (QG √regime) or much greater (FG 0 ; regime) than Rossby deformation radius Rd = fgH 0 • the balanced dynamics arises as a result of removal of resonances (secular terms) in the fast dynamics. The method was applied to the one- and two-layer RSW, as well as to the full stratified primitive equations (Dewar and Killworth, 1995; Reznik, Zeitlin and Ben Jelloul, 2001; Zeitlin, Reznik and Ben Jelloul, 2003). We explain below in detail the application of the method in the case of one-layer RSW and limit ourselves by stating the results for the two-layer RSW. The starting point of the method are full non-dimensional equations. We assume that initial data are characterized by a horizontal scale L, on which the velocity and depth change by their characteristic values U and H , respectively, and introduce the Rossby and Burger numbers: Ro ≡ ε =

U , f0 L

Bu ≡ s =

Rd2 , L2

(1.18)

and the nonlinearity parameter λ = H /H0 . The PV changes in time at the advective time scale Ta = L/U equal to the time required for a fluid particle moving with a typical velocity U to travel over the typical scale L of the initial disturbance. The typical time scale of the inertia– gravity waves is f0−1 . The condition of time-scale separation is (Ta f0 )−1 =

U = ε = o(1) f0 L

(1.19)

and filtering of the fast IGW may be achieved by using exclusively the scale (f0 ε)−1 . In this case a small parameter—Rossby number—appears in front of advective derivatives in the l.h.s. of the non-dimensionalized RSW equations and the geostrophic equilibrium (1.13) always results in the leading order in ε, provided the following relation between the parameters holds: λs = O(ε).

(1.20)

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

55

Under hypothesis (1.20) the non-dimensionalization with the rapid time scale f0−1 of the RSW equations gives: ∂t v + ε(v · ∇v) + zˆ ∧ v + ∇h = 0,

(1.21)

λ∂t h + ε(1 + λh)∇ · v + λεv · ∇h = 0, εζ − λh ∂t q + εv · ∇q = 0, q = . 1 + λh

(1.22) (1.23)

We introduce a hierarchy of slow times tn = ε n t, develop all variables in formal asymptotic series: v = v0 (x, y; t, t1 , t2 , . . .) + εv1 (x, y; t, t1 , t2 , . . .) + · · · , h = h0 (x, y; t, t1 , t2 , . . .) + εh1 (x, y; t, t1 , t2 , . . .) + · · · ,

(1.24)

and plug these expansions in (1.21), (1.22) and (1.23). Each field at any order has a unique decomposition into a slow part defined as the t-average of the whole field, and a residual fast part: hi = h¯ i (x, y; t1 , . . .) + h˜ i (x, y; t, t1 , . . .),

i = 0, 1, 2, . . . ,

(1.25)

and analogously for the velocity field. The relation between the nonlinearity parameter and the Rossby number should be fixed in order to render the procedure consistent. If the parameters are of the same order the QG regime results, and if nonlinearity is of order one, the FG regime arises. The intermediate case ε ∼ λ2 may be treated as well (Reznik, Zeitlin and Ben Jelloul, 2001) and will be not presented below. Once the relation between the parameters is fixed, at each order in ε an easily solvable linear problem for the velocity, height and PV fields arises. The evolution of the slow fields is obtained by fast-time averaging of the corresponding equations. The procedure is, thus, straightforward and algorithmic. The presentation below follows the papers by Reznik, Zeitlin and Ben Jelloul (2001), Zeitlin, Reznik and Ben Jelloul (2003), Reznik and Grimshaw (2002), and Le Sommer, Reznik and Zeitlin (2004), Reznik and Sutyrin (2005), where this procedure was applied to the geostrophic adjustment in laterally unbounded and bounded domains, and on the equator, respectively.

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW In this section, as in the next one, adjustment process of localized finite-energy perturbations on the whole horizontal plane is considered. Decaying boundary conditions are imposed for all fields.

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

56

2.1. The QG regime In this regime λ ∼ ε. We present the calculations and results order by order in ε. 2.1.1. The lowest order in Rossby number The problem is linear in this order and we recover the classical results, cf. Obukhov (1949), Monin and Obukhov (1958). It is often more convenient to use the PV equation, instead of the height equation. We thus get: ∂t v0 + zˆ ∧ v0 = −∇h0 ,

(2.1)

∂t (ζ0 − h0 ) = 0,

(2.2)

where ζ0 = zˆ · ∇ ∧ v0 is the zeroth-order relative vorticity. Initial conditions are taken as a localized, i.e. finite-energy, disturbance: u0 |t=0 = uI ,

v0 |t=0 = vI ,

h0 |t=0 = hI .

(2.3)

It is convenient to rewrite (2.1) in terms of relative vorticity ζ and divergence D: ∂t ζ0 + D0 = 0,

(2.4)

∂t D0 − ζ0 = −∇ h0 . 2

(2.5)

The last equation (2.2) may be immediately integrated in fast time giving ζ0 − h0 = Π0 ,

(2.6)

where Π0 is a yet unknown function of coordinates and slow times. Eliminating ζ0 and D0 results in the inhomogeneous linear equation for h0 : −

∂ 2 h0 − h0 + ∇ 2 h0 = Π0 (x, y; t1 , t2 , . . .). ∂t 2

(2.7)

Solution of this equation may be written as a combination of fast h˜ 0 and slow h¯ 0 components: h0 = h˜ 0 (x, y; t, . . .) + h¯ 0 (x, y; t1 , . . .)

(2.8)

satisfying the following equations ∂ 2 h˜ 0 − h˜ 0 + ∇ 2 h˜ 0 = 0; ∂t 2 −h¯ 0 + ∇ 2 h¯ 0 = Π0 −

(2.9) (2.10)

(Klein–Gordon and inhomogeneous Helmholtz equations, respectively). Note that Π0 corresponds to the balanced quasigeostrophic PV built from the slow component h¯ 0 . This is the LPV introduced in the previous section in the context of balanced models.

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

57

We, thus, get the fast–slow splitting on the level of the equations of motion. To get closed problems for slow and fast motions separately, the splitting should be as well accomplished on the level of initial conditions which constitutes the initialization problem. At the initial moment t = t1 = t2 = · · · = 0 one gets from the definition of Π0 (2.6) that Π0 (x, y; 0) = ∂x vI − ∂y uI − hI ≡ ΠI (x, y). (2.11) ¯ ¯ This allows to find an initial value h0I of h0 by inverting the Helmholtz operator in −h¯ 0I + ∇ 2 h¯ 0I = ΠI

(2.12)

(our choice of decaying boundary conditions on the plane guarantees that the inversion problem is solvable in a unique way). In turn, knowing h¯ 0I one obtains the initial condition for h˜ 0 : h˜ 0I = hI − h¯ 0I .

(2.13)

The second initial condition for h˜ 0 follows from the PV and vorticity–divergence equations (2.2), (2.4): ∂t h˜ 0 |t=0 = −DI ≡ ∂x uI + ∂y vI .

(2.14)

The Klein–Gordon (KG) equation (2.9) together with initial conditions (2.13), (2.14) form a closed problem for h˜ 0 . A similar decomposition takes place for the velocity field: v0 = v˜ 0 (x, y; t, . . .) + v¯ 0 (x, y; t1 , . . .),

(2.15)

where the slow components verify the geostrophic balance v¯ 0 = zˆ ∧ ∇ h¯ 0

(2.16)

and the fast ones obey the equations ∂t v˜ 0 + zˆ ∧ v˜ 0 = −∇ h˜ 0

(2.17)

with initial conditions v˜0I = vI − v¯0I , u˜ 0I = uI − u¯ 0I ; (2.18) where u¯ 0I , v¯0I , h¯ 0I satisfy (2.16). It may be readily checked that the LPV ζ˜0 − h˜ 0 of the fast component is identically zero. The fast component satisfying the KG equation with well-posed initial conditions describes the inertia–gravity waves due to the unbalanced (u˜ 0I , v˜0I , h˜ 0I ) part of the initial disturbance and propagating out of this latter. The corresponding solution is: ˜h0 (x; t) = dk H0(±) (k)ei(k·x±Ωk t) + h˜ 00 , (2.19) ±

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

58

where (±)

H0 (k) =

I (k) 1 ˆ˜ D h0I (k) ± i , 2 Ωk

(2.20)

hat notation is used for the Fourier-transforms, and h˜ 00 is a yet unknown slowtime dependent function which makes no contribution to the lowest-order initial conditions. This additional term is necessary to eliminate resonances in higher orders. Solution for the velocity field is most conveniently written with the help of the complex notation U = u + iv. The KG equation 0 ∂ 2U 0 = 0 − U˜0 + ∇ 2 U ∂t 2 with the initial conditions 0I , 0 = u˜ 0I + iv˜0I ≡ U U t=0

0 0I − ∂x h˜ 0I + i∂y h˜ 0I ≡ WI = −iU ∂t U t=0 −

results and is solved by 0 (x; t) = 00 U dk U0(±) (k)ei(k·x±Ωk t) + U

(2.21)

(2.22) (2.23)

(2.24)

±

with (±)

U0 (k) =

ˆ 1 0I (k) ± i WI (k) U 2 Ωk

(2.25)

00 analogous to h˜ 00 . The fast component and a slow time-dependent contribution U thus represents outgoing dispersive waves. At any fixed point x = (x, y) the following estimates hold (Reznik, Zeitlin and Ben Jelloul, 2001): 1 0 (x; t) = O 1 , t → ∞, h˜ 0 (x; t) = O (2.26) , U t t and, hence, the fast-time averages of the fast components at a given point of the plane (x, y) vanish

u˜ 0 (x, y) = v˜0 (x, y) = h˜ 0 (x, y) = 0. (2.27) Here and below for any function f (t) the average is defined as 1 f = lim T →∞ T

T f (t) dt.

(2.28)

0

Thus, in the zeroth order of the perturbation theory we get a fast–slow motion splitting. Both fast and slow components are defined in a unique way starting

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

59

from arbitrary localized initial conditions. Note that the procedure imposes no a priori limitations on the relative initial values of the fast and the slow components. The fast part of the flow is completely resolved while the slow part remains undetermined. Its evolution equation comes from the condition of the absence of secular growth of the next order solution. 2.1.2. The first order in Rossby number Momentum equations give at this order: ∂t v1 + zˆ ∧ v1 = −∇h1 + R(0) v ,

(0) (0) (0) Rv = Ru , Rv = −(∂t1 + v0 · ∇)v0 .

(2.29)

The first-order PV equation is

∂t (ζ1 − h1 ) − Π0 ∂t h˜ 0 + u˜ 0 ∂x Π0 + v˜0 ∂y Π0 = −∂t1 Π0 − J h¯ 0 , Π0 .

(2.30) A consistency condition for having bounded in time solutions of (2.30) is obtained by applying the fast-time averaging to the both sides of (2.30) and gives the standard quasigeostrophic PV equation:

∂t1 Π0 + J h¯ 0 , Π0 ≡ ∂t1 ∇ 2 h¯ 0 − h¯ 0 + J h¯ 0 , ∇ 2 h¯ 0 = 0. (2.31) This equation, together with the initial conditions obtained at the previous order in Rossby number, and the geostrophic balance relations defines completely the evolution of the slow motion. Using (2.31) and (2.17) we can integrate (2.30) once in t and get (0)

ζ1 − h1 = Π1 (x, y; t1 , . . .) + Rζ ,

(0) 0 , Π0 , Rζ = Π0 h˜ 0 − u˜ 0 ∂y Π0 + v˜0 ∂x Π0 − J H

with

(0)

Rζ

= 0.

(2.32)

(2.33)

Here Π1 (x, y; t1 , . . .) is a yet undetermined slow function and for technical convenience we have introduced the primitive of h˜ 0 and its variation with respect to the mean: t i i 0 ≡ H i − H . H0 = h˜ 0 (t ) dt , H (2.34) 0 0 0

i satisfies the inhomogeneous KG equation The function H 0 −

i ∂ 2H 0 i = −∂t h˜ 0 i + ∇ 2 H −H = DI 0 0 t=0 2 ∂t

(2.35)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

60

with the initial conditions i i = 0; ∂t H H 0 t=0

0 t=0

= h˜ 0I .

(2.36)

i we have For the time-average of H 0 i

+ ∇2 H i = DI −H 0 0

(2.37)

i = 0 if DI = 0. Correspondingly, H 0 satisfies the following and, hence, H 0 initial-value problem: −

0 ∂ 2H 0 = 0; 0 + ∇ 2 H −H ∂t 2

0 i ; H = −H 0 t=0

0 ∂t H = h˜ 0I . t=0 (2.38)

Thus, the first-order vorticity–divergence equations are: ∂t ζ1 + D1 = Z,

(2.39)

∂t D1 − ζ1 = −∇ h1 + D. 2

(2.40)

The expressions for Z, D are Z = −[∂t1 ζ0 + v0 · ∇ζ0 + ζ0 D0 ], D = − ∂t1 D0 + v0 · ∇D0 + (∂x u0 )2 + (∂y v0 )2 + 2∂x v0 ∂y u0 .

(2.41) (2.42)

We thus get the following equation for the first correction to the free-surface elevation: (0)

∂ 2 Rζ ∂ 2 h1 (0) 2 − h + ∇ h = + Rζ + D − ∂t Z + Π1 . 1 1 ∂t 2 ∂t 2 By splitting h1 into a sum of slow and fast components −

(2.43)

h1 = h˜ 1 (x, y; t, . . .) + h¯ 1 (x, y; t1 , . . .)

(2.44)

and averaging (2.43) with respect to t the following equations for h˜ 1 , h¯ 1 result (0)

∂ 2 Rζ ∂ 2 h˜ 1 ˜ 1 + ∇ 2 h˜ 1 = − h ∂t 2 ∂t 2 2 −h¯ 1 + ∇ h¯ 1 = Π1 + D , −

(0)

+ Rζ + D − D − ∂t Z,

(2.45) (2.46)

where from the definition of D and the geostrophic balance relation we get

2 2 D = 2 ∂x2 h¯ 0 ∂y2 h¯ 0 − ∂xy (2.47) h¯ 0 . The initial conditions for the full first-order fields are null, as follows from the fact that the initial conditions for velocity and free-surface elevation are already verified by the zero-order fields. From this fact and using the definition of Π1

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

61

(2.32) we get that (0) Π1 |t=0 = −Rζ t=0

0 , ΠI . = − ΠI h˜ 0I − u˜ 0I ∂y ΠI + v˜0I ∂x ΠI + J H

(2.48)

The initial value h¯ 1I = h¯ 1 |t=0 can be determined from −h¯ 1I + ∇ 2 h¯ 1I = D |t1 =0 + Π1 |t1 =0 ,

(2.49)

where the r.h.s. of this equation is a function of initial values uI , vI , hI only, as follows from (2.37), (2.47), (2.48). To solve the inhomogeneous KG equation (2.45) we need initial conditions for h˜ 1 and its time-derivative. The first one is, obviously, h˜ 1 t=0 ≡ h˜ 1I = −h¯ 1I . (2.50) The second one follows from (2.32), (2.39) using the fact that D1 |t=0 = 0: (0) ∂t h˜ 1 t=0 = Z|t=0 − ∂t Rζ t=0 (2.51) with the r.h.s. entirely expressed in terms of initial fields using, where necessary, the evolution equation for the slow component. We, thus, have an inhomogeneous linear initial-value problem for h˜ 1 with the source term which may be schematically rewritten in the form: −

∂ 2 h˜ 1 − h˜ 1 + ∇ 2 h˜ 1 ∂t 2 (f ) (f ) (f ) = −2∂t1 D0 + F (s) (x)F0 (x; t) + F1 (x; t)F2 (x; t),

(2.52)

where the superscript denotes slow and fast spatially localized functions, respec(f ) tively, with Fi (x; t), i = 0, 1, 2, being some solutions of the homogeneous KG equation—see (2.4), (2.32), (2.41), (2.45), (2.47) for their precise expressions. The initial problem (2.52), (2.50), (2.51) can be straightforwardly solved. The second and the third terms in the r.h.s. of (2.52) are not resonant, i.e. they are non-divergent while integrated in space and time with the inverse KG operator. The term with the slow-time derivative of the divergence is resonant if the func00 in (2.19), (2.24) do depend on the first slow time t1 . In order to tions h˜ 00 , U avoid resonance it is sufficient to suppose that they do not. Thus, the initial-value problem is well-posed and requires no additional constraints, i.e. there is no slow modulation of the wave envelope at this order. The solution represents the inertia– gravity waves generated by the source term, whose details are given by the r.h.s. of (2.45). Thus, the first order of the perturbation theory provides the evolution equation for the zeroth-order slow field, the initialization of the first-order fast and slow fields, and a correction to the zeroth-order fast field. Note that equation (2.52)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

62

shows that the fast–slow splitting observed in the linear approximation is, in fact, incomplete. Indeed, the slow-fast cross-term in this equation means that the interaction of the primary adjustment emission with the slow motion provides a secondary wave emission, although this is a higher-order correction to the primary field. Thus, the slow field does influence the fast one. On the contrary, no trace of the influence of the fast field upon the slow one is detected at this order. Note also that fast and slow motions are also coupled at the level of the initialization procedure, cf. (2.50). 2.1.3. The second order in Rossby number At this order the second slow time t2 enters and the horizontal momentum equations read ∂t v2 + zˆ ∧ v2 = −∇h2 + R(1) v , R(1) v = −(∂t2 + v1 · ∇)v0 − (∂t1 + v0 · ∇)v1 .

(2.53)

The PV equation gives ∂t q2 + ∂t1 q1 + ∂t2 q0 + v0 · ∇q1 + v1 · ∇q0 = 0,

(2.54)

where q0 = ζ0 − h0 = Π0 , q1 = q2 =

(0) ζ1 − h1 − h0 Π0 = Π1 − h0 Π0 + Rζ ,

ζ2 − h2 − h0 (ζ1 − h1 ) + h20 − h1 Π0 .

(2.55) (2.56) (2.57)

We concentrate on the PV equation (2.54). Let us average this equation in t and consider it term by term. We have ∂t q2 = 0. By virtue of (2.27), (2.33)

∂t1 q1 = ∂t1 Π1 − h¯ 0 Π0 .

(2.58)

(2.59)

By splitting velocity v0 and q1 into fast and slow components which are denoted, as usual, by tilde and overbar respectively, we get

v0 · ∇q1 = v¯ 0 · ∇ q¯1 + v˜ 0 · ∇ q˜1 . (2.60) The fast-time averaging of expressions containing rapidly oscillating wave factors may give non-zero (or divergent) result only if the expressions in question decay slow enough at t → ∞. As follows from (2.26) and (2.32), for any fixed point of the plane in this limit 1 1 (0) Rζ = O (2.61) ⇒ q˜1 = O t t

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

and, hence v˜ 0 · ∇ q˜1 = O

1 t2

⇒

v˜ 0 · ∇ q˜1 = 0.

63

(2.62)

Finally, v1 · ∇q0 = v1 · ∇Π0 . It readily follows from (2.45) that − h˜ 1 + ∇ 2 h˜ 1 = 0

(2.63)

(2.64)

and, hence, h˜ 1 = 0 with our choice of decaying boundary conditions on the plane. We, thus, get from (2.26), (2.29), (2.64)

v1 = v¯ 1 = zˆ ∧ ∇ h¯ 1 + ∂t1 + v¯ 0 · ∇ v¯ 0 . (2.65) Using (2.58)–( 2.65) we arrive at the following slow equation

∂t2 + v¯ 1 · ∇ Π0 + ∂t1 + v¯ 0 · ∇ Π1 − h¯ 0 Π0 = 0.

(2.66)

So, remarkably, there are no fast–fast (wave-drag) contributions to (2.66) which may be, thus, safely obtained by a direct expansion of (1.23) in ε considering all variables as being slow. In order to get a closed equation for h¯ 0 , h¯ 1 we express Π1 as Π1 = ∂x v¯1 − ∂y u¯ 1 − h¯ 1 ,

(2.67)

according to its definition. By using (2.65) which, with the help of the zeroth order geostrophic balance conditions may be rewritten as

v¯ 1 = zˆ ∧ ∇ h¯ 1 − ∂t1 ∇ h¯ 0 − J h¯ 0 , ∇ h¯ 0 (2.68) we get

Π1 = ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 .

(2.69)

With the help of (2.68) and the evolution equation for Π0 equation (2.66) takes the following form

∂t2 Π0 + ∂t1 ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 − h¯ 0 Π0 − ∇ h¯ 0 · ∇Π0

+ J h¯ 0 , ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 − h¯ 0 Π0 − ∇ h¯ 0 · ∇Π0 (∇ h¯ 0 )2 , Π0 = 0. + J h¯ 1 − (2.70) 2 This equation describes a next-order correction to (2.31) which is necessary to take into account while studying the slow evolution of the balanced component of the flow for times much longer than t1 . One can combine the two equations

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

64

by introducing the “full” slow elevation h¯ = h¯ 0 + ε h¯ 1 . The result, up to a O(ε 2 ) correction, is a following “improved” QGPV (IQGPV) equation

D 2¯ ¯ ¯ ∂y h¯ ∇ h − h − ε h¯ ∇ 2 h¯ − h¯ − ε∇ h¯ · ∇ ∇ 2 h¯ − h¯ − 2εJ ∂x h, Dt1 = 0, (2.71) where ¯ 2 (∇ h) D ¯ (. . .) := ∂t1 (. . .) + J h − ε ,... . Dt1 2

(2.72)

This equation is of the “generalized vorticity” kind. It should be solved with the initial conditions for the full height. One can readily show that the lowest- and the first-order slow evolution equations (2.31) and (2.70) are derived from (2.71) by expanding into the asymptotic series of the type (1.24) but omitting the fast time. Within the O(ε2 ) accuracy it coincides with the balanced dynamics equation obtained by imposing “height-slaving” (Warn, Bokhove, Shepherd and Vallis, 1995, equation (63)) which, in turn, reproduces the iterated geostrophic model IG2 by Allen (1993). We should emphasize that no slaving or other constraint was used in the above derivation which means that we prove validity of this balanced model assuming only the smallness of the Rossby number and of the typical relative elevation of the free surface. We should also stress that while providing the description of the slow motion within higher, with respect to the standard QG model, accuracy, our method gives the formal limits of validity of this latter. Indeed, by construction the QG equation (2.31) is valid for times O(f −1 Ro−1 ), while for longer times O(f −1 Ro−2 ) the improved QG equation (2.71) should be used. This is important to keep in mind, e.g. in applications of the balanced equations to climatic studies. 2.1.4. The resumé of the QG adjustment in RSW in unbounded domain Thus, up to the fourth order in Rossby number the motion in the QG regime, corresponding to the small Rossby number ε and the typical relative elevation λ ∼ ε, is split in a unique way into slow and fast components. The former is described by the IQGPV equation giving corrections to the standard QG dynamics for longer times, the latter represents inertia–gravity waves running away from the initial disturbance. Initial conditions for each component are obtained from the full initial data. A remarkable point is that the IQGPV equation (2.71) does not contain fast–fast (wave-drag) contributions, i.e. the fast IG waves do not influence the balanced slow component at least on times O(f −1 Ro−2 ). On the contrary, the IG waves are coupled to the slow component already at the first order due to the slow–fast coupling term in the right-hand side of (2.52).

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

65

In the above consideration the initial perturbation was assumed to be spatially localized (i.e. having finite energy in the unbounded domain). The fast–slow splitting was also proved for spatially periodic motion in the QG regime (e.g. Embid and Majda, 1996; Babin, Mahalov and Nicolaenko, 1998a, 1998b). The characteristic property of the spatially periodic configuration is the existence of quadruplet wave interactions found by Babin, Mahalov and Nicolaenko (1998a, 1998b); for the localized motion such interactions are absent. The physical reason for this difference is that in the localized motion case the fast waves are propagating out of the disturbance, and do not stay at a given space point long enough to produce a resonance, which is not true in a periodic box. 2.2. The FG regime 2.2.1. The lowest order in Rossby number Let us remind that in the FG regime the relative surface elevation is of order one (in what follows we take λ = 1 without loss of generality), so h in this subsection corresponds to the full depth. In the lowest order we have: ∂t v0 + zˆ ∧ v0 = −∇h0 ,

(2.73)

∂t h0 = 0,

(2.74)

with the same initial conditions (2.3) as before. Solutions to (2.73), (2.74) may be immediately split into fast and slow components, with the slow one verifying the geostrophic balance (2.16) and the fast one verifying the homogeneous version of (2.73). The initial conditions are also split as follows: (g) (a) v¯ |t=0 = vI , v˜ |t=0 = vI , h¯ 0 t=0 = hI , (2.75) (g)

(a)

where we separated the geostrophic vI and the ageostrophic vI initial conditions with respect to the initial free-surface elevation: (g)

vI

= zˆ ∧ hI ,

(a)

(g)

v I = vI − vI .

part of the (2.76)

The homogeneous version of equation (2.73) describes non-propagating inertial oscillations corresponding to the small k limit of the dispersion equation for IGW. Using the complex notation U, which is especially convenient in this case, the corresponding solution for the fast velocity field is t) = A0 (x; t1 , . . .)e−it , U(x;

(a)

A0 (x; t1 , . . .)|t=0 = UI ,

(2.77)

where A0 is an amplitude (possibly, slowly modulated) of the inertial oscillations. Thus, at the leading order of the perturbation theory in the FG regime, the fast and the slow components of motion are split, the first one describing inertial oscillations which result from the imbalanced part of the initial conditions, while the second one evolving from the geostrophically balanced part of initial conditions remains undetermined.

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

66

2.2.2. The first order in Rossby number At this order we have the same equations (2.29) as in the QG case for the velocity field. The evolution of the h-field is given by

(a) ∂t h1 = −∂t1 h¯ 0 − ∇ · h¯ 0 v0 . (2.78) Only unbalanced part of velocity gives rise to the r.h.s., the contribution of the balanced velocity vanishes identically. The t-independent first term in the r.h.s. is trivially resonant as it leads to a linear growth in time of h1 and, hence, should be equal to zero which means that the characteristic time-scale of h0 is, in fact, t2 , and not t1 . Equation (2.78) can be immediately integrated giving, in complex notation,

0 + i∂ξ ∗ h¯ 0 U ∗ + h¯ 1 . h1 = h˜ 1 + h¯ 1 = −i∂ξ h¯ 0 U (2.79) 0 Here the complex independent variables ξ = x + iy, ξ ∗ = x − iy are introduced. The following formulas will be used below: 1 1 (∂x − i∂y ), ∂ξ ∗ = (∂x + i∂y ), 2 2 ∇ 2 = 4∂ξ2ξ ∗ , J (ξ, ξ ∗ ) = −2i

∂ξ =

(2.80)

while vi · ∇ = Ui ∂ξ + Ui∗ ∂ξ ∗ ,

(2.81)

∂ξ Ui + ∂ξ ∗ Ui∗ .

(2.82)

∇ · vi =

Thus, we get the following equation for U1

0 + ∂ξ ∗ h¯ 0 U ∗ − ih¯ 1 ∂t U1 + iU1 = −2i∂ξ ∗ −∂ξ h¯ 0 U 0 − (∂t1 U0 + U0 ∂ξ U0 + U0∗ ∂ξ ∗ U0 ).

(2.83)

This is an inhomogeneous equation for inertial oscillations which are nonpropagative and non-dispersive, unlike the inertia–gravity waves appearing at the same stage in the QG regime above. Hence, if forced at its proper frequency, the equation exhibits a typical resonance behaviour. The resonant forcing ∼e−it corresponds to terms containing U0 only once in the r.h.s. of (2.83). These terms are to be eliminated in order to avoid a secular growth of U1 . Thus, we get the following modulation equation for the amplitude of inertial oscillations

∂t1 A0 − 2i∂ξ2ξ ∗ h¯ 0 A0 + 2iA0 ∂ξ2ξ ∗ h¯ 0 + 2i ∂ξ ∗ h¯ 0 ∂ξ A0 − ∂ξ h¯ 0 ∂ξ ∗ A0 = 0, (2.84) or, coming back to the real notation i

∂t1 A0 + J h¯ 0 , A0 − ∇ 2 h¯ 0 A0 − A0 ∇ 2 h¯ 0 = 0. 2

(2.85)

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

67

In the present context this equation (up to a change of variables) was first proposed by Falkovich (1992) and correctly derived by Falkovich, Kuznetsov and Medvedev (1994)—see the discussion below. After having eliminated the secular growth, the regular solution of (2.83) sat1 + U 1 , isfying zero initial conditions may be easily found in the form U1 = U where

0 + A∗ ∂ξ ∗ A0 , 1 = i 2∂ξ ∗ h¯ 1 + J h¯ 0 , U U (2.86) 0 i 1 = U + + U −− + U − ≡ − C1 eit + iC2 e−2it + A1 (x, y, t1 , . . .)e−it U 1 1 1 2 (2.87) with obvious notation, and, respectively,

i ¯ ∗ 2 ¯ 2 ¯ h¯ 0 , h0 A∗0 + A∗0 ∂xy h0 A0 + A∗0 P C1 = ∂xy h0 + P 2 1 C2 = − A0 (∂x A0 − i∂y A0 ) (2.88) 2 defined as P (f ) := ∂y2 f − ∂x2 f . with the operator P Thus, in the first order in Ro the slow component of motion remains undetermined, being, in fact, even slower than it was initially supposed, while the fast component—inertial oscillations—acquires a slow modulation described by a linear Schrödinger-type equation (2.85) with coefficients depending on the slow motion. An induced correction to slow motion also appears and is given by (2.86). It is worth emphasizing that this correction does contain a contribution arising from the self-interaction of the fast component. 2.2.3. The second order in Rossby number The horizontal momentum equations are ∂t v2 + (∂t2 + v1 · ∇)v0 + (∂t1 + v0 · ∇)v1 + zˆ ∧ v2 = −∇h2

(2.89)

and to close the system the height evolution equation should be added:

∂t h2 + ∂t1 h1 + ∂t2 h¯ 0 + ∇ h¯ 0 v1 + h1 v0 = 0.

(2.90)

By averaging this equation in t and supposing boundedness of h2 in time we get:

∂t1 h¯ 1 + ∂t2 h¯ 0 + ∇ · h¯ 0 v¯ 1 + h¯ 1 v¯ 0 + ∇ h˜ 1 v˜ 0 = 0. (2.91) Calculating consecutively the bilinear combinations of height and velocity variables in this equation we get:

(∇ h¯ 0 )2 1 2 − J h¯ 0 , A˜ 0 ∇ · h¯ 0 v¯ 1 = J h¯ 1 , h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + 2 4

∗ ∗ ¯ − i ∇ h0 · A0 ∇A0 − A0 ∇A0

(2.92) + h¯ 0 A∗0 ∇ 2 A0 − A0 ∇ 2 A∗0 ,

68

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

∇ · h¯ 1 v¯ 0 = J h¯ 0 , h¯ 1

(2.93)

where we used (2.86), and 1

∇ h˜ 1 v˜ 0 = J h¯ 0 , |A0 |2 4

− i ∇ h¯ 0 · A∗0 ∇A0 − A0 ∇A∗0 + h¯ 0 A∗0 ∇ 2 A0 − A0 ∇ 2 A∗0 , (2.94) where we used (2.79), (2.87). Thus, the contributions containing fast–fast terms remarkably cancel leaving a purely slow evolution equation for h0 : (∇ h¯ 0 )2 ∂t2 h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + = 0. 2

(2.95)

Thus, again as in the QG case, we have no fast motion drag and the slow component evolves by itself. On the contrary, the fast component evolves on the background of the slow one according to the linear modulation equation (2.85) which may produce non-trivial effects. Knowing the rich physics of the nonlinear Schrödinger equation it would be exciting to get a cubic in A0 correction to (2.85) at the present order. This, however, turns out to be impossible. Indeed, such correction could appear from elimination of resonances in the equation for the next order velocity U2 . Comparing with (2.89) written in complex notation we see that the resonant (i.e. ∼e−it ) cubic terms appear in the following expressions: −− ∗ ∂ξ ∗ U U 0 1

(2.96)

1 + U ∗ ∂ξ U 1 ∂ξ ∗ U 0 + U 0 . 0 ∂ξ U U 1

(2.97)

and

Combining them we get

∗ 0 0 ∂ξ U˜0 + iU 0 ∂ξ U ∂ξ ∗ U ∗ ∂ξ ∗ U −iU 0 0 0 ∂ξ U 0 − iU ∗ ∂ξ ∗ U 0 ≡ 0. 0 ∂ξ U ∗ ∂ξ ∗ U + iU 0 0

(2.98)

Therefore, there are no nonlinear corrections to the Schrödinger-type equation (2.85) up to order four in Ro. Evolution equation for the first correction A1 to the envelope of inertial oscillations may be easily obtained. We do not display it here because, basically, it contains nothing particularly interesting. Note that the absence of cubic corrections in the modulation equation means that there are no corresponding resonant quadruplets of quasi-inertial waves in the vicinity of the threshold frequency f .

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2.2.4. Discussion of the FG adjustment Thus, the FG regime is similar to the QG one in a sense that splitting is also taking place and the slow component of the flow evolves according to the balanced equation without being influenced by the fast one. However, in this case the behaviour of the fast component is sensible to the background slow component and is governed by the Schrödinger-type equation (2.85). Nevertheless, the energy of inertial oscillations = dx dy 1 |A0 |2 E (2.99) 2 is conserved, as well as dx dy F (h¯ 0 )|A0 |2 where F is an arbitrary function of h¯ 0 . Separately, the energy of the slow motion 2 1

E = dx dy h¯ 0 ∇ h¯ 0 (2.100) 2 is also conserved which achieves the demonstration of splitting. It is well known that the classical Schrödinger equation describes dispersion of the wave packets. We can make the resemblance between (2.85) and the Schrödinger equation even closer by the change of variables A0 → B = h¯ 0 A0 which yields

h¯ 0 ∇ 2 h¯ 0 i ∂t1 B + J h¯ 0 , B + ∇ 2 B − B = 0. 2 2

(2.101)

We, thus, get a standard Schrödinger equation with a variable “mass” h¯ 0 , a “potential” ∼∇ 2 h¯ 0 and additional advection by the geostrophic velocity field produced by h¯ 0 . The following qualitative picture, thus, follows in the FG regime: Any initial height perturbation evolves very slowly according to the FG dynamics (2.95). A non-geostrophic part of the velocity perturbation is rapidly oscillating with the inertial period and slowly dispersed according to (2.101) where the coefficients may be considered as being constant in time at the dispersion time-scale. It is simultaneously advected by the geostrophic part of velocity. The details of the dispersion process depend on the initial field (“front”) hI (x). As in the standard Schrödinger equation case, it is possible, for particular profiles, that some part of the initial wave-packet is trapped by the front (“bound states”) and evolves together with it. In any case, the presence of the inertial oscillations packet provides a sort of “fuzzyness” of the front and may have implications on the transport and mixing properties. As it was already mentioned, the Schrödinger-type equation (2.101) was first derived in (Falkovich, Kuznetsov and Medvedev, 1994). The procedure used in that paper consisted in perturbative expansion in Burger number, simultaneous decomposition of any field in the Fourier-series in time using the overtones of the inertial period and subsequent truncation. This result was confirmed by Reznik,

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Zeitlin and Ben Jelloul (2001) by the above-presented method with, in addition, the proof of absence of the inertial oscillations drag in the slow-motion equation. Thus, fast–slow splitting was demonstrated in the next order of the perturbation theory. Finally, let us note that propagation of near-inertial oscillations was studied by similar means, but in a different context, by Young and Ben Jelloul (1997). 2.3. Comments on the influence of the β-effect 2.3.1. The QG regime Let us introduce the non-dimensional β-parameter β¯ with an explicit small factor in front of it:

¯ . f = f0 + βy = f0 1 + εβ βy (2.102) Here

εβ = O

βL f0

1;

β¯ = O(1).

(2.103)

The third small parameter, thus, appears in the theory and its value should be fixed with respect to λ and ε. For the QG case we choose εβ ∼ ε 2 , i.e. we suppose that nonlinear terms dominate those induced by β in (1.1). This assumption is justified for the intense mesoscale structures (eddies and fronts) in the ocean and in the atmosphere (see e.g. Reznik and Grimshaw, 2001; Kamenkovich, Koshlyakov and Monin, 1986). The PV takes a form (cf. (1.23)) ¯ εζ − λh + ε 2 βy (2.104) 1 + λh and by expanding the fields in asymptotic series (1.24) and repeating the procedure of Section 2.1 we arrive to the same results in the two lowest orders of the perturbation theory. The parameter β appears at the third order. The slow-motion equation (2.71) becomes q=

D 2¯ ¯ ¯ − ε h¯ ∇ 2 h¯ − h¯ − ε∇ h¯ · ∇ ∇ 2 h¯ − h¯ ∇ h − h + ε βy Dt1

¯ ∂y h¯ = 0 − 2εJ ∂x h,

(2.105)

and we see that the β-term and the nonlinear ageostrophic terms give corrections of the same order to the QGPV equation. Therefore, it is (2.105) which is the proper equation to use while studying the impact of the β-effect on the intense mesoscale structures of general form. On the contrary, the standard QGPV equation on the β-plane is suitable only in cases where the initial state possesses a special symmetry, as for example an axisymmetric localized vortex. In this (specific, but practically important) case the β-term is

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the main factor of vortex evolution while the nonlinear ageostrophic contributions remain smaller for long times (cf. Reznik, Grimshaw and Benilov, 2000; Reznik and Grimshaw, 2001). We would like to stress here, however, that the nonlinear ageostrophic terms are crucial for understanding such important physical phenomena as, e.g., the cyclone–anticyclone asymmetry. Equation for the fast field h˜ 2 is derived along the same lines as (2.52) and has the following form −

∂ 2 h˜ 2 − h˜ 2 + ∇ 2 h˜ 2 ∂t 2

2 (1) ¯ 2 h˜ 0 . = R + β¯ ∂ty h˜ 0 + ∂y h˜ 0 − ∂x h˜ 0 + 2βy∇ h

(2.106)

(1) Here Rh does not depend on β¯ and contains non-resonant terms and linear resonant terms analogous to those in the r.h.s. of (2.52). These resonances, as before, may be eliminated by adding to h˜ 0 , h˜ 1 free-wave solutions with slow-time depending Fourier amplitudes, cf. (2.19). The same is true for the second group of terms in the r.h.s. of (2.106). On the contrary, the last y-dependent term poses a real problem as it is evident that this resonance, being spatially non-uniform, can not be removed within the single space-scale asymptotic theory (formally, this term gives the O(t 2 ) contribution to h˜ 2 when t → ∞). Hence, whenever the initial state deviates from the geostrophic balance and h˜ 0I = 0 the problem (2.106) is ill-posed on the β-plane. Moreover, if h˜ 0I = 0, but some initial h˜ n are not, where h˜ n is the nth order fast field, the incurable secular growth arises in the (n + 2)th order. Thus, we come to a conclusion that in the case of non-zero β and unbalanced initial conditions the one space and multiple time-scale asymptotic procedure adopted in the present study can not provide solution of the RSW equations with arbitrary accuracy. However, it still provides a solution within a certain accuracy which is determined by the “degree of imbalance” of the initial conditions. Physically, the problem is related to the fact that in times O( ε12 ) the fast inertia–gravity ¯ cannot waves travel a distance O( ε12 ) and the Coriolis parameter f0 (1 + ε 2 βy) be considered constant along this distance, which was implicitly assumed while writing (1.24). A self-consistent asymptotic procedure on the β-plane (to be developed yet) should take into account a distortion of the fast wave rays due to the space inhomogeneity produced by the β-effect. Another way to avoid this difficulty is to consider motion in a zonal channel on the β-plane. There are all reasons to believe, however, that a change of the perturbative scheme along these lines would affect only the fast component of the motion, the slow one remaining unchanged. This assertion rests on two observations: (1) the spectral gap persists in the presence of the β-effect, i.e. inertia–gravity waves are still much faster than Rossby waves; (2) already the “naive” asymptotic expansion used above provides

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a self-consistent derivation of the slow motion equations in the case of a balanced initial state. 2.3.2. The FG regime In the FG regime the characteristic scale is much greater than the deformation radius and we, therefore, take the β-parameter to be of the same order of magnitude as the Rossby number: εβ = O(ε).

(2.107)

In this case the lowest order of the perturbation theory does not change and the formulas of Section 2.2 remain valid. The β-effect manifests itself in the second order and the Schrödinger equation (2.85) acquires a contribution with an y-depending coefficient: i

¯ 0 = 0. ∂t1 A0 + J h¯ 0 , A0 − ∇ 2 h¯ 0 A0 − A0 ∇ 2 h¯ 0 + iβyA (2.108) 2 The slow-time evolution of the free surface elevation also changes (cf., e.g., Stegner and Zeitlin, 1995) and becomes (cf. (2.95)) (∇ h¯ 0 )2 = 0. ∂t2 h¯ 0 − β¯ h¯ 0 ∂x h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + (2.109) 2 Note, first, that we still find splitting on the β-plane (cf. Falkovich, Kuznetsov and Medvedev, 1994) where the presence of the fast component drag in the slow motion equation was claimed on heuristic level; this is not confirmed by our calculations). Second, the presence of the “simple wave” β-induced term in the FG equation (2.109) leads to a non-linear steepening of the h¯ 0 field in zonal (x) direction which has a tendency to decrease the characteristic space scale. This phenomenon may violate the self-consistency of the FG model if the characteristic scale goes down to the Rossby scale Rd in finite time. Precisely, this catastrophic scenario is realised in the case of one-dimensional profile of the initial h¯ 0 : h¯ 0I = h¯ 0I (ax + by).

(2.110)

In this case the profile remains one-dimensional forever, the Jacobian in (2.109) vanishes identically and this latter becomes a simple wave equation describing breaking in finite time, as is well-known. Let us stress, however, that initial condition (2.110) is not of the spatially localized (in both directions) form required by the present study. The steepening can not be catastrophic for spatially localized h¯ 0I of the form h¯ 0I = 1 + hd0 , where hd0 (x, y) is a decaying at infinity function. It is known (cf. Cushman-Roisin, 1986; Ben Jelloul and Zeitlin, 1999) that there are infinitely many conserved quantities in the dynamics described by (2.109).

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They are the so-called Casimir invariants CF = dx dy F (hd ) = const

(2.111)

and the following functional related to the kinetic energy (cf. (2.100)) ¯ 2d = const. E = dx dy (1 + hd )(∇hd )2 − βyh

(2.112)

Here F (hd ) is an arbitrary function of hd and we replace everywhere h¯ 0 by 1 + hd (x, y; t). Let us show that the constraints imposed by these conservation laws prohibit breaking. Suppose, for simplicity, that hd is a continuously differentiable function with compact support D. It readily follows from (2.109) that hd remains zero outsideD for all times. Therefore, by virtue of (2.111) with F (hd ) = h2d the value | dx dy yh2d | is bounded from above by some constant. Therefore, the positive-definite integral dx dy (1 + hd )(∇hd )2 is bounded as well and the development of infinite gradients of hd (breaking) is forbidden. Hence, one may believe that for spatially localized disturbances the Jacobian terms in (2.109) can stop (or, at least, decelerate) the nonlinear steepening due to the β-term. Of course, this semi-qualitative consideration should be verified by a direct high-resolution numerical simulation which is in progress and will be presented elsewhere. The β-term in the amplitude equation (2.108) does not violate the energy conservation of the inertial oscillations (2.99). As it is however clear from the standard quantum mechanical-type analysis of this equation (e.g., for h¯ 0 = const), the β-term will produce a systematic meridional shift of the wave packet and a tendency to decrease the spatial scale of A in time. Thus, the model may lose self-consistency (the details depend on the initial spatial distribution of the inertial oscillations) on times ∼ε −2 , in analogy and by the same reasons as in the QG case. Thus, multiple space-scales technique should be applied in order to avoid difficulties (cf. Pedlosky, 1984 in this relation).

3. Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW In the previous section we gave an exhaustive description of the geostrophic adjustment of localized disturbances in the RSW model. As was already discussed in Chapter 1, the one-layer RSW model is barotropic as the RSW dynamics may be thought as dynamics of fluid columns of variable depth where the horizontal motion of fluid particles within a column is synchronized. The simplest baroclinic model is the two-layer RSW with the rigid lid. Although the motion in each layer is still a motion of fluid columns, the columns in respective layers

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are not synchronized anymore (except for the specific case of purely barotropic motions), and velocity changes across the interface between the layers. The density also jumps across the interface. The role of the baroclinic effects in the geostrophic adjustment process is the subject of the present section. The calculations presented purposedly in detail in the previous section give a sufficient idea about technical implementation of the method, and we will not dwell into details in the present section. The interested reader may find them in the extended version of the paper by Zeitlin, Reznik and Ben Jelloul (2003) available at http://gershwin.ens.fr/zeitlin. 3.1. Two-layer RSW model, a reminder Let us remind the equations of motion of the two-layer RSW model: 1 ∇πi = 0, i = 1, 2; ρi

∂t Hi − (−1)i+1 η + ∇ · vi Hi − (−1)i+1 η = 0,

∂t vi + vi · ∇vi + f zˆ ∧ vi +

(3.1) i = 1, 2,

(3.2)

vi = (ui (x, y, t), vi (x, y, t)) are velocity fields in each layer, ρi are the densities of the layers, η is the vertical displacement of the interface. There is no summation here over i and i + 1 is understood modulo 2. πi are defined with the help of the full pressure fields Pi in each layer: Pi = −ρi gz + (i − 1)(ρ1 − ρ2 )gH1 + πi ,

(3.3)

1 g is the acceleration due to gravity (the so-called reduced gravity g = 2 ρρ22 −ρ +ρ1 g plays an important role below). From the dynamical boundary condition on the interface it follows that

(ρ2 − ρ1 )gη = π2 − π1 .

(3.4)

The PV is conserved in each layer: (∂t + vi · ∇)Πi = 0,

Πi =

ζi + f , Hi − (−1)i+1 η

(3.5)

where ζi = zˆ · ∇ ∧ vi is the relative vorticity of the layer i. The fast motions in the model are provided by internal IGW propagating at the interface between the layers (cf. Chapter 1). The geostrophic adjustment will be studied, as before, for localized initial perturbations of the interface and velocity in each layer. The method is the same as described in Section 1, although its realization is more technically involved by obvious reasons.

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3.2. The QG regime 3.2.1. Basic hypotheses and scaling Supposing in this subsection for the ratio of the layers’ depths d = O(1) we introduce the following QG-scaling: the horizontal velocity scale U , the horizontal spatial scale L ∼ RR = g He /f0 , where RR is the baroclinic Rossby deformation radius, the pressure scale P = ρf ¯ 0 U L, and the scale of the interface variations η∗ = εHe . He is the equivalent height introduced in Chapter 1. Timei , i = 1, 2, we scale is f0−1 . Introducing the (order one) parameters h¯ i = H1H+H 2 rewrite the horizontal momentum and mass conservation equations in the following non-dimensional form: ∂t vi + εvi · ∇vi + zˆ ∧ vi + ∇πi = 0, i = 1, 2;

∂t 1 − (−1)i+1 ε h¯ i+1 η + ∇ · 1 − (−1)i+1 ε h¯ i+1 η vi = 0,

(3.6)

i = 1, 2. (3.7) In order to simplify the formulae we put ourselves in the oceanographic context and suppose that the densities of the layers are close to each other (and to the mean density). The corresponding density ratios may be easily restored otherwise in front of the pressure gradient terms here and below. The non-dimensional PVs are εζi + 1 . Πi = (3.8) 1 − (−1)i+1 h¯ i+1 εη The non-dimensional relation between the interface displacement and pressures in the layers is π2 − π1 = η.

(3.9)

The solution of the equations of motion on the infinite f -plane is sought in the form of asymptotic expansions: (0)

(1)

vi = vi (x, y; t, t1 , t2 , . . .) + εvi (x, y; t, t1 , t2 , . . .) + · · · , η = η(0) (x, y; t, t1 , t2 , . . .) + εη(1) (x, y; t, t1 , t2 , . . .) + · · · ,

(3.10)

where each dynamical variable in each order may be uniquely split into the slow (denoted below by over-bar) and fast (denoted by tilde) part defined, correspondingly, as the average over the fast time t and the fluctuation around it. For example: vi =

∞ n=0

(n)

ε n v¯ i (x, y; t1 , t2 , . . .) +

∞

(n)

ε n v˜ i (x, y; t, t1 , t2 , . . .),

(3.11)

n=0

and the same for πi , η. The representation (3.11) is unique since the fast compo(n) nents v˜ i are defined to have zero mean over the fast time t.

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76

We are looking for a perturbative solution of the Cauchy problem with initial conditions vi |t=0 = vIi ,

η|t=0 = ηI

(3.12)

for the system (3.1), (3.2) under the QG scaling. The subscript I denotes initial values, as usual. The calculations are similar to those presented above for the one-layer case, although they are, obviously, more cumbersome, as they should be performed for each layer and coupled via the interface condition (3.9). 3.2.2. The main results The slow and the fast component of motion are dynamically split and noninteracting. The slow dynamics is described by the “improved” quasigeostrophic PV equations for π¯ i = π¯ i(0) + ε π¯ i(1) , η¯ = ηi(0) + εηi(1) with π¯ 2 − π¯ 1 = η: ¯

Di 2 ∇ π¯ i + (−1)i+1 h¯ i+1 η¯ + ε(−1)i+1 h¯ i+1 η¯ ∇ 2 π¯ i + (−1)i+1 h¯ i+1 η¯ Dt1

− ε∇ π¯ i · ∇ ∇ 2 π¯ i + (−1)i+1 h¯ i+1 η¯ − 2εJ (∂x π¯ i , ∂y π¯ i ) = 0, (3.13) where h¯ i =

Hi H1 +H2 ,

and

Di (∇ π¯ i )2 (. . .) := ∂t1 (. . .) + J π¯ i − ε ,... , Dt1 2

i = 1, 2.

(3.14)

The fast component is the internal inertia–gravity wave packet conditioned by the initial conditions and described by the wave equation −

∂ 2 η˜ − η˜ + ∇ 2 η˜ = εR(x, y; t, t1 , . . .), ∂t 2

(3.15)

where η˜ = η˜ (0) + ε η˜ (1) ,

(3.16)

and the r.h.s. of (3.15) is produced, as in the one-layer case, cf. (2.52), by nonlinear interactions of the lower-order fast field η˜ (0) with itself and with the slow component. Again, the key property of this source term is that it is not resonant having zero fast-time mean, and hence the field η˜ (and other fast fields) correspond to internal IGW propagating out of the localized initial perturbation and decaying in time at any given space location. Apart from the secondary IGW emission provided by the slow-fast interaction term in the r.h.s. of (3.15), the fast and the slow components of motion feel each other only on the level of initial conditions for their respective dynamical equations. If the terms O(ε) are omitted in (3.13), (3.14), the standard two-layer QG equations are recovered (cf. Chapter 1). Thus, as in the one-layer case, the by-product

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of the presented demonstration is establishing of the limits of validity of the standard baroclinic QG system. It is valid for times of the order (εf0 )−1 , while the improved QG equations are valid for much longer times O((εf0 )−2 ). 3.3. The FG regime 3.3.1. Basic hypotheses and scaling The FG scaling for the two-layer FG model was already described in Chapter 1 of the present volume where the corresponding balanced dynamics was reviewed. The same scaling is used below with an addition of the fast-time scale f0−1 , as usual. Using the complex notation we get the following non-dimensional equations for the FG regime: ∂t Ui + iUi + ε(Ui ∂ξ Ui + Ui∗ ∂ξ ∗ Ui ) = −2∂ξ ∗ πi ,

∂ξ (1 − η)U1 + d −1 + η U2 + c.c. = 0, ∂t η = ε∂ξ (1 − η)U1 + c.c.,

i = 1, 2,

π2 = π1 + η,

(3.17)

where U1,2 = u1,2 + iv1,2 are the complex velocities in respective layers and η is the interface displacement. It is convenient to rewrite (3.17) in terms of the barotropic and the baroclinic modes (cf. Benilov and Reznik, 1996):

εd 2 2 ∂t Ubt + iUbt + + ∂ξ ∗ |Ubt |2 + Φ|Ubc |2 ∂ξ Ubt + ΦUbc 1+d = −2∂ξ ∗ P , (3.18) εd ∗ ∗ ∗ ∂t Ubc + iUbc + ∂ξ ∗ Ubt + Ubt ∂ξ Ubc ∂ξ (Ubt Ubc ) + Ubc 1+d

−1 + Ubc ∂ξ d + η Ubc − (1 − η)∂ξ Ubc

−1 ∗ ∂ξ ∗ d + η Ubc − (1 − η)∂ξ ∗ Ubc = 2∂ξ ∗ η, + Ubc (3.19) ∂ξ Ubt + c.c. = 0, εd ∂t η = −Ubt ∂ξ η + ∂ξ (ΦUbc ) + c.c. 1+d Here

Ubt = (1 − η)U1 + d −1 + η U2 , P = π1 + d −1 π2 +

(3.20) (3.21)

Ubc = U1 − U2 ,

η2 2

and notation Φ = (1 − η)(d −1 + η) is used for compactness.

(3.22)

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3.3.2. The main results As in the QG regime all fields are split into slow and fast parts, cf. (3.11). Both the slow and the fast components evolve from uniquely defined respective initial conditions. The fast component consists of inertial oscillations with slowly changing amplitude. Two FG subregimes arise depending on the ratio of the layers’ depths: the FGH regime for comparable depths, and the FGI regime for shallow upper layer. The barotropic and the baroclinic complex velocities in both sub-regimes are expressed as follows: Ubt = 2i∂ξ ∗ P ,

Ubc = −2i∂ξ ∗ η + Ae−it ,

(3.23)

where the slow functions η, P express the leading-order interface displacement and the barotropic pressure, respectively, and A = A(ξ, ξ ∗ , t1 , . . .) is the slowly evolving envelope of the inertial oscillations. Correspondingly, the leading-order evolution is determined by two coupled equations for slow P and η and a separate equation for A. The evolution equations obtained by the method of multi time-scale asymptotic expansions in ε and removal of resonances are as follows: 3.3.2.1. The FGH sub-regime 1 (3.24) J (η, P ), 1 + d −1

1 J P , ∇ 2 P + ∇ · (1 − η) d −1 + η J (η, ∇η) = 0. ∂t1 ∇ 2 P + 1 + d −1 (3.25)

1 + d −1 ∂t1 A + J P − η2 − d −1 − 1 η, A

i

+ ∇ 2 P − η2 − d −1 − 1 η + (∇η)2 A 2

i

− ∇ 2 (1 − η) d −1 + η A = 0. (3.26) 2 ∂t1 η =

3.3.2.2. The FGI sub-regime (∇η)2 2 = 0, ∂t2 η + J (P , η) + J η, (1 − η)∇ η − 2

(∇η)2 ∂t2 ∇ 2 P + J P , ∇ 2 P + J η, (1 − η)∇ 2 η − = 0, 2 i i ∂t1 A − J (η, A) − ∇ 2 A + ∇ 2 (ηA) − A∇ 2 η = 0. 2 2

(3.27) (3.28) (3.29)

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The most important and non-trivial feature of both FG sub-regimes is that although the inertial oscillations do not run away as IGW in the preceding subsection, they still do not make any contribution to the evolution of the slow component. Hence, baroclinicity does not affect this crucial property of the FG dynamics observed previously in the barotropic one-layer case. So the fast oscillations exert no drag on the slow vortical motion. At the same time, the slow modulation of the inertial oscillations is guided by the vortical motion since the coefficients in the modulation equations (3.26), (3.29) depend on P , η. It is worth emphasizing the difference between the FGH and the FGI subregimes. In the former the slow component and the modulation amplitude of inertial oscillations evolve in the same slow time t1 , while in the latter the slow component evolves in the slow time t2 and the amplitude A—in the faster time t1 . This is a novel feature arising due to stratification: in the barotropic RSW model the single FG regime is analogous to the FGI sub-regime. 3.4. Discussion of the geostrophic adjustment in the two-layer RSW and comparison with the one-layer case Thus, the adjustment scenario in the baroclinic two-layer model in the QG regime is very similar to that of the barotropic one-layer case. Any localized perturbation emits the internal IGW running away along the interface between the layers out from the initial disturbance. The slow, balanced part obeys the baroclinic QG equations, which should be improved if considered for longer than (εf )−1 times. The balanced and unbalanced components are split, however a secondary IGW emission due to the interaction of the primary one with the slow component takes place. Note however that a qualitatively new with respect to the one-layer case phenomenon may appear due to baroclinicity: the balanced component may develop the baroclinic instability in course of its evolution. This instability develops, however, in the slow time-scale, and thus does not violate the fast–slow splitting. The baroclinicity is characterized both by the ratio of densities, and by the ratio of depths of the layers. The role of the second parameter is highlighted by the possibility to have two essentially different frontal regimes in the two-layer model: one with a shallow upper layer, and one with comparable layer depths. The first case is similar to the one-layer frontal regime: the fast component consists of inertial oscillations of the interface, and the slow one obeys the frontal dynamics equation with no fast-component drag. The inertial oscillations are guided by the slow component, their envelope dynamics, although slow, is faster than the underlying balanced motion dynamics. In the second case, although there is still no drag of the fast component upon the slow one, the envelope of inertial oscillations (the guided variable), and the balanced (guiding) motion evolve at the same slow time-scale. It should be stressed that similar analysis within continuously stratified primitive equations gives similar results (Zeitlin, Reznik and Ben Jelloul, 2003).

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4. Nonlinear geostrophic adjustment in the presence of a lateral boundary 4.1. Preliminaries In the present section the above-developed asymptotic multiple time-scale theory is applied, within the framework of the one-layer RSW, to nonlinear geostrophic adjustment in a laterally bounded domain. The motion of the fluid, which is described, in non-dimensional form, by equations (1.21)–(1.23), will be considered in a half-plane bounded by a rigid wall with free-slip (no cross-boundary flux) boundary conditions. This is the simplest, and analytically treatable, case of adjustment in bounded domains. A new effect in this case is appearance of the Kelvin waves which are trapped near the boundary and propagate in such a way that the boundary is on the right in the direction of propagation (see Chapter 1 of the present volume). As discussed in Chapter 1, the presence of the Kelvin waves destroys the spectral gap between the slow vortex motion and the fast wave motion, because Kelvin waves with arbitrarily small frequencies exist. Hence, one may expect alterations of the above-established adjustment scenario in the presence of boundaries. This expectation is comforted by a not very well-known fact that the standard derivation of the QG equations for a localized flow near an infinitely long boundary (e.g. in a channel or in a half-plane) encounters difficulties. Namely, it can not provide (1) the conservation of the geostrophic mass and (2) the locality of the successive approximations. To illustrate this point, let us consider the standard derivation of the QG equations starting from the full RSW equations (1.21)–(1.23) on the semi-infinite f -plane, and assuming that λ ∼ ε 1 and that the timedependence is slow, i.e. there is no dependence on the fast time t. The velocity obeys the no-flux boundary condition at the rigid wall situated at y = 0: v|y=0 = 0.

(4.1)

In the lowest order in Ro we get from (1.21), (1.23) the geostrophic relations and the quasigeostrophic PV equation: v¯0 = ∂x h¯ 0 , u¯ 0 = −∂y h¯ 0 ;

2

∂t1 ∇ h¯ 0 − h¯ 0 + J h¯ 0 , ∇ 2 h¯ 0 = 0. The boundary condition for h¯ 0 follows from (4.1), (4.2): ∂x h¯ 0 y=0 = 0.

(4.2) (4.3)

(4.4)

Let the initial field h|t=0 = hI (x, y)

(4.5)

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81

be localized, i.e. hI → 0,

y → +∞,

x → ±∞.

(4.6)

Obviously, for conditions (4.4), (4.6) to be consistent, it is necessary that hI |y=0 = 0.

(4.7)

We assume that the decay (4.6) is sufficiently rapid, so that the initial energy and enstrophy are finite. We are looking for the solution conserving energy and enstrophy (note that these quantities are conserved under condition (4.4). It means that conditions of locality (4.6) are valid at any time, i.e. h¯ 0 → 0,

y → +∞,

x → ±∞,

(4.8)

since in the opposite case the energy and enstrophy become infinite. Therefore, by virtue of (4.4) and (4.8) h¯ 0 y=0 = 0. (4.9) Thus, in the localized case the boundary streamfunction is always identically zero. It readily follows from (4.3) that the total geostrophic mass, which is defined as dx dy h¯ 0 M= (4.10) y>0

is conserved only if +∞

dx ∂yt1 h¯ 0 y=0 = 0.

(4.11)

−∞

But, as it is easy to see, the problem (4.3), (4.5), (4.9) is well-posed and, therefore, condition (4.11), generally, is not satisfied (see Section 4.3 below and Reznik and Grimshaw, 2002). The normal to the wall component of the first-order velocity is v¯1 = ∂x h¯ 1 + ∂t1 u¯ 0 + u¯ 0 ∂x u¯ 0 + v¯0 ∂y u¯ 0 . Therefore, from the first-order no-flux condition at the wall we get: ∂x h¯ 1 |y=0 = −(∂t1 u¯ 0 + u¯ 0 ∂x u¯ 0 ) y=0 .

(4.12)

(4.13)

For the asymptotic procedure to be self-consistent one has to require that h¯ 1 is also localized. By virtue of (4.2) and (4.13) this is possible only under the condition (4.11) which, generally, is not satisfied, as mentioned above. Thus, the “slow”

82

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

solution of problem (1.21)–(1.23), which is given by (4.1)–(4.6) does not conserve the total mass, and is not localized. The locality of solution and the mass conservation can be satisfied only in the framework of the full RSW model taking into account the fast Kelvin waves, as demonstrated below. As one can see, the locality condition (4.8) is crucial in the above consideration, since in the case of non-localized (periodic or non-periodic along the y-axis) motion the boundary streamfunction is, generally, non-zero being determined from the no-flux condition for the ageostrophic first-order velocity field (cf. Phillips, 1954; McWilliams, 1977). We emphasize also that the above consideration does not mean that the QG approximation is “deficient” or breaks down in the localized case: equations (4.2), (4.3), (4.5), (4.9) correctly describe the slow component of motion (see below). But, contrary to non-localized motion, in this case the lowest-order QG dynamics should be supplemented with the fast Kelvin wave to provide the locality of solutions and mass conservation. The key role of the Kelvin waves in the process of geostrophic adjustment near boundaries was investigated in a number of studies after the pioneering work by Gill (1976) (e.g. Hermann, Rhines and Johnson, 1989; Dorofeyev and Larichev, 1992; Tomasson and Melville, 1992; Helfrich and Pedlosky, 1995; Helfrich, Kuo and Pratt, 1999). In the rest of this section we consider the QG regime for the RSW system (1.21), (1.23) on a half-plane assuming that λ ∼ ε 1 and that the initial fields uI , vI , hI are not necessarily balanced. The initial motion is assumed to be bounded at infinity and localized in the y-direction. The initial fields are not entirely arbitrary and satisfy the following conditions at the wall y = 0: vI |y=0 = 0,

(uI = −∂y hI )y=0 ,

(4.14)

to be consistent with equations (1.21) and the no-flux condition (4.1). Special attention will be paid to two types of the initial conditions: • Periodic (in x) boundary conditions, (uI , vI , hI ) =

+∞

(uIn , vIn , hIn )einx ,

(4.15)

n=−∞

• Localized boundary conditions: (uI , vI , hI ) → 0,

x → ±∞.

(4.16)

Analysis of step-like initial conditions when the motion tends to some zonal flows as x → ±∞, can be found in Reznik and Grimshaw (2002). We will first derive the general formulae not depending on the type of the initial conditions, and then discuss the special cases (4.15), (4.16). Like in the unbounded case of Section 3 the solution to the system (1.21), (1.23), (4.1) is sought in the form of the multipletime-scale asymptotic expansions (1.24).

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

83

4.2. The lowest-order solution (the linear adjustment) 4.2.1. Fast–slow splitting Obviously, equations (2.1), (2.2), (2.6) and initial conditions (2.3) obtained on the unbounded f -plane remain valid in the bounded case. A new element appearing here is the no-flux condition v0 |y=0 = 0.

(4.17)

Again, the solution of (2.1) is represented as a sum of fast (tilde) and slow (bar) components,

(u0 , v0 , h0 ) = u˜ 0 , v˜0 , h˜ 0 (x, y, t, t1 , . . .) + u¯ 0 , v¯0 , h¯ 0 (x, y, t1 , . . .), (4.18) the fast component having zero fast-time average. Applying the t-averaging to (2.1), (2.6) we obtain the following equations for the slow and fast components: v¯0 = ∂x h¯ 0 ,

u¯ 0 = −∂y h¯ 0 ,

ζ¯0 − h¯ 0 = ∇ 2 h¯ 0 − h¯ 0 = Π0 ,

v¯0 |y=0 = 0.

(4.19) (4.20)

∂t u˜ 0 − v˜0 = −∂x h˜ 0 ,

∂t v˜0 + u˜ 0 = −∂y h˜ 0 ,

ζ˜0 − h˜ 0 = 0,

v˜0 |y=0 = 0.

(4.21) (4.22)

4.2.2. Initial conditions for fast and slow zonal velocities The fast–slow splitting (4.18) is incomplete unless the initial conditions for each component are determined. Using (4.19) the initial slow field v¯0I can be found from the equation: ∇ 2 v¯0I − v¯0I = ∂x ΠI ,

ΠI = ζI − hI ,

(4.23)

which should be solved under the boundary condition v¯0I |y=0 = 0.

(4.24)

At the same time, the initial slow field h¯ 0I cannot be determined at this stage since (4.24), (4.19) imply only that h¯ 0I is a constant at y = 0, (B) h¯ 0I y=0 = h¯ 0I = const. (4.25) To find the constant h¯ 0I , the analysis of Kelvin waves, which form a part of the response of the system for geostrophically unbalanced initial conditions, is necessary—see below. Knowing v¯0I from (4.23), (4.24) one can find the fast initial field: (B)

v˜0I = vI − v¯0I = V(x, y).

(4.26)

84

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

Another initial condition for v˜0 follows from equation (2.23): ∂t v˜0 |t=0 = −(uI + ∂y hI ) = Vt (x, y).

(4.27)

The fast field v˜0 obeys the KG equation readily following from (4.21): −∂tt v˜0 − v˜0 + ∇ 2 v˜0 = 0.

(4.28)

This equation together with the initial conditions (4.26), (4.27), and the boundary condition (4.22) allows us to determine the field v˜0 from any initial conditions (2.3). 4.2.3. Solution for the fast fields v˜0 , u˜ 0 , h˜ 0 To obtain the solution for v˜0 we define the odd counterparts of V(x, y), Vt (x, y): Vodd (x, y) y>0 = V(x, y); (4.29) Vodd (x, y) y0 dx dy h1 : ∞ ∂t M1 =

∞ dx R(0) u (x, 0, t)

=−

−∞

dx ∂t1 u0 |y=0 .

(4.141)

−∞

Substituting representation (4.18) for u0 into (4.141) and using (4.33), (4.34), (4.47), (4.121), and the fact that the function u˜ 01 does not depend on t1 (see Section 4.4.2), one obtains that ∞ ∂t M1 = −

dx ∂t1 u¯ 0 y=0 =

−∞

∞

dx ∂yt1 h¯ 0 y=0 .

(4.142)

−∞

We see that the changes of M1 compensate exactly the changes of the lowest-order geostrophic mass given by (4.81). Now instead of standard splitting of the first-order solution into the fast and the slow components, we use an alternative splitting into mass-conserving and mass non-conserving parts: (u1 , v1 , h1 ) = (uc , vc , hc ) + (unc , vnc , hnc ),

(4.143)

where the components conserving the corresponding total mass obey the equations: −y ¯ ∂t uc − vc + ∂x hc = R(0) (4.144) u − e ∂yt1 h0 y=0 , ∂t vc + uc + ∂y hc = R(0) v , ∂t (ζc − hc ) = vc |y=0 = 0,

R(0) ζ , (uc , vc , hc )t=0 = 0,

(4.145) (4.146) (4.147)

while the residual fields responsible for the mass changes (4.142) satisfy the system: ∂t unc − vnc + ∂x hnc = e−y ∂yt1 h¯ 0 y=0 , (4.148) ∂t vnc + unc + ∂y hnc = 0,

(4.149)

∂t (ζnc − hnc ) = 0,

(4.150)

vnc |y=0 = 0,

(4.151)

(unc , vnc , hnc )t=0 = 0.

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

101

Analysis of the system (4.144)–(4.147) analogous to the analysis presented already in this section above, reveals that the fields uc , vc , hc are localized; therefore they are not important unless one is interested in higher-order effects. The solution to the system (4.148)–(4.151) is sought in the form (unc , vnc , hnc ) = (φnc , 0, φnc )e−y ,

(4.152)

which satisfies the no-flux boundary condition (4.151). The evolution of φnc is described by the equation ∂t φnc + ∂x φnc = ∂yt1 h¯ 0 y=0 . (4.153) The solution satisfying zero initial condition is written as x

dx ∂yt1 h¯ 0 y=0 .

φnc (x, t, t1 ) =

(4.154)

x−t

As is seen from (4.153) the solution (4.152), (4.154) can be interpreted as a forced Kelvin wave generated by the along-wall slow velocity varying in time. This wave looks like an injected jet propagating along the wall to the right of the localized lowest-order disturbance as shown in Figs. 1, 2. The time derivative of the total mass of the jet is equal to dx dy e

∂t

−y

∞

dx ∂yt1 h¯ 0 y=0 = −∂t1 M.

φnc =

(4.155)

−∞

y>0

Comparing (4.81) to (4.155) we conclude that this compensating jet takes away of the mass from the lothe surplus or shortage (depending on the sign of ∂t1 M) decreases (increases) calized lowest-order disturbance; if the geostrophic mass M then the jet velocity unc is positive (negative) for sufficiently large x and t x. We now consider the dynamics of compensating jet (4.152) in some details by representing the amplitude φnc in the form x φnc =

∞

dx a(x , t1 ) +

dx ∂yt1 h¯ 0 y=0

−∞

x−t

x

dx φ(x ),

(4.156)

x−t

where a(x, t1 ) = ∂yt1 h¯ 0 − φ(x)

∞

−∞

dx ∂yt1 h¯ 0

, y=0

(4.157)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

102

Figure 1.

Schematic representation of the compensating jet in the oscillating regime. √ Plots of the (c) = π sin 0.5t. function φj given by (4.160) are shown for different times for the model case −∂t1 M 2 2 The function φ(x) is equal to √2 e−4x . Successive times are indicated on the corresponding curves. π

It is seen that the jet length increases monotonically with increasing time, but its intensity oscillates near zero changing its sign. The arrow indicates the direction of the jet velocity.

and φ(x) is an arbitrary localized function satisfying the normalization condition ∞

dx φ(x ) = 1.

(4.158)

−∞

The function a(x) is also localized in x and ∞

dx a(x ) = 0

(4.159)

−∞

by virtue of (4.158). One can readily show that the first term in the r.h.s. of (4.156) does not make any contribution to the integral in the l.h.s. of (4.155), and the mass conservation is provided entirely by the second term in the r.h.s. of (4.156): φj(c) =

∞ −∞

dx ∂yt1 h¯ 0 |y=0

x

x−t

dx φ(x ) = −∂t1 M

x

dx φ(x ).

(4.160)

x−t

For example, the function φ may be chosen to be Gaussian etc. The arbitrariness of φ is related to the fact that representation of a function as a sum of localized and

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

103

Figure 2.

Schematic representation of the compensating jet in the limiting regime. Plots √ of the func(c) = π e−0.5t . The tion φj given by (4.160) are shown for different times for the model case −∂t1 M 2 2 function φ(x) is equal to √2 e−4x . Successive times are indicated on the corresponding curves. It is π

seen that the jet length increases monotonically with increasing time, but its intensity gradually drops to zero. The arrow indicates the direction of the jet velocity.

x non-localized parts is not uniquely defined. The integral x−t dx φ(x ) represents the along-wall jet expanding with a constant speed of Kelvin waves to the right (the rate of of the initial perturbation. The slow time-dependent amplitude −∂t1 M change of the total mass, up to the sign) determines the sign and intensity of the jet at any moment. Reznik and Sutyrin (2005) studied the behaviour of the QG mass on the half-plane and showed that two regimes, the oscillating and the limiting oscillates near some constant value. ones, are possible. In the oscillating regime M This results in slow (in comparison with the jet propagation) oscillation of the tends jet sign and intensity as shown in Fig. 1. The limiting regime, when M monotonically to some constant value as t1 → ∞, is represented in Fig. 2. In this case the jet sign does not change and the jet amplitude slowly tends to zero. Note that the forced Kelvin wave (4.152) can be split into the slow and fast parts in the following way:

φnc e−y = φ¯ nc + φ˜ nc e−y , (4.161) φ¯ nc (x, t1 ) =

x −∞

dx ∂yt1 h¯ 0 y=0 ,

(4.162)

104

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

φ˜ nc (x, t1 ) = −

x−t

dx ∂yt1 h¯ 0 y=0 .

(4.163)

−∞

Here φ˜ nc e−y is a free Kelvin wave with zero fast-time average; both fast and slow components are not localized if the time-derivative of the QG mass (4.155) is non-zero. At the same time their sum (4.161) is localized. 4.7. The summary of the results on the adjustment in the laterally bounded domain The presence of the lateral boundary introduces into the f -plane RSW model a new important element which was absent in the unbounded case: the Kelvin waves trapped near the boundary. Unlike the IGW, the frequency gap between the Kelvin waves and the slow motion is absent since Kelvin waves with arbitrarily small frequencies can exist. Despite this fact the presence of the Kelvin waves does not forbid the fast– slow splitting. The possibility of splitting resides in the special structure of the Kelvin waves in the considered configurations. For the periodic initial conditions the Kelvin wave profile is also periodic, but it has zero mean which physically corresponds to the absence of a zero-frequency harmonic in the Kelvin wave spectrum. For the localized (and “step”-like, see Reznik and Grimshaw, 2002) initial conditions such harmonics can be present in the Kelvin wave spectrum, but the shape of the Kelvin wave profile is such that the Kelvin wave field rapidly decays at a fixed spatial point when the Kelvin wave propagates off this point. The theory provides simple formulae allowing to initialize the Kelvin wave starting from arbitrary initial conditions. With increasing time the Kelvin wave profile slowly distorts due to the Kelvin wave non-linear self-interaction, the distortion being described by the simple-wave equation. The resulting evolution of the initial Kelvin wave profile can be characterized as fast propagation accompanied by slow breaking. The breaking Kelvin waves presumably does not affect the slow mode in the localized and “step”-like cases since it happens far from the main regions of slow variability. For the periodic motion the effect of the Kelvin waves breaking on the geostrophic mode should be examined more thoroughly. An important role in this case should be played by the non-conservation of potential vorticity across a Kelvin front (see Section 5 below and, e.g. Helfrich, Kuo and Pratt, 1999). Generally, the geostrophic adjustment in the half f -plane is similar to that in the unbounded f -plane examined in Section 3. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time-scales f −1 and (εf )−1 , respectively. The slow component is not influenced by the fast one, at least for times (ε 2 f )−1 , and remains close to geostrophic balance. The fast component consists mainly of linear IG waves rapidly propagating

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

105

out of the initial disturbance and Kelvin waves confined near the boundary. Like in the unbounded case, the nonlinear interactions of IG waves with each other, with slow component and the Kelvin waves result only in a small correction to the fast field. The theory provides an algorithm to determine the initial slow and fast fields to any order in Rossby number. Evolution of the slow motion on times (εf0 )−1 is governed by the wellknown quasi-geostrophic potential vorticity equation for the elevation h¯ 0 . The boundary condition used in this non-viscous model is the no flux condition on the rigid wall. Being formulated in terms of h¯ 0 , this condition depends on the structure of initial fields. In the case of periodic (and “step-like”) initial fields the boundary value of h¯ 0 is equal to some arbitrary function h¯ (B) which is constant along the boundary but depends on the slow time (e.g. Kamenkovich and Reznik, 1978; Pedlosky, 1982). To remove this uncertainty the conservation of along-boundary circulation is used (cf. e.g. Pedlosky, 1982). If the fast component is absent then the slow circulation is assumed to be conserved, this conservation providing the energy and mass conservation of the lowest-order slow motion. However, if the initial conditions are not balanced, so that the fast component is present, then the total circulation (fast + slow) should be conserved. The question is if the fast and slow circulations are conserved separately. Our analysis demonstrates that this is really the case for the periodic (and step-like) initial conditions. The situation for localized initial fields is somewhat more complicated. The lowest-order slow motion in this case is also localized and the elevation is zero at the boundary, i.e. the problem for h¯ 0 is well-defined without using the conservation of slow circulation. Moreover, a simple analysis shows that, generally, the lowest-order slow circulation and, therefore, the total mass of slow localized motion are not conserved. Considering the first-order dynamics, we found that the conservation of the total mass and circulation is provided by a compensating jet taking away the surplus, or shortage of mass from the localized lowest-order slow disturbance. A simple representation for the compensating jet was obtained. It is seen from this representation that the along-wall jet expands with the fast speed of Kelvin wave to the right of the initial perturbation. The slow time-dependent i.e. the rate of change (modulo the sign) of the total lowest-order amplitude ∂t1 M, slow QG mass, determines the jet sign and intensity at each moment. Limits of validity of the considered model when localized disturbance interacts with infinitely long boundary were discussed by Reznik and Sutyrin (2005). It was shown that the approximation of infinite domain can be used if (1) the typical basin scale greatly exceeds typical size of the localized perturbation and the Rossby scale; (2) the time does not exceed the typical time which is required for the Kelvin wave to travel the typical basin scale. These conditions are typical for synoptic eddies of a ∼100 km spatial scale and temporal scale ∼10 days in the ocean of a basin scale of few thousands km, and baroclinic Kelvin wave speed ∼2 m/s.

106

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

On longer times the slow motion obeys the so-called “improved” quasigeostrophic potential vorticity (IQGPV) equation. The theory provides initial and boundary conditions for this equation. The IQGPV equation exactly coincides with the “improved” QGPV equation, derived in Section 3, in the localized (and step like) case. In the periodic case this equation contains an additional term due to the Kelvin wave self-interaction, this term depending on the initial Kelvin wave profile. In other words, in the presence of a boundary the IQGPV equation, in general, depends on initial conditions, which is not the case in the unbounded domain.

5. Nonlinear geostrophic adjustment in the equatorial region 5.1. The long-wave scaling and classification of possible dynamical regimes As was explained in Chapter 1 of the present volume, dynamics of the tropical atmosphere and ocean are special due to the change of sign of the Coriolis force at the equator. A response to a localized perturbation in the equatorial region will be therefore different as compared to the higher latitudes. The equator forms a waveguide with its proper waveguide modes, like e.g. the equatorial Kelvin mode which has much in common with the boundary Kelvin wave whose role in the adjustment process was studied in the previous section. In the present section we study the process of geostrophic adjustment and possibility of splitting of slow and fast motions on the equatorial β-plane in the framework of one-layer RSW. Our approach is based on the observation that the dispersion relation for equatorial waves possesses a spectral gap between the slow Kelvin and Rossby waves on the one hand and the fast inertia–gravity waves on the other hand, see Fig. 11 of Chapter 1. However, the situation on the equator differs significantly from that of the mid-latitude f -plane because the magnitude of the gap depends on the wavelength and tends to zero with increasing wavenumber. Therefore, an asymptotic theory similar to one developed by Reznik, Zeitlin and Ben Jelloul (2001) and exposed in Sections 2 and 3 of the present chapter may be consistently developed only for the long-wave perturbations. Correspondingly, an asymptotic (in Rossby number ε) multiple time-scale theory will be developed for geostrophic adjustment of localized perturbations with small aspect ratio δ measuring the typical ratio of meridional to zonal extensions and introduced in Chapter 1. The dynamical regimes arising for different relations between ε and δ will be classified. The slow and fast motions will be completely quantified, their respective equations of motion with proper initial conditions obtained, and their dynamical splitting demonstrated.

5. Nonlinear geostrophic adjustment in the equatorial region

107

The starting point of our analysis is the equatorial RSW equations under the scaling described in Section 6.1 of Chapter 1: ut + δεuux + εvuy − yv = −δhx ,

(5.1)

vt + δεuvx + εvvy + yu = −hy ,

(5.2)

ht + vy + δux + ε(hv)y + δε(hu)x = 0.

(5.3)

Both parameters ε and δ are supposed to be small, their relative values will be fixed below. The smallness of δ is equivalent to the long-wave approximation. We should emphasize that the meaning of the key parameter, Rossby number, is twofold under such scaling, as it has also the meaning of the Froude number (cf. Majda and Klein, 2003; Majda, 2003). Note also that smallness of the geopotential perturbations which is equivalent to the smallness of ε under this scaling, is a key ingredient of the weak temperature gradients approximation widely used in studies of the tropical circulation (cf. e.g. Majda and Klein, 2003; Bretherton and Sobel, 2003). We will, however, never consider below a forceddissipative problems typical for such studies. 5.1.1. The slow motion: long Rossby and Kelvin waves The long linear Rossby and Kelvin waves (cf. Section 6 of Chapter 1) are characterized by the small non-dimensional meridional velocity v ∼ δ and slow time evolution. Making in (5.1)–(5.3) the changes v → δv, t → t1 = δt we obtain the following equations (corresponding subscripts denote the partial derivatives in this section): ut1 + εuux + εvuy − yv = −hx ,

(5.4)

2

δ vt1 + δ εuvx + δ εvvy + yu = −hy ,

(5.5)

ht1 + vy + ux + ε(hv)y + ε(hu)x = 0.

(5.6)

2

2

In the lowest order in ε, δ we get the equations for slow-propagating linear Rossby and Kelvin waves: (0) u(0) + h(0) t1 − yv x = 0,

(5.7)

yu(0) + h(0) y = 0,

(5.8)

(0) ht1

+ u(0) x

+ vy(0)

= 0.

(5.9)

The next terms of the perturbation series are of the order of max(δ 2 , ε). Equations for these terms, with introduction of the corresponding slow time t2 = max(δ 2 , ε)t1 , are written as follows:

(0) ε (1) (0) (0) ut1 − yv (1) + h(1) (5.10) ut2 + u(0) u(0) x =− x + v uy , 2 max(δ , ε)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

108

δ2 (0) v , (5.11) max(δ 2 , ε) t1

(0) (0) (0) ε (1) (1) ht1 + u(1) ht2 + h u x + h(0) v (0) y . (5.12) x + vy = − 2 max(δ , ε) Already a qualitative analysis of these equations reveals different scenarios of the Rossby waves dynamics arising for different relations between ε and δ: yu(1) + h(1) y =−

(0)

• If δ 2 ε, equations (5.10)–(5.12) become linear. Because it is the term vt1 in the r.h.s. of (5.11) which provides dispersion in the system, it means that within this range of parameters dispersion overcomes nonlinearity and, in the context of the geostrophic adjustment, any initial packet of Rossby waves will be dispersed. • If δ 2 ε the nonlinearity overcomes dispersion. General properties of this regime were recently presented by Majda (2003). A non-compensated nonlinearity usually means breaking. However, the precise dynamical meaning of this process is not clear at this stage. For instance, it is known that equatorial modons may exist at strong nonlinearities (Boyd, 1985). Does equatorial adjustment with subsequent Rossby-wave “breaking” produce them in this regime? We give some elements of the answer below and in Chapter 5. • Finally, at δ 2 ∼ ε, the advective nonlinearity and dispersion may compensate each other and form solitary waves from the initial perturbation in the Rossbywave part of the spectrum (it should be remembered that long Rossby waves have weak dispersion). A typical equation combining the effects of weak nonlinearity and weak dispersion is the KdV (Korteweg–de Vries) or mKdV one and one may expect that the Rossby-wave packet will obey one of them in this regime and form solitons (Boyd, 1980a). As to the long Kelvin waves they should always break (Boyd, 1980b; Ripa, 1982), because they are defined by the condition v (0) = 0 and are, thus, nondispersive whatever the relation between δ and ε. Their breaking, therefore, consists in formation of a hydraulic jump. It is worth mentioning that breaking of Kelvin waves may be prevented by a fine-tuned mean zonal flow (Boyd, 1984; Le Sommer, Reznik and Zeitlin, 2004). 5.1.2. The fast motion: long inertia–gravity and Yanai waves For these waves u ∼ v and equations (5.10)–(5.12) remain unchanged. In the lowest order we get: (0) = 0, u(0) t − yv (0) vt (0) ht

+ yu

(0)

+ h(0) y

+ vy(0) = 0.

(5.13) = 0,

(5.14) (5.15)

5. Nonlinear geostrophic adjustment in the equatorial region

109

A single equation for v (0) follows from this system (cf. Gill, 1982): (0) vtt(0) + y 2 v (0) − vyy =0

(5.16)

which has the following solution in terms of the parabolic cylinder functions φn (y): v

(0)

=

∞

vn(0) (x, t)φn (y);

iσn t vn(0) = A+ + c.c.; σn = 0n (x, t1 )e

n=0

√ 2n + 1. (5.17)

Thus, in the lowest order we obtain non-propagating oscillations with frequencies σn which are well separated from the zero frequency (in the fast time t) of long Rossby and Kelvin waves (cf. Fig. 11 of dispersion curves for equatorial waves in Chapter 1). In the limit ε → 0, i.e. neglecting nonlinearity in (5.1)–(5.3), the next approximation in δ gives a non-dispersive eastward propagation of the envelope A0n in the slow time t1 = δt: t1 ± A± = A x − 0n 0n 2σn2

(5.18)

which is consistent with the group velocity estimate for long waves (cf. Fig. 9, Chapter 1). At the next order in δ dispersion will give rise to evolution in t2 = δ 2 t. If, on the contrary, the limit δ → 0 is taken in (5.1)–(5.3) (infinitely long waves), the resonances generated by nonlinearity appear only at the order O(ε2 )) because, as easily seen from (5.17), the O(ε) bilinear combinations of the oscillations of the form (5.17) are non-resonant: √ √ √ 2k + 1 = 2l + 1 ± 2m + 1

(5.19)

at any integer k, l, m. At the same time, the cubic combinations appearing at order O(ε 2 ) do produce resonances. Hence, in order to counter-balance the dispersion of the envelope of the fast waves by nonlinearity we need to have δ ∼ ε. In this regime one can expect that dynamics of the envelope A0n in t2 , as usual for a weakly nonlinear envelope evolution, obeys the nonlinear Schrödinger equation, or a system of coupled Schrödinger equations if simultaneous evolution of several modes of the type (5.17) is considered. Thus formation of the envelope solitons (cf. Boyd, 1983) is possible. Under condition of very weak dispersion, δ ε the nonlinear interactions give a small frequency shift (no breaking). Under condition of strong dispersion δ ε, the envelope A0n is slowly dispersed.

110

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

5.2. The resumé of possible weakly nonlinear dynamical regimes in the long-wave approximation In the absence of the mean flow the results of the preceding qualitative analysis may be summarized as follows • For Rossby numbers up to δ 2 Rossby waves are dispersed as well as the envelope of the gravity waves; Kelvin waves break down forming hydraulic jumps. • For Rossby numbers of the order of δ 2 Rossby waves form solitons, the envelope of the gravity waves is dispersed and Kelvin waves break down. • For Rossby numbers in the interval (δ 2 , δ) Rossby and Kelvin waves break down, the envelope of the gravity waves is dispersed. • For Rossby numbers of the order of δ, Rossby and Kelvin waves break down, the envelope of the gravity waves obeys the nonlinear Schrödinger equation and forms modulation solitons. • For Rossby numbers greater than δ Rossby and Kelvin waves break and the envelope of the gravity waves has a nonlinear frequency shift Asymptotic analysis of all the aforementioned regimes may be performed confirming these qualitative conclusions (Le Sommer, Reznik and Zeitlin, 2004, unpublished). The results of a detailed analysis of the regime ε ∼ δ 2 , which is, perhaps, the most interesting example due to appearance of Rossby solitons, are presented below. For typical Rossby numbers of the order 0.1 for the balanced motions in the equatorial atmosphere (cf. e.g. Majda and Klein, 2003) and of the order 0.3 for the ocean (cf. e.g. Boyd, 1980a, 1980b) the typical horizontal scales in this regime correspond to thousands of kilometers (planetary-scale perturbations) in the atmosphere and to hundreds of kilometers (mesoscale) in the ocean. They are, hence, appropriate e.g. for the oceanic perturbations due to equatorial westerly wind bursts (cf. Philander, 1990). 5.3. The long-wave adjustment at Ro = O(δ 2 ) In this regime, the basic equations (5.1)–(5.3) take the form: ut + δ 3 uux + δ 2 vuy − yv = −δhx ,

(5.20)

vt + δ 3 uvx + δ 2 vvy + yu = −hy ,

(5.21)

ht + vy + δux + δ (hv)y + δ (hu)x = 0.

(5.22)

2

3

The analysis of geostrophic adjustment in this system in the first four orders in δ is presented below. As usual, solution of (5.20)–(5.23) is sought in the form of an asymptotic series in δ: u = u(0) (x, y, t, t1 , t2 , . . .) + δu(1) (x, y, t, t1 , t2 , . . .) + · · · ,

(5.23)

5. Nonlinear geostrophic adjustment in the equatorial region

111

and similar for the fields v and h. Here the slow time-scales t1 = δt, t2 = δ 2 t, . . . are introduced. Each field f (x, y, t, t1 , t2 , . . .) is represented as a sum of the slow component f¯(x, y, t1 , t2 , . . .) defined as the average of f over the fast time t, and the fast component f˜ = f − f¯. At each order we describe the evolution of both slow and fast components of the flow and show how to split unambiguously the initial conditions and obtain well-posed Cauchy problems for both components. The fast and the slow variables are not mutually independent due to nonlinearity: the slow evolution of the fast fields depends on the slow ones (guiding) and, in principle, the self-interaction of the fast components should influence the slow fields. However, it may be shown (Le Sommer, Reznik and Zeitlin, 2004) that at least at the leading order there is now fast-component drag upon the slow one. The general scheme of calculations, which is entirely algorithmic, will be presented in the next subsection. Technical details may be found in Le Sommer, Reznik and Zeitlin (2004), as well as the initialization procedure which is well defined for both slow and fast fields. Note that results obtained at the order n pro1 vide a description of the system for times up to δ n+1 T , where T = (βRe )−1 is the equatorial time-scale, cf. Chapter 1. It should be stressed that the results on the slow motion alone, which are obtained below in parallel with the description of the fast motion, were known from the pioneering works of Boyd and Ripa: the KdV dynamics of weakly nonlinear equatorial Rossby waves (5.56) was discovered in Boyd (1980a). The overturning of the equatorial Kelvin waves (5.57) was also demonstrated by Boyd (1980b) and Ripa (1982). However, these studies started from a close to the geostrophic balance situation, i.e. they filtered the fast component of motion from the very beginning. The novelty of the approach of Le Sommer, Reznik and Zeitlin (2004) is that without any filtering it was proved that fast and slow motions are dynamically split and the old results are consistent. 5.3.1. The method Equations for the nth approximation in δ are taken in the form: (n−1) (n) u(n) + ut1 + h(n−1) = Pu(n) , t − yv x

(5.24)

(n) vt (n) ht

(5.25)

(n) + yu + h(n) y = Pv , (n−1) + ht1 + vy(n) + u(n−1) x (n)

=

(n) Ph .

(5.26)

Simultaneously the equation for v from the previous step is used: = Pv(n−1) . vt(n−1) + yu(n−1) + h(n−1) y

(5.27)

Here, by definition, u(−1) = v (−1) = h(−1) = 0;

(−1)

Pu(−1) = Pv(−1) = Ph

= 0,

(5.28)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

112

and the r.h.s. for the first three orders are: (0)

Pu(0) = Pv(0) = Ph Pu(1) = 0, Pu(2) (2) Ph

= =

Pu(3) = (3) Ph

=

= 0,

Pv(1) =

(5.29)

(0) −vt1 ,

(0) −ut2 (0) −ht2

− v (0) u(0) y ,

(0) −ut3 (0) −ht3

(1) − ut2 (1) − ht2

− h(0) v (0) y ,

(1) Ph

Pv(2)

= 0,

(0) (1) = − vt2 + vt1 + v (0) vy(0) ,

(5.30)

(5.31)

(0) (1) (0) (0) − v (1) u(0) y − v uy − u ux ,

(0) (1)

− h v + h(1) v (0) y − h(0) u(0) x .

(5.32) (5.33)

(3)

The explicit expression for Pv will not be needed in what follows. We get, by averaging (5.24)–(5.27), the following equations for the slow fields: u¯ (n−1) + h¯ (n−1) − y v¯ (n) = Pu(n) , t1 x (n) y u¯ (n) + h¯ (n) y = Pv ,

(5.34) (5.35)

(n−1) (n) h¯ t1 + u¯ (n−1) + v¯y(n) = Ph , x

y u¯

(n−1)

+ h¯ (n−1) y

=

(5.36)

Pv(n−1) .

(5.37)

Among these equations (5.34), (5.36) and (5.37) are sufficient to determine the slow motion, while (5.35) will be used in the next approximation. Correspondingly, for the fast fields we get: u˜ t

u(n) − u˜ (n−1) u(n) , − y v˜ (n) = P − h˜ (n−1) ≡R t1 x

(5.38)

(n) v˜t (n) h˜ t

(n) (n) + y u˜ + h˜ (n) y = Pv ≡ Rv , (n) − h˜ (n−1) + v˜y(n) = P − u˜ (n−1) t1 x h

(5.39)

(n)

(n)

(n) . ≡R h

(5.40)

The key idea of the algorithm is to solve for the variable v¯ (n) first, and then to determine u¯ (n−1) and h¯ (n−1) . By differentiating and combining equations (5.34), (5.36), (5.37) a single equation for v¯ (n) is obtained:

(n) v¯yy − y 2 v¯ (n) t + v¯x(n) 1

(n)

(n) (n) = y Pu + Phy − Pv(n−1) − y Ph + Pu(n) − Pv(n−1) , (5.41) t y x t x 1

1

(n) v¯I :

and an equation for its initial value

(n−1)

(n) (n) (n−1) (n) + y h¯ I + y Pu(n) + Phy − Pv(n−1) . (5.42) v¯Iyy − y 2 v¯I = u¯ Iy t x I 1

As usual, the slow evolution follows from removal of resonant terms in the r.h.s. of (5.41). We proceed by subsequent averaging of this equation in t2 , t3 to determine corresponding evolution in slow times of the previous-order slow components.

5. Nonlinear geostrophic adjustment in the equatorial region

113

After resonances are removed, the bounded solution v¯ (n) can be determined from (5.41). It should be stressed that the analysis of resonances in the r.h.s. of (5.41) allows to fix the slow evolution of the Rossby-wave part of the perturbation only, because the linear operator in the l.h.s. of this equation corresponds precisely to Rossby waves. To describe the slow evolution of the Kelvin-wave part we add equations (5.34) and (5.36) and obtain:

(n−1)

u¯ + h¯ (n−1) + u¯ (n−1) + h¯ (n−1) =

−v¯y(n)

+ y v¯

t1 (n)

x

+ Pu(n)

(n) + Ph .

(5.43)

The linear operator in the r.h.s. here corresponds to Kelvin waves and the resonances are absent if (n) (n) v¯y − y v¯ (n) (K) = Pu(n) + Ph (K) , (5.44) where the subscript (K) means that only the terms ∝ U (x −t1 ) (cf. (5.51) below) are considered in the bracketed expression. A bounded solution of this equation exists if +∞ dy φ0 Pu(n) + Ph(n) (K) = 0,

(5.45)

−∞

which gives a slow evolution of the Kelvin waves of preceding orders (see details in Le Sommer, Reznik and Zeitlin, 2004). Knowing v¯ (n) and having removed resonances from the r.h.s. of (5.34), (5.36) and (5.37) we may find u¯ (n−1) and h¯ (n−1) from (5.34) and (5.36), and thus determine the slow component of motion at the given order. The order n − 1 slow component is a sum of free Rossby and Kelvin waves and some “forced” terms generated by nonlinear interactions of preceding-order fields. The slow evolution of order n − 1 Rossby and Kelvin waves is determined from the analysis of the (n + 1)th order, etc. The same strategy is used for the fast component. From (5.38)–(5.40) we obtain an equation for v˜ (n) : (n) v(n) − R − yR u(n) , v˜tt + y 2 v˜ (n) − v˜yy =R hy t (n)

(n)

(5.46)

= −v¯I , (n) = − yu(n) + h(n) − P (n) v˜t

(5.47)

with initial conditions: (n)

(n)

v˜I

t=0

y

v

t=0

.

(5.48)

Removal of resonances from the r.h.s. of this equation gives slow evolution of the fast variables. The linear operator in the l.h.s. corresponds to inertia–gravity and Yanai waves. Hence, namely the dependence on slow times of these fast waves,

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

114

i.e. modulation, will be thus determined. After removal of resonances v˜ (n) can be determined from (5.46) and then u˜ (n) and h˜ (n) may be found from (5.38) and (5.40). It is easy to see that the fast and the slow variables are not mutually independent. The r.h.s. of (5.46) contains the terms of the form u¯ (k) v˜ (m) , k, m < n and, therefore, the slow evolution of the fast fields depends on slow fields (guiding). In turn, the r.h.s. of equations (5.34)–(5.37) contain terms of the type u˜ (k) v˜ (m) and, in principle, the fast motion should influence the slow ones. 5.3.2. The main results 5.3.2.1. Fast motion. The fast motion consists of (infinitely) long inertia– gravity and Yanai waves of the form (5.17). The slow-time dependence of the modulated amplitude A0n is given by (5.18). In the next approximation A0n acquires a nonlinear frequency shift and a frequency shift due to the mean zonal flow Φ (M) , see Le Sommer, Reznik and Zeitlin (2004) for details: i i 3 (M) ± A± (5.49) ± 1 − A± 0nt 0nxx ± 2σ Φnn A0n = 0. 2σn 4σn4 2 n The first correction to the amplitude obeys the following equation: A± 1n + t1

1 ± i (K) (R) Φnn (x, t1 ) + Φnn A1n = ∓ (x, t1 ) A± 0n , 2 x 2σn 2σn

(5.50)

where Φ (K) , Φ (R) represent contributions from the slow Kelvin and Rossby waves, respectively, see Le Sommer, Reznik and Zeitlin (2004) for details. Thus, the modulation of the fast component is guided by the slow one. 5.3.2.2. Slow motion. The full slow solution (u¯ (0) , v¯ (1) , h¯ (0) ) is a sum of three contributions. All of them are unambiguously defined. The first one, (0) (0) (u¯ (K) , 0, h¯ (K) ) is the eastward-propagating Kelvin-wave part: u¯ (K) = h¯ (K) = U0 (x − t1 )φ0 (y). (0)

(0)

(5.51)

(0) (1) (0) The second one is the westward-propagating Rossby-wave part (u¯ (R) , v¯(R) , h¯ (R) ): (0) u¯ (R)

∞

n=1

(1)

√

1

V1n x + c¯n t1 = 2

v¯(R) =

∞ n=1

√ 2(n + 1) 2n φn+1 (y) − φn−1 (y) , 1 + c¯n 1 − c¯n

v¯n(1) x + c¯n t1 φn (y),

(5.52) (5.53)

5. Nonlinear geostrophic adjustment in the equatorial region

h¯ (0) (R)

∞

115

√ √ 2(n + 1) 2n φn+1 (y) + φn−1 (y) , 1 + c¯n 1 − c¯n

1

V1n x + c¯n t1 = 2 n=1

1n = where V x

(1) v¯n ,

ically balanced

1 and c¯n = 2n+1 = σn2 . The (0) (0) zonal flow (u¯ (M) , 0, h¯ (M) ):

(5.54) third one is an arbitrary geostroph-

y u¯ (M) + h¯ (M)y = 0. (0)

(0)

(5.55)

The slow evolution of the Rossby-wave component is given by the KdV equation 1n + βn V 1n V 1n = 0. 1n + αn V V xxx x t 3

(5.56)

The coefficients αn , βn of the KdV equation are constants depending only on the meridional structure of the mode (see Le Sommer, Reznik and Zeitlin, 2004, for details). Hence, according to the well-known properties of the KdV equation, the Rossby-wave component of the initial localized perturbation will split into a sequence of westward-propagating solitons. The slow evolution of the Kelvin-wave component is given by the Riemann wave equation 3 1 U0 U0x = 0 U0t3 + 1/4 (5.57) 2 π leading to overturning in finite time (in terms of the third slow time t3 ). 5.3.2.3. Fast–slow splitting. The results obtained for the slow motion are rather surprising, because the evolution equations (5.56) and (5.57) do not contain any trace of the fast component, in spite of the aforementioned influence of the fast fields upon the slow ones. Le Sommer, Reznik and Zeitlin (2004) proved by a direct calculation that the averaged fast–fast terms are not resonant and, hence, give no contribution (drag) into the slow-fields evolution equations at least up to times O(δ −4 T ). Thus, the slow and the fast motions are dynamically split and justify the naive filtering (i.e. throwing away) of the fast waves which provides a “fast-track” derivation of (5.56) and (5.57). 5.3.3. The summary of the results of the asymptotic theory of equatorial adjustment Thus, the detailed analysis confirms the scenario proposed in Section 5.1. for the regime ε ∼ δ 2 . We have shown that an arbitrary long-wave equatorial perturbation with small characteristic Rossby number splits into fast and slow components. The former is a packet of inertia–gravity and Yanai waves, Doppler-shifted by

116

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

the slow component, and experiencing a nonlinear frequency shift. The fast component does not influence the slow one. The latter is a combination of Rossby and Kelvin waves propagating in opposite directions along the equator and obeying their own modulation equations. For Rossby waves the evolution equation is KdV which is known to produce a sequence of solitons from a localized initial disturbance. For Kelvin waves the evolution equation is a simple-wave equation which produces overturning in finite time. Thus, within the limits of validity of the non-dissipative asymptotic theory, i.e. for times of the order δ −3 T the dynamical splitting of slow and fast component is proved. Note that going farther in the asymptotic expansions does not make sense as the asymptotic theory breaks down once Kelvin waves overturn and generate smaller characteristic scales. Therefore, the proof is given that the filtering of the fast component applied in the works of Boyd (1980a, 1980b) and Ripa (1982) is legitimate and that the fast motion exerts no drag (up to times δ −3 T ) upon the slow motion. However, as was just mentioned, the non-dissipative asymptotic theory has intrinsic limitations due to the breaking of Kelvin waves. For localized initial disturbances on the infinite equatorial β-plane these limitations do not affect the westwardmoving Rossby-wave part of the perturbation much, as Kelvin waves move in the opposite direction. However, for the eastward-moving part of the perturbation the overturning of the Kelvin waves will, presumably, destroy the fast–slow splitting. The practical question of robustness of the asymptotic theory also arises (as its convergence is not proved), i.e. the question of the behaviour of the system for small but finite ε and δ. Another question beyond the limits of the present theory is whether the Rossby-wave breaking for strong nonlinearities can generate modons, which was already mentioned in Section 5.1. The predictions of the theory were successfully tested, and the aforementioned questions answered in high-resolution numerical simulations of the equatorial adjustment (Le Sommer, Reznik and Zeitlin, 2004). These results, along with the simulations of strongly nonlinear phenomena which allow to answer the aboveposed questions, will be presented in Chapter 5.

6. Summary and discussion Thus, the straightforward asymptotic analysis in Rossby number applied to the nonlinear geostrophic adjustment of localized perturbations in the RSW models on the mid-latitude f -plane shows that dynamical splitting of fast and slow motions does take place, in the sense that fast and slow components satisfy their proper dynamical equations with unambiguously defined initial conditions. While slow component does not “feel” the fast one at all in all of the considered regimes, the fast motion is influenced by the slow one beyond the leading order (secondary emission of IGW in the QG regime, guided inertial oscillations in the FD regime).

6. Summary and discussion

117

The initial conditions for the fast motion are also coupled to initial conditions for the slow motion. This coupling at the level of initial conditions becomes dramatic in the non-localized initial conditions at the boundary in the case of laterally bounded domains, where initial conditions of the fast component (Kelvin wave) enter explicitly in the evolution equation of the slow component. Although the boundary Kelvin waves inevitably appear in any event of adjustment in the presence of lateral boundary, and their presence is necessary to guarantee the mass conservation, they do not prevent splitting in the case of localized initial perturbations, in spite of the fact that their presence destroy the gap in the spectrum of linear excitations. In the laterally bounded domains the presence of non-dispersive Kelvin waves imposes a formal upper limit of validity of the asymptotic results in time, which is defined by the breaking time, beyond which the formal asymptotic theory is not valid anymore. However, the breaking event happens far from the location of the initial perturbation (due to the fast propagation of Kelvin waves) and does not influence much the localized slow-variability domain. Although the equatorial RSW dynamics possesses a highly nontrivial spectrum of linear excitations, the equatorial adjustment of localized finite-energy perturbations follow, roughly speaking, the same scenario (at least in the long-wave limit): the slow (Kelvin and Rossby modes) and fast (IGW and Yanai modes) are dynamically split, with initialization well-defined for each component. The slow motion experiences no fast-motion drag, while the fast motion, again is guided (Doppler-shifted) by the slow one. The proper dynamics of the vortex (Rossby) mode is found to be governed by KdV (or mKdV) equations, while equatorial Kelvin waves break down in finite time. This fact, as in the case of the boundary Kelvin waves imposes a formal time-limit of applicability of the asymptotic theory, although, as for the boundary Kelvin waves the influence of this phenomenon onto the vortex mode is weak. The detailed numerical analysis of nonlinear equatorial adjustment and Rossby–Kelvin interactions will be presented below in Chapter 5. Hence, we have a full and comprehensive theory of the geostrophic adjustment of localized perturbations at small Rossby numbers. It is to be stressed again that smallness of the Rossby number does not necessarily mean weak nonlinearities. It is instructive to see what kind of new phenomena could appear if Rossby number of initial perturbation increases. Some of these phenomena were already discussed in Chapter 1. The first one is Lighthill radiation (Lighthill, 1952) which requires, as already discussed in Chapter 1, large Rossby and Burger numbers. Due to this phenomenon, the adjusted state continues to emit the IGW. This emission mechanism is different from the adjustment emission described above. The Lighthill radiation is effective both in barotropic (one-layer) and baroclinic (twolayer) systems. The second phenomenon is purely baroclinic and does not exist in the barotropic system. It is a possible instability of the balanced motions with respect to unbalanced perturbations (see Chapter 1 and Sakai, 1989). In contradis-

118

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

tinction with the standard baroclinic instability, this instability is fast, may have growth rates higher than that of the baroclinic instability (see Sakai, 1989, and Chapter 1), and hence may drive the system out of the state to be reached in course of the geostrophic adjustment (this balanced state could be determined from the analysis of the integrals of motion, like in the pioneering paper by Rossby, 1938). In addition, such instability can be a source of a secondary IGW emission, either directly, or via formation of secondary intense vortices and Lighthill radiation. The symmetric instability, appearing necessarily at high Rossby numbers (see Chapter 5 below), is also of this kind. At even higher Rossby numbers the Kelvin– Helmholtz instability comes into play, which is another source of secondary IGW emission.

Acknowledgements The results presented above were obtained in collaboration with M. Ben Jelloul, R. Grimshaw, and G. Sutyrin which is gratefully acknowledged. This work was supported by the RFBR Grant No. 05-05-64212.

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Dewar, W.K., Killworth, P.D., 1995. Do fast gravity waves interact with geostrophic motions? DeepSea Research I 42, 1063–1081. Dorofeyev, V.L., Larichev, V.D., 1992. The exchange of fluid mass between quasi-geostrophic and ageostrophic motions during the reflection of Rossby waves from a coast. 1. The case of an infinite rectilinear coast. Dynam. Atmos. Oceans 16, 305–329. Embid, P.F., Majda, A.J., 1996. Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial Differential Equations 21, 619–658. Falkovich, G.E., 1992. Inverse cascade and wave condensate in mesoscale atmospheric turbulence. Phys. Rev. Lett. 69, 3173–3176. Falkovich, G.E., Kuznetsov, E.A., Medvedev, S.B., 1994. Nonlinear interaction between long inertio– gravity and Rossby waves. Nonlin. Proc. Geophys. 1, 168–172. Falkovich, G.E., Medvedev, S.B., 1992. Kolmogorov-like spectrum for turbulence of inertial-gravity waves. Europhys. Lett. 19, 279–284. Gent, P.R., McWilliams, J.C., 1983. Regimes of validity for balanced models. Dynam. Atmos. Ocean 7, 167–183. Gill, A.E., 1976. Adjustment under gravity in a rotating channel. J. Fluid Mech. 77, 603–621. Gill, A.E., 1982. Atmosphere–Ocean Dynamics. Academic Press, New York (Chapter 7.2). Helfrich, K.R., Kuo, A.C., Pratt, L.J., 1999. Nonlinear Rossby adjustment in a channel. J. Fluid Mech. 390, 187–222. Helfrich, K.R., Pedlosky, J., 1995. Large-amplitude coherent anomalies in baroclinic zonal flows. J. Atmos. Sci. 52, 1615–1629. Hermann, A.J., Rhines, P.B., Johnson, E.R., 1989. Nonlinear Rossby adjustment in a channel: beyond Kelvin waves. J. Fluid Mech. 205, 469–502. Hoskins, B.J., 1975. The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32, 233–242. Kamenkovich, V.M., Koshlyakov, M.N., Monin, A.S., 1986. Synoptic Eddies in the Ocean. Reidel, Dordrecht, 433 pp. Kamenkovich, V.M., Reznik, G.M., 1978. Rossby waves. In: Physics of the Ocean, vol. 2. Hydrodynamics of the Ocean. Nauka, Moscow, pp. 300–358 (in Russian). Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial β-plane. J. Fluid Mech. 515, 135–170. Lighthill, J., 1952. On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. London A 211, 564–587. Lighthill, J., 1980. Waves in Fluids. Cambridge Univ. Press, Cambridge, UK, 504 pp. Lorenz, E.N., 1980. Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37, 1685–1699. Lorenz, E.N., 1986. On the existence of the slow manifold. J. Atmos. Sci. 43, 1547–1557. Machenhauer, B., 1977. On the dynamics of gravity oscillations in a shallow water model, with application to normal-mode initialization. Contrib. Atmos. Phys. 50, 253–271. Majda, A.J., 2003. Introduction to PDEs and Waves for the Atmosphere and Ocean. Amer. Math. Soc., New York (Chapter 9). Majda, A.J., Klein, R., 2003. Systematic multiscale models for the tropics. J. Atmos. Sci. 60, 393–408. McWilliams, J.C., 1977. A note on a consistent quasigeostrophic model in a multiply connected domain. Dynam. Atmos. Oceans 1, 427–441. McWilliams, J.C., 1988. Vortex generation through balanced adjustment. J. Phys. Oceanogr. 18, 1178– 1192. Medvedev, S.B., 1999. The slow manifold for the shallow water equations on the f -plane. J. Atmos. Sci. 56, 1050–1054. Mohebalhojeh, A.R., Dritschel, D.G., 2001. Hierarchies of balance conditions for the f -plane shallowwater equations. J. Atmos. Sci. 58, 2411–2426. Monin, A.S., Obukhov, A.M., 1958. Small oscillations of the atmosphere and adjustment of meteorological fields. Izvestija—Geophysics 11, 1360–1373 (in Russian).

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Obukhov, A.M., 1949. On the problem of geostrophic wind. Izvestia—Geography and Geophysics 13, 281–306 (in Russian). Pedlosky, J., 1982. Geophysical Fluid Dynamics. Springer-Verlag, New York. Pedlosky, J., 1984. The equations for geostrophic motion in the ocean. J. Phys. Oceanogr. 14, 448– 455. Philander, S.G., 1990. El Nino, La Nina, and the Southern Oscillation. Academic Press, San Diego, CA. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286. Reznik, G.M., Grimshaw, R.H.G., 2001. Ageostrophic dynamics of an intense localized vortex on the β-plane. J. Fluid Mech. 443, 351–376. Reznik, G.M., Grimshaw, R.H.G., 2002. Nonlinear geostrophic adjustment in the presence of a boundary. J. Fluid Mech. 471, 257–283. Reznik, G.M., Grimshaw, R.H.G., Benilov, E.S., 2000. On the long-term evolution of an intense localized divergent vortex on the β-plane. J. Fluid Mech. 422, 249–280. Reznik, G.M., Sutyrin, G.G., 2005. Non-conservation of “geostrophic mass” in the presence of a long boundary and the related Kelvin wave. J. Fluid Mech. 527, 235–264. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow water model. J. Fluid Mech. 491, 207–228. Ripa, P., 1982. Nonlinear wave-wave interactions in a one-layer reduced-gravity model on the equatorial β-plane. J. Phys. Oceanography 12, 97–111. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems. J. Marine Res. 1, 239–263. Sakai, S., 1989. Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 445, 93–120. Saujani, S., Shepherd, T.G., 2002. Comments on “Balance and the slow quasimanifold: some explicit results”. J. Atmos. Sci. 59, 2874–2877. Stegner, A., Zeitlin, V., 1995. What can asymptotic expansions tell us about large-scale quasigeostrophic anticyclonic vortices? Nonlin. Proc. Geophys. 2, 186–193. Tomasson, G.G., Melville, W.K., 1992. Geostrophic adjustment in a channel: nonlinear and dispersive effects. J. Fluid Mech. 241, 23–58. Vautard, R., Legras, B., 1986. Invariant manifolds, quasi-geostrophy and initialization. J. Atmos. Sci. 43, 565–584. Warn, T., Bokhove, O., Shepherd, T.G., Vallis, G.K., 1995. Rossby number expansions, slaving principles, and balanced dynamics. Quart. J. R. Meteorol. Soc. 121, 723–739. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York (Chapter 11). Young, W.R., Ben Jelloul, M., 1997. Propagation of near-inertial oscillations through a geostrophic flow. J. Mar. Res. 55, 735–766. Zeitlin, V., Reznik, G.M., Ben Jelloul, M., 2003. Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations. J. Fluid Mech. 491, 207–228.

Chapter 3

The Method of Normal Forms and Fast–Slow Splitting S.B. Medvedev Institute of Computational Technologies, 6 Lavrentiev av., 630090 Novosibirsk, Russia Abstract In this chapter a development of normal form methods for special classes of partial differential equations is presented. A basic application of the methods is splitting of slow and fast wave motions and finding of equations for the slow wave motion. The rotating shallow water model is the main example for the application of the general theory.

Contents 1. Introduction 2. Slow manifold in the spatial variables on the f -plane 2.1. Slow manifold 2.2. Initialization

3. Slow manifold in the spectral variables on the f -plane 3.1. Fast and slow formal invariant manifolds 3.2. Normal forms

4. Poincaré normal forms 4.1. Basic theorems 4.2. Normal form for the RSW at large scales 4.3. Normal form of RSW equations on the mid-latitude β-plane 4.4. Normal form of RSW equations on the equatorial β-plane 4.5. Matrix elements for RSW equations

5. Skew-gradient normal forms for gradient systems 5.1. General construction 5.2. Skew-symmetric normal form for RSW equations

6. Hamiltonian normal forms 6.1. Darboux theorem for finite-dimensional systems 6.2. Darboux theorem for continuous systems Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02003-X 121

122 124 125 129 132 133 136 137 138 143 145 150 155 156 158 160 163 163 166 © 2007 Elsevier B.V. All rights reserved

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Chapter 3. The Method of Normal Forms and Fast–Slow Splitting 6.3. Rossby waves 6.4. RSW equations

7. Normal form of the Poisson bracket for one-dimensional fluid 7.1. Ideal fluid and non-rotating shallow water equations 7.2. RSW equations

8. Conclusion References

169 171 175 176 179 183 184

1. Introduction The method of normal forms is a powerful tool to study dynamical systems. The main idea of the method is to eliminate some non-essential nonlinear terms from a system by a change of variables and to keep other important nonlinear terms. Then one can study the system only with important terms. Often the definition of the important and non-essential terms is based on resonance conditions. In this case these terms are called resonant and non-resonant, respectively. This is a simplest situation. Many of the systems have special structures (for example, Hamiltonian structure) and it is reasonable to eliminate the non-essential terms and to keep the initial structure of the system. For such systems one has to develop a suitable change of variables. In this case the non-essentiality of the terms is defined by possibility to be eliminated by a special change of the variables. In this chapter three normal forms are considered. The first form is the Poincaré normal form. This form is an extension of the Poincaré normal form theory of the ordinary differential equations for partial differential equations, which have linear ordinary equations as a main part. The second form is a skew-symmetric normal form for the skew-gradient systems with a quadratic positive characteristic function. This form is an intermediate one between the Poincaré normal form and the Hamiltonian normal form. The third form is the Hamiltonian one. Actually this is the normal form for the Poisson brackets and the corresponding theorem is a version of the Darboux theorem for the Poisson brackets. All these normal forms arose in works of the author after attempts to split the slow and the fast wave motions for the rotating shallow water equations, which represent the simplest model of atmosphere and ocean. It is necessary to underline that in this context we are dealing with the wave motions. Any wave motion can be described by an amplitude and a phase. Here lies the difference with the averaging theory where usually the amplitude is a slow variable and the phase is a fast variable. If we talk about the fast (slow) wave motion then it means that the frequency of the wave motion is fast (slow). One of the aims of what follows is to present a general approach for getting equations for the slow motion at arbitrary

1. Introduction

123

order of approximation. The Charney–Obukhov equation1 is the first and main example of such equation, because it describes the slow motion of Rossby waves. However obtaining systematic approximations was an open problem. We present below the general theory of the normal forms to give a basis for future investigations in this direction. The applications of the general theory to the rotating shallow water equations give solution of the fast–slow splitting problem for this system and reveal strong and weak sides of the developed method. The plan of the chapter is the following. In Section 2 we construct the slow manifold and the corresponding equation on it for the rotating shallow water equations on the f -plane using an asymptotic expansion (Medvedev, 1999a). The approach using the spatial independent variables allows for representation of the slow manifold and of the corresponding equation in a simple form. The dynamical and static initializations are considered and a new balance equation, which is a generalization of the Charney balance equation, is obtained. In Section 3 we present a construction of the slow and fast manifolds using the spectral independent variables (Medvedev, 1996). This consideration allows for obtaining a general structure of these manifolds without knowledge of a partial form for matrix coefficients. Approximate splitting can be realized by an invertible transformation with a finite number of terms. In Section 4 the Poincaré normal forms are constructed for the rotating shallow water equations in different situations (Medvedev, 1999b). The first case is the normal form for large-scale motions (with respect to Rossby radius) with the positive Coriolis parameter. This situation is considered in spatial independent variables. The second and the third examples are considered in spectral variables on the mid-latitude and equatorial β-plane. In Section 5 a normal form for skew-gradient systems is suggested (Medvedev, 2003). A main advantage of this normal form is the conservation of a characteristic function after cancellation of higher order terms. The rotating shallow water equations are a particular example of the skew-gradient system. In Section 6 the asymptotic version of the Darboux theorem for continuous Hamiltonian system is stated (Medvedev, 2004). This theorem gives a method of factorization of Poisson brackets into symplectic and transversal parts. As the first example the Poisson bracket of the equation for the Rossby waves is transformed to a constant Poisson bracket. In the second example the Poisson bracket of the rotating shallow water equations is factorized. In the last Section 7 we study one-dimensional systems and select a partial set of the Poisson brackets which can be transformed into direct product of symplectic and zero Poisson brackets (Medvedev, 2005). The Poisson bracket of the onedimensional rotating shallow equations belongs to this class. In conclusion a discussion of the obtained results and a connection with the fast/slow splitting are presented. 1 The quasi-geostrophic barotropic equation on the β-plane (see Chapter 1).

124

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

2. Slow manifold in the spatial variables on the f -plane In this section functional equations defining the slow manifold are obtained for the shallow water equations on the f -plane. These equations are solved using an expansion in powers of nonlinearity. Solutions of the shallow water equations corresponding to initial conditions on the slow manifold evolve on this manifold during a long time and are devoid of oscillations at the frequency of inertia– gravity waves. The evolution equation on the slow manifold is obtained. The knowledge of explicit differential equations for the slow manifold allows one to solve the problem of initialization in a new manner. For the dynamical initialization, the sought fields are obtained as power series of the slow mode amplitude with known coefficients. For the static initialization in terms of the depth of the fluid, the velocity field is determined from strongly nonlinear equation. The problem of fast/slow splitting and the slow manifold arose from the practical problem of the initialization of the initial data for numerical models. Unfortunately, using the data without any preparation gives unrealistic fast oscillations for the initial period of numerical integration. The solution of this problem is to split the initial data into the slow and fast parts and to reduce the amplitude of the fast part. To understand this procedure we should know what the slow part is and how it evolves. This idea is based on the observation that the “real” weather is the slow part of the atmospheric motions. The problem of normal mode initialization has been widely studied (see Section 2.2 for definitions of initializations). Baer and Tribbia (1977) gave the solution of this problem by method of two time scales. A geometrical interpretation of solution of the initialization problem was suggested by Leith (1980) using the concept of the slow manifold. In the same work, the connection of the dynamical and static initializations was shown. Vautard and Legras (1986) proposed another initialization method based on the analytic expansion of local invariant manifolds. But in practice the Machenhauer (1977) initialization or the algorithm suggested by Tribbia (1984) are employed, because the realization complexity of the Baer– Tribbia scheme rapidly grows with the number of iterations (Temperton, 1988). The computational difficulties result from the necessity to compute a temporal derivative from a previous approximate. On the other hand, an initialization by the analytic expansion requires the computation and the storage of a large number of coefficients. See also Chapter 2 for a general discussion of balance and initialization. Vautard and Legras (1986) introduced their method for finite-dimensional models of the atmosphere. In this case one can use rigorous results from the theory of invariant manifold for systems of ordinary differential equations (Arnold and Il’yashenko, 1988; Carr, 1981). But a proof of existence or non-existence of a slow manifold for simplest five-dimensional model, suggested by Lorenz (1992), or any more complicated finite-dimensional models, cannot be employed for exam-

2. Slow manifold in the spatial variables on the f -plane

125

ination of infinite-dimensional models (Kreiss and Lorenz, 1994). It is naturally enough to begin such an examination by a straightforward application of the invariant manifold formalism to the shallow water equations on the f -plane. The method of constructing of balance dynamics, similar to the invariant manifold method employed here, was considered by Warn, Bokhove, Shepherd and Vallis (1995). They defined the slow manifold of the rotating shallow water equations for the slow variable. Their slow variable is the potential vorticity. This section is devoted to the construction of equations for an explicit description of the slow manifold and to the derivation of equations of motion on it. It is well known from classical fluid dynamics that if an ideal barotropic flow is potential initially, it remains so afterwards. This is not true for vortex flows, because flows, that are purely vortical initially, generate a potential part. For the shallow water equations with the Coriolis force the motions corresponding to the inertia–gravity (= fast mode) and the Rossby (= slow mode) waves are analogous to the potential and the pure vortical flows, respectively. The fast motion remains the same, if initially the slow motion is absent (Falkovich and Medvedev, 1992). However, it is not true for slow motion, because of the existence of nonlinear terms, depending only on the slow mode, that connect the fast and the slow modes in the equations for the fast motion. Below a change of variables is constructed, which eliminates these terms. It allows one to find the slow manifold and equations of motion on it. This method is called invariant or integral manifold method. The analytical description for the slow manifold allows us to solve the problem of initialization in a new fashion. For the dynamical initialization, the sought fields are obtained as a series of the slow mode with known coefficients. The first term of this series coincides with the main term of the solution obtained by the Machenhauer initialization (1977). For the static initialization, it is necessary to solve a highly nonlinear equation for the determination of the velocity from the depth of fluid. The linear part of the obtained equation coincides with the linear balance equation and the quadratic terms approach the corresponding terms of the Charney balance equation (Charney, 1962) on the scale much smaller then the Rossby radius. For the suitable variables, the forms of quadratic terms in these equations are equivalent. 2.1. Slow manifold The rotating shallow water equations in the standard form are ∂u ∂u ∂h ∂u +u +v −fv + g = 0, ∂t ∂x ∂y ∂x ∂v ∂v ∂h ∂v +u +v + fu + g = 0, ∂t ∂x ∂y ∂y

(2.1) (2.2)

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

126

∂h ∂ ∂ + (uh) + (vh) = 0, ∂t ∂x ∂y

(2.3)

where u, v are the components of velocity, h is the depth of the fluid, g the gravity constant and f the Coriolis parameter. For a discussion of geophysical references for these equations see Chapter 1 and the book of Pedlosky (1987). We introduce the dimensionless variables u , c0 x , x˜ = R0

u˜ =

v , c0 y y˜ = , R0

v˜ =

z˜ =

h − H0 , H0

t˜ = tf0 ,

(2.4)

where√H0 is the depth of the non-perturbed fluid over the flat bottom at rest, c0 = gH0 the gravity waves speed, f0 the mean value of the Coriolis parameter and R0 = c0 /f0 the Rossby deformation radius. Let us consider the case of constant Coriolis parameter f = f0 (the f -plane approximation). We write the shallow water equations without tilde for simplicity and transfer the nonlinear terms to the right-hand side to obtain ∂u ∂z ∂u ∂u −v+ = −u −v , ∂t ∂x ∂x ∂y ∂z ∂v ∂v ∂v +u+ = −u −v , ∂t ∂y ∂x ∂y ∂ ∂ ∂z ∂u ∂v + + = − uz − vz. ∂t ∂x ∂y ∂x ∂y

(2.5) (2.6) (2.7)

The linear part describes two kinds of motion: the slow and the fast. They can be separated by introducing the change of variables ∂χ ∂ξ ∂ϕ − + , ∂y ∂y ∂x ∂ξ ∂ϕ ∂χ + + , v= ∂x ∂x ∂y z = χ + ξ,

u=−

∂2

∂2

(2.8) (2.9) (2.10)

where = ∂x 2 + ∂y 2 is the Laplacian. The physical meaning of the new variables χ, ξ and ϕ can be easily understood from inverse relations ∂v −1 ∂u − +z , χ = (1 − ) (2.11) ∂y ∂x ∂u ∂v ξ = −−1 (1 − )−1 (2.12) − + z , ∂y ∂x

2. Slow manifold in the spatial variables on the f -plane

ϕ = −1

∂u ∂v + , ∂x ∂y

127

(2.13)

where −1 and (1 − )−1 are the inverse operators with vanishing boundary conditions at infinity. The variable χ defines the geostrophic component of velocity and describes stationary motion according to the linear approximation. The variable ϕ defines the potential part of velocity and the variable ξ is connected to the linear imbalance. The variables ξ and ϕ describe the fast motion corresponding to inertia–gravity waves. Rewriting the shallow water equation for χ, ξ and ϕ leads to ∂ ∂ ∂χ = N(χ, ξ, ϕ) ≡ u( − 1)χ + v( − 1)χ, (1 − ) (2.14) ∂t ∂x ∂y ∂ξ + (1 − )ϕ = R(χ, ξ, ϕ) ∂t ∂ ∂ (uz) − uΩ + (vz) − vΩ , ≡ ∂x ∂y

(1 − )

∂ϕ − (1 − )ξ = S(χ, ξ, ϕ) ∂t 2 2 ∂u ∂v ∂ϕ ∂u ∂v ∂ϕ +2 +v + + , ≡− u ∂x ∂y ∂x ∂y ∂x ∂y

(2.15)

(2.16)

∂v where Ω = ∂x − ∂u ∂y and u, v, z are expressed by (2.8)–(2.10). To decouple the slow and fast parts of the flow, we introduce the change of variables

χ = χ, ¯

ξ = ξ¯ + Ξ [χ¯ ],

ϕ = ϕ¯ + Φ[χ¯ ],

(2.17)

where Ξ [χ], Φ[χ] are some integro-differential operators applied to χ. The system (2.14)–(2.16) has the following form in the new variables: ∂χ = N χ, ξ¯ + Ξ [χ], ϕ¯ + Φ[χ] ≡ N¯ χ, ξ¯ , ϕ¯ , (2.18) ∂t ∂ ξ¯ + (1 − )ϕ¯ = R¯ χ, ξ¯ , ϕ¯ , (1 − ) (2.19) ∂t ∂ ϕ¯ (2.20) − (1 − )ξ¯ = S¯ χ, ξ¯ , ϕ¯ . ∂t The transformation (2.17) does not change the linear part of the equations, and therefore our task is to find Ξ [χ] and Φ[χ] such that the nonlinear part of system (2.18)–(2.20) has the following property (1 − )

¯ R(χ, 0, 0) = 0,

¯ S(χ, 0, 0) = 0.

(2.21)

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

128

Then assuming ξ¯ = 0, ϕ¯ = 0, (2.19)–(2.20) will be identically satisfied and the slow motion will be described by the equation (1 − )

∂χ ∂ = [χ, χ] + Ξ, ( − 1)χ + Φ( − 1)χ ∂t ∂x ∂ + Φ( − 1)χ, ∂y

(2.22)

∂g ∂f ∂g where [f, g] = ∂f ∂x ∂y − ∂y ∂x denotes the Jacobian of two functions f and g. Consequently, a solution sitting on a surface M = {χ, ¯ ξ¯ , ϕ: ¯ ξ¯ = 0, ϕ¯ = 0} stays on it always. In terms of the original variables, this means that solutions of the system with initial data

χ|t=0 = χ0 ,

ξ |t=0 = Ξ [χ0 ],

ϕ|t=0 = Φ[χ0 ]

(2.23)

depend on the slow variable χ only. Thus constructing an appropriate change of variables determines the slow manifold M and the projection of the shallow water equations on it. It is easy to note that the system (2.14)–(2.16) does not have the property (2.21), because R and S have terms depending on χ only R(χ, 0, 0) = [χ, χ], 2 2 2 ∂ χ ∂ 2χ ∂ χ S(χ, 0, 0) = 2 − . 2 2 ∂x∂y ∂x ∂y

(2.24) (2.25)

In the one-dimensional case the right-hand parts of (2.24)–(2.25) are equal to zero; therefore the variable transformation is required in the two-dimensional case only. We now derive conditions on Ξ [χ] and Φ[χ], which ensure that (2.21) is satisfied. We substitute (2.17) in (2.19)–2.20, equate terms not containing ξ¯ and ϕ¯ to zero, and obtain two functional equations for Ξ [χ] and Φ[χ], respectively, dΞ [χ] (1 − ) (2.26) + Φ[χ] = R χ, Ξ [χ], Φ[χ] , dt dΦ[χ] − (1 − )Ξ [χ] = S χ, Ξ [χ], Φ[χ] , (2.27) dt here dtd denotes calculation of the derivative with respect to t and substitution of ∂χ ∂t by its expression from (2.22). Therefore, (2.26) and (2.27) do not contain explicitly the temporal differentiation. It is difficult to solve the functional equations (2.26) and (2.27) exactly; therefore, assuming a weak nonlinearity of u, v, z 1 at the Rossby radius scale, we will solve them using an expansion in terms of the degree of the nonlinearity. The weak nonlinearity condition is realized for the atmosphere and it is equivalent to

2. Slow manifold in the spatial variables on the f -plane

129

smallness of variables χ, ξ¯ and ϕ¯ at the Rossby radius scale. At this scale the nonlinearity parameter is close to the Rossby number. Within the accuracy of cubic terms we find a formal solution of (2.26) and (2.27) given by Ξ [χ] = Ξ2 [χ] + Ξ3 [χ] + · · · ,

(2.28)

Φ[χ] = Φ2 [χ] + Φ3 [χ] + · · · ,

(2.29)

where Ξ2 [χ] = 2(1 − )−1 −1

∂ 2χ ∂x∂y

2 −

∂ 2χ ∂ 2χ , ∂x 2 ∂y 2

Φ2 = (1 − )−1 −1 [χ, χ], are quadratic terms and

∂χ ∂Φ2 ∂Ξ2 Ξ3 [χ] = (1 − ) , − [χ, Φ2 ] + 2 ∂x ∂x ∂y ∂χ ∂Φ2 ∂Ξ2 +2 , + ∂y ∂y ∂x −1 −2 + (1 − ) (1 − )−1 [χ, χ], χ + χ, (1 − )−1 [χ, χ] , Φ3 [χ] = −1 (1 − )−1 (1 − )[Ξ2 , χ] + [Ξ2 , χ] + [χ, Ξ2 ] ∂ ∂Φ2 ∂Φ2 ∂ ∂Φ2 ∂ ∂ ∂Φ2 χ + χ − χ + χ + ∂x ∂x ∂y ∂y ∂x ∂x ∂y ∂y 2 2 ∂ χ ∂ (1 − )−1 [χ, χ] − 2−1 (1 − )−1 2 ∂x∂y ∂x∂y ∂ 2χ ∂ 2 ∂ 2χ ∂ 2 −1 −1 (1 − ) [χ, χ] − 2 2 (1 − ) [χ, χ] − ∂x 2 ∂y 2 ∂y ∂x −1

−1

are the cubic terms in the expansion. In principle, we can find the equation for the slow manifold M = {χ, ξ, ϕ: ξ = F [χ], ϕ = G[χ]} and equations (2.22) of motion with an arbitrarily prescribed accuracy. 2.2. Initialization The explicit knowledge of equations (2.28)–(2.29) for the slow manifold allows to solve the problem of initialization in a new manner. For dynamical initialization, it is necessary to find an initialized state χ¯i , ξ¯i and ϕ¯i using the slow mode χ0 of the initial fields u, v and z, which will not lead to a fast oscillation. The initialized

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

130

fields are obtained from (2.28)–(2.29): χi = χ,

(2.30)

ξi = Ξ [χ],

(2.31)

ϕi = Φ[χ].

(2.32)

The initialized fields for physical variables ui , vi and zi are then derived from (2.8)–(2.10). We ask the following question: how many terms is it necessary to take into account in the expansion of Ξ [χ] and Φ[χ]? Assume that we make an estimate for the time T during which fast mode should remain negligible. Let us assume that the slow modes have an amplitude of order ε 1 during this time and let us truncate the series for Ξ [χ] and Φ[χ] in (2.31) and (2.32) to order εn . Then from (2.19)–(2.20) we derive the estimate for an amplitude of the fast mode ξ¯ , ϕ¯ ∼ (f0 T )ε n+1 . Therefore, if we want to have an order ε m for ξ¯ and ϕ, ¯ it is necessary to choose n from the condition ε m (f0 T )ε n+1 . For simple estimates we take ε = 1/10, f0 =

(2.33) 10−4 s−1 ,

then

n m − 1 + lg(f0 T ). For m = 2 and T = 1, 10 days we obtain n 2, 3, respectively. We now compare the solution (2.31) and (2.32) with the Machenhauer’s solution. We solve the Machenhauer’s conditions (1 − )ϕ = R(χ, ξ, ϕ),

(2.34)

−(1 − )ξ = S(χ, ξ, ϕ)

(2.35)

using an expansion in series of power of χ. This solution is accurate up to cubic terms ∂χ ∂Φ2 ∂Ξ2 −1 −1 ξ = Ξ2 [χ] + (1 − ) , − [χ, Φ2 ] + 2 ∂x ∂x ∂y ∂χ ∂Φ2 ∂Ξ2 +2 (2.36) , + , ∂y ∂y ∂x ϕ = Φ2 [χ] + −1 (1 − )−1 (1 − )[Ξ2 , χ] + [Ξ2 , χ] + [χ, Ξ2 ] ∂ ∂Φ2 ∂ ∂Φ2 ∂ ∂Φ2 ∂ ∂Φ2 + χ + χ − χ + χ . ∂x ∂x ∂y ∂y ∂x ∂x ∂y ∂y (2.37)

2. Slow manifold in the spatial variables on the f -plane

131

Comparing (2.31)–(2.32) with (2.36)–(2.37) we see that the quadratic terms are similar and the cubic terms are different. Consequently, the Machenhauer’s initialization gives n = 2 and it follows from estimate (2.33) for m = 2 and chosen f0 and ε that the period of the suppression of fast motion with given accuracy is less one day. The Machenhauer method is a first approximation to the complete Baer–Tribbia method. Therefore the previous comparison of higher approximations of the present method should be made for the corresponding approximations of the Baer–Tribbia (1977) or the Tribbia (1984) methods. Such comparison for the finite-dimensional model was presented by Vautard and Legras (1986). Consider the problem of static initialization. For static initialization it is necessary to determine a velocity field (ui , vi ) corresponding to slow motion, knowing the geopotential z. To solve this problem we should use (2.10) together with (2.31) and (2.32) to find of χi , ξi and ϕi . This system reduces to an equation for χi −1 (z − χi ) = Ξ [χi ],

(2.38)

ϕi being obtained from (2.32). Then ui and vi are determined by (2.8)–(2.9). In order to compare equation (2.38) and the Charney balance equation (Charney, 1962) 2 2 ∂ ψ ∂ 2ψ ∂ 2ψ (z − ψ) + 2 (2.39) − = 0, ∂x∂y ∂x 2 ∂y 2 where ψ = χ + ξ is the stream function, we rewrite the linear and quadratic terms in (2.38) in the following form 2 2 ∂ χ ∂ 2χ ∂ 2χ (z − ψ) + 2 (2.40) − = 0, ∂x∂y ∂x 2 ∂y 2 where χ = ψ + (1 − )−1 (z − ψ) and the index i is omitted. It appears that equations (2.39) and (2.40) have the same linear parts, while the nonlinear terms in (2.40) are equivalent to the corresponding terms in (2.39) for the scales much smaller than the Rossby radius because χ ≈ ψ for this scale. The quadratic terms in (2.39) have the same form as those in (2.40) with ψ substituted for χ. It also follows from (2.32) that it is necessary to take into account the divergent part of velocity at first order for the static initialization. To summarize, in this section a question of the slow manifold for the shallow water equations was considered. The linearized potential vorticity was chosen as the slow variable. For this slow variable, the functional equations of the slow manifold for the shallow water equations on the f -plane were obtained. Using these equations, an asymptotic formula for the slow manifold was found. The explicit formula of the slow manifold allowed us to write approximation equations

132

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

of motion on the slow manifold and to solve the problem of initialization. The comparison of the dynamical initialization by the invariant manifold method used above and by the Machenhauer method demonstrated that they are equivalent to the first order of the expansion. The comparison of the static initialization by the invariant manifold method and by the balance equation showed that they are close only at the scales much smaller then the Rossby radius.

3. Slow manifold in the spectral variables on the f -plane In the previous section the slow manifold and the equation of motion on it were constructed using the independent spatial variables. This can be useful for a practical realization. But the slow manifold can be also considered in the spectral domain. This allows to understand the general structure of the slow manifold. As the shallow water equations have constant coefficients, one can apply the Fourier transformation and obtain an infinite system of integral equations. For this infinite system, at least formally, the method of Poincaré normal forms from the theory of ordinary differential equations can be applied. It is easy to see in the spectral form of the shallow water equations that the resonant term formed by the amplitudes of the fast waves, is absent in the equation for the slow motion. This property leads to other special features of the interaction of the fast/slow variables. Let us consider the dimensionless shallow water equations (2.5)–(2.7) and assume a weak nonlinearity u, v, z 1. For the scales of the order of the Rossby radius this assumption is equal to the smallness of the Rossby number. See Chapter 1 for a discussion on the characteristic scales. After the Fourier transformation of the dependent variables

u(r), v(r), z(r) = (uk , vk , zk ) exp(ikr) dk, (3.1) where r = (x, y), k = (k1 , k2 ), and the integral is over the whole plane, and diagonalization of the linear part of the system is achieved by using the transformation ⎛ −ik2 k1√ωk +ik2 f0 −k√ 1 ωk +ik2 f0 ⎞ ωk 2ωk |k| 2ωk |k| uk ⎟ ak ⎜ ik1 −k√ k2√ ωk −ik1 f0 2 ωk −ik1 f0 ⎟ ⎜ (3.2) vk = ⎝ ωk bk , 2ωk |k| 2ωk |k| ⎠ zk ck f0 |k| |k|

√

ωk

2ωk

ωk = + |k|2 , f = f0 = 1. where |k| = From (3.2) for real-valued u, v, z we have the following symmetries k12

∗ ak = a−k ,

+ k22 ,

√

2ωk

∗ ck = b−k ,

f02

(3.3)

where ∗ denotes the complex conjugation, and one can consider only the equation for bk . The equation for ck is obtained by the complex conjugation.

3. Slow manifold in the spectral variables on the f -plane

133

As a result of the diagonalization we have the system ∂ak + − 0 ∗ al am +Uklm al bm + Uklm al bm , = Uklm ∂t

(3.4)

resonant

∂bk − 0 ∗ al am + V + al bm +Vklm al bm + iωk bk = Vklm klm ∂t resonant slow + 0 ∗ + Wklm bl bm + Wklm bl bm

− ∗ ∗ + Wklm bl bm ,

(3.5)

where the matrix elements are known (see next section) and the following compact notation is used

− − ∗ ∗ Uklm al bm = Uklm al bm δ(k − l + m) dl dm. Let us call the variable ak the slow variable, because of no evolution for this variable from the linear part of the system. And let us call the variable bk the fast variable, because it has the high frequency greater than f0 . These fast/slow definitions are different from the conventional definitions for the fast/slow variables from the averaging theory (e.g. Chapter 2). In the system (3.4)–(3.5) we separate the resonant terms and the slow terms, which are important for our further consideration. See the details below. 3.1. Fast and slow formal invariant manifolds Our method of splitting is based on the following definition of the formal invariant manifold (Arnold and Il’yashenko, 1988, p. 76). Consider a formal differential equation x˙ = Λx + f (x, y),

y˙ = My + g(x, y),

f (0, 0) = 0, g(0, 0) = 0, ∂f ∂f ∂g ∂g (0, 0) = (0, 0) = (0, 0) = (0, 0) = 0. ∂x ∂y ∂x ∂x

(3.6)

D EFINITION 1. The subspace y = 0 is called formal invariant manifold of equation (3.6), if g(x, 0) = 0. The equation x˙ = Λx + f (x, 0) is called the restriction of equation (3.6) to this formal invariant manifold. The results on the finite-dimensional formal invariant manifold can be found in Arnold and Il’yashenko (1988).

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Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

The aim of the following consideration is to use this definition for a description of the fast and slow invariant manifolds of the shallow water equations on the f -plane. Let us first define the fast manifold which is an analog of the potential motion for the inviscid fluid without the Coriolis force. Assuming ak = 0 we get the fast manifold F = (ak , bk ): ak = 0 and an equation on the fast manifold ∂bk + − ∗ ∗ 0 ∗ bl bm + Wklm bl bm + Wklm bl bm . + iωk bk = Wklm ∂t

(3.7)

This formula is exact because equation (3.4) for the slow variable ak does not ∗ . And as a consequence, the fast variable b contain terms of the form Uklm bl bm k does not generate any evolution of the slow variable ak . So if the initial state belongs to the fast manifold then the subsequent evolution takes place on this manifold for all times in future. The slow manifold cannot be determined so easy because equation (3.5) con0 a a which plays a role of an external force (a slow force) tains the term Vklm l m and generates the fast variable for an arbitrary non-zero slow variable ak . But an approximate slow manifold can be found in the case of weak nonlinearity. P ROPERTY 1. Slow terms of the form Vk1...i a1 . . . ai can be eliminated from equation (3.5) for the fast variable bk . Actually, any term Vk1...i a1 . . . ai has zero frequency and the frequency of the fast variable bk is positive ωk f0 , therefore the resonance conditions are not satisfied and this term can be eliminated using a transformation of the form bk → bk + Fk1...i a1 . . . ai .

(3.8)

This transformation is invertible and keeps a finite number of terms in the transformed system. (See the next section for a definition of the resonance conditions.) The original slow term from (3.5) is eliminated by the following transformation 0 bk → bk − iωk−1 Vk12 a 1 a 2 = b k + Fk .

(3.9)

The transformed equations take the form ∂ak + − 0 a1 a2 + Uk12 a1 b2 + Uk12 a1 b2∗ = Uk12 ∂t + − + Uk12 a1 F2 + Uk12 a1 F2∗ ,

(3.10)

3. Slow manifold in the spectral variables on the f -plane

135

∂bk + − a1 (b2 + F2 ) + Vk12 a1 (b2 + F2 )∗ + iωk bk = Vk12 ∂t 0 + Wk12 (b1 + F1 )(b2 + F2 )∗ + − + Wk12 (b1 + F1 )(b2 + F2 ) + Wk12 (b1 + F2 )∗ (b2 + F2 )∗ 0 + − 0 U234 a1 a3 a4 + U234 + 2iωk−1 Vk12 a1 a3 b4 + U234 a1 a3 b4∗ + − 0 + 2iωk−1 Vk12 U234 a1 a3 F4 + U234 a1 a3 F4∗ . (3.11)

Now we are able to compare the slow force of the initial system (3.4)–(3.5) 0 a1 a2 fk = Vk12

(3.12)

and the slow force of the obtained system (3.10)–(3.11) + − 0 0 a1 F2 + Vk12 a1 F2∗ + 2iωk−1 Vk12 U234 a1 a3 a4 fk = Vk12 + − 0 + Wk12 F1 F2∗ + Wk12 F1 F2 + Wk12 F2∗ F2∗ (0) + − + 2iωk−1 Vk12 U234 a1 a3 F4 + U234 a1 a3 F4∗ .

(3.13)

The slow force fk0 of the initial system is quadratic in the slow variable ak and the slow force fk1 is cubic in ak , therefore for weak nonlinearity one can say that fk1 is smaller than fk0 . Using this approach we can continue the procedure of reduction of the slow force. But this procedure generates an additional number of the terms in the expression of the slow force, therefore it is difficult to prove convergence of this series. For practical applications one can compare the order at the slow force of each step, and then choose a variable with a minimal slow force. The respective approximate slow manifolds and the slow equations can be found from the condition bk = 0. Therefore the two first approximations are S2 = (ak , bk ): bk = 0 , ∂ak 0 a1 a2 , = Uk12 ∂t 0 S3 = (ak , bk ): bk = −iωk−1 Vk12 a1 a2 , ∂ak + − 0 0 0∗ a1 a2 − iω2−1 Uk12 V234 a1 a3 a4 − iω2−1 Uk12 V234 a1 a3∗ a4∗ . = Uk12 ∂t

(3.14) (3.15) (3.16) (3.17)

From Property 1 it is easy to obtain the next change of variables eliminating the cubic slow terms from (3.11) + − 0 0 a1 F2 + Vk12 a1 F2∗ + 2iωk−1 Vk12 U234 a1 a3 a4 . bk → bk − iωk−1 Vk12

(3.18)

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Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

3.2. Normal forms As we saw from the previous subsection it is sufficient to eliminate only the terms of one type to split the slow and fast motions. It is remarkable that these transformations are finite and invertible. But we can also make the next step in the splitting procedure and eliminate all non-resonant terms. Our system has two frequencies: zero frequency for the slow motion and the frequency of the inertia–gravity waves

ωk = f02 + |k|2 for the fast motion. The dispersion law of the inertia–gravity waves is of non-decay type. It means that we cannot satisfy the three wave resonance conditions ω k = ω k1 + ω k2 ,

k = k1 + k2 .

A dispersion law is of decay type if this resonance condition can be satisfied. Therefore it is easy to write the main resonant terms of the shallow water equations ∂ak 0 (3.19) a1 a2 , = Uk12 ∂t ∂bk + + iωk bk = Vk12 (3.20) a1 b2 . ∂t It is well known that the slow motion is independent of the fast motion in the leading order. Let us study the connection of the fast and slow motions at the next orders. After a corresponding transformation the second order normal form takes the form ∂ak 0 1 2 = Uk12 a1 a2 + Uk123 a1 a2 a3 + Uk123 a1 b2 b3∗ , (3.21) ∂t ∂bk + 1 + iωk bk = Vk12 a1 b2 + Vk123 a1 a2 b3 + Tk123 b1∗ b2 b3 ∂t + 0 (3.22) + Sk123 a1 b2 b3∗ + Sk123 a1 b2 b3 . 1 a a a for the slow variIt is clear that we get cubic self nonlinearity Uk123 1 2 3 2 ∗ able and its modulation Uk123 a1 b2 b3 by the fast variables. The fast motion also acquires the modulation and the cubic self nonlinearity due to the four-wave interaction. If we continue to eliminate non-resonant terms in the higher orders then the following property of the our system becomes obvious.

P ROPERTY 2. The equation for the slow variable does not contain terms of the form Uk1...n (b1 b2∗ ) . . . (bn−1 bn∗ ). Actually, the right-hand side of the equation for ak contains only terms with the slow variable ak as a factor. Therefore all transformations, eliminating non-

4. Poincaré normal forms

137

resonant terms, has the same structure. As a result, the use of these transformations does not generate terms of the form Uk1...n (b1 b2∗ ) . . . (bn−1 bn∗ ). Thus the transformations eliminating non-resonant terms from the equation for bk do not change the structure of the slow equation. In this way we show that the equations for the slow variable ak contain only self nonlinearity and the modulation by the fast variable bk . The high-frequency force, which can excite the slow motion, is absent at any order. In the next section it will be shown that the high-frequency force does appears at the second order due to the (small) y-dependent β-term in the Coriolis parameter.

4. Poincaré normal forms In the two previous sections we considered the rotating shallow water equations on the f -plane. In this approximation the frequency of the Rossby wave is equal to zero and as a result the fast and slow motions can be separated at any order. To remove this degeneracy we introduce a variability of the Coriolis parameter. But in this situation we should understand how to eliminate variable non-resonant terms. Therefore in this section an extension of the Poincaré normal form theory for systems of partial differential equations which have linear ordinary equations as a main part, is obtained. Poincaré normal forms for the rotating shallow water equations on the β-plane and for large scales (with respect to Rossby radius) for the variable Coriolis parameter are found. The normal forms of the shallow water equations contain the Charney–Obukhov equation and higher approximation equations for Rossby waves and describe the nonlinear interaction of Rossby and inertia–gravity waves. A successful technique for dealing with many differential equations consists of the transformation of them to a simpler (normal, canonical) form. Poincaré suggested a theory of the normal forms for ODEs (see Arnold, 1983; Arnold and Il’yashenko, 1988). This theory produces these simple forms by means of changes of variables using power series in deviations from the equilibrium or periodic motion. This procedure is very effective in the theory of ODEs (see, for example, Arnold, 1983). Of course, the series are not always convergent (see Bruno, 1989), but already the first few terms of the Poincaré normal form can be useful for qualitative studies of the features of the original differential equation. The method of normal forms for PDEs is also useful (see, for example, Shatah, 1985; Nikolenko, 1986; McKean and Shatah, 1991; Ozawa, Tsutaya and Tsutsumi, 1995; Bruno, 2000), but in this case the method is exploited more effectively after a Fourier transformation when the initial PDEs become an infinite system of ODEs and the method is applicable directly. Homogeneous PDEs in a canonical Hamiltonian form give an excellent example of the

138

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

application of the normal forms method (see Zakharov and Schulman, 1993; Zakharov, Lvov and Falkovich, 1992). A general method analogous to the method of Poincaré normal forms for ODEs does not exist for inhomogeneous PDEs. In the present section the method of Poincaré normal forms is extended to PDEs which have a linear ODE as the main part, depending on independent spatial variables as parameters. In this case the definitions of the resonance conditions and the resonant monomials are extended directly and solutions of a homological equation can be presented in an explicit form. This homological equation (the equation for unknown change of variables) contains partial derivatives. Three examples of the application of the method are given for the rotating shallow water equations. These examples are chosen because the dispersion law has a gap between high- and low-frequency waves. This gap allows to separate easily the fast and slow wave motions. The first example is a long wave approximation to the shallow water equations with a variable Coriolis force. The long wave approximation means that motion with a typical scale that is greater than the Rossby radius is considered. The obtained normal form contains the simplest equations for Rossby and inertia–gravity waves. The variability of the Coriolis parameter is crucial for the influence of the inertia–gravity waves upon the Rossby waves. The second example is the shallow water equations on the β-plane. The main points of this consideration are that the β-term gives a partial derivative after the Fourier transformation, and a smallness of the β-term allows diagonalizing the main linear part of the system. Thus we place the rotating shallow water equations under a hypothesis of the general Poincaré theorem. The β-term plays the crucial role for the interaction of the Rossby and inertia–gravity waves too and generates an additional nonlinearity, a so-called scalar nonlinearity, for the Rossby waves. The presentation of the theory and the examples follow Medvedev (1996, 1999b). The third example is the separation of the equation for short equatorial inertia– gravity waves (Medvedev and Zeitlin, 2005). This example is similar the previous one but the main linear part does not contain the Coriolis term. 4.1. Basic theorems Consider a system of PDEs ∂ui (x, t)/∂t + λi (x)ui (x, t) = Fi (x, u),

u ∈ Rl , x ∈ Rm,

(4.1)

with the right-hand side of the form: Fi (x, u) =

∞

n=1 0 0, x > a,

(1.30)

with initial data U (0, x) = U 0 (x) for x > a. Then as in Example 1.1 one can look for a solution of the form U (t, x) = (1.31) αj0 (x − λj t)rj , j

where rj , λj are the eigenvectors and the eigenvalues of A. We would like to define this for t > 0, x > a. We can define αj0 (x) for x > a by decomposition of U 0 (x), U 0 (x) = j αj0 (x)rj . But then, αj0 (x − λj t) is well-defined for t > 0, x > a only if λj 0. Otherwise, for the j ’s such that λj > 0, formula (1.31) gives a solution if we impose arbitrarily the value of αj0 (x) for x < a. Thus we conclude that the following problem is well-posed: ⎧ ∂ U + A∂x U = 0 for t > 0, x > a, ⎪ ⎨ t U (0, x) = U 0 (x) for x > a, (1.32) ⎪ ⎩ lj U (t, a) = βj (t) for t > 0, λj > 0, where βj (t) are given boundary data. Indeed this defines in a unique way αj0 (y) = βj ( a−y λj ) for y < a. We conclude that in the linear case we have to give boundary conditions only for the incoming characteristics. Notice that one can also put

2. Finite volume schemes for conservative systems

197

boundary data βj that depend on the value of the outgoing characteristics, i.e. of the αj (t, a) for j such that λj 0. In the nonlinear case, one expects that a problem of the type ⎧ ⎪ ⎨ ∂t U + A(U )∂x U = 0 for t > 0, x > a, (1.33) for x > a, U (0, x) = U 0 (x) ⎪ ⎩ U (t, a) ∈ B(t) for t > 0, is well-posed, where B(t) is a submanifold of Rp of codimension the number of positive eigenvalues of A(U (t, a)). Not all such submanifold would give a wellposed problem, but indeed the main restriction is that we do not know in advance the number of incoming characteristics, it depends on the solution itself (and it can change with respect to time). Unfortunately there is no good theory that says that such problem is well-posed. In practice one merely uses physical arguments in order to ensure that (1.33) is well-posed. E XAMPLE 1.5. Consider the system of isentropic gas dynamics (1.14), that we set for t > 0, x > 0, with initial data and with wall boundary condition u(t, 0) = 0.

(1.34)

It involves one scalar boundary condition, thus it is a priori compatible with the case of a single incoming characteristics. However, since whenever u = 0, the eigenvalues of the system are ± p (ρ) (one positive and one negative eigenvalue), we are automatically in this situation, and this problem should be wellposed.

2. Finite volume schemes for conservative systems This section is devoted to the introduction of very basic tools for the numerical resolution of systems of conservation laws (without source term) by finite volume methods. The notions introduced here can be found in Godlewski and Raviart (1991, 1996), Bouchut (2004), Toro (1999), LeVeque (2002), and are not at all specific to shallow water systems. Let us consider a system of conservation laws ∂t U + ∂x F (U ) = 0, (2.1) where U (t, x) ∈ Rp , x ∈ R, t 0, is the unknown and F (U ) ∈ Rp is a given nonlinearity. We would like to approximate its solution by discrete values Uin , i ∈ Z, n ∈ N. In order to do so we consider a grid of points xi+1/2 , i ∈ Z, · · · < x−1/2 < x1/2 < x3/2 < · · · ,

(2.2)

198

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

and we define the cells (or finite volumes) and their lengths Ci = (xi−1/2 , xi+1/2 ),

xi = xi+1/2 − xi−1/2 > 0.

(2.3)

We shall denote also xi = (xi−1/2 +xi+1/2 )/2 the centers of the cells. We consider a constant timestep t > 0 and define the discrete times by tn = n t,

n ∈ N.

(2.4)

Uin

intend to be approximations of the averages of the exact The discrete values solutions over the cells,

1 U (tn , x) dx. Uin (2.5)

xi Ci

A finite volume conservative scheme for solving (2.1) is a formula of the form Uin+1 − Uin +

t (Fi+1/2 − Fi−1/2 ) = 0,

xi

(2.6)

telling how to compute the values Uin+1 at the next time level, knowing the values Uin at time tn . In this section we consider only first-order explicit three points schemes (“three points” means that Uin+1 in (2.6) depends only on three values n , U n , U n ) where Ui−1 i i+1 n Fi+1/2 = F Uin , Ui+1 (2.7) . The function F (Ul , Ur ) ∈ Rp is called the numerical flux, and determines the scheme. We shall often denote Ui instead of Uin , whenever there is no ambiguity. The formula (2.6) is very natural with respect to (2.1), because the exact solution U (t, x) to (2.1) satisfies

1 1 U (tn+1 , x) dx − U (tn , x) dx

xi

xi Ci

Ci

t + (F − F i−1/2 ) = 0,

xi i+1/2 where F i+1/2 is the exact flux F i+1/2

1 =

t

tn+1 F U (t, xi+1/2 ) dt.

(2.8)

(2.9)

tn

This is obtained by integrating equation (2.1) with respect to t and x over ]tn , tn+1 [ × Ci , and dividing the result by xi . Notice also that in (2.6), the interface flux Fi+1/2 represents the exchange of conserved quantities between cells Ci and Ci+1 .

2. Finite volume schemes for conservative systems

199

2.1. Consistency The numerical flux F(Ul , Ur ) is called consistent with (2.1) if F (U, U ) = F (U )

for all U.

(2.10)

We can see that this condition guarantees obviously that if for all i, Uin = U a constant, then also Uin+1 = U . A deeper motivation for this definition is indeed the Lax–Wendroff theorem, that states that if the numerical flux is consistent and if Uin defined by (2.6)–(2.7) converges to some bounded function U (t, x), then it is a weak solution to (2.1), see Godlewski and Raviart (1991). 2.2. CFL condition It is always necessary to impose what is called a CFL condition (for Courant, Friedrichs, Levy) on the timestep to prevent the blow up of the numerical values. It comes usually under the form

t ai+1/2 min( xi , xi+1 ),

i ∈ Z,

(2.11)

where ai+1/2 is an approximation of the speed of propagation, computed in terms of Ui and Ui+1 . A typical computation can be ai+1/2 = max λj (Ui ), λj (Ui+1 ) , (2.12) where λj (U ) are the eigenvalues of F (U ). The restriction (2.11) enables in practice to compute the timestep at each time level tn , in order to determine the new time level tn+1 = tn + t (within this view,

t is not constant, it is computed in an adaptive fashion). 2.3. Discrete entropy inequalities The stability of a numerical scheme can be analyzed in different ways. There are merely two classes of analysis: the linearized approach, which is always possible to perform, see for example Godlewski and Raviart (1991), and the nonlinear approach, that puts much more restrictions on the scheme, but which is much more reliable. In the latter approach, exposed in Bouchut (2004), two notions are available: the preservation of invariant domains (some natural bounds of the system must be respected, typically the nonnegativity of density), and the occurrence of discrete entropy inequalities. We shall only introduce here the latter property. Consider a convex entropy η associated to the system (2.1), as defined in Section 1.3. We say that the scheme (2.6)–(2.7) satisfies a discrete entropy inequality associated to η, if there exists a numerical entropy flux function G(Ul , Ur ) which is consistent with the exact entropy flux (in the sense that G(U, U ) = G(U )),

200

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

such that, under some CFL condition, the discrete values computed by (2.6)–(2.7) automatically satisfy

t (Gi+1/2 − Gi−1/2 ) 0, η Uin+1 − η Uin +

xi

(2.13)

n Gi+1/2 = G Uin , Ui+1 .

(2.14)

with

This definition comes very naturally as a local entropy balance, similarly as the conservative balance (2.6). The interest of such property is twofold. First, it ensures the computation of physically relevant discontinuous solutions, because again by the Lax–Wendroff theorem, if the scheme satisfies a discrete entropy inequality and if Uin converges to some function U (t, x), then U satisfies the entropy inequality (1.20). Second, (2.13) ensures a global a priori bound on the solution. Namely, under some suitable boundary conditions, summing up (2.13) with respect to i, one gets that i xi η(Uin ) does not increase with time. One can define also discrete entropy inequalities by interface, involving only two values Ui and Ui+1 instead of three values Ui−1 , Ui and Ui+1 in (2.13). These are useful in the analysis of the numerical flux F (Ui , Ui+1 ). We shall here only deal with the case of approximate Riemann solvers, and for the general case we refer to Bouchut (2004). 2.4. Godunov approach, approximate Riemann solvers Many methods exist to determine a numerical flux. The two main criteria that enter in its choice are its stability properties, and the precision qualities it has, which can be measured by the amount of viscosity it produces and by the property of exact computation of particular solutions. A general tool that enables to construct numerical fluxes is the notion of approximate Riemann solver in the sense of Harten, Lax and Van Leer (1983), that comes naturally within the Godunov approach. This section is devoted to introducing this concept, which is very intuitively related to the physical problem considered. We define the Riemann problem for (2.1) to be the problem of finding the solution to (2.1) with Riemann initial data Ul if x < 0, U 0 (x) = (2.15) Ur if x > 0, for two given constants Ul and Ur . By a simple scaling argument, this solution is indeed a function only of x/t. D EFINITION 2.1. An approximate Riemann solver for (2.1) is a vector function R(x/t, Ul , Ur ) that is an approximation of the solution to the Riemann problem,

2. Finite volume schemes for conservative systems

201

in the sense that it must satisfy the consistency relation R(x/t, U, U ) = U,

(2.16)

and the conservativity identity Fl (Ul , Ur ) = Fr (Ul , Ur ),

(2.17)

where the left and right numerical fluxes are defined by

0

R(v, Ul , Ur ) − Ul dv,

Fl (Ul , Ur ) = F (Ul ) − −∞

∞

Fr (Ul , Ur ) = F (Ur ) +

R(v, Ul , Ur ) − Ur dv.

(2.18)

0

It is called dissipative with respect to a convex entropy η for (2.1) if Gr (Ul , Ur ) − Gl (Ul , Ur ) 0,

(2.19)

where

0

η R(v, Ul , Ur ) − η(Ul ) dv,

Gl (Ul , Ur ) = G(Ul ) − −∞

∞

η R(v, Ul , Ur ) − η(Ur ) dv,

Gr (Ul , Ur ) = G(Ur ) +

(2.20)

0

and G is the entropy flux associated to η, G = η F . It is possible to prove that the exact solution to the Riemann problem satisfies these properties, but however the above definition is rather motivated by numerical schemes. Let us explain why. Consider a discrete sequence Uin , i ∈ Z, and define the function U n (x) that is piecewise constant over the mesh with value Uin in each cell Ci . Therefore, Uin is nothing but the cell average of the function U n (x). Then, the Godunov method consists in computing the solution U (t, x) to (2.1) for tn < t < tn+1 with U n (x) as initial data, and finally in defining Uin+1 as the average over Ci of U (tn+1 , x). According to the computation (2.8)–(2.9), this leads indeed to a numerical flux given by (2.9). Now, let us generalize the previous construction by replacing the exact solution to (2.1) by an approximate one. We still consider the initial data U n (x) that is piecewise constant, but we now consider the approximate solution Uapp (t, x) for

202

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 2. Construction of an approximate solution Uapp .

tn t < tn+1 (as illustrated in Figure 2) by x − xi+1/2 n n , Ui , Ui+1 Uapp (t, x) = R t − tn

if xi < x < xi+1 .

(2.21)

This formula is obtained by considering that close to each interface point xi+1/2 , we have to solve a translated Riemann problem. Since (2.1) is invariant under translation in time and space, we can think of sticking together the local approximate Riemann solutions, which leads to (2.21), at least for times such that these solutions do not interact. This is possible until time tn+1 under a CFL condition 1/2, in the sense that

xi 2 t

xi+1 x/t > 2 t

x/t < −

⇒

R(x/t, Ui , Ui+1 ) = Ui ,

⇒

R(x/t, Ui , Ui+1 ) = Ui+1 .

(2.22)

Then, we define Uin+1 to be the average over Ci of this approximate solution Uapp (t, x) at time tn+1 − 0. According to the definition (2.18) of Fl and Fr and by using (2.22), we get Uin+1

1 =

xi 1 =

xi

x i+1/2

Uapp (tn+1 − 0, x) dx xi−1/2

x

i /2

n R x/ t, Ui−1 , Uin dx

0

1 +

xi

0 − xi /2

n dx R x/ t, Uin , Ui+1

2. Finite volume schemes for conservative systems

=

Uin

1 +

xi

203

x

i /2

n R x/ t, Ui−1 , Uin − Uin dx

0

+

1

xi

= Uin −

0

n R x/ t, Uin , Ui+1 − Uin dx

− xi /2

n

t n n Fl Ui , Ui+1 − Fr Ui−1 , Uin .

xi

(2.23)

Therefore we see that with the conservativity assumption (2.17), this is a conservative scheme, with numerical flux F (Ul , Ur ) = Fl (Ul , Ur ) = Fr (Ul , Ur ).

(2.24)

The consistency assumption (2.16) ensures that this numerical flux is consistent, in the sense of Section 2.1. Then let us examine the discrete entropy inequality. Since η is convex, we can use Jensen’s inequality in (2.23), and we get

η

Uin+1

1

xi

x

i /2

n η R x/ t, Ui−1 , Uin dx

0

1 +

xi

n

0

n dx η R x/ t, Uin , Ui+1

− xi /2

= η Ui −

n

t n n Gl Ui , Ui+1 − Gr Ui−1 , Uin .

xi

(2.25)

Under assumption (2.19), we conclude that n

t n n G Ui , Ui+1 − G Ui−1 , Uin 0, η Uin+1 − η Uin +

xi

(2.26)

for any numerical entropy flux function G(Ul , Ur ) such that Gr (Ul , Ur ) G(Ul , Ur ) Gl (Ul , Ur ),

(2.27)

thus we recover the definition of Section 2.3, since (2.16) ensures that this numerical entropy flux is consistent. In this way, to any approximate Riemann solver R we can associate a conservative numerical scheme. If we use the exact Riemann solver, the scheme we get is called the (exact) Godunov scheme.

204

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

R EMARK 2.1. The approximate Riemann solver framework works as well with interface dependent solvers Ri+1/2 . This is used in practice to choose a solver adapted to the data Ui , Ui+1 , so as to produce a viscosity which is as small as possible. In practice, the exact resolution to the Riemann problem is too complicate and too expensive, especially for systems with large dimension. Thus we rather use approximate solvers. The most simple choice is the following. 2.5. Simple solvers We shall call simple solver an approximate Riemann solver consisting in a set of finitely many simple discontinuities. This means that there exists a finite number m 1 of speeds σ0 = −∞ < σ1 < · · · < σm < σm+1 = +∞,

(2.28)

and intermediate states U0 = Ul , U1 , . . . , Um−1 , Um = Ur

(2.29)

(depending on Ul and Ur ), such that as illustrated in Figure 3, R(x/t, Ul , Ur ) = Uk

if σk < x/t < σk+1 .

(2.30)

Then the conservativity identity (2.17) becomes m

σk (Uk − Uk−1 ) = F (Ur ) − F (Ul ),

(2.31)

k=1

and the entropy inequality (2.19) becomes m

σk η(Uk ) − η(Uk−1 ) G(Ur ) − G(Ul ).

k=1

Figure 3.

A simple solver.

(2.32)

2. Finite volume schemes for conservative systems

205

Conservativity thus enables to define the intermediate fluxes Fk , k = 0, . . . , m, by Fk − Fk−1 = σk (Uk − Uk−1 ),

F0 = F (Ul ),

Fm = F (Ur ),

(2.33)

which is a kind of generalization of the Rankine–Hugoniot relation. The numerical flux is then given by F(Ul , Ur ) = Fk ,

where k is such that σk 0 σk+1 .

(2.34)

We can observe that if it happens that σk = 0 for some k, there is no ambiguity in this definition since (2.33) gives in this case Fk = Fk−1 . An explicit formula for the numerical flux is indeed σk (Uk − Uk−1 ) F (Ul , Ur ) = F (Ul ) + σk 0

2.6. CFL condition for simple solvers For a simple solver we can define the local speed by ai+1/2 = a(Ui , Ui+1 ),

a(Ul , Ur ) = sup |σk |.

(2.36)

1km

Then the CFL condition (2.22) reads 1 (2.37) min( xi , xi+1 ). 2 This is called a CFL condition 1/2, and as the above computations show, it ensures the stability of the scheme (entropy inequalities). However, in practice, one uses rather the CFL 1 condition (as written in Section 2.2),

t ai+1/2

t ai+1/2 min( xi , xi+1 ).

(2.38)

It enables larger timesteps (and thus is less CPU consuming), and it gives sufficiently stable results. 2.7. Nonnegativity and vacuum Either in gas dynamics or shallow water models, the nonnegativity of density or water height is a key requirement for the numerical stability. This nonnegativity property is a particular case of what is called an invariant domain for a system of conservation laws. The corresponding analysis for approximate Riemann solvers

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

206

is performed in Bouchut (2004), but let us here say shortly how it works, in the case of nonnegativity of density. The key remark is that according to the definition (2.23) of Uin+1 as an average of values of Uapp and to the definition (2.21) of Uapp , it is enough that the approximate Riemann solver R(x/t, Ul , Ur ) always has a nonnegative density, whenever the arguments Ul and Ur have nonnegative densities. For simple solvers, it means that the only requirement is that the intermediate states Uk , k = 0, . . . , m, have nonnegative densities. Then, the next question is the possibility to treat data having vacuum (dry states for shallow water). In the computation of an approximate Riemann solver, if the two values Ul , Ur are vacuum data Ul = Ur = 0, there is no difficulty, we can simply set R = 0. The real problem occurs when one of the two values is zero and the other not. We shall say that an approximate Riemann solver can resolve the vacuum if in this case of two values Ul , Ur which are zero and nonzero, it gives a solution R(x/t, Ul , Ur ) with nonnegative density and with finite speeds of propagation σ1 , . . . , σm . This is necessary, otherwise the CFL condition (2.38) would give a zero timestep. The selection of solvers that are able to resolve vacuum is a main point for applications to flows in rivers with shallow water type equations. 2.8. Roe solver The Roe solver, introduced by Roe (1981), is an example of simple solver. It is obtained as follows. We need first to find a p ×p diagonalizable matrix A(Ul , Ur ) (called a Roe matrix), such that F (Ur ) − F (Ul ) = A(Ul , Ur )(Ur − Ul ),

A(U, U ) = F (U ).

(2.39)

Then we define R(x/t, Ul , Ur ) to be the solution to the linear problem ∂t U + A(Ul , Ur )∂x U = 0,

(2.40)

with initial Riemann data (2.15). Denoting by σ1 , . . . , σm the distinct eigenvalues of A(Ul , Ur ), we can decompose Ur − Ul along the eigenspaces Ur − Ul =

m

δUk ,

A(Ul , Ur )δUk = σk δUk ,

(2.41)

k=1

and the solution is given by R(x/t, Ul , Ur ) = Ul +

k0

δUk

if σk0 < x/t < σk0 +1 .

(2.42)

k=1

This defines a simple solver, the assumption (2.39) gives indeed the conservativity (2.31), since σk δUk = A(Ul , Ur )(Ur − Ul ) = F (Ur ) − F (Ul ). (2.43) k

2. Finite volume schemes for conservative systems

207

E XAMPLE 2.2. For the isentropic system (1.14), the classical Roe matrix is given by 0 1 A(Ul , Ur ) = p(ρr )−p(ρl ) (2.44) , − uˆ 2 2uˆ ρr −ρl with

√ uˆ =

√ ρl ul + ρr ur . √ √ ρl + ρr

(2.45)

The Roe method is in practice very accurate, but does suffer from incomplete stability properties. In particular, it is not entropy satisfying, entropy fixes have to be designed. We refer the reader to the literature (Godlewski and Raviart, 1996; Toro, 1999) for this class of schemes. For our purpose of designing good schemes for shallow water models, this method is a priori less interesting than the HLL or HLLC solvers because it can produce negative densities. 2.9. HLL solver A basic simple approximate Riemann solver is the HLL solver, that was introduced by Harten, Lax and Van Leer (1983), and was indeed the first example of approximate Riemann solver. It works for general system of conservation laws (2.1). We take two parameters σ1 < σ2 , and define the two speeds simple approximate Riemann solver by ⎧ if x/t < σ1 , ⎨ Ul R(x/t, Ul , Ur ) =

σ2 Ur −F (Ur ) σ2 −σ1

+

F (Ul )−σ1 Ul σ2 −σ1

if σ1 < x/t < σ2 , (2.46) if σ2 < x/t. Ur The consistency and conservativity conditions (2.16), (2.17) or (2.31) are easily checked for this solver. According to (2.33)–(2.34), the HLL numerical flux is given by ⎧ if 0 < σ1 , ⎨ F (Ul ) σ1 F (Ur ) σ1 σ2 F(Ul , Ur ) = σ2 F (Uσl )2− + (U − U ) if σ1 < 0 < σ2 , (2.47) r l −σ1 σ2 − σ1 ⎩ if σ2 < 0. F (Ur ) The invariant domains and entropy conditions can be analyzed and justified, under the subcharacteristic conditions ⎩

σ1 λj (U ) σ2 ,

(2.48)

where λj (U ) are the eigenvalues of F (u). According to Remark 2.1, a local optimization of (2.48) leads to the choice σ1 =

inf

inf λj (U ),

U =Ul ,Ur j

σ2 =

sup

sup λj (U ),

U =Ul ,Ur j

(2.49)

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

208

even if theoretically one should involve also intermediate values of U . In the case of isentropic gas dynamics, the solver preserves nonnegativity of density and can treat the vacuum. Indeed, according to Section 2.7, we only have to check the nonnegativity of the intermediate density in the approximate Riemann solver (2.46), which is given by σ2 − ur ul − σ1 (2.50) ρr + ρl . σ 2 − σ1 σ 2 − σ1 But (2.49) implies that σ1 ul − p (ρl ) ul and σ2 ur + p (ρr ) ur , which proves that ρ ∗ 0. In practice the HLL solver works quite well except an excessive numerical diffusion of waves associated to eigenvalues other than the lowest and largest ones (but this does not occur for isentropic gas dynamics since there are only two eigenvalues). ρ∗ =

2.10. HLLC solver The HLLC solver (also called acoustic solver) has been introduced for gas dynamics equations in order to restore the central wave traveling at velocity u, and better resolve the associated discontinuities. For the isentropic system (1.14), there is no such wave, and therefore there is no special interest in this solver, except that it is naturally expected to be used in systems that look like gas dynamics systems. The solver is defined as a three speeds simple solver ⎧ if x/t < σ1 , U ⎪ ⎪ l ⎪ ⎨ U ∗ if σ < x/t < σ , 1 2 l R(x/t, Ul , Ur ) = (2.51) ⎪ Ur∗ if σ2 < x/t < σ3 , ⎪ ⎪ ⎩ Ur if σ3 < x/t, where the speeds are defined by σ1 = ul − cl /ρl ,

σ2 = u∗ ,

σ3 = ur + cr /ρr ,

(2.52)

with Ul = (ρl , ρl ul ), Ur = (ρr , ρr ur ), Ul∗ = (ρl∗ , ρl∗ u∗ ), Ur∗ = (ρr∗ , ρr∗ u∗ ), and cl ul + cr ur + πl − πr , c l + cr 1 cr (ur − ul ) + πl − πr = + , ρl cl (cl + cr ) 1 cl (ur − ul ) + πr − πl = + , ρr cr (cl + cr )

u∗ = 1 ρl∗ 1 ρr∗

(2.53)

with πl = p(ρl ),

πr = p(ρr ).

(2.54)

2. Finite volume schemes for conservative systems

209

The consistency and conservativity conditions (2.16), (2.31) are satisfied, and the intermediate fluxes of (2.33) are Fr∗ = ρr∗ u∗r , ρr∗ (u∗r )2 + π ∗ , Fl∗ = ρl∗ u∗l , ρl∗ (u∗l )2 + π ∗ , (2.55) with π∗ =

cr πl + cl πr − cl cr (ur − ul ) . c l + cr

(2.56)

The numerical flux F (Ul , Ur ) is defined by (2.34). As in the HLL solver, one has two parameters cl , cr that need to be chosen satisfying the subcharacteristic conditions ρ 2 p (ρ) cl2 for any ρ between ρl and ρl∗ , and ρ 2 p (ρ) cr2 for any ρ between ρr and ρr∗ . It is proved in Bouchut (2004) via the interpretation by the Suliciu relaxation system that this subcharacteristic condition implies discrete entropy inequalities for the entropy η defined in (1.22), with numerical entropy fluxes Gl∗ = ρl∗ (u∗ )2 /2 + ρl∗ el∗ + π ∗ u∗ , Gr∗ = ρr∗ (u∗ )2 /2 + ρr∗ er∗ + π ∗ u∗ , (2.57) with el∗ = el − πl2 /2cl2 + (π ∗ )2 /2cl2 , er∗ = er − πr2 /2cr2 + (π ∗ )2 /2cr2 ,

(2.58)

and el = e(ρl ), er = e(ρr ). According to Bouchut (2004), the following choice of cl , cr enables to fully satisfy the nonnegativity of ρl∗ , ρr∗ and the subcharacteristic conditions, ⎧c πr −πl + ul − ur + , ⎨ ρll = p (ρl ) + α √ ρr p (ρr ) if πr − πl 0, (2.59) ⎩ cr = p (ρ ) + α πl −πr + u − u , r l r ρ cl + ⎧ cr πl −πr r √ + ul − ur + , ⎨ ρr = p (ρr ) + α ρl p (ρl ) if πr − πl 0, (2.60) ⎩ cl = p (ρ ) + α πr −πl + u − u , l l r + ρl cr with the notation x+ = max(0, x). For this to be valid, we have to assume that d ∀ρ > 0, (2.61) ρ p (ρ) > 0, dρ ρ p (ρ) → ∞ as ρ → ∞, (2.62) d ρ p (ρ) α p (ρ), for some constant α 1. (2.63) dρ In the case of shallow water p(ρ) = gρ 2 /2, this is satisfied with α = 3/2.

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

210

In conclusion, with the choice (2.59)–(2.60) and under assumptions (2.61)–(2.63), the solver preserves the nonnegativity of density, handles data with vacuum, and satisfies a discrete entropy inequality. 2.11. Passive transport If often occurs in fluid dynamics problems that one has to solve a transport equation for a material quantity φ, that follows the particle paths. In other words, φ satisfies ∂t (ρφ) + ∂x (ρuφ) = 0,

(2.64)

where ρ 0, u are assumed to be given by the resolution of a closed system of equations that include the continuity equation ∂t ρ + ∂x (ρu) = 0.

(2.65)

For smooth solutions, the two equations can be combined to give ∂t φ + u∂x φ = 0.

(2.66)

This equation means that φ is simply passively transported with the flow. The functions ρ, u can be obtained by solving a system of equations that can involve other quantities, but we need not specify how they are obtained for what we explain here. An important property of (2.64)–(2.65) is that it gives directly a family of entropy inequalities, because if we multiply (2.66) by S (φ) for any function S, we get ∂t (S(φ)) + u∂x (S(φ)) = 0, thus combining it with (2.65) we get ∂t (ρS(φ)) + ∂x (ρuS(φ)) = 0. Now, if S is convex, one can easily check that ρS(φ) is a convex function of (ρ, ρφ). Therefore, one expects for weak solutions a family of inequalities, ∂t ρS(φ) + ∂x ρuS(φ) 0, S convex. (2.67) In particular, taking S(φ) = (φ − k)+ ≡ max(0, φ − k) or S(φ) = (k − φ)+ ≡ max(0, k − φ), we deduce the maximum principle inf φ 0 (y) φ(t, x) sup φ 0 (y). y

(2.68)

y

Numerically, there is a very natural way to solve (2.64), being given a numerical flux for (2.65). This is done as follows. Assume that we have the discrete conservative formula

t 0 0 F = 0, − Fi−1/2 ρin+1 − ρi + (2.69)

xi i+1/2

2. Finite volume schemes for conservative systems

211

0 for some numerical flux Fi+1/2 that is computed in a way we do not need to specify. Then the natural scheme for solving (2.64) is

ρin+1 φin+1 − ρi φi +

t φ φ Fi+1/2 − Fi−1/2 = 0,

xi

(2.70)

with the by now classical passive transport flux 0 0 Fi+1/2 φi if Fi+1/2 0, φ Fi+1/2 = 0 0 Fi+1/2 φi+1 if Fi+1/2 0.

(2.71)

Denoting x+ = max(0, x) and x− = min(0, x), another way to write (2.71) is 0 0 φ φ + Fi+1/2 φ . Fi+1/2 = Fi+1/2 (2.72) + i − i+1 One can check (Bouchut, 2004) that the passive transport scheme (2.70)–(2.71) is consistent with (2.64), and under a natural CFL condition it satisfies the discrete maximum principle min(φi−1 , φi , φi+1 ) φin+1 max(φi−1 , φi , φi+1 )

(2.73)

and the discrete entropy inequalities

t S S F 0, − Fi−1/2 ρin+1 S φin+1 − ρi S(φi ) +

xi i+1/2 with

S Fi+1/2

=

0 Fi+1/2 S(φi )

0 if Fi+1/2 0,

0 Fi+1/2 S(φi+1 )

0 if Fi+1/2 0.

S convex, (2.74)

(2.75)

E XAMPLE 2.3. Consider the isentropic system in one and a half dimension, or isentropic system with transverse velocity ⎧ ⎪ ⎨ ∂t ρ + ∂x (ρu) = 0, (2.76) ∂t (ρu) + ∂x (ρu2 + p(ρ)) = 0, ⎪ ⎩ ∂t (ρv) + ∂x (ρuv) = 0. This conservative system is hyperbolic with eigenvalues λ1 = u − p (ρ), λ2 = u, λ3 = u + p (ρ), and it has an entropy (the physical total energy) η = ρ u2 + v 2 /2 + ρe(ρ), (2.77) with entropy flux G = ρ u2 + v 2 /2 + ρe(ρ) + p(ρ) u.

(2.78)

212

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

The system (2.76) can be considered as the coupling of the usual one-dimensional isentropic system with a passive transport equation on v. Therefore, a numerical flux for (2.76) can be obtained as l , U r , F v (Ul , Ur ) , F (Ul , Ur ) = F is U (2.79) l = (ρl , ρl ul ), U r = with Ul = (ρl , ρl ul , ρl vl ), Ur = (ρr , ρr ur , ρr vr ), U is (ρr , ρr ur ), F is any numerical flux for the usual isentropic system, and 0 F vl if F 0 0, F v (Ul , Ur ) = (2.80) F 0 vr if F 0 0, l , U r ) is the density component of F is (U l , U r ). The numerical with F 0 = F 0 (U flux (2.79) satisfies the discrete maximum principle on v, and if F is satisfies discrete entropy inequalities, it also satisfies discrete entropy inequalities, associated to η, obtained by addition with (2.74) with S(v) = v 2 /2. R EMARK 2.4. In the previous example, if F is is the numerical flux of the HLLC solver, then the numerical flux (2.79) is associated to a simple approximate Riemann solver obtained by completing the intermediate states by vl∗ = vl , vr∗ = vr . This is due to the fact that in the HLLC solver, F0 0

if and only if

σ2 = u∗ 0.

(2.81)

This property underlines the fact that the HLLC solver is particularly adapted to gas dynamics problems. 2.12. Numerical treatment of boundary conditions The numerical treatment of boundary conditions is usually done quite easily within the finite volume framework, even if we do not really know which solution we select, knowing that the mathematical theory is quite poor on the subject. Consider a system of conservation laws in a bounded interval (all we are discussing here would be valid also with source term or for a quasilinear system) t > 0, a < x < b, ∂t U + ∂x F (U ) = 0, (2.82) together with boundary conditions. To fix ideas, let us first assume that we wish to impose for t > 0 U (t, a) = Ua (t)

and U (t, b) = Ub (t),

(2.83)

even if in general only a part of the boundary values can be retained in the solution, see Section 1.4 and in particular (1.33). Assume that the interval (a, b) is divided into nx cells Ci = (xi−1/2 , xi+1/2 ), i = 1, . . . , nx , on which we compute discrete values Uin , i = 1, . . . , nx . The

2. Finite volume schemes for conservative systems

213

boundary is thus a = x1/2 , b = xnx +1/2 . Then the simple and classical way to discretize (2.82)–(2.83) is to set ghost values U0n and Unnx +1 by U0n = Ua (tn ),

Unnx +1 = Ub (tn ),

(2.84)

and then to apply the finite volume update Uin+1 − Uin +

t (Fi+1/2 − Fi−1/2 ) = 0,

xi

i = 1, . . . , nx ,

(2.85)

with as usual

n Fi+1/2 = F Uin , Ui+1 ,

i = 0, . . . , nx ,

(2.86)

for some given numerical flux. Then (2.84) is naturally taken into account in the computation of F1/2 and Fnx +1/2 . A more accurate choice would be indeed to take U0n and Unnx +1 so that (U0n + U1n )/2 = Ua (tn ), (Unnx + Unnx +1 )/2 = Ub (tn ), which means that we extend the values of U by odd symmetry. However, when doing such extrapolation, one has to care to get admissible values for U0n and Unnx +1 (nonnegative density, or other requirement depending on the system). Thus, formulas for doing this would be U0n = proj(2Ua (tn ) − U1n ), Unnx +1 = proj(2Ub (tn ) − Unnx ), where proj is a suitable projection onto admissible states. Assume now that we would like to solve (1.33) at the boundary x = a. Then one just has to set (2.84), where Ua (t) is chosen in such a way that Ua (t) ∈ B(t).

(2.87)

However, it is not sure at all that the numerical solution will converge to the expected one. It could converge to the solution to another boundary problem, for example U (t, a) ∈ C(t) for another set C(t). We refer to Godlewski and Raviart (1996) for a discussion. E XAMPLE 2.5. Consider the isentropic gas dynamics system (1.14), with the wall boundary condition u(t, a) = 0, as discussed in Example 1.5. Then, the most simple way to define the ghost value U0n is to symmetrize U1n by taking ρ0n = ρ1n , un0 = −un1 . E XAMPLE 2.6. For periodic boundary conditions, the natural choice is U0n = Unnx , Unnx +1 = U1n . E XAMPLE 2.7. For Neumann boundary conditions, i.e. ∂x U (t, a) = 0, one can take U0n = U1n . E XAMPLE 2.8. Assume the we would like to solve the isentropic system (1.14), with set incoming mass flux on the boundary ρ(t, a)u(t, a) = q(t) 0. Then

214

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

since the numerical update (2.85)–(2.86) involves a flux, an accurate way of proceeding is to look for a value U0n such that F 0 (U0n , U1n ) = q(tn ). A priori there is no unique solution to this problem, and of course it depends on the particular form of the numerical flux.

3. Finite volumes for systems with source terms A lot of progress has been made in the past years on the numerical treatment of source terms in systems of conservation laws, and in particular for the shallow water system. To fix ideas, in this section we consider systems of the form ∂t U + ∂x F (U, Z) + B(U, Z)Zx = 0, (3.1) where U (t, x) ∈ Rp is the unknown, Z(x) ∈ Rr is a given vector valued function, and Zx = ∂x Z. The nonlinearities are supposed to be smooth, F (U, Z) ∈ Rp , and B(U, Z) is a p × r matrix. The case when Z is scalar (r = 1) in (3.1) is already interesting, and it covers the special choice Z(x) = x, which writes if we take F = F (U ), B = B(U ), ∂t U + ∂x F (U ) = −B(U ). (3.2) The numerical challenge that occurs in the resolution of (3.1) is the accurate treatment of the balance between the differential term and the source during the time evolution. In this respect, the steady states play a crucial role. They are the solutions U (x) which are independent of time, and hence solve ∂x (F (U, Z)) + B(U, Z)Zx = 0. These solutions play an important role because they are usually obtained as limits when time tends to infinity of the general solutions to (3.1). In order to well resolve (3.1), in particular concerning the source term, main ideas were developed in Greenberg and LeRoux (1996), Gosse and LeRoux (1996), Greenberg, LeRoux, Baraille and Noussair (1997), Bermúdez and Vásquez (1994), LeRoux (1999), Gosse (2000, 2001), Jin (2001), Botchorishvili, Perthame and Vasseur (2003). In these works was introduced the notion of wellbalanced schemes, which are schemes that preserve exactly at the discrete level some discrete steady states solutions. Many schemes were afterwards developed (Vázquez-Cendón, 1999; García-Navarro and Vázquez-Cendón, 2000; Kurganov and Levy, 2002; Perthame and Simeoni, 2001; Chacón Rebollo, Domínguez Delgado and Fernández Nieto, 2003; Chinnayya, LeRoux and Seguin, 2004; Gallouët, Hérard and Seguin, 2003; Katsaounis, Perthame and Simeoni, 2004; Bale, LeVeque, Mitran and Rossmanith, 2002; Audusse, Bouchut, Bristeau, Klein and Perthame, 2004; Parés and Castro, 2004). Nonconservative systems with sources have been considered by Parés (2006) with systems of the form ∂t U + ∂x F (U, Z) + A(U, Z)∂x U + B(U, Z)Zx = 0. (3.3)

3. Finite volumes for systems with source terms

215

Note that the systems (3.1) or (3.3) can be interpreted as quasilinear systems in = (U, Z), by completing them with the trivial equation ∂t Z = 0. the variable U At this level, we have to explain a bit how (3.1) can be well posed, even if the mathematical theory is not very well settled. We shall always assume that the system is hyperbolic with respect to U , which means that FU (U, Z) is diagonalizable. The first case when (3.1) is well posed is when Z(x) is smooth (say Lipschitz continuous). Then this is just a conservative system with nonsingular source, and physically relevant discontinuous weak solutions are selected with the Rankine– Hugoniot conditions (1.18) (which do not involve the source), and with an entropy inequality ∂t η(U, Z) + ∂x G(U, Z) + ηU (FZ + B) − GZ Zx 0, (3.4) where indices U or Z denote partial differentiation, and η(U, Z) is an entropy with entropy flux G(U, Z) (i.e. GU = ηU FU ). Here η needs only be convex with respect to U . The second case when (3.1) is well posed is when Z can have discontinuities, but the eigenvalues of FU do not vanish (and we call resonant a point (U, Z) where 0 is an eigenvalue of FU ). Indeed, if we consider (3.1) as a quasilinear system (with the equation ∂t Z = 0), then the matrix of the system (in the sense of Section 1.1) is FU FZ + B . A(U, Z) = (3.5) 0 0 The eigenvalues of A(U, Z) are those of FU , to which we adjoin the value 0. At a noncritical point, FU is diagonalizable and does not have 0 as eigenvalue, thus obviously A(U, Z) is diagonalizable, and the system is hyperbolic. Then, the good property is that weak solutions to (3.1) are well-defined out of the resonant points, because the nonconservative product BZx could cause difficulties only when Z has discontinuities, and in this location the eigenvalue involved is 0 which is linearly degenerate. We recall that discontinuities associated with linearly degenerate eigenvalues are well-defined, without need of any Rankine– Hugoniot condition (Bouchut, 2004). Therefore, the only Rankine–Hugoniot condition needed is (1.18) when ξ˙ = 0. To complete the system we have to provide an entropy inequality for the quasilinear system in (U, Z), ∂t η(U, Z) + ∂x G(U, Z) 0, (3.6) with η(U, Z) convex with respect to U , GU = ηU FU , and GZ = ηU (FZ + B). In the general case of discontinuous Z with resonant points (which is the case for the shallow water system), there are some data for which several solutions exist, containing shocks or rarefactions attached to stationary contact discontinuities. All these solutions can be stable (limits of sequences of “good” unique

216

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

solutions). At the numerical level, this is even worse, one can have convergence to something that is even not a weak solution (but fortunately this happens rather rarely). Notice that contact discontinuities associated to the vanishing eigenvalue are in particular steady states, which explains the critical role played by these solutions. 3.1. The one-dimensional shallow water system The main example of system with source, resonance and nontrivial steady states is the Saint-Venant system for shallow water with topography. This system is naturally under the form (3.1). Denoting by h(t, x) 0 the water height, and u(t, x) the velocity, the system reads ∂t h + ∂x (hu) = 0, (3.7) ∂t (hu) + ∂x (hu2 + gh2 /2) + hgzx = 0, where g > 0 is the gravitational constant and z(x) is the topography. The system has the form (3.1) with U = (h, hu), Z = z, F (U, z) = F (U ) = F 0 (U ), F 1 (U ) = hu, hu2 + gh2 /2 , (3.8) B(U, z) = B(U ) = (0, gh).

(3.9)

The conservative part is nothing else than the isentropic gas dynamics sys2 tem (1.14) √ with ρ ≡ h and p(h) = gh /2. The resonant points are defined by u = ± gh. In order to obtain the steady states, we subtract u times the first equation in (3.7) to the second, and divide the result by h. We get ∂t u + ∂x u2 /2 + gh + gz = 0. (3.10) Therefore, the steady states are exactly the functions h(x), u(x) satisfying hu = cst, u2 /2 + gh + gz = cst.

(3.11)

Between these steady states, some play an important role, the steady state at rest for which the first constant is 0, or equivalently u = 0, (3.12) h + z = cst. An entropy can be obtained as follows. We multiply the first equation in (3.7) by u2 /2 + gh, we multiply (3.10) by hu, and add the results. This gives ∂t η + ∂x G + hugzx 0,

(3.13)

3. Finite volumes for systems with source terms

where η and G are defined according to (1.22) and (1.24), η(U ) = hu2 /2 + gh2 /2, G(U ) = hu2 /2 + gh2 u.

217

(3.14)

The inequality stands here just because of discontinuous solutions, as usual. Next, we add to (3.13) the first equation in (3.7) multiplied by gz, and since ∂t z = 0, this yields ∂t (η + hgz) + ∂x G + hugz 0. (3.15) Thus we have the entropy (the physical energy) and entropy flux η = η + hgz,

G = G + hugz.

(3.16)

The shallow water system has other specific properties that are worthwhile to state. The first is that, as in gas dynamics, the water height h needs to remain nonnegative. Also the total amount of water need to be preserved, which means that the first equation is conservative. Another property is that the system (3.7) is invariant under translations in z, in the sense that adding a constant to z does not modify the equations. The inequality (3.13) has the advantage to have this property also, in contrast with (3.15). A variant of the system is the one and a half dimensional shallow water system, which is (3.7) completed with the transverse velocity equation ∂t (hv) + ∂x (hvu) = 0.

(3.17)

As commented in Section 2.11, this extended system does not introduce any further difficulty. 3.2. Upwind interface schemes As explained above, the numerical challenge for (3.1) is to well resolve the balance between the conservative term ∂x (F (U, Z)) and the source B(U, Z)Zx . In order to do so, a nice way of formulating the schemes is to discretize these two terms at the same location, the interface xi+1/2 . This is the upwind interface scheme (UIS) approach, formulated in its generality by Katsaounis, Perthame and Simeoni (2004). We shall retain here only this formulation, which is rather general and flexible. It handles data Zi attached to each cell, instead of interface values Zi+1/2 used for example by Jin (2001). Similarly as in Section 2, let us consider a one-dimensional grid of points xi+1/2 , i ∈ Z, as in (2.2)–(2.3). In order to solve (3.1), we consider discrete data n = (U n , Zi ) over the mesh. The data Zi do not evolve with time, as expected, U i i while

t (Fi+1/2− − Fi−1/2+ ) = 0, Uin+1 − Uin + (3.18)

xi

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with n Fi+1/2− = Fl Uin , Ui+1 , Zi , Zi+1 , n , Zi , Zi+1 . Fi+1/2+ = Fr Uin , Ui+1

(3.19)

The functions l , U r , Fl (Ul , Ur , Zl , Zr ) ≡ Fl U l , U r Fr (Ul , Ur , Zl , Zr ) ≡ Fr U

(3.20)

are the left and right numerical fluxes. In this formulation we only consider firstorder three point nonconservative schemes. A more particular form occurs when solving (3.1) with F (U, Z) = F (U ) and B(U, Z) = B(U ) (which is the case for shallow water), because then there is invariance of the system by addition of a constant to Z. Therefore, in this case, Fl and Fr should depend only on Zr − Zl , and not on Zl and Zr independently. 3.3. Well-balancing A main feature that is desirable for the scheme (3.18)–(3.19) is that it preserves some discrete steady states, approximating the exact ones defined as smooth functions (U (x), Z(x)) satisfying ∂x F (U, Z) + B(U, Z)Zx = 0. (3.21) These discrete steady states are discrete sequences (Ui , Zi )i∈Z that satisfy an approximation of (3.21), under the form of a nonlinear relation at each interface, linking Ui , Ui+1 , Zi , Zi+1 , D(Ui , Ui+1 , Zi , Zi+1 ) = 0.

(3.22)

We shall often write this relation only locally, as D(Ul , Ur , Zl , Zr ) = 0. These discrete steady states can be defined in various ways. E XAMPLE 3.1. For a scalar law U ∈ R, and if F (U, Z) = F (U ), B(U, Z) = B(U ) > 0, we can define D(U ) by D (U ) =

F (U ) . B(U )

Then (3.21) becomes ∂x D(U ) + Z = 0.

(3.23)

(3.24)

Therefore, we can take for discrete steady states the relation D(Ul ) + Zl = D(Ur ) + Zr .

(3.25)

3. Finite volumes for systems with source terms

219

E XAMPLE 3.2. For the shallow water system (3.7), since the continuous steady states are those solving (3.11), we can take for discrete steady states the relations hl ul = hr ur , (3.26) u2l /2 + ghl + gzl = u2r /2 + ghr + gzr . In particular, the discrete steady states at rest are those for which ul = ur = 0, hl + zl = hr + zr .

(3.27)

The relations (3.27) will be chosen in our applications. Once some discrete steady states are selected, we define the well-balanced schemes as follows. D EFINITION 3.1. The scheme (3.18)–(3.19) is well-balanced relatively to some discrete steady states defined by a relation D if one has for any Ul , Ur , Zl , Zr satisfying D(Ul , Ur , Zl , Zr ) = 0 the identities Fl (Ul , Ur , Zl , Zr ) = F (Ul , Zl ), Fr (Ul , Ur , Zl , Zr ) = F (Ur , Zr ).

(3.28)

According to (3.18)–(3.19), this property guarantees obviously that if at time tn we start with a steady state sequence (Ui ), then it remains unchanged at the next time level. 3.4. Consistency The consistency of (3.18)–(3.19) with (3.1) comes naturally after the discussion of well-posedness in the beginning of Section 3 in the case of discontinuous Z. D EFINITION 3.2. We say that the scheme (3.18)–(3.19) is consistent with (3.1) if the numerical fluxes satisfy the consistency with the exact flux Fl (U, U, Z, Z) = Fr (U, U, Z, Z) = F (U, Z) for any (U, Z) ∈ Rp × Rr ,

(3.29)

and the asymptotic conservativity/consistency with the source Fr (Ul , Ur , Zl , Zr ) − Fl (Ul , Ur , Zl , Zr ) = −B(U, Z)(Zr − Zl ) + o(Zr − Zl ), as Ul , Ur → U and Zl , Zr → Z.

(3.30)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

We recall here that the notation φ = o(Zr −Zl ) means that φ = |Zr −Zl |φ, with φ → 0. We have to mention that condition (3.30) only requires a property when Zl and Zr are asymptotically close, thus it only concerns continuous functions Z(x). This is coherent with our discussion above: one should impose consistency in the weak sense for Z = cst, and consistency for smooth solutions, which is exactly what (3.29)–(3.30) means (notice that taking Zl = Zr in (3.30) gives in that case the conservativity Fl = Fr ). Further justifications can be found in Bouchut (2004), Perthame and Simeoni (2003). 3.5. Discrete entropy inequalities Stability can be analyzed, as in the conservative case, via invariant domains and entropy. Let us only write explicitly the definition concerning entropy inequalities. D EFINITION 3.3. We say that the scheme (3.18)–(3.19) satisfies a discrete entropy inequality associated to the convex entropy η for (3.1) (in the sense of (3.6)), if there exists a numerical entropy flux function G(Ul , Ur , Zl , Zr ) which is consistent with the exact entropy flux (in the sense that G(U, U, Z, Z) = G(U, Z)), such that, under some CFL condition, the discrete values computed by (3.18)– (3.19) automatically satisfy

t η Uin+1 , Zi − η Uin , Zi + (Gi+1/2 − Gi−1/2 ) 0,

xi

(3.31)

n , Zi , Zi+1 . Gi+1/2 = G Uin , Ui+1

(3.32)

with

3.6. Roe and F-wave methods The most general efficient method that enables to treat source terms is the Roe method, even if it has its usual weaknesses which are the need of entropy fix and the failure of nonnegativity. The use of the Roe method in the context of source and with eventually nonconservative terms, as in (3.3), has been studied in Bermúdez and Vásquez (1994), Vázquez-Cendón (1999), García-Navarro and Vázquez-Cendón (2000), Parés and Castro (2004), see also the references therein. The approach there is to apply the Roe method to the quasilinear system in (U, Z) obtained by completing (3.1) or (3.3) with the equation ∂t Z = 0. A treatment of wet/dry fronts is proposed in Castro, Ferreiro, García-Rodríguez, González-Vida, Macías, Parés and Vázquez-Cendón (2005). The two layer shallow water is also treated with this approach in Castro, Macías and Parés (2001), Castro, GarcíaRodríguez, González-Vida, Macías, Parés and Vázquez-Cendón (2004). A related method is Gallouët, Hérard and Seguin (2003). Independently, the F-wave decom-

3. Finite volumes for systems with source terms

221

position method has been proposed by Bale, LeVeque, Mitran and Rossmanith (2002). It appears that indeed both methods give the same basic scheme, that we describe here. The solver can be put under the form of a simple approximate Riemann solver, as formulated in Section 2.5, generalized to the case of source. Knowing the form (3.5) of the matrix of our system (3.1), and assuming no resonance, one has to take one speed σm0 = 0, and the other speeds nonzero. An approximation of the jump in the flux across the stationary discontinuity can be taken as r − Zl ), δFm0 = −B(Z

(3.33)

of B(U, Z) that needs to be chosen. Then, noticing for some approximation B that for the waves with nonzero eigenvalues, Z has no jump, we can write for the flux jump δFk = σk δUk

for k = m0 .

Now we write F (Ur , Zr ) − F (Ul , Zl ) =

(3.34)

m

k=1 δFk ,

r − Zl ) = F (Ur , Zr ) − F (Ul , Zl ) + B(Z

so that with (3.33) it gives

δFk .

(3.35)

k=m0

It is natural to take δUk eigenvector of Aˆ for k = m0 , where Aˆ is a diagonalizable ˆ k = σk δUk . matrix (with nonzero eigenvalues) approximating FU (U, Z), i.e. AδU Therefore, by (3.34), δFk is also an eigenvector, ˆ k = σk δFk AδF

for k = m0 .

(3.36)

The relations (3.35)–(3.36) define in a unique way the values σk for k = m0 , ˆ and the δFk for k = m0 . The δUk are also which are the distinct eigenvalues of A, determined by (3.34), thus we recover the intermediate states by Uk − Uk−1 = δUk ,

k = m0 .

(3.37)

Finally, the numerical fluxes are given by δFk , Fl (Ul , Ur , Zl , Zr ) = F (Ul , Zl ) + σk 0

The name F-wave decomposition method comes from the decomposition (3.35) ˆ that directly of the jump in fluxes into components δFk in the eigenspaces of A, come into the definition of the numerical fluxes (3.38). The consistency condition (3.29) is obvious since if Zl = Zr and Ul = Ur , all δFk for k = m0 vanish by (3.35), thus the δUk also vanish by (3.34), and

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Uk = Ul = Ur by (3.37). The consistency with the source (3.30) is also easy to get since by (3.38) and (3.35), Fr −Fl = F (Ur , Zr )−F (Ul , Zl )− k=m0 δFk = r − Zl ), which reduces therefore to the consistency of B. −B(Z We notice that in the case without source Zl = Zr , the method reduces m0 −1 to the Roe method only if Um0 = Um0 −1 , i.e. Ul + k=1 δUk = Ur − m ˆ ˆ r− δU , and applying A, which is supposed invertible, this gives A(U k k=m0 +1 ˆ Ul ) = F (Ur , Zr ) − F (Ul , Zl ) i.e. the condition on A to be a Roe matrix. There a Roe matrix obtained by freezing Z fore, a possible choice is Aˆ = A(Ul , Ur , Z), to a value Z. The well-balancing property is also easy to obtain from (3.38), it means that r −Zl ) = 0. δFk = 0 for all k = m0 , or equivalently F (Ur , Zr )−F (Ul , Zl )+ B(Z In the case of the shallow water system (3.7), one can take hl + hr B = 0, g (3.39) . 2 This gives the well-balanced property with respect to the steady states at being 0, this gives the conservrest (3.27). Moreover the first component of B ativity of the water height. However, as in the Roe method, it is not possible to analyze the nonnegativity of water height and the entropy inequality. 3.7. Hydrostatic reconstruction scheme The hydrostatic reconstruction scheme is especially designed to solve shallow water type problems. It has been introduced by Audusse, Bouchut, Bristeau, Klein and Perthame (2004), and widely used in Audusse and Bristeau (2005), Bouchut, Le Sommer and Zeitlin (2004), Le Sommer, Reznik and Zeitlin (2004), Bouchut, Le Sommer and Zeitlin (2005), Mangeney-Castelnau, Bouchut, Vilotte, Lajeunesse, Aubertin and Pirulli (2005), Noelle, Pankratz, Puppo and Natvig (2006), Marche (2005). Its strengths are its simplicity and its robustness. Additionally, the hydrostatic reconstruction scheme has the property to be easily adaptable to systems of the same type as the shallow water problem. The construction directly involves the steady states are rest (3.27). Denote as before U = (h, hu). Then the numerical fluxes are defined by 0 Fl (Ul , Ur , zl , zr ) = F(Ul∗ , Ur∗ ) + , p(hl ) − p(h∗l ) 0 Fr (Ul , Ur , zl , zr ) = F (Ul∗ , Ur∗ ) + (3.40) , p(hr ) − p(h∗r ) where p(h) = gh2 /2 and F(Ul , Ur ) is a given consistent numerical flux for the shallow water problem without source. Good choices are for example the HLLC

3. Finite volumes for systems with source terms

223

numerical flux of Section 2.10 (our favorite choice), or the HLL numerical flux of Section 2.9. The name “hydrostatic” comes from the fact that in (3.40) and (3.42), only the pressure part of the system is really involved, the advection part being neglected. The reconstructed states Ul∗ , Ur∗ are defined by Ul∗ = (h∗l , h∗l ul ), h∗l

Ur∗ = (h∗r , h∗r ur ), ∗

= (hl + zl − z )+ ,

h∗r

(3.41) ∗

= (hr + zr − z )+ .

(3.42)

The positive parts (recall that x+ = max(0, x)) in the right-hand sides of (3.42) are just to ensure that we get some nonnegative heights h∗l , h∗r . The value z∗ is defined by z∗ = max(zl , zr ), and another way of writing (3.42)–(3.43) is, with z = zr − zl , h∗r = hr − (− z)+ + , h∗l = hl − ( z)+ + ,

(3.43)

(3.44)

which shows that Fl , Fr do indeed depend only on the difference z, as expected from the invariance z → z + c. P ROPOSITION 3.4. Consider a consistent numerical flux F for the homogeneous shallow water problem (i.e. with z = cst), that preserves nonnegativity of the water height and satisfies a semi-discrete entropy inequality corresponding to the entropy η in (3.14). Then the scheme defined by the numerical fluxes (3.40), (3.41), (3.44) (0) (i) (ii) (iii) (iv)

is conservative in water height, preserves the nonnegativity of h, is well-balanced, i.e. it preserves the discrete steady states at rest (3.27), is consistent with the shallow water system (3.7), satisfies a semi-discrete entropy inequality associated to the entropy η in (3.16).

P ROOF. These results have been proved in Audusse, Bouchut, Bristeau, Klein and Perthame (2004), but for completeness we rewrite here the arguments, except about the semi-discrete entropy inequality (iv). Here, semi-discrete refers to the limit scheme obtained as t → 0. Denote the components of the numerical fluxes by upper indices 0 and 1, Fl = (Fl0 , Fl1 ), Fr = (Fr0 , Fr1 ). Then the conservativity property (0) is obvious since (3.40) gives that Fl0 = Fr0 . For property (ii) of well-balancing, consider discrete steady states at rest, i.e. ul = ur = 0, hl + zl = hr + zr . Then, (3.42) gives h∗l = h∗r , and by (3.41),

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Ul∗ = Ur∗ . Therefore, by consistency of F and since ul = ur = 0, (3.40) gives 0 Fl = F (Ul∗ ) + (3.45) = F (Ul ), p(hl ) − p(h∗l ) 0 ∗ Fr = F (Ur ) + (3.46) = F (Ur ), p(hr ) − p(h∗r ) which gives (3.28). The consistency (iii) has to be looked at carefully, it is the less obvious property. We have to check the two properties of Definition 3.2. The consistency with the exact flux Fl (U, U, z, z) = Fr (U, U, z, z) = F (U ) is obvious since Ul∗ = Ul and Ur∗ = Ur whenever zr = zl . For consistency with the source (3.30), we write 0 . Fr − Fl = (3.47) p(h∗l ) − p(hl ) + p(hr ) − p(h∗r ) ∗∗ Now, we can write p(h∗l ) − p(hl ) = g(h∗l − hl )h∗∗ l for some hl between hl and ∗ ∗ ∗ ∗∗ ∗∗ hl , and p(hr ) − p(hr ) = g(hr − hr )hr for some hr between hr and h∗r . Then, assuming h > 0, the positive parts in (3.44) play no role if hl − h, hr − h and z ∗ are small enough. Thus we have p(h∗l )−p(hl ) = −h∗∗ l (g z)+ , p(hr )−p(hr ) = ∗∗ −hr (−g z)+ , which gives (3.30). In the special case h = 0, the positive parts in (3.44) can play a role only when hl = O( z), or respectively hr = O( z), and we conclude that (3.30) always holds, proving (iii). Let us now prove the nonnegativity statement (i). It is a consequence of the property

h∗l hl ,

h∗r hr ,

(3.48)

that comes directly from the definition (3.44). By assumption the numerical flux F preserves the nonnegativity or h, and here it is meant it holds interface by interface. According to Bouchut (2004), this means that there exists some σl (Ul , Ur ) < 0 < σr (Ul , Ur ) such that hl +

F 0 (Ul , Ur ) − hl ul 0, σl (Ul , Ur )

hr +

F 0 (Ul , Ur ) − hr ur 0, σr (Ul , Ur )

(3.49)

for any Ul and Ur (with nonnegative densities hl , hr ). This implies in particular that h∗l +

F 0 (Ul∗ , Ur∗ ) − h∗l ul 0, σl (Ul∗ , Ur∗ )

h∗r +

F 0 (Ul∗ , Ur∗ ) − h∗r ur 0. σr (Ul∗ , Ur∗ )

(3.50)

3. Finite volumes for systems with source terms

225

But since necessarily 1−ul /σl (Ul∗ , Ur∗ ) 0, 1−ur /σr (Ul∗ , Ur∗ ) 0, we deduce with (3.48) that hl +

Fl0 (Ul , Ur , zl , zr ) − hl ul 0, σl (Ul∗ , Ur∗ )

hr +

Fr0 (Ul , Ur , zl , zr ) − hr ur 0, σr (Ul∗ , Ur∗ )

(3.51)

which means that the scheme preserves the nonnegativity of h by interface. Note in particular that the speeds are σl (Ul∗ , Ur∗ ) and σr (Ul∗ , Ur∗ ), thus the CFL condition associated to the scheme is the one of F corresponding to the data Ul∗ and Ur∗ ,

t max −σl Ul∗ , Ur∗ , σr Ul∗ , Ur∗ min( xl , xr ). (3.52) In particular, if the numerical flux F is able to treat the vacuum (see Section 2.7), the hydrostatic scheme can also handle the vacuum since there is no blow up in (3.52) in the limit of vanishing Ul or Ur . We have to remark that the entropy inequality is only semi-discrete, thus does not give any a priori bound on the total energy (see Section 2.3) In practice we do not observe instabilities, as long as hi remains nonnegative, which is the case under the CFL condition (3.52). 3.8. Other efficient first-order schemes for shallow water Several other well-balanced methods were proposed to solve the shallow water system (3.7), like Jin (2001), Kurganov and Levy (2002). However, very few were able to treat correctly the vacuum, and at the same time to work for all type of data, including resonance. The only methods that enjoy similar properties as the hydrostatic reconstruction scheme are the kinetic method (Perthame and Simeoni, 2001) and the Suliciu relaxation method (Bouchut, 2004). Both overall satisfy a fully discrete entropy inequality, but on the counterpart they are quite expensive in terms of CPU and complexity. 3.9. Additional source terms: the apparent topography method In order to treat additional zero-order source terms with the well-balanced property, a general approach is as follows, explained here for the shallow water system. The particular case of Coriolis force can be treated within this method, see next section. Consider the shallow water system with topography and external force E, ∂t h + ∂x (hu) = 0, (3.53) ∂t (hu) + ∂x (hu2 + gh2 /2) + ghzx = hE,

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

where z = z(x), E = E(t, x). One can think in particular of a nonlinear coupling like E(t, x) = ϕ(h, u). Now the velocity equation writes ∂t u + ∂x u2 /2 + gh + gz = E, (3.54) and the steady states at rest are given by u = 0,

∂x (gh + gz) = E.

(3.55)

The idea to solve (3.53) is to use an apparent topography, which means to identify the system as the usual shallow water problem with a new topography z+b, where gbx = −E. Now, b depends also on time while it should be time independent, but when using discrete times tn , we can freeze its value on a time interval, thus we take gbxn = −E n and solve the shallow water system on the time interval (tn , tn+1 ) with topography z + bn . Note that for stationary solutions, this approximation is exact, thus the stationary solutions are preserved by this procedure. At the fully discrete level, this is done as follows. We define n n = −Ei+1/2

xi+1/2 , g bi+1/2

(3.56)

where xi+1/2 = xi+1 − xi , and update U = (h, hu) via Uin+1 − Uin +

t (Fi+1/2− − Fi−1/2+ ) = 0,

xi

(3.57)

with n , Fi+1/2− = Fl1d Ui , Ui+1 , zi+1/2 + bi+1/2 n Fi+1/2+ = Fr1d Ui , Ui+1 , zi+1/2 + bi+1/2 ,

(3.58)

where zi+1/2 = zi+1 − zi and the numerical fluxes Fl1d (Ul , Ur , z), Fr1d (Ul , Ur , z) are associated to the usual shallow water problem. If the numerical fluxes Fl1d , Fr1d are consistent with the shallow water system (3.7), then the new scheme (3.56)–(3.58) is consistent with (3.53) in the sense of Definition 3.2, writing (3.53) as hE = hEφx , φ(x) = x, and taking Z = (z, φ). Moreover, if the numerical fluxes Fl1d , Fr1d are well-balanced with respect to steady states at rest, then the new scheme (3.56)–(3.58) automatically lets invariant the data satisfying ui = 0,

ghi+1 + gzi+1 = ghi + gzi + Ei+1/2 xi+1/2 ,

(3.59)

which can be considered as discrete steady states approximating the relations (3.55). Concerning discrete entropy inequalities, there is no reason in general to have a special compatibility with this method, but however if E is bounded, this should not be a problem because the right-hand side is a lower-order term that should not influence the global stability.

3. Finite volumes for systems with source terms

227

3.10. The shallow water system with Coriolis force and transverse velocity Consider now the shallow water system with Coriolis force and transverse velocity ⎧ ⎪ ⎨ ∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x (hu2 + gh2 /2) + ghzx − f hv = 0, (3.60) ⎪ ⎩ ∂t (hv) + ∂x (huv) + f hu = 0, where z = z(x), f = f (x). Since the two first equations of (3.60) identify with (3.53) with E = f v, according to (3.55) the steady states at rest are given by u = 0,

∂x (gh + gz) = f v.

(3.61)

Since the Coriolis force is orthogonal to the velocity (u, v), the system has the same entropy inequality as the shallow water system with transverse velocity, ∂t η(h, u, v, z) + ∂x G(h, u, v, z) 0, (3.62) with η(h, u, v, z) = h u2 + v 2 /2 + gh2 /2 + hgz, G(h, u, v, z) = h u2 + v 2 /2 + gh2 + hgz u.

(3.63)

The particular structure of (3.60) enables to derive an additional conservative quantity, which is obtained as follows. Combining the third and first equations in (3.60) gives ∂t v + u∂x v + f u = 0. Then, differentiating with respect to x gives ∂t (∂x v + f ) + ∂x (∂x v + f )u = 0,

(3.64)

(3.65)

thus the potential vorticity (∂x v + f )/ h is transported at velocity u. A related equation comes by defining a potential Ω(x) by ∂x Ω = f,

∂t Ω = 0.

(3.66)

Then the last equation in (3.60) writes ∂t (hv) + ∂x (hvu) + huΩx = 0,

(3.67)

or in conservative form ∂t h(v + Ω) + ∂x h(v + Ω)u = 0.

(3.68)

Notice that the definition (3.66) of Ω allows to write down the system (3.60) under the form (3.1) with U = (h, hu, hv), Z = (z, Ω). Compared to the usual shallow

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

water system, we have here an extra eigenvalue u associated to the equation on v, and thus an extra resonance occurs when u = 0. This resonance does not imply further difficulty since the eigenvalue u is linearly degenerate. In order to discretize the system (3.60), a natural thing to do is to use the apparent topography method of Section 3.9 for the two first equations. When doing n so, we need a value for the interface force Ei+1/2 in (3.56), and it can be taken simply n Ei+1/2 = fi+1/2

n vin + vi+1

(3.69) , 2 where fi+1/2 is a discretization of f at the point xi+1/2 , that can be taken for example fi+1/2 = f (xi+1/2 ). Then, according to (3.68), in order to get the update for v, a natural thing to do is to apply the passive transport scheme of Section 2.11 to the variable v + Ω. Rewriting it in terms of the physical variable hv and denoting Ωi+1/2 = Ωi+1 − Ωi , this gives finally the scheme

t h h Fi+1/2 − Fi−1/2 = 0,

xi

t hu hu hn+1 Fi+1/2− − Fi−1/2+ = 0, un+1 − hni uni + i i

xi

t hv hv hn+1 F = 0, vin+1 − hni vin + − Fi−1/2+ i

xi i+1/2− − hni + hn+1 i

(3.70)

with h hu n Fi+1/2 , Fi+1/2− = Fl1d hi , ui , hi+1 , ui+1 , zi+1/2 + bi+1/2 , h hu 1d n Fi+1/2 , Fi+1/2+ = Fr hi , ui , hi+1 , ui+1 , zi+1/2 + bi+1/2 , h h Fi+1/2 vi if Fi+1/2 0, hv Fi+1/2− = h h (vi+1 + Ωi+1/2 ) if Fi+1/2 0, Fi+1/2 h h Fi+1/2 (vi − Ωi+1/2 ) if Fi+1/2 0, hv = Fi+1/2+ h h vi+1 if Fi+1/2 0, Fi+1/2 n

bi+1/2 = −fi+1/2

n vin + vi+1

2

xi+1/2 /g.

(3.71) (3.72)

(3.73) (3.74)

Here, Fl1d and Fr1d denote left and right numerical fluxes associated to the onedimensional shallow water equations, as the hydrostatic reconstruction fluxes of Section 3.7. We recall that zi+1/2 = zi+1 − zi and xi+1/2 = xi+1 − xi . We complete (3.72)–(3.73) by setting

Ωi+1/2 = fi+1/2 xi+1/2 .

(3.75)

3. Finite volumes for systems with source terms

229

It is easy to check that whenever Fl1d and Fr1d have the suitable properties (similarly as in Proposition 3.4), the scheme (3.70)–(3.75) is consistent with (3.60), is conservative in water height, preserves nonnegativity of h and is able to compute dry states, and is well-balanced, in the sense that it lets invariants the data satisfying ui = 0,

ghi+1 + gzi+1 = ghi + gzi + fi+1/2

vi + vi+1

xi+1/2 , (3.76) 2

which are obviously a discrete version of (3.61). There is no special property concerning the transport of the potential vorticity (3.65), but numerical experiments show very low numerical diffusion in that equation (Bouchut, Le Sommer and Zeitlin, 2004; Le Sommer, Reznik and Zeitlin, 2004; Bouchut, Le Sommer and Zeitlin, 2005). Concerning entropy inequalities, as commented in the previous section it is enough that the scheme is entropy satisfying when f = 0, since the Coriolis terms are nonsingular zero-order terms. This is the case if Fl1d and Fr1d satisfy a discrete entropy inequality, according to the analysis of Section 2.11. Nevertheless, the structure of the scheme (3.70)–(3.75) enables to prove that an entropy inequality holds, under some condition of sign on the potential vorticity (∂x v + f )/ h. P ROPOSITION 3.5. Assume that the numerical fluxes Fl1d and Fr1d satisfy a discrete entropy inequality, and consider data for which for all i one has fi+1/2 (vi+1 − vi + fi+1/2 xi+1/2 ) 0.

(3.77)

Then there holds a discrete entropy inequality associated to (3.62)–(3.63). n P ROOF. Denote zi+1/2 = zi+1/2 + bi+1/2 . According to Bouchut (2004), 1d the discrete entropy inequality associated with Fl , Fr1d can be written 2

2

hn+1 un+1 /2 + ghn+1 /2 − hi u2i /2 − gh2i /2 i i i

zi+1/2 h

t Fi+1/2 ϑ 1d hi , ui , hi+1 , ui+1 , zi+1/2 + g +

xi 2

zi−1/2 h − ϑ 1d hi−1 , ui−1 , hi , ui , zi−1/2 + g Fi−1/2 0, 2

(3.78)

where ϑ 1d is a numerical entropy flux satisfying ϑ 1d (h, u, h, u, 0) = (hu2 /2 + gh2 )u. But the passive transport scheme applied to v + Ω satisfies the entropy inequality (2.74)–(2.75) with S(φ) = φ 2 /2, thus rewriting it in terms of v gives 2

vin+1 /2 − hi vi2 /2 + hn+1 i

t v v ϑ 0, − ϑi−1/2+

xi i+1/2−

(3.79)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with

v ϑi+1/2−

=

v ϑi+1/2+

=

h if Fi+1/2 0,

h Fi+1/2 vi2 /2

h h (vi+1 + Ωi+1/2 )2 /2 if Fi+1/2 0, Fi+1/2 h Fi+1/2 (vi − Ωi+1/2 )2 /2

h if Fi+1/2 0,

h 2 /2 vi+1 Fi+1/2

h if Fi+1/2 0.

(3.80)

(3.81)

Adding (3.78) and (3.79) yields n+1 2 2 2 hn+1 ui + vin+1 /2 + ghn+1 /2 − hi u2i + vi2 /2 − gh2i /2 i i

t (ϑi+1/2− − ϑi−1/2+ ) 0, +

xi

(3.82)

with

zi+1/2 h v , Fi+1/2 + ϑi+1/2− 2

zi+1/2 h 1d v ϑi+1/2+ = ϑi+1/2 −g , Fi+1/2 + ϑi+1/2+ 2 1d ϑi+1/2 = ϑ 1d hi , ui , hi+1 , ui+1 , zi+1/2 .

1d ϑi+1/2− = ϑi+1/2 +g

(3.83) (3.84)

We claim that this implies a discrete entropy inequality as soon as v v h h ϑi+1/2+ − ϑi+1/2− − g zi+1/2 Fi+1/2 + g zi+1/2 Fi+1/2 0.

Indeed if (3.85) is satisfied, then one can find

v ϑi+1/2

(3.85)

such that

zi+1/2 h

zi+1/2 h v +g Fi+1/2 + ϑi+1/2+ Fi+1/2 2 2

zi+1/2 h

zi+1/2 h v v ϑi+1/2 g −g Fi+1/2 + ϑi+1/2− Fi+1/2 , 2 2 and with (3.83) it implies that −g

(3.86)

zi+1/2 h Fi+1/2 , 2

zi+1/2 h 1d v + ϑi+1/2 +g Fi+1/2 , ϑi+1/2− ϑi+1/2 (3.87) 2 and thus adding to (3.82) gzi times the first equation of (3.70) if gives a discrete entropy inequality associated to (3.62)–(3.63) with numerical entropy flux 1d v ϑi+1/2+ ϑi+1/2 + ϑi+1/2 −g

1d v + ϑi+1/2 +g Gi+1/2 = ϑi+1/2

zi + zi+1 h Fi+1/2 . 2

(3.88)

4. Second-order well-balanced schemes

231

Now let us examine condition (3.85). The left-hand side can be computed using (3.80), (3.81), (3.74) and (3.75), v v n h ϑi+1/2+ − ϑi+1/2− − g bi+1/2 Fi+1/2 h 2 n Fi+1/2 + = Ωi+1/2 /2 − vi Ωi+1/2 − g bi+1/2 h 2 n Fi+1/2 − + − Ωi+1/2 /2 − vi+1 Ωi+1/2 − g bi+1/2 vi + vi+1 h = Ωi+1/2 Ωi+1/2 /2 − vi + Fi+1/2 + 2 vi + vi+1 h + Ωi+1/2 − Ωi+1/2 /2 − vi+1 + Fi+1/2 − 2 h 1 = Ωi+1/2 ( Ωi+1/2 + vi+1 − vi ) Fi+1/2 + 2 h 1 + Ωi+1/2 (− Ωi+1/2 + vi − vi+1 ) Fi+1/2 − 2 h 1 . = Ωi+1/2 ( Ωi+1/2 + vi+1 − vi )Fi+1/2 (3.89) 2 Recalling that Ωi+1/2 = fi+1/2 xi+1/2 , we conclude that this is nonpositive whenever condition (3.77) holds.

4. Second-order well-balanced schemes There are many methods to go to higher-order in the context of finite volume schemes. In the case of source terms with the well-balanced property, several works have been performed in the past few years. A classical second-order method with slope reconstruction has been developed in Gallouët, Hérard and Seguin (2003), Katsaounis and Simeoni (2003), Audusse, Bouchut, Bristeau, Klein and Perthame (2004), while second-order Roe schemes are proposed in Bale, LeVeque, Mitran and Rossmanith (2002), Parés and Castro (2004), Chacón Rebollo, Domínguez Delgado and Fernández Nieto (2003). Higher-order methods are developed in Noelle, Pankratz, Puppo and Natvig (2006), Vukovic and Sopta (2002), Xing and Shu (2005), and the nonconservative context is studied in Castro, Gallardo and Parés (2006). Related works are a method for unstructured mesh in Audusse and Bristeau (2005), a residual distribution scheme in Ricchiuto, Abgrall and Deconinck (2006), and discontinuous Galerkin schemes in Xing and Shu (2006), Lukacova-Medvidova, Noelle and Kraft (2006). We shall describe here the most classical method derived from Katsaounis and Simeoni (2003), Audusse, Bouchut, Bristeau, Klein and Perthame (2004), Bouchut (2004). Since second-order accuracy in time can be obtained as usual by the Heun method, see below, it is enough to build a scheme that is second-order in

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

space only. Starting from a first-order method for solving (3.1), the new ingredient to go to second-order in space is a reconstruction operator, which is attached to the mesh defined as previously (2.2)–(2.3). We shall denote by δ = sup xi

(4.1)

i

the size of the mesh. D EFINITION 4.1. A second-order reconstruction is an operator which to a sequence (Ui , Zi ) associates values Ui+1/2− , Zi+1/2− , Ui+1/2+ , Zi+1/2+ for i ∈ Z, in such a way that it is conservative in U , Ui−1/2+ + Ui+1/2− = Ui , 2 and it is second-order in the sense that whenever for all i,

1 1 Ui = U (x) dx, Zi = Z(x) dx,

xi

xi Ci

(4.2)

(4.3)

Ci

for some smooth functions U (x), Z(x), then Ui+1/2+ = U (xi+1/2 ) + O δ 2 , Ui+1/2− = U (xi+1/2 ) + O δ 2 , Zi+1/2+ = Z(xi+1/2 ) + O δ 2 . (4.4) Zi+1/2− = Z(xi+1/2 ) + O δ 2 , We recall that the notation φ = O(δ 2 ) means that φ = δ 2 φ, with φ bounded as δ tends to 0. Given consistent first-order fluxes Fl , Fr in the sense of Definition 3.2, the second-order scheme is defined by Uin+1 − Uin + with

t (Fi+1/2− − Fi−1/2+ − Fi ) = 0,

xi

n n n n Fi+1/2− = Fl Ui+1/2− , , Ui+1/2+ , Zi+1/2− , Zi+1/2+ n n n n , , Zi+1/2− , Zi+1/2+ Fi+1/2+ = Fr Ui+1/2− , Ui+1/2+ n n n n Fi = Fc Ui−1/2+ , Ui+1/2− , Zi−1/2+ , Zi+1/2− ,

(4.5)

(4.6)

where the arguments are obtained from the reconstruction operator applied to the sequence (Uin , Zin ), and the centered flux function Fc needs to be chosen. As for first-order, Zi does not change with time, but however the interface values n can be time dependent if the reconstruction operator is not performed Zi+1/2± componentwise. We define n n n = Zi+1/2+ − Zi+1/2− ,

Zi+1/2

n n

Zin = Zi+1/2− − Zi−1/2+ .

(4.7)

4. Second-order well-balanced schemes

233

4.1. Achievement of second-order accuracy The scheme (4.5)–(4.6) generalizes the first-order scheme (3.18)–(3.19), the latter being a formal particular case of the former if the reconstruction operator is trivial, Ui+1/2− = Ui , Ui+1/2+ = Ui+1 , Zi+1/2− = Zi , Zi+1/2+ = Zi+1 , and if Fc (U, U, Z, Z) = 0. The idea in (4.5)–(4.6) is to distribute the variation of the solution between interfaces and centers of the cells, depending on the smoothness of the computed solution. One hand, if the solution has large discontinuities, the reconstruction step should give (U, Z)ni+1/2− = (U, Z)ni−1/2+ = (Uin , Zi ), thus reducing to the first-order scheme, with source discretized at the interfaces. On the other hand, if the solution is smooth, the interface jumps (U, Z)i+1/2+ − (U, Z)i+1/2− should be O(δ 2 ), and the source cannot appear at the interface, because (3.30) gives then O(δ 2 ). It has to be handled by the centered term Fi . In order that this term gives a second-order resolution, we require that Ul + Ur Zl + Zr Fc (Ul , Ur , Zl , Zr ) = − B , 2 2 2 2 + O |Ur − Ul | + |Zr − Zl | (4.8) (Zr − Zl ), as Ur − Ul → 0 and Zr − Zl → 0. This condition implies that Fc (Ul , Ur , Z, Z) = 0. A slight restriction on the reconstruction is also necessary, which is that whenever the sequence (Ui , Zi ) is realized as the cell averages of smooth functions (U (x), Z(x)), as in (4.3), then Zi+1/2+ − Zi+1/2− = O ( xi + xi+1 )δ , Zi+1/2− − Zi−1/2+ = O( xi ).

(4.9)

These last assumptions (4.9) are very weak and of local nature, since in general one would have according to the definition of a second-order reconstruction operator errors in δ 2 and δ respectively on the right-hand sides of (4.9) (thus (4.9) is true for a mesh of uniform size x). It is proved in Bouchut (2004) that if the numerical fluxes Fl , Fr are consistent and sufficiently smooth, and if (4.8) and (4.9) are satisfied, then the scheme is second-order accurate in space in the weak sense, for smooth solutions. 4.2. Well-balanced property The well-balanced property can be achieved for the second-order scheme if all the ingredients involved in the formulas (4.5)–(4.6) satisfy a natural condition. We assume here that a family of discrete steady states is chosen, as described in Section 3.3.

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

D EFINITION 4.2. We say that the second-order reconstruction operator is well-balanced if to a sequence (Ui , Zi ) that is a discrete steady state (i.e. D(Ui , Ui+1 , Zi , Zi+1 ) = 0 for all i), it associates a sequence of interface values . . . , (U, Z)i−1/2− , (U, Z)i−1/2+ , (U, Z)i+1/2− , (U, Z)i+1/2+ , . . . (4.10) that is again a discrete steady state (i.e. D(Ui−1/2+ , Ui+1/2− , Zi−1/2+ , Zi+1/2− ) = 0 and D(Ui+1/2− , Ui+1/2+ , Zi+1/2− , Zi+1/2+ ) = 0 for all i). We remark that here it is important to consider global steady states sequences, since the reconstruction operator can have a large dependency stencil. P ROPOSITION 4.3. Assume that the reconstruction operator is well-balanced, that the numerical fluxes Fl , Fr are well-balanced in the sense of Definition 3.1, and that the centered flux Fc satisfies that whenever (Ul , Ur , Zl , Zr ) is a local steady state (i.e. D(Ul , Ur , Zl , Zr ) = 0), one has Fc (Ul , Ur , Zl , Zr ) = F (Ur , Zr ) − F (Ul , Zl ).

(4.11)

Then the second-order scheme (4.5)–(4.6) is well-balanced, in the sense that steady states sequences are let invariant. The proof of this result is obvious by the formulas (4.5)–(4.6). 4.3. Choice of centered flux The centered flux Fc has to satisfy (4.8) for second-order accuracy, and (4.11) for well-balancing. In practice we define Fc directly by an algebraic formula. For Example 3.1 and if F (U ) > 0, we can take Fc (Ul , Ur , Z) = −

F (Ur ) − F (Ul )

Z. D(Ur ) − D(Ul )

(4.12)

For the shallow water system (3.7) and if we retain only steady states at rest, we have the simple choice hl + hr g z . Fc (Ul , Ur , z) = 0, − (4.13) 2 An important point is that the first component in (4.13) vanishes, thus the water height remains conservative.

4. Second-order well-balanced schemes

235

4.4. Choice of reconstruction operator We shall not give general considerations on second-order reconstructions here, we rather refer the reader to Godlewski and Raviart (1991) for a general introduction. We would like here only to give the most simple and well known reconstructions for scalar conservative equations, in order to be able afterwards to discuss the shallow water system. E XAMPLE 4.1. In the case of a scalar function U ∈ R, the second-order minmod reconstruction is defined as follows, Ui−1/2+ = Ui − with

xi DUi , 2

DUi = minmod

Ui+1/2− = Ui +

xi DUi , 2

Ui − Ui−1 Ui+1 − Ui , , ( xi−1 + xi )/2 ( xi + xi+1 )/2

(4.14)

(4.15)

and

⎧ ⎨ min(x, y) if x, y 0, minmod(x, y) = max(x, y) if x, y 0, ⎩ 0 otherwise. A good property of this reconstruction is the maximum principle, inf Uj Ui−1/2+ , Ui+1/2− sup Uj . j

(4.16)

(4.17)

j

E XAMPLE 4.2. In the previous example, a loss of accuracy in the derivative can occur. This is corrected by the second-order ENO (essentially non oscillatory) reconstruction. For a uniform mesh, it is defined as follows for a scalar sequence Ui , Ui−1/2+ = Ui −

x Deno Ui , 2

Ui+1/2− = Ui +

x Deno Ui , 2

(4.18)

where

x 2 Ui − Ui−1 + θeno D Ui−1/2 , Deno Ui = minmod

x 2

x 2 Ui+1 − Ui − θeno D Ui+1/2 , (4.19)

x 2 Ui+1 − 2Ui + Ui−1 Ui+2 − 2Ui+1 + Ui D 2 Ui+1/2 = minmod , , (4.20)

x 2

x 2

with θeno = 1.

(4.21)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

This reconstruction does however somehow produce some instabilities (oscillations), thus one can prefer a value 0 θeno < 1, even if it kills the second-order accuracy of the slopes. A value θeno = 0.25 is a good value in practice. Notice that θeno = 0 gives the minmod reconstruction (4.15). A counterpart of the increased accuracy in the ENO reconstruction is the loss of the maximum principle, which is often needed, for example for positiveness of density or water height. A possible way to obtain this is to consider the modified ENO reconstruction, defined still by (4.18) but with slopes Denom Ui = minmod(Deno Ui , 2θenom Dmm Ui ),

(4.22)

where Dmm Ui is the minmod slope of (4.15), and 0 θenom 1. The value θenom = 1 is enough to recover the maximum principle, but in order to avoid arithmetic exceptions we prefer to use a slightly stronger limitation θenom = 0.9.

(4.23)

The maximum principle holds for this ENOm reconstruction, at the price of loosing the second-order accuracy of the slope close to local extrema. E XAMPLE 4.3. A second-order reconstruction for the shallow water system (3.7) can be performed on the vector (Ui , zi ) = (hi , hi ui , zi ) as follows. Let us first describe the reconstruction for U , ignoring the variable z. Let us denote Ui+1/2± ≡ (hi+1/2± , hi+1/2± ui+1/2± ) the reconstructed values. Then the conservation constraint (4.2) reads hi−1/2+ + hi+1/2− = hi , 2 hi−1/2+ ui−1/2+ + hi+1/2− ui+1/2− = hi ui . 2 It is easily seen to be equivalent to the representation

xi

xi hi+1/2− = hi + Dhi , Dhi , 2 2 hi+1/2− xi = ui − Dui , hi 2 hi−1/2+ xi = ui + Dui , hi 2

(4.24)

hi−1/2+ = hi − ui−1/2+ ui+1/2−

(4.25)

for some slopes Dhi , Dui . Moreover, the second-order accuracy (4.4) means that Dhi , Dui have to be consistent with dh/dx and du/dx respectively. Thus we can take for Dhi and Dui the minmod reconstruction (4.15), where we put the values hi or ui respectively in the right-hand side. A more accurate choice is to use the ENOm slope (4.22) for Dhi and the ENO slope (4.19)–(4.20) for Dui . This last

4. Second-order well-balanced schemes

237

choice is the one we select as our favorite reconstruction. It keeps the water height h nonnegative Concerning the variable z, we perform the ENOm reconstruction on ζ = h + z. This means that we set ζi = hi + zi , from these values we compute ζi+1/2± by the ENOm reconstruction, and we finally set zi+1/2± = ζi+1/2± − hi+1/2± .

(4.26)

This reconstruction is obviously well-balanced with respect to steady states at rest. Consequently, according to Proposition 4.3, with this choice of second-order reconstruction and with Fc defined by (4.13), the whole second-order scheme is well-balanced. This choice of reconstructing ζ to get z has the advantage to treat correctly interfaces between dry and wet cells, contrarily to other choices (like reconstructing ζ and z and getting h as h = ζ − z, or reconstructing h and z independently), see Audusse, Bouchut, Bristeau, Klein and Perthame (2004), Marche (2005). 4.5. Stability and CFL condition Entropy inequalities are not available at second order, only very special schemes can verify such criterion. In practice this is not really a problem since the schemes usually degenerate at first order whenever the computed solution has sharp gradients. Therefore, the stability of second-order schemes can be analyzed only by two main approaches. The first is the study of the non oscillatory behavior, which can be really justified only for scalar equations (see Godlewski and Raviart (1991) for example). The second is the study of invariant domains, like nonnegativity of density or water height. With the approach presented here (4.5)–(4.6), the invariant domains are preserved if one assumes that the first-order numerical fluxes Fl , Fr and the reconstruction operator do, and if the centered flux Fc satisfies a suitable requirement (it has to “look like” Fr − Fl ). This result uses strongly the conservativity of the reconstruction (4.2), see Bouchut (2004). The analysis shows that at secondorder, the CFL condition has to be divided by 2 with respect to the one used at first-order. In the case of shallow water with the reconstruction proposed above in Example 4.3, with the centered flux (4.13), and with the first-order fluxes given by the hydrostatic reconstruction method, we have conservativity and nonnegativity of water height, the interface between dry and wet is well resolved under half the CFL condition (3.52). 4.6. Second-order boundary conditions Boundary conditions are treated at second-order very similarly to the first-order case discussed in Section 2.12: one has to define suitable ghost values. Assume

238

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

that the interval of computation (a, b) is divided into cells Ci = (xi−1/2 , xi+1/2 ), i = 1, . . . , nx , on which we have discrete values Uin , Zi , i = 1, . . . , nx . The boundary is a = x1/2 , b = xnx +1/2 . The first step is to define ghost values U0n , Z0 , Unnx +1 , Znx +1 , and if necessary also ghost values for i = −1, i = nx + 2 . . . Then, n n one applies the second-order reconstruction to get the values of Ui+1/2+ , Zi+1/2+ , n n i = 0, . . . , nx − 1, and of Ui+1/2− , Zi+1/2− , i = 1, . . . , nx . The next step is to n n , Z1/2− , Unnx +1/2+ , Znnx +1/2+ . Finally, one applies define ghost values for U1/2− the update (4.5)–(4.6) for i = 1, . . . , nx . E XAMPLE 4.4. Consider the isentropic gas dynamics system (1.14), with the wall boundary condition u(t, a) = 0, as discussed in Example 1.5. Then, one can n = ρn, define the ghost values by even/odd symmetry, ρ0n = ρ1n , un0 = −un1 , ρ−1 2 n n u−1 = −u2 , . . . (as many values as needed in order to be able to get the reconn n structed values at 1/2+). Finally, we define ρ1/2− = ρ1/2+ , un1/2− = −un1/2+ . E XAMPLE 4.5. For periodic boundary conditions, one takes U0n = Unnx , Unnx +1 = n = Un n n n n n U1n , U−1 nx −1 , Unx +2 = U2 , . . . , and U1/2− = Unx +1/2− , Unx +1/2+ = n U1/2+ . E XAMPLE 4.6. For Neumann boundary conditions ∂x U (t, a) = 0, we proceed by even symmetry. 4.7. Second-order accuracy in time The second-order accuracy in time can be classically obtained by the Heun method, which reads as follows. The second-order method in x can be written as U n+1 = U n + tΦ U n , (4.27) where U = (Ui )i∈Z , and Φ is a nonlinear operator depending on the mesh. Then the second-order scheme in time is n+1 = U n + tΦ U n , U n+1 n+2 = U n+1 + tΦ U U , n+2 Un + U (4.28) . 2 It is easy to see that if the operator Φ does not depend on t, this procedure gives a fully second-order scheme in space and time. The convex average in (4.28) enables to ensure the stability without any further limitation on the CFL. Indeed, because of this average, the above second-order algorithm is more stable than the first-order one (4.27). U n+1 =

4. Second-order well-balanced schemes

239

4.8. The shallow water system with Coriolis force and transverse velocity An extension to second-order of the scheme proposed in Section 3.10 for the system (3.60) is performed according to the principle above: one needs to define a centered numerical flux, and a reconstruction operator. Denote U = (h, hu, hv). The first-order scheme (3.70)–(3.75) can be written

t n n n Uin+1 − Uin + Fl Ui , Ui+1 , zi+1/2 + bi+1/2 , Ωi+1/2

xi n n , Ωi−1/2 = 0, − Fr Ui−1 , Uin , zi−1/2 + bi−1/2

(4.29)

with n

bi+1/2 = −fi+1/2

n vin + vi+1

2

xi+1/2 /g,

Ωi+1/2 = fi+1/2 xi+1/2 ,

(4.30) (4.31)

and zi+1/2 = zi+1 − zi , xi+1/2 = xi+1 − xi . The second-order scheme then takes the form

t n n n n n Fl Ui+1/2− , Ui+1/2+ , zi+1/2 + bi+1/2 , Ωi+1/2 Uin+1 − Uin +

xi n n n n n , zi−1/2 + bi−1/2 , Ωi−1/2 − Fr Ui−1/2− , Ui−1/2+ n n , Ui+1/2− , zin + bin , Ωin = 0, − Fc Ui−1/2+ (4.32) with now n n n

zi+1/2 = zi+1/2+ − zi+1/2− ,

n n

zin = zi+1/2− − zi−1/2+ .

(4.33)

For the centered flux Fc , we take similarly as in (4.13) Fc (Ul , Ur , z + b, Ω) hl + hr hl ul + hr ur = 0, − g( z + b), −

Ω , 2 2

(4.34)

and it remains to explain how we perform the reconstruction of U , z, and how we n n define bi+1/2 , Ωi+1/2 , bin , Ωin . The reconstruction of h, u, z is performed exactly as in the shallow water case of Example 4.3. The transverse velocity v is reconstructed similarly as u, by setting hi+1/2− xi Dvi , hi 2 hi−1/2+ xi = vi + Dvi , hi 2

vi−1/2+ = vi − vi+1/2−

(4.35)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with slope Dvi computed by the ENO reconstruction involving the values vi . Concerning Ω, we indeed make a time independent reconstruction, using directly the relation (3.66), and the formulation (3.67). Given the values fi+1/2 , i ∈ Z, we can define up to a constant the values of Ωi , i ∈ Z, by Ωi+1 − Ωi = fi+1/2 xi+1/2 ,

(4.36)

with as before xi+1/2 = xi+1 −xi . From these values Ωi , we perform the ENOm reconstruction to get interface values Ωi+1/2− , Ωi+1/2+ . We finally set

Ωi+1/2 = Ωi+1/2+ − Ωi+1/2− ,

Ωi = Ωi+1/2− − Ωi−1/2+ . (4.37)

The variable b is treated similarly, defining values bin by n bi+1 − bin = −fi+1/2

n vin + vi+1

(4.38)

xi+1/2 /g. 2 From these values bin , we perform the ENOm reconstruction to get interface values n n bi+1/2− , bi+1/2+ , and we finally set n n n = bi+1/2+ − bi+1/2− ,

bi+1/2

n n

bin = bi+1/2− − bi−1/2+ .

(4.39)

bin

Notice that in this procedure, an arbitrary constant added to does not modify the differences (4.39). Moreover, according to the formulas defining the ENOm n − bn , the result can be computed reconstruction that involve only differences bi+1 i n locally without really computing bi . The scheme we obtain is second-order, conservative and nonnegative in water height, and computes dry states. It is well-balanced with respect to the steady states at rest (3.76), because this relation can be written hi+1 + zi+1 + bi+1 = hi + zi + bi , and it is easily seen that for such data the reconstruction also satisfies h + z + b = cst.

5. Two-dimensional finite volumes on a rectangular grid This section is devoted to the extension of the previously developed finite volume schemes with source terms to two dimensions on a rectangular grid. The case of unstructured mesh is more involved, and we refer to Audusse and Bristeau (2005). The generalization of (3.1) that we are going to consider is a two-dimensional system of the form ∂t U + ∂x F1 (U, Z) + ∂y F2 (U, Z) + B1 (U, Z)∂x Z + B2 (U, Z)∂y Z = 0,

(5.1)

where the space variable is x = (x, y) ∈ R2 , and the partial derivatives ∂x , ∂y refer to these variables. The unknown is U (t, x) ∈ Rp , Z = Z(x) ∈ Rr is given, and

5. Two-dimensional finite volumes on a rectangular grid

241

the nonlinearities F1 , F2 , B1 , B2 are smooth. As in the one-dimensional case, this system can be written as a quasilinear system in two dimensions for the variable = (U, Z), by completing the system with the trivial equation ∂t Z = 0. U By two-dimensional quasilinear system we mean a system of the form ∂x U ∂y U + A1 U + A2 U = 0. ∂t U (5.2) For such a system we can consider planar solutions, i.e. solutions of the form (t, x) = U (t, ζ ) with ζ = x · n and n = (n1 , n2 ) is any unit vector in R2 , which U leads to ∂ζ U + An U = 0, ∂t U (5.3) with = n1 A1 U + n2 A2 U . An U

(5.4)

Via this reduction, the notions introduced in Section 1 can be applied to the one-dimensional quasilinear system (5.3), and thus one defines hyperbolicity, entropies, and other notions for (5.2) by requiring the one-dimensional properties for all directions n. In particular, the system (5.2) is hyperbolic if for any unit , An (U ) is diagonalizable. The reader is referred to the vector n in R2 and any U literature for more details, for example Serre (1999). 5.1. Resolution interface by interface One of the basic principles of finite volume methods is the very simple way to solve a multidimensional quasilinear system like (5.2) by solving onedimensional problems (5.3)–(5.4) at each interface between two cells, n being the normal to the interface. In the case of a rectangular grid, the formulation comes very simply, as follows. Consider a two-dimensional mesh made of rectangles Cij = (xi−1/2 , xi+1/2 ) × (yj −1/2 , yj +1/2 ),

i ∈ Z, j ∈ Z,

(5.5)

and denote their lengths by

xi = xi+1/2 − xi−1/2 > 0,

yj = yj +1/2 − yj −1/2 > 0.

(5.6)

The centers of the cells are denoted by xij = (xi , yj ),

(5.7)

with xi =

xi−1/2 + xi+1/2 , 2

yj =

yj −1/2 + yj +1/2 . 2

(5.8)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

We would like to approximate a solution U (t, x) to (5.1) by discrete values Uijn that are approximations of the mean value of U over the cell Cij at time tn = n t, 1 Uijn

xi yj

x i+1/2 y j +1/2

U (tn , x, y) dx dy.

(5.9)

xi−1/2 yj −1/2

A finite volume method for solving (5.1) takes the form Uijn+1 − Uij + +

t (Fi+1/2−,j − Fi−1/2+,j )

xi

t (Fi,j +1/2− − Fi,j −1/2+ ) = 0,

yj

(5.10)

where as before, Uij stands for Uijn . In this formula, the terms Fi+1/2±,j are exchange terms between the cells Cij and Ci+1,j , and the terms Fi,j +1/2± are exchange terms between the cells Cij and Ci,j +1 . As in the one-dimensional case (3.18)–(3.19), these exchange terms are computed by Fi+1/2−,j = Fl1 (Uij , Ui+1,j , Zij , Zi+1,j ), Fi+1/2+,j = Fr1 (Uij , Ui+1,j , Zij , Zi+1,j ), Fi,j +1/2− = Fl2 (Uij , Ui,j +1 , Zij , Zi,j +1 ), Fi,j +1/2+ = Fr2 (Uij , Ui,j +1 , Zij , Zi,j +1 ), Fl1 ,

(5.11)

Fr1 , Fl2 ,

for some numerical fluxes Fr2 . With the formulation (5.10)–(5.11), the consistency with (5.1) can be defined obviously, by saying that the x difference in (5.10) must be consistent with terms with x derivatives in (5.1) in the sense of (3.29)–(3.30), and similarly for the y difference and the y derivatives. Entropy inequalities and nonnegativity properties can be formulated very similarly to the one-dimensional case for the scheme (5.10)–(5.11), and incell properties are deduced from interface properties at the price of diminishing the CFL condition. In practice one has only to use the CFL condition attached to each of the two directions,

tai+1/2,j min( xi , xi+1 ),

tai,j +1/2 min( yj , yj +1 ),

(5.12)

where ai+1/2,j and ai,j +1/2 are estimates of the speeds in x and y directions respectively. We shall not give the details here. 5.2. Well-balancing Since the above formulation means that we are solving (5.1) independently in the directions x and y, it follows that the steady states that can be preserved by such a

5. Two-dimensional finite volumes on a rectangular grid

243

method are those that can be seen locally as one-dimensional steady states in the directions x or y. In particular, more general steady states where the dependence in x and y are nontrivially balanced cannot be exactly resolved. Nevertheless, the generalization of equations (3.28) that enable to preserve a discrete steady state is that the fluxes computed by (5.11) must give the value Fi+1/2−,j = F1 (Uij , Zij ),

Fi+1/2+,j = F1 (Ui+1,j , Zi+1,j ),

Fi,j +1/2− = F2 (Uij , Zij ),

Fi,j +1/2+ = F2 (Ui,j +1 , Zi,j +1 ).

(5.13)

It is obvious from (5.10)–(5.11) that if at time tn we start with a sequence (Uij ) that satisfies (5.13), then it remains unchanged at the next time level. 5.3. Second-order accuracy For rectangular grids, the extension to second-order is straightforward, it follows directly the one-dimensional case developed in Section 4. Given first-order numerical fluxes Fl1 , Fr1 , Fl2 , Fr2 , the second-order version of (5.10)–(5.11) writes

t 1 Fi+1/2−,j − Fi−1/2+,j − Fi,j

xi

t 2 Fi,j +1/2− − Fi,j −1/2+ − Fi,j = 0, +

yj

Uijn+1 − Uij +

(5.14)

with Fi+1/2−,j = Fl1 (Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j ), Fi+1/2+,j = Fr1 (Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j ), Fi,j +1/2− = Fl2 (Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ ), Fi,j +1/2+ = Fr2 (Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ ), 1 = Fc1 (Ui−1/2+,j , Ui+1/2−,j , Zi−1/2+,j , Zi+1/2−,j ), Fi,j 2 Fi,j = Fc2 (Ui,j −1/2+ , Ui,j +1/2− , Zi,j −1/2+ , Zi,j +1/2− ).

(5.15)

The centered numerical fluxes Fc1 , Fc2 correspond respectively to the x and y parts of (5.1). The reconstructed values can be obtained from the one-dimensional case. Indeed, for fixed j , from Uij , Zij , i ∈ Z, we can get the values Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j , and for fixed i, from Uij , Zij , j ∈ Z, we can get the values Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ . The CFL condition has to be restricted by a factor 1/2 with respect to (5.12).

244

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

5.4. Two-dimensional shallow water system with Coriolis force Consider now the two-dimensional shallow water system with Coriolis force ⎧ ∂ h + ∂x (hu) + ∂y (hv) = 0, ⎪ ⎨ t ∂t (hu) + ∂x (hu2 + gh2 /2) + ∂y (huv) + hg∂x z − f hv = 0, (5.16) ⎪ ⎩ 2 2 ∂t (hv) + ∂x (huv) + ∂y (hv + gh /2) + hg∂y z + f hu = 0, where h(t, x) 0 is the water height, u(t, x), v(t, x) are the two components of the velocity field, g > 0 is the gravity constant, z(x) is the topography, and f (x) is the Coriolis force. Apart from the Coriolis terms, the system can be written as (5.1) with Z = z, and ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ hu hv h ⎠, huv U = ⎝hu⎠ , F1 = ⎝hu2 + gh2 /2⎠ , F2 = ⎝ 2 2 hv huv hv + gh /2 (5.17) ⎛ ⎞ ⎛ ⎞ 0 0 B1 = ⎝gh⎠ , (5.18) B2 = ⎝ 0 ⎠ . 0 gh Denoting u = (u, v) the vector velocity, the equations of (5.16) can be combined to get, for smooth solutions, ∂t u + u · ∇u + ∇(gh + gz) + f u⊥ = 0

(5.19)

with u⊥ = (−v, u), or, using the identity u · ∇u = ∇(|u|2 /2) + u⊥ curl u with curl u = ∂x v − ∂y u, ∂t u + ∇ |u|2 /2 + gh + gz + (curl u + f )u⊥ = 0. (5.20) From this we deduce the entropy inequality ∂t h|u|2 /2 + gh2 /2 + hgz + div h|u|2 /2 + gh2 + hgz u 0.

(5.21)

Taking the curl of (5.20) we also get the potential vorticity equation ∂t (curl u + f ) + div (curl u + f )u = 0.

(5.22)

The steady states at rest are characterized by u = v = 0,

h + z = cst.

(5.23)

A property of the system (5.16) is that it is invariant under rotation. The consequence is that x and y play the same role, one has only to exchange the role of u and v. This invariance enables to define numerical fluxes by applying the same one-dimensional procedure in x and y directions. This is done as follows, by using the same approach as in Section 3.10. Consider one-dimensional numerical

5. Two-dimensional finite volumes on a rectangular grid

fluxes Fl (Ul , Ur , z), Fr (Ul , Ur , z) associated to the problem ⎧ ⎪ ⎨ ∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x (hu2 + gh2 /2) + hg∂x z = 0, ⎪ ⎩ ∂t (hv) + ∂x (huv) = 0.

245

(5.24)

Our choice is indeed to take the hydrostatic reconstruction fluxes for the two first equations, and the passive transport fluxes for the last equation, according to Sections 3.7 and 2.11. Then, denoting the exchange of x and y components by = (h, hv, hu) U

if U = (h, hu, hv),

the two-dimensional scheme is defined by (5.10) with numerical fluxes 1 , Fi+1/2−,j = Fl Uij , Ui+1,j , zi+1/2,j + bi+1/2,j 1 Fi+1/2+,j = Fr Uij , Ui+1,j , zi+1/2,j + bi+1/2,j , 2 l U ij , U i,j +1 , zi,j +1/2 + bi,j Fi,j +1/2− = F +1/2 , 2 r U ij , U i,j +1 , zi,j +1/2 + bi,j Fi,j +1/2+ = F +1/2 ,

(5.25)

(5.26)

with

zi+1/2,j = zi+1,j − zij ,

zi,j +1/2 = zi,j +1 − zij ,

(5.27)

vij + vi+1,j

xi+1/2 /g, 2 uij + ui,j +1 2

bi,j (5.28)

yj +1/2 /g, +1/2 = fi,j +1/2 2 and xi+1/2 = xi+1 − xi , yj +1/2 = yj +1 − yj , where fi+1/2,j and fi,j +1/2 are discretized values for f (x) at the locations (xi+1/2 , yj ), and (xi , yj +1/2 ) respectively. As in the one-dimensional case, whenever Fl and Fr have the suitable properties (which is the case for the hydrostatic reconstruction fluxes with passive transport), the two-dimensional scheme is consistent with (5.16), is conservative in water height, preserves nonnegativity of h and is able to compute dry states, and is well-balanced, in the sense that it lets invariants the data satisfying 1

bi+1/2,j = −fi+1/2,j

uij = 0,

vij = 0,

hij + zij = cst.

(5.29)

There is no special property concerning the transport of the potential vorticity (5.22), but numerical experiments show very low numerical diffusion in that equation (Le Sommer, Reznik and Zeitlin, 2004; Bouchut, Le Sommer and Zeitlin, 2005). The comments of Section 3.10 are still valid concerning entropy inequalities.

246

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

The second-order version works similarly as the one-dimensional case explained in Section 4.8. The scheme writes as (5.14), with 1 Fi+1/2−,j = Fl Ui+1/2−,j , Ui+1/2+,j , zi+1/2,j + bi+1/2,j , 1 , Fi+1/2+,j = Fr Ui+1/2−,j , Ui+1/2+,j , zi+1/2,j + bi+1/2,j 2 l U i,j +1/2− , U i,j +1/2+ , zi,j +1/2 + bi,j +1/2 , Fi,j +1/2− = F 2 r U i,j +1/2− , U i,j +1/2+ , zi,j +1/2 + bi,j Fi,j +1/2+ = F +1/2 , 1 1 1 , = Fc1 Ui−1/2+,j , Ui+1/2−,j , zi,j + bi,j Fi,j 2 2 2 2 Fi,j = Fc Ui,j −1/2+ , Ui,j +1/2− , zi,j + bi,j , (5.30) hl + hr g( z + b), 0 , Fc1 (Ul , Ur , z + b) = 0, − 2 hl + hr 2 Fc (Ul , Ur , z + b) = 0, 0, − (5.31) g( z + b) . 2 The variations of z, b1 and b2 are now defined by

zi+1/2,j = zi+1/2+,j − zi+1/2−,j , 1

zij

= zi+1/2−,j − zi−1/2+,j ,

1 1 1 = bi+1/2+,j − bi+1/2−,j ,

bi+1/2,j 1 1 1 = bi+1/2−,j − bi−1/2+,j ,

bij

zi,j +1/2 = zi,j +1/2+ − zi,j +1/2− , 2

zij

= zi,j +1/2− − zi,j −1/2+ ,

(5.32)

2 2 2

bi,j +1/2 = bi,j +1/2+ − bi,j +1/2− , 2 2 2

bij = bi,j +1/2− − bi,j −1/2+ .

(5.33)

The reconstruction of U , z, b1 , b2 is performed as follows. A reconstruction in x is performed on U , z, b1 exactly as it was done in the one-dimensional case, see Section 4.8. The same is done in y for U , z, b2 . The values of b1 and b2 are defined by vij + vi+1,j

xi+1/2 /g, 2 uij + ui,j +1 = fi,j +1/2

yj +1/2 /g, 2 = xi+1 − xi , yj +1/2 = yj +1 − yj .

1 1 bi+1,j − bi,j = −fi+1/2,j 2 2 bi,j +1 − bi,j

with still xi+1/2

(5.34)

6. Numerical tests The usual shallow water system (3.7) has already been intensively tested with the hydrostatic reconstruction method (Audusse, Bouchut, Bristeau, Klein and Perthame, 2004; Marche, 2005). In this section we provide numerical results for the one-dimensional shallow water system with Coriolis force (3.60). We consider

6. Numerical tests

247

a few test cases that are representative of the difficulties that can be encountered for this system. We use the scheme proposed in Section 3.10, with Fl1d , Fr1d the hydrostatic reconstruction fluxes (with HLLC homogeneous flux F ). The second-order version described in Section 4.8 is also tested. 6.1. Test 1: Stability of steady states In this test we consider a steady state which is partly at rest, i.e. of the form (3.61), and partly with h = 0. The numerical stability of this steady state is tested, by observing the discrepancy with the exact solution for large times. The interface dry/wet is also tested concerning its stability. The space variable x lies in (0, 25), the topography is defined by 0.2 − 0.05(x − 10)2 if 8 < x < 12, z(x) = (6.1) 0 otherwise, the initial data are ⎧ ⎪ ⎨ max(0, 0.1 − z(x) + 0.1(25 − x)/g) h(x) = max(0, 0.1 − z(x) + 0.1(x − 5)/g) ⎪ ⎩ max(0, 0.1 − z(x))

if x 15, if 5 x < 15,

(6.2)

otherwise,

u(x) = 0, and ⎧ ⎨ −0.1/2π if x 15, v(x) = 0.1/2π (6.3) if 5 x < 15, ⎩ 0 otherwise. The constant g is taken g = 9.81, and f (x) is a constant, f = 2π. The boundary conditions are Neumann on both sides, and the final time is tmax = 100. We use 200 points in space. The first run is at first-order in space. Figure 4 shows the water level at final time. The computation for first-order in time at CFL = 1 is unstable, while if either we take second-order in time or CFL = 0.5 we recover the exact initial data. Notice that the instability seems to be generated by the point x = 15 where f (∂x v + f ) < 0. This situation allows to apply Proposition 3.5, ensuring the stability. But in order for this to apply rigorously one needs the half CFL condition, which corresponds to the numerical experiment. We then run the second-order scheme, with also second-order in time (firstorder in time is unstable, even for CFL = 0.25), at CFL = 0.5. The steady state is well preserved. Figure 5 shows a zoom around the bump of the water level, to compare with the projection on the grid of the exact solution. Figure 6 shows a zoom of the transverse velocity. The oscillations in the center are meaningless

248

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

since they correspond to the dry region where h = 0, and there only hv is meaningful and not v itself. The computation takes 2285 timesteps.

Figure 4.

Figure 5.

Water level h + z for Test 1, first-order.

Water level h + z for Test 1, second-order (zoom).

6. Numerical tests

Figure 6.

249

Transverse velocity v for Test 1, second-order (zoom).

6.2. Test 2: Accuracy This test is devoted to a numerical measurement of the accuracy of the first-order and second-order schemes. The space variable x lies in (0, 40), and the topography is given as z(x) =

2 if |x − 20| 4, 0.48 1 − x−20 4 0

(6.4)

otherwise.

The initial data are constant, h0 = 4, u0 = 10/4, v 0 = 0, the constant g is taken g = 9.81, and f (x) is chosen as f (x) =

2 π 1 − x−22 if |x − 22| 10, 10 0

(6.5)

otherwise.

The final time is tmax = 1. We use 50 points in space. We run both the first-order scheme in space and time at CFL = 1, and the second-order scheme in space and time at CFL = 0.5. We compare the results to a reference solution computed with 5000 cells. Figure 7 shows of the water level, Figure 8 shows the velocity, and Figure 9 shows the transverse velocity. The second-order computation takes 22 timesteps.

250

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 7. Water level h + z for Test 2.

Figure 8.

Velocity u for Test 2.

Then, we compute the L1 error for several grid sizes. The results are shown on Table 1, together with the numerical order of accuracy. The accuracy corresponds to what can be expected taking into account the regularity of the solution.

6. Numerical tests

Figure 9.

Table 1

251

Transverse velocity v for Test 2.

L1 error and numerical order of accuracy for Test 2

Cells

First-order

50 100 200 400 800

48.0 24.0 12.0 6.07 3.07

Second-order / 1.00 1.00 0.98 0.98

5.13 1.95 0.670 0.243 0.0866

/ 1.40 1.54 1.46 1.49

6.3. Test 3: Stability of steady states with nonzero velocity In this test we would like to show the stability of the method and convergence to a steady state with u = 0. Such a steady state is characterized by the relations hu = cst, ∂x u2 /2 + g(h + z) − f v = 0, ∂x v + f = 0. (6.6) The two last equations can indeed be combined to get u2 /2 + v 2 /2 + g(h + z) = cst.

(6.7)

As in Test 1, the space variable x lies in (0, 25), and the topography is as in (6.1). The initial data are constant, h = 0.33, u = 0.18/0.33, v = 0. The constant g is

252

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

taken g = 9.81, and f (x) is a constant, f = 2π/50. The boundary conditions are hu(x = 0) = 0.18,

v(x = 0) = 0,

h(x = 25) = 0.33,

∂x v(x = 25) = 0.

The final time is tmax = 200. We use 200 points in space.

Figure 10.

Water level h + z for Test 3.

Figure 11.

Velocity u for Test 3.

(6.8)

6. Numerical tests

253

We use the second-order scheme in space and time, at CFL = 0.5. Figures 10– 14 show respectively the water level h+z, the velocity u, the transverse velocity v, the discharge hu, and the invariant u2 /2 + v 2 /2 + g(h + z). We observe that

Figure 12.

Transverse velocity v for Test 3.

Figure 13.

Discharge hu for Test 3.

254

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 14.

Invariant u2 /2 + v 2 /2 + g(h + z) for Test 3.

the solution has converged to a steady state. The computation takes 9574 timesteps.

References Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B., 2004. A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065. Audusse, E., Bristeau, M.-O., 2005. A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206, 311–333. Bale, D.S., LeVeque, R.J., Mitran, S., Rossmanith, J.A., 2002. A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24, 955–978. Bermúdez, A., Vásquez, M.E., 1994. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071. Botchorishvili, R., Perthame, B., Vasseur, A., 2003. Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp. 72, 131–157. Bouchut, F., 2004. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Frontiers in Mathematics Series. Birkhäuser, Basel. Bouchut, F., Le Sommer, J., Zeitlin, V., 2004. Frontal geostrophic adjustment and nonlinear-wave phenomena in one dimensional rotating shallow water. Part 2: High-resolution numerical simulations. J. Fluid Mech. 514, 35–63. Bouchut, F., Le Sommer, J., Zeitlin, V., 2005. Breaking of balanced and unbalanced equatorial waves. Chaos 15, 13503–13521.

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Castro, M.J., Ferreiro, A.M., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C., Vázquez-Cendón, M.E., 2005. The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. Math. Comput. Modelling 42, 419–439. Castro, M.J., Gallardo, J.M., Parés, C., 2006. High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math. Comp. 75, 1103–1134. Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C., Vázquez-Cendón, M.E., 2004. Numerical simulation of two-layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195, 202–235. Castro, M.J., Macías, J., Parés, C., 2001. A Q-scheme for a class of systems of coupled conservation laws with source term, application to a two-layer 1-D shallow water system. M2AN Math. Model. Numer. Anal. 35, 107–127. Chacón Rebollo, T., Domínguez Delgado, A., Fernández Nieto, E.D., 2003. A family of stable numerical solvers for the shallow water equations with source terms. Comput. Methods Appl. Mech. Engrg. 192, 203–225. Chinnayya, A., LeRoux, A.-Y., Seguin, N., 2004. A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Internat. J. Finite Volume 1, 1–33 (electronic). Gallouët, T., Hérard, J.-M., Seguin, N., 2003. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513. García-Navarro, P., Vázquez-Cendón, M.E., 2000. On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29, 17–45. Godlewski, E., Raviart, P.-A., 1991. Hyperbolic Systems of Conservation Laws. Mathématiques & Applications, vol. 3/4. Ellipses, Paris. Godlewski, E., Raviart, P.-A., 1996. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer-Verlag, New York. Gosse, L., 2000. A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39, 135–159. Gosse, L., 2001. A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11, 339–365. Gosse, L., LeRoux, A.-Y., 1996. A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. 323, 543–546. Greenberg, J.M., LeRoux, A.-Y., 1996. A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16. Greenberg, J.-M., LeRoux, A.-Y., Baraille, R., Noussair, A., 1997. Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34, 1980–2007. Harten, A., Lax, P.D., Van Leer, B., 1983. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61. Jin, S., 2001. A steady-state capturing method for hyperbolic systems with geometrical source terms. Math. Model. Numer. Anal. 35, 631–645. Katsaounis, T., Perthame, B., Simeoni, C., 2004. Upwinding sources at interfaces in conservation laws. Appl. Math. Lett. 17, 309–316. Katsaounis, T., Simeoni, C., 2003. Second order approximation of the viscous Saint-Venant system and comparison with experiments. In: Hyperbolic Problems: Theory, Numerics, Applications. Springer-Verlag, Berlin, pp. 633–644. Kurganov, A., Levy, D., 2002. Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425. LeRoux, A.-Y., 1999. Riemann solvers for some hyperbolic problems with a source term. ESAIM Proc. 6, 75–90. Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170.

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LeVeque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge Univ. Press, Cambridge, UK. Lukacova-Medvidova, M., Noelle, S., Kraft, M., 2006. Well-balanced finite volume evolution Galerkin methods for the shallow water equations. Preprint Univ. Aachen. Mangeney-Castelnau, A., Bouchut, F., Vilotte, J.-P., Lajeunesse, E., Aubertin, A., Pirulli, M., 2005. On the use of Saint Venant equations to simulate the spreading of a granular mass. J. Geophys. Res. 110 (B9), B09103. Marche, F., 2005. Theoretical and numerical study of shallow water models. Applications to nearshore hydrodynamics. PhD thesis, Univ. Bordeaux. Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R., 2006. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499. Parés, C., 2006. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321. Parés, C., Castro, M.J., 2004. On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. M2AN Math. Model. Numer. Anal. 38, 821–852. Perthame, B., Simeoni, C., 2001. A kinetic scheme for the Saint Venant system with a source term. Calcolo 38, 201–231. Perthame, B., Simeoni, C., 2003. Convergence of the upwind interface source method for hyperbolic conservation laws. In: Hyperbolic Problems: Theory, Numerics, Applications. Springer-Verlag, Berlin, pp. 61–78. Ricchiuto, M., Abgrall, R., Deconinck, H., 2006. Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J. Comput. Phys, in press. Roe, P.L., 1981. Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372. Serre, D., 1999. Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves. Translated from the 1996 French original by I.N. Sneddon. Cambridge Univ. Press, Cambridge, UK. Serre, D., 2000. Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and InitialBoundary Value Problems. Translated from the 1996 French original by I.N. Sneddon. Cambridge Univ. Press, Cambridge, UK. Toro, E.F., 1999. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, second ed. Springer-Verlag, Berlin/New York. Vázquez-Cendón, M.E., 1999. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148, 497–526. Vukovic, S., Sopta, L., 2002. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys. 179, 593–621. Xing, Y., Shu, C.-W., 2005. High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227. Xing, Y., Shu, C.-W., 2006. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598.

Chapter 5

Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction 2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW 2.1. General features of the model 2.2. Lagrangian approach to 1dRSW 2.3. Theoretical predictions for the geostrophic adjustment in the 1dRSW model 2.4. Wave breaking and shock formation in the 1dRSW model

258 259 259 260 264 268

2.5. High-resolution numerical simulations of fully nonlinear geostrophic adjustment

270 274

in 1dRSW 2.6. Finite-amplitude wave solutions of the 1dRSW equations

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW 3.1. General features of the model 3.2. The existence and uniqueness of the adjusted state 3.3. The role of baroclinicity: obstacles to the standard adjustment scenario 3.4. Lagrangian approach to 2L1dRSW and symmetric instability 3.5. Stationary waves of finite amplitude in the 2-layer model

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide 4.1. Equatorial waves and geostrophic adjustment in the equatorial waveguide 4.2. Wave-breaking and shock formation by equatorial waves 4.3. Solitons and modons formed by equatorial Rossby waves 4.4. Transport and mixing phenomena during the equatorial geostrophic adjustment

5. Summary and conclusions Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02005-3 257

281 281 285 286 291 292 297 298 300 305 314 317

© 2007 Elsevier B.V. All rights reserved

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Acknowledgements References

319 319

1. Introduction The inspection of the dispersion curve for inertia–gravity waves (IGW) in the RSW model on the f -plane, see Chapter 1 of this volume, shows that IGW are dispersionless in the short-wave limit and strongly dispersive in the long-wave limit. So, at least at the heuristic level, the short IGW behave like acoustic waves which is not surprising, as in the absence of rotation the model is equivalent to the acoustics of a two-dimensional barotropic gas. One can therefore anticipate that at finite amplitude IGW can break and form shocks. The question then arises, whether rotation could prevent breaking, and if not, what is the role of this phenomenon in the fundamental process of geostrophic adjustment, and what are its consequences for the dynamical separation of fast and slow motions. On the other hand, if the dispersion due to rotation does prevent breaking at least for some configurations, one can expect existence of finite-amplitude nonlinear stationary waves in the model. In what follows, we will study essentially nonlinear wave phenomena in the RSW model, most often within the adjustment context. Weakly nonlinear phenomena in RSW involving resonant interactions of wave triads or quartets, or their ensembles (wave, or weak turbulence) will be not addressed below (see Falkovich and Medvedev, 1992, and Medvedev and Zeitlin, 2005, in this relation). We start our study (Section 2) within the framework of the simplified “one-and-a-half” dimensional version of the RSW model on the f -plane (1dRSW). The dependence of any field on one of the spatial coordinates will be prohibited in such model, and the dynamics will be reduced to that of straight fronts/jets and plane-parallel waves. The advantage of this reduced model is that the slow manifold is here unambiguously defined, as it is just an “infinitely slow” steady state. Some exact results on the fully nonlinear geostrophic adjustment may be obtained due to this simplification. The use of Lagrangian variables turns to be advantageous because of the quasi one-dimensional character of the model, and allows to treat the fully nonlinear adjustment process and to prove that rotation cannot prevent breaking for a large class of initial conditions. On the other hand, we show that for periodic boundary conditions, exact solutions in the form of finite-amplitude propagating waves of special form appear in the model thus proving that rotation does equilibrate nonlinearity for some sets of initial conditions. Similar analysis will be performed in the framework of the “one-and-a-half” dimensional two-layer RSW model on the f -plane (Section 3) in order to study the role of the baroclinic effects. We will show that new phenomena appear due

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

259

to baroclinicity, such as wave-trapping, symmetric and Kelvin–Helmholtz instabilities, the latter being related to the loss of hyperbolicity of the system. New peculiar types of finite-amplitude propagating waves become possible due to baroclinicity. The differential rotation effects arising beyond the f -plane approximation in the RSW model introduce new dispersive phenomena, which manifest themselves in a specific form on the equatorial β-plane, where the leading constant-rotation term in the Coriolis parameter vanishes. Some of the equatorial waves are strictly dispersionless (Kelvin waves), some of them strongly dispersive in the shortwave limit, and weakly dispersive in the long-wave limit (Rossby waves), some of them have weak or strong dispersion depending of the propagation direction (Yanai waves), and some of them are close to the f -plane waves (inertia–gravity waves)—see e.g. Gill (1982) and Chapter 1. In addition, waves in the RSW model on the equatorial β-plane are quasi-one-dimensional, especially the long ones, as they are strongly localized in the vicinity of the equator and have a discrete spectrum in the meridional direction (cf. Chapters 1 and 2). One can therefore anticipate nontrivial nonlinear phenomena both in the sector of non-dispersive waves (breaking and shock formation) and in the sector of weakly dispersive waves (formation of solitons) in the equatorial wave-guide. We will display these phenomena in Section 4.

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW 2.1. General features of the model The RSW equations on the f -plane with no dependence on one of the coordinates (y) are: ∂t u + u∂x u − f v + g∂x h = 0, ∂t v + u∂x v + f u = 0, ∂t h + u∂x h + h∂x u = 0.

(2.1)

Here u, v are the two components of the horizontal velocity, h is the total fluid’s depth, f is the constant Coriolis parameter, and g is the acceleration due to gravity. Here and below ∂t , ∂x , . . . denote the partial derivatives with respect to the corresponding arguments. As seen from these equations, the model is “one-and-a-half” dimensional, because although the y-dependence is forbidden, flows in y-direction are allowed. In what follows we often consider localized jets (localized distributions of v(x)) and/or fronts (localized distributions of ∂x h(x)). A configuration is front-like if

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

260

u, v, ∂x h have a common compact support in x. It is worth noting that the 1dRSW model was first considered in the pioneering work of Rossby (1938) where the very concept of geostrophic adjustment was introduced in the context of the relaxation of the oceanic jet to the state of geostrophic equilibrium. The model possesses two Lagrangian invariants: the generalized (sometimes called geostrophic) momentum M = v +fx

(2.2)

and the potential vorticity (PV) Q = (∂x v + f )/ h, (∂t + u∂x )M = 0,

(2.3) (∂t + u∂x )Q = 0.

(2.4)

Linearization around the rest state h = H = const gives a zero-frequency mode and surface inertia–gravity waves with the standard dispersion law: 1/2 ω = ± c02 k 2 + f 2 (2.5) . √ Here c0 = gH , ω is the wave-frequency and k is the wavenumber. As usual, one may evoke a parallel between the shallow-water dynamics and the gas dynamics (modified by rotation), so c0 is analogous to the sound speed. The conceptual advantage of this reduced model with respect to the full twodimensional one is that the geostrophic equilibria (cf. Chapter 1): f v = g∂x h,

u = 0,

(2.6)

are the exact stationary solutions of equations (2.1). Hence, in this model the slow manifold (cf. Chapter 2) is just a steady state, i.e. it is “infinitely slow”. In the state of geostrophic equilibrium velocity is entirely determined by the height perturbation (and vice versa) and the geostrophic PV is given by Q(g) =

f + g ∂2 h f xx h

.

(2.7)

2.2. Lagrangian approach to 1dRSW The system (2.1) may be reduced to a single nonlinear partial differential equation by using Lagrangian variables following Zeitlin, Medvedev and Plougonven (2003). Although physical Lagrangian particles are moving both in x and in y directions in the model, we introduce the coordinates of Lagrangian ‘quasiparticles’ X(x, t) moving in x direction via the mapping x → X(x, t), where x is position of a quasi-particle at t = 0 and X—its position at time t. From the Eulerian point of view X are just the Eulerian coordinates (we use the notation

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

261

x for the Lagrangian labels because the standard notation a is reserved for another purpose—see below). Therefore X˙ = u(X, t) and the over-dot will denote the material time-derivative from now on, while the prime notation will be used for differentiation with respect to x for compactness, e.g. X = ∂x X. The two momentum equations in (2.1) are then rewritten as X¨ − f v + g∂X h = 0, v˙ + f X˙ = 0,

(2.8)

where v is considered as a function of x and t. By virtue of mass conservation the variable h in Eulerian variables (X, t) may be obtained from its initial distribution hI (x) via the Jacobi transformation: h(X, t) = hI (x)∂X x.

(2.9)

By time differentiating the corresponding equation for h−1 it is easy to see that it is equivalent to the continuity equation h˙ + h∂X X˙ = 0.

(2.10)

The second of equations (2.8) may be immediately integrated giving v(x, t) + f X(x, t) = M(x),

(2.11)

where the conserved generalized momentum M(x) is to be determined from the initial conditions. If vI (x) is an initial distribution of the transverse velocity then, as X(x, 0) = x, M(x) = f x + vI (x).

(2.12)

By applying the chain differentiation rule to (2.9) we may express the partial derivative ∂X as follows ∂X h = ∂X hI (x)∂x X = hI (X )−2 − hI (x)X (X )−3 , (2.13) and, thus, get a closed equation for X: ghI −2 X¨ + f 2 X + ghI (X )−2 + (2.14) = f M. (X ) 2 This equation may be rewritten in terms of the deviations φ(x, t) of quasi-particles from their initial positions X(x, t) = x + φ(x, t): ghI (2.15) (1 + φ )−2 = f vI . 2 ˙ It is to be solved with initial conditions φ(x, 0) = 0; φ(x, 0) = uI (x), where uI is the initial distribution of the velocity in the x-direction. φ¨ + f 2 φ + ghI (1 + φ )−2 +

262

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

It should be emphasized that in the absence of rotation, f = 0, the nonlinear equation (2.14) contains only derivatives of X and not X itself. It may be reduced to a linear problem and completely integrated in this case by the hodograph ˙ X or, equivalently, u and h as new variables, and swapping demethod using X, pendent and independent variables: x = x(u, h), t = t (u, h). A linear PDE for x then results—cf. e.g. Landau and Lifshits (1975). The presence of rotation makes the hodograph transformation ineffective. Equation (2.14) may be obtained from an action principle. Indeed, by multiplying the l.h.s. by hI we get 2 ghI 2 ¨ hI X + f X − f M + (2.16) = 0. 2X 2 Equation (2.16) is an Euler–Lagrange equation following from the variational principle with the action S = dt dx L, (2.17) where the Lagrangian density is given by ˙2 gh2 1 X2 X − f2 + f MX − I . L = hI 2 2 2 X

(2.18)

˙ X we get the corresponding HamilIntroducing canonical variables P = hI X, tonian 2 gh2 1 X2 P + hI f 2 − f MX + I . H = dx (2.19) 2hI 2 2 X As usual, vanishing the first variation of the Hamiltonian δH gives the equation for the steady state: 2 ghI hI f 2 X − f M + (2.20) = 0; P = 0. 2X 2 This is the state of geostrophic equilibrium (2.6) expressed in Lagrangian coordinates. A straightforward calculation of the second variation δ 2 H shows that it is always positive-definite and, thus, the geostrophic equilibria are formally stable. It is important to note that the equation for the adjusted geostrophic equilibrium solution may be obtained by simply omitting the term with time derivatives in (2.14). Equations (2.14), (2.15) have advantage of being physically transparent and of incorporating explicitly arbitrary initial conditions. The latter fact leads, however, to the appearance of variable coefficients in the equations. Further simplification may be achieved by additional change of variables x = x(a), which “straightens” the initial elevation profile hI (x). We then get a following relation between J and h (we remind that h is everywhere positive): J = ∂X/∂a = H / h(X, t), where

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

263

the uniform mean height H is introduced. By the chain differentiation rule we get g ∂X h = ∂a P , where P = gH /(2J 2 ) is the pressure variable. The Lagrangian equations of motion in a-coordinates take the form: u˙ − f v + ∂a P = 0,

(2.21)

v˙ + f u = 0, J˙ − ∂a u = 0.

(2.22) (2.23)

This system is equivalent to a single equation, which may be as well obtained by differentiation and change of variables in (2.14): 2 P = f H Q. J¨ + f 2 J + ∂aa

(2.24)

Here Q(a) is the potential vorticity in a-coordinates: Q(a) = H1 (∂a v(a, t) + ˙ = 0. Note, that equations (2.21)–(2.23) f J (a, t)) = H1 (∂a vI (a) + f JI (a)), Q may be deduced, as well, from the original Eulerian system (2.1) by applying the transformation of independent variables mixing the independent and the dependent variables which is known in the theory of partial differential equations (see for details Rozhdestvenskii and Janenko, 1983) da = h(x, t) dx − h(x, t)u(x, t) dt,

dτ = dt,

(2.25)

where a is the so-called Lagrangian mass variable. For positive h one gets ∂x = h∂a ,

∂t = ∂τ − hu∂a = ∂τ − u∂x

(2.26)

and the system (2.21)–(2.23) results. Again, the nonperturbative slow manifold is a stationary solution Js of (2.24). The advantage of the present formulation with respect to equation (2.14) is that it highlights the role of PV: for a given set of (localized) initial conditions the adjusted state is a stationary state with the same potential vorticity as the initial one. It is given by the equation: gH d2 1 (2.27) + f Js (a) = H Q(a). f da 2 2Js2 (a) This equation is nonlinear and can be rewritten as a non-autonomous ODE describing the motion of a material point in a given potential under the action of a “time”-dependent force, where a is “time”. Using the non-dimensional pressure variable p = P /(gH ) and introducing the Rossby deformation radius Rd2 = gH /f 2 one gets: f 1 1 d2 p = Q. + 2√ 2 g da Rd 2p

(2.28)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

264

The corresponding boundary conditions are: the “external force” in the r.h.s. exactly equilibrates the “potential force” (the second term in the l.h.s.) at a = ±∞. Hence, a stationary solution, if it exists, is a separatrix relating two states of unstable equilibrium in the phase-space of this simple one-dimensional system, and thus resembles the soliton or instanton solutions in a number of physical models. 2.3. Theoretical predictions for the geostrophic adjustment in the 1dRSW model 2.3.1. Existence and uniqueness of the adjusted state The first fundamental question which arises in the adjustment context is whether the adjusted state, i.e. the steady state with the same values of Lagrangian invariants as the initial state, exists, and whether it is unique. The simplest proof of the existence and uniqueness theorem for the adjusted state follows from the reformulation of the problem in terms of the variables X: dX = J da. This gives the following equation for the adjusted state: −

g d2 h(X) + h(X)Q(X) = f. f dX 2

(2.29)

Here PV is considered as a function of X given by the inverse mapping x = 1 (f + ∂vI (x(X)) ). x(X, t) (or a = a(X)): Q(X) = hI (x(X)) ∂x The following theorem giving sufficient conditions for existence and uniqueness of the adjusted state was proved by Zeitlin, Medvedev and Plougonven (2003) using the standard tools of the theory of ordinary differential equations: T HEOREM 1. For positive Q(X) with compact support derivatives and arbitrary constant asymptotics (frontal case) equation (2.29) has unique bounded and everywhere positive solution h(X) on R1 . The proof of Theorem 1 is not constructive in the sense that it uses the mapping x = x(X, t) which is not known explicitly. However, the Lagrangian conservation of PV guarantees that this mapping preserves the positiveness of PV and its asymptotics at infinity (provided the infinity is a fixed point). It should be noted that positiveness of Q is a sufficient condition for the absence of inertial/symmetric instability in the full 3d primitive equations models (cf. e.g. Holton, 1979). As shown below, this instability is absent in 1d RSW. 2.3.2. Analysis of small perturbations about the adjusted state The second question in the adjustment context, once the existence and uniqueness of the adjusted state is proved, is whether this state is attainable. Indeed, in the absence of dissipation, the 1dRSW model is energy-preserving and, although the adjusted state is a minimal energy state, it cannot be an attractor in the usual

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sense. However, the evacuation of energy can be provided by outgoing IGW if the radiation boundary conditions are imposed. In order to understand this process, we consider the small perturbations around the adjusted state (Zeitlin, Medvedev and Plougonven, 2003). If the initial conditions vI and hI are in geostrophic balance ghI = f vI ,

(2.30)

then φ = 0 is an exact solution of (2.15). Therefore, for small deviations from the exact geostrophy equation (2.15) may be linearized and solved perturbatively. A necessary condition for this is smallness of the initial imbalance AI = f vI − ghI . It is easy to see that this condition is equivalent to the smallness of the Rossby number: Ro = fUL 1, where U and L are characteristic velocity and scale in the x-direction. An advantage of the Lagrangian approach is that contrary to the Eulerian one, where it is necessary to attribute the initial imbalance either to vI or to hI , the imbalance AI enters the perturbative equations as a whole. Note that another source of imbalance is a non-zero uI . The linearized equation (2.15) is φ¨ + f 2 φ − 2ghI φ − ghI φ = AI .

(2.31)

We represent the solution as a combination of the slow and the fast components: φ = φ¯ + φ˜ where the over-bar denotes the time mean and the tilde corresponds to fluctuations around it. For the time-mean (slow part), corresponding to the shift of Lagrangian particles which is necessary to arrive to the new equilibrium positions, we get f 2 φ¯ − 2ghI φ¯ − ghI φ¯ = AI ,

(2.32)

a linear second-order inhomogeneous differential equation. Introducing a new variable Φ¯ = ghI φ¯ and dividing by ghI we get: 2 f + ghI ¯ Φ¯ = AI , −Φ + (2.33) ghI (g)

where the geostrophic PV constructed from the initial height perturbation QI , cf. (2.7), explicitly enters. Equation (2.33) is a linear inhomogeneous ODE with variable coefficients and its solution may be obtained by the method of variation of constants once solutions of the homogeneous equation are known. We are mostly interested in the case of localized fronts where AI is a compact support function and hI is monotonous and has constant asymptotics. Solutions of the homogeneous equation are then exponentially growing/decaying at both spatial infinities. However, it is not obvious that a decaying at both spatial infinities so(g) lution of the inhomogeneous equation may be found for arbitrary QI (x). By the same method which is used in the full nonperturbative proof above it may be

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shown that unique solution φs (x) of (2.32) exists for non-negative geostrophic (g) PV: QI 0. Hence, at least for non-negative initial PVs one can always find a corrected adjusted state. For the variable in time fast part of the solution a homogeneous equation φ¨˜ + f 2 φ˜ − 2ghI φ˜ − ghI φ˜ = 0

(2.34)

results. By introducing a new variable Φ˜ = ghI φ˜ this equation may be rewritten as (2.35) Φ¨˜ + f 2 + ghI Φ˜ − ghI Φ˜ = 0. Solution of the Cauchy problem for arbitrary initial conditions φ˜ I , uI = φ˙˜ I can ∞ +iω ˜ be obtained via e.g. the Fourier transform Ψ (x, ω) = −∞ e t Φ(x, t) dt for which we get:

f + ghI f ω2 f − Ψ = 0. −Ψ + g hI ghI

(2.36)

(g)

Note that QI plays a rôle of potential in this Schrödinger-type equation. As φI = φ˜ I + φ¯ I = 0 the initial condition for φ˜ and, hence, Φ˜ follows once φ¯ ¯ is found: φ˜ I = −φ. The solution of (2.35) represents a packet of inertia–gravity waves propagating out of the initial localized perturbation plus, possibly, some bound states (trapped modes). The presence of the trapped modes would mean the non-attainability of the adjusted state, as part of the energy of the initial perturbation which is projected onto these modes cannot be evacuated due to radiation. It may be proved however (Zeitlin, Medvedev and Plougonven, 2003) that trapping by an isolated front is impossible The following argument shows that the trapped modes should be sub-inertial, i.e. having a frequency below f . Consider the Fourier-transform ˜ φ˜ = dω(ψ(ω, x)e−iωt + c.c.). Then for each Fourier-component ψ(ω, x) of φ: we get from (2.34): ghI ψ + 2ghI ψ + ω2 − f 2 ψ = 0, (2.37) which is equivalent to (2.36). In the case of a front-like initial configuration the asymptotics of φ˜ I and uI at infinity are zero and those of hI and Q(g) are constant: hI |±∞ = h± ,

f Q(g) ±∞ = . h±

(2.38)

Hence, at x → ±∞ equation (2.36) becomes −Ψ +

f 2 − ω2 Ψ =0 gh±

(2.39)

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and in order to have decaying at spatial infinity bound states we should have ω < f . However, this is impossible, as seen from the following simple consideration. By multiplying (2.37) by ghI ψ ∗ , where the asterisk denotes the complex conjugation, we get 2 2 ∗ g hI ψ ψ − g 2 h2I ψ ∗ ψ + ω2 − f 2 ghI ψ ∗ ψ = 0 (2.40) and for decaying at ±∞ states an estimate +∞ dx g 2 h2I |ψ |2 2 2 f2 ω = f + −∞ +∞ 2 dx gh |ψ| I −∞

(2.41)

follows by integration. This contradicts the initial hypothesis and, hence, subinertial trapped modes do not exist, and the frequency spectrum is continuous. ˜ Therefore, all of the initial φ-perturbation will be dispersed leaving only the stationary part φs in the vicinity of the initial perturbation. As shown in Reznik, Zeitlin and Ben Jelloul (2001) the outgoing waves do not exert any drag upon the stationary state in lowest orders in Ro and, thus, slow and fast variables are split in the perturbation theory, at least for non-negative PVs. The speed of the relaxation toward the adjusted state will depend on further details of the potential Q(g) in the Schrödinger equation (2.36). If quasi-stationary states, i.e. those which decay only by a sub-barrier tunneling, are present the decay rate will be exponential, as is well-known from quantum mechanics (cf. e.g. Migdal (1977)). Otherwise the decay will be dispersive according to the t −1/2 law. Here and below we mean by decay the time-decrease of the amplitude of a spatially localized perturbation. It should be also emphasized that equation (2.41) also shows that negative values of ω2 are impossible in the 1dRSW model. This will not be the case in the two-layer 1dRSW model treated in the next section. There both sub-inertial and negative values of the frequency square may exist, the latter case corresponding to the appearance of the so-called symmetric instability. 2.3.3. Low-resolution numerical simulations of the geostrophic adjustment in Lagrangian variables Equation (2.15) is ready for studies of the Cauchy problem which is equivalent to the fully nonlinear geostrophic adjustment problem. The advantage of having a single PDE with two independent variables is that it is readily solvable by the PDE solvers available in standard software packages. An example of low-resolution calculation using the standard MATHEMATICA routine NDSolve (Wolfram, 2004) is presented in Figure 1 giving a qualitative picture of adjustment. This example is taken from Zeitlin, Medvedev and Plougonven (2003). The 2 Gaussian profile of initial elevation: hI (x) = 1 + e−x is chosen. The along-jet 2 velocity is slightly imbalanced vI (x) = −2(x + 0.2 sin(x))e−x , and non-zero

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 1. Fully nonlinear geostrophic adjustment of the double-jet configuration, see the main text for explanation.

cross-front initial velocity is imposed uI (x) = 0.1e−x . The horizontal axes correspond to time and initial positions of Lagrangian particles, respectively, and the particle displacement X − x is plotted on the vertical axis. Emitted fast gravity waves and slowly dispersing quasi-inertial oscillations concentrated in the vicinity of the jet are clearly seen, as well as systematic displacements of fluid particles necessary to reach the final adjusted state. A zone of steep gradients forms and propagates toward the left. It indicates a shock formation but such process cannot be properly resolved by a generic solver and results in the fuzzy region on the left. The shock formation thus needs attention in the model. If confirmed, being an essentially dissipative process the shock formation would provide an alternative sink of energy due to waves. 2

2.4. Wave breaking and shock formation in the 1dRSW model The formulation of the 1dRSW equations using the independent variable a allows to understand the shock formation, which manifests itself in the low-resolution simulations presented above. For this purpose the Lax method of analysis of the quasi-linear hyperbolic systems (Lax, 1973; Engelberg, 1996) is applied.

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Following Zeitlin, Medvedev and Plougonven (2003), let us rewrite equations (2.21)–(2.23) as a system of two equations u˙ + ∂a p = v, J˙ − ∂a u = 0,

(2.42) (2.43)

where v is not an independent variable and should be determined from the relation ∂a v = Q(a) − J . To avoid cumbersome formulae, all dimensional parameters are taken to be equal to unity in this subsection; the correct dimensions are easy to recover. Equations (2.42), (2.43) represent a quasi-linear system in a standard form (cf. Whitham, 1974): u v u˙ 0 −J −3 ∂a = . (2.44) + −1 0 J 0 J˙ The left eigenvectors of the advection matrix in the l.h.s. are (1, ±J −3/2 ) and the corresponding eigenvalues are μ± = ±J −3/2 . Hence, the Riemann invariants are w± = u ± 2J −1/2 and obey the following equations: w˙ ± + μ± ∂a w± = v.

(2.45)

Expressions of original variables in terms of w± are given by 1 (w+ + w− ), (2.46) 2 16 > 0, J = (2.47) (w+ − w− )2 w + − w− 3 μ± = ± (2.48) . 4 The following equations for the derivatives of the Riemann invariants r± = ∂a w± readily follow: u=

∂μ± ∂μ± r+ r± + r− r± = ∂a v = Q(a) − J. (2.49) ∂w+ ∂w− These equations may be rewritten, using the “double” Lagrangian derivatives along the characteristics dtdr± = r˙ + μ± ∂a r, as r˙± + μ± ∂a r± +

dr± ∂μ± ∂μ± (2.50) + r+ r± + r− r± = Q(a) − J. dt± ∂w+ ∂w− Wave-breaking and shock formation correspond to the loss of smoothness by the Riemann invariants in finite time, i.e. to r± going to ±∞ in finite time. Equations (2.50) are the generalized Ricatti equations on the characteristics. The qualitative analysis of such equations (see Lemmas 1 and 2 in Engelberg, 1996) gives both conditions of shock appearance, and of their absence:

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

(1) If the initial relative vorticity Q − J = ∂a v is sufficiently negative breaking always takes place whatever the initial conditions are. (2) If the relative vorticity is positive, as well as the derivatives of the Riemann invariants at the initial moment, breaking never takes place. In the adjustment context this result means that shocks should be more easy to produce on the anticyclonic (negative relative vorticity) side of the jet. As one of the Riemann invariants has always a negative derivative for the stepwise height profiles, shocks should be always produced by the pure height (no vI ) adjustment. The shock production in course of adjustment of front-like perturbations is likely, because these latter never satisfy the second condition. 2.5. High-resolution numerical simulations of fully nonlinear geostrophic adjustment in 1dRSW High-resolution numerical simulations of fully nonlinear geostrophic adjustment in 1dRSW were undertaken by Bouchut, Le Sommer and Zeitlin (2004) in order to check the theoretical predictions and to test the finite-volume numerical method by Bouchut (Chapter 4 in this volume). We first remind the general framework for the shock description in the RSW model in Eulerian variables, and then display some of the results of the numerical simulations following this paper. 2.5.1. Hydraulic theory as applied to the 1dRSW model Equations (2.1) may be reformulated in the form of conservation laws for the two components of momentum and the mass (although momentum is not conserved due to the presence of the Coriolis acceleration in (2.1) we use the term “conservation laws” for the equations below, as is frequently done in the literature, cf. e.g. (Rozhdestvenskii and Janenko, 1983)): 1 2 2 (hu)t + hu + gh − f hv = 0, 2 x (hv)t + (huv)x + f hu = 0, ht + (hu)x = 0.

(2.51)

The analogy with the rotating compressible gas with a specific √ heat ratio γ = 2, h as a density variable, and the equivalent sound speed c0 = gH , gives a mathematically rigorous framework for a description of the hydraulic jumps as gas dynamics shock-waves, see Lighthill (1978), and allows to use powerful computational methods developed in gas dynamics. The basic properties required in order to define the weak solutions to (2.51) are the conservation of mass and momentum across a supposed discontinuity, see e.g. Lighthill (1978). The weak solutions may be also obtained as solutions of

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a dissipative set of equations corresponding to (2.51) in the limit of vanishing viscosity, cf. e.g. Schär and Smith (1993). The standard discontinuity calculus holds for shallow water. (For non-rotating case see Whitham, 1974, for rotating case see Pratt, 1983, 1984; and Nof, 1984.) Weak solutions are known to be completely determined if one specifies the Rankine–Hugoniot (RH) conditions together with the entropy condition. In our case, the RH conditions are: −U [hu] + hu2 + gh2 /2 = 0, −U [hv] + [huv] = 0, −U [h] + [hu] = 0,

(2.52)

where U is the speed of the discontinuity and [A] is the jump, following a fluid particle, of any quantity A across the discontinuity. These conditions do not depend on the Coriolis parameter f and express, respectively, the conservation of momentum in the x and y directions and mass conservation across the discontinuity. Yet, physically relevant solutions are not specified unless the dissipation of the total energy E across the discontinuity is ensured (cf. Whitham, 1974): h E + gh2 /2 u 0 with E = gh + u2 + v 2 , (2.53) 2 which is an analog of the gas dynamics entropy condition. As a direct consequence, this implies that [h] > 0 across a shock. Note that the conditions (2.52) are satisfied by any finite-volume numerical scheme whereas the fulfillment of the condition (2.53), which is not straightforward, requires special care. From the RH and entropy conditions it follows that the rate of the energy dissipation in every material volume V (t) which contains a discontinuity depends on the amplitude of the jump only g d E dx = [h]2 [u]. (2.54) dt 4 −U [E] +

V (t)

However, the PV field is not affected by dissipation due to the passage of shocks. Following Pratt (1983, 1984) and Peregrine (1998), we consider the jump in the PV equation −U [hq] + [huq] = 0 and continuity equation [(U − u)h] = 0 in the absence of contact discontinuities i.e. for [v] = 0, and obtain (U −u)h[q] = 0. Hence [q] = 0 in one dimension. Only transverse variations in shock strength can affect the PV in this framework, cf. Pratt (1983). Hence in the y-independent reduction of the shallow water equations no PV deposit by shocks is possible. As shown by Houghton (1969), shocks are modified by rotation. This implies a decrease in shock strength with time, but, as just explained, shocks do not affect the vortex part of the flow.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

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2.5.2. The results of high-resolution numerical simulations of the adjustment process The high-resolution finite-volume numerical method by Bouchut (2004) (see also Chapter 4 of the present volume) applied to the system (2.51) perfectly resolves shocks, automatically satisfies the RH conditions including the entropy condition (2.53), and allows for the efficient treatment of the Coriolis source terms. It was extensively tested in Bouchut, Le Sommer and Zeitlin (2004) and shown to perform better than other comparable methods (see Appendix A of this paper). The shock formation was studied in a number of cases, and found to be ubiquitous. For example for the classical Rossby problem of the adjustment of the unbalanced jet with initial conditions of the form:

h(x, 0) = H = const, (2.55) u(x, 0) = 0, v(x, 0) = V NL (x), where (1 + tanh(4x/L + 2)) · (1 − tanh(4x/L − 2)) NL (x) = (2.56) (1 + tanh(2))2 and the relevant parameters are Rossby and Burger numbers, respectively: gH V , Bu = 2 2 , (2.57) fL f L the shocks are formed in the wide range of parameters, as shown in Figure 2. A typical evolution of the height field h(x, t) in the Rossby adjustment is shown in Figure 3 for Ro = 1 and Bu = 0.25. (The figures here and below are taken from Bouchut, Le Sommer and Zeitlin, 2004.) As expected, the height field adjusts to the momentum imbalance by emitting IGW which propagate out from the jet. As seen in Figure 3, and in agreement with the results of the previous subsection and Zeitlin, Medvedev and Plougonven (2003), two discontinuities form rapidly at the wave front. The formation of shocks in RSW, i.e. the fact that rotation does not inhibit wave-breaking, was first predicted by Houghton (1969) and observed by Kuo and Polvani (1997) in their numerical study of the rotating dam-break problem. As shown in Bouchut, Le Sommer and Zeitlin (2004), the energy is conserved with high accuracy before the appearance of shocks, and drops up to 10% when a shock forms. Thus the energy evacuation in the adjustment process due to shocks, which is important for understanding the scenario of adjustment, is substantial. In spite of such sink of energy, the adjusted state is not rapidly attained due to the quasi-inertial oscillations remaining for the long time at the location of the initial jet, as shown in Figure 4. Although persistent quasi-inertial (i.e. having frequencies close to the inertial one) oscillations were observed in all of the numerical simulations of Bouchut, Le Sommer and Zeitlin (2004), confirming the earlier results by Kuo and Polvani (1997, 1999), no significant deviations of their decay Ro =

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Figure 2. Statistics of the shock formation in the Rossby adjustment problem on the Bu–Ro plane: each point corresponds to a run with the corresponding parameters Bu and Ro. Breaking happens in t < π/f : shaded circles; breaking event in π/f < t < 2π/f : open circles; breaking in t > 2π/f : crosses. Appearance of transonic shocks, i.e. those with propagation velocity changing sign in course of evolution, are marked by the superscript t. Drying was observed for large Ro and small Bu: squares.

law from the t 1/2 were found. These numerical simulations allowed to complete the theoretical results on existence of the adjusted state. Thus, it was observed, cf. Figure 5, that negative PV-states adjust as well. The wave-breaking criteria exposed in the previous subsection were also tested and confirmed numerically. The adjustment of a PV-less wave perturbation super-imposed onto a simple balanced jet was performed in order to check how the vorticity in the jet region influences shock formation. The parameters are the Rossby and the Burger numbers of the jet, Rojet , Bujet and the Rossby and the Froude numbers which specify the perturbation, Rop , Frp . The typical evolution of the perturbation is shown in Figure 6. The perturbation splits into a left and a right propagating parts but, in this case, only the former breaks inside the jet. Let us remind that, as shown above, two factors lead to breaking: strong enough gradients of the Riemann invariants and strong enough anticyclonic shear. The perturbation is small enough in Figure 6, and the derivatives R± of the Riemann invariants rewritten in Eulerian variables

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 3. The Rossby adjustment of a jet. Two shocks are formed at t = 0.3 and propagate to the left and to the right from the jet, respectively. One of the shocks is formed immediately within the jet core.

(these expressions are obtained by a change of variables from corresponding expressions in Lagrangian coordinates calculated before): 3/4 1/4 g h H ux ± hx , R± = 8 (2.58) h H H are dominated by the derivative of the height field (the second term). Since the sign of this term is fixed by the geostrophic balance of the basic state, the part of the perturbation propagating toward the small h region is more likely to break. Hence, as observed, breaking happens at the cyclonic part of the balanced jet as the influence of anticyclonicity is overcome by the strength of pressure gradients in this case. 2.6. Finite-amplitude wave solutions of the 1dRSW equations In the previous subsections we explored the previously theoretically established fact that rotation does not prevent shock formation for a large class of initial con-

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Figure 4. Check of balance between f v and ghx in the Rossby adjustment: although a close to the geostrophic balance mean state (second panel) is rapidly achieved, oscillations persist in the jet core. The amplitude of oscillations is decreasing with time and depends on the parameters Ro and Bu. The period of oscillations is close to Tf . The scale of the graphs is c02 /L.

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 4.

(Continued.)

Figure 5. PV-shift during the Rossby adjustment process. The initial distribution of PV and the mean distribution at t = 34.2Tf are shown. Note that the initial PV distribution contains a negative-PV part.

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Figure 6. Wave breaking in a simple jet: Rojet = 1, Bujet = 1 and Rop = 0.8. Left column: h, u in the case up > 0. Right column: h, u in the case up < 0. In both cases, the fields are plotted for successive times 0, 0.1, 0.2, 0.3Tf (from top to bottom). The arrows indicate the shock locations.

ditions. We now turn to the opposite case of initial conditions when it does. It is known (Ostrovsky, 1978; Shrira, 1981, 1986; Grimshaw, Ostrovsky, Shrira and Stepanyants, 1998) that the dispersion due to rotation is sufficient to warrant the existence of periodic steady-propagating finite-amplitude plane-parallel waves in RSW. Their particular feature is that amplitudes are bound from above by some limiting value, and that solution tends to form cusps at it crests while approaching the limiting amplitude. These solutions may be obtained using Lagrangian variables as well (Bühler, 1993; Zeitlin, Medvedev and Plougonven, 2003). However, in view of subsequent generalization to the two-layer RSW model below, we present here the derivation in Eulerian coordinates following Plougonven and Zeitlin (2003). We start from the system (2.1) and look for stationary propagating solutions depending on ξ = x − ct only; the prime will denote the derivative with respect to ξ . Therefore, −cu + uu − f v + gh = 0,

(2.59a)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

278

−cv + u(f + v ) = 0,

−ch + (uh) = 0.

(2.59b) (2.59c)

Integration of (2.59c) gives u = c + K/ h, where K is an integration constant. We look for spatially periodic solutions with a given (unknown) wavelength λ and suppose that there is no overall mass-flux in the x direction. Then integration of uh over one wavelength λ gives zero: λ (K + ch) dξ = 0,

(2.60)

0

and the value of the integration constant is K = −cH , where H is the mean (rest) height of the fluid. The expression for u results: H u=c 1− (2.61) . h Note that solutions in question do not carry PV anomaly. Indeed, the PV (2.3) by virtue of (2.59b) and (2.61) is equal to q=

f + v f = , h H

(2.62)

on the solutions, i.e. is equal to the background PV of the unperturbed fluid layer. This is consistent with the result of Falkovich and Medvedev (1992) showing that the invariant definition of the wave component of motion in RSW is that of the absence of the PV anomaly. Hence, the solutions we are looking for are finiteamplitude generalizations of the linear inertia–gravity waves. By introducing the dimensionless elevation χ: h = H χ and the Mach number √ M = c/ gH the following equation for χ follows from (2.61) and (2.59a)– (2.59c):

2 f2 M + χ − (χ − 1) = 0. (2.63) 2χ 2 c02 Considering this equation at the wave-crest which is supposed to be regular, χ = 0, we get 1−

M2 f2 χ − 1 = . χ3 c02 χ

(2.64)

At the wave-crest χ − 1 > 0, χ < 0, and hence the r.h.s. of this expression is negative. Therefore, 1 < χ 3 < M 2 . Hence, the wave solutions exist only for M > 1.

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

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2

By multiplication by [ M2 χ12 + χ] equation (2.63) may be integrated and transformed into a “particle-in-a-well” equation at zero energy level of the “particle”: χ2 +

f2 V(χ; M, A) = 0. c02

(2.65)

Note that only non-negative values of χ are physically acceptable. Zero values are excluded as well, as they lead to infinite u, cf. (2.61). The expression for the “potential” is: 2

− 1) − A] [(χ − 1)2 ( M N (χ; M, A) χ2 V(χ; M, A) = = , 2 (D(χ; M, A))2 (M − 1)2 χ3

(2.66)

where A is an integration constant. It can be easily seen that A is necessarily nonnegative (it is zero for the fluid at rest, strictly positive otherwise, and increases with the amplitude of the wave solutions to be obtained). Stationary waves of finite amplitude correspond to the oscillations of the particle around the rest position χ = 1. They exist if there is a potential well which is situated between two strictly positive zeros, χl and χr , 0 < χl < 1 < χr , where the potential is negative and bounded from below. It is easy to show (Plougonven and Zeitlin, 2003) that: The denominator (D)2 is always positive; thus the sign of the potential is determined by the numerator; D has a unique zero for χ0 = M 2/3 > 1 The numerator N tends to +∞ as χ → 0; for χ = 1, it is equal to −A and, hence, is strictly negative for positive A. Therefore, there is a zero of N in the interval ]0, 1[. For χ → +∞, N → −∞, and N has a single maximum in ]1, +∞[. This maximum occurs for the same value of χ as the zero of the denominator: χ0 = M 2/3 > 1. Therefore, if N (χ0 ; M, A) is strictly positive, the numerator has one zero in the interval ]1, χ0 [ : the potential then has a well and stationary waves exist. This situation is illustrated in Figure 7. For A close to zero, it is easy to check that the numerator has two zeros close to 1, and waves of small amplitude always exist. For small deviations from the rest-state, (2.66) becomes the

Figure 7.

The numerator N (dashed), the denominator (D)2 (gray) of the potential V, and the potential itself (plain) for M = 2 and A = 0.18.

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harmonic oscillator equation with “frequency” (= wavenumber) 1 . (2.67) −1 This expression is equivalent to the dispersion relation for linear surface inertia– gravity waves (2.5). Thus there exist a smooth transition from infinitesimal IGW to finite-amplitude non-harmonic waves with increasing amplitude, so that the solution we found may be called finite-amplitude IGW. As A increases, the graph of N descends and there exists a critical value Ac for which the maximum of the numerator in ]1, +∞[ is zero: N (χ0 ; M, A) = 0. Asymptotic analysis in the vicinity of χ0 = M 2/3 shows that the potential itself is continuous and has no well: hence, waves exist only for 0 < A < Ac . Note that as A increases, the potential becomes more and more asymmetric. Correspondingly, the wave-crests sharpen, and the waves tend to develop cusps as A → Ac . However, as it was shown by Zeitlin, Medvedev and Plougonven (2003) a true cusp is an asymptotic envelope of the solutions and is not attainable. Summarizing, stationary periodic waves exist if the equation 2 2 M −1 −A=0 N (χ; M, A) = (χ − 1) (2.68) χ2 k2 =

M2

has two distinct roots in the interval ]1, +∞[ or, equivalently, if N (M 2/3 ; M, A) > 0. The height of the crests of the waves is bounded from above by the limiting value M 2/3 . The difference between the roots of the numerator gives the amplitude of the wave; its wavelength depends on the value of the deformation radius f 2 /c02 . Thus obtained nonlinear waves are therefore the plane-parallel surface inertia–gravity waves of finite amplitude. Note that they are not strictly one-dimensional as the cross-propagation velocity v is non-zero and varies periodically (cf. equation (2.59b)). A limiting-amplitude nonlinear wave solution is presented in Figure 8 (Bouchut, Le Sommer and Zeitlin, 2004). The stability analysis of the finite-amplitude waves is a nontrivial technical problem. Some idea of their robustness, and their emergence from the arbitrary periodic initial conditions may be made on the basis of direct numerical simulations by Bouchut, Le Sommer and Zeitlin (2004). Figure 9 illustrates stability of the limiting-amplitude wave with respect to localized perturbations. No destabilization of the finite-amplitude wave were found within a large class of superimposed perturbations. In Figure 10 spontaneous emergence of nonlinear periodic waves during the evolution of a finite amplitude harmonic wave is displayed (Bouchut, Le Sommer and Zeitlin, 2004). An exact harmonic solution to the linearized equations (2.1) with Ro = 0.1 and k = π/2 is taken as initial condition in this simulation. As seen from Figure 10, the initial wave breaks down and, after a stage of geostrophic adjustment, a quasi-steady propagative state is reached which resembles closely

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

Figure 8.

281

Exact nonlinear-wave solution; the limiting-amplitude configuration with the phase-speed c = 2c0 .

the cusp pattern of the exact propagative nonlinear solution. The adjustment is achieved through shock formation and related energy dissipation. The energy decay due to dissipative processes in the shock is displayed in Figure 11 and shows that in this case the energy tends to a nearly steady value after a rapid stage of adjustment. These simulations allow to conjecture that finite-amplitude wave solutions are attracting in the constant PV sector of periodic initial data.

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW 3.1. General features of the model The 2-layer 1dRSW (2L1dRSW) is obtained by superposition of shallow-water layers of different densities (see Chapter 1 of this volume), with a constraint of no variations in one of the spatial directions. In order to reduce the number of the degrees of freedom, the rigid-lid vertical boundary conditions are used. Although

282

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 9. Stability of the nonlinear wave solution of limiting amplitude: localized perturbation is added to the nonlinear wave solution—left panel; the resulting profile after ten inertial periods—right panel. Between these two snapshots, the wave propagated at the speed c = 2c0 so that 12 recurrences in the computation domain were observed.

some features of the model are close to the 1dRSW, with the displacements of the interface between the layers replacing the free-surface elevation, it will be shown below that new essentially baroclinic effects due to the difference in motions of the layers appear in the 2L1dRSW dynamics (Le Sommer, Medvedev, Plougonven and Zeitlin, 2003). The equations of the 2L1dRSW model are: ∂t u1 + u1 ∂x u1 − f v1 + ρ1−1 ∂x π = 0,

(3.1a)

∂t v1 + u1 (f + ∂x v1 ) = 0,

(3.1b)

∂t u2 + u2 ∂x u2 − f v2 + ρ2−1 ∂x π

+ g ∂x η = 0,

∂t v2 + u2 (f + ∂x v2 ) = 0, ∂t (H1 − η) + ∂x (H1 − η)u1 = 0, ∂t (H2 + η) + ∂x (H2 + η)u2 = 0,

(3.1c) (3.1d) (3.1e) (3.1f)

where (u1 , v1 ) are two components of velocity in the upper layer, (u2 , v2 ) are two components of velocity in the lower layer; π is the barotropic pressure imposed by the rigid lid and equal to the pressure field in the upper layer π1 ; η is the

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

Figure 10.

283

Time evolution of a finite-amplitude harmonic wave.

interface displacement, H1 and H2 are the heights of the two fluid layers at rest; H = H1 + H2 = const is the total height, finally, g is the reduced gravity: g = g(ρ2 − ρ1 )/ρ2 . If the full layers’ heights h1,2 = H1,2 ∓ η are used as dynamical variables, the pressure field in the lower layer is expressed as π2 = π1 + g(ρ1 h1 + ρ2 h2 ); The approximation of the rigid lid leads to a dynamical constraint following from combining equations (3.1e) and (3.1f): (H1 − η)u1 + (H2 + η)u2 = H Ub (t). (3.2) Here Ub is the barotropic velocity in the x-direction. In what follows we will impose Ub = 0 which corresponds to vanishing of the overall x-momentum of the system. The potential vorticities in each layer are: Q1 =

f + ∂x v1 h1

and Q2 =

f + ∂x v2 . h2

(3.3)

They are Lagrangian invariants of (3.1a)–(3.1f). Linearizing the equations of motion about the rest-state gives: ∂t u1 − f v1 + ρ1−1 ∂x π = 0,

(3.4a)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

284

Figure 11. Energy decay during the evolution of a high-amplitude harmonic wave. The evolution of the non-dimensional total energy e/e(0), of the non-dimensional available potential energy ep /e(0) and of the non-dimensional kinetic energy ec /e(0), with ep (t) = dx g(h − H )2 /2, ec (t) = dx h(u2 + v 2 )/2 and e = ep + ec computed in the whole periodic domain.

∂t v1 + f u1 = 0, ∂t u2 − f v2 + ρ2−1 ∂x π

(3.4b)

+ g ∂x η = 0,

(3.4c)

∂t v2 + f u2 = 0,

(3.4d)

−∂t η + H1 ∂x u1 = 0,

(3.4e)

∂t η + H2 ∂x u2 = 0.

(3.4f)

Two kinds of solutions of these equations are easily found. They are stationary flows in geostrophic equilibrium and linear inertia–gravity waves. The stationary solutions corresponding to the geostrophic equilibria are, as in the 1dRSW case, the exact solutions of the full nonlinear equations and are given by: 1 ∂x π, fρ1 1 g v2 = ∂x π + ∂x η. fρ2 f

v1 =

(3.5a) (3.5b)

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

285

The stationary (balanced) solutions are, therefore, baroclinic in the general case, i.e. v1 = v2 . The fast motions in the linear approximation are internal inertia–gravity waves propagating along the interface between the layers; their form can be deduced from the equation for the variable H U = H2 u2 = −H1 u1 following from (3.4a)– (3.14f): 2 ∂tt + f 2 U − ce2 ∂xx U = 0. (3.6) Here ce2 = gHe is the phase speed of the waves, He =

H1 H2 ρ H1 + ρ1 H2

is the equivalent

2

height for the baroclinic modes of the model. The dispersion relation for the waves with frequency ω and wavenumber k reads: ω2 = f 2 + ce2 k 2 .

(3.7)

It is important to note that it is possible to eliminate the barotropic pressure in the full system (3.1a)–(3.1f) or, which is equivalent, to eliminate both pressures π1 , π2 , by expressing them in terms of other variables. From (3.1a), (3.1c) and (3.2) the following expressions result:

h2 −1 h1 + f (h1 v1 + h2 v2 ) − ρ1 ρ2 gh2 ∂ (ρ1 h1 + ρ2 h2 ) , − ρ2 ∂x ∂π2 h2 −1 h1 + = f (h1 v1 + h2 v2 ) − ∂x ρ1 ρ2 gh1 ∂ (ρ1 h1 + ρ2 h2 ) . + ρ1 ∂x

∂π1 = ∂x

∂ h1 u21 + h2 u22 ∂x (3.8) ∂ h1 u21 + h2 u22 ∂x (3.9)

To summarize, the structure of the stationary (balanced) states in the 2L1dRSW model is more complex than in the 1dRSW one. The dynamics is more complex, too, as the elimination of the barotropic pressure via (3.8), (3.9) generates new nonlinearities, see below. The geostrophic adjustment process will be more complex, as well, and will display qualitative differences as compared to the 1dRSW adjustment studied in the previous section. 3.2. The existence and uniqueness of the adjusted state As in the one-layer model, conditions for existence and uniqueness of the adjusted state can be obtained directly in the Eulerian framework as conditions for existence and uniqueness of solutions to the PV equations (Le Sommer, Medvedev,

286

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Plougonven and Zeitlin, 2003): f + ∂x v1 = Q1 (x ), h1 f + ∂x v2 = Q2 (x ). h2

(3.10a) (3.10b)

Here x is an Eulerian coordinate, which is related to the x-coordinate in the original equations via some (unknown explicitly) change of variables (see Section 2 and Zeitlin, Medvedev and Plougonven, 2003). Q1 and Q2 are the potential vorticity distributions in the layers, and v1 and v2 are given by (3.5a) and (3.5b). These equations can be combined to give two ordinary differential equations for the depths of the layers: g h1 − (Q2 + rQ1 )h1 = − −f (1 − r) + H Q2 , f g h2 − (Q2 + rQ1 )h2 = − f (1 − r) + rH Q1 , f

(3.11a) (3.11b)

where notation r = ρ1 /ρ2 for the density ratio of the layers has been introduced and the prime denotes the x -differentiation. An essential difference of these equations from their one-layer counterpart is that the forcing terms in the r.h.s. are not constant. They may be, nevertheless, analyzed by the same method (cf. Appendix A in Zeitlin, Medvedev and Plougonven, 2003). For an equation of the form h − R(x)h = −S(x), the existence and uniqueness of solutions are guaranteed for R and S with constant asymptotics at ±∞. Furthermore, the solution is positive if R and S are positive. Hence, for the initial states with localized PV anomalies such that Q1 0 and Q2 (1 − r)f/H,

(3.12)

the above equations have unique solutions h1 and h2 that are everywhere positive. Therefore, positiveness of the PV subject to the additional constraint (3.12) is sufficient for existence and uniqueness of the adjusted state in the baroclinic two-layer model. 3.3. The role of baroclinicity: obstacles to the standard adjustment scenario Although positiveness of the PV was also found in 1dRSW to be a formal criterion of existence of the adjusted state, the numerical experiments carried by Bouchut, Le Sommer and Zeitlin (2004) and presented in the previous section showed that negative-PV initial configuration were also adjustable. The constraint of positiveness of the PV is more drastic in 2L1dRSW than in the 1dRSW, and has a clear physical meaning, because negative PVs correspond to the appearance of the trapped modes and so-called symmetric instability in the baroclinic

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

287

systems. Both phenomena are absent in the barotropic one-layer 1dRSW model. On the other hand, the baroclinicity gives rise to the classical Kelvin–Helmholtz instability, if the initial velocity shear is strong enough. These two typically baroclinic instabilities constitute obstacles to the standard scenario of adjustment in the two-layer system with strong enough initial shears. We briefly describe them below. Note that essentially two-dimensional (“non-symmetric”) Rossby–Kelvin instability of the two-layer system discussed in Chapter 1 does not appear in the 1.5-dimensional model we are considering here. 3.3.1. Trapped modes and symmetric instability An entirely new phenomenon with respect to the 1dRSW model arises in the 2LdRSW one due to the baroclinic effects. Namely, some part of the initial perturbation can be trapped inside the jet in the form of fast oscillations, thus rendering adjustment incomplete. Consider a background state consisting of a balanced jet with nonzero velocities in both layers. The jet (cf. (3.5a)–(3.5b)) is characterized by its velocity Vig in each layer, the barotropic pressure Π and the variable interface height: V1g = ∂x Π,

(3.13a)

V2g = r∂x Π + Bu∂x h2g .

(3.13b)

Here we use the non-dimensional form of equations and introduce the Burger 1 ,H2 ) . Note that h1g and h2g are not independent as h1g + number Bu = g Min(H f 2 L2 h2g = 1. Linearizing around the jet gives: ∂t u1 − v1 + ∂x Π = 0,

(3.14a)

∂t v1 + u1 (1 + ∂x V1g ) = 0,

(3.14b)

∂t u2 − v2 + r∂x Π + Bu∂x η = 0,

(3.14c)

∂t v2 + u2 (1 + ∂x V2g ) = 0,

(3.14d)

∂t η − ∂x (h1g u1 ) = 0,

(3.14e)

∂t η + ∂x (h2g u2 ) = 0.

(3.14f)

The above equations can be combined into a single one for the variable U = h2g u2 = −h1g u1 :

2 Bu∂xx U−

= 0.

rh2g + h1g 2 ∂ 2 h2g 1 2 Π ∂tt + 1 + r∂xx + Bu xx U h1g h2g h1g h2g h2g (3.15)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

288

It is easy to check that for h1g → ∞, we recover from this equation the corresponding equation for the one-layer model and that in the absence of the jet the equation is reduced to that of free inertia–gravity waves at the interface (3.6). The existence of trapped modes may be analyzed following Le Sommer, Medvedev, Plougonven and Zeitlin (2003). As in the one-layer model the localized modes necessarily have frequencies below the inertial frequency f (which is equal to 1 in non-dimensional terms). By introducing the following auxiliary functions rh2g + h1g F (x) = (3.16) , h1g h2g ∂xx h2g 1 2 + Bu G(x) = r∂xx Π (3.17) h1g h2g h2g (ω, x)e−iωt + c.c. we and considering the Fourier-transform of U (x, t) = dω U get for each of the Fourier-components U (ω, x): 2 = 0. Bu∂xx (3.18) U − F 1 − ω2 + G U ∗ and integration over the whole domain the following After multiplication by U estimate for the frequencies ω of the localized modes is obtained: |2 dx + G|U |2 dx Bu |∂x U 2 ω =1+ (3.19) . |2 dx F |U Here F is, by definition, positive while G may be negative, particularly in the 2 Π < 0, cf. (3.16), (3.17). Thus, contrary to the anticyclonic regions where ∂xx situation in the one-layer model, sub-inertial frequencies are possible if the anticyclonic shear is strong enough. Note that anticyclonic shear means negative 2 Π is the relative vorticity in relative vorticity. According to (3.13a), (3.13b), ∂xx the upper layer, while h2g G(x) is the relative vorticity in the lower layer. When their values become lower than minus one (lower than −f in dimensionful units), the respective PVs become negative. To demonstrate the existence of the trapped modes we choose the simplest case of the barotropic jet with no displacement of the interface (η = 0). Equation (3.15) is then reduced to: 2 2 Bu∂xx (3.20) U − ∂tt2 + 1 He−1 + r∂xx Π(H1 H2 )−1 U = 0. eiωt + c.c. we get: Seeking solutions in the form U 1 2 −1 −1 2 = 0. Π U ω He − He + (H1 H2 )−1 r∂xx Bu This is the quantum-mechanical Schrödinger equation: 2 ψ + E − V (x) ψ = 0 ∂xx 2 ∂xx U+

(3.21)

(3.22)

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

289

for a particle having the energy E = ω2 (He Bu)−1 and moving in the potential 2 Π). It is known (cf. e.g. Migdal, 1977) V (x) = Bu−1 (He−1 + (H1 H2 )−1 r∂xx that in the case of potential well this equation has both propagating solutions corresponding to the continuous spectrum ω2 1 and localized, trapped in the well, solutions corresponding to the discrete spectrum Min(V (x))He < ω2 < 1. The potential well corresponds to the region of anticyclonic shear and the related trapped modes will be localized there, oscillating at sub-inertial frequencies. It is worth noting that small Burger numbers makes trapping easier, cf. Chapter 2 of this volume. Furthermore, if the potential is deep enough (i.e. the anticyclonic shear is strong enough) modes with ω2 < 0, i.e. non-oscillatory unstable modes, are possible thus giving rise to the symmetric instability, which bears this name precisely due to its one-dimensional, independent of the along-jet direction character. Whether this instability is saturated by interactions with e.g. continuous spectrum (in this case a secondary source of wave emissions during adjustment would appear), or leads to some sort of breaking inside the jet is an open problem. Some hints to its solution may be provided by the Lagrangian approach to the 2L1dRSW presented below. 3.3.2. The loss of hyperbolicity and Kelvin–Helmholtz instability In order to study wave-breaking and shock formation in the 2L1dRSW model it is useful, following Le Sommer, Medvedev, Plougonven and Zeitlin (2003), to rewrite the system (3.1a)–(3.1f) in the canonical hyperbolic form. In doing this we use (3.8) and (3.9) and reduce the system to four equations for four independent variables u2 , h2 , v2 and v1 , i.e. lower(heavier)-layer variables plus upper-layer jet velocity: ∂u2 ∂u2 ρ1 + u2 − f v2 + f (h1 v1 + h2 v2 ) ∂t ∂x ρ2 h1 + ρ1 h2 gρ ∂h2 ∂ − (3.23) h1 h1 u21 + h2 u22 + = 0, ∂x ρ1 ∂x ∂h2 ∂u2 ∂h2 + u2 + h2 = 0, (3.24) ∂t ∂x ∂x ∂v2 ∂v2 + u2 + f u2 = 0, (3.25) ∂t ∂x ∂v1 ∂v1 ∂v1 + u2 + (u1 − u2 ) + f u1 = 0, (3.26) ∂t ∂x ∂x where u1 =

h2 u2 , h2 − H

h1 = H − h2 .

(3.27)

290

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

As in the one-layer case, useful insights may be gained rewriting these equations in terms of the mass Lagrangian variable a (h2 = ∂a/∂x): ∂ ρ1 du2 − f v2 + h1 u21 + h2 u22 f (h1 v1 + h2 v2 ) − h2 dt ρ2 h1 + ρ1 h2 ∂a ∂h2 gρ h1 h2 = 0, + (3.28) ρ1 ∂a dh2 ∂u2 (3.29) + h22 = 0, dt ∂a dv2 (3.30) + f u2 = 0, dt dv1 ∂v1 (3.31) + (u1 − u2 )h2 + f u1 = 0. dt ∂a This system may be rewritten in the matrix form ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ 0 0 0 0 −f u2 v2 v2 d ⎜ v1 ⎟ ⎜ 0 T 0 0 ⎟ ∂ ⎜ v1 ⎟ ⎜ −f u1 ⎟ ⎝ ⎠+⎝ ⎝ ⎠=⎝ ⎠, 2 ⎠ 0 0 0 h h h 0 dt ∂a 2 2 2 ρ2 v2 −ρ1 v1 u2 0 0 M 2N u2 f h1 ρ2 h1 +ρ1 h2 (3.32) where we introduce, for compactness, the following auxiliary variables: H u2 T = (u1 − u2 )h2 = h2 , H − h2 H 2 u22 ρ1 h2 ρ M= (H − h2 ) − g , ρ2 h1 + ρ1 h2 ρ1 (H − h2 )2 H h2 ρ1 h2 u2 . N =− ρ2 h1 + ρ1 h2 H − h2 In order to reduce (3.32) to the canonical for the hyperbolic systems form (cf. e.g. Whitham, 1974), the matrix in the l.h.s. of (3.32) should be diagonalized and the left eigenvectors found. The equation for the nontrivial eigenvalues (the trivial ones are 0 and T ) is −λ h22 det (3.33) = λ2 − 2N λ − h22 M = 0 M 2N − λ and its solution is λ± = N ± N 2 + Mh22 . The discriminant of this equation is H 2 u22 ρ1 ρ2 h1 h32 h2 h1 + gρ − . D= ρ1 ρ2 (ρ2 h1 + ρ1 h2 )2 h21

(3.34)

(3.35)

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

291

The eigenvalues (3.33) are, thus, real (and, hence, the system is hyperbolic) only when D is positive. Therefore, if h1 h2 2 (u2 − u1 ) > gρ (3.36) + ρ1 ρ2 then D < 0 and the system loses hyperbolicity. We, thus, see that, unlike its onelayer counterpart, the 2L1dRSW model changes type if the vertical shear of the velocity transverse to the front is sufficiently strong. One may recognize in (3.36) the condition for Kelvin–Helmholtz (KH) instability. The KH instability is known to produce breaking of the growing interface wave with the subsequent formation of the KH billows (cf. e.g. Gossard and Hooke, 1975). This expected singularity is of a different nature as compared to shock formation which should be also present in the hyperbolic system (3.32). The left eigenvectors l(α) , α = 1, 2, 3, 4, corresponding to the eigenvalues λ(α) = (0, T , λ± ) may be easily found. They are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, M, λ± ) and (3.32) may be rewritten in the canonical form, required for the application of numerical Riemann solvers: ∂ ∂ (3.37) U + λ(α) l(α) · U = l(α) · B, ∂t ∂x where U, B denote the 4-vector of dynamical variables and the 4-vector in the r.h.s. of (3.32), respectively. However, finding explicit expressions for Riemann invariants r(α) ; dr(α) = l(α) · U is a non-trivial technical task. Its complexity prohibits the analytical study of the shock formation along the lines of the previous section, although we have all reasons to expect the existence of initial configurations leading to shock formation in the 2-layer model. l(α) ·

3.4. Lagrangian approach to 2L1dRSW and symmetric instability Thus, the 2L1dRSW model displays a number of qualitatively new phenomena. As in the 1dRSW one, sufficient conditions for existence and uniqueness of the adjusted state are established in terms of the initial distributions of PV in both layers. They prove to be more restrictive than in 1dRSW and the question of adjustment in situations not covered by these criteria remains open. The configurations with strong anticyclonic shears corresponding to negative PV are of particular interest, as we showed that they may support trapped states which render adjustment incomplete. For strong enough anticyclonic anomalies unstable localized states are possible, whose evolution will complicate considerably the adjustment scenario. For strong enough transverse velocity shears the developing KH instability will make the adjustment deviate from the standard scenario, too. In the prospect of investigating the fully nonlinear adjustment in the model including the nonlinear stage of the instabilities, especially the symmetric one

292

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

which is rather poorly understood, the Lagrangian approach similar to that developed for 1dRSW is useful. We start from the system (3.23)–(3.27), taken for simplicity in the frequently used in oceanography limit r → 1, and introduce the Lagrangian coordinate X(x, t) corresponding to the positions of the fluid particles in the lower layer. In terms of displacements φ with respect to initial positions ∂ X(x, t) = x+φ(x, t). The corresponding Lagrangian derivative is dtd = ∂t∂ +u2 ∂x and will be denoted by the overdot, as usual. The dependence of the height variable h2 on the Lagrangian labels, and transformation of its derivatives are obtained via the mass-conservation in the lower layer: h2I dx = h(X(x, t), t) dX, where the index I means initial value, as usual. Again, as in the one-layer case, equation (3.25) expresses the conservation of the geostrophic momentum in the lower layer, and allows to eliminate v2 in terms of its initial value and φ: v2 (x, t) + f φ(x, t) = v2I (x).

(3.38)

For simplicity, we will consider the particular case of the barotropic initial flow, which was already treated above to illustrate the mechanism of symmetric instability. Hence h2I = H2 = const, v2I = v1I = vI (x) and, by introducing the H2 1 shorthand notation α1 = H H0 , α2 = H0 , α1 + α2 = 1 we obtain: φ˙ 2 α2 α2 α1 + φ 2 ¨ (v1 − vI ) − · φ+f 1− φ+f 1+φ 1+φ α1 + φ 1 + φ α 1 − g H2 φ = 0, (3.39) (1 + φ )4 φ˙ α2 v1 − v˙1 − (3.40) φ˙ = 0, α1 + φ α1 + φ where prime denotes x-differentiation, as usual, and g is the reduced gravity. ˙ The equations are to be solved with initial conditions φ(x, 0) = 0, φ(x, 0) = u2I , v1 (x, 0) = vI . These equations are, certainly, more complicated than the corresponding onelayer equation for φ, but still may be solved by standard package solvers. For instance, the nonlinear stage of symmetric instability of the barotropic jet may be studied with the help of (3.39), (3.40) by imposing a profile of vI (x) having strong enough anticyclonic vorticity. It is easy to see that for vI (x) satisfying (3.13a) the system (3.39), (3.40) in the linear approximation reproduces equation (3.20) at r → 1. 3.5. Stationary waves of finite amplitude in the 2-layer model In spite of various singularities which lead to wave breaking in the 2L1dRSW model, as in the one-layer 1dRSW model there still exist exact solutions in the form of propagating finite-amplitude waves (Plougonven and Zeitlin, 2003),

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

293

where nonlinearity and dispersion due to rotation are compensated. These waves, again, are fully-nonlinear counterparts of the linear inertia–gravity waves on the interface between the layers. They also carry no anomaly of potential vorticity, and give standard linear waves in the limit of small amplitudes. However, their variety is richer due to additional parameters (the density ratio and the depth ratio of the layers) present in the model. Three families of finite-amplitude waves thus appear. 3.5.1. General equation for the stationary waves in the two-layer model The stationary wave solutions of (3.1a)–(3.1f) which are functions of ξ = x − ct should verify the following system of ordinary differential equations (the prime denotes the ξ -derivative, as usual): −cu1 + u1 u1 − f v1 + ρ1−1 π = 0,

(3.41a)

−cv1 + u1 (f + v1 ) = 0,

(3.41b)

− f v2 + ρ2−1 π + g h + v2 ) = 0, + u1 (H − h) = 0,

−cu2 + u2 u2 −cv2 + u2 (f −c(H − h)

−ch + (u2 h) = 0.

= 0,

(3.41c) (3.41d) (3.41e) (3.41f)

We use here the height of the lower layer h (the absolute interface position) as a dependent variable. The last two equations can be directly integrated: u1 = c +

C1 , H −h

u2 = c +

C2 , h

(3.42)

where C1 and C2 are the integration constants. The zero-flux condition u1 (H − h) + u2 h = 0 yields: C1 + C2 = −cH.

(3.43)

The expressions (3.42) are then inserted into (3.41a) and (3.41c), and v1 and v2 are eliminated using (3.41b) and (3.41d). The following single equation for h follows: 2

2 r r C1 1 1 C2 2 − + g h + cf + h 2 h 2 H −h C2 C1 cH +f 2 1 − r − r (3.44) = 0, C1 where r = ρ1 /ρ2 is the ratio of the densities of the layers. By integrating (3.44) over one period of the supposed periodic solution, and by using the fact that the

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

294

integral of the interface position h over the period is H2 by definition, the following constraint results: cH2 cH1 1+ (3.45) −r 1+ =0 C2 C1 which, together with (3.43), allows to fix the values of C1 and C2 : C1 = −cH1 , C2 = −cH2 , and to determine u1 and u2 : u1 = c

−(h − H2 ) , H −h

u2 = c

h − H2 . h

Equation (3.44) then becomes:

2 2 2 c H1 f2 H2 −r (h − H2 ) = 0, + gh − 2 h H −h He

(3.46)

(3.47)

1 H2 is introduced. As in the one-layer case, where the equivalent depth He = HH1 +rH 2 the PV-anomaly in each layer is zero, as may be easily seen by using (3.41b), (3.41d) and (3.46):

q1 =

f , H1

q2 =

f . H2

(3.48)

Introducing the non-dimensional variable χ = h/H , and the notation γ = H2 /H , M = c/ g He , ϕ(r, γ ) = He /H = (1 − γ )γ /(rγ + 1 − γ ) equation (3.47) may be rewritten in the form:

2 M ϕ(r, γ ) γ 2 (1 − γ )2 f2 (3.49) − r − (χ − γ ) = 0. + χ 2 χ2 (1 − χ)2 ce2 This equation is similar to (2.63), but the first term here have singularities both at χ → 0 and χ → 1, i.e. when the interface approaches each of the boundaries. It is easy to check that in the limit H1 → ∞, equation (3.47) takes the form of the equation for nonlinear waves in the one-layer model with the replacements H → H2 , g → g . It can also be seen that the linearization of (3.47) for χ close to γ , η = χ − γ → 0 gives g He − c2 η − f 2 η = 0, (3.50) with the dispersion relation for linear internal inertia–gravity waves at the interface (3.7) which readily follows. After multiplication by the first derivative of the expression in the square brackets, (3.49) may be integrated once and gives: χ2 +

f2 V2 (χ; M, r, γ , A) = 0 ce2

(3.51)

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295

where 1−γ (χ − γ )2 [ϕM 2 ( χγ2 + r (1−χ) 2 ) − 1] − A N2 = , V2 (χ; M, r, γ , A) = 2 2 2 γ (1−γ ) (D2 ) (ϕM 2 ( χ 3 + r (1−χ)3 ) − 1)2

(3.52) is the effective potential, playing the same role as V in the one-layer case. Only the values of χ between 0 and 1 are physically meaningful. As in the one-layer case A > 0 and M > 1. 3.5.2. Analysis of the effective potential The structure of the effective potential is similar to that of the potential V in the one-layer case. However, there are two additional parameters, the ratios of the heights and densities of the two layers, respectively γ and r, and the numerator and denominator now diverge at two points: χ → 0 and χ → 1. In order to have stationary waves, a well in the potential is required about the rest state χ = γ . The following properties of the effective potential may be established (Plougonven and Zeitlin, 2003): The denominator (D2 )2 is always positive; the sign of the potential is given by the sign of the numerator. The numerator N2 tends to +∞ for χ → 0 and for χ → 1. Furthermore, for χ → γ N2 (χ, M, A, r, γ ) ∼ (χ − γ )2 M 2 − 1 − A, (3.53) with A > 0. Hence, N2 has at least one zero in each interval ]0, γ [ and ]γ , 1[. Only the zeros closest to γ , χl and χr , respectively, matter. The possibility of having waves in the system depends on the behavior of the denominator: in order to have wave solutions it is sufficient that the denominator does not have zeros in the interval [χl , χr ]. Note that the derivative of N2 with respect to χ is equal to 2(χ − γ )D2 . Therefore, as in the one-layer case, local extrema of the numerator other than χ = γ coincide with the zeros of D2 . The square root of the denominator, D2 , tends to +∞ at χ → 0 and χ → 1, and has a single minimum in between. Elementary analysis shows that the minimum corresponds to χm (r, γ ) =

1 1/2 1 + r 1/4 ( 1−γ γ )

.

(3.54)

Note that the location of this minimum is a function of parameters r and γ only. If D2 (χm ; M, r, γ ) > 0, the denominator never vanishes in ]0, 1[, and hence wave solutions always exist, whatever the value of A. If D2 (χm ; M, r, γ ) = 0, the denominator vanishes only at the point χm , while if D2 (χm ; M, r, γ ) < 0 it has two zeros, one on each side of χm . It is then necessary to investigate the location of roots of the numerator relative to those of the denominator.

296

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

The existence (non-existence) of wave solutions follows from the existence (non-existence) of zeros in the denominator. The fact that the denominator at the rest position χ = γ is equal to (M 2 −1)2 > 0 is crucial. Two important properties of the denominator are: (1) For any given value of r, there always exists a value of γ such that the denominator never vanishes: this value γ0 (r) is given by χm (r, γ0 ) = γ0 . In this case stationary wave solutions exist for any A. For example, for layers of close densities waves exist for any A at least for the values of γ close to 0.5. (2) If D2 has two zeros, both are situated either in the interval ]0, γ [ or in the interval ]γ , 1[. More precisely, if γ < γ0 (r), both are in ]γ , 1[, and if γ > γ0 (r), both are in ]0, γ [. The location of zeros of D2 determines the orientation of the cusps at the wave extrema. For γ greater than γ0 (r) (or equivalently χm (r, γ ) < γ ), cusps are downward-oriented, for γ below γ0 (r), cusps are upward-oriented. 3.5.3. Wave families and transitions between them with the change of parameters The investigation of different configurations of zeros of the numerator and denominator in (3.52) is given in Plougonven and Zeitlin (2003). Three families of finite-amplitude wave solution are possible: 3.5.3.1. Family A: D2 (χm (r, γ ); M, r, γ ) 0 and γ < γ0 (r). The corresponding waves behave qualitatively in the same manner as waves of the one-layer RSW: there exists a limiting amplitude, the wave profiles tend to develop cusps at their crests as the amplitude approaches the critical one. The typical form and the response of the waves of this family to the increase of amplitude is shown in Figure 12. The cusp is an asymptotic envelope and the solutions are continuous with their derivatives. Qualitatively, waves of the family A arise when the depth of the upper level is considerably larger than the depth of the lower layer (what “considerably” precisely means depends on the values of other parameters). 3.5.3.2. Family B: D2 (χm (r, γ ); M, r, γ ) > 0. In the absence of zeros in the denominator, the potential always has a well around χ = γ , whatever the value of A. Hence, formally, there is no limiting amplitude for the waves other than that imposed by the boundaries. Profiles of the waves of this family for given values of M and r, and for various amplitudes are shown in Figure 13. Note the sharpening of the gradients of the interface displacement with increasing amplitude (a kinklike structure). 3.5.3.3. Family C: D2 (χm (r, γ ); M, r, γ ) 0 and γ > γ0 (r). Qualitatively, these waves are the upside-down version of the waves of the family A. They

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

297

Figure 12. Waves of the family A and their response to the increase of amplitude (from top to bottom). Panel d) shows a wave approaching the limiting amplitude. The values of parameters are H1 = 1, H2 = 2.8, r = 0.1 and M = 2 (Plougonven and Zeitlin, 2003).

are limited by a critical amplitude and tend to form cusps at the troughs as their amplitude increases. They correspond to the situation when the depth of the lower layer is considerably larger than the depth of the upper one. For a given value of r, the domains corresponding to the families A, B, and C in the parameter space (M, γ ) may be obtained by plotting the curve corresponding to the zero level of D2 (χm ; M, r, γ ) as shown in Figure 14. Crossing this curve corresponds to a bifurcation. For example, for given values of γ and r, a transition from the family A to the family B is produced by increasing the phase speed, i.e. the value of M, cf. Figures 15 and 12.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide Due to the absence of the constant part of the Coriolis parameter, the only true geostrophically balanced states at the equator are mean zonal flows. Hence, the geostrophic adjustment of any perturbation localized in the zonal (x-) direction will be wave-dominated, the net result of it being the transformation of the initial perturbation into packets of waves of different nature. Increasing the amplitude of initial perturbation will result in essentially nonlinear wave phenomena. The equatorial waves exhibit a variety of dispersion properties, and nonlinearity will manifest itself differently for, e.g. strongly dispersive and weakly dispersive

298

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 13. Waves of the family B and their response to the increase of amplitude (from top to bottom): these waves ‘feel’ the presence of the rigid lid and are bounded only by the lower and upper boundaries. Parameters are: H1 = 1, H2 = 1, r = 0.9 and M = 2 (Plougonven and Zeitlin, 2003).

waves. We show below that different kinds of waves display different patterns of breaking behavior, where breaking is understood as irreversible changes in the flow due to propagation of a finite-amplitude nonlinear wave. As in the previous sections, in spite of the fact that resonant triadic interactions of weakly nonlinear equatorial waves produce a number of interesting phenomena (Majda, Rosales, Tabak and Turner, 1999; Majda and Biello, 2003; Medvedev and Zeitlin, 2005; Reznik and Zeitlin, 2006) we will limit ourselves below only by strongly nonlinear phenomena. 4.1. Equatorial waves and geostrophic adjustment in the equatorial waveguide As shown in Chapter 1 of the present volume, the equatorial Kelvin waves are dispersionless, the long Rossby waves are weakly dispersive, while the short Rossby waves are strongly dispersive, the short inertia–gravity are almost dispersionless, and eastward-propagating and long westward-propagating Yanai waves are weakly dispersive (cf. Figure 11 of Chapter 1). This variety of dispersion properties indicates that different patterns of behavior may be expected for non-

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

299

Figure 14. Parameter regimes on the (M, γ )-plane corresponding to the waves of family A (bottom region), B (center) and C (top region), for a) r = 0.9 and b) r = 0.1. As the difference between the densities of the two layers increases, the domain in parameter space corresponding to family C shrinks.

linear equatorial waves of different nature. For example, breaking and shock formation can be expected for nonlinear Kelvin waves (Boyd, 1980a; Ripa, 1982). Soliton formation can be expected for long Rossby waves (Boyd, 1980b; Ripa, 1982). Breaking can be expected for short enough inertia–gravity waves, and for short eastward-propagating Yanai waves. Already this first guess indicates that shock formation should be rather common in the dynamics of nonlinear waves in the equatorial RSW model. Moreover, the argument of Section 2 showing that PV distribution cannot be changed by one-dimensional shock is no more applicable at the equator, as the waves are not one-dimensional anymore. So one can expect nontrivial phenomena produced by the PV “deposits” by shocks. Illustrations of nonlinear wave phenomena in the equatorial waveguide are given in the next section. To give a synthetic idea of various nonlinear phenomena in the equatorial waveguide, an example of fully nonlinear geostrophic adjustment of a long-wave initial perturbation is given in Figure 16. The spectral gap in the equatorial wave spectrum, which was discussed in Chapter 1 means that slow (Kelvin and Rossby waves) and fast (inertia–gravity) parts of motion are dynamically well-separated (cf. Le Sommer, Reznik and Zeitlin, 2004) and (almost,

300

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 15. Nonlinear waves of family B for the same values of the model parameters as in Figure 12, except for a slightly higher phase speed (M = 2.1 instead of M = 2), which corresponds to a longer wavelength. The amplitude of the wave on panel a) is deliberately chosen close to that of Figure 12c). The response of the wave to the increase of amplitude, panels b)–e), differs significantly from that on Figure 12. In particular, there is no cusp formation anymore (Plougonven and Zeitlin, 2003).

see below) non-interacting. The antisymmetric Yanai wave signal does not appear for symmetric initial perturbations. In addition, Rossby and Kelvin wave-packets travelling in the opposite directions do not interact (see also below). 4.2. Wave-breaking and shock formation by equatorial waves 4.2.1. Kelvin-wave breaking Shock formation is straightforward in the Kelvin-wave sector, as nothing can prevent breaking in the absence of dispersion. This phenomenon was pinpointed in the early works by Boyd (1980a), and Ripa (1982), and studied numerically by Fedorov and Melville (2000). We give here results of high-resolution numerical simulations following Le Sommer, Reznik and Zeitlin (2004) which used the method exposed in Chapter 4 of the present volume. The evolution of the Kelvin-wave packets corresponding to positive and negative height anomalies, respectively, is presented in Figures 17 and 18. Time (T ) and length (Re ) units in all

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

301

Figure 16. Adjustment of a localized large-scale height anomaly symmetric with respect to equator. Rossby number is 0.3. The Rossby-wave part of initial perturbation is moving west forming a dipolar soliton, the Kelvin-wave part is moving east and breaks down in finite time forming a shock. A packet of inertia–gravity waves centered at the location of the initial perturbation’s maximum (x = 30) is slowly dispersed.

simulation were chosen according to the equatorial scaling explained in Chapter 1, Section 6. The mechanism of the wake formation, observed in Figure 17 was explained in Fedorov and Melville (2000) by a direct resonance of the nonlinear Kelvin wave and inertia–gravity waves. Indeed, due to nonlinearity, the phasespeed of positive-anomaly Kelvin wave is larger than that of linear waves. Hence the effective slope of the dispersive curve for nonlinear waves is greater than for linear waves, and it can intersect the dispersion curves of inertia–gravity waves (see the dispersion curves in Figure 11 of Chapter 1). On the contrary, negativeanomaly nonlinear waves are slower than linear ones, and the effect cannot take place. As seen from these figures, the Kelvin shock-front is curved, and hence one may expect a redistribution of PV due to the shock formation. The idea of a dissipative vorticity change due to discontinuities in gas dynamics has been studied by Hayes (1957) and Berndt (1966). More recently this analysis was generalized by Kevlahan (1997). For the shallow-water equations, a potential vorticity jump

302

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 17. Evolution of a strongly nonlinear Kelvin wave with positive height anomaly (initial Rossby number 0.3). The wave breaks down in finite time (∼10T ) forming a shock. Inertia–gravity wave emission starts at this moment behind the front and continues on.

formula was derived by Pratt (1983) in the case of straight shocks moving with a constant speed. In all of these works the along-shock momentum equation was written in a local reference frame in order to calculate the vorticity jump. Following this prescription we consider a shock (S) propagating through a material volume V (t) and we introduce a local reference frame with normal and tangential unit vectors ( n, s) with respect to the shock. This frame is moving at the local speed of the shock Cn n. The velocity and Bernoulli function in the moving frame are: (n) (n) u − Cn u¯ = gh + u¯ 2 /2. = , B u¯ = (4.1) u¯ (s) loc u(s) loc With a notation [A] = Afront − Arear the Rankine–Hugoniot conditions are: −Cn [h] + hu(n) = 0,

(n) 1 2 (n)2 −Cn hu + gh = 0, + hu 2

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

303

Figure 18. Evolution of a strongly nonlinear Kelvin wave with negative height anomaly (initial Rossby number −0.3). The wave breaks down in finite time (∼10T ) forming a shock. No inertia–gravity wave emission is observed.

−Cn hu(s) + hu(n) u(s) = 0,

(4.2)

where u(n,s) are the velocity components normal and tangential with respect to the is obtained from the first and the second of the Rankine– shock. The jump in B Hugoniot conditions (4.2); see e.g. Lighthill (1978), where the subscript f (r) denotes the front (the rear) state: 3 = − g [h] . B 4 hf hr

(4.3)

Following the aforementioned works, we obtain the potential vorticity jump across the shock from the along-front momentum equation: . hu¯ (n) [q] = −∂s B (4.4) For moving shocks of any shape: [q] =

g [h]3 ∂ . s 4hu¯ (n) hf hr

(4.5)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

304

Figure 19.

A sketch of a shock front which propagates in a material volume V at local speed Cn n, and the local reference frame ( n, s).

Note that for shocks with ∂s [h] no change of potential vorticity is possible. The rate of change of the amount of vorticity contained in any material volume V may be, thus, calculated: d − B . hq dV = − hu[q] ¯ dS = B (4.6) A B dt V

(S)

Figure 20 taken from Le Sommer, Reznik and Zeitlin (2004) shows the breaking of a Kelvin wave of depression and the subsequent PV-deposit. Choosing V to be the whole southern half-plane, the formula (4.6) suggests that the overall negative potential vorticity will arise there. The right panel of the figure confirms this result. It should be emphasized that such dipolar deposit of PV results in the corresponding zonal jet behind the shock. A theory of vorticity transport due to breaking waves in rotating shallow water on the f -plane was developed by Bühler (2000). It was based on generalized Lagrangian-mean theory (GLM) and confirmed by numerical simulations. However, its generalization to fully nonlinear processes on the (equatorial) β-plane is difficult due to the problem of extending the standard GLM pseudomomentum definition beyond the f -plane (cf. Bühler and McIntyre, 1998). 4.2.2. Yanai and inertia–gravity waves breaking Direct numerical simulations with finite-amplitude sinusoidal waves of a given family taken as initial conditions confirm the qualitative predictions made at the beginning of this subsection. Thus, eastward propagating Yanai wave, as well as eastward and westward propagating inertia–gravity waves break and form specific

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 20.

305

Potential vorticity deposit in the wake of a broken Kelvin wave at Ro = 0.6. Left panel: v −u +βy

the height field. Right panel represents the PV-anomaly q = x Hy the interval (−0.2, 0.2) are shown.

βy − H . The isolines of q in 0

patterns of shocks displayed in Figures 21, 22, 23 taken from Bouchut, Le Sommer and Zeitlin (2005). The shock zone is defined by the enhanced dissipation and grey-shadowed in the figures. In these and subsequent figures the height anomaly isolines (solid for positive and dashed for negative anomalies), and velocity field (arrows) are shown. The measure of nonlinearity is the Rossby number ε. It should be emphasized that breaking and shock formation are not limited to spatially periodic waves. They are ubiquitous at already moderate nonlinearities and appear during the evolution of localized wave-packets as well, as shown in Figure 24 where a snapshot of the evolution of a packet of eastward-propagating inertia–gravity waves with a non-dimensional amplitude ε = 0.3 is presented. Shocks are formed and propagate across the whole wave pattern. 4.3. Solitons and modons formed by equatorial Rossby waves The soliton formation by long equatorial Rossby waves was predicted by Boyd (1980b), and Ripa (1982). Boyd (1985) has also predicted the existence of equa-

306

Figure 21.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Finite-amplitude eastward-propagating Yanai wave—upper panel, which breaks and forms an asymmetric chevron-type shock pattern—lower panel.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 22.

307

Finite-amplitude eastward-propagating inertia–gravity wave—upper panel, which breaks and forms a symmetric chevron-type shock pattern—lower panel.

308

Figure 23.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Finite-amplitude westward-propagating inertia–gravity wave—upper panel, which breaks and forms a horseshoe-type shock pattern—lower panel.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 24.

309

Snapshot of an eastward-propagating inertia–gravity wave packet with k0 = 1.0/Rd at time t = 4.0. The shock-front area is shaded.

torial modons, the finite-amplitude Rossby solitons. The analytic description of adjustment of the long-wave equatorial perturbations given in Chapter 2 of this volume (see also Le Sommer, Reznik and Zeitlin, 2004) shows that the Rossby-wave part of the response to a localized perturbation is described by the Korteweg–de Vries (KdV) equation and, thus, leads to appearance of the solitons. High-resolution numerical simulations confirm that a dipolar initial perturbation forms a typical soliton sequence consistent with the predictions of the KdV theory, Figure 25. Note that, in full agreement with the KdV theory results, the soliton tail is formed only for positive initial height anomalies. Negative height anomalies disperse and do not form coherent structures (Bouchut, Le Sommer and Zeitlin, 2005). It should be emphasized that such typically KdV behaviour holds for strong nonlinearities here, in spite of the fact that the KdV theory is formally valid for weak nonlinearities only. What is more surprising, a build-up in amplitude for the leading soliton with respect to the initial perturbation is observed. As shown below, at such amplitudes the solitons become modons, i.e. closed streamlines are formed in the dipole core and, therefore, a trapping of material within the vor-

310

Figure 25.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Long-time evolution of a nonlinear dipolar perturbation. Initial perturbation forms a sequence of modons of diminishing amplitude and phase-speed.

tex core, and subsequent anomalous transport are taking place—see also the next subsection. The transition between solitons and modons may be demonstrated by considering nonlinear equatorial Rossby waves in a fluid layer of varying depth. Let us consider, following Le Sommer and Zeitlin (2005), and Bouchut, Le Sommer and Zeitlin (2005) the evolution of an equatorial Rossby soliton in a fluid layer with for x > 0 smoothly decreasing thickness. We take the linear slope between h = H for x −20. The initial condition is a zeroth order Boyd (1980b) and h = 0.4H soliton centered at x = 15 with B = 0.25. In order to capture eventual fluid particle trapping in the soliton core, the evolution of the passive tracer field is calculated during numerical simulation together with the fluid velocity and elevation. The tracer field is taken to be complex (i.e. there are, in fact, two real tracer fields ar and ai ), and initialized in order that initial tracer contours form a rectangular grid easy to follow in the subsequent evolution: a(x, y, 0) = ar + iai = x + iy. The evolution of the height (solid contours), velocity (arrows) and passive tracer (gray contours) is shown in Figure 26. The soliton is initially propagating westward, as expected. At this stage, the tracer field is advected reversibly. Then the

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Propagation of a solitary Rossby wave on a sloping topography and related Lagrangian tracer transport.

311

Figure 26.

312

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

soliton enters the sloping topography region. Both its phase speed and meridional extent are weakening as may be expected from the WKB analysis (Yang and Yu, 1992). During this phase a partial reflection in a form of low-amplitude Kelvin wave takes place. (It is below the threshold of the displayed h-perturbation and not visible in the figure.) The tracer contours are getting more elongated as the wave propagates. Finally the wave enters the left flat region and keeps a quasisteady shape shown at time t = 150. The soliton is now enclosing a recirculation region. It has been transformed into a modon. The details of the mass-trapping process are visible. Contrary to the intuitive expectations, mass (passive tracer) trapping does not occur at the maxima of h around which velocity vectors are circling due to the geostrophic balance, but rather in the vicinity of the saddle point at the equator where h has a local minimum. Thus, when a solitary Rossby wave propagates over a sloping topography, it “breaks” and forms a modon which encloses a recirculation region. This recirculation region is trapped within the wave pattern so that the underlying particle distribution is irreversibly modified. This allows to qualify such process as breaking, where breaking is understood as irreversible changes of the flow due to nonlinearity. Such breaking is of a totally different nature than shock formation observed above for Kelvin, inertia–gravity and Yanai waves. The difference between the Rossby waves and the other types of equatorial waves is that Rossby waves are essentially balanced, i.e. their meridional velocity is in geostrophic balance with the height field, while other types of waves (except for short westward-moving Yanai waves) are essentially unbalanced, as shown in Bouchut, Le Sommer and Zeitlin (2005). Hence, balanced and unbalanced waves display qualitatively different patterns of breaking. 4.3.1. Elastic scattering of Rossby solitons and Kelvin fronts Perhaps the most striking phenomenon observed in the numerical simulations of Bouchut, Le Sommer and Zeitlin (2005) was quasi-elastic scattering of Rossby solitons and Kelvin fronts showing the robustness of the patterns emerging from breaking equatorial waves. The initial condition is taken as a superposition of two distinct patterns. A Kelvin wave-packet with Gaussian zonal structure is centered at x = 10.0. Its zonal extent is Lx = 3Rd and its non-dimensional amplitude is ε = −0.3. A Kelvin wave-packet with negative height anomaly was deliberately chosen in order to have no inertia–gravity waves tail appearing for positive height anomalies (cf. Figure 17). A solitary Rossby wave was initialized at x = 45.0 with amplitude ε ∼ 0.18. Snapshots of the evolution of this configuration are shown on Figure 27. At time t = 15.0, a shock front has formed at the rear of the Kelvin wave packet as in Figure 18. Dissipation in the shock induces a slow decrease of the Kelvin wave amplitude as it is propagating eastward. On the other hand, the solitary Rossby wave is moving westward with no change of shape. At time t = 30.0, the two wave patterns collide but the interaction is quasi-elastic, as seen

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 27.

313

Interaction between a Kelvin wave and a Rossby solitary wave. See the text for explanation.

at time t = 45.0. The elastic character of collision can be checked by comparing the results at time t = 45.0 of two different runs respectively with and without the initial Kelvin wave (or with and without initial Rossby wave). The comparisons show that results are practically identical. Such almost-non-interaction between Kelvin and Rossby wave was observed for a wide range of parameters. The only observed effect was a slight phase-lag in the solitary Rossby wave propagation for high enough amplitudes ε > 0.5. 4.3.2. Interaction of Rossby solitons with inertia–gravity waves In order to test the interactions of Rossby solitons and inertia–gravity waves the evolution of a “soliton” with non-dimensional amplitude ∼0.3 arising in the adjustment process of the type shown in Figure 16 was followed in Le Sommer, Reznik and Zeitlin (2004). A typical evolution of the height field of this part of the perturbation is presented in Figure 28. A field of inertia–gravity waves of weak intensity is seen in front of the soliton, while a (mostly Rossby-wave)

314

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 28. Inertia–gravity waves and a Rossby-wave soliton with Ro ∼ 0.3. Upper panel—the height field with isolines h = 0.99, 1.01, 1.1, 1.2, 1.3 shown. Lower panel: a section of the height field along the equator. Inertia–gravity waves in front of the westward-propagating soliton are clearly seen. Behind the soliton there is a wake of weak Rossby waves

wake is formed behind. Although the modons or “solitons” like the one presented on Figure 28 are quite stationary, this simulation suggests that they emit or capture inertia–gravity waves. The precise origin of the observed IGW cannot be determined at this stage. Several mechanisms may be at work, and to distinguish among them further investigations are necessary. As possible explanations we should mention a direct resonance of the soliton with the west-moving inertia– gravity waves (like in the Kelvin front case; however the time-evolution of the field presented in Figure 28 suggests that it is the group velocity of the inertia– gravity wave packet which coincides with the speed of the Rossby solitons, while phase velocity of the inertia–gravity waves is higher), a non-local resonant triad interaction of the type described by Ripa (1982), and a synchronization of IGW produced by adjustment of the initial perturbation while they pass through the modon. 4.4. Transport and mixing phenomena during the equatorial geostrophic adjustment The mixing and transport properties by nonlinear equatorial waves formed during the adjustment process may be studied by the same method which led

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

315

to the identification of mass-trapping by the equatorial modons (Le Sommer and Zeitlin, 2005). The passive tracer behaviour during the adjustment of localized mass anomalies and current anomalies at the equator may be thus obtained. 4.4.1. Transport and mixing during adjustment of a symmetric mass anomaly A Gaussian mass anomaly centered at the equator at x = 0 with the amplitude ε = 0.5 is taken as initial condition. Its zonal extent is Lx = 3 and the meridional extent is Ly = 1. The subsequent evolution of this anomaly is illustrated in Figure 29. The initial field is immediately split up into a Kelvin wave, a Rossby wave and a gravity wave packet, as seen at time t = 15. These three components then evolve quasi-independently. The Kelvin wave is propagating eastward. During this propagation its gradients are steepening and a Kelvin front is formed. This wave is rapidly leaving the computation domain and the resulting effect on the tracer field consists in a reversible Stokes drift with the same meridional structure as predicted by Li, Chang and Pacanowski (1996). The Rossby wave is propagating westward. As expected in this parameter regime, it encloses a recirculation region so that the tracer isolines are modified irreversibly during the adjustment, see the snapshot at t = 30. Furthermore, the tracer field never attains a steady structure, which is significantly different from the behavior predicted by the small amplitude Stokes drift analysis. Finally, a gravity wave packet is left at the location of the initial perturbation. Since it is mainly composed of inertia–gravity waves propagating eastward, its effect should be to induce an eastward drift at the equator. This effect is not visible in the figure due to its small amplitude It is worth noting that the Kelvin-wave induced drift and the Rossby wave trapping are acting in the opposite directions so that the tracer at the location of the perturbation is stretched in the x-direction as seen at time t = 45. Another interesting fact is the complicated spatial structure of the isolines of the tracer in the core of the Rossby wave. We can therefore expect a strong mixing to occur in this region if diffusion is added. Hence, the three main transport phenomena due to adjustment are (i) the ballistic transport of the tracer toward the west, (ii) tracer stretching at the perturbation location and (iii) an intense stirring in the Rossby wave pattern. 4.4.2. Transport and mixing during adjustment of a zonal current anomaly A localized zonal current anomaly centered at the equator (produced e.g. by a sudden wind blast) is considered. This situation is typical for the ocean which is submitted to intense zonal winds in the tropics. The initial condition is √ chosen to be a Gaussian zonal velocity patch with maximum velocity u = −0.5 gH . As

316

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 29.

Transport and mixing during adjustment of a mass anomaly centered on the equator. See the text for details.

5. Summary and conclusions

317

in the previous simulations the zonal and meridional extents are set to be Lx = 3 and Ly = 1, respectively. Figure 30 shows the evolution of the dynamical variables and the advection of the tracer field. As previously, the perturbation is split up into a Kelvin wave, a Rossby wave and a gravity wave packet. The Kelvin wave is now an upwelling wave, which signature is a reduction of the depth. Therefore, the gradients are concentrated at its rear front (cf. Le Sommer, Reznik and Zeitlin, 2004, and above). At time t = 15, it induces a reversible westward drift. This drift was not predicted by previous studies which considered only the drift induced by harmonic waves. This fact emphasizes the importance of considering finite amplitude localized wave patterns while studying the transport properties. Again, the Rossby wave is trapping fluid which is then advected westward. As in the previous simulations, the tracer field is stretched along the equator and stirred at the Rossby wave location.

5. Summary and conclusions Two archetype phenomena characterize the essentially nonlinear wave dynamics in the RSW model on the f -plane, where only a single type of waves (IGW) is present. The first is shock formation due to overturning. We have rigorously demonstrated that rotation does not prevent shock formation for large classes of initial data. Shock formation is ubiquitous in the geostrophic adjustment, i.e. initial value problems. It leads to efficient energy evacuation, allowing to reach the adjusted states, whose existence and uniqueness was proved for non-negative PV configurations. Shock formation does not qualitatively change the classical adjustment scenario in the simple 1.5-dimensional configurations. The second is existence of exact nonlinear stationary wave solutions of specific form, and their emergence from periodic signals. These waves were shown to be the finite-amplitude counterparts of the infinitesimal harmonic IGW. Numerical evidence was presented in favor of the hypothesis that they are attractors in the constant PV-sector of the model. New effects are introduced by baroclinicity in the two-layer RSW model. Although existence and uniqueness of the adjusted states may be proved for nonnegative PV configurations with some minor additional constraints in the 1.5d version of the model, the attainability of such states is compromised by appearance of the trapped modes for strong enough horizontal anticyclonic shears. These modes become unstable for negative PV and such symmetric instability may give rise to new kind of breaking. In any case, one cannot expect that negative PVconfigurations in the two-layer case follow the same scenario of adjustment as the positive PV ones, unlike the one-layer case, where this fact was demonstrated by direct numerical simulations. The Kelvin–Helmholtz instability which appears

318

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 30.

Adjustment of a zonal current anomaly centered on the equator. See the text for details.

References

319

for strong enough vertical shears (at positive PV) leads to the loss of hyperbolicity of the system and a new kind of breaking, too. The exact nonlinear wave solutions in the two-layer models appear in form of three different families, two of them (up to reflection) similar to the one-layer ones, and a new one which has a kink-like form with narrow zones of strong gradients of the interface displacement. Beyond the f -plane approximation, the differential rotation effects, appearing in their most spectacular form in the equatorial region, introduce new type of waves, Rossby waves, with specific dispersion which allows to compensate weak nonlinearities for long enough waves through the KdV mechanism, and leads to the soliton formation. Such solitons, however, survive even at severe nonlinearities by becoming modons through the process of “dispersive breaking”, which leads to reconnection of the streamlines and formation of closed recirculation zones moving with the wave. At the same time, weakly dispersive equatorial waves exhibit “standard” shock formation by overturning. This process is most pronounced for specifically equatorial Kelvin waves which are strictly dispersionless. The geometry of the shock pattern depends on the nature of the waves. Shock fronts are, generally, curved and produce recirculation zones, i.e. the PV deposit, behind them. Nonlinear wave phenomena at the equator are accompanied by nontrivial transport and mixing processes: ballistic transport by modons and specific mixing patterns related to the wave-breaking.

Acknowledgements This review is based on the work done in collaboration with F. Bouchut, J. Le Sommer, S. Medvedev, R. Plougonven, and G. Reznik, which is gratefully acknowledged. Special thanks are to J. Le Sommer and R. Plougonven for their help in preparation of the figures.

References Berndt, S.B., 1966. The vorticity jump across flow discontinuity. J. Fluid Mech. 26, 433–436. Bouchut, F., 2004. Nonlinear Stability of Finite Volume Methods for Hyperbolic Systems of Conservation Laws, and Well Balanced Schemes for Sources. Frontiers in Mathematics. Birkhäuser, Basel. Bouchut, F., Le Sommer, J., Zeitlin, V., 2004. Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. Numerical simulations. J. Fluid Mech. 514, 35–63. Bouchut, F., Le Sommer, J., Zeitlin, V., 2005. Breaking of balanced and unbalanced equatorial waves. Chaos 15, 013503-1–013503-19. Boyd, J.P., 1980a. Equatorial solitary waves. Part 1: Rossby solitons. J. Phys. Oceanogr. 10, 1699– 1717. Boyd, J.P., 1980b. The nonlinear equatorial Kelvin wave. J. Phys. Oceanogr. 10, 1–11.

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Boyd, J.P., 1985. Equatorial solitary waves. Part 3: Westward travelling modons. J. Phys. Oceanogr. 15, 46–54. Bühler, O., 1993. A nonlinear wave in rotating shallow water, GFD Summer School preprint (Woods Hole), unpublished. Bühler, O., 2000. On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263. Bühler, O., McIntyre, M.E., 1998. On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343. Engelberg, S., 1996. Formation of singularities in the Euler and Euler–Poisson equations. Physica D 98, 67–74. Falkovich, G.E., Medvedev, S.B., 1992. Europhys. Lett. 19, 279–284. Fedorov, A.V., Melville, W.K., 2000. Kelvin fronts on the equatorial thermocline. J. Phys. Oceanogr. 30, 1692–1705. Gill, A.E., 1982. Atmosphere–Ocean Dynamics. Academic Press, New York (Chapter 7.2). Gossard, E.E., Hooke, W.H., 1975. Waves in the Atmosphere. Elsevier, Amsterdam. Grimshaw, R.H.G., Ostrovsky, L.A., Shrira, V.I., Stepanyants, Yu.A., 1998. Long nonlinear surface and internal gravity waves in a rotating ocean. Surveys in Geophys. 19, 289–338. Hayes, W.D., 1957. The vorticity jump across a gas-dynamic discontinuity. J. Fluid Mech. 2, 595–600. Holton, J.R., 1979. An Introduction to Dynamic Meteorology. Academic Press, San Diego, CA. Houghton, D.D., 1969. Effect of rotation on the formation of hydraulic jumps. J. Geoph. Res. 74, 1351–1360. Kevlahan, N.K.-R., 1997. The vorticity jump across a shock in a non-uniform flow. J. Fluid Mech. 341, 371–384. Kuo, A.C., Polvani, L.M., 1997. Time-dependent fully nonlinear geostrophic adjustment. J. Phys. Oceanogr. 27, 1614–1634. Kuo, A.C., Polvani, L.M., 1999. Wave–vortex interactions in rotating shallow water. Part 1. One space dimension. J. Fluid Mech. 394, 1–27. Landau, L.D., Lifshits, E.M., 1975. Hydrodynamics. Academic Press, San Diego, CA. Lax, P.D., 1973. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, PA. Le Sommer, J., Medvedev, S.B., Plougonven, R., Zeitlin, V., 2003. Singularity formation during the relaxation of jets and fronts towards the state of geostrophic equilibrium. Commun. Nonlinear Sci. Numer. Simul. 8, 415–442. Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170. Le Sommer, J., Zeitlin, V., 2005. Tracer transport during the geostrophic adjustment in the equatorial ocean. In: Collet, P., et al. (Eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems. In: NATO Science Series C. Kluwer, Amsterdam, pp. 413–429. Li, X., Chang, P., Pacanowski, R.C., 1996. A wave-induced stirring mechanism in the mid-depth equatorial ocean. J. Mar. Res. 54, 487–520. Lighthill, J., 1978. Waves in Fluids. Cambridge Univ. Press, Cambridge, UK. Majda, A.J., Biello, J.A., 2003. The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci. 60, 1809–1821. Majda, A.J., Rosales, R.R., Tabak, E.G., Turner, C.V., 1999. Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances. J. Atmos. Sci. 56, 4118–4133. Medvedev, S.B., Zeitlin, V., 2005. Weak turbulence of short equatorial waves. Phys. Lett. A 342, 217–227. Migdal, A.B., 1977. Qualitative Methods in Quantum Theory. Frontiers in Physics, vol. 48. Benjamin, Reading, MA. Nof, D., 1984. J. Phys. Oceanogr. 14, 1683–1702.

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Ostrovsky, L.A., 1978. Nonlinear internal waves in the rotating ocean. Oceanology 18, 119–124. Peregrine, D.H., 1998. Surf zone currents. Theoret. Comput. Fluid Dyn. 10, 295–309. Plougonven, R., Zeitlin, V., 2003. On finite-amplitude stationary inertia–gravity waves propagating without change of form along the sharp density gradients. Phys. Lett. A 314, 140–149. Pratt, L.J., 1983. On inertial flow over topography. Part 1. Semi-geostrophic adjustment to an obstacle. J. Fluid Mech. 131, 195–218. Pratt, L.J., 1984. On inertial flow over topography. Part 2. Rotating channel flow near the critical speed. J. Fluid Mech. 145, 95–110. Reznik, G.M., Zeitlin, V., 2006. Resonant excitation of baroclinic Rossby waves in the equatorial waveguide and their nonlinear evolution. Phys. Rev. Lett. 96 (3), 034502. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part I. Rotating shallow water. J. Fluid Mech. 445, 93–120. Ripa, P., 1982. Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial β-plane. J. Phys. Oceanogr. 12, 97–111. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res. 1, 239–263. Rozhdestvenskii, B.L., Janenko, N.N., 1983. Systems of Quasilinear Equations and Their Applications to Gas Dynamics. AMS Translation of Mathematical Monographs, vol. 55. Amer. Math. Soc., Providence, RI. Schär, C., Smith, R.B., 1993. Shallow water flow past isolated topography. Part 2: Transition to vortex shedding. J. Atmos. Sci. 50, 1401–1412. Shrira, V.I., 1981. Propagation of long nonlinear waves in the layer of rotating fluid. Sov. Phys.— Izvestija, Atmos. Ocean Phys. 17 (1), 55–59. Shrira, V.I., 1986. On the long strongly nonlinear waves in rotating ocean. Sov. Phys.—Izvestija, Atmos. Ocean Phys. 22 (4), 298–305. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Wolfram, S., 2004. The Mathematica Book. Cambridge Univ. Press, Cambridge, UK. Yang, J., Yu, L., 1992. Propagation of equatorially trapped waves on a sloping thermocline. J. Phys. Oceanogr. 22, 573–582. Zeitlin, V., Medvedev, S.B., Plougonven, R., 2003. Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory. J. Fluid Mech. 481, 269–290.

Chapter 6

Experimental Reality of Geostrophic Adjustment A. Stegner Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. The Holy Graal of rotating shallow-water flows

323 324 329 331 334 339 340 343 350 352 352 358 368 375 376 377

1.1. Single layer f -plane configuration 1.2. Influence of the centrifugal force 1.3. Non-hydrostatic wave modes 1.4. Two-layer stratification

2. Potential vorticity measurements: a new challenge 2.1. Particle image velocimetry and vorticity field measurements 2.2. Height field measurements 2.3. Potential vorticity measurements

3. Simple case studies of geostrophic adjustment 3.1. “Warm-core” lens 3.2. Cyclonic and anticyclonic PV patches 3.3. Uniform PV front

4. What do we learn from laboratory experiments? Acknowledgements References

1. The Holy Graal of rotating shallow-water flows The rotating shallow water model (RSW) is probably the most pedagogical and useful model to understand geophysical fluid dynamics. Even if the RSW equations are based on drastic assumptions (hydrostatic balance, quasibidimensionality, weak dissipation) it is a surprisingly good model of many Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02006-5 323

© 2007 Elsevier B.V. All rights reserved

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phenomena in the atmosphere and the ocean. Nevertheless, as far as laboratory experiments are concerned, one should keep in mind that the dynamics of a rotating and stratified fluid is given by the full three-dimensional Navier–Stokes equations at the final place. As it was shown in Chapter 1, the RSW equations can be derived from the primitive equations according to an asymptotic expansion which indeed remains valid for some restricted range of dynamical parameters. However, this derivation starts from the hydrostatic and non-dissipative primitive equations (Chapter 1), while both hydrostatic and dissipative effects could play a role in the laboratory. Hence, we recall in this section the derivation of the RSW model from the Navier–Stokes equations. The main purpose is to understand here which dynamical processes are filtered out by the RSW model while they occur sometimes in real experiment. Moreover, we will try to specify the values of the dynamical parameters needed to be achieved in laboratory experiments in order to be close to the RSW dynamics. 1.1. Single layer f -plane configuration Let us consider first a single barotropic and incompressible fluid layer in a rotating tank with a flat bottom and a free upper surface, as shown in Figure 1. In order to get a dimensionless set of equations we use: L and H0 as horizontal and vertical scales, T the characteristic time-scale for the flow evolution, U and U (H0 /L) as horizontal and vertical velocity scales, ρgH0 as the characteristic hydrostatic pressure scale and ρf U L the scale of pressure deviation from hydrostatic balance (f = 2Ω0 the Coriolis parameter). Using this dimensionless formulation, the Navier–Stokes equations can be written as follows: ε∂t u + RoDu − v = −∂x π + Ek u,

(1)

ε∂t v + RoDv + u = −∂y π + Ek v,

(2)

Figure 1. Single water layer on a rotating turntable H0 = 10 cm and D = 90 cm.

1. The Holy Graal of rotating shallow-water flows

α 2 [ε∂t w + RoDw] = −∂z π + α 2 Ek w

325

(3)

where D = u∂x + v∂y + w∂z and = ∂z2 + α 2 (∂x 2 + ∂y 2 ). Besides, in this formulation we decouple the hydrostatic pressure PH corresponding to the fluid at rest and the dynamical pressure π (pressure deviation induced by the fluid motion) according to: Ro (4) π(x, y, z, t) Bu where PH (z) = 1−z+P0 and P0 is the dimensionless pressure at the free-surface. In addition one should consider the continuity equation P = PH (z) +

∂x u + ∂y v + ∂z w = 0

(5)

with upper (z1 = 1 + λη) and lower (z0 = 0) boundary conditions: u(z0 ) = v(z0 ) = w(z0 ) = 0, (6) Ro w(z1 ) = λ ε∂t η + Ro(u∂x η + v∂y η) , (7) λBu π(x, y, z1 , t) = (8) η(x, y, t) Ro where η(x, y, t) is the dimensionless deviation of the free-surface. We have introduced in this formulation the following non-dimensional parameters: √ √ ν • The Ekman number Ek = f H Ek H0 = ν/f 2 fixes the vertical scale δE = of the viscous Ekman layer, where ν is the fluid viscosity. According to the standard boundary layer theory, this viscous layer cannot be neglected at the bottom boundary where the no-slip condition (6) must be satisfied (Gill, 1982; Pedlosky, 1987; Vallis, 2006). In the laboratory, the thickness of this boundary layer is fixed only by the rotation rate Ω0 = 2π/T0 . For typical values of Ω0 1–10 rpm we get δE 1–2 mm. Hence, as far as H0 δE , we usually neglect viscous effects in the upper part of the fluid layer (z 2 − 3δE ). Nevertheless, the Ekman layer forces a secondary re-circulation which induces an efficient transfer of angular momentum from the boundary to the whole fluid domain (Greenspan, 1968). For a fluid layer close to the geostrophic balance √ the √ characteristic time of this Ekman pumping is TE = H0 / νf = T0 /(4π Ek ) (Pedlosky, 1987). Therefore, if we want to neglect this dissipative process over at least several rotation periods T0 , the Ekman number should be quite small Ek 10−4 . Such values can be easily reached if the fluid layer is thick enough. • The aspect ratio parameter α = H0 /L. While this parameter is generally small for synoptic atmospheric or oceanic structures (α 10−2 –10−3 ), this is not always the case in laboratory experiments. Indeed, we could hardly work with

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ultra-thin layers. The first limitation is due to the surface tension which acts on a millimeter scale. The second constraint is due to the Ekman pumping described above. Hence, typical layer depths in rotating tank experiments are about few tens of centimeters H0 10–50 cm. Therefore, in order to get a small aspect ratio (at least α 0.1) the characteristic horizontal scales should be about L 1–5 m. To study the dynamics of multiple structures or to avoid the end effects, the experiment should then be done on a very large turntable D 10 m. A unique installation reaches such large scale (D = 13 m), the Coriolis turntable1 in Grenoble, France. However, for medium size experiments (D 1–2 m) the aspect ratio parameter cannot be asymptotically small and is often close to unity α 1. • The classical Rossby number Ro = fUL characterizes the importance of rotation in the fluid layer. In order to be consistent, the horizontal scale L in the Rossby number Ro should correspond to the characteristic scale of the horizontal velocity gradient. In other words, the Rossby number quantifies the ratio of the relative vorticity ζ = ∂x v − ∂y u with respect to the planetary vorticity (Ro ζ /f ). For large-scale flows, the Rossby number is generally small or finite in the atmosphere and the ocean, leading to the geostrophic balance. It will exceed unity only for very intense vortices, such as hurricanes. This parameter is usually well controlled in a rotating experiment and both small and large values could be obtained. Nevertheless, to reach large values which would exceed unity an external forcing is generally needed. Indeed, without external energy source, from any initial state the geostrophic adjustment process will quickly lead to a mean flow in geostrophic balance which implies small or finite Rossby numbers (Ro 1). • We introduce here the time evolution parameter ε = f1T . This parameter quantifies the dynamical evolution of the flow. It depends on the flow response to the initial condition or to the external forcing. Hence, this parameter cannot be fixed by the experimental setup. Classical textbooks (Gill, 1982; Pedlosky, 1987) usually consider the case of large-scale and slow advective motion and therefore the time-evolution parameter and the Rossby number are fixed to be of the same order ε Ro. However, for high frequency linear waves (i.e. short gravity-waves) ε 1 and Ro 1 while for intense cyclones Ro 1 and ε 1. Hence, as far as experiments on geostrophic adjustment are concerned, it is useful to consider both the cases of slow (ε 1) advective motion or fast (ε 1) wave motion independently of the√Rossby number value. • The Burger number Bu = (Rd /L)2 where Rd = gH0 /f is the Rossby deformation radius. As far as we consider a relatively thick layer H0 10–50 cm and a relatively slow rotation rate Ω0 1–10 rpm we get a large deformation radius Rd 50 cm which is usually close to the size of the experimental apparatus. Hence, with a single barotropic layer, we can hardly obtain 1 http://www.coriolis-legi.org.

1. The Holy Graal of rotating shallow-water flows

327

small Burger number values. Only few experiments using high rotation speed (Ω0 60 rpm) reached small Burger number value within a single layer configuration. However, in such case a parabolic vessel is needed to compensate the resulting parabolic shape of the free-surface (Nezlin and Snezhkin, 1993; Stegner and Zeitlin, 1998; van de Konijnenberg, Nielsen, Rasmussen and Stenum, 1999). For such setups the strong curvature of the fluid layer induces, as in the spherical planetary geometry, a strong beta effect. Hence, such parabolic configurations are relevant for modelling large-scale planetary flows, as the Jovian atmosphere for instance. • The relative elevation parameter λ = η0 /H0 where η0 is the characteristic amplitude of the free surface deviation. When the flow is close to geostrophic balance, namely when the dynamical pressure gradient ∇π is balanced at the same order by the Coriolis force, the relative geopotential deviation λ depends on both the Rossby and the Burger number λ ∼ Ro/Bu (cf. Chapters 1 and 2). The RSW model is based on three main approximations: weak dissipation, hydrostatic balance and quasi-bidimensionality of the horizontal velocity. We discuss, in what follows, when and in which range of dynamical parameters these approximations could be valid or not. We first assume the hydrostatic balance for the whole pressure field. The vertical acceleration in (3) could be neglected if both α 2 Ro 1 and α 2 ε 1. Note that for rotating flows, the shallow-water constraint (α 1) is not necessary to get the hydrostatic balance. Indeed, a weakly viscous (Ek 1) slow (ε 1) and geostrophic (Ro 1) flow will follow the hydrostatic balance even if the aspect ratio parameter α is finite. Hence, hopefully for the experimentalists, quasi-geostrophic motions can be accurately reproduced in a rotating tank while α 1. Moreover, the shallow-water constraint is not a sufficient condition that guarantees the hydrostatic balance. Indeed, if the system supports high-frequency waves (ε 1) they could be a source of non-hydrostatic motion or instability. Besides, the case of intense (Ro 1) shallow-water (α 1) vortices or jets is also complex. These intense structures could exhibit in anticyclonic vorticity region an inertial or centrifugal instability which generates three-dimensional and nonhydrostatic perturbations within the large scale flow (Teinturier, Stegner, Viboud, Didelle and Ghil, 2006). Such short-wavelength instabilities could amplify smallscale perturbations (having a finite aspect ratio αp 1) with a rapid growth rate (εp 1). In such case, the cyclonic vorticity regions may satisfy the hydrostatic balance, while intense non-hydrostatic motion occurs in the anticyclonic regions (cf. Figure 5 below). However, when the evolution of a shallow-water flow (or its unstable perturbations) is not fast (ε 1) the hydrostatic balance is satisfied at the first order of approximation and equation (3) leads to: ∂z π = 0.

(9)

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Then according to (8) the dynamical pressure becomes directly proportional to the free-surface geopotential deviation: π(x, y, t) = η(x, y, t).

(10)

We then assume that the fluid layer experiences a weak dissipation. The viscous terms in equations (1)–(2) could be neglected at a first order of approximation if the Ekman number is small enough Ek 1. However, we cannot totally suppress the no-slip condition at the bottom and we should introduce an Ekman layer. This layer will then change the bottom boundary conditions for the upper inviscid layer. It will allow a free-slip condition for the horizontal velocities (u(z0 ), v(z0 )) but will induce a non-zero vertical velocity. In the case of hydrostatic and slow geostrophic motions this vertical velocity is proportional to the horizontal flow vorticity (Pedlosky, 1987; Vallis, 2006) and the boundary conditions (6) should be replaced by: Ek Ek (∂x v − ∂y u) = ζ w(z0 ) = (11) 2 2 where ζ is the vertical component of vorticity. Hence, the√ horizontal dissipation and the Ekman pumping mechanism could be neglected if Ek 1. Practically, in laboratory experiments the dissipation will be a second order process when Ek 10−4 at least. The third approximation assumes a quasi-bidimensional horizontal flow, in other words, the vertical derivatives ∂z u and ∂z v are expected to be negligible. This assumption corresponds to the Taylor–Proudman theorem which is valid only in the limit of small Rossby number (geostrophic flows). A similar approximation is made by the closure hypothesis (22) given in the introduction, which decorrelates the vertical averaging of the horizontal velocity field. Then integrating the continuity equation (22) along the vertical and using the boundary conditions (7) and (11) we finally obtain the following dimensionless formulation of the RSW model: ε∂t u + RoDh u − v = −∂x η, ε∂t v + RoDh v + u = −∂y η, λ[ε∂t η + RoDh η] + (1 + λη)Ro[∂x u + ∂y v] = Ro

(12)

(13) Ek ζ 2

(14)

where Dh = u∂x + v∂y and ζ = ∂x v − ∂y u. Strictly speaking, these RSW equations are valid in the asymptotic limit of slow quasi-geostrophic flows even if the aspect ratio parameter is finite α 1. However, this model is often accurate beyond its limit of validity for finite Rossby numbers (Ek 1; ε 1; Ro 1) and could be applied to a wide variety of laboratory experiments. This will be indeed the case if the vertical motions remain

1. The Holy Graal of rotating shallow-water flows

329

weak enough (w 1). This latter condition implies both the hydrostatic balance (9) and a quasi-bidimensional horizontal flow. Nevertheless, in such case, highorder terms should be added in (11) to account for non-linear Ekman pumping (Zavala Sanson and van Heijst, 2000; Hart, J.E., 2000). 1.2. Influence of the centrifugal force Since the Newton’s bucket experiment (1689) it is well known that the freesurface of a fluid layer in solid body rotation is deformed under the action of the centrifugal force. The surfaces of constant pressure for a fluid at rest in the rotating frame (i.e. equipotential surfaces Φ = cst) are given by the potential function: 1 Φ(R, Z) = − Ω02 R 2 + gZ. (15) 2 Hence, the free-surface of a rotating fluid layer takes a parabolic shape. Moreover, according to (15), all the equipotential surfaces corresponds to the same paraboloid simply translated along the rotation axis (Figure 2(a)). We use in what follows a dimensionless formulation where H0 is the mean thickness of the layer, D is the tank diameter and Zc = g/Ω02 is the curvature radius at the center of the parabola. In cylindrical coordinates, the equation for the unperturbed free-surface can be written as: 1 1 2 Z 1β 2 1 2 2 =1+ r − d . h(r) = (16) R − D =1+ H0 2H0 Zc 8 2α 8

Figure 2. Parabolic deformation of the equipotential surfaces (thin lines) due to the centrifugal acceleration in rotating laboratory experiments (a). Schematic description of the tangent plane approximation when the corrective terms of order β 2 d could be neglected (b).

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Chapter 6. Experimental Reality of Geostrophic Adjustment

We have introduced here two dimensionless parameters: • The dimensionless tank diameter d = D/L. The experimentalists tend to use a large tank d 1 in order to satisfy the shallow-water constraint and to avoid the boundary effects. However, for such case, the influence of the centrifugal force could become non negligible close to the wall. • The curvature parameter β = L/Zc quantifies the influence of the curved equipotential surfaces on a dynamical structure of horizontal size L. For the atmosphere or the oceans an equivalent parameter is induced by the spherical geopotential where Zc is replaced by the earth radius RE . It is therefore natural to chose a coordinate system so that the unperturbed water surface, or any equipotential surface, is given by z = cst. Hence, paraboloidal coordinates should be used for rotating laboratory experiments (Nycander, 1993) while spherical coordinates are used for planetary flows (Pedlosky, 1987). However, for small values of the curvature parameter (β 1) the tangent plane approximation is generally made. In other words, a Cartesian system of coordinates is used locally and the corrective terms induced by the parabolic curvature will appear only at the order β 2 d (Nycander, 1993; Stegner and Zeitlin, 1998). If we consider a medium scale experiment (D 100 cm) and a typical horizontal scale L 10 cm, these corrective terms could be neglected for moderate rotation rate (Ω0 10 rpm). The main difference between rotating laboratory experiments and planetary flows is that the effective gravity g e is variable in both directions and amplitude in the laboratory. Indeed, when the centrifugal force is not negligible, it induces a tilting of the effective gravity but also a change in its amplitude (Figure 2(a)). The latitudinal dependence of the effective gravity (also called the γ -effect) in paraboloidal coordinates and in the tangent plane approximation is given by: 1 1 ge = = 1 + β 2r 2 1 + β 2r 2. (17) g cos θ 2 For paraboloidal equipotential surfaces this latitudinal dependence of the effective gravity is of the same order as the latitudinal dependence of the Coriolis force (classical β-effect). Ωz 1 f = = cos θ 1 − β 2 r 2 . (18) f0 Ω0 2 Under the tangent plane approximation the equipotential surface are assumed to be locally flat (Figure 2(b)) while the radial variations of the effective gravity g e , the local component of rotation Ωz and the unperturbed layer depth h are expanded at the first order in y = r − a the local latitudinal coordinate centered at the radial position r a. If we consider the center of the rotating tank (a = 0 and y = r), according to (16)–(18) the latitudinal variations of h(y), Ωz (y) or g e (y) are all quadratic

1. The Holy Graal of rotating shallow-water flows

331

and when β/α = L2 /(Zc H0 ) 1 the fluid layer respects the f -plane configuration at the first order of approximation. This will be generally the case for moderate rotation rate (Ω0 10 rpm) in a central region of several tens of centimeters (r 10–20 cm). However, out from the center, the latitudinal variations could be linearly expended in y and they reach their extremal values at the tank wall. Therefore, in order to quantify the relative influence of the β-effect, the γ effect or the topography, we estimate (when a = d) the magnitude of the first derivatives as follows: ∂y h/ h ∝ βd/α, ∂y f/f ∝ β 2 d and ∂y g e /g e ∝ β 2 d. Hence, for standard experimental configurations (β 1, α 1, d 1) the y-dependence of the equilibrium layer depth induced by the parabolic free-surface deformation is the dominant effect. Due to this topographic effect, the equilibrium fluid layer can support topographic-Rossby waves. The linear dispersion relation of these low-frequency waves is analogous to planetary Rossby waves and they are strongly coupled to the slow geostrophic motion. This effect may induce, for instance, a significant drift velocity (Vd f L(∂y h/ h) when L Rd ) and the dispersion of localized vortices (Masuda, Marubayashi and Ishibashi, 1990; Carnevale, Kloosterziel and van Heijst, 1991; Flor and Eames, 2002). Hence, the dynamical influence of the topographic variations could be neglected in the whole tank, from the center to the wall, if ∂y h/ h ∝ βd/α ε. The latter criterion will be generally satisfied in a medium scale experiment (D 100 cm) if the rotation rate is weak enough (Ω0 4 rpm). However, if the ratio βd/α becomes too large, the variation of the layer thickness induced by the centrifugal force could be compensated by a parabolic bottom topography or a parabolic vessel adjusted to the rotation rate (Nezlin and Snezhkin, 1993; Stegner and Zeitlin, 1998; van de Konijnenberg, Nielsen, Rasmussen and Stenum, 1999). 1.3. Non-hydrostatic wave modes Focusing on the geostrophic adjustment problem, where both slow geostrophic motion and fast waves are generated, we look here more carefully at the wave motion that may occur in a rotating fluid layer. We linearize the primitives equations (1)–(8) assuming small amplitudes for the velocity Ro 1 and the free surface displacement λ 1. We neglect all dissipative terms (Ek 1) and take the deformation radius as a characteristic horizontal scale L = Rd of the unperturbed rotating fluid layer, therefore Bu = 1 and α = H0 /Rd . Besides, we keep in mind that the aspect ratio α cannot be asymptotically small for laboratory experiment and we keep the vertical acceleration in (3). Hence, we get: ε∂t u − v = −∂x π,

(19)

ε∂t v + u = −∂y π,

(20)

α ε∂t w = −∂z π,

(21)

2

Chapter 6. Experimental Reality of Geostrophic Adjustment

332

∂x u + ∂y v + ∂z w = 0

(22)

with upper (z1 = 1) and lower (z0 = 0) boundary conditions w(z0 ) = 0,

(23)

w(z1 ) = ε∂t η,

(24)

π(z1 ) = η.

(25)

According to the space and time shift invariance of the system, we use the following Fourier decomposition A(x, y, z, t) = A0 (z)ei(t−kx−ly) for all variables. In this case the temporal evolution parameter corresponds to a dimensionless wave frequency ε = ω/f . This linear system finally leads to: ∂z2 π0 +

α 2 ε2 K 2 π0 = 0 1 − ε2

(26)

where K 2 = k 2 + l 2 , with the boundary conditions ∂z π0 (0) = 0,

(27)

∂z π0 (1) = α ε π0 (1).

(28)

2 2

For inertia–gravity waves (ε > 1) we obtain the following dispersion relation: α 2 ε2 K 2 (29) ε2 − 1 corresponding to a single vertical mode π0 (z) = ch(γ z). We can see here that we will recover the dispersion relation of Poincaré waves γ tanh(γ ) = α 2 ε 2 ;

ε2 = 1 + K 2

γ2 =

(30)

only if αK 1, in such case short inertia–gravity waves are dispersionless. This condition is more restrictive than the shallow-water constraint α 1, and indeed short enough gravity waves will always deviate from the RSW model. We have plotted in Figure 3 the deviation from the Poincaré dispersion relation for various values of the aspect ratio parameter α that could be found in laboratory experiments. For a finite value of the aspect ratio parameter, α = 0.3 for instance as it is shown in Figure 4, high-frequency waves (ε 1) or, in other words, short-waves (K 1) will satisfy the dispersion relation of non-rotating surface gravity waves (SGW). K (31) tanh(αK). α It can be shown that the same dispersion relation (31) applies for boundary Kelvin waves propagating along a lateral wall of the tank. Hence, unlike the RSW model, ε2 =

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Figure 3. Log–log plot of the dimensionless dispersion relation for inertia–gravity waves. The curves correspond to different values of the aspect ratio parameter: RSW model or α = 0 (thick line), α = 0.1 (thick dashed line), α = 0.3 (thin dashed line), α = 1 (thin dotted line).

Figure 4. Dimensionless dispersion relation of inertia–gravity waves in a non-rotating (thin line), a rotating fluid layer (thick dashed line) and for the RSW model (thick line) for the fixed value of α = H0 /Rd = 0.3.

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both inertia–gravity waves and Kelvin waves will become dispersive in the shortwave limit if the aspect ratio parameter is not small enough. Nevertheless, these non-hydrostatic effects could be neglected for a wide range of the inertia–gravity wave spectrum if αK 1, which corresponds to λ˜ 2πH0

(32)

where λ˜ is the characteristic wavelength. For inertial waves (ε < 1) we obtain a discrete spectrum of n vertical modes which correspond to the dispersion relations: −γn tan(γn ) = α 2 εn2 ;

γn2 =

α 2 εn2 K 2 1 − εn2

(33)

where (1 + n)π/2 < γn < (1 + n)π. These non-hydrostatic waves exhibit strong variations of pressure and velocity along the vertical axis (Figure 5(b)). When the vertical wavenumber becomes large (n 1) the wave frequency approaches the Coriolis frequency ε 1 for a wide range of horizontal wavenumber components. According to Figure 5(a), for a given horizontal wavelength k, the short wavelength perturbations along the vertical will have the highest horizontal phase speed. Hence, the non-hydrostatic inertial waves play a crucial role in the vertical alignment and the rapid formation of Taylor columns in a rotating fluid layer. If now we add a mean flow component, such type of non-hydrostatic modes will lead to inertial instability in anticyclonic vorticity regions (Johnson, 1963; Yanase, Flores, Metais and Riley, 1993). Such instabilities occur when the Rossby number Ro exceeds unity and the growth rates for these three-dimensional modes exceed significantly the two-dimensional unstable Rossby modes when Ro 2. Hence, for finite Rossby numbers, starting initially from a two-dimensional flow the three-dimensional perturbations could grow exponentially and break both the hydrostatic and the geostrophic balance (Afanasyev and Peltier, 1998; Stegner, Pichon and Beunier, 2005). As it can be seen in Figure 6, such small-scale instability may occur in shallow-water anticyclonic vortices when the Rossby number is large enough. 1.4. Two-layer stratification We have seen previously that with a single barotropic layer experiment the deformation radius is generally close to the tank size. In other words, for a single layer f -plane experiment we are restricted to large Burger number dynamics. Nevertheless, if we use a two-layer stratification we introduce a baroclinic deformation

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(a)

(b) Figure 5. Dimensionless dispersion relation of non-hydrostatic inertial waves (a) corresponding to various vertical modes (b). All the wave modes are calculated for the aspect ratio parameter α = H0 /Rd = 1.

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Figure 6. Dye visualization of von Karman wake in a rotating shallow-water layer with α 0.07, Ro 2 and Re 20000. Small-scale instability is visible in anticyclonic vortices (black dye), while cyclonic vortices (red dye) remain stable (Teinturier, Stegner, Viboud, Didelle and Ghil, 2006).

radius which could be much smaller than the barotropic one. Besides, the Ekman pumping affects only the lower layer, and for an appropriate set of parameters the upper layer dissipation could be strongly reduced. To create a density stratification in water, salt or sugar are generally used instead of temperature. Indeed, the thermal diffusivity (κT 10−7 m2 s−1 ) is a hundred times larger than the salt diffusivity (κS 10−9 m2 s−1 ). In a motionless fluid layer, an initial salt perturbation will diffuse over a 1 cm distance in half a day instead of ten minutes for a thermal perturbation. Hence, for typical layer depths about few to tens of centimeters an initial salt or sugar stratification will remain robust for at least several hours. To obtain a sharp density jump corresponding to a well-defined two-layer stratification we generally proceed as follows. The tank is first filled with the deep and dense lower layer. Then, we start to spin up the rotating table and when the solid-rotation is reached we slowly inject the light upper layer ρ1 at the surface of the dense bottom layer ρ2 . To reduce the vertical

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mixing during the injection we could use floating Hele-shaw cells or small tubes to inject the light fluid horizontally at the free-surface. Another method consists in injecting very slowly the upper layer through floating porous plates. Let us consider a two-layer salt stratification. According to the classical dimensional analysis we add to the previous ones at least four new dimensionless parameters: • The thickness ratio parameter δ = H1 /(H1 + H2 ). This parameter controls the dynamical interaction between the two layers. For equivalent depth layers δ 0.5 the two layers are strongly coupled and baroclinic instability may occurs even if the vertical velocity shear is weak. On the other hand, according to the standard two-layer Phillips model (Pedlosky, 1987), for small ratios δ 1, the baroclinic growth rates tend to vanish. Hence, to avoid a strong baroclinic destabilisation of the flow, we will first consider laboratory experiments with a thin upper layer and a deep lower layer having δ 0.1. Besides, in order to keep the Ekman number small enough in the lower layer (Ek(2) = (δE /H2 )2 10−4 Ek ), we generally fix H2 = 10–20 cm, and therefore the upper layer thickness is about H1 2 cm. • The density ratio parameter N = 2(ρ2 − ρ1 )/(ρ2 + ρ1 ). With salt stratification, we can easily obtain N small up to 10−3 . We then introduce the reduced gravity g = Ng g which controls the dynamics of internal gravity waves at the interface between the layers. • The internal Burger number Bu = (Rd /L)2 corresponding to the baroclinic deformation radius Rd = g H1 H2 (H1 + H2 )/f . We can see here that for small N 10−3 and δ 0.1 the baroclinic deformation radius could be two orders of magnitude smaller than the barotropic deformation radius Rd √ g H1 /f 10−2 Rd , where Rd = g(H1 + H2 )/f . Hence, for a thin upper layer with H1 2 cm and a weak density difference ρ2 − ρ1 2–10 g l−1 we can reach deformation radius as small as Rd 1 cm. Therefore, with a two-layer stratification, the internal Burger number could be easily varied from small to large values 0.01 Bu 100. • We introduce the stratification parameter ES = (δE /dS )2 in order to quantify the dissipation induced by the fluid–fluid interface. This parameter is an equivalent Ekman number for a continuously stratified fluid where dS is the characteristic scale of the vertical density gradient. Indeed, for salt stratification, due to the molecular diffusion and the injection process, the density gradient is always continuous between the upper and the lower layer. Even with very slow laminar injection of both layers the characteristic size dS cannot be infinitely small, and we generally get a density gradient thickness of dS = 3–5 mm (Stegner, Bouruet-Aubertot and Pichon, 2004). For geostrophic flows, the vertical gradient of the horizontal velocity will be directly proportional to the vertical density gradient. Hence, the dissipative terms in the

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right-hand side of the horizontal momentum equations (12) should scale with ES = (δE /dS )2 . This parameter is larger than the Ekman number defined using the upper layer thickness Ek(1) = (δE /H1 )2 . However, due to the absence of the no-slip boundary condition, there is no boundary Ekman layer for the upper layer. The fluid–fluid interface will then induce (if any) a much weaker recirculation than a classical bottom Ekman layer if the vertical stratification is not too sharp ES = (δE /dS )2 1. We get a dimensionless set of equations for the two-layer RSW model with rigid lid and bottom boundary condition using: L as horizontal scale and T the characteristic time-scale for both layers, U (i) as the horizontal velocity, Hi the vertical thickness and ρi f U (i) L the pressure deviation from hydrostatic balance in each layer. (i)

ε∂t u(i) + Ro(i) Dh u(i) − v (i) = −∂x π (i) , ε∂t v

(i)

+ Ro

(i)

(i) Dh v (i)

+u

(i)

= −∂y π

(i)

(34) (35)

where Dh(i) = u(i) ∂x +v (i) ∂y and the superscript i = 1, 2 corresponds respectively to the upper and the lower layer. The pressure continuity at the interface gives: λBu η = (1 − N )Ro(1) π (1) − (1 + N )Ro(2) π (2) 1−δ and the mass conservation in each layer leads to: (1) λ ε∂t η + Ro(1) Dh η + (1 + λη)Ro(1) ∂x u(1) + ∂y v (1) = 0, (2) λδ ε∂t η + Ro(2) Dh η − (1 − δ − λδη)Ro(2) ∂x u(2) + ∂y v (2)

(2) Ek (2) = (1 − δ)Ro(2) ζ 2

(36)

(37)

(38)

where the relative elevation parameter λ corresponds here to the characteristic deviation of the internal interface rescaled by the upper layer thickness H1 . According to the above equations, if the thickness ratio parameter δ and the density ratio parameter N are small enough and if the motion has a strong baroclinic component (intense velocities in the thin upper layer while the deep lower layer remains almost at rest Ro(2) δRo(1) ), the interface deviation η is controlled by the upper layer pressure only η π (1) . In such case, at the first order of approximation, the upper layer motion is not affected by the lower layer which acts as a neutral layer (Cushman-Roisin, Sutyrin and Tang, 1992). Hence, the upper layer dynamics can be described by the shallow-water reduced-gravity model. Namely, a one-layer RSW model where the gravity g is replaced by the reduced gravity g induced by the two-layer stratification.

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Table 1 Vortical motion

Wave motion

Single layer

εα 2 1; Ro α 2 1; Ro < 1

kH0 1

Two layers

δ 1; Ro(2) δRo(1) ; Ro(i) (α (i) )2 1; Ro(1) < 1

kH2 1

However, as for the single-layer case, non-hydrostatic wave motions or inertial instability may occur in the two-layer experiment when respectively λ˜ 2πH2 or Ro(1) > 1. Note that the hydrostatic constraint on the wave activity is fixed here by the deep layer thickness H2 and not the thin upper layer H1 . Recent laboratory experiments performed in a two-layer configuration (α 0.66; δ 0.2) exhibit non-hydrostatic wave behaviour for λ˜ 80 cm wavelength, while H1 12.5 cm and H2 50 cm (Thivolle-Cazat, 2003). Taking into account the above mentioned laboratory constraints, the physical modelling of rotating shallow-water flows looks like a Holy Graal for experimentalist. Nevertheless, for a specific range of the dynamical parameters, the motion in rotating fluid layers could be close to the one layer RSW model. We recall in Table 1, both for single-layer and the two-layer configurations, the distinct conditions needed to be satisfied, respectively, for the slow vortical motion and the fast wave motion in order to follow the RSW dynamics.

2. Potential vorticity measurements: a new challenge Both vorticity and potential vorticity play an important role in the dynamics of rotating fluid layers. The application of the Kelvin theorem to a non-dissipative rotating shallow-water flow implies the Lagrangian conservation of potential vorticity (Chapter 2) in each layer: D 1 + Roζ = 0. (39) Dt 1 + λη An elementary fluid parcel (i.e. fluid column) moving within a layer could be stretched or compressed. These changes in the height of the fluid parcel during its motion will be accompanied by a change in its vorticity. In other words, for a purely incompressible two-dimensional flow when the free-surface or the interface deviations are negligible (λ Ro), we recover the Lagrangian conservation of vorticity: Dζ = 0. (40) Dt In this case, vorticity will be generated in the flow only if there is an external source (boundary layer or fluid injection, for instance). For a rotating fluid layer,

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if the layer thickness varies sufficiently, relative vorticity could be generated from an adjustment process without any external source. The potential vorticity conservation is a key concept for adjustment processes even in the presence of dissipative forces. Hence, as far as laboratory experiments on geostrophic adjustment are concerned, it is crucial to perform quantitative measurements of the potential vorticity field. Such measurements in a rotating fluid layer are indeed not simple. Both the vorticity field ζ and the height field η should be measured simultaneously. If such measurements are now possible, it is mainly due to recent progress in computers, lasers and cameras technology. Besides, additional difficulties are encountered on a rotating turntable where sufficiently compact devices (especially lasers) and remote control of the whole setup are needed. Therefore, direct measurement of the potential vorticity field is always challenging for experimentalists. We give below some details on the non-intrusive methods which can be used to achieve such measurements for specific experimental configurations. 2.1. Particle image velocimetry and vorticity field measurements The particle image velocimetry (PIV) was developed since 1994 to perform accurate and quantitative measurements of fluid velocity vectors at a very large number of points simultaneously (Adrian, 2005). Presently, the 2D PIV method consist to add small neutrally buoyant beads to the working fluid and lightened them with a laser sheet. The 2D particle motion along this plane are recorded with a digital video camera. Cross-correlation image processing is then performed to measure the mean particle displacement in small box region between two successive images (Fincham and Spedding, 1997). Standard systems are sold by commercial companies and it is now the most efficient and non-intrusive technique used in fluid mechanics to obtain a vorticity map in a given region of the flow field. Nevertheless, some technical limitations appear which restrict the spatial resolution of such measurements in rotating fluid layers. The energy necessary to illuminate fine particles and produce images of sufficient exposure and clarity is the first limitation of PIV. The maximum size of the measurement window is then fixed by the laser intensity and the camera exposure time. Hopefully, vertical motions are strongly damped in a rotating fluid layer, therefore neutrally buoyant particles could stay for a relatively long time inside the horizontal laser sheet. Besides, the horizontal velocities of geostrophic motions are usually not too large (V 1–10 cm s−1 ) and the camera exposure time could be optimized to 10–20 ms. But nevertheless, if the intensity per unit area is too small the clarity of recorded images may not be sufficient enough with classical digital cameras. On a medium size turntable (D 1 m), high power lasers which require cooling systems are generally excluded. However, the last

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Figure 7. Horizontal velocity field of a cyclone obtained from particle image velocimetry. For clarity, only half of the vectors (47 × 35) are displayed instead of the full (94 × 70) field. The measurement was made in the thin upper layer of a two-layer stratified fluid corresponding to: Bu 0.4, λ 0.5, α 0.75, δ 0.1.

generation of compact high power laser diodes2 can generate an uniform intensity line (non-Gaussian) with an output power up to 1 W. With such system we could easily detect the horizontal particle motion (V < 10 cm s−1 ) from small (10 cm × 10 cm) to large (1 m × 1 m) areas of investigation. The second limitation is induced by the pixel resolution of the camera. Indeed, to obtain a precise cross-correlation between two interrogation windows, a minimum number of particles (∼5) should be present in the interrogation box (Raffel, Willert and Kompenhans, 1998). This constraint induces practically a minimum size of a 12 × 12 pixel box. Hence, with a standard 750 × 550 pixels camera we usually get a velocity field of 95×70 vectors as it is shown in Figure 7. Using digital cameras with higher resolution (3000 × 2000 pixels) we could, for instance, reach a 370 × 250 vector grid field. However, even with very high resolution camera and optimized software, PIV measurements will always give a coarse-grid resolution in comparison to direct dye tracer visualizations (3000 × 2000 pixels) or high-resolution numerical simulations (4096×4096 for two-dimensional flows (Bracco, McWilliams, Murante, Provenzale and Weiss, 2000)). The third limitation comes from the limited precision of the velocity field. Even with hierarchical correlation methods, where correlations deduced from a large interrogation box are used to guide correlation analysis at smaller boxes (Hart, D.P., 2 Lasiris Magnum Laser (www.laser2000.fr).

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Figure 8. Cyclonic (red) and anticyclonic (blue) dimensionless vorticity ζ /f calculated from an instantaneous velocity field (a) or calculated from the time averaged velocity field (b) shown in Figure 7. The Rossby number deduced from the maximum vorticity is about Ro = ζmax /f 0.6. The measurement area is a rectangular window of 280 mm × 220 mm.

2000), the available dynamical velocity range is about 100:1. In other words, the method cannot detect fluctuations in the velocity field below 1%. Besides, experimental noise on recorded images could easily produce 5% error in the velocity field. This can be a serious problem, because a weak noise in the velocity field

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induces a stronger noise in the derivatives and therefore strongly influences the vorticity measurement. Typical errors of order of 20% (or higher) in the vorticity field could be frequent, and strong efforts on the improvement of the image quality and the software used in the PIV process are needed to reach such precision on the vorticity field measurements. However, if the flow evolves slowly in comparison with the frequency of the PIV acquisition system, time averaging of the velocity field is a simple and efficient way to reduce the experimental noise. For the case of geostrophic adjustment, when a quasi-steady motion is reached, time averaging will lead to sufficiently precise vorticity measurements. For instance, the velocity field shown in Figure 8 is an average of ten velocity fields separated by a time interval of 120 ms. Hence, this corresponds to a time averaged velocity measurements over 1.2 s which is smaller √ than the inertial period Tf = 12 s or the characteristic decay time TE = H2 / νf 200 s. Such time averaging reduces the noise in the vorticity field (Figure 8(b)) by a factor 10 in comparison with the instantaneous vorticity measurements (Figure 8(a)). According to the above comments, we should emphasize that even if we can easily obtain a vorticity map from the standard PIV system the accuracy of such measurements should be checked carefully. Let us recall, that if PIV measurements have a coarse grid resolution corresponding to 12 × 12 pixels on the digitized image, this will be even more pronounced for the vorticity field. Indeed, to resolve accurately a velocity gradient, at least 3–5 grid points are needed. Therefore, quantitative vorticity measurements will not be possible if the dynamical structure under consideration is too small. We can roughly estimate a limiting value as 36 × 36 pixels on the digitized image. Thus, small vortices or thin vorticity filaments are generally smoothened by the PIV process. In such case it could be useful to use two cameras with a wide and a zoom angle in order to quantify accurately the large scale flow and smaller vortical structures (Perret, Stegner, Farge and Pichon, 2006). Hence, the standard PIV method is well suited to quantify slow and large-scale structures in rotating fluid layers. However, for fast and small-scale structures such as high-frequency waves, this velocimetry method can hardly provide quantitative measurements, unless an expensive high-speed PIV technology is used. 2.2. Height field measurements Laboratory techniques for measuring the velocity field, such as the PIV method described above, are quite advanced. However, methods for making accurate measurements of the height field of a fluid layer have remained relatively elusive. As far as we know, four non-invasive techniques were used to detect or to measure the height field fluctuations in rotating fluid layers: light absorption, optical altimetry from the parabolic free-surface, optical rotation of the working fluid and laser induced visualization (LIV).

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2.2.1. Light absorption The light absorption technique is based on the optical density of a dyed layer. It consists in measuring the light intensity after absorption through a uniformly dyed fluid layer (Holford and Dalziel, 1996). The fluid layer is usually lightened from below through a transparent vessel while a video camera records the intensity fluctuations from the top (Figure 9(a)). A specific pass-band-filter, which is centered at the maximum absorption of the dye, is put on the video camera to

Figure 9. Light absorption technique for a parabolic vessel experiment (a) (Stegner and Zeitlin, 1998). Intensity fluctuations view from the top of the experiment (b) post-processed calibrated image corresponding to a relative elevation λ 1 of the layer depth (c).

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increase the sensitivity. With this method a local increase of the layer thickness induces a higher absorption and this region will appear darker on the video image (Figure 9(b)). This altimetric measurements was used successfully on small-scale parabolic vessel where small (10%) and large (60–100%) relative free-surface deviations were detected with an accuracy of less than one mm (Stegner and Zeitlin, 1998). Nevertheless, caustic effects induce systematic errors of order 5–10% of the layer thickness which limits the precision on the height measurements. 2.2.2. Optical altimetry of the parabolic free-surface The free-surface fluctuations of a rotating fluid layer can be imaged and analysed using its parabolic free-surface as a Newtonian telescope mirror. Parallel light rays from a source high above the rotating table reflects from the water surface and converges on the parabolic focus Zf = 12 Zc (Figure 10(a)). However, parallel light rays can hardly be obtained on a rotating table. The image of a point-light source located at Zc (the radius of curvature at the center) no longer have sharp focus but converge through a small disk located at the same height Zc (Figure 10(b)). For practical purpose the light source and the camera are symmetrically displayed off the axis. Then, putting a knife-edge barrier in the middle of this singular disk, where all the rays converge, can partially obscure the image giving great sensitivity to slight imperfections of the reflecting surface. This optical altimetry technique is one of the most sensitive methods used so far in geophysical fluid dynamics experiments. Indeed it is potentially able to detect free-surface fluctuations with a one micron precision, independently of the mean thickness

Figure 10. Focusing of parallel light-rays reflected by the parabolic free-surface (a). Sharp convergence of a point light-source located close to Zc the center of curvature of the apex (b). A knife-edge barrier induces a contrasted black and white image of the free-surface. It increases the sensitivity to deviations from a perfect paraboloid.

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Figure 11. Optical altimetry visualization of inertia–gravity waves (50 µm in amplitude) interacting with localized vortices. The wave maker is on the bottom left, while the vortices appear in the center. Courtesy Y. Afanasyev.

of the parabolic layer. Therefore, this method is particularly suited for the investigations of small amplitude waves (less than 0.1% fluctuation) which are often difficult to detect by other methods. For instance, an inertia–gravity wave having an amplitude of 50 µm (0.04% of the mean layer thickness) could be visualized in Figure 11. Qualitative observations of a large variety of dynamical features such as gravity waves, inertial waves, Rossby waves and small-scale convection could then be performed (Rhines, Lindahl and Mendez, 2006; Rhines, 2006). At the same time, a quantitative method of determination of the slope variation using speckle patterns is possible. A reference image of the fluid layer in solid-body rotation is first made. The slope is measured by comparing the original pattern and a reflected image of this pattern distorted by the surface perturbation induced by the relative flow motion. The procedure is analogous to PIV process where correlations are computed between the small areas of the image and the reference. Nevertheless, the speckle method is limited by large-amplitude deformations and have a limited spatial resolution due to the minimum size of the correlation boxes (Rhines, Lindahl and Mendez, 2006; Afanasyev, Rhines and Lindahl, 2006). A different quantitative method based on optical color coding was also developed using every pixel of the image. A color slide is fixed just below the light

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source. For a given rotation rate (the null point) the entire surface of water is illuminated by only one color. Any perturbation of the free surface results in the appearance of color different from the null point. It is then possible to measure from each pixel of the image the x and the y component of the slope with a 0.1% sensitivity (more details are given in (Afanasyev, Rhines and Lindahl, 2006). Hence, by integrating the slope field quantitative height measurements of the parabolic fluid layer could be achieved. Besides, for a rotating fluid layer satisfying the geostrophic balance, the local slope is directly proportional to the geostrophic velocity. Hence, the primary purpose of optical altimetry is to perform an Altimetric Imaging Velocimetry (AIV) having a much higher spatial resolution than PIV. 2.2.3. Optical rotation Another sophisticated remote sensing method for measuring the thickness of a fluid layer relies upon the optical rotation properties of the working fluid. The liquid is chosen to be optically active (limonene and CFC-113 for instance), so that plane-polarized white light propagating vertically through the fluid layer has its plane of polarization rotated by an angle which depends upon both the wavelength and the layer depth. After leaving the fluid, the angulardispersed white light passes through a sheet of polaroid. For a given layer depth, only light of a certain wavelength has its polarization axis rotated into exact alignment with the polaroid. Light of other wavelengths is either partially or fully extinguished by the polaroid, giving a correlation between the interface height and colour registered by the camera (Hart and Kittelman, 1986; Williams, Read and Haine, 2004). A high-sensitivity up to 1–2% of the layer height could be reached with this technique. Both the large-scale geostrophic flow and small-scale waves could be accurately measured with the technique (Williams, Read and Haine, 2003). According to Figure 12 small-scale fluctuations in the two-layer interface having 1 to 5 mm amplitude are quantitatively detected. However, the sensitivity will be optimal when the mean rotation angle is about 90◦ and this implies for the limonene/CFC mixture that the fluid layer should be relatively thick H = 10–15 cm. Besides, specific precautions should be taken to prevent harmful limonene vapours from evaporating into the laboratory. 2.2.4. Laser induced visualization Laser induced visualization (LIV) technique could also be used in rotating experiments to measure with precision the fluid layer thickness along a line. Initially, the working fluid is uniformly mixed with a fluorescent dye. A vertical laser sheet crosses the horizontal fluid layer and induces the fluorescence of the dye within this plane (Figures 13(a) and 14). In order to optimize the fluorescence, the maximum of dye absorption λabs should be close to the laser wavelength. Hence, we choose the fluoresceine (λabs = 490 nm) or the Rodhamine 6G (λabs = 530 nm)

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Figure 12. Color calibrated visualization of the internal interface of a two-layer fluid using optically active CFC-13/limonene for the lower layer. Small-scale waves (5 mm amplitude and 2 cm wavelength) are visible during one cycle of a large-scale unstable baroclinic mode 2. Courtesy P.D. Williams.

Figure 13. Side view visualization (a) of the fluorescent upper layer. The lower layer appears dark because it does not contain any fluorescent dye and remains therefore transparent to the laser sheet. Edge detection processing (b) allows for a precise measurement of the layer thickness corresponding here to a cyclonic depression: Bu 0.4, λ 0.5, α 0.75, δ 0.1.

if we use, respectively, an argon laser (488 nm) or Nd:Yag laser (532 nm). A video camera, fixed on the side of the tank and perpendicular to the laser sheet could then record the fluorescence of the fluid layer. Using an adequate

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Figure 14. Measurement of the vertical salinity profile from the fluorescent light emission (Stegner, Bouruet-Aubertot and Pichon, 2004). Initially, the upper fluid is uniformly mixed with a fluorescent dye. (a) The upper water initially confined in a transparent bottomless cylinder appears white owing to the fluorescent light emission while the dense water is black. (b) Vertical distribution of the light intensity in the central rectangle shown in (a). (c) The salinity profile can be deduced from (b) if we perform a careful calibration of the laser sheet intensity along the vertical plane. The disturbance at z = −3.3 cm is due to light reflection at the bottom of the cylinder.

image processing we then detect the position of the interface between the light fluorescent and the dark transparent fluid (Figure 13(b)). With this non-intrusive technique we were able, for the two-layer configuration, to measure the displacement of the fluid interface between the fresh and the dense water with an accuracy of 2% at fast acquisition rates (Stegner, Bouruet-Aubertot and Pichon, 2004; Perret, Stegner, Farge and Pichon, 2006). The acquisition frequency is limited by the acquisition rate of the camera and the transfer capacity of the video card. A frequency of 100 Hz could be easily reached nowadays with standard firewire cameras. Note in Figure 15 that the LIV camera is not exactly perpendicular to the fluid layers, in order to reduce the image distortion due to the ray diffraction through the stratified interface between the two layers. Unlike the previous techniques which estimate the height field or its fluctuations in the whole layer, the LIV method gives a measurement of the height field only along a line. Hence, the position of the vertical laser sheet should be

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Figure 15. A horizontal red (670 nm) laser sheet with a vertical green (532 nm) laser sheet are used simultaneously in order to couple PIV and LIV measurements in the upper layer.

carefully chosen. However, this limitation is compensated by the possibility to detect small-scale and three-dimensional structures along the vertical. Indeed, this method measures precisely the dye distribution at each point (x, z) of a vertical plane and does not integrate the information along a vertical ray path. Besides, we could also measure the density field from the fluorescent dye emission (Figure 14(a)). Indeed, on short time scale (i.e. few minutes) the mixing of the initial uniform concentration of both dye and salt is expected to be driven mainly by convection (i.e. turbulent mixing). Hence, dye and salt gradient are not affected by relative diffusion and they are therefore proportional. In a first step, we measured the relative fluorescent light emission (Figure 14(b)) which depends mainly on the dye concentration and the laser sheet intensity. Then, taking into account the vertical distribution of the laser sheet intensity, we correlate the light intensity with the local salinity (i.e. density) as shown in Figure 14(c). This could be, in the next future, an efficient non-invasive technique to measure the density field in a continuously stratified fluid. 2.3. Potential vorticity measurements In order to measure the potential vorticity field according to (39) we chose to use both PIV and LIV measurements simultaneously. However, due to the restrictions of a LIV technique, which gives the height field only along a line, the combination of these non-invasive methods is best suited for unidirectional flows. It is then possible from a line measurement to estimate a global potential vorticity field for either circular (Stegner, Bouruet-Aubertot and Pichon, 2004) or parallel flows (Perret, Stegner, Farge and Pichon, 2006). A typical experimental setup for

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Figure 16. Plots of the averaged vorticity profile measured by PIV (a) and the averaged height profile (b) measured by LIV corresponding to the mean steady state at t = 2Tf . The potential vorticity (c) is deduced from these two profiles. These measurements correspond to the same cyclonic PV anomaly as shown in Figures 8(b) and 13.

a circular cyclonic PV anomaly is shown in Figure 15. Two lasers having different wavelengths are used in addition with specific optical filters, fixed on each camera, in order to detect the dye emission only in the vertical plane and the buoyant particles only in the horizontal plane. Time averaged vorticity and height profiles along a diameter are displayed in Figure 18. The profiles correspond to the PIV and LIV measurements shown in Figures 8(b) and 13. The temporal averaging (over one inertial period Tf = 2π/f ) filters out the fast wave motion from both the density interface and the azimuthal horizontal velocity. Hence, these profiles correspond to the mean adjusted state of the system which evolves slowly in comparison with the wave motion.

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From these data we can then easily quantify the potential vorticity (Figure 16(c)) of the cyclonic PV anomaly. The PV profile is rescaled here by Q0 = f/H1 , the intrinsic PV of the unperturbed upper fluid layer (solid line in Figure 16(c)). For this case, the initial circular PV anomaly was a constant Q patch with Q/Q0 = 2. As far as we know, this technique is the first attempt for direct and quantitative measurements of the potential vorticity field in a rotating shallow-water layer experiment. In other laboratory studies, either the height field or the velocity field were measured but not both of them simultaneously. In such case, the “missing” field could be estimated according to the geostrophic or cyclo-geostrophic balance and the potential vorticity field reconstructed. However, these indirect methods could induce significant errors especially when ageostrophic or non-hydrostatic motions become non negligible. A more refined method based on data assimilation was used recently (Thivolle-Cazat, Sommeria and Galmiche, 2005). The experimental results were compared with a two-layer isopycnal model and data assimilation was used to extrapolate from PIV measurements both the interface position and the potential vorticity field. However, such PV extrapolation depends strongly on the underlying assumptions of the numerical model used and on the assimilation scheme. Therefore, we do believe that coupled measurements is the best way to quantify the PV. Nevertheless, data assimilation will fully benefit from these coupled measurements and it could become an optimal method to test the limits of validity of the shallow-water modelisation for real flows.

3. Simple case studies of geostrophic adjustment We describe in what follows few cases of geostrophic adjustment based on lock released experiments performed in rotating fluids. In a two-layer configuration, vertical boundaries (i.e. locks) are used to fix initial height (or density) steps in the upper layer. For such cases when there is no relative motion in the layers the initial PV field is precisely controlled by the layer thickness. If the release of the vertical walls is rapid enough, we could then follow the geostrophic adjustment of a well defined initial condition corresponding to discontinuous profiles of constant PV. The simplicity of the initial condition makes these experiments easily reproducible. 3.1. “Warm-core” lens 3.1.1. Initial state and experimental configuration The term warm-core lens is generally used for mesoscale vortices which contain a finite volume of warm and light water at the ocean surface. A simple experimental configuration leads to similar dynamical structure (Griffiths and Linden, 1981;

3. Simple case studies of geostrophic adjustment

Figure 17.

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Initial configuration of the “warm-core” lens: the setup (a), and the initial profile of the corresponding potential vorticity (b).

Rubino and Brandt, 2003). A fixed volume of buoyant water is initially confined within a bottomless cylinder of radius Rc on the top of a dense rotating fluid (Figure 17(a)). Assuming that the thin upper layer follows the reduced-gravity RSW equations (cf. Chapter 1), the initial PV distribution is constant for r < Rc and exhibit a singularity at r = Rc (Figure 17(b)) due to the vanishing layer thickness. Similar experiments were performed to study the baroclinic instability of a density front leading to meanders and eddies (Griffiths and Linden, 1981; Bouruet-Aubertot and Linden, 2002). The present experiment was made with a smaller thickness ratio parameter δ 0.1 to reduce the growth rate of baroclinic disturbances and focus the study on the adjustment process (Stegner, BouruetAubertot and Pichon, 2004). 3.1.2. Dynamical stages Three stages were observed during the adjustment process. Just after the rapid withdrawal of the transparent cylinder, the fresh water spreads radially as a gravity current. During this initial stage, the flow is fully three-dimensional (Figure 18(a)) and the effects of rotation are expected to be weak. After approximately half of the inertial period, the radial extension of the lens is stopped (Ungarish and Huppert, 1998). The second stage corresponds to a radial contraction of the lens where steep jumps at the interface may appear (Figure 18(b)). Then, after about two in-

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Figure 18. Dynamical evolution of the interface for the initial configuration corresponding to Bu = (Rd /Rc )2 0.4; α 0.76; δ 0.08 (Stegner, Bouruet-Aubertot and Pichon, 2004). The snapshots are taken at t = 0.5Tf (a), t = 0.8Tf (b), t = 3Tf (c) and t = 0.5Tf (d). The dark rays on the right-hand side of the image are experimental shadows produced by the upper fixations of the cylinder. The full movie could be downloaded from http://gershwin.ens.fr/stegner.

ertial periods, the density front reaches an equilibrium characterized by a standing wave mode superimposed on the mean state (Figures 18(c), (d)). In all our experiments, this third stage is rapidly reached after approximately one or two inertial periods. These results agree with previous studies (Mahalov, Pacheco, Voropayev, Fernando and Hunt, 2000) who also found that the inertial period is the characteristic time of the transition from a density current to a geostrophic front. 3.1.3. Rotating shallow-water predictions We present here the approximations and the calculation of classical Rossby adjustment theory for the axisymmetric warm-core lens configuration (Figure 17). According to the small thickness ratio parameter δ 0.1 and the weak motion observed in the lower layer Ro(2) δRo(1) (Stegner, Bouruet-Aubertot and Pichon, 2004) the reduced gravity RSW equations are expected to provide, in first order of approximation, an accurate description of the upper layer dynamics. Besides, we assume that viscosity and dissipative effects are negligible and that the system reaches a final steady state. Using the deformation radius as the characteristic horizontal scale (i.e. Ro = 1 and λ = 1) we get the following dimensionless

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cyclo-geostrophic balance for an axisymmetric steady state: v2 + v = ∂r η. (41) r The Lagrangian conservation of PV implies a constant value for all fluid parcels within the upper lens: Q(r rf ) = 1 =

1 + ∂r v + v/r 1+η

(42)

where rf is the final radius of the density lens. According to (41) and (42) we get: v v2 1 ∂r (r∂r v) − 2 − v + =0 (43) r r r with the boundary conditions v(0) = 0, h(rf ) = 1 + η(rf ) = 0

⇒

v ∂r v + + 1 (rf ) = 0. r

(44) (45)

For a given radius rf , we can solve (43), (44) and (45) numerically with standard shooting methods. Then, the angular momentum conservation or mass conservation or mass conservation both give the same implicit relation between rf and the initial radius of the cylinder rc : rc2 = rf2 + 2rf v(rf )

(46)

The velocity v(r) and the height h(r) profiles of the steady adjusted density lens are fixed by a single parameter rc = Rc /Rd = Bu−1/2 . Examples of velocities and height profiles are given in Figure 19 for the same initial state and two different deformation radii corresponding to Bu = 0.05 and Bu = 5. For small Burger number we expect an axisymmetric jet (or large-scale ring) whereas for large Burgers number an eddy (close to solid rotation) is expected. 3.1.4. Mean adjusted state In the warm-core lens configuration, the interface between the two fluids intersects the free surface. Hence, unlike the standard Rossby adjustment problem (Gill, 1982; Vallis, 2006) inertia–gravity waves cannot propagate away from the region of the initial density anomaly. Therefore, the separation between the adjusted state and the wave motion is not direct. Hence, we used time averaging over one or two Tf , as described in Section 2, in order to extract the slow dynamics of the height profile and the velocity field. We first observe that the averaged height profile, displayed in Figure 20(a), remains almost constant between t 1.5Tf and t 7Tf . During that time, the averaged velocity profile experiences a slow

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Figure 19. Final steady state according to the standard Rossby adjustment. Two adjusted velocity (dashed line) and height (solid line) profiles resulting from the same initial density anomaly (h = H1 , r = Rc ) are plotted for two different deformation radii Bu = 0.05 and Bu = 5.

dissipation (Figure 20(b)). Therefore, even if a strong wave activity is present according to Figure 18, the averaged mean state remains quasi-steady after one or two inertial periods. Moreover, in the central region, this quasi-steady state is relatively close to the cyclo-geostrophic adjusted state predicted by the PV conservation in the RSW framework. According to this inviscid adjustment model the velocity reaches its maximum and is discontinuous at the edge of the lens. This is obviously unrealistic in a physical system where dissipative processes occur. Indeed, according to Figure 20(b) the maximum velocity is almost three times smaller than the predicted one. Hence, both the velocity and the potential vorticity of these anticyclonic lenses are smoothened near the edge front over a characteristic distance equal to the deformation radius (in the present case Rd = 3.2 cm while Rc = 5.25 cm). 3.1.5. Small-scale instabilities Detailed analysis of the velocity field evolution shows that strong and localized dissipation occurs in the very initial stage of adjustment (t 2Tf ) while the flow experiences only a weak dissipation afterwards. This rapid dissipation which occurs at the edge of the anticyclonic lenses induces a significant deficit in the

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Figure 20. Mean height (a), velocity (b) and PV (c) profiles averaged over one inertial period. The initial density anomaly confined within the bottomless cylinder is plotted with a thin line in (a). The thick solid line corresponds to the cyclo-geostrophic adjusted state predicted by the geostrophic adjustment scenario of the inviscid RSW model.

kinetic energy of the adjusted flow up to 50% or 80% (Stegner, Bouruet-Aubertot and Pichon, 2004). Dye visualization reveals that transient and rapid three-dimensional instabilities occur in the very first stage of adjustment (Figure 21). A first unstable perturbation having a short wavelength grows very quickly, then

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Figure 21. Dye visualization of the three-dimensional perturbations at the edge of the anticyclonic lens (a) t = 0.3Tf , (b) t = 0.5Tf , (c) t = 0.7Tf and (d) t = 1.7Tf (Stegner, Bouruet-Aubertot and Pichon, 2004). The movie is available on http://gershwin.ens.fr/stegner.

spiralling arms appear with a larger wavelength. The first instability scales with the viscous diffusion length Lv = νTf 3–4 mm and does not depend on the Burger number while the secondary mode corresponding to the spiralling arms does scale with the deformation radius. These three-dimensional instabilities localized in time (less than one inertial period) and space (the edge of the anticyclonic lens) provide an efficient mechanism of turbulent dissipation which cascades energy toward small scales in the frontal region. However, outside the outcropping region the potential vorticity conservation is well verified. 3.2. Cyclonic and anticyclonic PV patches 3.2.1. Initial state and experimental configuration We used the term “PV patches” for localised positive or negative potential vorticity anomalies of constant values within a uniform PV layer. The “PV patch” model is the generalisation of the Rankine vortex (cylindrical vorticity patch) for a rotating shallow-water layer. It is the simplest description of potential vortic-

3. Simple case studies of geostrophic adjustment

Figure 22.

359

Initial configuration of the experimental setup corresponding to an anticyclonic (a) and a cyclonic (b) PV-patch and their respective PV profiles (c) and (d).

ity front with no outcropping. It could be, for instance, a simplified description of the cyclonic polar vortex in the stratosphere. The corresponding experimental configurations for anticyclonic and cyclonic “PV-patches” are shown respectively in Figures 23(a) and (b). A two-layer stratification with a small thickness parameter δ = 0.125 is first realized. Then, a bottomless cylinder is used to produce a height step at the two-layer interface. Assuming that the thin upper layer follows the reduced-gravity RSW equations, the initial PV distribution is uniform inside (r < Rc ) and outside (r > Rc ) the cylinder. Unlike the “warm core” lens configuration (Figure 17) the potential vorticity exhibits a discontinuity (but not a singularity) at r = Rc due to the finite jump in the layer thickness. Besides, both positive and negative circular PV jumps could be obtained (Figures 23(b), (c)). A positive (negative) thickness anomaly in the upper layer corresponds to a negative (positive) PV jump and will generally lead to a localized anticyclonic (cyclonic) circular ring or vortex. 3.2.2. Dynamical stages The very initial stage of adjustment differs from the warm-core lens configuration. Just after the withdrawal of the transparent cylinder, the vertical density jump gets tilted and a local overturning motion is initiated at the initial position of the cylindrical wall. However, due to the rotation, the overturning motion is stopped after

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Figure 23. Dynamical evolution of the interface of an anticyclonic PV-patch corresponding to Bu = (Rd /Rc )2 0.084; λ = 0.5; α (1) 1.6; δ 0.125. The snapshots are taken at (a) t = 0, (b) t = Tf , (c) t = 2Tf , (d) t = 3Tf and (e) t = 10Tf . The movie is available on http://gershwin.ens.fr/stegner.

one inertial period and a localized shock (steep density front) occurs as shown in Figures 23(b) and (b). Due to the absence of outcropping front no gravity current head is visible for the PV-patch configuration. Afterwards, the thickness anomaly reaches an equilibrium. Even if small fluctuations could be detected, this mean adjusted state holds for a relatively long time. According to Figures 23(c)–(e) and 24(c)–(e), for a small Burger number configuration (here Bu = 0.084) the amplitude and size of the mean adjusted state remain close to the initial unbalanced height profile. Besides, the thickness anomaly remains almost unchanged from t = 2Tf to t = 20Tf . Hence, the system reaches a quasi-steady state in a very short time, approximately one or two inertial periods. For higher Burger numbers, the amplitude of the fluctuations is larger and the system seems to be far from an equilibrium. However, using an accurate time averaging to filter out

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Figure 24. Dynamical evolution of the interface of an cyclonic PV-patch corresponding to Bu = (Rd /Rc )2 0.084; λ = −0.5; α (1) 1.6; δ 0.125. The snapshots are taken at (a) t = 0, (b) t = Tf , (c) t = 2Tf , (d) t = 3Tf and (e) t = 10Tf . The movie is available on http://gershwin.ens.fr/stegner.

the fast wave motion (see below), an averaged mean state is reached with the same rapidity. This characteristic time for adjustment (one or two inertial periods) does not depend on the size or the amplitude of the initial PV-patch. 3.2.3. Rotating shallow-water predictions We present here the calculation of classical Rossby adjustment theory for the cylindrical “PV-patch” configuration (Figure 22). As for the “warm-core” lens case, the small thickness ratio and weak motions in the lower layer justify the use of the reduced gravity RSW equations for the upper layer dynamics. Here again we neglect, at the first order of approximation, the dissipation. Therefore, we use the same set of dimensionless equations as the “warm-core” lens configuration, but we need to consider two distinct regions of uniform PV. We will use the index 0 for the inner PV anomaly region (r < rf ) and the index 1 for the outer region (r > rf ) where rf is the radial position of the PV jump in the final adjusted state. For the case of a anticyclonic PV-patch (Figure 22(c)) we expect a radial extension of the PV front (rc < rf ) while for the cyclonic PV-patch (Figure 22(d)) we expect a radial contraction (rf < rc ). The Lagrangian conservation of potential vorticity implies a constant but distinct value of PV for all fluid parcels within each region

Chapter 6. Experimental Reality of Geostrophic Adjustment

362

of the upper fluid layer. Hence, for the inner PV anomaly region we have: Q0 (r < rf ) =

1 1 + ∂r v0 + v0 /r = 1+λ h0

(47)

while for the outer region we have: Q1 (r > rf ) = 1 =

1 + ∂r v1 + v1 /r h1

(48)

where the relative PV anomaly is given initially by λ = η1 /H1 . Then, looking for a steady adjusted state, implies the cyclo-geostrophic balance (41) and according to (47)–(48) we get the second order non-linear ordinary differential equations: v2 1 vi (49) ∂r (r∂r vi ) − 2 − Qi vi + i = 0 r r r with the boundary conditions: v0 (0) = 0,

(50)

v0 (rf ) = v1 (rf ),

(51)

h0 (rf ) = h1 (rf ) ⇒ v0 v1 + 1 (rf ) = ∂r v1 + + 1 (rf ). (1 + λ) ∂r v0 + r r

(52)

Besides, far away from the potential vorticity front (r rf ), a localized solution satisfies the geostrophic balance which implies neglecting the non-linear terms in (49). In such case, the general solution of the linearized equation (49) is expressed through Bessel functions. The outer velocity of a localized adjusted state should then decay at infinity as a modified Bessel function of the second kind v1 (r → +∞) ∝ K1 (r).

(53)

For a given radius rf , we can solve equations (49)–(53) numerically by appropriate shooting methods. Then, as for the “warm-core” lens configuration, the angular momentum conservation leads to the same implicit relation (46) between rf and the initial position of the front rc (i.e. the dimensionless cylinder radius). The velocity and the height profiles of the steady adjusted PV-patch are then fixed by two dimensionless parameters: rc = Rc /Rd = Bu−1/2 and λ = η1 /H1 . Examples of velocities and height profiles for both cyclonic and anticyclonic PV patches are given in Figure 25. For large Burger numbers, in other words for a small cylinder radius in comparison with the deformation radius, the adjusted state corresponds to a localized vortex (Figure 25(a)). The velocity profile exhibits a core solid rotation analogous to Rankine vortices. For small Burger numbers (Figure 25(b)), the adjusted state corresponds to a circular jet (i.e. a circular veloc-

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363

Figure 25. Velocity (dashed lines) and thickness profiles (solid lines) predicted by the standard geostrophic adjustment for two different sizes of the initial PV-patch: rc = Rc /Rd 0.45 (a) and rc = Rc /Rd 4.47 (b). The anticyclonic (thin line) and the cyclonic (thick line) profiles are given respectively for λ = 0.5 and λ = −0.5.

ity ring). For all these cases, the maximum velocity radius corresponds to rf , the final position of the PV jump. Unlike, the “warm-core” lens configuration, the velocity profiles for PV patches are always continuous. Besides, in agreement with previous studies (Kuo and Polvani, 2000), the geostrophic adjustment process

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Chapter 6. Experimental Reality of Geostrophic Adjustment

Figure 26. Profiles of the upper layer thickness for a cyclonic (a) and an anticyclonic (b) vortex resulting from the initial PV-patch rc = Rc /Rd 3.47 (Bu = 0.083) and λ = 0.5 or λ = −0.5. The corresponding velocity profiles are displayed in (c) and (d). All the profiles were time-averaged over one inertial period Tf . These mean profiles are shown at various time: t = 2Tf (filled circle), t = 5Tf (filled triangle), t = 10Tf (open circle) and t = 20Tf (open square).

induces a cyclone–anticyclone asymmetry. For the same amplitude of the initial potential energy fluctuation, the cyclonic structures will be here more intense than the anticyclonic ones. Indeed, according to Figure 25, for the same relative amplitude of the initial thickness anomaly, the maximum velocity of cyclonic vortices (vmax /(f Rd ) = 0.3 for λ = −0.5 and rc = 4.47) will always be higher than of the anticyclonic ones (vmax /(f Rd ) = −0.2 for λ = 0.5 and rc = 4.47). 3.2.4. Mean adjusted state As for the warm-core lens configuration, we used time averaging over one or two inertial periods Tf in order to extract the slow dynamics of the height profile and

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365

Figure 27. Spatio-temporal diagram of the relative fluctuations of the upper layer thickness for an anticyclonic (a) and a cyclonic (b) PV-patch. The time evolves along the y axis from top (t = 0) to bottom (t = 7Tf ), while the layer thickness is plotted along the x axis corresponding to a full length of 260 mm. The grayscale levels were decalibrated and intensified in order to enhance the contrast. The white rectangular area on both panels corresponds to the initial positive (λ = 0.5) or negative (λ = −0.5) height anomaly.

the velocity field. According to Figure 26, we observe that both the mean height and the velocity profiles remain almost constant for several inertial periods, at least up to 20Tf for PV patches having small Burger number values (Bu = 0.083 in Figure 26). Hence, the time-averaged state have reached an equilibrium even though small wave motions could be detected both in the inner region of the PV anomaly (Figure 27) and the outer region. For the cyclonic structure, the mean adjusted state coincides perfectly with the predictions of standard geostrophic adjustment (Figures 26(a), (c)). However, for the anticyclonic structure a significant discrepancy occurs for the velocity field (Figure 26(d)). The maximum velocity is at least two times smaller than the predicted one. Such anticyclonic

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dissipation, in comparison with the non-dissipative predicted state, was observed in all the experiments from small to large Burger numbers Bu = rc−2 . Hence, the cyclone-anticyclone asymmetry becomes even more pronounced with this unpredicted dissipation. As for the warm-core lens configuration (which corresponds to the asymptotic limit λ → +∞), we could suspect that this rapid dissipation of kinetic energy is due to a transient three-dimensional instability which affects only the anticyclonic PV-fronts. However, dye visualisations appeared to be less efficient for the PV-patch experiments and we could hardly capture small-scale perturbations. We should note that an outcropping PV front (PV singularity) leads to intense velocities (Ro 1) in comparison with a PV-step front (PV discontinuity) which induces continuous velocity field close to the geostrophic balance. Indeed, the Rossby number never exceed Ro = 0.3 in the PV-patch experiment, therefore ageostrophic motions and related instabilities are expected to be weaker than for the outcropping configuration. 3.2.5. Inertial and sub-inertial wave activity The geostrophic adjustment process is expected to transfer a small (large) amount of the initial potential energy to the fast wave motion for PV patches having small (large) Burger number corresponding to rc = Rc /Rd > 1 (rc < 1). Hence, for the small Burger number case described above Bu = 0.083, the amplitude of the wave fluctuations were about few percents of the mean upper layer thickness. The sensitivity of the LIF technique was high enough to quantify this wave activity in the inner region (inside the PV anomaly) and in the outer region. Spatio-temporal diagrams (Hovmöller plots) of the wave oscillations within the cyclonic and the anticyclonic PV patch are rendered in Figure 27. This plot shows qualitatively the temporal variations (y axis) of the upper layer thickness across a diameter (x axis). The grayscale levels were decalibrated and intensified in order to enhance the contrast for a better visualisation. Unlike the outcropping configuration, the two-layer interface extends here to the whole experimental domain and the inertiagravity waves could freely propagate in the outer region outside the PV anomaly. Nevertheless, a significant wave activity remains for a long time (several inertial periods) inside the PV anomaly even if the mean steady state is already adjusted (Figure 26). A similar behavior was found in previous theoretical (Plougonven and Zeitlin, 2005) and numerical (Kuo and Polvani, 2000) studies dealing with sharp PV fronts in the RSW dynamics. The most striking result is a strong cyclone–anticyclone asymmetry in the wave frequency. According to the spatio-temporal plots, the oscillation is faster for the positive PV anomaly (Figure 27(b)) in comparison with the negative PV anomaly (Figure 27(a)). Indeed, if we measure the relative fluctuations of the upper layer thickness at the center (r = 0), the frequency is sub-inertial (ω/f 0.7) in the anticyclonic PV patch while the inertial (ω f ) frequency is found in the cyclonic PV patch (Figure 28).

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Figure 28. Evolution of the relative amplitude of the upper layer thickness at the center (r = 0) of the initial height anomaly. The case of a cyclonic (anticyclonic) PV-patch is displayed in the upper (lower) panel.

In the rotating shallow-water configuration, the appearance of sub-inertial modes (ω f ) corresponds to trapped modes in other words, these modes must have an evanescent structure outside the PV patch. If the relative vorticity is strong enough, a finite number of trapped modes could appear in anticyclonic vorticity region only (Kunze, 1985; Klein and Treguier, 1993; Young and Ben Jelloul, 1997; Llewellyn Smith, 1999; Plougonven and Zeitlin, 2005). The present experiment shows, for the first time in the laboratory, the existence of sub-inertial modes within an anticyclonic PV-patch. However, according to Figure 28, these modes have a finite lifetime. Unlike, the long-lived trapped modes these sub-inertial waves probably radiate their energy to the lower layer. According to Figure 28, there is no cyclone–anticyclone asymmetry in the lifetime of the inner wave modes. Besides, according to the spatio-temporal graph

368

Figure 29.

Chapter 6. Experimental Reality of Geostrophic Adjustment

Experimental three-layer setup (a) and initial distribution of the potential vorticity (b).

displayed in Figure 27, the characteristic size of the inner wave structure, both the cyclonic (inertial) and the anticyclonic (sub-inertial) one, decays with time. This is a signature of dispersive effects, which could be induced by the high value of the wave aspect ratio α (2) = H2 /L 10 in the lower layer. 3.3. Uniform PV front 3.3.1. Initial state and experimental configuration The geostrophic adjustment of a motionless horizontal density gradient generally leads to a baroclinic tilted front corresponding to a simplified model of synoptic atmospheric fronts. However, recent studies (Ou, 1984; Blumen and Wu, 1995; Kalashnik, 2004; Plougonven and Zeitlin, 2005) have shown that even if the initial unbalanced state is smooth, a well defined continuous adjusted state may no longer exist. Indeed, for the case of uniform potential vorticity when the horizontal density gradient is sharp enough the steady adjusted solution exhibits discontinuities in both the density and the velocity field when the front outcrop the top or the bottom boundary. We use a three layer setup to study the adjustment of a uniform PV front (Figure 29(a)). Two upper fluid layers having different density ρ1 and ρ2 but the same thickness H1 are initially separated by a bottomless cylinder. A third deep and dense lower layer acts as a neutral layer which separates the thin upper layers from the bottom boundary. According to the small thickness parameter δ = 0.125 and the weak motion in the lower layer we assume that two upper layers follow the reduced-gravity RSW equations. Hence, if the upper layers have exactly the

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369

same thickness, the initial PV distribution is uniform and have the same value inside (r < Rc ) and outside (r > Rc ) the cylinder (Figure 29(b)). However, as for the warm-core lens configuration, the PV distribution exhibits a singularity at r = Rc for both the inner layer ρ2 and the outer layer ρ1 due to the vanishing of the layer thickness. = √ We can define two baroclinic deformation radii namely: Rd ((ρ2 − ρ1 )/ρ2 )gh/f related to the tilted density interface between the two √ upper layers, RD = ((ρ3 − ρ2 )/ρ3 )gh/f related to the horizontal density interface between the upper layers and one barotropic deformation radius correspond√ ing to the dense bottom layer RB = gH /f 1 m. The density differences between the layers (ρ2 − ρ1 = 3–25 g l−1 ; ρ3 − ρ2 100 g l−1 ) were adjusted, in the present experiment, in order to get Rd = 2–3 cm RD 12 cm. Besides, the size of the rotating tank L = 45 cm was large enough (L Rc Rd ) to neglect side wall effects. More details on the experimental procedure are given in (Mitkin, Stegner, Zeitlin and Pichon, 2006). 3.3.2. Dynamical stages Several dynamical stages were observed during the adjustment process. Just after the withdrawal of the separating cylinder, the inner dense fluid spreads radially at the bottom interface. During this very initial stage, the flow exhibits strong three-dimensional motions (Figure 30(b)) identical to those in a gravity current’s head (Paterson, Simpson, Dalziel and van Heijst, 2006). At this stage horizontal vorticity is generated at the interface between the inner and the outer upper fluid layers. After half of the inertial period the radial tilting of the density front is stopped and a reverse flow occurs. Then, in about one inertial period this tilted baroclinic front reaches an equilibrium characterized by an oscillating mean state. Theses oscillations can be seen in the fluctuations of the extremal positions rin and rout of the tilted front (Figures 30(c) and (d)). At longer time (t = 5 − 10Tf ) this tilted front experiences a large-scale baroclinic instability. The initial volume of dense fluid loses its axial symmetry and splits in two vortices which move away from the center of the tank. Hence, the vertical laser sheet does not capture the central cross-section of the density field any more (Figure 30(e)). 3.3.3. Rotating shallow-water predictions We assume here that: viscosity and dissipative effects are negligible (Re 1, Ek 1); each layer follows the rotating shallow-water dynamics (α 1); top and bottom boundary conditions are free-slip and rigid lid (δ 1, Bu∗ 1). Under these assumptions, the geostrophic adjustment of the density front is now controlled by a single parameter, namely the Burger number Bu. We look here for an axisymmetric steady state, solution of the RSW equations in both the light outer layer 1 and the dense inner layer 2. For simplicity, we

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Figure 30. Vertical cross-section of the density front between the inner layer ρ2 (white region) and the outer layer ρ1 (dark region) visualized by LIV. The snapshots were taken at t = 0 (a), t = 0.5Tf (b), t = Tf (c), t = 1.5Tf (d) and t = 5Tf (e) were Tf = 2π/f = 5 s is the inertial period. This experiment corresponds to rc = Rc /Rd = 2.1 (Bu = 0.22; α = 1).

neglect the cyclostrophic terms in the horizontal momentum equations. Such approximation is valid when Rc Rd . In this case, the steady state satisfies an exact geostrophic balance and therefore using the pressure continuity at the interface we get v1 − v2 = ∂r η

(54)

where η is the dimensionless thickness of the inner layer 2. The Lagrangian PV conservation leads to a constant PV value for all fluid parcels in both layers: 1 + ζ1 = 1, 1−η 1 + ζ2 Q2 (r rout ) = =1 η

Q1 (r rin ) =

(55) (56)

where ζi = 1r ∂r (rvi ) is the relative vorticity and rin (rout ) is the position of the upper (lower) intersection of the tilted density front with the top (bottom) boundary (Figure 31(b)). Note that, the boundary conditions η(rin ) = 1 and η(rout ) = 0 im-

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371

Figure 31. Density front (a), velocity (b) and vorticity (c) fields in the inner and outer layers of the adjusted steady-state according to the Rossby adjustment theory when rc = 3.33 (Bu = 0.09).

ply a singularity in the PV field at the ends of both layers, even if the PV have the same constant value within the layers. Such singularities will be the source of discontinuities in the vorticity and velocity field of the adjusted state (Figures 31(b) and (c)). Then, the angular momentum conservation or the mass conservation gives both the same implicit relations between (rin , rout ) and the initial radius of the density front rc = Rc /Rd = Bu−1/2 : 2 2 + 2rin v1 (rin ) = rout + 2rout v2 (rout ). rc2 = rin

(57)

Besides, outside of the region of the tilted front (r rin and r rout ) there is no radial displacement of fluid parcels. Therefore, the angular momentum conservation implies: v2 (r rin ) = v1 (r rout ) = 0.

(58)

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Chapter 6. Experimental Reality of Geostrophic Adjustment

Then, according to equations (54), (55) and (56) we obtain the following system of linear equations: 1 1 ∂r 2 φ + ∂r φ − 2 + 2 φ = 0, (59) r r ∂r (rψ) = −r (60) where according to (57) and (58) φ(r) = v1 − v2 satisfies the following boundary conditions 1 r2 1 rc2 − rout φ(rin ) = (61) rin − c ; φ(rout ) = 2 rin 2 rout and ψ(r) = v1 (r) + v2 (r) =

1 rc2 −r . 2 r

(62)

For a given initial radius rc = Bu−1/2 we can solve numerically (59) with (61) using a standard shooting method. An example of height, velocity and vorticity profiles in both layers are given in Figure 33 corresponding to rc = 3.33 (Bu = 0.09). Due to the volume conservation in cylindrical geometry, the front displacement in the outer layer (rc − rin ) is not identical to the front displacement in the inner layer (rout − rc ). This leads to higher velocity amplitude in the outer layer (Figure 31(b)) according to (57). Even if the velocity field is strongly baroclinic (of opposite direction in the upper and the lower layer) the vorticity is anticyclonic in both layers (Figure 31(c)) and reaches the extreme value ζ = −f at both ends of the tilted density front. 3.3.4. Mean adjusted state As for the previous cases, we use a time averaging over one inertial period, in order to separate the slow dynamics of the mean front and the fast dynamics of the oscillations. This temporal averaging filters out the fast dynamics from both the density front and the azimuthal velocity field. According to Figure 32 the qualitative structure of the mean adjusted state measured in the experiment is in agreement with the geostrophic adjustment predicted by a simple two-layer RSW model. The averaged velocity profile, measured close to the upper free surface, is displayed in Figure 32(b). According to the standard inviscid adjustment model (solid line) the velocity is expected to be discontinuous in the outer layer at the upper edge of the front. This is obviously unrealistic in a physical system where dissipative processes occur. Hence, during the adjustment process a strong but continuous cyclonic shear is formed instead of the discontinuous velocity jump predicted by the inviscid theory. The width of this cyclonic shear is much smaller than the deformation radius (0.2 − 0.3Rd ).

3. Simple case studies of geostrophic adjustment

373

Figure 32. Comparison between the mean experimental adjusted state (dots) and the Rossby adjustment model (solid line). The vertical cross-section of the density profile (a) corresponds to Rc /Rd = 2.1 (Bu = 0.22) while the horizontal azimuthal velocity (b) measured close to the upper free surface corresponds to Rc /Rd = 3.33 (Bu = 0.09).

Besides, the vorticity in such thin shear layer exceeds the planetary vorticity (ζ = 3 − 4f in Figure 32(b)) and may induce fast small-scale instabilities. However, the spatio-temporal resolution of the particle image velocimetry we used could hardly capture such small-scale instability patterns. 3.3.5. Small-scale instabilities By means of LIV, we could visualize a horizontal cross-section of the sharp density gradient just below the free surface. The dynamical evolution of this sharp gradient is shown in Figure 33. After the release of the bottomless cylinder the upper front experiences a rapid radial contraction. Due to the angular momentum conservation, this radial contraction generates at the same time a strong azimuthal flow. During this very initial stage, small disturbances appear at the edge of the front. Using an edge detection image processing we could accurately measure the initial wavelength λ of this instability (Figure 33(b)). Due to its rapid growth rate this instability is probably not affected by the rotation and the wavelength of

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Chapter 6. Experimental Reality of Geostrophic Adjustment

Figure 33. Visualization of small-scale disturbances at t = 0 (a), t = 0.25Tf (b), t = 0.5Tf (c) and t = 0.75Tf (d) in a horizontal cross-section of the density front (Bu = 0.09) just bellow the upper free surface. Local image processing of edge front detection are shown in (b) and (c), the black pixels corresponds to high values of the intensity gradient. The movie is available on http://gershwin.ens.fr/stegner.

the small-scale perturbations does not depend on the deformation radius (Mitkin, Stegner, Zeitlin and Pichon, 2006). In the present case, unlike the outcropping lens configuration, no secondary instability occurs and the non-linear saturation of the initial perturbation leads to the formation of strong cusps and small cyclones appear according to Figure 33(c). Here again the size of the intense cyclones could be much smaller that the deformation radius and remain independent from this latter. These cat-eye patterns look like a classical horizontal shear instability. Nevertheless, due to the baroclinic structure of the flow, the vertical extension of these small cyclones is limited and they should be formed preferentially at the top or the bottom edge of the density front. Besides, these vortices are transient features of the adjustment process. Indeed, after half an inertial period the front reaches its maximal contraction and the small cyclones disappear during the reverse oscillation (Figure 33(d)).

4. What do we learn from laboratory experiments?

375

4. What do we learn from laboratory experiments? Laboratory experiments can hardly reproduce the complex thermodynamics (moisture, turbulent boundary layer convection, evaporation, air-sea fluxes. . .) and the wide range of dynamical regimes (Re 1; δ 1) encounters in the atmosphere or the ocean. However, the physical modeling of rotating shallow water flows is very useful especially for the geostrophic adjustment process, where several dynamical features occur at various temporal and spatial scales. Previous laboratory experiments have shown that the geostrophic adjustment is a rapid process (Ungarish, Hallworth and Huppert, 2001; Bouruet-Aubertot and Linden, 2002; Rubino and Brandt, 2003; Thivolle-Cazat, Sommeria and Galmiche, 2005). But very few studies investigate the characteristic time of this process, especially when strong wave activity is present. According to all the cases we studied, a mean adjusted state is reached after approximately one or two inertial periods Tf . The rapidity of the geostrophic adjustment does not depend on the size or the amplitude of the initial unbalance state. The mean state is obtained from a simple time-averaging over Tf , in order to filter out the fast wave motion. We say that this averaged state reaches an equilibrium (i.e. gets adjusted) when the inverse scale of its temporal evolution remain small in comparison with the characteristic wave frequency. Hence, even if a strong wave activity is present in the initial region of unbalance, the mean flow could nevertheless be adjusted. This experimental observation is in good agreement with the standard hypothesis of dynamical splitting between the fast (ε 1) and the slow (ε 1) component of motion (Reznik, Zeitlin and Ben Jelloul, 2001; Zeitlin, Reznik and Ben Jelloul, 2003). In the limit of small Rossby numbers, the asymptotic analysis shows that the slow component of motion does not feel the fast one (Chapter 2). Therefore, the existence of a mean adjusted state does not depend on the presence (or not) of fast wave motion. Besides, the experimental results for both the warm-core lens configuration and the uniform PV-front configurations showed that a mean adjusted state could be extracted from the wave motion with a simple time-averaging even for finite Rossby numbers. Hence, according to the whole set of experiments, the fast component of motion seems to have only a weak influence (if any) on the evolution of the mean adjusted state for a wide range of parameters (Ro < 1, Bu 0.1–10, −0.5 < λ < 0.5). These two-layers or three-layers experiments also show that the PV conservation remain robust even if the initial state does not satisfy the assumptions of the rotating shallow-water model. Indeed, in almost all the cases, three-dimensional and non-hydrostatic motions (shocks or gravity current head) could occur in the early stage of adjustment. Nevertheless, the prediction of the RSW model based on the PV conservation gives a correct estimate of the mean adjusted state. A very

376

Chapter 6. Experimental Reality of Geostrophic Adjustment

good agreement is found for the cases of cyclonic PV front when there is no outcropping. However, the PV conservation can be locally broken in the case of outcropping fronts when the initial PV profile exhibits a singularity (i.e. the layer thickness vanish at a given position). In such case, all the experiments exhibit transient and three-dimensional instabilities localized around the PV singularity. These instabilities are an efficient mechanism of turbulent dissipation which rapidly cascades energy toward small scales in the frontal region. For the uniform PV front configuration, small and intense cyclones are formed in a very short time (∼0.5Tf ) during the adjustment of a large-scale anticyclonic front. The rapid formation of these structures, which are much smaller than the deformation radius, were not predicted by the standard scenario of adjustment and they could hardly be captured by standard numerical simulations which have limited spatial resolution. The laboratory experiment shows here a new mechanism of formation of small and intense structures within a large-scale synoptic front. The relaxation of any unbalanced initial state in a rotating shallow-water model will always lead to the emission of Poincaré waves (away from lateral boundaries). In a real laboratory experiment, both hydrostatic inertia–gravity waves and non-hydrostatic inertial waves could be emitted at the same time and the spectral gap between the fast and the slow component of motion could then be filled. However, according to our experiments and previous studies (Bouruet-Aubertot and Linden, 2002; Rubino and Brandt, 2003; Thivolle-Cazat, 2003) the energy released due to the wave modes during the adjustment is mainly concentrated around the inertial frequency. A significant wave activity remain for a long time (several inertial period) inside both the cyclonic and the anticyclonic structures even if the mean steady state is already adjusted. For some specific configurations the anticyclonic structures may exhibit sub-inertial oscillations. Such wave activity is in good agreement with the RSW model predictions (Chapters 2 and 5) and confirm the dynamical splitting between the fast waves and the slow adjusted motion. However, the inertia–gravity waves detected in the experiment have a dispersive behavior due to the finite value of the aspect ratio parameter α. This could explain why we did not see any evidence of the wave breaking events predicted in the RSW framework (Chapter 5). The small-scale shocks we observed in the very initial stage of adjustment seems to be due to a local overturning event rather than a propagating wave leading to breaking.

Acknowledgements The studies on geostrophic adjustment presented in this chapter were done in collaboration with P. Bouruet-Aubertot, T. Pichon, V. Mitkin and V. Zeitlin which are warmly acknowledged.

References

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List of Contributors

F. Bouchut, Département de Mathématiques et Applications, Ecole Normale Supérieure et CNRS, UMR 8553, 45, rue d’Ulm, 75230 Paris Cedex 05, France; [email protected] (Ch. 4) S.B. Medvedev, Institute of Computational Technologies, 6 Lavrentiev av., 630090 Novosibirsk, Russia; [email protected] (Ch. 3) G.M. Reznik, P.P. Shirshov Institute of Oceanology, Moscow, Russia; [email protected] yahoo.com (Ch. 2) A. Stegner, Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France; [email protected] (Ch. 6) V. Zeitlin, Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France; [email protected] (Chs. 1, 2, 5)

381

Author Index

Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages.

Abgrall, R., 231, 256 Abramowitz, M., 15, 31, 35, 43 Adrian, R.J., 340, 377 Afanasyev, Y.D., 346, 347, 377 Afanasyev, Ya.D., 334, 377 Allen, J.S., 9, 10, 43, 51, 64, 118 Arnold, V.I., 124, 133, 137, 157, 163, 184 Aubertin, A., 222, 256 Audusse, E., 214, 222, 223, 231, 237, 240, 246, 254

Boyd, J.P., 108–111, 116, 118, 299, 300, 305, 310, 319, 320 Bracco, A., 341, 377 Brandt, P., 353, 375, 376, 378 Bretherton, C.S., 107, 118 Bretherton, F.P., 4, 44 Bristeau, M.-O., 214, 222, 223, 231, 237, 240, 246, 254 Bruno, A.D., 137, 185 Bühler, O., 33, 43, 277, 304, 320

Babin, A., 65, 118, 154, 184 Baer, F., 41, 43, 52, 118, 124, 131, 184 Bakai, A.S., 148, 184 Bale, D.S., 214, 221, 231, 254 Balkovsky, E., 175, 184 Baraille, R., 214, 255 Ben Jelloul, M., 16, 23, 25, 44, 45, 54, 55, 58, 69, 70, 72, 74, 79, 106, 118, 120, 267, 321, 367, 375, 378, 379 Benilov, E.S., 24, 25, 43, 71, 77, 118, 120 Bermúdez, A., 214, 220, 254 Berndt, S.B., 301, 319 Beunier, M., 334, 378 Bialynicki-Birula, I., 175, 185 Biello, J.A., 298, 320 Birkhoff, G.D., 157, 185 Blumen, W., 19, 43, 48, 118, 368, 377 Bogaevskii, V.N., 154, 185 Bokhove, O., 10, 43, 50–52, 64, 118, 120, 125, 187 Botchorishvili, R., 214, 254 Bouchut, F., 197, 199, 200, 206, 209, 211, 214, 215, 220, 222–225, 229, 231, 233, 237, 245, 246, 254, 256, 270, 272, 280, 286, 305, 309, 310, 312, 319 Bouruet-Aubertot, P., 337, 349, 350, 353, 354, 357, 358, 375, 376, 377, 378

Camassa, A., 42, 43 Carnevale, G.F., 331, 377 Carr, J., 124, 185 Castro, M.J., 214, 220, 231, 255, 256 Chacón Rebollo, T., 214, 231, 255 Chang, P., 315, 320 Charney, J.G., 14, 43, 125, 131, 185 Chinnayya, A., 214, 255 Cho, H.-R., 182, 185 Cicogna, G., 154, 185 Cushman-Roisin, B., 14, 24, 43, 72, 118, 338, 377 Dalziel, S.B., 344, 369, 377, 378 Deconinck, H., 231, 256 Dewar, W.K., 54, 119 Didelle, H., 327, 336, 378 Domínguez Delgado, A., 214, 231, 255 Dorofeyev, V.L., 82, 119 Dritschel, D.G., 15, 43, 50, 51, 119 Dubrovin, B.A., 166, 167, 175, 185 Eames, I., 331, 377 Embid, P.F., 65, 119, 154, 185 Engelberg, S., 268, 269, 320 383

384

Author Index

Falkovich, G.E., 12, 43, 50, 67, 69, 72, 119, 125, 138, 145, 147, 148, 150, 163, 170, 173–175, 185, 187, 258, 278, 320 Farge, M., 343, 349, 350, 378 Fedorov, A.V., 300, 301, 320 Fernández Nieto, E.D., 214, 231, 255 Fernando, H.J.S., 354, 377 Ferreiro, A.M., 220, 255 Fincham, A.M., 340, 377 Flierl, G.R., 14, 26, 43, 44 Flor, J.B., 331, 377 Flores, C., 334, 378 Fomenko, A.T., 158, 185 Ford, R., 31, 32, 43 Gaeta, G., 154, 185 Gallardo, J.M., 231, 255 Gallouët, T., 214, 220, 231, 255 Galmiche, M., 352, 375, 378 García-Navarro, P., 214, 220, 255 García-Rodríguez, J.A., 220, 255 Ge, Z., 175, 185 Gent, P.R., 51, 119 Ghil, M., 327, 336, 378 Giacaglia, G.E.O., 154, 163, 185 Gill, A.E., 2, 19, 21, 25, 34, 35, 41, 43, 82, 109, 119, 150, 185, 259, 320, 325, 326, 355, 377 Godlewski, E., 191, 193, 194, 196, 197, 199, 207, 213, 235, 237, 255 Goncharov, V., 163, 185 González-Vida, J.M., 220, 255 Gossard, E.E., 291, 320 Gosse, L., 214, 255 Greenberg, J.-M., 214, 255 Greenberg, J.M., 214, 255 Greenspan, G.P., 325, 377 Griffiths, R.W., 352, 353, 377 Grimshaw, R.H.G., 55, 70, 71, 81, 82, 84, 86, 94, 96–98, 104, 120, 277, 320 Gryanik, V., 29, 31, 43 Haine, T.W.N., 347, 378, 379 Hallworth, M.A., 375, 378 Hart, D.P., 341, 377 Hart, J.E., 329, 347, 377 Harten, A., 200, 207, 255 Hayes, W.D., 301, 320 Helfrich, K.R., 82, 104, 119 Hérard, J.-M., 214, 220, 231, 255

Hermann, A.J., 82, 119 Holford, J.M., 344, 377 Holm, D.D., 9–11, 15, 42, 43, 44 Holton, J.R., 2–5, 9, 13, 14, 24, 33, 44, 146, 185, 264, 320 Hooke, W.H., 291, 320 Hoskins, B.J., 4, 44, 51, 119 Houghton, D.D., 271, 272, 320 Hunt, J.C.R., 354, 377 Huppert, H.E., 353, 375, 378 Il’yashenko, Yu.S., 124, 133, 137, 184 Ishibashi, M., 331, 377 Janenko, N.N., 263, 270, 321 Jeffreys, H., 5, 44 Jin, S., 214, 217, 225, 255 John, F., 139, 140, 185 Johnson, E.R., 82, 119 Johnson, J.A., 334, 377 Kalashnik, M.V., 368, 377 Kamenkovich, V.M., 70, 89, 105, 119 Karasev, M.V., 174, 175, 185 Kats, E.I., 157, 185 Katsaounis, T., 214, 217, 231, 255 Kevlahan, N.K.-R., 301, 320 Killworth, P.D., 54, 119 Kittelman, S., 347, 377 Klein, P., 367, 377 Klein, R., 107, 110, 119, 214, 222, 223, 231, 237, 246, 254 Kloosterziel, R.C., 331, 377 Kompenhans, J., 341, 378 Koshlyakov, M.N., 70, 119 Kraft, M., 231, 256 Krasitskii, V.P., 173, 185 Kreiss, H.-O., 125, 185 Kruse, H.P., 175, 185 Kunze, E., 367, 377 Kuo, A.C., 82, 104, 119, 272, 320, 363, 366, 377 Kurganov, A., 214, 225, 255 Kuznetsov, E.A., 67, 69, 72, 119, 145, 148, 150, 175, 185, 187 Lajeunesse, E., 222, 256 Lamb, H.H., 28, 44 Landau, L.D., 262, 320

Author Index Larichev, V.D., 18, 44, 82, 119 Lax, P.D., 200, 207, 255, 268, 320 Le Sommer, J., 34, 44, 55, 108, 110, 111, 113–116, 119, 222, 229, 245, 254, 255, 270, 272, 280, 282, 285, 286, 288, 289, 299, 300, 304, 305, 309, 310, 312, 313, 315, 317, 319, 320 Lebedev, V.V., 157, 185 LeBlond, P.H., 21, 44 Legras, B., 51, 52, 120, 124, 131, 187 Leith, C.E., 124, 185 LeRoux, A.-Y., 214, 255 LeVeque, R.J., 197, 214, 221, 231, 254, 256 Levermore, C.D., 42, 43 Levy, D., 214, 225, 255 Li, X., 315, 320 Lifshits, E.M., 262, 320 Lighthill, J., 28, 33, 44, 97, 117, 119, 270, 303, 320 Lindahl, E.G., 346, 347, 377, 378 Linden, P.F., 352, 353, 375, 376, 377 Llewellyn Smith, S., 367, 377 Lorenz, E.N., 52, 119, 124, 185 Lorenz, J., 125, 185 Lukacova-Medvidova, M., 231, 256 Lvov, V.S., 138, 147, 148, 163, 170, 173, 175, 187 Machenhauer, B., 41, 44, 52, 119, 124, 125, 185 Macías, J., 220, 255 Mahalov, A., 65, 118, 154, 184, 354, 377 Majda, A.J., 34, 44, 65, 107, 108, 110, 119, 154, 185, 298, 320 Mangeney-Castelnau, A., 222, 256 Marche, F., 222, 237, 246, 256 Marchuk, G.I., 157, 185 Margolin, L.G., 42, 44 Marsden, J.E., 11, 44, 175, 185 Marubayashi, K., 331, 377 Maslov, V.P., 174, 175, 185 Masuda, A., 331, 377 McIntyre, M.E., 32, 33, 43, 304, 320 McKean, H.P., 137, 186 McKenzie, J.F., 145, 150, 186 McWilliams, J.C., 51, 53, 82, 119, 341, 377 Medvedev, S.B., 12, 43, 50–52, 67, 69, 72, 119, 123, 125, 138, 145, 147, 148, 150, 153, 154, 160, 169, 173, 174, 176, 177, 179, 180, 184, 185–187, 258, 260,

385

264–267, 269, 272, 277, 278, 280, 282, 285, 286, 288, 289, 298, 320, 321 Melville, W.K., 82, 120, 300, 301, 320 Mendez, A.J., 346, 378 Metais, O., 334, 378 Migdal, A.B., 267, 289, 320 Mikhailov, A.V., 175, 185 Miles, J., 42, 44 Milnor, J., 158, 186 Mitkin, V., 369, 374, 377 Mitran, S., 214, 221, 231, 254 Mohebalhojeh, A.R., 50, 51, 119 Mokhov, O.I., 170, 171, 186 Monin, A.S., 56, 70, 119 Morrison, P.J., 175, 185 Murante, G., 341, 377 Mysak, L.A., 21, 44 Nadiga, B.T., 42, 44 Natvig, J.R., 222, 231, 256 Nezlin, M.V., 18, 44, 144, 145, 149, 186, 327, 331, 378 Nicolaenko, B., 65, 118, 154, 184 Nielsen, A.H., 327, 331, 378 Nikolenko, N.V., 137, 186 Noelle, S., 222, 231, 256 Nof, D., 271, 320 Nore, C., 169, 172, 186 Norton, W.A., 32, 43 Noussair, A., 214, 255 Novikov, S.P., 166, 167, 175, 185, 186 Nycander, J., 330, 378 Obukhov, A.M., 5, 44, 56, 119, 120 Olver, P.J., 163, 167, 175, 186 Ostrovsky, L.A., 277, 320, 321 Ou, H.W., 368, 378 Ozawa, T., 137, 186 Pacanowski, R.C., 315, 320 Pacheco, J.R., 354, 377 Pankratz, N., 222, 231, 256 Parés, C., 214, 220, 231, 255, 256 Paterson, M.D., 369, 378 Pavlov, V., 163, 185 Pedlosky, J., 2, 4, 13, 14, 16, 18, 22, 24, 25, 44, 51, 73, 82, 89, 105, 119, 120, 126, 145, 186, 325, 326, 328, 330, 337, 378 Peltier, W.R., 334, 377

386

Author Index

Peregrine, D.H., 271, 321 Perret, G., 343, 349, 350, 378 Perthame, B., 214, 217, 220, 222, 223, 225, 231, 237, 246, 254–256 Petviashvili, V.I., 145, 149, 150, 154, 186 Philander, S.G., 110, 120 Phillips, N.A., 25, 44, 82, 120 Pichon, T., 334, 337, 343, 349, 350, 353, 354, 357, 358, 369, 374, 377, 378 Pirulli, M., 222, 256 Piterbarg, L.I., 169, 170, 187 Plougonven, R., 184, 187, 260, 264–267, 269, 272, 277, 279, 280, 282, 285, 286, 288, 289, 292, 295–298, 300, 320, 321, 366–368, 378 Pokhotelov, O.A., 145, 149, 150, 154, 186 Polvani, L.M., 272, 320, 363, 366, 377 Poulin, F.J., 26, 44 Povzner, A., 154, 185 Pratt, L.J., 82, 104, 119, 271, 302, 321 Provenzale, A., 341, 377 Puppo, G., 222, 231, 256 Raffel, M., 341, 378 Rasmussen, J.J., 327, 331, 378 Ratiu, T., 11, 44 Raviart, P.-A., 191, 193, 194, 196, 197, 199, 207, 213, 235, 237, 255 Read, P.L., 347, 378, 379 Reznik, G.M., 16, 18, 23–25, 34, 43–45, 54, 55, 58, 69–71, 74, 77, 79, 81, 82, 84, 86, 89, 91, 94, 96–98, 103–106, 108, 110, 111, 113–116, 118–120, 222, 229, 245, 255, 267, 298–300, 304, 309, 313, 317, 320, 321, 375, 378, 379 Rhines, P.B., 82, 119, 346, 347, 377, 378 Ricchiuto, M., 231, 256 Riley, J., 334, 378 Ripa, P., 108, 111, 116, 120, 299, 300, 305, 314, 321 Roe, P.L., 206, 256 Romanova, N.N., 14, 44 Rosales, R.R., 298, 320 Rossby, C.-G., 13, 44, 118, 120, 260, 321 Rossmanith, J.A., 214, 221, 231, 254 Rozhdestvenskii, B.L., 263, 270, 321 Rubenchik, A.M., 148, 174, 187 Rubino, A., 353, 375, 376, 378

Sakai, S., 26, 27, 44, 117, 118, 120 Salmon, R., 11, 42, 44 Saujani, S., 32, 44, 51, 120 Schär, C., 271, 321 Schulman, E.I., 138, 155, 187 Scovel, C., 175, 185 Seguin, N., 214, 220, 231, 255 Serre, D., 191, 193, 241, 256 Shatah, J., 137, 186 Shepherd, T.G., 32, 44, 50–52, 64, 118, 120, 125, 151, 169, 172, 176, 180, 182, 185–187 Shrira, V.I., 277, 320, 321 Shu, C.-W., 231, 256 Shukla, P.K., 145, 150, 186 Simeoni, C., 214, 217, 220, 225, 231, 255, 256 Simpson, J.E., 369, 378 Smith, R.B., 271, 321 Smolarkevich, P.K., 42, 44 Snezhkin, E.N., 18, 44, 144, 145, 149, 186, 327, 331, 378 Sobel, A.H., 107, 118 Sommeria, J., 352, 375, 378 Sopta, L., 231, 256 Spedding, G.R., 340, 377 Stegner, A., 14, 44, 72, 120, 327, 330, 331, 334, 336, 337, 343–345, 349, 350, 353, 354, 357, 358, 369, 374, 377, 378 Stegun, I.A., 15, 31, 35, 43 Stenflo, L., 145, 150, 186 Stenum, B., 327, 331, 378 Stepanyants, Yu.A., 277, 320 Sutyrin, G.G., 24, 43, 55, 91, 103, 105, 120, 338, 377 Tabak, E.G., 298, 320 Tang, B., 24, 43, 338, 377 Teinturier, S., 327, 336, 378 Temperton, C., 124, 186 Thivolle-Cazat, E., 339, 352, 375, 376, 378 Tomasson, G.G., 82, 120 Toro, E.F., 197, 207, 256 Treguier, A.M., 367, 377 Tribbia, J.J., 41, 43, 52, 118, 124, 131, 184, 187 Tsutaya, K., 137, 186 Tsutsumi, Y., 137, 186 Turner, C.V., 298, 320

Author Index Ungarish, M., 353, 375, 378 Vallis, B., 325, 328, 355, 378 Vallis, G.K., 50–52, 64, 120, 125, 187 van de Konijnenberg, J.A., 327, 331, 378 van Heijst, G.J.F., 329, 331, 369, 377–379 Van Leer, B., 200, 207, 255 Vásquez, M.E., 214, 220, 254 Vasseur, A., 214, 254 Vautard, R., 51, 52, 120, 124, 131, 187 Vázquez-Cendón, M.E., 214, 220, 255, 256 Viboud, S., 327, 336, 378 Vilotte, J.-P., 222, 256 Vladimirov, V.A., 182, 185, 187 Voropayev, S.I., 354, 377 Vukovic, S., 231, 256 Warn, T., 50–52, 64, 120, 125, 187 Weinstein, A., 11, 44, 158, 163, 176, 187 Weiss, J.B., 341, 377 Whitham, G.B., 3, 33, 44, 97, 120, 269, 271, 290, 321 Willert, C.E., 341, 378 Williams, G.P., 14, 44 Williams, P.D., 347, 378, 379

387

Wolfram, S., 267, 321 Wu, R., 368, 377 Xing, Y., 231, 256 Yakhot, V., 175, 187 Yamagata, T., 14, 44 Yanase, S., 334, 378 Yang, J., 312, 321 Young, W.R., 70, 120, 367, 378 Yu, L., 312, 321 Zakharov, V.E., 138, 147, 148, 150, 155, 163, 169, 170, 173–175, 187 Zavala Sanson, L., 329, 379 Zeitlin, V., 14–16, 23, 25, 29, 31, 34, 44, 45, 54, 55, 58, 69, 70, 72, 74, 79, 106, 108, 110, 111, 113–116, 118–120, 138, 150, 151, 153, 154, 171, 184, 186, 187, 222, 229, 245, 254, 255, 258, 260, 264–267, 269, 270, 272, 277, 279, 280, 282, 285, 286, 288, 289, 292, 295–300, 304, 305, 309, 310, 312, 313, 315, 317, 319–321, 327, 330, 331, 344, 345, 366–369, 374, 375, 377–379

Subject Index

complex coordinates, 29, 66 complex potential, 29 complex velocity, 29, 65, 77 conservation laws, 193 conservative, 193, 198, 203, 217, 232 consistency, 199, 201, 219 Coriolis parameter, 3 critical layers, 33 cusps, 280, 296, 297, 300

1.5d model, 258, 259 2d Euler equations, 15, 28 β-plane, 4, 16, 18, 70, 72, 121, 123, 137, 138, 145, 148, 150, 151, 171, 183 β-plane approximation, 9 A absolute vorticity, 9 acoustic waves, 258 advective time scale, 13, 54 ageostrophic instabilities, 26 ageostrophic velocity, 14, 17

D density-weighted vertical average, 6 diagnostic equation, 14, 51 discontinuity calculus, 271 dispersion equation, 39 dispersion relation, 11, 17, 23, 285 drift waves, 18

B balanced dynamic, 64 balanced manifold, 52 balanced model, 48, 64, 69 baroclinic, 8, 77, 78, 285 baroclinic deformation radius, 23 baroclinic instability, 25, 287 barotropic, 8, 77, 78 barotropic gas, 8, 258 barotropic jet, 288 Bernoulli function, 302 Bessel equation, 30 boundary conditions, 4, 196, 212, 237 boundary Kelvin wave, 20, 21, 27, 80, 87, 94–96, 98, 101, 103–105 Burger number, 13, 18, 23, 24, 31, 54, 69, 272, 287, 289

E entropy, 194 entropy condition, 271, 272 entropy inequalities, 194, 199, 203, 220 equatorial β-plane, 33, 259 equatorial deformation radius, 34 equatorial inertia–gravity waves, 38, 108, 113, 298, 301, 304, 307 equatorial Kelvin wave, 35, 106, 107, 110, 113, 115, 298, 300, 315 equatorial Rossby wave, 38, 107, 110, 113, 115, 298, 305, 312, 315 equatorial waveguide, 35, 297, 299 equivalent depth, 294 equivalent height, 23, 285

C Casimir invariant, 73, 176, 177 Cayley formula, 160, 162 center of vorticity, 29 centered flux, 232, 234 CFL condition, 199, 202, 205, 237, 242 characteristic function, 122, 123, 157–160 characteristics, 269 Charney balance equation, 123, 125, 131, 183 Charney–Obukhov equation, 18 compensating jet, 101–103, 105

F f -plane, 3, 9, 11, 17, 49, 106, 121, 123–126, 131, 134, 137, 183, 258, 259 fast manifold, 123, 134 fast motion, 11, 19, 55, 56, 60, 61, 75, 83, 106, 111, 114, 265, 285 fast–slow splitting, 16, 48, 57, 72, 88, 96, 104, 106, 111, 115, 116, 258 fast-time average, 54, 58, 83 389

390

Subject Index

finite volumes, 198, 242 finite-amplitude wave, 274, 279, 281, 292 finite-volume numerical method, 272 frontal geostrophic (FG) dynamics, 14, 15, 24, 65, 77 Froude number, 26, 29, 34 G gas dynamics, 193, 195 geostrophic adjustment, 12, 13, 19, 48, 54, 74, 104, 106, 258, 260, 268, 299 geostrophic balance, 12, 13, 49, 57, 59, 60, 104, 260, 262, 265, 284, 312 geostrophic mass, 81, 91, 101, 104 geostrophic momentum, 260, 261, 292 geostrophic velocity, 14 Green function, 15 H Hamiltonian, 10, 262 Hasegawa–Mima equation, 18 Helmholtz equation, 56 hodograph method, 262 homological equation, 138, 140, 142, 148, 154 I improved quasi-geostrophic PV equation, 64, 76, 98, 106 inertia–gravity wave (IGW), 11, 48, 64, 74, 76, 104, 124, 125, 127, 136–138, 145, 147–150, 152, 153, 173, 174, 258, 260, 280, 284, 294, 314 inertial instability, 334 inertial oscillations, 12, 65, 67, 78, 79, 268, 272 initialization, 16, 52, 62, 111, 121, 123–125, 129, 131, 132, 183 instanton, 264 internal inertia–gravity waves, 23 invariant domains, 199, 205, 207, 237 invariant manifold, 121, 124, 125, 132–134, 163, 183, 184 J Jacobi transformation, 261 Jacoby identity, 168, 169 K Kelvin circulation theorem, 9

Kelvin front, 312, 314 Kelvin wave, 150, 151 Kelvin–Helmholtz instability, 26, 259, 287, 289, 291 kink, 296 Kircchoff vortex, 29, 31 Klein–Gordon equation, 56, 57 Korteweg–de Vries equation, 108, 115, 309 L Lagrangian conservation, 9 Lagrangian variables, 260, 261 linearized equation, 11, 17, 18, 20, 23, 25, 35, 49, 265, 283 linearized potential vorticity, 49 loss of hyperbolicity, 259, 291 M mass Lagrangian variable, 263, 290 mass trapping, 312, 315 material surfaces, 5 mixing, 315, 316 modons, 18, 309, 310, 312, 314, 315, 319 multipolar expansion, 29 N no-flux boundary condition, 19, 80 nonlinear frequency shift, 114 nonlinearity parameter, 13, 34, 54 numerical flux, 198, 201, 203, 218 P parabolic cylinder functions, 35, 109 plane-parallel wave, 258 planetary vorticity, 9, 17 point vortex, 15, 28 Poisson bracket, 10, 175–177, 179–182, 184 potential vorticity measurements, 339, 340, 350, 352 potential vorticity (PV), 9, 14, 20, 260, 263, 283, 286, 301, 305 primitive equations, 3 prognostic equations, 14, 51 pseudo-height, 4 PV anomaly, 9, 12, 17, 49, 50, 278 PV conservation, 11, 14, 16, 22, 74 Q QG potential vorticity, 24 quasi-geostrophic PV, 14, 22, 260, 265

Subject Index quasi-geostrophic PV equation, 59 quasi-geostrophic (QG) dynamics, 14, 15, 24, 25, 56, 75, 82 quasi-linear systems, 191, 215, 241, 268 quasi-stationary states, 267 R Rankine–Hugoniot conditions, 194, 195, 215, 271, 302, 303 rapid time scale, 55 recirculation region, 312, 315 reconstruction operator, 232, 234, 235 reduced gravity, 23, 283, 292 relative vorticity, 9 resonance condition, 134, 136, 138, 144 resonant interactions of wave triads, 258 Reynolds number, 4 Ricatti equation, 269 Riemann invariants, 269, 273 Riemann solvers, 200, 204 Rossby deformation radius, 13, 51, 75, 123, 125, 126, 128, 129, 131, 132, 137, 138, 144–146, 150, 183, 263 Rossby number, 13, 14, 16, 18, 23, 27, 28, 32, 34, 50, 54, 59, 64, 110, 132, 272 Rossby wave, 17, 18, 27, 122, 123, 125, 137, 138, 145, 147–153, 163, 169, 170, 172–174, 183 Rossby–Kelvin instability, 26, 287 rotating shallow water model, 2 S scalar nonlinearity, 18 Schrödinger equation, 67–69, 109, 266, 288 screening, 15 shock, 32, 258, 268–273, 291, 299, 301, 302, 304, 312 slaved variable, 51 slaving variable, 51 slow manifold, 16, 53, 121, 124, 125, 127–129, 131–136, 183, 258, 260, 263

391

slow motion, 11, 15, 16, 19, 24, 55, 56, 59–61, 70, 75, 83, 104–106, 111, 114, 122, 125, 128, 131–137, 145, 265 solitons, 108–110, 115, 264, 301, 305, 309, 310, 312, 313, 319 sound speed, 260, 270 spectral gap, 11, 30, 42, 104, 299 stratification parameter, 23 sub-barrier tunneling, 267 sub-inertial modes, 366, 367, 376 superbalance condition, 52 symmetric instability, 259, 264, 267, 286, 289, 291 synoptic-scale motions, 2, 13, 16 T tangent plane approximation, 3 three-dimensional instabilities, 357, 358, 366, 376 tracer transport, 311, 314–316 transonic shocks, 273 V variational principle, 262 vertical averaging, 5 vortex pair, 28, 29 vorticity equation, 9, 28 W wave emission, 28, 62 wave-breaking, 97, 108, 110, 116, 258, 269, 270, 289, 291, 298–300, 305, 312, 317 wave-drag, 63, 68, 115 wave-trapping, 259, 266, 286, 289, 317 weak solution, 193, 195, 215, 270 weak turbulence, 258 well-balanced schemes, 214, 218, 233, 242 Y Yanai wave, 36, 108, 113, 150, 151, 298, 304, 306

The book is devoted to the rotating shallow water model (RSW). This model, in spite of its simplicity and physical transparency, contains essential ingredients of large-scale atmospheric and oceanic dynamics, which explains its frequent use in applications. Besides, nonlinear dynamics of the model is of general physical interest, as it embraces vortex dynamics, wave–vortex interaction, and nonlinear dynamics of both dispersive and almost dispersionless waves. For several years the contributors of the present volume were actively working in this area, and the book is mostly based on the progress achieved by their, often joint, work. This explains the choice of accents in the book. We apologize in advance for not giving an exhaustive list of references on RSW studies, which is an almost impossible task. Recent advances in analytical, numerical, and experimental treatment of the nonlinear RSW dynamics will be presented. The volume is organized as follows. In Chapter 1 an introduction to geophysical fluid dynamics and the derivation of the model are given in order to make the whole volume self-contained. Although a large part of material is standard and is presented for convenience of a non-specialist reader, some of it is not widely known, such as Lighthill radiation or ageostrophic baroclinic instability. In Chapter 2 the analytic approach based on multi-scale asymptotic expansions is explained and recent results on the geostrophic adjustment and decoupling of wave and vortex motions are presented. Chapter 3 contains a description of the method of Poincaré normal forms and its ramifications with applications to the RSW dynamics. Chapter 4 gives an introduction to the finite-volume numerical methods and presents new efficient numerical schemes for RSW. Chapter 5 deals with essentially nonlinear RSW dynamics. Some exact results on shock formation and existence of the adjusted state are given. Exact solutions for nonlinear travelling waves are also presented. These analytic advances are combined with direct numerical simulations using the new methods of Chapter 4. Finally, Chapter 6 gives a critical review of the experimental methods for studying the RSW dynamics and the results of laboratory experiments. V. Zeitlin

v

Contents

Preface

v

Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective

1

V. Zeitlin 1. 2. 3. 4. 5. 6. 7.

Introduction Derivation of the model General properties of the RSW model An overview of the two-layer RSW model Wave–vortex interactions RSW model on the equatorial β-plane Concluding remarks Acknowledgements References

Asymptotic Methods with Applications to the Fast–Slow Splitting and the Geostrophic Adjustment

2 3 8 22 28 33 41 43 43

47

G.M. Reznik and V. Zeitlin 1. 2. 3. 4. 5. 6.

Introduction Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW Nonlinear geostrophic adjustment in the presence of a lateral boundary Nonlinear geostrophic adjustment in the equatorial region Summary and discussion Acknowledgements References

The Method of Normal Forms and Fast–Slow Splitting

48 55 73 80 106 116 118 118 121

S.B. Medvedev 1. 2. 3. 4. 5.

Introduction Slow manifold in the spatial variables on the f -plane Slow manifold in the spectral variables on the f -plane Poincaré normal forms Skew-gradient normal forms for gradient systems vii

122 124 132 137 156

Contents

viii

6. Hamiltonian normal forms 7. Normal form of the Poisson bracket for one-dimensional fluid 8. Conclusion References

163 175 183 184

Efficient Numerical Finite Volume Schemes for Shallow Water Models 189 François Bouchut 1. A few notions on hyperbolic systems 191 2. Finite volume schemes for conservative systems 197 3. Finite volumes for systems with source terms 214 4. Second-order well-balanced schemes 231 5. Two-dimensional finite volumes on a rectangular grid 240 6. Numerical tests 246 References 254 Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment V. Zeitlin 1. 2. 3. 4.

Introduction Nonlinear wave phenomena and geostrophic adjustment in 1dRSW Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide 5. Summary and conclusions Acknowledgements References

257 258 259 281 297 317 319 319

Experimental Reality of Geostrophic Adjustment A. Stegner

323

1. 2. 3. 4.

323 339 352 375 376 377

The Holy Graal of rotating shallow-water flows Potential vorticity measurements: a new challenge Simple case studies of geostrophic adjustment What do we learn from laboratory experiments? Acknowledgements References

List of Contributors Author Index Subject Index

381 383 389

Chapter 1

Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction 2. Derivation of the model 2.1. Primitive equations model of GFD 2.2. Vertical averaging of the primitive equations and multi-layer RSW models

3. General properties of the RSW model 3.1. The conservation laws 3.2. Hamiltonian formulation of the RSW model

2 3 3 5 8 8 10

3.3. The spectrum of linear excitations in the RSW model on the f -plane and the first

11

notion of the geostrophic adjustment 3.4. Characteristic parameters of the RSW model on the f -plane. Geostrophic balance and slow motions 3.5. Slow motions on the β-plane. Rossby waves 3.6. The fast-slow splitting and geostrophic adjustment 3.7. Lateral boundaries and Kelvin waves 3.8. Topographic effects in RSW

4. An overview of the two-layer RSW model 4.1. Conservation laws. Linearization. Fast and slow motions 4.2. Characteristic parameters and scaling 4.3. Slow motion: the QG regime 4.4. Slow motion: the FG regimes 4.5. Baroclinic instabilities in the two-layer RSW

5. Wave–vortex interactions Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02001-6 1

13 16 19 19 20 22 22 23 24 24 25 28 © 2007 Elsevier B.V. All rights reserved

2

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model 5.1. Wave emission by vortex motions (Lighthill radiation) in the shallow water model 5.2. Brief comments on the wave–mean flow interactions

6. RSW model on the equatorial β-plane 6.1. General properties of the equatorial RSW 6.2. A reminder on equatorial waves and their dispersion properties in the RSW model

7. Concluding remarks Acknowledgements References

28 32 33 33 34 41 43 43

1. Introduction The oceans and the atmosphere are thin, with respect to the Earth radius, stratified fluid layers on the rotating sphere with variable bottom topography under the influence of gravity. The effects of sphericity (the small Earth’s non-sphericity is usually neglected) depend on the horizontal scale of the motions of interest. They are essential for the planetary-scale motions, like tides, but for description of the so-called synoptic-scale motions with characteristic scales of the order of thousand kilometers in the atmosphere, and hundreds kilometers in the ocean, the tangent plane approximation is largely sufficient (Gill, 1982; Holton, 1979; Pedlosky, 1982). The effects of sphericity may be then introduced via the variable Coriolis parameter. Although the vertical motions may be of considerable intensity in the oceans and the atmosphere, like e.g. deep convection, on the synoptic scale average the typical horizontal velocities are at least two orders of magnitude larger than the vertical ones. Moreover, on average at these scales the vertical accelerations are negligible, and the motion within a good accuracy is in the hydrostatic balance (cf. e.g. Holton, 1979). These observations explain why the rotating shallow water model (RSW in what follows) is widely used in geophysical fluid dynamics (GFD). Taken literally, the model describes the motion of quasi two-dimensional incompressible thin fluid layer of constant density with a free surface in hydrostatic balance on the rotating plane with constant or variable Coriolis parameter, and with the centrifugal acceleration absorbed into (effective) gravity. The key ingredients of the geophysical fluid dynamics at synoptic scales, such as rotation, quasi-bidimensionality, and hydrostatic balance, are, thus, incorporated into the model. The stratification is rudimentary: the model describes a single isopycnal surface, the free surface of the fluid layer. Stratification may be taken into account in a more realistic manner by superimposing several shallowwater layers of different densities. Bottom topography is easily introduced. Thus, the model is plausible already at such purely heuristic level. We will demonstrate below how it may be obtained from the full, so called primitive GFD equations, and remind its basic properties.

2. Derivation of the model

3

The literature on RSW and its GFD applications is vast. Needless to say that beyond GFD the shallow water model has numerous and practically important applications in hydraulics. The goal of the present chapter is not to review all of the shallow-water literature (which is an impossible task already in the GFD context), but to introduce the concepts and notions necessary for understanding of the subsequent chapters and, thus, to make the whole volume self-contained. The accents in what follows are correspondingly made on the problems addressed in the following chapters, as for example the problem of splitting of balanced and non-balanced motions, which is one of the main topics of the book. Part of material of this chapter is standard and may be found in the cited textbooks, it is given below for completeness.

2. Derivation of the model The standard derivation of the (non-rotating) shallow-water equations from the Euler equations for incompressible constant-density fluid with a free surface (Whitham, 1974) is based on the expansion in vertical to horizontal scale ratio. Rotation may be easily incorporated and we will not repeat this derivation here. Instead, we will show how the RSW and multi-layer RSW equations can be derived from the full equations of the stratified rotating fluid (“primitive equations”). 2.1. Primitive equations model of GFD The hydrostatic primitive equations can be written in standard notation as follows: ∂t vh + v · ∇vh + f zˆ ∧ vh +

1 ∇h p = 0, ρ

∂z p + ρg ∗ = 0, ∂t ρ + v · ∇ρ = 0,

(2.1) (2.2)

∇ · v = 0.

(2.3)

Here ρ is density, p is pressure, the subscript “h” denotes the horizontal part of the fields, or operators which depend on r = (x, y, z) and t. The velocity field is three-dimensional: v = (vh , w) = (u, v, w), as well as the operator nabla ∇ = (∇h , ∂z ) = (∂x , ∂y , ∂z ). These equations are written on the tangent plane to a sphere rotating with angular velocity Ω. The coordinate in the meridional direction is y (northward in the geophysical context), the coordinate in the zonal direction (eastward) is x, and z is the vertical coordinate in the direction normal to the tangent plane, i.e. opposite to the effective gravity g ∗ , which includes a (small) correction due to the centrifugal acceleration (see e.g. Holton, 1979; the asterisk will be omitted from now on). zˆ is the normal unit vector in z-direction. The Coriolis parameter f entering the horizontal momentum equation is equal to f0 = 2Ω sin φ = const, if the f -plane approximation is used, and to f = f0 +βy,

4

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

if the β-plane approximation is used. Here φ is the contact point latitude which is considered as fixed or variable in the first and second approximations, respectively. Equations (2.1)–(2.3) are non-dissipative. They express momentum and mass conservation of the fluid. Molecular (or turbulent) viscosity and diffusivity may be easily introduced in the r.h.s. of the momentum and mass equations. It should be noted, however that the Reynolds numbers for free (i.e. far from boundary layers) atmospheric and oceanic motions of synoptic scale are extremely high (Pedlosky, 1982; Holton, 1979). This is why, being interested in what follows mostly in the intrinsic dynamics of the system, and not in the forced-dissipative one, we will almost never use the dissipative form of equations. We will work for simplicity with a rigid lid and flat bottom boundary conditions for equations (2.1)–(2.3): w|z=H = w|z=0 = 0,

(2.4)

which are of frequent use in GFD. Other upper boundary conditions, like free surface boundary, or prescribed form of the pressure (geopotential) function at a given vertical level (cf. e.g. Holton, 1979) may be used as well. In (2.3) the continuity equation ∂t ρ + ∇ · (ρv) = 0

(2.5)

is replaced by the incompressibility condition and mass conservation by each fluid parcel. Such (Boussinesq) approximation is totally justified in the oceanic context taking into account the negligible compressibility of water. Further simplification may be made in this case, using the fact that in the ocean ρ = ρ0 + ρs (z) + ρ (x, y, z; t),

ρ0 ρs ρ .

(2.6) ρ

Here ρ0 is a constant, ρs is a background oceanic stratification, and represents the spatio-temporal density variations. Then, the variable part of the density may be neglected in the horizontal momentum equations, and we get ∂t vh + v · ∇vh + f zˆ ∧ vh + ∇Φ = 0, ρ ∂z Φ + g = 0, ρ0 ∂t ρ + v · ∇ρ = 0, ∇ · v = 0,

(2.7)

is introduced. Remarkably, the same equations, where the geopotential Φ = up to the change of sign in the hydrostatic equation, arise in the atmospheric context if the so-called pseudo-height Z, which was first introduced by Hoskins and Bretherton (1972), is used as the vertical coordinate: (γ −1)/γ p . Z = Za 1 − (2.8) Pr 1 ρ0 p

2. Derivation of the model

5

Pseudo-height is a modified pressure coordinate (see Holton, 1979, for pressure coordinates in the atmosphere). The constants are given by Za = γ γ−1 ρPrrg , and c

γ = cpv . In these definitions, Pr and ρr are (constant) surface pressure and density, cv and cp are the specific heats of air at constant volume and constant pressure respectively. The variable ρ has a different physical meaning in the atmosphere: it becomes up to a sign the so-called potential temperature θ which is directly related to the entropy density of air parcels (cf. Holton, 1979). The pseudo-height Z and the physical height z are related as follows: θ dZ = θr dz. 2.2. Vertical averaging of the primitive equations and multi-layer RSW models The derivation of the RSW model by vertical averaging was proposed in the classical works of Jeffreys (1925) in the linear approximation, and Obukhov (1949) in the nonlinear case. We follow below the method of Obukhov by generalizing it to the multi-layer case. To lose no generality we will consider a general case of equations (2.1), together with the continuity equation (2.5), and the hydrostatic balance equation (2.2). Equations (2.1), (2.5) may be rewritten as evolution equations of the horizontal momentum density: ∂t (ρu) + ∂x ρu2 + ∂y (ρvu) + ∂z (ρwu) − fρv = −∂x p, (2.9) 2 ∂t (ρv) + ∂x (ρuv) + ∂y ρv + ∂z (ρwv) + fρu = −∂y p, (2.10) As a preliminary step we integrate these equations in the vertical direction between a pair of material surfaces z1,2 . By definition w|zi =

dzi = ∂t zi + u∂x zi + v∂y zi , dt

i = 1, 2.

(2.11)

By using (2.11) and the formula z2

z2 dz Fx = ∂x

z1

dz F − ∂x z2 F |z2 + ∂x z1 F |z1

(2.12)

z1

which is valid for any function F , we get from (2.9) z2

z2 dz ρu + ∂x

∂t z1

z2 dz ρu + ∂y

z1

z2 dz ρuv − f

2

z1

dz ρv z1

z2 = −∂x

dz p − ∂x z1 p|z1 + ∂x z2 p|z2 . z1

(2.13)

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

6

Analogously, from (2.10) we get z2

z2 dz ρv + ∂x

∂t z1

z2 dz ρuv + ∂y

z1

z2 dz ρv + f 2

z1

dz ρu z1

z2 dz p − ∂y z1 p|z1 + ∂y z2 p|z2 .

= −∂y

(2.14)

z1

From (2.5) and (2.11) we obtain z2

z2 dz ρ + ∂x

∂t z1

z2 dz ρu + ∂y

z1

dz ρv = 0.

(2.15)

z1

We introduce the integrated layer density: z2 dz ρ = −

μ=

1 p|z2 − p|z1 , g

(2.16)

z1

and define for any quantity F its density-weighted vertical average over the layer: 1 F = μ

z2 dz ρF.

(2.17)

z1

We thus obtain the averaged horizontal momentum and continuity equations from (2.13)–(2.15): ∂t μu + ∂x μ u2 + ∂y μuv − f μv z2 = −∂x

dz p − ∂x z1 p|z1 + ∂x z2 p|z2 ,

(2.18)

z1

∂t μv + ∂x μuv + ∂y μ v 2 + f μu z2 = −∂y

dz p − ∂y z1 p|z1 + ∂y z2 p|z2 ,

(2.19)

z1

∂t μ + ∂x μu + ∂y μv = 0.

(2.20)

With the help of hydrostatic relation (2.2), the pressure at any point inside the layer (z1 , z2 ) is expressed in terms of the pressure at the lower material surface

2. Derivation of the model

7

and the position of this latter as follows: z p(x, y, z, t) = −g

dz ρ(x, y, z , t) + p|z1 .

(2.21)

z1

Up to now no approximation was made whatsoever, and no advantage gained with respect to original equations. In order to advance, as usual, some closure hypothesis should be made. Supposing that all variations of all the variables are weak in the vertical direction (the quasi-bidimensionality hypothesis), the correlations may be decoupled: 2

2

v ≈ vv. uv ≈ uv, u ≈ uu, (2.22) By introducing the mean density ρ: ¯ 1 ρ¯ = (z2 − z1 )

z2 dz ρ,

μ = ρ(z ¯ 2 − z1 ),

(2.23)

z1

the integral in (2.21) may be expressed in terms of ρ: ¯ p(x, y, z, t) ≈ −g ρ(z ¯ − z1 ) + p|z1 .

(2.24)

An interesting possibility is to consider ρ¯ as a given function of horizontal coordinates and/or time. We will adopt however a conventional approach and consider only the case ρ¯ = const below. Note, that what is just described is the mean field approximation. The neglected “Reynolds stresses” may be added, if needed, in the r.h.s. of the equations via e.g. the turbulent-viscosity like closures. Omitting the angle brackets in the mean fields from now on we get from (2.18), (2.19), (2.22), (2.24), with the help of (2.20), (2.23): ρ(z ¯ 2 − z1 )(∂t vh + vh · ∇vh + f zˆ ∧ vh ) (z2 − z1 )2 + (z2 − z1 )p|z1 − ∇h z1 p|z1 + ∇h z2 p|z2 . = −∇h −g ρ¯ 2 (2.25) All dependent variables in this equation are functions of horizontal coordinates and time only. We apply now the general equations (2.25) to the fluid contained between a flat bottom and a rigid lid, with boundary conditions (2.4). We choose a material surface z = z2 (x, y, t) ≡ h(x, y, t) inside the fluid, and consider two layers, one above and one below this surface. The generalization to the N -layer case is straightforward. The vertical boundaries are material surfaces. Hence, for the lower layer its lower boundary is constant: z1 = 0, while for the upper layer its upper boundary is constant: z3 = H . The generalization to the nontrivial bottom

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

8

topography is provided by taking z1 = b(x, y), a given topography function. Hence, μ1 = ρ1 (z2 − z1 ), μ2 = ρ2 (H − z2 ). From (2.20) the equations for the evolution of the interface between the layers follow, while the equations for the evolution of the horizontal momentum of the layers arise from (2.25). The two-layer rotating shallow water model thus results: ∂t vi + vi · ∇vi + f zˆ ∧ vi +

1 ∇πi = 0, ρ¯i

i = 1, 2,

(2.26)

∂t h + ∇ · (v1 h) = 0, ∂t (H − h) + ∇ · v2 (H − h) = 0,

(2.27)

π1 = (ρ¯1 − ρ¯2 )gh + π2 ,

(2.29)

(2.28)

where the subscripts 1 (2) denote the lower (the upper) layer, respectively, π2 is the pressure value at the upper boundary, the bottom topography is not introduced, for simplicity, and the subscript h is omitted in velocities and nabla. The one-layer RSW model arises in the limit ρ¯2 → 0: ∂t v + v · ∇v + f zˆ ∧ v + g∇h = 0,

(2.30)

∂t h + ∇ · (vh) = 0.

(2.31)

In the presence of nontrivial bottom topography h should be replaced by h − b in the second equation. Equations (2.30), (2.31) are barotropic, in the sense that the fluid layer moves horizontally as a whole, while equations (2.26)–(2.29) are baroclinic, as vertical shears of the horizontal velocity are possible across the whole fluid layer (0, H ). Below in this chapter, and throughout this volume, the study is centered most of the time at the one-layer RSW model, which will be simply called RSW. However, on several occasions we will illustrate the role of the baroclinic effects by using the two-layer RSW model. Note also that equations (2.30), (2.31) are equivalent to the two-dimensional barotropic gas dynamics in the presence of the Coriolis force, where h plays the role of the gas density ρ, and the equation of state is: P ∝ ρ 2 . This analogy is one of the advantages of the RSW model as it allows, for example, to apply the powerful computation methods developed in gas dynamics (see below, and Chapter 4 of the volume).

3. General properties of the RSW model 3.1. The conservation laws By construction equations (2.30), (2.31) express the (local) conservation of the horizontal momentum and mass. It is easy to see that the energy density e=h

v2 h2 +g 2 2

(3.1)

3. General properties of the RSW model

9

obeys the local conservation equation 2 v + gh = 0, ∂t e + ∇ · vh (3.2) 2 and hence the total energy, E = dx dy e, is constant for the isolated system. A peculiar Lagrangian conservation law is present in the RSW model. Its existence is, perhaps, the most specific feature of the rotating stratified systems which alone is sufficient to understand many of their dynamical properties. The Lagrangian invariant is potential vorticity q (PV in what follows) which is constructed from the vorticity ζ = vx − uy , the Coriolis parameter f , and the height h as follows: q=

ζ +f . h

(3.3)

The Lagrangian conservation of this quantity: (∂t + v · ∇)q = 0,

(3.4)

follows by combining the equation for vorticity: (∂t + v · ∇)(ζ + f ) + (ζ + f )∇ · v = 0,

(3.5)

and the continuity equation (2.31). Note that in the two- (or more) layer RSW, PV is conserved layer-wise. In Eulerian terms the PV conservation is expressed as time-independence of any integral of the form dx dy hF (q), (3.6) over the domain of the flow, if proper boundary conditions are used. Here F is an arbitrary function. The PV conservation is a consequence of the Kelvin circulation theorem (Allen and Holm, 1996). ζ is called relative vorticity in GFD, and ζ +f is called absolute vorticity because, as it is easy to see (cf. e.g. Holton, 1979), f is the planetary vorticity, i.e. vorticity due to the Earth rotation. It is constant in the f -plane approximation, and varies linearly with y in the β-plane approximation. A slight inconvenience of the PV is that it is non-zero even in the absence of motion. That is why the PV-anomaly defined by subtracting from the full PV the background PV equal to Hf00 , where H0 is the rest height, is frequently used. Thus the intuitive picture of the dynamics in the RSW model is that of fluid columns of variable height moving horizontally in such a way that each change of height is compensated by the change of vorticity, in order to preserve the PV.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

10

3.2. Hamiltonian formulation of the RSW model The RSW model possesses the Hamiltonian structure. The Hamiltonian is given by the following expression following from the Lagrangian description of the model (see Allen and Holm, 1996 for details, and also Chapter 3 of the present volume; for Hamiltonian formulation of the two-layer RSW see Bokhove, 2005): v2 h2 ¯ ·v+h +g H = dx dy (m − m) (3.7) . 2 2 Here ¯ = Rh + hv, m

(3.8)

is the Eulerian momentum density corrected due to rotation entering via the vector R defined as ∇ ∧ R = f zˆ .

(3.9)

The Hamiltonian (3.7) is constrained as δH m − hR (3.10) = . δm h The Hamiltonian equation of motion are (summation over repeated indices is understood): δH ∂t h = {h, H} = −∂i h (3.11) , i = 1, 2, δmi δH δH ∂t mi = {h, H} = −(∂j mi + mj ∂i ) (3.12) − h∂i , δmj δh v=

where the Lie–Poisson bracket for RSW is defined as (Allen and Holm, 1996): δA δB δB {A, B}(m, h) = − dx dy + h∂i (∂j mi + mj ∂i ) δmi δmj δh δA δB + (3.13) ∂i h . δh δmi Thus, equation (3.11) gives ∂t h = −∇ · (hv), and equation (3.12) together with (3.10) gives v2 ∂t v + (f zˆ + ∇ ∧ v) ∧ v + ∇ gh + = 0, 2

(3.14)

(3.15)

3. General properties of the RSW model

11

which, using the vector analysis identity: (∇ ∧ A) ∧ B + ∇(A · B) = (B · ∇)A + ai ∇bi ,

(3.16)

is equivalent to the horizontal momentum equation (2.30) in RSW. The Lagrangian conservation of PV follows from the symmetry of the Hamiltonian (or of the action) with respect to relabelling of Lagrangian particles (Salmon, 1998). The Hamiltonian formulation of the RSW model allows to apply the standard nonlinear stability analysis (Holm, Marsden, Ratiu and Weinstein, 1985) to the stationary RSW flows, as the stationary solutions realize the extrema of the Hamiltonian. 3.3. The spectrum of linear excitations in the RSW model on the f -plane and the first notion of the geostrophic adjustment The first idea of the dynamics of the RSW model may be obtained, as usual, by considering small perturbations about the rest state v = 0, h = H0 = const. The linearized equations (2.30), (2.31) in the f -plane approximation f = f0 = const are: ∂t u − f v + g∂x η = 0, ∂t v + f u + g∂y η = 0, ∂t η + H0 (∂x u + ∂y v) = 0,

(3.17)

where u, v are two components of the velocity perturbation, and η is the free surface perturbation. By looking for solutions in the form of harmonic waves with phases ωt −k·x, where ω and k are wave frequency and wave vector, respectively, we obtain the solvability condition of (3.17) in the form: ω ω2 − gH0 k2 − f 2 = 0. (3.18) The three roots of this equation correspond to the stationary solutions ω = 0 and propagative waves with the dispersion relation ω2 − gH0 k2 − f 2 = 0.

(3.19)

The polarisation relation for these waves, which are called inertia–gravity waves (IGW in what follows) are easy to obtain from (3.17), (3.19). A remarkable feature of the RSW model becomes evident from (3.18), namely that two different kinds of motion exist—the slow one and the fast one, which are separated by the spectral gap. Indeed, the minimal frequency of the fast waves is f , and the frequency of the slow motions is zero (they are, thus, infinitely slow in this approximation). Obviously the zero root of (3.18) is related to the PV conservation, because the PV conservation law (3.4) is reduced in the linear approximation to time-independence of the PV linearized about the rest state, as

12

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

it is easy to check. By using the polarisation relations for IGW, it may be seen that the waves carry no PV anomaly. Hence, the slow motions are PV-bearing, vortex ones, while the fast ones are the PV-less IGW. Moreover, nonlinear wave motions in the RSW model can be defined in an invariant way as motions with zero PV-anomaly (Falkovich and Medvedev, 1992). Examples of nonlinear waves verifying this property will be given in Chapter 5 of this volume. In the short-wave limit the IGW become dispersionless, as their phase velocity √ tends to a constant: c = gH0 . In the acoustic analogy perspective, these are just sound waves. Rotation introduces dispersion in the long-wave limit. Indeed, for the waves with frequencies close to the inertial frequency f (inertial oscillations), the group velocity is almost zero. This means that the packets of such waves, once created, will tend to stay at the same location for a long time. The solutions of the dispersion equation (3.18) are presented in Figure 1. The linear analysis above suggests that the solution of the initial-value problem with localized initial data in RSW in linear approximation will consist, in general, of IGW going out of the initial perturbation and leaving the steady vortex component at the location of the initial perturbation. We will see in the next subsection that the vortex component is in geostrophic balance in the leading order. Thus, the just described process represents adjustment to the geostrophic equilibrium, or simply geostrophic adjustment. This process, which was first introduced

√ Figure 1. Dispersion curves for inertia–gravity waves. c = gH0 and f are taken to be one, k is the absolute value of the wavenumber, and only the upper branch of the dispersion curve is presented. The ω = 0 solution is also shown in order to illustrate the spectral gap.

3. General properties of the RSW model

13

in the pioneering paper by Rossby (1938) in the RSW context is universal and of fundamental importance in GFD. 3.4. Characteristic parameters of the RSW model on the f -plane. Geostrophic balance and slow motions A peculiar property of the RSW model is the existence of the internal length-scale which is called the Rossby deformation radius: √ gH0 Rd = (3.20) . f For motions with a single typical velocity scale U and a single typical length scale L, the following non-dimensional parameters may be introduced: Ro =

U , f0 L

Bu =

Rd2 , L2

(3.21)

which are called Rossby and Burger numbers, respectively, together with the nonlinearity parameter λ = H /H0 , where H is the characteristic free-surface displacement. The characteristic time scale for vortex-type motions is T = L/U (∼ vortex turnover time) and, hence, for slow motions the Rossby number, which is the ratio of the vortex turnover frequency to the double frequency of the Earth rotation, is small. The non-dimensional RSW equations for velocity and freesurface displacement η for the motions with the typical scales L, T , U are λ Bu Ro(∂t v + v · ∇v) + zˆ ∧ v = ∇η, Ro λ∂t η + ∇ · v(1 + λη) = 0.

(3.22) (3.23)

Hence, for small Rossby numbers the Coriolis acceleration dominates in the l.h.s. of (3.22). In order to balance this term by the acceleration due to the pressure force the following relation among parameters should hold: λ Bu = O(Ro).

(3.24)

In this case the geostrophic balance f zˆ ∧ v + g∇η = 0

(3.25)

arises in the leading order in the Rossby number. In this sense the slow motions are balanced in the RSW. As is well known, the synoptic-scale motions in the atmosphere and oceans verify the geostrophic balance within a good accuracy (Holton, 1979; Pedlosky, 1982). The physical mechanism leading to balance is the geostrophic adjustment through the IGW emission. This process will be studied in detail in the next chapter.

14

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

The relation (3.25) is diagnostic as it predicts no time-evolution allowing only to express the velocity of the balanced motions via the pressure anomaly (the geostrophic velocity, or “geostrophic wind” relation). In order to get prognostic relations equations (3.22), (3.23) should be considered in the next-to-leading order in Rossby number. For this purpose the freedom in parameters ratios remaining after imposing (3.24) should be removed. This opens a possibility to have different dynamical regimes depending on the ratio of nonlinearity to Rossby number (Charney and Flierl, 1981; Romanova and Zeitlin, 1984; Williams and Yamagata, 1984). Two main regimes are possible: the standard (cf. Holton, 1979; Pedlosky, 1982) quasi-geostrophic one (QG) corresponding to the case λ ∼ Ro 1 and, hence, to L ∼ Rd . The frontal dynamics (FD), or frontal geostrophic (FG) regime (Romanova and Zeitlin, 1984; Williams and Yamagata, 1984; Cushman-Roisin, 1986) is characterized by strong displacements of the free surface λ = O(1) and, consequently, by large scales L Rd , according to (3.24). The intermediate regime called nonlinear geostrophic (NLQG) in Stegner and Zeitlin (1995), also exists and corresponds to Ro λ 1 and to large but smaller than in FD scales. The experimental confirmation of the existence of these regimes was given by Stegner and Zeitlin (1998). We start from the conventional QG regime. By calculating the first-order correction to the geostrophic velocity: va = zˆ ∧ (∂t + vg · ∇)vg ,

(3.26)

and its divergence ∇ · va = −(∂t + vg · ∇)∇ 2 η,

(3.27)

where vg is given by (3.25), and plugging this result in (3.23) we obtain the following slow-motion equation: ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η = 0. (3.28) Here and below all parameters of the order one are taken to be equal to unity for simplicity of notation and J (A, B) denotes the Jacobian of two functions A, B. This balanced equation expresses the PV conservation with PV being calculated in the leading order in the Rossby number. Indeed, the leading-order potential vorticity anomaly in non-dimensional variables is: qqg = ∇ 2 η − η,

(3.29)

and the physical meaning of equation (3.28) is that fluid particles moving with the QG-velocity (3.25) transport their QG potential vorticity (3.29). This “QG-Lagrangian” PV conservation is expressed in Eulerian terms as t-independence of any function of qqg integrated over the domain of the flow—cf. (3.6).

3. General properties of the RSW model

15

It is instructive to compare equations (3.28), (3.29) with the vorticity equation, and vorticity–streamfunction relation for the two-dimensional incompressible Euler equations. In the latter case we have ∂t ζ + J (η, ζ ) = 0,

ζ = ∇ 2 η,

(3.30)

where ζ denotes vorticity and we deliberately used the same notation and the same conventions for the streamfunction as in (3.25). The additional with respect to (3.30) terms in (3.28), (3.29) arise due to the presence of the finite deformation radius. Indeed, by restoring dimensions a factor 12 appears in front of the terms Rd

containing ∂t η in (3.28) and η in (3.29), respectively. Thus, slow-motion dynamics in the QG regime of the RSW model is vortex dynamics with modified vorticity– streamfunction relation, which tends to the standard one as Rd → ∞. While for the two-dimensional Euler equations in the unbounded domain the Green function, i.e. the streamfunction corresponding to a point vortex situated at x = x1 , is given by: 1 log|x − x1 |; ∇ 2 G(x − x1 ) = −δ(x − x1 ), 2π the related expressions for the QG equations are: 1 |x − x1 | K0 ; G(x − x1 ) = 2π Rd 1 ∇ 2 − 2 G(x − x1 ) = −δ(x − x1 ), Rd G(x − x1 ) = −

(3.31)

(3.32)

where K0 (x) is the modified Bessel function of the second kind. It has logarithmic asymptotics at small values of the argument, and exponentially decreasing asymptotics at large values of the argument (Abramowitz and Stegun, 1964): 2 −x K0 (x)|x→0 = − log x, (3.33) K0 (x)|x→∞ = e . x Hence, there is a screening of QG-vortices at distances larger than the deformation radius, while for distances smaller than Rd they behave as usual ones. This difference is however not important for the application of the powerful numerical methods based on the Lagrangian point-wise conservation of vorticity to the QG dynamics, like for example the contour advection method (Dritschel, 1997). It is worth noting that, as well as the parent RSW, or multi-layer RSW model, the QG equations admit the variational principle and Hamiltonian formulation (Holm and Zeitlin, 1998). It should not be forgotten, however, that the QG equation, as well as the FG one below, is not exact and that corrections to these equations should be taken into account e.g. for times much longer than T . Such corrections, as well as conditions

16

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

of applicability of the QG and FG models were analysed in Reznik, Zeitlin and Ben Jelloul (2001) (see also Chapter 2 of the present volume). Another remark is that IGW were completely filtered out by the choice of the time-scale T above. The consistency of this procedure will be discussed in Chapter 2. It should be emphasized that the problem of initialization, i.e. of projection of arbitrary initial data onto the slow manifold arises if equation (3.28) alone is to be used to describe the evolution of the system. It was shown by Reznik, Zeitlin and Ben Jelloul (2001) (see also Chapter 2 of the present volume) how this may be done order by order in (small) Rossby number. It should be remembered also that the derivation of the balanced QG equations above (and FG equations below) is based on the smallness of the Rossby number. At large Rossby numbers new physical mechanisms arise preventing the fast–slow splitting, such as Lighthill radiation (see Section 5 below). In the FG regime the nonlinearity is of the order one and it makes no sense to distinguish between the full height h of the free surface and its perturbation η. The formulas for the ageostrophic velocity are the same as (3.26), (3.27), with the replacement η → h. It is immediately seen from equation (3.23) that the timescale T is not appropriate for the FG regime, as ∂t h = 0 and, hence a slower time τ = Ro t should be used. The slow-motion equation in the FG regime follows: 2 h (∇h)2 ∂τ h − J (3.34) , ∇ 2 h − J h, = 0. 2 2 The physical meaning of this equation is, again, the PV conservation. Indeed, at λ ∼ 1, the leading-order contribution to PV is 1/ h in non-dimensional terms, and equation (3.34) may be rewritten in the form of advection equation for h (or, as a consequence, of any function of h): (∇h)2 ,h . ∂τ h = −J h∇ 2 h + (3.35) 2 As in the QG case, this “FG-Lagrangian” conservation of h gives rise to the infinity of Eulerian conservation laws: any function of h integrated over the domain of the flow is τ -independent. The intermediate NLQG regime may be treated along the same lines, see Reznik, Zeitlin and Ben Jelloul (2001) for details. 3.5. Slow motions on the β-plane. Rossby waves On the β-plane f = f0 + βy in (2.30), and the fourth parameter, the nondimensional β, appears in the model. If it is of the same order as Ro (which is roughly the case for synoptic-scale atmospheric and oceanic motions (Pedlosky, 1982)), the geostrophic balance still arises in the leading order, and the β-terms

3. General properties of the RSW model

17

enter only in the expressions for ageostrophic velocity: ∇ · va = −(∂t + vg · ∇)∇ 2 η − yvg .

(3.36)

(As before the order one ratios of the non-dimensional parameters are absorbed into dynamical variables.) Correspondingly, the slow-motion equation also changes: ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η − ∂x η = 0. (3.37) It still expresses the advection of the geostrophic PV anomaly, but takes into account the variable part of the planetary vorticity, y in non-dimensional terms. This new effect, with respect to the f -plane dynamics, leads to a qualitatively new phenomenon, the appearance of Rossby waves. Indeed, a formal linearization of (3.37) gives the equation ∂t η − ∂t ∇ 2 η − ∂x η = 0,

(3.38)

which has propagating wave solutions with the dispersion relation: ω = −β

kx kx2

+ ky2 +

1 Rd2

,

(3.39)

where ω is the wave frequency, and k = (kx , ky ) the wave vector, and the dimensions were restored. The dispersion curve for Rossby waves is shown in Figure 2.

Figure 2.

Dispersion curve for Rossby waves. Rd and β are taken to be one, k is the zonal component of the wave vector, and the meridional component is taken to be 0.5.

18

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Note that this curve is well separated from the dispersion curve of the IGW, if both are plotted in dimensionful units and smallness of β is taken into account. A peculiar property of Rossby waves is that their zonal phase velocity is always negative. In contradistinction with inertia–gravity waves, these waves are slow and “vortical”, as they are waves of vorticity whose propagation is possible due to the presence of the underlying gradient of the planetary vorticity (see e.g. Pedlosky (1982), for explanation of the physical mechanism of the Rossby wave propagation). As seen from (3.39) Rossby waves are strongly dispersive and strongly anisotropic. It is worth emphasizing that plane-parallel harmonic Rossby waves of arbitrary amplitude are exact solutions of (3.37). Remarkably, equation (3.37) which is called Charney–Obukhov equation coincides with the so-called Hasegawa–Mima equation which arises in the physics of drift waves in plasma (cf. e.g. Nezlin and Snezhkin, 1993). Another remarkable property of this equation is existence of localized steady-moving exact dipolar solutions, the modons which may be obtained in the way similar to the Lamb dipole in two-dimensional hydrodynamics by matching a dipolar solution and a decaying solution across a circle on the β-plane (Larichev and Reznik, 1976). Another way to introduce the Rossby waves is to consider the limit of small nonlinearities λ → 0, and large Rossby and Burger numbers in (3.22), (3.23) considered on the β-plane. In this limit the continuity equation (3.23) is reduced to the incompressibility condition for velocity, and hence the latter may be expressed in terms of a streamfunction ψ. The r.h.s. of equations (3.22) becomes negligible, and they become equivalent to the vorticity equation (3.5), i.e. the system is reduced to incompressible two-dimensional Euler equations in the presence of (non-uniform) rotation. After linearization, the equation ∂t ∇ 2 ψ + ∂x ψ = 0

(3.40)

results. This equation is equivalent to (3.38) in the limit Rd → ∞, which is consistent with the hypothesis of large Burger numbers. Finally, the corrections to the FG regime on the β-plane lead to the following equation: 2 h (∇h)2 ∂τ h − J (3.41) , ∇ 2 h − J h, − h∂x h = 0. 2 2 A peculiar combination of the so-called scalar nonlinearity hhx and the advective (Jacobian) one in this equation is to be noted. A formal linearization of this equation about the rest state h = 1 leads to the dispersionless wave equation: ∂τ η − ∂x η = 0.

(3.42)

This equation is nothing else than the long Rossby waves equation, because, as seen from (3.39), the dispersion of these latter is given by ω ∝ −kx . This result is consistent with the hypothesis of small Bu underlying (3.41).

3. General properties of the RSW model

19

3.6. The fast-slow splitting and geostrophic adjustment Thus, our semi-heuristic analysis (see the next chapter for the rigorous one) shows that the slow vortex and the fast inertia–gravity wave motions in the RSW model are dynamically split, at least for small Rossby numbers. The refined, with respect to the simple linear version discussed above, notion of the geostrophic adjustment then arises, see Blumen (1972) for a review. Any initial perturbation in RSW (and in the atmosphere or oceans, in general) will produce an inertia–gravity waves response which will rapidly propagate out of the (localized) perturbation leaving the geostrophically adjusted, or balanced part to slowly evolve. This process, at least for small Ro, is universal and accompanies the relaxation of any perturbation. Up to now we considered the RSW model in the infinite domain and in the absence of topography. We will now briefly sketch the new phenomena appearing in the presence of these new elements, which we will still treat in the dissipationless limit. 3.7. Lateral boundaries and Kelvin waves The principal modification introduced by lateral boundaries concerns the sector of the fast motions of the model, as a new type of waves, the Kelvin waves, appear in this case (see e.g. Gill, 1982). Let us consider the RSW model in a semi-infinite domain with a straight boundary along the y-axis. The boundary condition v · nˆ = 0

(3.43)

should be imposed, where nˆ is the normal vector to the boundary. Hence u|x=0 = 0.

(3.44)

By considering the linearized RSW equations (3.17) and looking for the planewave solutions with wave-frequency ω, and wavevector l in the y-direction, while keeping the x-dependence arbitrary, we reduce the system to a single equation, e.g. for the variable η: 2 ω − f 2 − c02 l 2 η (x) + (3.45) η(x) = 0, c02 where prime denotes x-differentiation. The boundary condition (3.44) takes the form:

lf η − ωη

(3.46) = 0. ω2 − f 2 x=0 Hence, two situations are possible.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

20

• If ω2 − f 2 − c02 l 2 c02

= k 2 > 0,

(3.47)

then the standard IGW solution results, with reflection at the boundary, and the dispersion relation (3.19). • If ω2 − f 2 − c02 l 2 c02

= −k 2 < 0,

(3.48)

then we get an exponentially decaying off the boundary solution η ∝ e−kx , with ω2 = c02 l 2 ,

k=

1 , Rd

(3.49)

and u ≡ 0.

(3.50)

The latter solution is the boundary Kelvin wave. It is dispersionless and propagating along the boundary. Note that its presence destroys the spectral gap, because it is spectrally well separated from the IGW only for small wavenumbers, cf. Figure 3. It follows from (3.50), (3.17) that the velocity field and the free surface elevation are in geostrophic balance in the Kelvin wave and, hence, this wave belongs to the sector of balanced motions. The dispersion curves for RSW with a straight lateral boundary are presented in Figure 3, showing the IGW and the Kelvin-wave branches. 3.8. Topographic effects in RSW The RSW equations in the presence of the bottom topography b(x, y) are written as follows: ∂t v + v · ∇v + f zˆ ∧ v + g∇h = 0, ∂t (h − b) + ∇ · v(h − b) = 0.

(3.51) (3.52)

The PV in this case is expressed as ζ +f . (3.53) h−b It is obvious from these equations that topography will influence both wave and vortex components of the flow. Thus, linearization of equations (3.51), (3.52) about the rest state h = H0 gives a linear system with variable coefficients, q=

3. General properties of the RSW model

21

√ Figure 3. Dispersion curves for inertia–gravity waves and boundary Kelvin waves. c = gH0 , and f are taken to be one, l is the meridional wavenumber, and only the upper branch of the dispersion curves for positive l is presented.

which may be solved in a number of simple cases, like idealized sea-shelves, canyons, beaches, sea-mountains, etc., see LeBlond and Mysak (1978) and references therein, leading to a variety of topographically trapped waves. Reviewing this subject, which is well represented in the book of LeBlond and Mysak (1978), is out of the scope of our presentation. Let us only mention a simplification frequently used in the oceanic context (Gill, 1982) for motions/topographies with a typical scale corresponding to a large Burger number. In this case the variations of the free surface in equation (3.52) may be neglected in the first approximation, and this equation becomes: ∇ · (H v) = 0,

(3.54)

where H (x, y) is the full variable depth of the water layer. Hence a streamfunction ψ may be introduced: H u = −∂y ψ,

H v = ∂x ψ,

(3.55)

and injected into vorticity equation (3.30) to give an equation for ψ. For instance, for one-dimensional topography H = H (x) on the f -plane this equation reads: H 1 ∂x ψ + ∂yy ψ + f 2 ∂y ψ = 0, ∂t ∂x (3.56) H H H and may be solved by the method of separation of variables, see LeBlond and Mysak (1978) for examples.

22

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

If the topography is weak, i.e. the typical variations of the bottom topography are of the order of the Rossby number, supposed to be small, then the topographic effects in the balanced motion may be obtained by expanding the PV in Rossby number. Thus, in the QG regime the PV modified by topography is: qqg = ∇ 2 η − η + b(x, y). And the QG equation is ∂t η − ∂t ∇ 2 η − J η, ∇ 2 η − J (η, b) = 0.

(3.57)

(3.58)

Thus, topography introduces a “topographic beta-effect” (Pedlosky, 1982), whence the existence, for example, of the topographic Rossby waves on the linear bottom slope becomes evident.

4. An overview of the two-layer RSW model 4.1. Conservation laws. Linearization. Fast and slow motions As before, we will consider the two-layer RSW with a rigid lid and a flat bottom on the f -plane, for simplicity. We rewrite equations (2.26)–(2.29) in a more symmetric form by introducing the deviation of the interface η between the layers, with unperturbed depths H1 and H2 (upper and lower layer, respectively: H1 + H2 = H ). The equations of motion are the horizontal momentum equations: 1 ∂t vi + vi · ∇vi + f zˆ ∧ vi + ∇πi = 0, i = 1, 2; (4.1) ρi (no summation over i; the combination i + 1 is understood modulo 2 everywhere below), the mass conservation equations in each layer: ∂t Hi − (−1)i+1 η + ∇ · vi Hi − (−1)i+1 η = 0, i = 1, 2; (4.2) and the equation of continuity of pressure at the interface: (ρ2 − ρ1 )gη = π2 − π1 .

(4.3)

The PV conservation equations in each layer readily follow: ζi + f , Hi − (−1)i+1 η where ζi = zˆ · ∇ ∧ vi is the relative vorticity in each layer. The conservation of energy: v2 η2 E = dx dy ρi Hi − (−1)i+1 η i + g 2 2 (∂t + vi · ∇)qi = 0,

i=1,2

may be easily established.

qi =

(4.4)

(4.5)

4. An overview of the two-layer RSW model

23

The straightforward linearization of the equations of motion with the harmonic wave ansatz for the solution ∝ ei(ωt−k·x) leads to the dispersion relation ω2 ω2 − gHe k2 − f 2 = 0 (4.6) between the wave frequency and wavenumber, where He =

H1 H2 ρ H1 + ρ1 H2

is called

2

the equivalent height. The two zero roots of this equation correspond to the linearized PVs (4.4). The remaining opposite roots give the dispersion relation of the internal inertia–gravity waves propagating along the interface between the layers. Their phase speed is less than the phase speed of the inertia–gravity waves propagating on the surface of the shallow water layer of the depth H1 + H2 . This is a general property of the baroclinic waves which are always slower than their barotropic counterparts. Otherwise, the properties of these waves are the same as for the surface waves in the one-layer RSW. 4.2. Characteristic parameters and scaling By introducing a typical horizontal velocity scale U and a typical horizontal scale of the motion L, the Rossby number is defined as usual. The nonlinearity parameter λ measures typical amplitudes of η. The natural characteristic scale of the vertical displacements is He , and that of the pressures Pi = ρi U Lf . Two new parameters, with respect to the one-layer RSW, appear in the model: N =2

ρ2 − ρ1 , ρ2 + ρ1

d=

H1 , H2

(4.7)

the stratification parameter, and the layers depth ratio, respectively. Correspondingly, the baroclinic deformation radius arises: Rd = g He /f, (4.8) where g = gN is the reduced gravity. It gives an internal length-scale for the baroclinic motions. Obviously, the baroclinic deformation radius is smaller than the barotropic one constructed from the full gravity and the total depth of the fluid. The baroclinic deformation radius appears naturally while non-dimensionalizing equation (4.3). Note, that in the oceanic context the two densities ρ1,2 are usually taken to be close. The stratification parameter then tends to zero and is retained only in the reduced gravity arising in (4.3), cf. e.g. Zeitlin, Reznik and Ben Jelloul (2003). R2

The Burger number, Bu = Ld2 , hence, should be defined with the help of the baroclinic deformation radius. Similarly to the one-layer case, if the (vortex turnover) time-scale T = L/U is chosen, the Rossby number arises in front of the acceleration terms in non-dimensionalized horizontal momentum equations (4.1),

24

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

and the geostrophic balance in each layer follows at small Ro, provided relation (3.24) is verified. Thus, as in the one-layer case, the spectral gap guarantees the fast-slow motions splitting at small Rossby numbers. 4.3. Slow motion: the QG regime Following the same lines as in the one-layer case the QG equations are straightforwardly derived: Di 2 ∇ πi + (−1)i+1 h¯ i+1 η = 0, Dt

i = 1, 2.

(4.9)

Di (. . .) := ∂t (. . .) + J (πi , . . .), Dt

i = 1, 2,

(4.10)

Here

i h¯ i = H1H+H , i = 1, 2, for simplicity it is supposed that d = O(1), and the 2 hypothesis of small stratification parameter is made, such that

π2 − π1 ≈ η

(4.11)

in non-dimensional terms. As in the one-layer case, the physical meaning of the QG equations is the geostrophic PV advection by the geostrophic velocity in each layer. As follows from (4.9) the dynamics of the upper and lower layers is coupled. This coupling leads to the appearance of the crucially important phenomenon of baroclinic instability which manifests itself already in this simple two-layer model and is of primary importance in oceanic and atmospheric dynamics (Holton, 1979; Pedlosky, 1982)—see below. 4.4. Slow motion: the FG regimes The FG regime corresponds to the interface displacements of the order one. The FG scaling, thus, differs from the QG one used before. It is as follows. The interface displacement η is scaled as H1 . Choosing the characteristic length-scale L we rescale the pressure perturbations πi by ρi f Ui L, where Vi , i = 1, 2, are the velocity scales in each layer. The velocity scales U1 and U2 are of the same order when the parameter d is of the order one, which corresponds to the so-called FGH (homogeneous FG) sub-regime (cf. Benilov and Reznik, 1996) or are chosen to be U2 ∼ Ro U1 for the FGI (inhomogeneous FG) sub-regime where the parameter d is small, d ∼ ε2 (cf. Cushman-Roisin, Sutyrin and Tang, 1992). The consistency of these scalings with the dynamical boundary condition at the interface (4.3) requires that in order to have order one interface displacements the Burger number Bu = ( RLd )2 should be small Bu = O(ε). Here the Rossby deformation radius is

4. An overview of the two-layer RSW model

defined with the help of H1 : Rd =

25

g H1 /f . In terms of the new variable

η2 (4.12) , 2 and η the following slow-motion equations result in the respective sub-regimes (Benilov and Reznik, 1996; Zeitlin, Reznik and Ben Jelloul, 2003): P = π1 + d −1 π2 +

• FGH sub-regime 1 J (η, P ), (4.13) 1 + d −1 1 ∂t ∇ 2 P + J P , ∇ 2 P + ∇ · (1 − η) d −1 + η J η, ∇η = 0. −1 1+d (4.14) • FGI sub-regime (∇η)2 ∂τ η + J (P , η) + J η, (1 − η)∇ 2 η − (4.15) = 0, 2 (∇η)2 = 0, ∂τ ∇ 2 P + J P , ∇ 2 P − J η, (1 − η)∇ 2 η + (4.16) 2 ∂t η =

where, as in the FG regime in the one-layer case, the slower time τ is necessary to introduce. 4.5. Baroclinic instabilities in the two-layer RSW The most drastic consequence of the baroclinicity of the two-layer RSW is the appearance of specific baroclinic instabilities. The most famous one, playing a crucial role in the atmosphere and ocean dynamics (Gill, 1980; Pedlosky, 1982) is the classical quasigeostrophic baroclinic instability. 4.5.1. The standard baroclinic instability The standard configuration to illustrate the baroclinic instability is the geostrophically balanced flow with a vertical shear of horizontal velocity between the layers: vi = (Ui , 0), Ui = ∂y π¯ i : π¯ i = −Ui y + φi (x, y, t) (the Phillips, 1954, model). The two-layer QG equations on the f -plane (4.9) are linearized around this state. Imposing periodic boundary conditions in the direction of the mean flow and periodic or channel boundary conditions in the transverse direction and looking for solutions in the form φi = Ai ei(k·x−ωt) , the condition of solvability of the algebraic equations for Ai gives for c = kωx (Pedlosky, 1982): U1 (k2 + 2h¯ 1 ) + U2 (k2 + 2h¯ 2 ) ± (U2 − U1 )2 (k4 − 4h¯ 1 h¯ 2 ) c= . (4.17) 2(k2 + h¯ 1 + h¯ 2 )

26

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Thus, the flow is unstable for perturbations with long enough wavelengths. This is the instability of balanced flow with respect to balanced slow perturbations, as all calculations are done within the framework of the balanced QG model. 4.5.2. Ageostrophic instabilities of the geostrophic flow The stability of the same initial configuration may be analyzed within the full 2-layer RSW equations. The corresponding stability analysis in the channel configuration with free-slip lateral boundaries was performed in the pioneering work by Sakai (1989) who reproduced the standard baroclinic instability for small Rossby numbers and small-k perturbations, whereas a new instability with respect to unbalanced perturbations arises at larger Rossby numbers and larger wavenumbers. It was called Rossby–Kelvin (RK) by Sakai. The classical Kelvin– Helmholtz (KH) instability of stratified shear flows arises at even larger Ro and smaller k. In Figures 4 and 5 the instability zones in the wavenumber – Froude number plane, and typical growth rates are shown. The results were obtained following Sakai in the configuration of equal layers depths by the collocation method.

Figure 4. The instability zones in the wavenumber – Froude number plane for the two-layer RSW flow in the channel with constant and different velocities in the upper and lower layer. The isolines of the growth rates are shown. The Froude number is defined as a ratio of Rossby and Burger numbers. The calculations are made following Sakai (1989) with greater resolution using the improved version of the spectral collocation method (Poulin and Flierl, 2003).

4. An overview of the two-layer RSW model

Figure 5.

27

The growth rates along the section 3 Fr = k on the previous figure. Note that the growth rate of the RK instability may exceed that of the standard baroclinic one.

There are different kinds of linear waves in the system: Rossby waves propagating due to the equivalent beta-effect produced by the inclined interface between the layers, Kelvin waves appearing due to the presence of lateral boundaries (see the preceding section), and IGW. The dynamical origin of the displayed instabilities is resonance between these waves propagating in each layer (Sakai, 1989): respectively, Rossby–Rossby for the standard baroclinic instability, Rossby–Kelvin for the RK instability, and Kelvin–Kelvin, or IGW–Kelvin, or IGW–IGW for KH instability. While the RK instability is produced by the presence of boundaries, which in any case destroy the spectral gap in the RSW (see Section 3.8), other ageostrophic instabilities persist even in the absence of boundaries. Apart from the classical KH one, characterized by high Rossby numbers, the “Rossby–Gravity” instability seen as the descending from the KH region branch in Figure 4 and corresponding to the resonance between the Rossby and the IG wave is presumably not sensitive to the presence of lateral boundaries. (Its typical growth rates are of the same order as for RK, not shown.) As in the one-layer RSW, the formal validity of the QG and FG equations and, hence, of the dynamical splitting of fast and slow variables in the two-layer RSW may be established by using the asymptotic expansions in Rossby number, see the next chapter. The corrections to these equations for longer times t ∼ Ro12 f

28

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

may be derived as well, although the fast component (waves) still does not influence the slow one at this time-scale. The just displayed ageostrophic instabilities of geostrophic flows indicate that there are limits to dynamical splitting of fast and slow variables at large Rossby numbers. Another obstacles for this splitting at large Rossby numbers are provided by wave–vortex interactions becoming effective in this limit, and in particular the Lighthill radiation.

5. Wave–vortex interactions As we have seen previously, at small Rossby numbers the vortex motions are decoupled from the wave motions. This is not longer the case at large Rossby numbers. Even in the absence of the baroclinic effects discussed in the previous section, waves and vortices do interact in this limit. We will consider below a specific wave emission by vortices, which is less familiar than the classical wavemean flow interactions due to the wave scattering on vortex structures, which we will sketch after. 5.1. Wave emission by vortex motions (Lighthill radiation) in the shallow water model The derivation of the QG and FD equations in the preceding sections indicates that, at least for small Rossby numbers, vortex and wave motions are dynamically split and non-interacting. It is, however, intuitively clear that vortex motions which stir the fluid at frequencies larger than the inertia–gravity waves threshold frequency f cannot avoid generating waves. We will show below that it is indeed the case. To introduce the phenomenon of wave radiation by vortex motions (Lighthill, 1952) in the shallow-water context we begin with the non-rotating shallow water model which is, as already said, equivalent to the two-dimensional acoustics. If the free surface is “frozen”, the motion obeys the two-dimensional Euler equations, which are equivalent to the vorticity equation (3.30). The elementary solutions of this equation are point vortices with the streamfunction given by (3.31) times the intensity κ of the vortex. Dynamics of arbitrary vorticity distributions can be modelled by representing them as superpositions of point vortices. The dynamics of a system of point vortices is essentially Lagrangian, because vorticity is a Lagrangian invariant of the two-dimensional Euler equations, and hence, any point vortex preserves its own intensity and is advected by the velocity field obtained by superposition of velocities due to the other vortices (cf. e.g. Lamb, 1932). A vortex pair is the simplest exact solution of the vorticity equation. If vortex intensities are not equal and opposite (the case of a steady moving vortex

5. Wave–vortex interactions

29

dipole), the vortices rotate around the center of vorticity with a constant angular speed: κ1 + κ2 . Ω= (5.1) 2πa 2 This vortex system has, thus, a single proper frequency and, following Gryanik (1983), we will use it to construct an archetype example of Lighthill radiation. The singular character of point vortices is not important in this context. The results are very similar if the distributed vorticity configuration is chosen, like an elliptic Kircchoff vortex (Zeitlin, 1991). We present below the technically simpler vortex pair example. For spatially two-dimensional problems, like shallow-water ones, it is convenient to introduce the complex coordinates on the (x, y)-plane: z = x + iy, z∗ = x −iy. The complex velocity field V = u−iv of a point vortex of intensity κ situated at point z0 may be described with the help of the complex potential φ(z): κ log(z − z0 ). V = ∂z φ, φ1 = i (5.2) 2π The complex potential of a vortex pair is obtained by superposition: κ2 κ1 log z − a1 eiΩt + i log z − a2 eiΩt , φ2 = i (5.3) 2π 2π where a1,2 are the algebraic distances of the vortices 1 and 2, respectively, from the origin chosen at the center of vorticity, i.e. a1 κ1 + a2 κ2 = 0,

(5.4)

and it is taken into account that the pair rotates uniformly with angular velocity (5.1). Introducing a, the distance between the vortices, and using the polar representation of z: z = reiθ , we obtain the following multipolar expansion of the complex potential (5.3) at distances much larger than the distance a between the vortices: 2 i κ1 κ2 κ1 + κ2 a φ2 |r→∞ = i e2i(Ωt−θ) + · · · . (5.5) log reiθ − 2π 4π κ1 + κ2 r Note that the dipolar term ∝ 1r is absent due to (5.4), and hence the first timedependent term in this expansion is quadrupolar. The vortex-pair solution is relevant for the full shallow water equations, provided the distance a between the vortices is much smaller than the typical wavelength of the gravity waves. In the context of the gravity-wave emission by the pair, the frequency of √ the wave is, obviously, Ω. Therefore, the wavelength is λ= Ω , where c = gH0 is the phase velocity of the gravity waves. Hence the 0 c0 characteristic Froude (Mach) number of the pair is M=

a aΩ , =√ λ gH0

(5.6)

30

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

and for small M the vortex-pair solution may be used at small distances from the origin. The linear gravity waves are solutions of equations (3.17) at f = 0. These equations can be easily reduced to a single equation for η in this case: ∂tt η − gH0 ∇ 2 η = 0.

(5.7)

Thus, in the absence of f the waves are strictly dispersionless (they are sound waves) and have constant phase velocity c0 . Note that there is no spectral gap anymore in the absence of rotation, although ω = 0 is still a solution of the dispersion relation corresponding to Lagrangian conservation of the (rotationless) PV. In what follows the flux of energy carried by the gravity waves is important. It may be obtained from the local form of the energy conservation law (3.2) in the lowest order in wave amplitudes: v2 η2 ∂t H0 + g (5.8) = −H0 ∇ · (vgη). 2 2 The total flux of energy due to waves over any closed contour on the plane can be obtained by integrating the r.h.s. of (5.8) with the help of the Stokes theorem. The physical picture of the presumable wave emission by the vortex pair is as follows. Close to the origin, where the pair is situated (inner region), the vortex pair solution is a good approximation for small M, cf. (5.6). Further off (the intermediate region) the vortex solution should be matched with the solution of the SW equation with non-frozen free surface. As the time-dependent part of the vortex solution rapidly decays with the distance to the origin, cf. (5.5), it is reasonable to take this full solution at small amplitude, i.e. in the form of gravity waves. In the outer region far from the vortex, the (non-stationary) solution is the outgoing gravity wave field carrying the energy towards infinity. If matching respecting these conditions is possible, the wave emission by the vortex is proved. Let us consider the wave solution in the outer region. In the polar coordinates which are convenient for description of the wave field produced by the localized source, η can be sought in the form: η(r, θ, t) = (5.9) ηn (r)ein(Ωt−θ) , n

and in terms of the new variable ρ = Bessel equation, as usual: 1 n2 ηn + ηn + 1 − 2 ηn = 0. ρ ρ

nΩ c0 r,

equation (5.7) takes the form of the

(5.10) (2)

The Hankel function of the second kind Hn is a solution corresponding to the outgoing waves with the energy flux (5.8) directed outward. This function has the

5. Wave–vortex interactions

following asymptotics (Abramowitz and Stegun, 1964): −n

i ρ , Hn(2) (ρ) ρ→0 → n! π 2

2 −i(ρ−n π − π ) (2) 2 4 . e Hn (ρ) ρ→∞ → πρ

31

(5.11) (5.12)

According to the general idea exposed above, the near wave field should be matched to the far vortex field. The continuity of pressure should be used for the purpose. (We do not pay attention to the monopolar time-independent part of the vortex field, which is the zero-mode solution of the wave equation (5.10), as well.) The pressure is the time derivative of the complex potential for the wave field. Hence the matching condition is

(vortex)

gη(waves) r→0 = −∂t φ2 (5.13) . r→∞ Comparing (5.5) with (5.11), we see that only the n = 2 harmonic should be present in the wave field, and the following choice of its amplitude ensures the matching (5.13): i κ1 κ2 (κ1 + κ2 )2 (2) 2Ω (waves) = H2 r e2i(Ωt−θ) . gη (5.14) c0 8π 3 c0 a4 The far-field asymptotics of the wave field can be easily calculated with the help of (5.12) giving the total energy flux at infinity ∝ M 4 (Gryanik, 1983). As was shown by Gryanik (1983) such energy loss results in adiabatic change of a (the backreaction of the wave radiation), the only variable parameter in the vortex system as the conservation of vortex intensities is guaranteed by the Kelvin circulation theorem, and gives rise to a slow separation of vortices of the same sign, and to their closing and collapse in finite time for vortices of opposite signs. A similar analysis was performed by Zeitlin (1991) for the distributed Kircchoff vortex. The behavior close to that of the vortex pair of the same sign vortices was demonstrated, with an important new phenomenon of stability loss by the vortex due to the wave radiation. The phenomenon of destabilization of vortices by wave radiation at further stages was investigated by Ford (1994). Thus, the process of wave-emission by vortices, although weak, exists and its backreaction may lead to qualitative changes in the vortex system at long enough times. What changes in the presence of rotation? In the rotating case at large Burger numbers the vortex dynamics is still described by equation (3.30), cf. (3.33) and the discussion of the Rossby waves in Section 3. Hence, if the same reasoning as above is to be applied to the rotating case, the vortex part may be considered the same if its characteristic scale is much less than the deformation radius. The ratio of the double vortex pair turnover

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

32

frequency to f is the Rossby number of the system: Ro2 =

2Ω . f

(5.15)

In what concerns the wave part, equations (3.17) are equivalent to a single equation for η: −

1 ∂ 2η − 2 η + ∇ 2 η = 0, ∂t 2 Rd

(5.16)

where the deformation radius is purposedly reintroduced in explicit form. If solutions of this system are sought in the form (5.9), the same equation (5.10) results with a modified variable ρ for the nth angular mode: nΩ nΩ ρ= (5.17) r Ro2n − 1; Ron = , c0 f where the Rossby number Ron corresponding to a given mode is introduced. An immediate consequence of this formula is that the propagating waves exist only for Ron > 1, which is consistent with the existence of the spectral gap (see discussion in Section 3). A second conclusion is that the same procedure as in the non-rotating case above (up to the constant arising in (5.17)) will allow to match the near wave field with the far field of the vortex at r such that a r Rd , and thus obtain the wave-radiation from the vortex. Therefore, the necessary condition of the wave radiation is Ro > 1, while Bu 1 is a sufficient condition. The technical condition M 1 appearing in the irrotational argument becomes Bu Ro22 , as it is easy to see. This simple example illustrates the mechanism of Lighthill radiation in the RSW model. A more elaborated study, with similar results, was performed by Ford, McIntyre and Norton (2000, 2002). The role of the spectral gap in suppression of this process at small Ro was emphasized by Saujani and Shepherd (2002). 5.2. Brief comments on the wave–mean flow interactions The Lighthill radiation is an example of how vortices engender waves at high Rossby numbers. Note that this mechanism is purely dissipationless. The opposite, i.e. vorticity generation by waves is also possible at large Rossby numbers, although it happens through dissipative effects. Indeed, as was explained at the beginning of this chapter, waves do not carry the PV anomaly. Hence, an explicit source in the PV conservation equation (3.4) is needed to produce it. Such source can be provided by enhanced dissipation due to the wave breaking. As was already stressed, the RSW model is equivalent to the (rotating) gas dynamics. Hence, shock formation can be expected for waves of large enough amplitude (and, hence, necessarily of high enough Rossby numbers), or for strong enough

6. RSW model on the equatorial β-plane

33

initial gradients. Of course, formally speaking the RSW model, e.g. as it was derived earlier in this chapter from the primitive equations, reaches the limits of its validity at the threshold of shock formation. Nevertheless, extending, as usual, the model to include weak solutions allows to treat the shocks. Those may be considered just as zones of enhanced dissipation and mixing (cf. Whitham, 1974). It is known (e.g. Lighthill, 1978) that shock fronts of variable strength and/or curvature lead to local production (and global redistribution) of vorticity. This process in the RSW context was analysed by Bühler (2000). It will be studied in detail below in Chapter 5, where, in particular, it will be proved that rotation does not prevent shock formation and specific wave-breaking patterns of equatorial waves will be presented. The second wave-breaking mechanism is wave encounter with critical layers, i.e. the locations where the phase-speed of the wave is equal to the local speed of the mean flow (e.g. Lighthill, 1978). Indeed, linearizing the RSW equations about nontrivial coordinate-dependent mean (vortex) flow shows that there are non-removable singularities at these locations, and dissipative effects are necessary to treat them. The momentum exchange between the wave-field and the mean flow due to the presence of critical layers is at the origin of fundamental atmospheric and oceanic processes (e.g. Holton, 1979). They are, however, out of the scope of the present book. (For a comprehensive introduction into the subject see Bühler, 2005.) It should be noted that dissipationless wave–vortex interactions, leading to changes in vortex flow due to the wave propagation, are also possible within the shallow-water dynamics, as was shown by Bühler and McIntyre (2003). This mechanism was not studied in the presence of rotation but, presumably, the vortex screening due to finite deformation radius will make it effective only at high enough Burger and Rossby numbers, as it was the case with Lighthill radiation.

6. RSW model on the equatorial β-plane 6.1. General properties of the equatorial RSW The tangent plane approximation to the atmospheric and oceanic motions on the sphere, which is at the base of the primitive and, as a consequence, of the RSW equations has particular properties on the equator. Indeed, the axis of rotation lying in the equatorial tangent plane, the Coriolis parameters has no constant part and its expansion in coordinates starts from βy. Thus, the equatorial β-plane RSW equations are: ∂t v + v · ∇v + βy zˆ ∧ v + g∇h = 0,

(6.1)

∂t h + ∇ · (vh) = 0.

(6.2)

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

34

The absence of f0 necessitates a redefinition of the deformation radius, which becomes Re =

(gH0 )1/4 , √ β

(6.3)

which is a natural scale in the meridional (y-) direction: Ly ∼ Re . Due to the x–y anisotropy of equations (6.1), (6.2), an important parameter is the aspect ratio of the equatorial motions: δ=

Ly . Lx

(6.4)

The nonlinearity parameter may be introduced, as usual, via the ratio of the typical perturbation of the free surface to its unperturbed position: λ=

H . H0

(6.5)

Natural time- and velocity scales are related to Re : T =

1 ; βRe

U=

gH . βRe2

(6.6)

In terms of this velocity scale, the nonlinearity parameter has also a meaning of the Rossby or Froude number: λ=

U U = Ro = √ = Fr. 2 βRe gH0

(6.7)

Another consequence of the absence of f0 at the equator is that the natural geostrophically balanced states are zonal flows with v = 0, as the inspection of Coriolis and pressure terms in (6.1), (6.2) shows. Hence the notion of slow vortex dynamics, as it was used on the midlatitude tangent plane, is not quite adapted to the equatorial region. On the contrary, the wave dynamics is very rich. The equatorial dynamics is basically the wave dynamics. As will be shown below and in Chapter 5, the fast-slow separation takes place at the equator between the fast and the slow waves, which is distinctively seen in particular in the long-wave sector, i.e. for δ 1. 6.2. A reminder on equatorial waves and their dispersion properties in the RSW model We remind below the standard linear theory of the equatorial waves (Gill, 1980; Majda, 2003) with special attention paid to initialization of different types of waves (Le Sommer, Reznik and Zeitlin, 2004), which will be used later in Chapter 5 of the present volume.

6. RSW model on the equatorial β-plane

35

The non-dimensional linearized RSW equations are: ut − yv + hx = 0,

(6.8)

vt + yu + hy = 0,

(6.9)

ht + ux + vy = 0.

(6.10)

(We use the subscript notation for partial derivatives hereafter.) Wave solutions are sought in the equatorial waveguide, by imposing exponential decay boundary condition in the meridional y-direction. The complete base of parabolic cylinder functions φn (y) (Abramowitz and Stegun, 1964): φn (y) + 2n + 1 − y 2 φn (y) = 0, (6.11) Hn (y)e−y /2 φn (y) = √ , 2n n! π 2

(6.12)

where Hn are Hermite polynomials, is used while separating independent variables x, t and y. 6.2.1. Kelvin waves By a change of variables (Gill, 1982): 1 1 (u + h); g = (u − h) 2 2 equations (6.8)–(6.10) simplify: f =

(6.13)

1 ft + fx + (vy − yv) = 0, 2 1 gt − gx − (vy + yv) = 0, 2 vt + y(f + g) + (f − g)y = 0.

(6.14) (6.15) (6.16)

They admit a particular solution with v = 0 and f = F (x − t, y);

g = G(x + t, y).

(6.17)

For solutions bounded in the meridional direction G = 0 and the equatorial Kelvin wave solution is given by u = F0 (x − t)e−y

2 /2

;

h = F0 (x − t)e−y

2 /2

;

v = 0.

(6.18)

In the context of initial-value problems (geostrophic adjustment) the function F0 is determined by expanding the initialcondition fI = 12 (uI + hI ) in a series in parabolic cylinder functions: fI = ∞ n=0 fIn φn (y). Hence, the function F0 is

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

36

Figure 6.

The Kelvin mode.

given by 1 F0 (x) = fI0 (x) = 2

+∞ (uI + hI )φ0 (y) dy.

(6.19)

−∞

The initial conditions for the Kelvin-wave part of arbitrary initial perturbation uI , vI , hI are (K)

uI

= fI0 φ0 ,

(K)

hI

= fI0 φ0 ,

(K)

vI

= 0.

(6.20)

The velocity and height pattern corresponding to the Kelvin wave is presented in Figure 6. 6.2.2. Yanai waves The Yanai-wave solution corresponds to g = 0, f = 0, v = 0. In this case from (6.8)–(6.10) it follows that 1 ft + fx + (vy − yv) = 0, 2 vy + yv = 0,

(6.21) (6.22)

6. RSW model on the equatorial β-plane

37

vt + yf + fy = 0,

(6.23)

v = v0 (x, t)φ0 (y),

(6.24)

and by expanding the function f in series in φn we get f = F1 (x, t)φ1 (y).

(6.25)

Hence 1 F1t + F1x − √ v0 = 0, 2

v0t +

√ 2F1 = 0.

(6.26)

The initial conditions for the Yanai-wave part of arbitrary initial perturbation are 1 (uI + hI1 )φ1 , 2 1

(Y)

=

(Y)

= vI0 φ0 ,

uI vI

(Y)

hI

=

1 (uI + hI1 )φ1 , 2 1 (6.27)

where, as above, we suppose that initial conditions are expanded in φn . In terms of the variable v the Yanai wave is a solution of the following problem, cf. (6.24)–(6.26), (6.27): (Y) (Y) vtt(Y) + vxt + v (Y) = 0, vI = vI0 φ0 , 1 (Y) vt t=0 = − √ (uI1 + hI1 )φ0 . 2

(6.28)

The solution has the form (6.24) with v0 given by v0 =

√ √ |σ2 |v˜I0 − i 2f˜I1 i(kx−σ1 t) σ1 v˜I0 + i 2f˜I1 i(kx+|σ2 |t) e e dk + dk , √ √ k2 + 4 k2 + 4 (6.29)

where k σ1 = + 2

k2 + 1, 4

k |σ2 | = − + 2

k2 + 1, 4

and we defined the Fourier-transforms of the initial fields: 1 ikx vI0 (x) = dk v˜I0 (k)e , (uI + hI1 ) = dk f˜I1 (k)eikx . 2 1

(6.30)

(6.31)

The velocity and height pattern corresponding to a Yanai wave is presented in Figure 7.

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

38

Figure 7.

The lowest westward-propagating Yanai mode.

6.2.3. Rossby and inertia–gravity waves The use of the full equation for the v-field which follows from (6.8)–(6.10) by eliminating u and h is necessary for determining this part of the spectrum: ∂t v − y 2 v − ∂tt v + ∂x v = 0.

(6.32)

The three initial conditions for this equation are: (RG)

v|t=0 = vI

, (RG)

vt |t=0 = − yuI (RG)

vtt |t=0 = vIyy

(RG)

+ hIy

(RG)

− y 2 vI

,

(RG) (RG) , + ∂x uIy + yhI

(6.33)

where (RG)

uI

(K)

= uI − uI

(Y)

− uI

∞

=

1 1 uIn (x)φn (y), (uI0 − hI0 )φ0 + (uI1 − hI1 )φ1 + 2 2 n=2

(6.34)

6. RSW model on the equatorial β-plane (RG)

hI

(K)

= hI − hI

39

(Y)

− hI

∞

1 1 = − (uI0 − hI0 )φ0 − (uI1 − hI1 )φ1 + hIn (x)φn (y), 2 2

(6.35)

n=2

(RG)

vI

(Y)

= vI − vI

=

∞

(6.36)

vIn (x)φn (y).

n=1

By expanding v in a series in φn , v = n vn (x, t)φn (y) we get for vn 2 ∂t ∂xx vn − (2n + 1)vn − ∂tt2 vn + ∂x vn = 0, while for its Fourier-transform v˜n (k, t) = dx e−ikx vn (x, t) we get 3 ∂ttt v˜n + k 2 + 2n + 1 ∂t v˜n − ik v˜n = 0.

(6.37)

(6.38)

Hence v˜n = vn1 (k)e−iσn1 t + vn2 (k)e−iσn2 t + vn3 (k)e−iσn3 t where σnα , α = 1, 2, 3, are the three roots of the dispersion equation σn3α − k 2 + 2n + 1 σnα − k = 0.

(6.39)

(6.40)

By introducing the projections vIn , vtIn , vttIn of the initial conditions vI , vtI , vttI , respectively, onto the corresponding φn , and their Fourier-transforms, we get the following system of algebraic equations for vn(α) 3

vnα = v˜nI ,

α=1

3

σnα vnα = iv˜tnI ,

α=1

3

σn2α vnα = −v˜ttnI .

(6.41)

α=1

The lowest eigenvalue σ1 defines the Rossby mode, the other two—the inertia– gravity wave modes. For the Rossby mode we get (1)

v1 =

1 2 k v ˜ . + iσ v ˜ − σ v ˜ n t n tt n n n I 1 1 I I 2σn31

(6.42)

Therefore, the part of the initial conditions corresponding to this wave may be determined, ∞ φn (y) dk eikx vn1 (k) vI = (6.43) n=1

and similarly for the gravity-wave modes. The Rossby wave and inertia–gravity velocity and height distributions are presented in Figures 8–10. Note that dispersion relations for Kelvin and Yanai modes follow from (6.40) at n = −1 and n = 0, respectively, while the spectrum of Rossby and inertia–gravity waves starts at n = 1.

40

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Figure 8.

The lowest Rossby mode.

Figure 9. The lowest eastward-propagating inertia–gravity mode.

7. Concluding remarks

Figure 10.

41

The lowest westward-propagating inertia–gravity mode.

6.2.4. Dispersion and spectral separation of equatorial waves The dispersion curves for lowest meridional eigenmodes of the equatorial waves are displayed in Figure 11 with special emphasis on the long waves, whose dynamics will be studied in more detail in Chapters 2, 5 of the present volume. The spectral gap between the fast and the slow long waves is indicated. The curves show the spectral separation between the long Rossby and Kelvin waves, which represent the slow motion in the equatorial waveguide, and the long Yanai and inertia–gravity waves, representing the fast motion.

7. Concluding remarks We hope that it is clear from the preceding material that the RSW model is a useful conceptual tool for understanding fundamental processes on GFD. It should be emphasized that the model is far from being academic and some crucial advances in atmospheric and ocean dynamics were achieved with its help. Without reviewing them, which is an impossible task in the limited volume, let us mention as an example a classical paper by Gill (1980), treating tropical convection with the help of the RSW model. Another example of utmost importance is the use of RSW model for developing the initialization algorithms for weather prediction (Baer and Tribbia, 1977; Machenhauer, 1977, see also Chapter 3 of this volume).

42

Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

√ Figure 11. Dispersion curves for equatorial waves. c = gH0 , k is the zonal wavenumber. The domain of long waves and the spectral gap between fast and slow waves is indicated. Only two first Rossby modes and two first inertia–gravity modes are shown.

Finally, let us mention for completeness that models beyond the standard RSW may be derived by vertical averaging of the full three-dimensional equations. Thus, starting from the three-dimensional non-hydrostatic Euler equations the socalled Green–Naghdi (GN) equations may be obtained in the leading order in small deviations from hydrostaticity by using the variational principle and vertical averaging (Miles and Salmon, 1985); these equations were rediscovered many times, see Camassa, Holm and Levermore (1996) for the history of the question. The GN equations take into account the dispersive effects for short gravity waves which are absent in RSW. The small-amplitude limit of the GN equations gives the so-called “great lake” equations (Camassa, Holm and Levermore, 1996). Although the GN equations do give useful insights in some circumstances (see e.g. Nadiga, Margolin and Smolarkevich, 1996), the SW model remains a universal leading-order approximate model, which is, in addition, “computation-friendly” (see Chapter 4 of this volume). In the subsequent chapters some recent theoretical, computational, and experimental advances in understanding the RSW dynamics will be presented.

References

43

Acknowledgements The author thanks J. Gula and J. Le Sommer for help with calculations and preparation of illustrations. Part of the presentation stems from the collaboration with G. Reznik which is gratefully acknowledged.

References Abramowitz, M., Stegun, I.A., 1964. Handbook of Mathematical Functions. Dover, New York. Allen, J.S., Holm, D.D., 1996. Extended geostrophic Hamiltonian models for rotating shallow water motion. Physica D 98, 229–248. Baer, F., Tribbia, J.J., 1977. On complete filtering of gravity modes through nonlinear initialization. Monthly Weather Rev. 105, 1536–1539. Benilov, E.S., Reznik, G.M., 1996. The complete classification of large-amplitude geostrophic flows in a two-layer fluid. Geophys. Astrophys. Fluid Dynam. 82, 1–22. Blumen, W., 1972. Geostrophic adjustment. Rev. Geophys. Space Phys. 10, 485–528. Bokhove, O., 2005. Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations. Appl. Math. Lett. 18, 1418–1425. Bühler, O., 2000. On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263. Bühler, O., 2005. Wave-mean interaction theory. In: Grimshaw, R. (Ed.), Nonlinear Waves in Fluids: Recent Advances and Modern Applications. Springer-Verlag, Vienna/New York. Bühler, O., McIntyre, M.E., 2003. Remote recoil: a new wave–mean interaction effect. J. Fluid Mech. 492, 207–230. Camassa, A., Holm, D.D., Levermore, C.D., 1996. Long-time effects of bottom topography in shallow water. Physica D 98, 258–286. Charney, J.G., Flierl, G.R., 1981. Oceanic analogs of large-scale atmospheric motions. In: Warren, B.A., Wunsch, C. (Eds.), Evolution of Physical Oceanography. MIT Press, Cambridge, MA, pp. 504–548. Cushman-Roisin, B., 1986. Frontal geostrophic dynamics. J. Phys. Oceanogr. 16, 132–143. Cushman-Roisin, B., Sutyrin, G.G., Tang, B., 1992. Two-layer geostrophic dynamics. Part I: Governing equations. J. Phys. Oceanogr. 22, 117–127. Dritschel, D.G., 1997. Introduction to “contour dynamics for the Euler equations in two dimensions”. J. Comp. Phys. 135, 217–219. Falkovich, G.E., Medvedev, S.B., 1992. Kolmogorov-like spectrum for turbulence of inertia–gravity waves. Europhys. Lett. 19, 279–284. Ford, R., 1994. The response of the rotating ellipse of uniform potential vorticity to gravity wave radiation. Phys. Fluids 6, 3694–3704. Ford, R., McIntyre, M.E., Norton, W.A., 2000. Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57, 1236–1254. Ford, R., McIntyre, M.E., Norton, W.A., 2002. Reply to Saujani and Shepherd. J. Atmos. Sci. 59, 2878–2882. Gill, A.E., 1980. Some simple solutions for heat-induced tropical circulation. Quart. J. R. Meteorolog. Soc. 106, 447–462. Gill, A.E., 1982. Atmosphere–Ocean Dynamics. Academic Press, New York. Gryanik, V., 1983. Emission of sound by linear vortex filaments. Izvestija—Atmosphere Ocean Phys. 19, 150–153.

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Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model

Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A., 1985. Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116. Holm, D.D., Zeitlin, V., 1998. Hamilton’s principle for quasi-geostrophic dynamics. Phys. Fluids 10, 800–806. Holton, J.R., 1979. An Introduction to Dynamic Meteorology. Academic Press, New York. Hoskins, B.J., Bretherton, F.P., 1972. Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29, 11–37. Jeffreys, H., 1925. On the dynamics of geostrophic winds. Quart. J. R. Meteorolog. Soc. 52, 85–104. Lamb, H.H., 1932. Hydrodynamics. Dover, New York. Larichev, V.D., Reznik, G.M., 1976. 2-dimensional solitary Rossby waves. Sov. Phys. Dokl. 231, 1077–1079. LeBlond, P.H., Mysak, L.A., 1978. Waves in the Ocean. Elsevier, New York. Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170. Lighthill, J., 1952. On sound generated aerodynamically, I. General theory. Proc. R. Soc. London A 211, 564–571. Lighthill, J., 1978. Waves in Fluids. Cambridge Univ. Press, Cambridge. Machenhauer, B., 1977. On the dynamics of gravity oscillations in a shallow water model, with application to normal-mode initialization. Contrib. Atmos. Phys. 50, 253–271. Majda, A.J., 2003. Introduction to PDEs and Waves for the Atmosphere and Ocean. Amer. Math. Soc., New York. Miles, J., Salmon, R., 1985. Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531. Nadiga, B.T., Margolin, L.G., Smolarkevich, P.K., 1996. Different approximations of shallow fluid flow over an obstacle. Phys. Fluids 8, 2066–2077. Nezlin, M.V., Snezhkin, E.N., 1993. Rossby Vortices, Spiral Structures, Solitons. Astrophysics and Plasma Physics in Shallow Water Experiments. Springer-Verlag, Berlin. Obukhov, A.M., 1949. On the problem of geostrophic wind. Izvestija—Geography and Geophysics 13, 281–306 (in Russian). Pedlosky, J., 1982. Geophysical Fluid Dynamics. Springer-Verlag, New York. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286. Poulin, F.J., Flierl, G.R., 2003. The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173–2192. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part I. Rotating shallow-water model. J. Fluid Mech. 445, 93–120. Romanova, N.N., Zeitlin, V., 1984. On quasigeostrophic motions in barotropic and baroclinic fluids. Atmos. Ocean Physics—Izvestija 20, 85–91. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res. 1, 239–263. Sakai, S., 1989. Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176. Salmon, R., 1998. Geophysical Fluid Dynamics. Oxford Univ. Press, New York. Saujani, S., Shepherd, T.G., 2002. Comments on “Balance and the slow quasimanifold. Some explicit results”, by Ford, McIntyre and Norton. J. Atmos. Sci. 59, 2874–2877. Stegner, A., Zeitlin, V., 1995. What can asymptotic expansions tell us about large-scale quasigeostrophic anticyclonic vortices? Nonlin. Proc. Geophys. 2, 186–193. Stegner, A., Zeitlin, V., 1998. From quasi-geostrophic to strongly nonlinear monopolar vortices in paraboloidal shallow-water-layer experiment. J. Fluid Mech. 356, 1–24. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Williams, G.P., Yamagata, T., 1984. Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci. 41, 453–478.

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Zeitlin, V., 1991. On the back-reaction of acoustic radiation for distributed vortex structures. Phys. Fluids A 3, 1677–1680. Zeitlin, V., Reznik, G.M., Ben Jelloul, M., 2003. Nonlinear theory of the geostrophic adjustment. Part II: Two-layer and continuously stratified models. J. Fluid. Mech. 491, 207–228.

Chapter 2

Asymptotic Methods with Applications to the Fast–Slow Splitting and the Geostrophic Adjustment G.M. Reznik P.P. Shirshov Institute of Oceanology, Moscow, Russia

V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction

48 48 54

1.1. Basic notions and general context 1.2. A synopsis of the method

2. Nonlinear geostrophic adjustment in the unbounded domain. Onelayer RSW 2.1. The QG regime 2.2. The FG regime 2.3. Comments on the influence of the β-effect

3. Nonlinear geostrophic adjustment in the unbounded domain. Twolayer RSW 3.1. Two-layer RSW model, a reminder 3.2. The QG regime 3.3. The FG regime

55 56 65 70 73 74 75 77

3.4. Discussion of the geostrophic adjustment in the two-layer RSW and comparison with the one-layer case

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary 4.1. Preliminaries 4.2. The lowest-order solution (the linear adjustment) 4.3. Dynamics of the lowest-order slow motion 4.4. The first-order solution Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02002-8 47

79 80 80 83 88 91

© 2007 Elsevier B.V. All rights reserved

48

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting 4.5. Improved QG equation in the presence of the lateral boundary 4.6. The problem of mass conservation in the localized case 4.7. The summary of the results on the adjustment in the laterally bounded domain

5. Nonlinear geostrophic adjustment in the equatorial region 5.1. The long-wave scaling and classification of possible dynamical regimes 5.2. The resumé of possible weakly nonlinear dynamical regimes in the long-wave approximation 5.3. The long-wave adjustment at Ro = O(δ 2 )

6. Summary and discussion Acknowledgements References

97 99 104 106 106 110 110 116 118 118

1. Introduction 1.1. Basic notions and general context The one- and multi-layer rotating shallow water models, as well as their parent primitive equation model, possess a gap in the spectrum of linear excitations which separate the fast inertia–gravity wave (IGW) motions and slow vortex motions (see Chapter 1 of this volume). The vortex motions at small Rossby numbers are in geostrophic balance, i.e. close to the equilibrium between the Coriolis force and the pressure force. As discussed in Chapter 1, already these observations allow to anticipate the ubiquitous character of the geostrophic adjustment process i.e. the process of relaxation of arbitrary initial perturbations to a geostrophic equilibrium state via emission of inertia–gravity waves. The oceanic and atmospheric data provide the confirmation of this fact (Blumen, 1972). At the same time, these properties of the spectrum are at the origin of the notion of dynamical splitting of the fast and slow variables. Under hypothesis of splitting, i.e. (almost) noninteraction of fast and slow motions, useful balanced models may be derived from the full dynamical equations by using exclusively slow time-scales. The QG and FG models discussed in Chapter 1 arise in this way. Their advantage is that dynamics is reduced to that of the balanced component alone, which allows for conceptual and computational simplifications, as compared to the full original (“primitive”) equations. Especially the QG model is of wide and frequent use in applications ranging from the analysis of particular oceanic and atmospheric processes to studies of global climate dynamics. It is worth mentioning that the first successful weather forecasts were accomplished with the help of the QG model. However, the QG model is not exact, as ageostrophic velocity component (cf. Chapter 1) gives rise to corrections at higher orders in Rossby number. Going beyond the standard QG, but preserving the concept of balance (possibly modified with respect to the simplest geostrophic balance) is the philosophy of the entire

1. Introduction

49

branch of research in GFD dedicated to balance and balanced models. Reviewing this activity is beyond the scope of the present chapter. We will, however, briefly expose the main ideas in order to put in this context the results of the direct asymptotic approach, which will be developed further in this chapter. We illustrate the balanced models approach using the simplest one-layer RSW model. We remind that the equations of the model are: ∂t v + v · ∇v + f zˆ ∧ u + g∇H = 0,

(1.1)

∂t H + ∇ · (vH ) = 0.

(1.2)

The Lagrangian conservation of potential vorticity (PV), or its anomaly, q=

ζ +f f , − H H0

(1.3)

is expressed by ∂t q + v · ∇q = 0.

(1.4)

The PV anomaly proves to be often more useful than the full PV (we will make no distinction between the two when it does not lead to confusion). Here ζ = ∂x v − ∂y u is the relative vorticity, H = H0 + h(x, y, t) and H0 are the full fluid depth and its unperturbed value, respectively (no topographic effects will be considered in this chapter). Another useful quantity is the linearized potential vorticity (LPV) which is equal to Ω=ζ−

f h H0

(1.5)

and satisfies in the f -plane approximation the conservation equation ∂t Ω + ∇ · (vΩ) = 0.

(1.6)

As was shown in Chapter 1, the spectrum of linear perturbations in RSW consists of zero-frequency vortex mode and the IGW spectrum: ω = ± gH0 k2 + f 2 . (1.7) The solution of the linearized RSW equations can be represented as a sum (v, h) = (vg , hg ) + (va , ha ),

(1.8)

where (vg , hg ) correspond to the vortex mode and satisfy the geostrophic balance relation: f zˆ ∧ vg + g∇hg = 0.

(1.9)

50

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

The ageostrophic part represents the fast IGW and is characterized by zero PV-anomaly and zero LPV: qa = Ωa = 0.

(1.10)

If at the initial moment PV is zero then the slow geostrophic part of the solution is absent and, vice versa, if the initial conditions are geostrophically balanced then the fast IGW do not appear in the linearized model. As to the full nonlinear system (1.1), (1.2), the condition of zero PV completely filters out the vortex component, just as in the linear case. Under this condition equations of conservation of PV and LPV are satisfied identically and the resulting equations describe uniquely the fast ageostrophic motion (Falkovich, 1992; Falkovich and Medvedev, 1992). On the contrary, the complete filtering of the wave motions from the non-linear system (1.1), (1.2) is generally impossible (excluding, of course, the case of exact stationary vortex solutions). The simplest way to see this is to consider equations for the velocity divergence D and the so called geostrophic departure ξ , sometimes called ageostrophic vorticity (cf. e.g. Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999; Mohebalhojeh and Dritschel, 2001): D = ∇ · v,

ξ = f ζ − g∇ 2 h.

(1.11)

Both are equal to zero for the linear vortex mode and non-zero for linear IGW. Equations for D, ξ show that even if both quantities are zero at the initial moment, the nonlinear self-interactions of the slow vortex mode generate these fields at subsequent times. However, in the practically important case when the typical Rossby number is small: Ro =

U = ε 1, f0 L

(1.12)

where the typical horizontal velocity scale is U , and the typical horizontal length scale is L, thus induced IGW contributions are small. The smallness of the Rossby number means smallness of the ratio of the typical wave period to the vortex advective time-scale (see Chapter 1). The QG model derived along the standard lines for small Rossby numbers in Chapter 1 may be introduced in an alternative way. Assuming that all fields change at the slow time-scale L/U f −1 , equations (1.1), (1.5) give, up to small corrections: f zˆ ∧ v + g∇h = 0, h=H

−1

Ω,

(1.13) (1.14)

equation (1.6) remaining unchanged. Here H is the Helmholtz operator ∇ 2 −

1 Rd2

and H−1 is its inverse (proper boundary conditions are understood), Rd =

1. Introduction √

51

gH0 f

is the Rossby deformation radius. The first of these equations represents the balance condition, the second gives the diagnostic relation between the height field and the prognostic variable Ω. The evolution equation for Ω is provided by the conservation law (1.6). The QG equations rewritten in the form (1.6), (1.13), (1.14) give an archetype example of the balanced model consisting, generally, of two components (Warn, Bokhove, Shepherd and Vallis, 1995): • Diagnostic balance relations filtering out the fast oscillations (in the present case this is the geostrophic relation (1.13)). • Prognostic equations determining the slow time evolution (in the present case this is equation (1.6) for LPV). Vast literature exists on balance models and their properties (see e.g. Hoskins, 1975; Gent and McWilliams, 1983; Pedlosky, 1984; Allen, 1993; Warn, Bokhove, Shepherd and Vallis, 1995; Mohebalhojeh and Dritschel, 2001, and references therein). Balance conditions and diagnostic relations vary in different models. Reviewing this literature is beyond the scope of the present chapter. Let us only mention in this context that the QG system (1.6), (1.13), (1.14) contains no IGW contributions at all, since the functions D, ξ in (1.11) are identically zero there. However, in the higher-order QG models (e.g. Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999) the fields D, ξ are, generally, non-zero and in this sense the contribution of IGW in the balanced dynamics is also non-zero. To comment on this important point we note that “slowness” of a function generally does not mean the absence of harmonics with arbitrarily high frequency in its spectrum. For example,√the Fourier transformation of the function exp(−ε 2 t 2 ) is equal to exp(−ω2 /ε 2 )/ 2ε, i.e. all harmonics are present in the spectrum of this function. For small ε 1, however, the contribution of harmonics with ω = O(1) is very small, of the order of exp(−1/ε 2 ). Similarly, the slow QG dynamics, generally, contains the contributions from the fast IGW but these contributions are exponentially small and do not appear in the straightforward asymptotic expansions in the Rossby number (see in this context also Saujani and Shepherd, 2002, and references therein). Note that the notion of slowness of the motion we used in the previous argument is well defined only for small Rossby numbers. In the jargon of balanced models the slow “prognostic” variable is called slaving variable (Warn, Bokhove, Shepherd and Vallis, 1995) or master variable (Vautard and Legras, 1986), and the variables determined from the geostrophic relation—the slaved or subordinate variables. The choice of the slaved and slaving variables is not unique. In principle any variable can be taken as a slaving one (Warn, Bokhove, Shepherd and Vallis, 1995). The general idea (applicable to general dynamical systems) is that the slaving variables are described by

52

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

non-stationary equations, and that slaving and slaved variables are related by diagnostic balance relations. In symbolic form these relations can be written in the form (Warn, Bokhove, Shepherd and Vallis, 1995): F (x, y) = 0,

(1.15)

where the slaved y and the slaving x variables constitute a complete set of the dependent variables. Equation (1.15) is called superbalance condition (Lorenz, 1980) and determines some invariant manifold in (generally, infinite-dimensional) phase space x, y. Note that in general the superbalance condition (1.15) does not require smallness of any parameter of the considered problem, i.e. some “fast” motions can belong to the balanced manifold. However, if the slaving variables are slow then the invariant manifold determines a set of slow solutions of the systems. In the RSW case one would choose x = Ω and try to find the balance conditions in the form: y = Y(Ω),

(1.16)

where y corresponds to the horizontal velocity v, or to the pair D, ξ , and Y(Ω) is some functional. In the limit ε → 0, (1.16) together with (1.6), guarantees the slowness of motion. If the slow manifold could be found, the problem of initialization, i.e. that of constructing the initial conditions for the full non-balanced equations, in such a way that the contribution of the fast IGW is minimized, would be solved. Needless to say that such initialization is of extreme importance in the meteorological applications (see the pioneering works of Baer and Tribbia, 1977 and Machenhauer, 1977 which used the RSW model for constructing the initialization algorithms). The problem with this tempting approach is that the superbalance conditions (1.15) can be written and analyzed in explicit form only for a finite dimensional systems of ordinary differential equations (cf. Vautard and Legras, 1986; Bokhove and Shepherd, 1996). Such systems may be obtained, for example, in the case of a doubly-periodic motion when each variable is expanded in a double Fourier series which is truncated then in a proper way (see e.g. Lorenz, 1986). In the general case of a system with infinite number of degrees of freedom the existence of explicit conditions of the type (1.15), and therefore the existence of the invariant slow manifold, is not proved. One can, however, construct approximate balance equations for small Rossby numbers. To do this the special asymptotic technique is used when the slaved variables are expanded in the series in ε, and the slaving variables are not (see Chapter 3 of this volume and Warn, Bokhove, Shepherd and Vallis, 1995; Medvedev, 1999). The condition (1.16) is then represented in the form: y = Y0 (Ω) + εY1 (Ω) + · · · ,

(1.17)

1. Introduction

53

where Yi (Ω), i = 0, 1, . . . , are some integro-differential operators. Convergence of this series would mean that an invariant slow manifold exists. Unfortunately, nothing is known about such convergence. However, using the expansion (1.17) one can construct a hierarchy of balanced models where the influence of the IGW is minimized. In this case the n-term expansion (1.17) provides the absence of IGW contributions within accuracy O(ε n+1 ) at least on times of the order of (f ε)−1 . We, thus, see that the problem of existence of the balanced slow manifold, which is, hypothetically, the endpoint of the geostrophic adjustment process, remains open. Some exact results may be obtained in this context within the simplified models using strictly rectilinear frontal configurations, see Chapter 5 of the present volume. Even if the slow manifold does exist, a question of its attainability in course of the geostrophic adjustment process arises (McWilliams, 1988). This is a question to be answered by studying the joint evolution of balanced and unbalanced components of motion in the relaxation (i.e. adjustment) process. The question of possible influence of the unbalanced (fast) component upon the balanced (slow) one, i.e. the question of eventual fast component drag, if the nonlinear effects are taken into account, is of primary importance in the context of such joint evolution. We will show below how the straightforward approach based on multiple time-scale expansions in Rossby number and on timeaveraging allows to answer these questions. The slow manifold will be described order by order in Rossby number with initialization of the slow fields well defined at each order. Up to the fourth order in Rossby number the problem of nonlinear geostrophic adjustment of single-scale (vortex-like) perturbations will be fully resolved. The advantage of such approach is that balance nor diagnostic relations are never imposed ad hoc. They arise from the time-averaging and removal of resonances in the equations of the fast motion. The whole procedure is thus completely algorithmic. The key element of the approach is the time-scale separation, which is well defined for small Rossby numbers because, as already explained, the Rossby number gives the ratio of fast to slow time-scales. There is no need to separate the flow in vortex and wave parts, which can not be done unambiguously in the presence of nonlinear interactions—only the well-defined separation in slow and fast components should be applied. To have a unique Rossby number characterizing the motion an assumption of a single spatial scale will be made. We will consider below the nonlinear geostrophic adjustment in unbounded domains, and then show what are the modifications introduced by the presence of lateral boundaries (as shown in Chapter 1, lateral boundaries destroy the spectral gap). A particular case of the geostrophic adjustment arises in the equatorial region with its specific spectrum of linear excitations (see Chapter 1). This case will be also treated below.

54

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

1.2. A synopsis of the method The geostrophic adjustment problem will be considered as initial value problem for arbitrary initial disturbance and treated by multi time-scale perturbation theory in Rossby number Ro. The main features of the method are summarized as follows: • technical tools are asymptotic expansions in Ro and fast-time averaging with subsequent separation of slow and fast parts of the flow; 1 ,...; • multiple time scales are used: T0 ∼ f10 , T1 ∼ Rof 0 • arbitrary surface elevations may be treated; • typical spatial scales are of the order of (QG √regime) or much greater (FG 0 ; regime) than Rossby deformation radius Rd = fgH 0 • the balanced dynamics arises as a result of removal of resonances (secular terms) in the fast dynamics. The method was applied to the one- and two-layer RSW, as well as to the full stratified primitive equations (Dewar and Killworth, 1995; Reznik, Zeitlin and Ben Jelloul, 2001; Zeitlin, Reznik and Ben Jelloul, 2003). We explain below in detail the application of the method in the case of one-layer RSW and limit ourselves by stating the results for the two-layer RSW. The starting point of the method are full non-dimensional equations. We assume that initial data are characterized by a horizontal scale L, on which the velocity and depth change by their characteristic values U and H , respectively, and introduce the Rossby and Burger numbers: Ro ≡ ε =

U , f0 L

Bu ≡ s =

Rd2 , L2

(1.18)

and the nonlinearity parameter λ = H /H0 . The PV changes in time at the advective time scale Ta = L/U equal to the time required for a fluid particle moving with a typical velocity U to travel over the typical scale L of the initial disturbance. The typical time scale of the inertia– gravity waves is f0−1 . The condition of time-scale separation is (Ta f0 )−1 =

U = ε = o(1) f0 L

(1.19)

and filtering of the fast IGW may be achieved by using exclusively the scale (f0 ε)−1 . In this case a small parameter—Rossby number—appears in front of advective derivatives in the l.h.s. of the non-dimensionalized RSW equations and the geostrophic equilibrium (1.13) always results in the leading order in ε, provided the following relation between the parameters holds: λs = O(ε).

(1.20)

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

55

Under hypothesis (1.20) the non-dimensionalization with the rapid time scale f0−1 of the RSW equations gives: ∂t v + ε(v · ∇v) + zˆ ∧ v + ∇h = 0,

(1.21)

λ∂t h + ε(1 + λh)∇ · v + λεv · ∇h = 0, εζ − λh ∂t q + εv · ∇q = 0, q = . 1 + λh

(1.22) (1.23)

We introduce a hierarchy of slow times tn = ε n t, develop all variables in formal asymptotic series: v = v0 (x, y; t, t1 , t2 , . . .) + εv1 (x, y; t, t1 , t2 , . . .) + · · · , h = h0 (x, y; t, t1 , t2 , . . .) + εh1 (x, y; t, t1 , t2 , . . .) + · · · ,

(1.24)

and plug these expansions in (1.21), (1.22) and (1.23). Each field at any order has a unique decomposition into a slow part defined as the t-average of the whole field, and a residual fast part: hi = h¯ i (x, y; t1 , . . .) + h˜ i (x, y; t, t1 , . . .),

i = 0, 1, 2, . . . ,

(1.25)

and analogously for the velocity field. The relation between the nonlinearity parameter and the Rossby number should be fixed in order to render the procedure consistent. If the parameters are of the same order the QG regime results, and if nonlinearity is of order one, the FG regime arises. The intermediate case ε ∼ λ2 may be treated as well (Reznik, Zeitlin and Ben Jelloul, 2001) and will be not presented below. Once the relation between the parameters is fixed, at each order in ε an easily solvable linear problem for the velocity, height and PV fields arises. The evolution of the slow fields is obtained by fast-time averaging of the corresponding equations. The procedure is, thus, straightforward and algorithmic. The presentation below follows the papers by Reznik, Zeitlin and Ben Jelloul (2001), Zeitlin, Reznik and Ben Jelloul (2003), Reznik and Grimshaw (2002), and Le Sommer, Reznik and Zeitlin (2004), Reznik and Sutyrin (2005), where this procedure was applied to the geostrophic adjustment in laterally unbounded and bounded domains, and on the equator, respectively.

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW In this section, as in the next one, adjustment process of localized finite-energy perturbations on the whole horizontal plane is considered. Decaying boundary conditions are imposed for all fields.

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

56

2.1. The QG regime In this regime λ ∼ ε. We present the calculations and results order by order in ε. 2.1.1. The lowest order in Rossby number The problem is linear in this order and we recover the classical results, cf. Obukhov (1949), Monin and Obukhov (1958). It is often more convenient to use the PV equation, instead of the height equation. We thus get: ∂t v0 + zˆ ∧ v0 = −∇h0 ,

(2.1)

∂t (ζ0 − h0 ) = 0,

(2.2)

where ζ0 = zˆ · ∇ ∧ v0 is the zeroth-order relative vorticity. Initial conditions are taken as a localized, i.e. finite-energy, disturbance: u0 |t=0 = uI ,

v0 |t=0 = vI ,

h0 |t=0 = hI .

(2.3)

It is convenient to rewrite (2.1) in terms of relative vorticity ζ and divergence D: ∂t ζ0 + D0 = 0,

(2.4)

∂t D0 − ζ0 = −∇ h0 . 2

(2.5)

The last equation (2.2) may be immediately integrated in fast time giving ζ0 − h0 = Π0 ,

(2.6)

where Π0 is a yet unknown function of coordinates and slow times. Eliminating ζ0 and D0 results in the inhomogeneous linear equation for h0 : −

∂ 2 h0 − h0 + ∇ 2 h0 = Π0 (x, y; t1 , t2 , . . .). ∂t 2

(2.7)

Solution of this equation may be written as a combination of fast h˜ 0 and slow h¯ 0 components: h0 = h˜ 0 (x, y; t, . . .) + h¯ 0 (x, y; t1 , . . .)

(2.8)

satisfying the following equations ∂ 2 h˜ 0 − h˜ 0 + ∇ 2 h˜ 0 = 0; ∂t 2 −h¯ 0 + ∇ 2 h¯ 0 = Π0 −

(2.9) (2.10)

(Klein–Gordon and inhomogeneous Helmholtz equations, respectively). Note that Π0 corresponds to the balanced quasigeostrophic PV built from the slow component h¯ 0 . This is the LPV introduced in the previous section in the context of balanced models.

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

57

We, thus, get the fast–slow splitting on the level of the equations of motion. To get closed problems for slow and fast motions separately, the splitting should be as well accomplished on the level of initial conditions which constitutes the initialization problem. At the initial moment t = t1 = t2 = · · · = 0 one gets from the definition of Π0 (2.6) that Π0 (x, y; 0) = ∂x vI − ∂y uI − hI ≡ ΠI (x, y). (2.11) ¯ ¯ This allows to find an initial value h0I of h0 by inverting the Helmholtz operator in −h¯ 0I + ∇ 2 h¯ 0I = ΠI

(2.12)

(our choice of decaying boundary conditions on the plane guarantees that the inversion problem is solvable in a unique way). In turn, knowing h¯ 0I one obtains the initial condition for h˜ 0 : h˜ 0I = hI − h¯ 0I .

(2.13)

The second initial condition for h˜ 0 follows from the PV and vorticity–divergence equations (2.2), (2.4): ∂t h˜ 0 |t=0 = −DI ≡ ∂x uI + ∂y vI .

(2.14)

The Klein–Gordon (KG) equation (2.9) together with initial conditions (2.13), (2.14) form a closed problem for h˜ 0 . A similar decomposition takes place for the velocity field: v0 = v˜ 0 (x, y; t, . . .) + v¯ 0 (x, y; t1 , . . .),

(2.15)

where the slow components verify the geostrophic balance v¯ 0 = zˆ ∧ ∇ h¯ 0

(2.16)

and the fast ones obey the equations ∂t v˜ 0 + zˆ ∧ v˜ 0 = −∇ h˜ 0

(2.17)

with initial conditions v˜0I = vI − v¯0I , u˜ 0I = uI − u¯ 0I ; (2.18) where u¯ 0I , v¯0I , h¯ 0I satisfy (2.16). It may be readily checked that the LPV ζ˜0 − h˜ 0 of the fast component is identically zero. The fast component satisfying the KG equation with well-posed initial conditions describes the inertia–gravity waves due to the unbalanced (u˜ 0I , v˜0I , h˜ 0I ) part of the initial disturbance and propagating out of this latter. The corresponding solution is: ˜h0 (x; t) = dk H0(±) (k)ei(k·x±Ωk t) + h˜ 00 , (2.19) ±

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

58

where (±)

H0 (k) =

I (k) 1 ˆ˜ D h0I (k) ± i , 2 Ωk

(2.20)

hat notation is used for the Fourier-transforms, and h˜ 00 is a yet unknown slowtime dependent function which makes no contribution to the lowest-order initial conditions. This additional term is necessary to eliminate resonances in higher orders. Solution for the velocity field is most conveniently written with the help of the complex notation U = u + iv. The KG equation 0 ∂ 2U 0 = 0 − U˜0 + ∇ 2 U ∂t 2 with the initial conditions 0I , 0 = u˜ 0I + iv˜0I ≡ U U t=0

0 0I − ∂x h˜ 0I + i∂y h˜ 0I ≡ WI = −iU ∂t U t=0 −

results and is solved by 0 (x; t) = 00 U dk U0(±) (k)ei(k·x±Ωk t) + U

(2.21)

(2.22) (2.23)

(2.24)

±

with (±)

U0 (k) =

ˆ 1 0I (k) ± i WI (k) U 2 Ωk

(2.25)

00 analogous to h˜ 00 . The fast component and a slow time-dependent contribution U thus represents outgoing dispersive waves. At any fixed point x = (x, y) the following estimates hold (Reznik, Zeitlin and Ben Jelloul, 2001): 1 0 (x; t) = O 1 , t → ∞, h˜ 0 (x; t) = O (2.26) , U t t and, hence, the fast-time averages of the fast components at a given point of the plane (x, y) vanish

u˜ 0 (x, y) = v˜0 (x, y) = h˜ 0 (x, y) = 0. (2.27) Here and below for any function f (t) the average is defined as 1 f = lim T →∞ T

T f (t) dt.

(2.28)

0

Thus, in the zeroth order of the perturbation theory we get a fast–slow motion splitting. Both fast and slow components are defined in a unique way starting

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

59

from arbitrary localized initial conditions. Note that the procedure imposes no a priori limitations on the relative initial values of the fast and the slow components. The fast part of the flow is completely resolved while the slow part remains undetermined. Its evolution equation comes from the condition of the absence of secular growth of the next order solution. 2.1.2. The first order in Rossby number Momentum equations give at this order: ∂t v1 + zˆ ∧ v1 = −∇h1 + R(0) v ,

(0) (0) (0) Rv = Ru , Rv = −(∂t1 + v0 · ∇)v0 .

(2.29)

The first-order PV equation is

∂t (ζ1 − h1 ) − Π0 ∂t h˜ 0 + u˜ 0 ∂x Π0 + v˜0 ∂y Π0 = −∂t1 Π0 − J h¯ 0 , Π0 .

(2.30) A consistency condition for having bounded in time solutions of (2.30) is obtained by applying the fast-time averaging to the both sides of (2.30) and gives the standard quasigeostrophic PV equation:

∂t1 Π0 + J h¯ 0 , Π0 ≡ ∂t1 ∇ 2 h¯ 0 − h¯ 0 + J h¯ 0 , ∇ 2 h¯ 0 = 0. (2.31) This equation, together with the initial conditions obtained at the previous order in Rossby number, and the geostrophic balance relations defines completely the evolution of the slow motion. Using (2.31) and (2.17) we can integrate (2.30) once in t and get (0)

ζ1 − h1 = Π1 (x, y; t1 , . . .) + Rζ ,

(0) 0 , Π0 , Rζ = Π0 h˜ 0 − u˜ 0 ∂y Π0 + v˜0 ∂x Π0 − J H

with

(0)

Rζ

= 0.

(2.32)

(2.33)

Here Π1 (x, y; t1 , . . .) is a yet undetermined slow function and for technical convenience we have introduced the primitive of h˜ 0 and its variation with respect to the mean: t i i 0 ≡ H i − H . H0 = h˜ 0 (t ) dt , H (2.34) 0 0 0

i satisfies the inhomogeneous KG equation The function H 0 −

i ∂ 2H 0 i = −∂t h˜ 0 i + ∇ 2 H −H = DI 0 0 t=0 2 ∂t

(2.35)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

60

with the initial conditions i i = 0; ∂t H H 0 t=0

0 t=0

= h˜ 0I .

(2.36)

i we have For the time-average of H 0 i

+ ∇2 H i = DI −H 0 0

(2.37)

i = 0 if DI = 0. Correspondingly, H 0 satisfies the following and, hence, H 0 initial-value problem: −

0 ∂ 2H 0 = 0; 0 + ∇ 2 H −H ∂t 2

0 i ; H = −H 0 t=0

0 ∂t H = h˜ 0I . t=0 (2.38)

Thus, the first-order vorticity–divergence equations are: ∂t ζ1 + D1 = Z,

(2.39)

∂t D1 − ζ1 = −∇ h1 + D. 2

(2.40)

The expressions for Z, D are Z = −[∂t1 ζ0 + v0 · ∇ζ0 + ζ0 D0 ], D = − ∂t1 D0 + v0 · ∇D0 + (∂x u0 )2 + (∂y v0 )2 + 2∂x v0 ∂y u0 .

(2.41) (2.42)

We thus get the following equation for the first correction to the free-surface elevation: (0)

∂ 2 Rζ ∂ 2 h1 (0) 2 − h + ∇ h = + Rζ + D − ∂t Z + Π1 . 1 1 ∂t 2 ∂t 2 By splitting h1 into a sum of slow and fast components −

(2.43)

h1 = h˜ 1 (x, y; t, . . .) + h¯ 1 (x, y; t1 , . . .)

(2.44)

and averaging (2.43) with respect to t the following equations for h˜ 1 , h¯ 1 result (0)

∂ 2 Rζ ∂ 2 h˜ 1 ˜ 1 + ∇ 2 h˜ 1 = − h ∂t 2 ∂t 2 2 −h¯ 1 + ∇ h¯ 1 = Π1 + D , −

(0)

+ Rζ + D − D − ∂t Z,

(2.45) (2.46)

where from the definition of D and the geostrophic balance relation we get

2 2 D = 2 ∂x2 h¯ 0 ∂y2 h¯ 0 − ∂xy (2.47) h¯ 0 . The initial conditions for the full first-order fields are null, as follows from the fact that the initial conditions for velocity and free-surface elevation are already verified by the zero-order fields. From this fact and using the definition of Π1

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

61

(2.32) we get that (0) Π1 |t=0 = −Rζ t=0

0 , ΠI . = − ΠI h˜ 0I − u˜ 0I ∂y ΠI + v˜0I ∂x ΠI + J H

(2.48)

The initial value h¯ 1I = h¯ 1 |t=0 can be determined from −h¯ 1I + ∇ 2 h¯ 1I = D |t1 =0 + Π1 |t1 =0 ,

(2.49)

where the r.h.s. of this equation is a function of initial values uI , vI , hI only, as follows from (2.37), (2.47), (2.48). To solve the inhomogeneous KG equation (2.45) we need initial conditions for h˜ 1 and its time-derivative. The first one is, obviously, h˜ 1 t=0 ≡ h˜ 1I = −h¯ 1I . (2.50) The second one follows from (2.32), (2.39) using the fact that D1 |t=0 = 0: (0) ∂t h˜ 1 t=0 = Z|t=0 − ∂t Rζ t=0 (2.51) with the r.h.s. entirely expressed in terms of initial fields using, where necessary, the evolution equation for the slow component. We, thus, have an inhomogeneous linear initial-value problem for h˜ 1 with the source term which may be schematically rewritten in the form: −

∂ 2 h˜ 1 − h˜ 1 + ∇ 2 h˜ 1 ∂t 2 (f ) (f ) (f ) = −2∂t1 D0 + F (s) (x)F0 (x; t) + F1 (x; t)F2 (x; t),

(2.52)

where the superscript denotes slow and fast spatially localized functions, respec(f ) tively, with Fi (x; t), i = 0, 1, 2, being some solutions of the homogeneous KG equation—see (2.4), (2.32), (2.41), (2.45), (2.47) for their precise expressions. The initial problem (2.52), (2.50), (2.51) can be straightforwardly solved. The second and the third terms in the r.h.s. of (2.52) are not resonant, i.e. they are non-divergent while integrated in space and time with the inverse KG operator. The term with the slow-time derivative of the divergence is resonant if the func00 in (2.19), (2.24) do depend on the first slow time t1 . In order to tions h˜ 00 , U avoid resonance it is sufficient to suppose that they do not. Thus, the initial-value problem is well-posed and requires no additional constraints, i.e. there is no slow modulation of the wave envelope at this order. The solution represents the inertia– gravity waves generated by the source term, whose details are given by the r.h.s. of (2.45). Thus, the first order of the perturbation theory provides the evolution equation for the zeroth-order slow field, the initialization of the first-order fast and slow fields, and a correction to the zeroth-order fast field. Note that equation (2.52)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

62

shows that the fast–slow splitting observed in the linear approximation is, in fact, incomplete. Indeed, the slow-fast cross-term in this equation means that the interaction of the primary adjustment emission with the slow motion provides a secondary wave emission, although this is a higher-order correction to the primary field. Thus, the slow field does influence the fast one. On the contrary, no trace of the influence of the fast field upon the slow one is detected at this order. Note also that fast and slow motions are also coupled at the level of the initialization procedure, cf. (2.50). 2.1.3. The second order in Rossby number At this order the second slow time t2 enters and the horizontal momentum equations read ∂t v2 + zˆ ∧ v2 = −∇h2 + R(1) v , R(1) v = −(∂t2 + v1 · ∇)v0 − (∂t1 + v0 · ∇)v1 .

(2.53)

The PV equation gives ∂t q2 + ∂t1 q1 + ∂t2 q0 + v0 · ∇q1 + v1 · ∇q0 = 0,

(2.54)

where q0 = ζ0 − h0 = Π0 , q1 = q2 =

(0) ζ1 − h1 − h0 Π0 = Π1 − h0 Π0 + Rζ ,

ζ2 − h2 − h0 (ζ1 − h1 ) + h20 − h1 Π0 .

(2.55) (2.56) (2.57)

We concentrate on the PV equation (2.54). Let us average this equation in t and consider it term by term. We have ∂t q2 = 0. By virtue of (2.27), (2.33)

∂t1 q1 = ∂t1 Π1 − h¯ 0 Π0 .

(2.58)

(2.59)

By splitting velocity v0 and q1 into fast and slow components which are denoted, as usual, by tilde and overbar respectively, we get

v0 · ∇q1 = v¯ 0 · ∇ q¯1 + v˜ 0 · ∇ q˜1 . (2.60) The fast-time averaging of expressions containing rapidly oscillating wave factors may give non-zero (or divergent) result only if the expressions in question decay slow enough at t → ∞. As follows from (2.26) and (2.32), for any fixed point of the plane in this limit 1 1 (0) Rζ = O (2.61) ⇒ q˜1 = O t t

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

and, hence v˜ 0 · ∇ q˜1 = O

1 t2

⇒

v˜ 0 · ∇ q˜1 = 0.

63

(2.62)

Finally, v1 · ∇q0 = v1 · ∇Π0 . It readily follows from (2.45) that − h˜ 1 + ∇ 2 h˜ 1 = 0

(2.63)

(2.64)

and, hence, h˜ 1 = 0 with our choice of decaying boundary conditions on the plane. We, thus, get from (2.26), (2.29), (2.64)

v1 = v¯ 1 = zˆ ∧ ∇ h¯ 1 + ∂t1 + v¯ 0 · ∇ v¯ 0 . (2.65) Using (2.58)–( 2.65) we arrive at the following slow equation

∂t2 + v¯ 1 · ∇ Π0 + ∂t1 + v¯ 0 · ∇ Π1 − h¯ 0 Π0 = 0.

(2.66)

So, remarkably, there are no fast–fast (wave-drag) contributions to (2.66) which may be, thus, safely obtained by a direct expansion of (1.23) in ε considering all variables as being slow. In order to get a closed equation for h¯ 0 , h¯ 1 we express Π1 as Π1 = ∂x v¯1 − ∂y u¯ 1 − h¯ 1 ,

(2.67)

according to its definition. By using (2.65) which, with the help of the zeroth order geostrophic balance conditions may be rewritten as

v¯ 1 = zˆ ∧ ∇ h¯ 1 − ∂t1 ∇ h¯ 0 − J h¯ 0 , ∇ h¯ 0 (2.68) we get

Π1 = ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 .

(2.69)

With the help of (2.68) and the evolution equation for Π0 equation (2.66) takes the following form

∂t2 Π0 + ∂t1 ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 − h¯ 0 Π0 − ∇ h¯ 0 · ∇Π0

+ J h¯ 0 , ∇ 2 h¯ 1 − h¯ 1 − 2J ∂x h¯ 0 , ∂y h¯ 0 − h¯ 0 Π0 − ∇ h¯ 0 · ∇Π0 (∇ h¯ 0 )2 , Π0 = 0. + J h¯ 1 − (2.70) 2 This equation describes a next-order correction to (2.31) which is necessary to take into account while studying the slow evolution of the balanced component of the flow for times much longer than t1 . One can combine the two equations

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

64

by introducing the “full” slow elevation h¯ = h¯ 0 + ε h¯ 1 . The result, up to a O(ε 2 ) correction, is a following “improved” QGPV (IQGPV) equation

D 2¯ ¯ ¯ ∂y h¯ ∇ h − h − ε h¯ ∇ 2 h¯ − h¯ − ε∇ h¯ · ∇ ∇ 2 h¯ − h¯ − 2εJ ∂x h, Dt1 = 0, (2.71) where ¯ 2 (∇ h) D ¯ (. . .) := ∂t1 (. . .) + J h − ε ,... . Dt1 2

(2.72)

This equation is of the “generalized vorticity” kind. It should be solved with the initial conditions for the full height. One can readily show that the lowest- and the first-order slow evolution equations (2.31) and (2.70) are derived from (2.71) by expanding into the asymptotic series of the type (1.24) but omitting the fast time. Within the O(ε2 ) accuracy it coincides with the balanced dynamics equation obtained by imposing “height-slaving” (Warn, Bokhove, Shepherd and Vallis, 1995, equation (63)) which, in turn, reproduces the iterated geostrophic model IG2 by Allen (1993). We should emphasize that no slaving or other constraint was used in the above derivation which means that we prove validity of this balanced model assuming only the smallness of the Rossby number and of the typical relative elevation of the free surface. We should also stress that while providing the description of the slow motion within higher, with respect to the standard QG model, accuracy, our method gives the formal limits of validity of this latter. Indeed, by construction the QG equation (2.31) is valid for times O(f −1 Ro−1 ), while for longer times O(f −1 Ro−2 ) the improved QG equation (2.71) should be used. This is important to keep in mind, e.g. in applications of the balanced equations to climatic studies. 2.1.4. The resumé of the QG adjustment in RSW in unbounded domain Thus, up to the fourth order in Rossby number the motion in the QG regime, corresponding to the small Rossby number ε and the typical relative elevation λ ∼ ε, is split in a unique way into slow and fast components. The former is described by the IQGPV equation giving corrections to the standard QG dynamics for longer times, the latter represents inertia–gravity waves running away from the initial disturbance. Initial conditions for each component are obtained from the full initial data. A remarkable point is that the IQGPV equation (2.71) does not contain fast–fast (wave-drag) contributions, i.e. the fast IG waves do not influence the balanced slow component at least on times O(f −1 Ro−2 ). On the contrary, the IG waves are coupled to the slow component already at the first order due to the slow–fast coupling term in the right-hand side of (2.52).

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65

In the above consideration the initial perturbation was assumed to be spatially localized (i.e. having finite energy in the unbounded domain). The fast–slow splitting was also proved for spatially periodic motion in the QG regime (e.g. Embid and Majda, 1996; Babin, Mahalov and Nicolaenko, 1998a, 1998b). The characteristic property of the spatially periodic configuration is the existence of quadruplet wave interactions found by Babin, Mahalov and Nicolaenko (1998a, 1998b); for the localized motion such interactions are absent. The physical reason for this difference is that in the localized motion case the fast waves are propagating out of the disturbance, and do not stay at a given space point long enough to produce a resonance, which is not true in a periodic box. 2.2. The FG regime 2.2.1. The lowest order in Rossby number Let us remind that in the FG regime the relative surface elevation is of order one (in what follows we take λ = 1 without loss of generality), so h in this subsection corresponds to the full depth. In the lowest order we have: ∂t v0 + zˆ ∧ v0 = −∇h0 ,

(2.73)

∂t h0 = 0,

(2.74)

with the same initial conditions (2.3) as before. Solutions to (2.73), (2.74) may be immediately split into fast and slow components, with the slow one verifying the geostrophic balance (2.16) and the fast one verifying the homogeneous version of (2.73). The initial conditions are also split as follows: (g) (a) v¯ |t=0 = vI , v˜ |t=0 = vI , h¯ 0 t=0 = hI , (2.75) (g)

(a)

where we separated the geostrophic vI and the ageostrophic vI initial conditions with respect to the initial free-surface elevation: (g)

vI

= zˆ ∧ hI ,

(a)

(g)

v I = vI − vI .

part of the (2.76)

The homogeneous version of equation (2.73) describes non-propagating inertial oscillations corresponding to the small k limit of the dispersion equation for IGW. Using the complex notation U, which is especially convenient in this case, the corresponding solution for the fast velocity field is t) = A0 (x; t1 , . . .)e−it , U(x;

(a)

A0 (x; t1 , . . .)|t=0 = UI ,

(2.77)

where A0 is an amplitude (possibly, slowly modulated) of the inertial oscillations. Thus, at the leading order of the perturbation theory in the FG regime, the fast and the slow components of motion are split, the first one describing inertial oscillations which result from the imbalanced part of the initial conditions, while the second one evolving from the geostrophically balanced part of initial conditions remains undetermined.

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

66

2.2.2. The first order in Rossby number At this order we have the same equations (2.29) as in the QG case for the velocity field. The evolution of the h-field is given by

(a) ∂t h1 = −∂t1 h¯ 0 − ∇ · h¯ 0 v0 . (2.78) Only unbalanced part of velocity gives rise to the r.h.s., the contribution of the balanced velocity vanishes identically. The t-independent first term in the r.h.s. is trivially resonant as it leads to a linear growth in time of h1 and, hence, should be equal to zero which means that the characteristic time-scale of h0 is, in fact, t2 , and not t1 . Equation (2.78) can be immediately integrated giving, in complex notation,

0 + i∂ξ ∗ h¯ 0 U ∗ + h¯ 1 . h1 = h˜ 1 + h¯ 1 = −i∂ξ h¯ 0 U (2.79) 0 Here the complex independent variables ξ = x + iy, ξ ∗ = x − iy are introduced. The following formulas will be used below: 1 1 (∂x − i∂y ), ∂ξ ∗ = (∂x + i∂y ), 2 2 ∇ 2 = 4∂ξ2ξ ∗ , J (ξ, ξ ∗ ) = −2i

∂ξ =

(2.80)

while vi · ∇ = Ui ∂ξ + Ui∗ ∂ξ ∗ ,

(2.81)

∂ξ Ui + ∂ξ ∗ Ui∗ .

(2.82)

∇ · vi =

Thus, we get the following equation for U1

0 + ∂ξ ∗ h¯ 0 U ∗ − ih¯ 1 ∂t U1 + iU1 = −2i∂ξ ∗ −∂ξ h¯ 0 U 0 − (∂t1 U0 + U0 ∂ξ U0 + U0∗ ∂ξ ∗ U0 ).

(2.83)

This is an inhomogeneous equation for inertial oscillations which are nonpropagative and non-dispersive, unlike the inertia–gravity waves appearing at the same stage in the QG regime above. Hence, if forced at its proper frequency, the equation exhibits a typical resonance behaviour. The resonant forcing ∼e−it corresponds to terms containing U0 only once in the r.h.s. of (2.83). These terms are to be eliminated in order to avoid a secular growth of U1 . Thus, we get the following modulation equation for the amplitude of inertial oscillations

∂t1 A0 − 2i∂ξ2ξ ∗ h¯ 0 A0 + 2iA0 ∂ξ2ξ ∗ h¯ 0 + 2i ∂ξ ∗ h¯ 0 ∂ξ A0 − ∂ξ h¯ 0 ∂ξ ∗ A0 = 0, (2.84) or, coming back to the real notation i

∂t1 A0 + J h¯ 0 , A0 − ∇ 2 h¯ 0 A0 − A0 ∇ 2 h¯ 0 = 0. 2

(2.85)

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

67

In the present context this equation (up to a change of variables) was first proposed by Falkovich (1992) and correctly derived by Falkovich, Kuznetsov and Medvedev (1994)—see the discussion below. After having eliminated the secular growth, the regular solution of (2.83) sat1 + U 1 , isfying zero initial conditions may be easily found in the form U1 = U where

0 + A∗ ∂ξ ∗ A0 , 1 = i 2∂ξ ∗ h¯ 1 + J h¯ 0 , U U (2.86) 0 i 1 = U + + U −− + U − ≡ − C1 eit + iC2 e−2it + A1 (x, y, t1 , . . .)e−it U 1 1 1 2 (2.87) with obvious notation, and, respectively,

i ¯ ∗ 2 ¯ 2 ¯ h¯ 0 , h0 A∗0 + A∗0 ∂xy h0 A0 + A∗0 P C1 = ∂xy h0 + P 2 1 C2 = − A0 (∂x A0 − i∂y A0 ) (2.88) 2 defined as P (f ) := ∂y2 f − ∂x2 f . with the operator P Thus, in the first order in Ro the slow component of motion remains undetermined, being, in fact, even slower than it was initially supposed, while the fast component—inertial oscillations—acquires a slow modulation described by a linear Schrödinger-type equation (2.85) with coefficients depending on the slow motion. An induced correction to slow motion also appears and is given by (2.86). It is worth emphasizing that this correction does contain a contribution arising from the self-interaction of the fast component. 2.2.3. The second order in Rossby number The horizontal momentum equations are ∂t v2 + (∂t2 + v1 · ∇)v0 + (∂t1 + v0 · ∇)v1 + zˆ ∧ v2 = −∇h2

(2.89)

and to close the system the height evolution equation should be added:

∂t h2 + ∂t1 h1 + ∂t2 h¯ 0 + ∇ h¯ 0 v1 + h1 v0 = 0.

(2.90)

By averaging this equation in t and supposing boundedness of h2 in time we get:

∂t1 h¯ 1 + ∂t2 h¯ 0 + ∇ · h¯ 0 v¯ 1 + h¯ 1 v¯ 0 + ∇ h˜ 1 v˜ 0 = 0. (2.91) Calculating consecutively the bilinear combinations of height and velocity variables in this equation we get:

(∇ h¯ 0 )2 1 2 − J h¯ 0 , A˜ 0 ∇ · h¯ 0 v¯ 1 = J h¯ 1 , h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + 2 4

∗ ∗ ¯ − i ∇ h0 · A0 ∇A0 − A0 ∇A0

(2.92) + h¯ 0 A∗0 ∇ 2 A0 − A0 ∇ 2 A∗0 ,

68

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

∇ · h¯ 1 v¯ 0 = J h¯ 0 , h¯ 1

(2.93)

where we used (2.86), and 1

∇ h˜ 1 v˜ 0 = J h¯ 0 , |A0 |2 4

− i ∇ h¯ 0 · A∗0 ∇A0 − A0 ∇A∗0 + h¯ 0 A∗0 ∇ 2 A0 − A0 ∇ 2 A∗0 , (2.94) where we used (2.79), (2.87). Thus, the contributions containing fast–fast terms remarkably cancel leaving a purely slow evolution equation for h0 : (∇ h¯ 0 )2 ∂t2 h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + = 0. 2

(2.95)

Thus, again as in the QG case, we have no fast motion drag and the slow component evolves by itself. On the contrary, the fast component evolves on the background of the slow one according to the linear modulation equation (2.85) which may produce non-trivial effects. Knowing the rich physics of the nonlinear Schrödinger equation it would be exciting to get a cubic in A0 correction to (2.85) at the present order. This, however, turns out to be impossible. Indeed, such correction could appear from elimination of resonances in the equation for the next order velocity U2 . Comparing with (2.89) written in complex notation we see that the resonant (i.e. ∼e−it ) cubic terms appear in the following expressions: −− ∗ ∂ξ ∗ U U 0 1

(2.96)

1 + U ∗ ∂ξ U 1 ∂ξ ∗ U 0 + U 0 . 0 ∂ξ U U 1

(2.97)

and

Combining them we get

∗ 0 0 ∂ξ U˜0 + iU 0 ∂ξ U ∂ξ ∗ U ∗ ∂ξ ∗ U −iU 0 0 0 ∂ξ U 0 − iU ∗ ∂ξ ∗ U 0 ≡ 0. 0 ∂ξ U ∗ ∂ξ ∗ U + iU 0 0

(2.98)

Therefore, there are no nonlinear corrections to the Schrödinger-type equation (2.85) up to order four in Ro. Evolution equation for the first correction A1 to the envelope of inertial oscillations may be easily obtained. We do not display it here because, basically, it contains nothing particularly interesting. Note that the absence of cubic corrections in the modulation equation means that there are no corresponding resonant quadruplets of quasi-inertial waves in the vicinity of the threshold frequency f .

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

69

2.2.4. Discussion of the FG adjustment Thus, the FG regime is similar to the QG one in a sense that splitting is also taking place and the slow component of the flow evolves according to the balanced equation without being influenced by the fast one. However, in this case the behaviour of the fast component is sensible to the background slow component and is governed by the Schrödinger-type equation (2.85). Nevertheless, the energy of inertial oscillations = dx dy 1 |A0 |2 E (2.99) 2 is conserved, as well as dx dy F (h¯ 0 )|A0 |2 where F is an arbitrary function of h¯ 0 . Separately, the energy of the slow motion 2 1

E = dx dy h¯ 0 ∇ h¯ 0 (2.100) 2 is also conserved which achieves the demonstration of splitting. It is well known that the classical Schrödinger equation describes dispersion of the wave packets. We can make the resemblance between (2.85) and the Schrödinger equation even closer by the change of variables A0 → B = h¯ 0 A0 which yields

h¯ 0 ∇ 2 h¯ 0 i ∂t1 B + J h¯ 0 , B + ∇ 2 B − B = 0. 2 2

(2.101)

We, thus, get a standard Schrödinger equation with a variable “mass” h¯ 0 , a “potential” ∼∇ 2 h¯ 0 and additional advection by the geostrophic velocity field produced by h¯ 0 . The following qualitative picture, thus, follows in the FG regime: Any initial height perturbation evolves very slowly according to the FG dynamics (2.95). A non-geostrophic part of the velocity perturbation is rapidly oscillating with the inertial period and slowly dispersed according to (2.101) where the coefficients may be considered as being constant in time at the dispersion time-scale. It is simultaneously advected by the geostrophic part of velocity. The details of the dispersion process depend on the initial field (“front”) hI (x). As in the standard Schrödinger equation case, it is possible, for particular profiles, that some part of the initial wave-packet is trapped by the front (“bound states”) and evolves together with it. In any case, the presence of the inertial oscillations packet provides a sort of “fuzzyness” of the front and may have implications on the transport and mixing properties. As it was already mentioned, the Schrödinger-type equation (2.101) was first derived in (Falkovich, Kuznetsov and Medvedev, 1994). The procedure used in that paper consisted in perturbative expansion in Burger number, simultaneous decomposition of any field in the Fourier-series in time using the overtones of the inertial period and subsequent truncation. This result was confirmed by Reznik,

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

70

Zeitlin and Ben Jelloul (2001) by the above-presented method with, in addition, the proof of absence of the inertial oscillations drag in the slow-motion equation. Thus, fast–slow splitting was demonstrated in the next order of the perturbation theory. Finally, let us note that propagation of near-inertial oscillations was studied by similar means, but in a different context, by Young and Ben Jelloul (1997). 2.3. Comments on the influence of the β-effect 2.3.1. The QG regime Let us introduce the non-dimensional β-parameter β¯ with an explicit small factor in front of it:

¯ . f = f0 + βy = f0 1 + εβ βy (2.102) Here

εβ = O

βL f0

1;

β¯ = O(1).

(2.103)

The third small parameter, thus, appears in the theory and its value should be fixed with respect to λ and ε. For the QG case we choose εβ ∼ ε 2 , i.e. we suppose that nonlinear terms dominate those induced by β in (1.1). This assumption is justified for the intense mesoscale structures (eddies and fronts) in the ocean and in the atmosphere (see e.g. Reznik and Grimshaw, 2001; Kamenkovich, Koshlyakov and Monin, 1986). The PV takes a form (cf. (1.23)) ¯ εζ − λh + ε 2 βy (2.104) 1 + λh and by expanding the fields in asymptotic series (1.24) and repeating the procedure of Section 2.1 we arrive to the same results in the two lowest orders of the perturbation theory. The parameter β appears at the third order. The slow-motion equation (2.71) becomes q=

D 2¯ ¯ ¯ − ε h¯ ∇ 2 h¯ − h¯ − ε∇ h¯ · ∇ ∇ 2 h¯ − h¯ ∇ h − h + ε βy Dt1

¯ ∂y h¯ = 0 − 2εJ ∂x h,

(2.105)

and we see that the β-term and the nonlinear ageostrophic terms give corrections of the same order to the QGPV equation. Therefore, it is (2.105) which is the proper equation to use while studying the impact of the β-effect on the intense mesoscale structures of general form. On the contrary, the standard QGPV equation on the β-plane is suitable only in cases where the initial state possesses a special symmetry, as for example an axisymmetric localized vortex. In this (specific, but practically important) case the β-term is

2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW

71

the main factor of vortex evolution while the nonlinear ageostrophic contributions remain smaller for long times (cf. Reznik, Grimshaw and Benilov, 2000; Reznik and Grimshaw, 2001). We would like to stress here, however, that the nonlinear ageostrophic terms are crucial for understanding such important physical phenomena as, e.g., the cyclone–anticyclone asymmetry. Equation for the fast field h˜ 2 is derived along the same lines as (2.52) and has the following form −

∂ 2 h˜ 2 − h˜ 2 + ∇ 2 h˜ 2 ∂t 2

2 (1) ¯ 2 h˜ 0 . = R + β¯ ∂ty h˜ 0 + ∂y h˜ 0 − ∂x h˜ 0 + 2βy∇ h

(2.106)

(1) Here Rh does not depend on β¯ and contains non-resonant terms and linear resonant terms analogous to those in the r.h.s. of (2.52). These resonances, as before, may be eliminated by adding to h˜ 0 , h˜ 1 free-wave solutions with slow-time depending Fourier amplitudes, cf. (2.19). The same is true for the second group of terms in the r.h.s. of (2.106). On the contrary, the last y-dependent term poses a real problem as it is evident that this resonance, being spatially non-uniform, can not be removed within the single space-scale asymptotic theory (formally, this term gives the O(t 2 ) contribution to h˜ 2 when t → ∞). Hence, whenever the initial state deviates from the geostrophic balance and h˜ 0I = 0 the problem (2.106) is ill-posed on the β-plane. Moreover, if h˜ 0I = 0, but some initial h˜ n are not, where h˜ n is the nth order fast field, the incurable secular growth arises in the (n + 2)th order. Thus, we come to a conclusion that in the case of non-zero β and unbalanced initial conditions the one space and multiple time-scale asymptotic procedure adopted in the present study can not provide solution of the RSW equations with arbitrary accuracy. However, it still provides a solution within a certain accuracy which is determined by the “degree of imbalance” of the initial conditions. Physically, the problem is related to the fact that in times O( ε12 ) the fast inertia–gravity ¯ cannot waves travel a distance O( ε12 ) and the Coriolis parameter f0 (1 + ε 2 βy) be considered constant along this distance, which was implicitly assumed while writing (1.24). A self-consistent asymptotic procedure on the β-plane (to be developed yet) should take into account a distortion of the fast wave rays due to the space inhomogeneity produced by the β-effect. Another way to avoid this difficulty is to consider motion in a zonal channel on the β-plane. There are all reasons to believe, however, that a change of the perturbative scheme along these lines would affect only the fast component of the motion, the slow one remaining unchanged. This assertion rests on two observations: (1) the spectral gap persists in the presence of the β-effect, i.e. inertia–gravity waves are still much faster than Rossby waves; (2) already the “naive” asymptotic expansion used above provides

72

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

a self-consistent derivation of the slow motion equations in the case of a balanced initial state. 2.3.2. The FG regime In the FG regime the characteristic scale is much greater than the deformation radius and we, therefore, take the β-parameter to be of the same order of magnitude as the Rossby number: εβ = O(ε).

(2.107)

In this case the lowest order of the perturbation theory does not change and the formulas of Section 2.2 remain valid. The β-effect manifests itself in the second order and the Schrödinger equation (2.85) acquires a contribution with an y-depending coefficient: i

¯ 0 = 0. ∂t1 A0 + J h¯ 0 , A0 − ∇ 2 h¯ 0 A0 − A0 ∇ 2 h¯ 0 + iβyA (2.108) 2 The slow-time evolution of the free surface elevation also changes (cf., e.g., Stegner and Zeitlin, 1995) and becomes (cf. (2.95)) (∇ h¯ 0 )2 = 0. ∂t2 h¯ 0 − β¯ h¯ 0 ∂x h¯ 0 − J h¯ 0 , h¯ 0 ∇ 2 h¯ 0 + (2.109) 2 Note, first, that we still find splitting on the β-plane (cf. Falkovich, Kuznetsov and Medvedev, 1994) where the presence of the fast component drag in the slow motion equation was claimed on heuristic level; this is not confirmed by our calculations). Second, the presence of the “simple wave” β-induced term in the FG equation (2.109) leads to a non-linear steepening of the h¯ 0 field in zonal (x) direction which has a tendency to decrease the characteristic space scale. This phenomenon may violate the self-consistency of the FG model if the characteristic scale goes down to the Rossby scale Rd in finite time. Precisely, this catastrophic scenario is realised in the case of one-dimensional profile of the initial h¯ 0 : h¯ 0I = h¯ 0I (ax + by).

(2.110)

In this case the profile remains one-dimensional forever, the Jacobian in (2.109) vanishes identically and this latter becomes a simple wave equation describing breaking in finite time, as is well-known. Let us stress, however, that initial condition (2.110) is not of the spatially localized (in both directions) form required by the present study. The steepening can not be catastrophic for spatially localized h¯ 0I of the form h¯ 0I = 1 + hd0 , where hd0 (x, y) is a decaying at infinity function. It is known (cf. Cushman-Roisin, 1986; Ben Jelloul and Zeitlin, 1999) that there are infinitely many conserved quantities in the dynamics described by (2.109).

3. Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW

73

They are the so-called Casimir invariants CF = dx dy F (hd ) = const

(2.111)

and the following functional related to the kinetic energy (cf. (2.100)) ¯ 2d = const. E = dx dy (1 + hd )(∇hd )2 − βyh

(2.112)

Here F (hd ) is an arbitrary function of hd and we replace everywhere h¯ 0 by 1 + hd (x, y; t). Let us show that the constraints imposed by these conservation laws prohibit breaking. Suppose, for simplicity, that hd is a continuously differentiable function with compact support D. It readily follows from (2.109) that hd remains zero outsideD for all times. Therefore, by virtue of (2.111) with F (hd ) = h2d the value | dx dy yh2d | is bounded from above by some constant. Therefore, the positive-definite integral dx dy (1 + hd )(∇hd )2 is bounded as well and the development of infinite gradients of hd (breaking) is forbidden. Hence, one may believe that for spatially localized disturbances the Jacobian terms in (2.109) can stop (or, at least, decelerate) the nonlinear steepening due to the β-term. Of course, this semi-qualitative consideration should be verified by a direct high-resolution numerical simulation which is in progress and will be presented elsewhere. The β-term in the amplitude equation (2.108) does not violate the energy conservation of the inertial oscillations (2.99). As it is however clear from the standard quantum mechanical-type analysis of this equation (e.g., for h¯ 0 = const), the β-term will produce a systematic meridional shift of the wave packet and a tendency to decrease the spatial scale of A in time. Thus, the model may lose self-consistency (the details depend on the initial spatial distribution of the inertial oscillations) on times ∼ε −2 , in analogy and by the same reasons as in the QG case. Thus, multiple space-scales technique should be applied in order to avoid difficulties (cf. Pedlosky, 1984 in this relation).

3. Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW In the previous section we gave an exhaustive description of the geostrophic adjustment of localized disturbances in the RSW model. As was already discussed in Chapter 1, the one-layer RSW model is barotropic as the RSW dynamics may be thought as dynamics of fluid columns of variable depth where the horizontal motion of fluid particles within a column is synchronized. The simplest baroclinic model is the two-layer RSW with the rigid lid. Although the motion in each layer is still a motion of fluid columns, the columns in respective layers

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are not synchronized anymore (except for the specific case of purely barotropic motions), and velocity changes across the interface between the layers. The density also jumps across the interface. The role of the baroclinic effects in the geostrophic adjustment process is the subject of the present section. The calculations presented purposedly in detail in the previous section give a sufficient idea about technical implementation of the method, and we will not dwell into details in the present section. The interested reader may find them in the extended version of the paper by Zeitlin, Reznik and Ben Jelloul (2003) available at http://gershwin.ens.fr/zeitlin. 3.1. Two-layer RSW model, a reminder Let us remind the equations of motion of the two-layer RSW model: 1 ∇πi = 0, i = 1, 2; ρi

∂t Hi − (−1)i+1 η + ∇ · vi Hi − (−1)i+1 η = 0,

∂t vi + vi · ∇vi + f zˆ ∧ vi +

(3.1) i = 1, 2,

(3.2)

vi = (ui (x, y, t), vi (x, y, t)) are velocity fields in each layer, ρi are the densities of the layers, η is the vertical displacement of the interface. There is no summation here over i and i + 1 is understood modulo 2. πi are defined with the help of the full pressure fields Pi in each layer: Pi = −ρi gz + (i − 1)(ρ1 − ρ2 )gH1 + πi ,

(3.3)

1 g is the acceleration due to gravity (the so-called reduced gravity g = 2 ρρ22 −ρ +ρ1 g plays an important role below). From the dynamical boundary condition on the interface it follows that

(ρ2 − ρ1 )gη = π2 − π1 .

(3.4)

The PV is conserved in each layer: (∂t + vi · ∇)Πi = 0,

Πi =

ζi + f , Hi − (−1)i+1 η

(3.5)

where ζi = zˆ · ∇ ∧ vi is the relative vorticity of the layer i. The fast motions in the model are provided by internal IGW propagating at the interface between the layers (cf. Chapter 1). The geostrophic adjustment will be studied, as before, for localized initial perturbations of the interface and velocity in each layer. The method is the same as described in Section 1, although its realization is more technically involved by obvious reasons.

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75

3.2. The QG regime 3.2.1. Basic hypotheses and scaling Supposing in this subsection for the ratio of the layers’ depths d = O(1) we introduce the following QG-scaling: the horizontal velocity scale U , the horizontal spatial scale L ∼ RR = g He /f0 , where RR is the baroclinic Rossby deformation radius, the pressure scale P = ρf ¯ 0 U L, and the scale of the interface variations η∗ = εHe . He is the equivalent height introduced in Chapter 1. Timei , i = 1, 2, we scale is f0−1 . Introducing the (order one) parameters h¯ i = H1H+H 2 rewrite the horizontal momentum and mass conservation equations in the following non-dimensional form: ∂t vi + εvi · ∇vi + zˆ ∧ vi + ∇πi = 0, i = 1, 2;

∂t 1 − (−1)i+1 ε h¯ i+1 η + ∇ · 1 − (−1)i+1 ε h¯ i+1 η vi = 0,

(3.6)

i = 1, 2. (3.7) In order to simplify the formulae we put ourselves in the oceanographic context and suppose that the densities of the layers are close to each other (and to the mean density). The corresponding density ratios may be easily restored otherwise in front of the pressure gradient terms here and below. The non-dimensional PVs are εζi + 1 . Πi = (3.8) 1 − (−1)i+1 h¯ i+1 εη The non-dimensional relation between the interface displacement and pressures in the layers is π2 − π1 = η.

(3.9)

The solution of the equations of motion on the infinite f -plane is sought in the form of asymptotic expansions: (0)

(1)

vi = vi (x, y; t, t1 , t2 , . . .) + εvi (x, y; t, t1 , t2 , . . .) + · · · , η = η(0) (x, y; t, t1 , t2 , . . .) + εη(1) (x, y; t, t1 , t2 , . . .) + · · · ,

(3.10)

where each dynamical variable in each order may be uniquely split into the slow (denoted below by over-bar) and fast (denoted by tilde) part defined, correspondingly, as the average over the fast time t and the fluctuation around it. For example: vi =

∞ n=0

(n)

ε n v¯ i (x, y; t1 , t2 , . . .) +

∞

(n)

ε n v˜ i (x, y; t, t1 , t2 , . . .),

(3.11)

n=0

and the same for πi , η. The representation (3.11) is unique since the fast compo(n) nents v˜ i are defined to have zero mean over the fast time t.

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

76

We are looking for a perturbative solution of the Cauchy problem with initial conditions vi |t=0 = vIi ,

η|t=0 = ηI

(3.12)

for the system (3.1), (3.2) under the QG scaling. The subscript I denotes initial values, as usual. The calculations are similar to those presented above for the one-layer case, although they are, obviously, more cumbersome, as they should be performed for each layer and coupled via the interface condition (3.9). 3.2.2. The main results The slow and the fast component of motion are dynamically split and noninteracting. The slow dynamics is described by the “improved” quasigeostrophic PV equations for π¯ i = π¯ i(0) + ε π¯ i(1) , η¯ = ηi(0) + εηi(1) with π¯ 2 − π¯ 1 = η: ¯

Di 2 ∇ π¯ i + (−1)i+1 h¯ i+1 η¯ + ε(−1)i+1 h¯ i+1 η¯ ∇ 2 π¯ i + (−1)i+1 h¯ i+1 η¯ Dt1

− ε∇ π¯ i · ∇ ∇ 2 π¯ i + (−1)i+1 h¯ i+1 η¯ − 2εJ (∂x π¯ i , ∂y π¯ i ) = 0, (3.13) where h¯ i =

Hi H1 +H2 ,

and

Di (∇ π¯ i )2 (. . .) := ∂t1 (. . .) + J π¯ i − ε ,... , Dt1 2

i = 1, 2.

(3.14)

The fast component is the internal inertia–gravity wave packet conditioned by the initial conditions and described by the wave equation −

∂ 2 η˜ − η˜ + ∇ 2 η˜ = εR(x, y; t, t1 , . . .), ∂t 2

(3.15)

where η˜ = η˜ (0) + ε η˜ (1) ,

(3.16)

and the r.h.s. of (3.15) is produced, as in the one-layer case, cf. (2.52), by nonlinear interactions of the lower-order fast field η˜ (0) with itself and with the slow component. Again, the key property of this source term is that it is not resonant having zero fast-time mean, and hence the field η˜ (and other fast fields) correspond to internal IGW propagating out of the localized initial perturbation and decaying in time at any given space location. Apart from the secondary IGW emission provided by the slow-fast interaction term in the r.h.s. of (3.15), the fast and the slow components of motion feel each other only on the level of initial conditions for their respective dynamical equations. If the terms O(ε) are omitted in (3.13), (3.14), the standard two-layer QG equations are recovered (cf. Chapter 1). Thus, as in the one-layer case, the by-product

3. Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW

77

of the presented demonstration is establishing of the limits of validity of the standard baroclinic QG system. It is valid for times of the order (εf0 )−1 , while the improved QG equations are valid for much longer times O((εf0 )−2 ). 3.3. The FG regime 3.3.1. Basic hypotheses and scaling The FG scaling for the two-layer FG model was already described in Chapter 1 of the present volume where the corresponding balanced dynamics was reviewed. The same scaling is used below with an addition of the fast-time scale f0−1 , as usual. Using the complex notation we get the following non-dimensional equations for the FG regime: ∂t Ui + iUi + ε(Ui ∂ξ Ui + Ui∗ ∂ξ ∗ Ui ) = −2∂ξ ∗ πi ,

∂ξ (1 − η)U1 + d −1 + η U2 + c.c. = 0, ∂t η = ε∂ξ (1 − η)U1 + c.c.,

i = 1, 2,

π2 = π1 + η,

(3.17)

where U1,2 = u1,2 + iv1,2 are the complex velocities in respective layers and η is the interface displacement. It is convenient to rewrite (3.17) in terms of the barotropic and the baroclinic modes (cf. Benilov and Reznik, 1996):

εd 2 2 ∂t Ubt + iUbt + + ∂ξ ∗ |Ubt |2 + Φ|Ubc |2 ∂ξ Ubt + ΦUbc 1+d = −2∂ξ ∗ P , (3.18) εd ∗ ∗ ∗ ∂t Ubc + iUbc + ∂ξ ∗ Ubt + Ubt ∂ξ Ubc ∂ξ (Ubt Ubc ) + Ubc 1+d

−1 + Ubc ∂ξ d + η Ubc − (1 − η)∂ξ Ubc

−1 ∗ ∂ξ ∗ d + η Ubc − (1 − η)∂ξ ∗ Ubc = 2∂ξ ∗ η, + Ubc (3.19) ∂ξ Ubt + c.c. = 0, εd ∂t η = −Ubt ∂ξ η + ∂ξ (ΦUbc ) + c.c. 1+d Here

Ubt = (1 − η)U1 + d −1 + η U2 , P = π1 + d −1 π2 +

(3.20) (3.21)

Ubc = U1 − U2 ,

η2 2

and notation Φ = (1 − η)(d −1 + η) is used for compactness.

(3.22)

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Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

3.3.2. The main results As in the QG regime all fields are split into slow and fast parts, cf. (3.11). Both the slow and the fast components evolve from uniquely defined respective initial conditions. The fast component consists of inertial oscillations with slowly changing amplitude. Two FG subregimes arise depending on the ratio of the layers’ depths: the FGH regime for comparable depths, and the FGI regime for shallow upper layer. The barotropic and the baroclinic complex velocities in both sub-regimes are expressed as follows: Ubt = 2i∂ξ ∗ P ,

Ubc = −2i∂ξ ∗ η + Ae−it ,

(3.23)

where the slow functions η, P express the leading-order interface displacement and the barotropic pressure, respectively, and A = A(ξ, ξ ∗ , t1 , . . .) is the slowly evolving envelope of the inertial oscillations. Correspondingly, the leading-order evolution is determined by two coupled equations for slow P and η and a separate equation for A. The evolution equations obtained by the method of multi time-scale asymptotic expansions in ε and removal of resonances are as follows: 3.3.2.1. The FGH sub-regime 1 (3.24) J (η, P ), 1 + d −1

1 J P , ∇ 2 P + ∇ · (1 − η) d −1 + η J (η, ∇η) = 0. ∂t1 ∇ 2 P + 1 + d −1 (3.25)

1 + d −1 ∂t1 A + J P − η2 − d −1 − 1 η, A

i

+ ∇ 2 P − η2 − d −1 − 1 η + (∇η)2 A 2

i

− ∇ 2 (1 − η) d −1 + η A = 0. (3.26) 2 ∂t1 η =

3.3.2.2. The FGI sub-regime (∇η)2 2 = 0, ∂t2 η + J (P , η) + J η, (1 − η)∇ η − 2

(∇η)2 ∂t2 ∇ 2 P + J P , ∇ 2 P + J η, (1 − η)∇ 2 η − = 0, 2 i i ∂t1 A − J (η, A) − ∇ 2 A + ∇ 2 (ηA) − A∇ 2 η = 0. 2 2

(3.27) (3.28) (3.29)

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79

The most important and non-trivial feature of both FG sub-regimes is that although the inertial oscillations do not run away as IGW in the preceding subsection, they still do not make any contribution to the evolution of the slow component. Hence, baroclinicity does not affect this crucial property of the FG dynamics observed previously in the barotropic one-layer case. So the fast oscillations exert no drag on the slow vortical motion. At the same time, the slow modulation of the inertial oscillations is guided by the vortical motion since the coefficients in the modulation equations (3.26), (3.29) depend on P , η. It is worth emphasizing the difference between the FGH and the FGI subregimes. In the former the slow component and the modulation amplitude of inertial oscillations evolve in the same slow time t1 , while in the latter the slow component evolves in the slow time t2 and the amplitude A—in the faster time t1 . This is a novel feature arising due to stratification: in the barotropic RSW model the single FG regime is analogous to the FGI sub-regime. 3.4. Discussion of the geostrophic adjustment in the two-layer RSW and comparison with the one-layer case Thus, the adjustment scenario in the baroclinic two-layer model in the QG regime is very similar to that of the barotropic one-layer case. Any localized perturbation emits the internal IGW running away along the interface between the layers out from the initial disturbance. The slow, balanced part obeys the baroclinic QG equations, which should be improved if considered for longer than (εf )−1 times. The balanced and unbalanced components are split, however a secondary IGW emission due to the interaction of the primary one with the slow component takes place. Note however that a qualitatively new with respect to the one-layer case phenomenon may appear due to baroclinicity: the balanced component may develop the baroclinic instability in course of its evolution. This instability develops, however, in the slow time-scale, and thus does not violate the fast–slow splitting. The baroclinicity is characterized both by the ratio of densities, and by the ratio of depths of the layers. The role of the second parameter is highlighted by the possibility to have two essentially different frontal regimes in the two-layer model: one with a shallow upper layer, and one with comparable layer depths. The first case is similar to the one-layer frontal regime: the fast component consists of inertial oscillations of the interface, and the slow one obeys the frontal dynamics equation with no fast-component drag. The inertial oscillations are guided by the slow component, their envelope dynamics, although slow, is faster than the underlying balanced motion dynamics. In the second case, although there is still no drag of the fast component upon the slow one, the envelope of inertial oscillations (the guided variable), and the balanced (guiding) motion evolve at the same slow time-scale. It should be stressed that similar analysis within continuously stratified primitive equations gives similar results (Zeitlin, Reznik and Ben Jelloul, 2003).

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4. Nonlinear geostrophic adjustment in the presence of a lateral boundary 4.1. Preliminaries In the present section the above-developed asymptotic multiple time-scale theory is applied, within the framework of the one-layer RSW, to nonlinear geostrophic adjustment in a laterally bounded domain. The motion of the fluid, which is described, in non-dimensional form, by equations (1.21)–(1.23), will be considered in a half-plane bounded by a rigid wall with free-slip (no cross-boundary flux) boundary conditions. This is the simplest, and analytically treatable, case of adjustment in bounded domains. A new effect in this case is appearance of the Kelvin waves which are trapped near the boundary and propagate in such a way that the boundary is on the right in the direction of propagation (see Chapter 1 of the present volume). As discussed in Chapter 1, the presence of the Kelvin waves destroys the spectral gap between the slow vortex motion and the fast wave motion, because Kelvin waves with arbitrarily small frequencies exist. Hence, one may expect alterations of the above-established adjustment scenario in the presence of boundaries. This expectation is comforted by a not very well-known fact that the standard derivation of the QG equations for a localized flow near an infinitely long boundary (e.g. in a channel or in a half-plane) encounters difficulties. Namely, it can not provide (1) the conservation of the geostrophic mass and (2) the locality of the successive approximations. To illustrate this point, let us consider the standard derivation of the QG equations starting from the full RSW equations (1.21)–(1.23) on the semi-infinite f -plane, and assuming that λ ∼ ε 1 and that the timedependence is slow, i.e. there is no dependence on the fast time t. The velocity obeys the no-flux boundary condition at the rigid wall situated at y = 0: v|y=0 = 0.

(4.1)

In the lowest order in Ro we get from (1.21), (1.23) the geostrophic relations and the quasigeostrophic PV equation: v¯0 = ∂x h¯ 0 , u¯ 0 = −∂y h¯ 0 ;

2

∂t1 ∇ h¯ 0 − h¯ 0 + J h¯ 0 , ∇ 2 h¯ 0 = 0. The boundary condition for h¯ 0 follows from (4.1), (4.2): ∂x h¯ 0 y=0 = 0.

(4.2) (4.3)

(4.4)

Let the initial field h|t=0 = hI (x, y)

(4.5)

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

81

be localized, i.e. hI → 0,

y → +∞,

x → ±∞.

(4.6)

Obviously, for conditions (4.4), (4.6) to be consistent, it is necessary that hI |y=0 = 0.

(4.7)

We assume that the decay (4.6) is sufficiently rapid, so that the initial energy and enstrophy are finite. We are looking for the solution conserving energy and enstrophy (note that these quantities are conserved under condition (4.4). It means that conditions of locality (4.6) are valid at any time, i.e. h¯ 0 → 0,

y → +∞,

x → ±∞,

(4.8)

since in the opposite case the energy and enstrophy become infinite. Therefore, by virtue of (4.4) and (4.8) h¯ 0 y=0 = 0. (4.9) Thus, in the localized case the boundary streamfunction is always identically zero. It readily follows from (4.3) that the total geostrophic mass, which is defined as dx dy h¯ 0 M= (4.10) y>0

is conserved only if +∞

dx ∂yt1 h¯ 0 y=0 = 0.

(4.11)

−∞

But, as it is easy to see, the problem (4.3), (4.5), (4.9) is well-posed and, therefore, condition (4.11), generally, is not satisfied (see Section 4.3 below and Reznik and Grimshaw, 2002). The normal to the wall component of the first-order velocity is v¯1 = ∂x h¯ 1 + ∂t1 u¯ 0 + u¯ 0 ∂x u¯ 0 + v¯0 ∂y u¯ 0 . Therefore, from the first-order no-flux condition at the wall we get: ∂x h¯ 1 |y=0 = −(∂t1 u¯ 0 + u¯ 0 ∂x u¯ 0 ) y=0 .

(4.12)

(4.13)

For the asymptotic procedure to be self-consistent one has to require that h¯ 1 is also localized. By virtue of (4.2) and (4.13) this is possible only under the condition (4.11) which, generally, is not satisfied, as mentioned above. Thus, the “slow”

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solution of problem (1.21)–(1.23), which is given by (4.1)–(4.6) does not conserve the total mass, and is not localized. The locality of solution and the mass conservation can be satisfied only in the framework of the full RSW model taking into account the fast Kelvin waves, as demonstrated below. As one can see, the locality condition (4.8) is crucial in the above consideration, since in the case of non-localized (periodic or non-periodic along the y-axis) motion the boundary streamfunction is, generally, non-zero being determined from the no-flux condition for the ageostrophic first-order velocity field (cf. Phillips, 1954; McWilliams, 1977). We emphasize also that the above consideration does not mean that the QG approximation is “deficient” or breaks down in the localized case: equations (4.2), (4.3), (4.5), (4.9) correctly describe the slow component of motion (see below). But, contrary to non-localized motion, in this case the lowest-order QG dynamics should be supplemented with the fast Kelvin wave to provide the locality of solutions and mass conservation. The key role of the Kelvin waves in the process of geostrophic adjustment near boundaries was investigated in a number of studies after the pioneering work by Gill (1976) (e.g. Hermann, Rhines and Johnson, 1989; Dorofeyev and Larichev, 1992; Tomasson and Melville, 1992; Helfrich and Pedlosky, 1995; Helfrich, Kuo and Pratt, 1999). In the rest of this section we consider the QG regime for the RSW system (1.21), (1.23) on a half-plane assuming that λ ∼ ε 1 and that the initial fields uI , vI , hI are not necessarily balanced. The initial motion is assumed to be bounded at infinity and localized in the y-direction. The initial fields are not entirely arbitrary and satisfy the following conditions at the wall y = 0: vI |y=0 = 0,

(uI = −∂y hI )y=0 ,

(4.14)

to be consistent with equations (1.21) and the no-flux condition (4.1). Special attention will be paid to two types of the initial conditions: • Periodic (in x) boundary conditions, (uI , vI , hI ) =

+∞

(uIn , vIn , hIn )einx ,

(4.15)

n=−∞

• Localized boundary conditions: (uI , vI , hI ) → 0,

x → ±∞.

(4.16)

Analysis of step-like initial conditions when the motion tends to some zonal flows as x → ±∞, can be found in Reznik and Grimshaw (2002). We will first derive the general formulae not depending on the type of the initial conditions, and then discuss the special cases (4.15), (4.16). Like in the unbounded case of Section 3 the solution to the system (1.21), (1.23), (4.1) is sought in the form of the multipletime-scale asymptotic expansions (1.24).

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

83

4.2. The lowest-order solution (the linear adjustment) 4.2.1. Fast–slow splitting Obviously, equations (2.1), (2.2), (2.6) and initial conditions (2.3) obtained on the unbounded f -plane remain valid in the bounded case. A new element appearing here is the no-flux condition v0 |y=0 = 0.

(4.17)

Again, the solution of (2.1) is represented as a sum of fast (tilde) and slow (bar) components,

(u0 , v0 , h0 ) = u˜ 0 , v˜0 , h˜ 0 (x, y, t, t1 , . . .) + u¯ 0 , v¯0 , h¯ 0 (x, y, t1 , . . .), (4.18) the fast component having zero fast-time average. Applying the t-averaging to (2.1), (2.6) we obtain the following equations for the slow and fast components: v¯0 = ∂x h¯ 0 ,

u¯ 0 = −∂y h¯ 0 ,

ζ¯0 − h¯ 0 = ∇ 2 h¯ 0 − h¯ 0 = Π0 ,

v¯0 |y=0 = 0.

(4.19) (4.20)

∂t u˜ 0 − v˜0 = −∂x h˜ 0 ,

∂t v˜0 + u˜ 0 = −∂y h˜ 0 ,

ζ˜0 − h˜ 0 = 0,

v˜0 |y=0 = 0.

(4.21) (4.22)

4.2.2. Initial conditions for fast and slow zonal velocities The fast–slow splitting (4.18) is incomplete unless the initial conditions for each component are determined. Using (4.19) the initial slow field v¯0I can be found from the equation: ∇ 2 v¯0I − v¯0I = ∂x ΠI ,

ΠI = ζI − hI ,

(4.23)

which should be solved under the boundary condition v¯0I |y=0 = 0.

(4.24)

At the same time, the initial slow field h¯ 0I cannot be determined at this stage since (4.24), (4.19) imply only that h¯ 0I is a constant at y = 0, (B) h¯ 0I y=0 = h¯ 0I = const. (4.25) To find the constant h¯ 0I , the analysis of Kelvin waves, which form a part of the response of the system for geostrophically unbalanced initial conditions, is necessary—see below. Knowing v¯0I from (4.23), (4.24) one can find the fast initial field: (B)

v˜0I = vI − v¯0I = V(x, y).

(4.26)

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Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

Another initial condition for v˜0 follows from equation (2.23): ∂t v˜0 |t=0 = −(uI + ∂y hI ) = Vt (x, y).

(4.27)

The fast field v˜0 obeys the KG equation readily following from (4.21): −∂tt v˜0 − v˜0 + ∇ 2 v˜0 = 0.

(4.28)

This equation together with the initial conditions (4.26), (4.27), and the boundary condition (4.22) allows us to determine the field v˜0 from any initial conditions (2.3). 4.2.3. Solution for the fast fields v˜0 , u˜ 0 , h˜ 0 To obtain the solution for v˜0 we define the odd counterparts of V(x, y), Vt (x, y): Vodd (x, y) y>0 = V(x, y); (4.29) Vodd (x, y) y0 dx dy h1 : ∞ ∂t M1 =

∞ dx R(0) u (x, 0, t)

=−

−∞

dx ∂t1 u0 |y=0 .

(4.141)

−∞

Substituting representation (4.18) for u0 into (4.141) and using (4.33), (4.34), (4.47), (4.121), and the fact that the function u˜ 01 does not depend on t1 (see Section 4.4.2), one obtains that ∞ ∂t M1 = −

dx ∂t1 u¯ 0 y=0 =

−∞

∞

dx ∂yt1 h¯ 0 y=0 .

(4.142)

−∞

We see that the changes of M1 compensate exactly the changes of the lowest-order geostrophic mass given by (4.81). Now instead of standard splitting of the first-order solution into the fast and the slow components, we use an alternative splitting into mass-conserving and mass non-conserving parts: (u1 , v1 , h1 ) = (uc , vc , hc ) + (unc , vnc , hnc ),

(4.143)

where the components conserving the corresponding total mass obey the equations: −y ¯ ∂t uc − vc + ∂x hc = R(0) (4.144) u − e ∂yt1 h0 y=0 , ∂t vc + uc + ∂y hc = R(0) v , ∂t (ζc − hc ) = vc |y=0 = 0,

R(0) ζ , (uc , vc , hc )t=0 = 0,

(4.145) (4.146) (4.147)

while the residual fields responsible for the mass changes (4.142) satisfy the system: ∂t unc − vnc + ∂x hnc = e−y ∂yt1 h¯ 0 y=0 , (4.148) ∂t vnc + unc + ∂y hnc = 0,

(4.149)

∂t (ζnc − hnc ) = 0,

(4.150)

vnc |y=0 = 0,

(4.151)

(unc , vnc , hnc )t=0 = 0.

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

101

Analysis of the system (4.144)–(4.147) analogous to the analysis presented already in this section above, reveals that the fields uc , vc , hc are localized; therefore they are not important unless one is interested in higher-order effects. The solution to the system (4.148)–(4.151) is sought in the form (unc , vnc , hnc ) = (φnc , 0, φnc )e−y ,

(4.152)

which satisfies the no-flux boundary condition (4.151). The evolution of φnc is described by the equation ∂t φnc + ∂x φnc = ∂yt1 h¯ 0 y=0 . (4.153) The solution satisfying zero initial condition is written as x

dx ∂yt1 h¯ 0 y=0 .

φnc (x, t, t1 ) =

(4.154)

x−t

As is seen from (4.153) the solution (4.152), (4.154) can be interpreted as a forced Kelvin wave generated by the along-wall slow velocity varying in time. This wave looks like an injected jet propagating along the wall to the right of the localized lowest-order disturbance as shown in Figs. 1, 2. The time derivative of the total mass of the jet is equal to dx dy e

∂t

−y

∞

dx ∂yt1 h¯ 0 y=0 = −∂t1 M.

φnc =

(4.155)

−∞

y>0

Comparing (4.81) to (4.155) we conclude that this compensating jet takes away of the mass from the lothe surplus or shortage (depending on the sign of ∂t1 M) decreases (increases) calized lowest-order disturbance; if the geostrophic mass M then the jet velocity unc is positive (negative) for sufficiently large x and t x. We now consider the dynamics of compensating jet (4.152) in some details by representing the amplitude φnc in the form x φnc =

∞

dx a(x , t1 ) +

dx ∂yt1 h¯ 0 y=0

−∞

x−t

x

dx φ(x ),

(4.156)

x−t

where a(x, t1 ) = ∂yt1 h¯ 0 − φ(x)

∞

−∞

dx ∂yt1 h¯ 0

, y=0

(4.157)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

102

Figure 1.

Schematic representation of the compensating jet in the oscillating regime. √ Plots of the (c) = π sin 0.5t. function φj given by (4.160) are shown for different times for the model case −∂t1 M 2 2 The function φ(x) is equal to √2 e−4x . Successive times are indicated on the corresponding curves. π

It is seen that the jet length increases monotonically with increasing time, but its intensity oscillates near zero changing its sign. The arrow indicates the direction of the jet velocity.

and φ(x) is an arbitrary localized function satisfying the normalization condition ∞

dx φ(x ) = 1.

(4.158)

−∞

The function a(x) is also localized in x and ∞

dx a(x ) = 0

(4.159)

−∞

by virtue of (4.158). One can readily show that the first term in the r.h.s. of (4.156) does not make any contribution to the integral in the l.h.s. of (4.155), and the mass conservation is provided entirely by the second term in the r.h.s. of (4.156): φj(c) =

∞ −∞

dx ∂yt1 h¯ 0 |y=0

x

x−t

dx φ(x ) = −∂t1 M

x

dx φ(x ).

(4.160)

x−t

For example, the function φ may be chosen to be Gaussian etc. The arbitrariness of φ is related to the fact that representation of a function as a sum of localized and

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

103

Figure 2.

Schematic representation of the compensating jet in the limiting regime. Plots √ of the func(c) = π e−0.5t . The tion φj given by (4.160) are shown for different times for the model case −∂t1 M 2 2 function φ(x) is equal to √2 e−4x . Successive times are indicated on the corresponding curves. It is π

seen that the jet length increases monotonically with increasing time, but its intensity gradually drops to zero. The arrow indicates the direction of the jet velocity.

x non-localized parts is not uniquely defined. The integral x−t dx φ(x ) represents the along-wall jet expanding with a constant speed of Kelvin waves to the right (the rate of of the initial perturbation. The slow time-dependent amplitude −∂t1 M change of the total mass, up to the sign) determines the sign and intensity of the jet at any moment. Reznik and Sutyrin (2005) studied the behaviour of the QG mass on the half-plane and showed that two regimes, the oscillating and the limiting oscillates near some constant value. ones, are possible. In the oscillating regime M This results in slow (in comparison with the jet propagation) oscillation of the tends jet sign and intensity as shown in Fig. 1. The limiting regime, when M monotonically to some constant value as t1 → ∞, is represented in Fig. 2. In this case the jet sign does not change and the jet amplitude slowly tends to zero. Note that the forced Kelvin wave (4.152) can be split into the slow and fast parts in the following way:

φnc e−y = φ¯ nc + φ˜ nc e−y , (4.161) φ¯ nc (x, t1 ) =

x −∞

dx ∂yt1 h¯ 0 y=0 ,

(4.162)

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Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

φ˜ nc (x, t1 ) = −

x−t

dx ∂yt1 h¯ 0 y=0 .

(4.163)

−∞

Here φ˜ nc e−y is a free Kelvin wave with zero fast-time average; both fast and slow components are not localized if the time-derivative of the QG mass (4.155) is non-zero. At the same time their sum (4.161) is localized. 4.7. The summary of the results on the adjustment in the laterally bounded domain The presence of the lateral boundary introduces into the f -plane RSW model a new important element which was absent in the unbounded case: the Kelvin waves trapped near the boundary. Unlike the IGW, the frequency gap between the Kelvin waves and the slow motion is absent since Kelvin waves with arbitrarily small frequencies can exist. Despite this fact the presence of the Kelvin waves does not forbid the fast– slow splitting. The possibility of splitting resides in the special structure of the Kelvin waves in the considered configurations. For the periodic initial conditions the Kelvin wave profile is also periodic, but it has zero mean which physically corresponds to the absence of a zero-frequency harmonic in the Kelvin wave spectrum. For the localized (and “step”-like, see Reznik and Grimshaw, 2002) initial conditions such harmonics can be present in the Kelvin wave spectrum, but the shape of the Kelvin wave profile is such that the Kelvin wave field rapidly decays at a fixed spatial point when the Kelvin wave propagates off this point. The theory provides simple formulae allowing to initialize the Kelvin wave starting from arbitrary initial conditions. With increasing time the Kelvin wave profile slowly distorts due to the Kelvin wave non-linear self-interaction, the distortion being described by the simple-wave equation. The resulting evolution of the initial Kelvin wave profile can be characterized as fast propagation accompanied by slow breaking. The breaking Kelvin waves presumably does not affect the slow mode in the localized and “step”-like cases since it happens far from the main regions of slow variability. For the periodic motion the effect of the Kelvin waves breaking on the geostrophic mode should be examined more thoroughly. An important role in this case should be played by the non-conservation of potential vorticity across a Kelvin front (see Section 5 below and, e.g. Helfrich, Kuo and Pratt, 1999). Generally, the geostrophic adjustment in the half f -plane is similar to that in the unbounded f -plane examined in Section 3. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time-scales f −1 and (εf )−1 , respectively. The slow component is not influenced by the fast one, at least for times (ε 2 f )−1 , and remains close to geostrophic balance. The fast component consists mainly of linear IG waves rapidly propagating

4. Nonlinear geostrophic adjustment in the presence of a lateral boundary

105

out of the initial disturbance and Kelvin waves confined near the boundary. Like in the unbounded case, the nonlinear interactions of IG waves with each other, with slow component and the Kelvin waves result only in a small correction to the fast field. The theory provides an algorithm to determine the initial slow and fast fields to any order in Rossby number. Evolution of the slow motion on times (εf0 )−1 is governed by the wellknown quasi-geostrophic potential vorticity equation for the elevation h¯ 0 . The boundary condition used in this non-viscous model is the no flux condition on the rigid wall. Being formulated in terms of h¯ 0 , this condition depends on the structure of initial fields. In the case of periodic (and “step-like”) initial fields the boundary value of h¯ 0 is equal to some arbitrary function h¯ (B) which is constant along the boundary but depends on the slow time (e.g. Kamenkovich and Reznik, 1978; Pedlosky, 1982). To remove this uncertainty the conservation of along-boundary circulation is used (cf. e.g. Pedlosky, 1982). If the fast component is absent then the slow circulation is assumed to be conserved, this conservation providing the energy and mass conservation of the lowest-order slow motion. However, if the initial conditions are not balanced, so that the fast component is present, then the total circulation (fast + slow) should be conserved. The question is if the fast and slow circulations are conserved separately. Our analysis demonstrates that this is really the case for the periodic (and step-like) initial conditions. The situation for localized initial fields is somewhat more complicated. The lowest-order slow motion in this case is also localized and the elevation is zero at the boundary, i.e. the problem for h¯ 0 is well-defined without using the conservation of slow circulation. Moreover, a simple analysis shows that, generally, the lowest-order slow circulation and, therefore, the total mass of slow localized motion are not conserved. Considering the first-order dynamics, we found that the conservation of the total mass and circulation is provided by a compensating jet taking away the surplus, or shortage of mass from the localized lowest-order slow disturbance. A simple representation for the compensating jet was obtained. It is seen from this representation that the along-wall jet expands with the fast speed of Kelvin wave to the right of the initial perturbation. The slow time-dependent i.e. the rate of change (modulo the sign) of the total lowest-order amplitude ∂t1 M, slow QG mass, determines the jet sign and intensity at each moment. Limits of validity of the considered model when localized disturbance interacts with infinitely long boundary were discussed by Reznik and Sutyrin (2005). It was shown that the approximation of infinite domain can be used if (1) the typical basin scale greatly exceeds typical size of the localized perturbation and the Rossby scale; (2) the time does not exceed the typical time which is required for the Kelvin wave to travel the typical basin scale. These conditions are typical for synoptic eddies of a ∼100 km spatial scale and temporal scale ∼10 days in the ocean of a basin scale of few thousands km, and baroclinic Kelvin wave speed ∼2 m/s.

106

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

On longer times the slow motion obeys the so-called “improved” quasigeostrophic potential vorticity (IQGPV) equation. The theory provides initial and boundary conditions for this equation. The IQGPV equation exactly coincides with the “improved” QGPV equation, derived in Section 3, in the localized (and step like) case. In the periodic case this equation contains an additional term due to the Kelvin wave self-interaction, this term depending on the initial Kelvin wave profile. In other words, in the presence of a boundary the IQGPV equation, in general, depends on initial conditions, which is not the case in the unbounded domain.

5. Nonlinear geostrophic adjustment in the equatorial region 5.1. The long-wave scaling and classification of possible dynamical regimes As was explained in Chapter 1 of the present volume, dynamics of the tropical atmosphere and ocean are special due to the change of sign of the Coriolis force at the equator. A response to a localized perturbation in the equatorial region will be therefore different as compared to the higher latitudes. The equator forms a waveguide with its proper waveguide modes, like e.g. the equatorial Kelvin mode which has much in common with the boundary Kelvin wave whose role in the adjustment process was studied in the previous section. In the present section we study the process of geostrophic adjustment and possibility of splitting of slow and fast motions on the equatorial β-plane in the framework of one-layer RSW. Our approach is based on the observation that the dispersion relation for equatorial waves possesses a spectral gap between the slow Kelvin and Rossby waves on the one hand and the fast inertia–gravity waves on the other hand, see Fig. 11 of Chapter 1. However, the situation on the equator differs significantly from that of the mid-latitude f -plane because the magnitude of the gap depends on the wavelength and tends to zero with increasing wavenumber. Therefore, an asymptotic theory similar to one developed by Reznik, Zeitlin and Ben Jelloul (2001) and exposed in Sections 2 and 3 of the present chapter may be consistently developed only for the long-wave perturbations. Correspondingly, an asymptotic (in Rossby number ε) multiple time-scale theory will be developed for geostrophic adjustment of localized perturbations with small aspect ratio δ measuring the typical ratio of meridional to zonal extensions and introduced in Chapter 1. The dynamical regimes arising for different relations between ε and δ will be classified. The slow and fast motions will be completely quantified, their respective equations of motion with proper initial conditions obtained, and their dynamical splitting demonstrated.

5. Nonlinear geostrophic adjustment in the equatorial region

107

The starting point of our analysis is the equatorial RSW equations under the scaling described in Section 6.1 of Chapter 1: ut + δεuux + εvuy − yv = −δhx ,

(5.1)

vt + δεuvx + εvvy + yu = −hy ,

(5.2)

ht + vy + δux + ε(hv)y + δε(hu)x = 0.

(5.3)

Both parameters ε and δ are supposed to be small, their relative values will be fixed below. The smallness of δ is equivalent to the long-wave approximation. We should emphasize that the meaning of the key parameter, Rossby number, is twofold under such scaling, as it has also the meaning of the Froude number (cf. Majda and Klein, 2003; Majda, 2003). Note also that smallness of the geopotential perturbations which is equivalent to the smallness of ε under this scaling, is a key ingredient of the weak temperature gradients approximation widely used in studies of the tropical circulation (cf. e.g. Majda and Klein, 2003; Bretherton and Sobel, 2003). We will, however, never consider below a forceddissipative problems typical for such studies. 5.1.1. The slow motion: long Rossby and Kelvin waves The long linear Rossby and Kelvin waves (cf. Section 6 of Chapter 1) are characterized by the small non-dimensional meridional velocity v ∼ δ and slow time evolution. Making in (5.1)–(5.3) the changes v → δv, t → t1 = δt we obtain the following equations (corresponding subscripts denote the partial derivatives in this section): ut1 + εuux + εvuy − yv = −hx ,

(5.4)

2

δ vt1 + δ εuvx + δ εvvy + yu = −hy ,

(5.5)

ht1 + vy + ux + ε(hv)y + ε(hu)x = 0.

(5.6)

2

2

In the lowest order in ε, δ we get the equations for slow-propagating linear Rossby and Kelvin waves: (0) u(0) + h(0) t1 − yv x = 0,

(5.7)

yu(0) + h(0) y = 0,

(5.8)

(0) ht1

+ u(0) x

+ vy(0)

= 0.

(5.9)

The next terms of the perturbation series are of the order of max(δ 2 , ε). Equations for these terms, with introduction of the corresponding slow time t2 = max(δ 2 , ε)t1 , are written as follows:

(0) ε (1) (0) (0) ut1 − yv (1) + h(1) (5.10) ut2 + u(0) u(0) x =− x + v uy , 2 max(δ , ε)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

108

δ2 (0) v , (5.11) max(δ 2 , ε) t1

(0) (0) (0) ε (1) (1) ht1 + u(1) ht2 + h u x + h(0) v (0) y . (5.12) x + vy = − 2 max(δ , ε) Already a qualitative analysis of these equations reveals different scenarios of the Rossby waves dynamics arising for different relations between ε and δ: yu(1) + h(1) y =−

(0)

• If δ 2 ε, equations (5.10)–(5.12) become linear. Because it is the term vt1 in the r.h.s. of (5.11) which provides dispersion in the system, it means that within this range of parameters dispersion overcomes nonlinearity and, in the context of the geostrophic adjustment, any initial packet of Rossby waves will be dispersed. • If δ 2 ε the nonlinearity overcomes dispersion. General properties of this regime were recently presented by Majda (2003). A non-compensated nonlinearity usually means breaking. However, the precise dynamical meaning of this process is not clear at this stage. For instance, it is known that equatorial modons may exist at strong nonlinearities (Boyd, 1985). Does equatorial adjustment with subsequent Rossby-wave “breaking” produce them in this regime? We give some elements of the answer below and in Chapter 5. • Finally, at δ 2 ∼ ε, the advective nonlinearity and dispersion may compensate each other and form solitary waves from the initial perturbation in the Rossbywave part of the spectrum (it should be remembered that long Rossby waves have weak dispersion). A typical equation combining the effects of weak nonlinearity and weak dispersion is the KdV (Korteweg–de Vries) or mKdV one and one may expect that the Rossby-wave packet will obey one of them in this regime and form solitons (Boyd, 1980a). As to the long Kelvin waves they should always break (Boyd, 1980b; Ripa, 1982), because they are defined by the condition v (0) = 0 and are, thus, nondispersive whatever the relation between δ and ε. Their breaking, therefore, consists in formation of a hydraulic jump. It is worth mentioning that breaking of Kelvin waves may be prevented by a fine-tuned mean zonal flow (Boyd, 1984; Le Sommer, Reznik and Zeitlin, 2004). 5.1.2. The fast motion: long inertia–gravity and Yanai waves For these waves u ∼ v and equations (5.10)–(5.12) remain unchanged. In the lowest order we get: (0) = 0, u(0) t − yv (0) vt (0) ht

+ yu

(0)

+ h(0) y

+ vy(0) = 0.

(5.13) = 0,

(5.14) (5.15)

5. Nonlinear geostrophic adjustment in the equatorial region

109

A single equation for v (0) follows from this system (cf. Gill, 1982): (0) vtt(0) + y 2 v (0) − vyy =0

(5.16)

which has the following solution in terms of the parabolic cylinder functions φn (y): v

(0)

=

∞

vn(0) (x, t)φn (y);

iσn t vn(0) = A+ + c.c.; σn = 0n (x, t1 )e

n=0

√ 2n + 1. (5.17)

Thus, in the lowest order we obtain non-propagating oscillations with frequencies σn which are well separated from the zero frequency (in the fast time t) of long Rossby and Kelvin waves (cf. Fig. 11 of dispersion curves for equatorial waves in Chapter 1). In the limit ε → 0, i.e. neglecting nonlinearity in (5.1)–(5.3), the next approximation in δ gives a non-dispersive eastward propagation of the envelope A0n in the slow time t1 = δt: t1 ± A± = A x − 0n 0n 2σn2

(5.18)

which is consistent with the group velocity estimate for long waves (cf. Fig. 9, Chapter 1). At the next order in δ dispersion will give rise to evolution in t2 = δ 2 t. If, on the contrary, the limit δ → 0 is taken in (5.1)–(5.3) (infinitely long waves), the resonances generated by nonlinearity appear only at the order O(ε2 )) because, as easily seen from (5.17), the O(ε) bilinear combinations of the oscillations of the form (5.17) are non-resonant: √ √ √ 2k + 1 = 2l + 1 ± 2m + 1

(5.19)

at any integer k, l, m. At the same time, the cubic combinations appearing at order O(ε 2 ) do produce resonances. Hence, in order to counter-balance the dispersion of the envelope of the fast waves by nonlinearity we need to have δ ∼ ε. In this regime one can expect that dynamics of the envelope A0n in t2 , as usual for a weakly nonlinear envelope evolution, obeys the nonlinear Schrödinger equation, or a system of coupled Schrödinger equations if simultaneous evolution of several modes of the type (5.17) is considered. Thus formation of the envelope solitons (cf. Boyd, 1983) is possible. Under condition of very weak dispersion, δ ε the nonlinear interactions give a small frequency shift (no breaking). Under condition of strong dispersion δ ε, the envelope A0n is slowly dispersed.

110

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

5.2. The resumé of possible weakly nonlinear dynamical regimes in the long-wave approximation In the absence of the mean flow the results of the preceding qualitative analysis may be summarized as follows • For Rossby numbers up to δ 2 Rossby waves are dispersed as well as the envelope of the gravity waves; Kelvin waves break down forming hydraulic jumps. • For Rossby numbers of the order of δ 2 Rossby waves form solitons, the envelope of the gravity waves is dispersed and Kelvin waves break down. • For Rossby numbers in the interval (δ 2 , δ) Rossby and Kelvin waves break down, the envelope of the gravity waves is dispersed. • For Rossby numbers of the order of δ, Rossby and Kelvin waves break down, the envelope of the gravity waves obeys the nonlinear Schrödinger equation and forms modulation solitons. • For Rossby numbers greater than δ Rossby and Kelvin waves break and the envelope of the gravity waves has a nonlinear frequency shift Asymptotic analysis of all the aforementioned regimes may be performed confirming these qualitative conclusions (Le Sommer, Reznik and Zeitlin, 2004, unpublished). The results of a detailed analysis of the regime ε ∼ δ 2 , which is, perhaps, the most interesting example due to appearance of Rossby solitons, are presented below. For typical Rossby numbers of the order 0.1 for the balanced motions in the equatorial atmosphere (cf. e.g. Majda and Klein, 2003) and of the order 0.3 for the ocean (cf. e.g. Boyd, 1980a, 1980b) the typical horizontal scales in this regime correspond to thousands of kilometers (planetary-scale perturbations) in the atmosphere and to hundreds of kilometers (mesoscale) in the ocean. They are, hence, appropriate e.g. for the oceanic perturbations due to equatorial westerly wind bursts (cf. Philander, 1990). 5.3. The long-wave adjustment at Ro = O(δ 2 ) In this regime, the basic equations (5.1)–(5.3) take the form: ut + δ 3 uux + δ 2 vuy − yv = −δhx ,

(5.20)

vt + δ 3 uvx + δ 2 vvy + yu = −hy ,

(5.21)

ht + vy + δux + δ (hv)y + δ (hu)x = 0.

(5.22)

2

3

The analysis of geostrophic adjustment in this system in the first four orders in δ is presented below. As usual, solution of (5.20)–(5.23) is sought in the form of an asymptotic series in δ: u = u(0) (x, y, t, t1 , t2 , . . .) + δu(1) (x, y, t, t1 , t2 , . . .) + · · · ,

(5.23)

5. Nonlinear geostrophic adjustment in the equatorial region

111

and similar for the fields v and h. Here the slow time-scales t1 = δt, t2 = δ 2 t, . . . are introduced. Each field f (x, y, t, t1 , t2 , . . .) is represented as a sum of the slow component f¯(x, y, t1 , t2 , . . .) defined as the average of f over the fast time t, and the fast component f˜ = f − f¯. At each order we describe the evolution of both slow and fast components of the flow and show how to split unambiguously the initial conditions and obtain well-posed Cauchy problems for both components. The fast and the slow variables are not mutually independent due to nonlinearity: the slow evolution of the fast fields depends on the slow ones (guiding) and, in principle, the self-interaction of the fast components should influence the slow fields. However, it may be shown (Le Sommer, Reznik and Zeitlin, 2004) that at least at the leading order there is now fast-component drag upon the slow one. The general scheme of calculations, which is entirely algorithmic, will be presented in the next subsection. Technical details may be found in Le Sommer, Reznik and Zeitlin (2004), as well as the initialization procedure which is well defined for both slow and fast fields. Note that results obtained at the order n pro1 vide a description of the system for times up to δ n+1 T , where T = (βRe )−1 is the equatorial time-scale, cf. Chapter 1. It should be stressed that the results on the slow motion alone, which are obtained below in parallel with the description of the fast motion, were known from the pioneering works of Boyd and Ripa: the KdV dynamics of weakly nonlinear equatorial Rossby waves (5.56) was discovered in Boyd (1980a). The overturning of the equatorial Kelvin waves (5.57) was also demonstrated by Boyd (1980b) and Ripa (1982). However, these studies started from a close to the geostrophic balance situation, i.e. they filtered the fast component of motion from the very beginning. The novelty of the approach of Le Sommer, Reznik and Zeitlin (2004) is that without any filtering it was proved that fast and slow motions are dynamically split and the old results are consistent. 5.3.1. The method Equations for the nth approximation in δ are taken in the form: (n−1) (n) u(n) + ut1 + h(n−1) = Pu(n) , t − yv x

(5.24)

(n) vt (n) ht

(5.25)

(n) + yu + h(n) y = Pv , (n−1) + ht1 + vy(n) + u(n−1) x (n)

=

(n) Ph .

(5.26)

Simultaneously the equation for v from the previous step is used: = Pv(n−1) . vt(n−1) + yu(n−1) + h(n−1) y

(5.27)

Here, by definition, u(−1) = v (−1) = h(−1) = 0;

(−1)

Pu(−1) = Pv(−1) = Ph

= 0,

(5.28)

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

112

and the r.h.s. for the first three orders are: (0)

Pu(0) = Pv(0) = Ph Pu(1) = 0, Pu(2) (2) Ph

= =

Pu(3) = (3) Ph

=

= 0,

Pv(1) =

(5.29)

(0) −vt1 ,

(0) −ut2 (0) −ht2

− v (0) u(0) y ,

(0) −ut3 (0) −ht3

(1) − ut2 (1) − ht2

− h(0) v (0) y ,

(1) Ph

Pv(2)

= 0,

(0) (1) = − vt2 + vt1 + v (0) vy(0) ,

(5.30)

(5.31)

(0) (1) (0) (0) − v (1) u(0) y − v uy − u ux ,

(0) (1)

− h v + h(1) v (0) y − h(0) u(0) x .

(5.32) (5.33)

(3)

The explicit expression for Pv will not be needed in what follows. We get, by averaging (5.24)–(5.27), the following equations for the slow fields: u¯ (n−1) + h¯ (n−1) − y v¯ (n) = Pu(n) , t1 x (n) y u¯ (n) + h¯ (n) y = Pv ,

(5.34) (5.35)

(n−1) (n) h¯ t1 + u¯ (n−1) + v¯y(n) = Ph , x

y u¯

(n−1)

+ h¯ (n−1) y

=

(5.36)

Pv(n−1) .

(5.37)

Among these equations (5.34), (5.36) and (5.37) are sufficient to determine the slow motion, while (5.35) will be used in the next approximation. Correspondingly, for the fast fields we get: u˜ t

u(n) − u˜ (n−1) u(n) , − y v˜ (n) = P − h˜ (n−1) ≡R t1 x

(5.38)

(n) v˜t (n) h˜ t

(n) (n) + y u˜ + h˜ (n) y = Pv ≡ Rv , (n) − h˜ (n−1) + v˜y(n) = P − u˜ (n−1) t1 x h

(5.39)

(n)

(n)

(n) . ≡R h

(5.40)

The key idea of the algorithm is to solve for the variable v¯ (n) first, and then to determine u¯ (n−1) and h¯ (n−1) . By differentiating and combining equations (5.34), (5.36), (5.37) a single equation for v¯ (n) is obtained:

(n) v¯yy − y 2 v¯ (n) t + v¯x(n) 1

(n)

(n) (n) = y Pu + Phy − Pv(n−1) − y Ph + Pu(n) − Pv(n−1) , (5.41) t y x t x 1

1

(n) v¯I :

and an equation for its initial value

(n−1)

(n) (n) (n−1) (n) + y h¯ I + y Pu(n) + Phy − Pv(n−1) . (5.42) v¯Iyy − y 2 v¯I = u¯ Iy t x I 1

As usual, the slow evolution follows from removal of resonant terms in the r.h.s. of (5.41). We proceed by subsequent averaging of this equation in t2 , t3 to determine corresponding evolution in slow times of the previous-order slow components.

5. Nonlinear geostrophic adjustment in the equatorial region

113

After resonances are removed, the bounded solution v¯ (n) can be determined from (5.41). It should be stressed that the analysis of resonances in the r.h.s. of (5.41) allows to fix the slow evolution of the Rossby-wave part of the perturbation only, because the linear operator in the l.h.s. of this equation corresponds precisely to Rossby waves. To describe the slow evolution of the Kelvin-wave part we add equations (5.34) and (5.36) and obtain:

(n−1)

u¯ + h¯ (n−1) + u¯ (n−1) + h¯ (n−1) =

−v¯y(n)

+ y v¯

t1 (n)

x

+ Pu(n)

(n) + Ph .

(5.43)

The linear operator in the r.h.s. here corresponds to Kelvin waves and the resonances are absent if (n) (n) v¯y − y v¯ (n) (K) = Pu(n) + Ph (K) , (5.44) where the subscript (K) means that only the terms ∝ U (x −t1 ) (cf. (5.51) below) are considered in the bracketed expression. A bounded solution of this equation exists if +∞ dy φ0 Pu(n) + Ph(n) (K) = 0,

(5.45)

−∞

which gives a slow evolution of the Kelvin waves of preceding orders (see details in Le Sommer, Reznik and Zeitlin, 2004). Knowing v¯ (n) and having removed resonances from the r.h.s. of (5.34), (5.36) and (5.37) we may find u¯ (n−1) and h¯ (n−1) from (5.34) and (5.36), and thus determine the slow component of motion at the given order. The order n − 1 slow component is a sum of free Rossby and Kelvin waves and some “forced” terms generated by nonlinear interactions of preceding-order fields. The slow evolution of order n − 1 Rossby and Kelvin waves is determined from the analysis of the (n + 1)th order, etc. The same strategy is used for the fast component. From (5.38)–(5.40) we obtain an equation for v˜ (n) : (n) v(n) − R − yR u(n) , v˜tt + y 2 v˜ (n) − v˜yy =R hy t (n)

(n)

(5.46)

= −v¯I , (n) = − yu(n) + h(n) − P (n) v˜t

(5.47)

with initial conditions: (n)

(n)

v˜I

t=0

y

v

t=0

.

(5.48)

Removal of resonances from the r.h.s. of this equation gives slow evolution of the fast variables. The linear operator in the l.h.s. corresponds to inertia–gravity and Yanai waves. Hence, namely the dependence on slow times of these fast waves,

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

114

i.e. modulation, will be thus determined. After removal of resonances v˜ (n) can be determined from (5.46) and then u˜ (n) and h˜ (n) may be found from (5.38) and (5.40). It is easy to see that the fast and the slow variables are not mutually independent. The r.h.s. of (5.46) contains the terms of the form u¯ (k) v˜ (m) , k, m < n and, therefore, the slow evolution of the fast fields depends on slow fields (guiding). In turn, the r.h.s. of equations (5.34)–(5.37) contain terms of the type u˜ (k) v˜ (m) and, in principle, the fast motion should influence the slow ones. 5.3.2. The main results 5.3.2.1. Fast motion. The fast motion consists of (infinitely) long inertia– gravity and Yanai waves of the form (5.17). The slow-time dependence of the modulated amplitude A0n is given by (5.18). In the next approximation A0n acquires a nonlinear frequency shift and a frequency shift due to the mean zonal flow Φ (M) , see Le Sommer, Reznik and Zeitlin (2004) for details: i i 3 (M) ± A± (5.49) ± 1 − A± 0nt 0nxx ± 2σ Φnn A0n = 0. 2σn 4σn4 2 n The first correction to the amplitude obeys the following equation: A± 1n + t1

1 ± i (K) (R) Φnn (x, t1 ) + Φnn A1n = ∓ (x, t1 ) A± 0n , 2 x 2σn 2σn

(5.50)

where Φ (K) , Φ (R) represent contributions from the slow Kelvin and Rossby waves, respectively, see Le Sommer, Reznik and Zeitlin (2004) for details. Thus, the modulation of the fast component is guided by the slow one. 5.3.2.2. Slow motion. The full slow solution (u¯ (0) , v¯ (1) , h¯ (0) ) is a sum of three contributions. All of them are unambiguously defined. The first one, (0) (0) (u¯ (K) , 0, h¯ (K) ) is the eastward-propagating Kelvin-wave part: u¯ (K) = h¯ (K) = U0 (x − t1 )φ0 (y). (0)

(0)

(5.51)

(0) (1) (0) The second one is the westward-propagating Rossby-wave part (u¯ (R) , v¯(R) , h¯ (R) ): (0) u¯ (R)

∞

n=1

(1)

√

1

V1n x + c¯n t1 = 2

v¯(R) =

∞ n=1

√ 2(n + 1) 2n φn+1 (y) − φn−1 (y) , 1 + c¯n 1 − c¯n

v¯n(1) x + c¯n t1 φn (y),

(5.52) (5.53)

5. Nonlinear geostrophic adjustment in the equatorial region

h¯ (0) (R)

∞

115

√ √ 2(n + 1) 2n φn+1 (y) + φn−1 (y) , 1 + c¯n 1 − c¯n

1

V1n x + c¯n t1 = 2 n=1

1n = where V x

(1) v¯n ,

ically balanced

1 and c¯n = 2n+1 = σn2 . The (0) (0) zonal flow (u¯ (M) , 0, h¯ (M) ):

(5.54) third one is an arbitrary geostroph-

y u¯ (M) + h¯ (M)y = 0. (0)

(0)

(5.55)

The slow evolution of the Rossby-wave component is given by the KdV equation 1n + βn V 1n V 1n = 0. 1n + αn V V xxx x t 3

(5.56)

The coefficients αn , βn of the KdV equation are constants depending only on the meridional structure of the mode (see Le Sommer, Reznik and Zeitlin, 2004, for details). Hence, according to the well-known properties of the KdV equation, the Rossby-wave component of the initial localized perturbation will split into a sequence of westward-propagating solitons. The slow evolution of the Kelvin-wave component is given by the Riemann wave equation 3 1 U0 U0x = 0 U0t3 + 1/4 (5.57) 2 π leading to overturning in finite time (in terms of the third slow time t3 ). 5.3.2.3. Fast–slow splitting. The results obtained for the slow motion are rather surprising, because the evolution equations (5.56) and (5.57) do not contain any trace of the fast component, in spite of the aforementioned influence of the fast fields upon the slow ones. Le Sommer, Reznik and Zeitlin (2004) proved by a direct calculation that the averaged fast–fast terms are not resonant and, hence, give no contribution (drag) into the slow-fields evolution equations at least up to times O(δ −4 T ). Thus, the slow and the fast motions are dynamically split and justify the naive filtering (i.e. throwing away) of the fast waves which provides a “fast-track” derivation of (5.56) and (5.57). 5.3.3. The summary of the results of the asymptotic theory of equatorial adjustment Thus, the detailed analysis confirms the scenario proposed in Section 5.1. for the regime ε ∼ δ 2 . We have shown that an arbitrary long-wave equatorial perturbation with small characteristic Rossby number splits into fast and slow components. The former is a packet of inertia–gravity and Yanai waves, Doppler-shifted by

116

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

the slow component, and experiencing a nonlinear frequency shift. The fast component does not influence the slow one. The latter is a combination of Rossby and Kelvin waves propagating in opposite directions along the equator and obeying their own modulation equations. For Rossby waves the evolution equation is KdV which is known to produce a sequence of solitons from a localized initial disturbance. For Kelvin waves the evolution equation is a simple-wave equation which produces overturning in finite time. Thus, within the limits of validity of the non-dissipative asymptotic theory, i.e. for times of the order δ −3 T the dynamical splitting of slow and fast component is proved. Note that going farther in the asymptotic expansions does not make sense as the asymptotic theory breaks down once Kelvin waves overturn and generate smaller characteristic scales. Therefore, the proof is given that the filtering of the fast component applied in the works of Boyd (1980a, 1980b) and Ripa (1982) is legitimate and that the fast motion exerts no drag (up to times δ −3 T ) upon the slow motion. However, as was just mentioned, the non-dissipative asymptotic theory has intrinsic limitations due to the breaking of Kelvin waves. For localized initial disturbances on the infinite equatorial β-plane these limitations do not affect the westwardmoving Rossby-wave part of the perturbation much, as Kelvin waves move in the opposite direction. However, for the eastward-moving part of the perturbation the overturning of the Kelvin waves will, presumably, destroy the fast–slow splitting. The practical question of robustness of the asymptotic theory also arises (as its convergence is not proved), i.e. the question of the behaviour of the system for small but finite ε and δ. Another question beyond the limits of the present theory is whether the Rossby-wave breaking for strong nonlinearities can generate modons, which was already mentioned in Section 5.1. The predictions of the theory were successfully tested, and the aforementioned questions answered in high-resolution numerical simulations of the equatorial adjustment (Le Sommer, Reznik and Zeitlin, 2004). These results, along with the simulations of strongly nonlinear phenomena which allow to answer the aboveposed questions, will be presented in Chapter 5.

6. Summary and discussion Thus, the straightforward asymptotic analysis in Rossby number applied to the nonlinear geostrophic adjustment of localized perturbations in the RSW models on the mid-latitude f -plane shows that dynamical splitting of fast and slow motions does take place, in the sense that fast and slow components satisfy their proper dynamical equations with unambiguously defined initial conditions. While slow component does not “feel” the fast one at all in all of the considered regimes, the fast motion is influenced by the slow one beyond the leading order (secondary emission of IGW in the QG regime, guided inertial oscillations in the FD regime).

6. Summary and discussion

117

The initial conditions for the fast motion are also coupled to initial conditions for the slow motion. This coupling at the level of initial conditions becomes dramatic in the non-localized initial conditions at the boundary in the case of laterally bounded domains, where initial conditions of the fast component (Kelvin wave) enter explicitly in the evolution equation of the slow component. Although the boundary Kelvin waves inevitably appear in any event of adjustment in the presence of lateral boundary, and their presence is necessary to guarantee the mass conservation, they do not prevent splitting in the case of localized initial perturbations, in spite of the fact that their presence destroy the gap in the spectrum of linear excitations. In the laterally bounded domains the presence of non-dispersive Kelvin waves imposes a formal upper limit of validity of the asymptotic results in time, which is defined by the breaking time, beyond which the formal asymptotic theory is not valid anymore. However, the breaking event happens far from the location of the initial perturbation (due to the fast propagation of Kelvin waves) and does not influence much the localized slow-variability domain. Although the equatorial RSW dynamics possesses a highly nontrivial spectrum of linear excitations, the equatorial adjustment of localized finite-energy perturbations follow, roughly speaking, the same scenario (at least in the long-wave limit): the slow (Kelvin and Rossby modes) and fast (IGW and Yanai modes) are dynamically split, with initialization well-defined for each component. The slow motion experiences no fast-motion drag, while the fast motion, again is guided (Doppler-shifted) by the slow one. The proper dynamics of the vortex (Rossby) mode is found to be governed by KdV (or mKdV) equations, while equatorial Kelvin waves break down in finite time. This fact, as in the case of the boundary Kelvin waves imposes a formal time-limit of applicability of the asymptotic theory, although, as for the boundary Kelvin waves the influence of this phenomenon onto the vortex mode is weak. The detailed numerical analysis of nonlinear equatorial adjustment and Rossby–Kelvin interactions will be presented below in Chapter 5. Hence, we have a full and comprehensive theory of the geostrophic adjustment of localized perturbations at small Rossby numbers. It is to be stressed again that smallness of the Rossby number does not necessarily mean weak nonlinearities. It is instructive to see what kind of new phenomena could appear if Rossby number of initial perturbation increases. Some of these phenomena were already discussed in Chapter 1. The first one is Lighthill radiation (Lighthill, 1952) which requires, as already discussed in Chapter 1, large Rossby and Burger numbers. Due to this phenomenon, the adjusted state continues to emit the IGW. This emission mechanism is different from the adjustment emission described above. The Lighthill radiation is effective both in barotropic (one-layer) and baroclinic (twolayer) systems. The second phenomenon is purely baroclinic and does not exist in the barotropic system. It is a possible instability of the balanced motions with respect to unbalanced perturbations (see Chapter 1 and Sakai, 1989). In contradis-

118

Chapter 2. Asymptotic Methods with Applications to the Fast–Slow Splitting

tinction with the standard baroclinic instability, this instability is fast, may have growth rates higher than that of the baroclinic instability (see Sakai, 1989, and Chapter 1), and hence may drive the system out of the state to be reached in course of the geostrophic adjustment (this balanced state could be determined from the analysis of the integrals of motion, like in the pioneering paper by Rossby, 1938). In addition, such instability can be a source of a secondary IGW emission, either directly, or via formation of secondary intense vortices and Lighthill radiation. The symmetric instability, appearing necessarily at high Rossby numbers (see Chapter 5 below), is also of this kind. At even higher Rossby numbers the Kelvin– Helmholtz instability comes into play, which is another source of secondary IGW emission.

Acknowledgements The results presented above were obtained in collaboration with M. Ben Jelloul, R. Grimshaw, and G. Sutyrin which is gratefully acknowledged. This work was supported by the RFBR Grant No. 05-05-64212.

References Allen, J.S., 1993. Iterated geostrophic intermediate models. J. Phys. Oceanogr. 23, 2447–2461. Babin, A., Mahalov, A., Nicolaenko, B., 1998a. Global splitting and regularity of rotating shallowwater equations. Europ. J. Mech. B/Fluids 15 (3), 291–300. Babin, A., Mahalov, A., Nicolaenko, B., 1998b. Regularity and integrability of rotating shallow-water equations. C. R. Acad. Sci. Paris Ser. I 324, 593–598. Baer, F., Tribbia, J.J., 1977. On complete filtering of gravity modes through nonlinear initialization. Monthly Weather Rev. 105, 1536–1539. Ben Jelloul, M., Zeitlin, V., 1999. Remarks on stability of the rotating shallow water vortices in the frontal regime. Nuovo Cim. C 22, 931–941. Benilov, E.S., Reznik, G.M., 1996. The complete classification of large-amplitude geostrophic flows in a two-layer fluid. Geoph. Astrophys. Fluid Dynam. 82, 1–22. Blumen, W., 1972. Geostrophic adjustment. Rev. Geophys. Space Phys. 10, 485–528. Bokhove, O., Shepherd, T.G., 1996. On Hamiltonian balanced dynamics and the slowest invariant manifold. J. Atmos. Sci. 53, 276–297. Boyd, J.P., 1980a. Equatorial solitary waves. Part 1. Rossby solitons. J. Phys. Oceanography 10, 1–11. Boyd, J.P., 1980b. The nonlinear equatorial Kelvin waves. J. Phys. Oceanogr. 10, 1699–1717. Boyd, J.P., 1983. Equatorial solitary waves. Part 2. Envelope solitons. J. Phys. Oceanogr. 10, 1699– 1717. Boyd, J.P., 1984. Equatorial solitary waves. Part 4. Kelvin solitons in a shear flow. Dynam. Atmos. Ocean 8, 173–184. Boyd, J.P., 1985. Equatorial solitary waves. Part 3. Westward traveling modons. J. Phys. Oceanogr. 15, 46–54. Bretherton, C.S., Sobel, A.H., 2003. The Gill model and the weak temperature gradient approximation. J. Atmos. Sci. 60, 451–460. Cushman-Roisin, B., 1986. Frontal geostrophic dynamics. J. Phys. Oceanogr. 16, 132–143.

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Obukhov, A.M., 1949. On the problem of geostrophic wind. Izvestia—Geography and Geophysics 13, 281–306 (in Russian). Pedlosky, J., 1982. Geophysical Fluid Dynamics. Springer-Verlag, New York. Pedlosky, J., 1984. The equations for geostrophic motion in the ocean. J. Phys. Oceanogr. 14, 448– 455. Philander, S.G., 1990. El Nino, La Nina, and the Southern Oscillation. Academic Press, San Diego, CA. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286. Reznik, G.M., Grimshaw, R.H.G., 2001. Ageostrophic dynamics of an intense localized vortex on the β-plane. J. Fluid Mech. 443, 351–376. Reznik, G.M., Grimshaw, R.H.G., 2002. Nonlinear geostrophic adjustment in the presence of a boundary. J. Fluid Mech. 471, 257–283. Reznik, G.M., Grimshaw, R.H.G., Benilov, E.S., 2000. On the long-term evolution of an intense localized divergent vortex on the β-plane. J. Fluid Mech. 422, 249–280. Reznik, G.M., Sutyrin, G.G., 2005. Non-conservation of “geostrophic mass” in the presence of a long boundary and the related Kelvin wave. J. Fluid Mech. 527, 235–264. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow water model. J. Fluid Mech. 491, 207–228. Ripa, P., 1982. Nonlinear wave-wave interactions in a one-layer reduced-gravity model on the equatorial β-plane. J. Phys. Oceanography 12, 97–111. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems. J. Marine Res. 1, 239–263. Sakai, S., 1989. Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 445, 93–120. Saujani, S., Shepherd, T.G., 2002. Comments on “Balance and the slow quasimanifold: some explicit results”. J. Atmos. Sci. 59, 2874–2877. Stegner, A., Zeitlin, V., 1995. What can asymptotic expansions tell us about large-scale quasigeostrophic anticyclonic vortices? Nonlin. Proc. Geophys. 2, 186–193. Tomasson, G.G., Melville, W.K., 1992. Geostrophic adjustment in a channel: nonlinear and dispersive effects. J. Fluid Mech. 241, 23–58. Vautard, R., Legras, B., 1986. Invariant manifolds, quasi-geostrophy and initialization. J. Atmos. Sci. 43, 565–584. Warn, T., Bokhove, O., Shepherd, T.G., Vallis, G.K., 1995. Rossby number expansions, slaving principles, and balanced dynamics. Quart. J. R. Meteorol. Soc. 121, 723–739. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York (Chapter 11). Young, W.R., Ben Jelloul, M., 1997. Propagation of near-inertial oscillations through a geostrophic flow. J. Mar. Res. 55, 735–766. Zeitlin, V., Reznik, G.M., Ben Jelloul, M., 2003. Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations. J. Fluid Mech. 491, 207–228.

Chapter 3

The Method of Normal Forms and Fast–Slow Splitting S.B. Medvedev Institute of Computational Technologies, 6 Lavrentiev av., 630090 Novosibirsk, Russia Abstract In this chapter a development of normal form methods for special classes of partial differential equations is presented. A basic application of the methods is splitting of slow and fast wave motions and finding of equations for the slow wave motion. The rotating shallow water model is the main example for the application of the general theory.

Contents 1. Introduction 2. Slow manifold in the spatial variables on the f -plane 2.1. Slow manifold 2.2. Initialization

3. Slow manifold in the spectral variables on the f -plane 3.1. Fast and slow formal invariant manifolds 3.2. Normal forms

4. Poincaré normal forms 4.1. Basic theorems 4.2. Normal form for the RSW at large scales 4.3. Normal form of RSW equations on the mid-latitude β-plane 4.4. Normal form of RSW equations on the equatorial β-plane 4.5. Matrix elements for RSW equations

5. Skew-gradient normal forms for gradient systems 5.1. General construction 5.2. Skew-symmetric normal form for RSW equations

6. Hamiltonian normal forms 6.1. Darboux theorem for finite-dimensional systems 6.2. Darboux theorem for continuous systems Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02003-X 121

122 124 125 129 132 133 136 137 138 143 145 150 155 156 158 160 163 163 166 © 2007 Elsevier B.V. All rights reserved

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7. Normal form of the Poisson bracket for one-dimensional fluid 7.1. Ideal fluid and non-rotating shallow water equations 7.2. RSW equations

8. Conclusion References

169 171 175 176 179 183 184

1. Introduction The method of normal forms is a powerful tool to study dynamical systems. The main idea of the method is to eliminate some non-essential nonlinear terms from a system by a change of variables and to keep other important nonlinear terms. Then one can study the system only with important terms. Often the definition of the important and non-essential terms is based on resonance conditions. In this case these terms are called resonant and non-resonant, respectively. This is a simplest situation. Many of the systems have special structures (for example, Hamiltonian structure) and it is reasonable to eliminate the non-essential terms and to keep the initial structure of the system. For such systems one has to develop a suitable change of variables. In this case the non-essentiality of the terms is defined by possibility to be eliminated by a special change of the variables. In this chapter three normal forms are considered. The first form is the Poincaré normal form. This form is an extension of the Poincaré normal form theory of the ordinary differential equations for partial differential equations, which have linear ordinary equations as a main part. The second form is a skew-symmetric normal form for the skew-gradient systems with a quadratic positive characteristic function. This form is an intermediate one between the Poincaré normal form and the Hamiltonian normal form. The third form is the Hamiltonian one. Actually this is the normal form for the Poisson brackets and the corresponding theorem is a version of the Darboux theorem for the Poisson brackets. All these normal forms arose in works of the author after attempts to split the slow and the fast wave motions for the rotating shallow water equations, which represent the simplest model of atmosphere and ocean. It is necessary to underline that in this context we are dealing with the wave motions. Any wave motion can be described by an amplitude and a phase. Here lies the difference with the averaging theory where usually the amplitude is a slow variable and the phase is a fast variable. If we talk about the fast (slow) wave motion then it means that the frequency of the wave motion is fast (slow). One of the aims of what follows is to present a general approach for getting equations for the slow motion at arbitrary

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123

order of approximation. The Charney–Obukhov equation1 is the first and main example of such equation, because it describes the slow motion of Rossby waves. However obtaining systematic approximations was an open problem. We present below the general theory of the normal forms to give a basis for future investigations in this direction. The applications of the general theory to the rotating shallow water equations give solution of the fast–slow splitting problem for this system and reveal strong and weak sides of the developed method. The plan of the chapter is the following. In Section 2 we construct the slow manifold and the corresponding equation on it for the rotating shallow water equations on the f -plane using an asymptotic expansion (Medvedev, 1999a). The approach using the spatial independent variables allows for representation of the slow manifold and of the corresponding equation in a simple form. The dynamical and static initializations are considered and a new balance equation, which is a generalization of the Charney balance equation, is obtained. In Section 3 we present a construction of the slow and fast manifolds using the spectral independent variables (Medvedev, 1996). This consideration allows for obtaining a general structure of these manifolds without knowledge of a partial form for matrix coefficients. Approximate splitting can be realized by an invertible transformation with a finite number of terms. In Section 4 the Poincaré normal forms are constructed for the rotating shallow water equations in different situations (Medvedev, 1999b). The first case is the normal form for large-scale motions (with respect to Rossby radius) with the positive Coriolis parameter. This situation is considered in spatial independent variables. The second and the third examples are considered in spectral variables on the mid-latitude and equatorial β-plane. In Section 5 a normal form for skew-gradient systems is suggested (Medvedev, 2003). A main advantage of this normal form is the conservation of a characteristic function after cancellation of higher order terms. The rotating shallow water equations are a particular example of the skew-gradient system. In Section 6 the asymptotic version of the Darboux theorem for continuous Hamiltonian system is stated (Medvedev, 2004). This theorem gives a method of factorization of Poisson brackets into symplectic and transversal parts. As the first example the Poisson bracket of the equation for the Rossby waves is transformed to a constant Poisson bracket. In the second example the Poisson bracket of the rotating shallow water equations is factorized. In the last Section 7 we study one-dimensional systems and select a partial set of the Poisson brackets which can be transformed into direct product of symplectic and zero Poisson brackets (Medvedev, 2005). The Poisson bracket of the onedimensional rotating shallow equations belongs to this class. In conclusion a discussion of the obtained results and a connection with the fast/slow splitting are presented. 1 The quasi-geostrophic barotropic equation on the β-plane (see Chapter 1).

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2. Slow manifold in the spatial variables on the f -plane In this section functional equations defining the slow manifold are obtained for the shallow water equations on the f -plane. These equations are solved using an expansion in powers of nonlinearity. Solutions of the shallow water equations corresponding to initial conditions on the slow manifold evolve on this manifold during a long time and are devoid of oscillations at the frequency of inertia– gravity waves. The evolution equation on the slow manifold is obtained. The knowledge of explicit differential equations for the slow manifold allows one to solve the problem of initialization in a new manner. For the dynamical initialization, the sought fields are obtained as power series of the slow mode amplitude with known coefficients. For the static initialization in terms of the depth of the fluid, the velocity field is determined from strongly nonlinear equation. The problem of fast/slow splitting and the slow manifold arose from the practical problem of the initialization of the initial data for numerical models. Unfortunately, using the data without any preparation gives unrealistic fast oscillations for the initial period of numerical integration. The solution of this problem is to split the initial data into the slow and fast parts and to reduce the amplitude of the fast part. To understand this procedure we should know what the slow part is and how it evolves. This idea is based on the observation that the “real” weather is the slow part of the atmospheric motions. The problem of normal mode initialization has been widely studied (see Section 2.2 for definitions of initializations). Baer and Tribbia (1977) gave the solution of this problem by method of two time scales. A geometrical interpretation of solution of the initialization problem was suggested by Leith (1980) using the concept of the slow manifold. In the same work, the connection of the dynamical and static initializations was shown. Vautard and Legras (1986) proposed another initialization method based on the analytic expansion of local invariant manifolds. But in practice the Machenhauer (1977) initialization or the algorithm suggested by Tribbia (1984) are employed, because the realization complexity of the Baer– Tribbia scheme rapidly grows with the number of iterations (Temperton, 1988). The computational difficulties result from the necessity to compute a temporal derivative from a previous approximate. On the other hand, an initialization by the analytic expansion requires the computation and the storage of a large number of coefficients. See also Chapter 2 for a general discussion of balance and initialization. Vautard and Legras (1986) introduced their method for finite-dimensional models of the atmosphere. In this case one can use rigorous results from the theory of invariant manifold for systems of ordinary differential equations (Arnold and Il’yashenko, 1988; Carr, 1981). But a proof of existence or non-existence of a slow manifold for simplest five-dimensional model, suggested by Lorenz (1992), or any more complicated finite-dimensional models, cannot be employed for exam-

2. Slow manifold in the spatial variables on the f -plane

125

ination of infinite-dimensional models (Kreiss and Lorenz, 1994). It is naturally enough to begin such an examination by a straightforward application of the invariant manifold formalism to the shallow water equations on the f -plane. The method of constructing of balance dynamics, similar to the invariant manifold method employed here, was considered by Warn, Bokhove, Shepherd and Vallis (1995). They defined the slow manifold of the rotating shallow water equations for the slow variable. Their slow variable is the potential vorticity. This section is devoted to the construction of equations for an explicit description of the slow manifold and to the derivation of equations of motion on it. It is well known from classical fluid dynamics that if an ideal barotropic flow is potential initially, it remains so afterwards. This is not true for vortex flows, because flows, that are purely vortical initially, generate a potential part. For the shallow water equations with the Coriolis force the motions corresponding to the inertia–gravity (= fast mode) and the Rossby (= slow mode) waves are analogous to the potential and the pure vortical flows, respectively. The fast motion remains the same, if initially the slow motion is absent (Falkovich and Medvedev, 1992). However, it is not true for slow motion, because of the existence of nonlinear terms, depending only on the slow mode, that connect the fast and the slow modes in the equations for the fast motion. Below a change of variables is constructed, which eliminates these terms. It allows one to find the slow manifold and equations of motion on it. This method is called invariant or integral manifold method. The analytical description for the slow manifold allows us to solve the problem of initialization in a new fashion. For the dynamical initialization, the sought fields are obtained as a series of the slow mode with known coefficients. The first term of this series coincides with the main term of the solution obtained by the Machenhauer initialization (1977). For the static initialization, it is necessary to solve a highly nonlinear equation for the determination of the velocity from the depth of fluid. The linear part of the obtained equation coincides with the linear balance equation and the quadratic terms approach the corresponding terms of the Charney balance equation (Charney, 1962) on the scale much smaller then the Rossby radius. For the suitable variables, the forms of quadratic terms in these equations are equivalent. 2.1. Slow manifold The rotating shallow water equations in the standard form are ∂u ∂u ∂h ∂u +u +v −fv + g = 0, ∂t ∂x ∂y ∂x ∂v ∂v ∂h ∂v +u +v + fu + g = 0, ∂t ∂x ∂y ∂y

(2.1) (2.2)

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

126

∂h ∂ ∂ + (uh) + (vh) = 0, ∂t ∂x ∂y

(2.3)

where u, v are the components of velocity, h is the depth of the fluid, g the gravity constant and f the Coriolis parameter. For a discussion of geophysical references for these equations see Chapter 1 and the book of Pedlosky (1987). We introduce the dimensionless variables u , c0 x , x˜ = R0

u˜ =

v , c0 y y˜ = , R0

v˜ =

z˜ =

h − H0 , H0

t˜ = tf0 ,

(2.4)

where√H0 is the depth of the non-perturbed fluid over the flat bottom at rest, c0 = gH0 the gravity waves speed, f0 the mean value of the Coriolis parameter and R0 = c0 /f0 the Rossby deformation radius. Let us consider the case of constant Coriolis parameter f = f0 (the f -plane approximation). We write the shallow water equations without tilde for simplicity and transfer the nonlinear terms to the right-hand side to obtain ∂u ∂z ∂u ∂u −v+ = −u −v , ∂t ∂x ∂x ∂y ∂z ∂v ∂v ∂v +u+ = −u −v , ∂t ∂y ∂x ∂y ∂ ∂ ∂z ∂u ∂v + + = − uz − vz. ∂t ∂x ∂y ∂x ∂y

(2.5) (2.6) (2.7)

The linear part describes two kinds of motion: the slow and the fast. They can be separated by introducing the change of variables ∂χ ∂ξ ∂ϕ − + , ∂y ∂y ∂x ∂ξ ∂ϕ ∂χ + + , v= ∂x ∂x ∂y z = χ + ξ,

u=−

∂2

∂2

(2.8) (2.9) (2.10)

where = ∂x 2 + ∂y 2 is the Laplacian. The physical meaning of the new variables χ, ξ and ϕ can be easily understood from inverse relations ∂v −1 ∂u − +z , χ = (1 − ) (2.11) ∂y ∂x ∂u ∂v ξ = −−1 (1 − )−1 (2.12) − + z , ∂y ∂x

2. Slow manifold in the spatial variables on the f -plane

ϕ = −1

∂u ∂v + , ∂x ∂y

127

(2.13)

where −1 and (1 − )−1 are the inverse operators with vanishing boundary conditions at infinity. The variable χ defines the geostrophic component of velocity and describes stationary motion according to the linear approximation. The variable ϕ defines the potential part of velocity and the variable ξ is connected to the linear imbalance. The variables ξ and ϕ describe the fast motion corresponding to inertia–gravity waves. Rewriting the shallow water equation for χ, ξ and ϕ leads to ∂ ∂ ∂χ = N(χ, ξ, ϕ) ≡ u( − 1)χ + v( − 1)χ, (1 − ) (2.14) ∂t ∂x ∂y ∂ξ + (1 − )ϕ = R(χ, ξ, ϕ) ∂t ∂ ∂ (uz) − uΩ + (vz) − vΩ , ≡ ∂x ∂y

(1 − )

∂ϕ − (1 − )ξ = S(χ, ξ, ϕ) ∂t 2 2 ∂u ∂v ∂ϕ ∂u ∂v ∂ϕ +2 +v + + , ≡− u ∂x ∂y ∂x ∂y ∂x ∂y

(2.15)

(2.16)

∂v where Ω = ∂x − ∂u ∂y and u, v, z are expressed by (2.8)–(2.10). To decouple the slow and fast parts of the flow, we introduce the change of variables

χ = χ, ¯

ξ = ξ¯ + Ξ [χ¯ ],

ϕ = ϕ¯ + Φ[χ¯ ],

(2.17)

where Ξ [χ], Φ[χ] are some integro-differential operators applied to χ. The system (2.14)–(2.16) has the following form in the new variables: ∂χ = N χ, ξ¯ + Ξ [χ], ϕ¯ + Φ[χ] ≡ N¯ χ, ξ¯ , ϕ¯ , (2.18) ∂t ∂ ξ¯ + (1 − )ϕ¯ = R¯ χ, ξ¯ , ϕ¯ , (1 − ) (2.19) ∂t ∂ ϕ¯ (2.20) − (1 − )ξ¯ = S¯ χ, ξ¯ , ϕ¯ . ∂t The transformation (2.17) does not change the linear part of the equations, and therefore our task is to find Ξ [χ] and Φ[χ] such that the nonlinear part of system (2.18)–(2.20) has the following property (1 − )

¯ R(χ, 0, 0) = 0,

¯ S(χ, 0, 0) = 0.

(2.21)

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

128

Then assuming ξ¯ = 0, ϕ¯ = 0, (2.19)–(2.20) will be identically satisfied and the slow motion will be described by the equation (1 − )

∂χ ∂ = [χ, χ] + Ξ, ( − 1)χ + Φ( − 1)χ ∂t ∂x ∂ + Φ( − 1)χ, ∂y

(2.22)

∂g ∂f ∂g where [f, g] = ∂f ∂x ∂y − ∂y ∂x denotes the Jacobian of two functions f and g. Consequently, a solution sitting on a surface M = {χ, ¯ ξ¯ , ϕ: ¯ ξ¯ = 0, ϕ¯ = 0} stays on it always. In terms of the original variables, this means that solutions of the system with initial data

χ|t=0 = χ0 ,

ξ |t=0 = Ξ [χ0 ],

ϕ|t=0 = Φ[χ0 ]

(2.23)

depend on the slow variable χ only. Thus constructing an appropriate change of variables determines the slow manifold M and the projection of the shallow water equations on it. It is easy to note that the system (2.14)–(2.16) does not have the property (2.21), because R and S have terms depending on χ only R(χ, 0, 0) = [χ, χ], 2 2 2 ∂ χ ∂ 2χ ∂ χ S(χ, 0, 0) = 2 − . 2 2 ∂x∂y ∂x ∂y

(2.24) (2.25)

In the one-dimensional case the right-hand parts of (2.24)–(2.25) are equal to zero; therefore the variable transformation is required in the two-dimensional case only. We now derive conditions on Ξ [χ] and Φ[χ], which ensure that (2.21) is satisfied. We substitute (2.17) in (2.19)–2.20, equate terms not containing ξ¯ and ϕ¯ to zero, and obtain two functional equations for Ξ [χ] and Φ[χ], respectively, dΞ [χ] (1 − ) (2.26) + Φ[χ] = R χ, Ξ [χ], Φ[χ] , dt dΦ[χ] − (1 − )Ξ [χ] = S χ, Ξ [χ], Φ[χ] , (2.27) dt here dtd denotes calculation of the derivative with respect to t and substitution of ∂χ ∂t by its expression from (2.22). Therefore, (2.26) and (2.27) do not contain explicitly the temporal differentiation. It is difficult to solve the functional equations (2.26) and (2.27) exactly; therefore, assuming a weak nonlinearity of u, v, z 1 at the Rossby radius scale, we will solve them using an expansion in terms of the degree of the nonlinearity. The weak nonlinearity condition is realized for the atmosphere and it is equivalent to

2. Slow manifold in the spatial variables on the f -plane

129

smallness of variables χ, ξ¯ and ϕ¯ at the Rossby radius scale. At this scale the nonlinearity parameter is close to the Rossby number. Within the accuracy of cubic terms we find a formal solution of (2.26) and (2.27) given by Ξ [χ] = Ξ2 [χ] + Ξ3 [χ] + · · · ,

(2.28)

Φ[χ] = Φ2 [χ] + Φ3 [χ] + · · · ,

(2.29)

where Ξ2 [χ] = 2(1 − )−1 −1

∂ 2χ ∂x∂y

2 −

∂ 2χ ∂ 2χ , ∂x 2 ∂y 2

Φ2 = (1 − )−1 −1 [χ, χ], are quadratic terms and

∂χ ∂Φ2 ∂Ξ2 Ξ3 [χ] = (1 − ) , − [χ, Φ2 ] + 2 ∂x ∂x ∂y ∂χ ∂Φ2 ∂Ξ2 +2 , + ∂y ∂y ∂x −1 −2 + (1 − ) (1 − )−1 [χ, χ], χ + χ, (1 − )−1 [χ, χ] , Φ3 [χ] = −1 (1 − )−1 (1 − )[Ξ2 , χ] + [Ξ2 , χ] + [χ, Ξ2 ] ∂ ∂Φ2 ∂Φ2 ∂ ∂Φ2 ∂ ∂ ∂Φ2 χ + χ − χ + χ + ∂x ∂x ∂y ∂y ∂x ∂x ∂y ∂y 2 2 ∂ χ ∂ (1 − )−1 [χ, χ] − 2−1 (1 − )−1 2 ∂x∂y ∂x∂y ∂ 2χ ∂ 2 ∂ 2χ ∂ 2 −1 −1 (1 − ) [χ, χ] − 2 2 (1 − ) [χ, χ] − ∂x 2 ∂y 2 ∂y ∂x −1

−1

are the cubic terms in the expansion. In principle, we can find the equation for the slow manifold M = {χ, ξ, ϕ: ξ = F [χ], ϕ = G[χ]} and equations (2.22) of motion with an arbitrarily prescribed accuracy. 2.2. Initialization The explicit knowledge of equations (2.28)–(2.29) for the slow manifold allows to solve the problem of initialization in a new manner. For dynamical initialization, it is necessary to find an initialized state χ¯i , ξ¯i and ϕ¯i using the slow mode χ0 of the initial fields u, v and z, which will not lead to a fast oscillation. The initialized

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

130

fields are obtained from (2.28)–(2.29): χi = χ,

(2.30)

ξi = Ξ [χ],

(2.31)

ϕi = Φ[χ].

(2.32)

The initialized fields for physical variables ui , vi and zi are then derived from (2.8)–(2.10). We ask the following question: how many terms is it necessary to take into account in the expansion of Ξ [χ] and Φ[χ]? Assume that we make an estimate for the time T during which fast mode should remain negligible. Let us assume that the slow modes have an amplitude of order ε 1 during this time and let us truncate the series for Ξ [χ] and Φ[χ] in (2.31) and (2.32) to order εn . Then from (2.19)–(2.20) we derive the estimate for an amplitude of the fast mode ξ¯ , ϕ¯ ∼ (f0 T )ε n+1 . Therefore, if we want to have an order ε m for ξ¯ and ϕ, ¯ it is necessary to choose n from the condition ε m (f0 T )ε n+1 . For simple estimates we take ε = 1/10, f0 =

(2.33) 10−4 s−1 ,

then

n m − 1 + lg(f0 T ). For m = 2 and T = 1, 10 days we obtain n 2, 3, respectively. We now compare the solution (2.31) and (2.32) with the Machenhauer’s solution. We solve the Machenhauer’s conditions (1 − )ϕ = R(χ, ξ, ϕ),

(2.34)

−(1 − )ξ = S(χ, ξ, ϕ)

(2.35)

using an expansion in series of power of χ. This solution is accurate up to cubic terms ∂χ ∂Φ2 ∂Ξ2 −1 −1 ξ = Ξ2 [χ] + (1 − ) , − [χ, Φ2 ] + 2 ∂x ∂x ∂y ∂χ ∂Φ2 ∂Ξ2 +2 (2.36) , + , ∂y ∂y ∂x ϕ = Φ2 [χ] + −1 (1 − )−1 (1 − )[Ξ2 , χ] + [Ξ2 , χ] + [χ, Ξ2 ] ∂ ∂Φ2 ∂ ∂Φ2 ∂ ∂Φ2 ∂ ∂Φ2 + χ + χ − χ + χ . ∂x ∂x ∂y ∂y ∂x ∂x ∂y ∂y (2.37)

2. Slow manifold in the spatial variables on the f -plane

131

Comparing (2.31)–(2.32) with (2.36)–(2.37) we see that the quadratic terms are similar and the cubic terms are different. Consequently, the Machenhauer’s initialization gives n = 2 and it follows from estimate (2.33) for m = 2 and chosen f0 and ε that the period of the suppression of fast motion with given accuracy is less one day. The Machenhauer method is a first approximation to the complete Baer–Tribbia method. Therefore the previous comparison of higher approximations of the present method should be made for the corresponding approximations of the Baer–Tribbia (1977) or the Tribbia (1984) methods. Such comparison for the finite-dimensional model was presented by Vautard and Legras (1986). Consider the problem of static initialization. For static initialization it is necessary to determine a velocity field (ui , vi ) corresponding to slow motion, knowing the geopotential z. To solve this problem we should use (2.10) together with (2.31) and (2.32) to find of χi , ξi and ϕi . This system reduces to an equation for χi −1 (z − χi ) = Ξ [χi ],

(2.38)

ϕi being obtained from (2.32). Then ui and vi are determined by (2.8)–(2.9). In order to compare equation (2.38) and the Charney balance equation (Charney, 1962) 2 2 ∂ ψ ∂ 2ψ ∂ 2ψ (z − ψ) + 2 (2.39) − = 0, ∂x∂y ∂x 2 ∂y 2 where ψ = χ + ξ is the stream function, we rewrite the linear and quadratic terms in (2.38) in the following form 2 2 ∂ χ ∂ 2χ ∂ 2χ (z − ψ) + 2 (2.40) − = 0, ∂x∂y ∂x 2 ∂y 2 where χ = ψ + (1 − )−1 (z − ψ) and the index i is omitted. It appears that equations (2.39) and (2.40) have the same linear parts, while the nonlinear terms in (2.40) are equivalent to the corresponding terms in (2.39) for the scales much smaller than the Rossby radius because χ ≈ ψ for this scale. The quadratic terms in (2.39) have the same form as those in (2.40) with ψ substituted for χ. It also follows from (2.32) that it is necessary to take into account the divergent part of velocity at first order for the static initialization. To summarize, in this section a question of the slow manifold for the shallow water equations was considered. The linearized potential vorticity was chosen as the slow variable. For this slow variable, the functional equations of the slow manifold for the shallow water equations on the f -plane were obtained. Using these equations, an asymptotic formula for the slow manifold was found. The explicit formula of the slow manifold allowed us to write approximation equations

132

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

of motion on the slow manifold and to solve the problem of initialization. The comparison of the dynamical initialization by the invariant manifold method used above and by the Machenhauer method demonstrated that they are equivalent to the first order of the expansion. The comparison of the static initialization by the invariant manifold method and by the balance equation showed that they are close only at the scales much smaller then the Rossby radius.

3. Slow manifold in the spectral variables on the f -plane In the previous section the slow manifold and the equation of motion on it were constructed using the independent spatial variables. This can be useful for a practical realization. But the slow manifold can be also considered in the spectral domain. This allows to understand the general structure of the slow manifold. As the shallow water equations have constant coefficients, one can apply the Fourier transformation and obtain an infinite system of integral equations. For this infinite system, at least formally, the method of Poincaré normal forms from the theory of ordinary differential equations can be applied. It is easy to see in the spectral form of the shallow water equations that the resonant term formed by the amplitudes of the fast waves, is absent in the equation for the slow motion. This property leads to other special features of the interaction of the fast/slow variables. Let us consider the dimensionless shallow water equations (2.5)–(2.7) and assume a weak nonlinearity u, v, z 1. For the scales of the order of the Rossby radius this assumption is equal to the smallness of the Rossby number. See Chapter 1 for a discussion on the characteristic scales. After the Fourier transformation of the dependent variables

u(r), v(r), z(r) = (uk , vk , zk ) exp(ikr) dk, (3.1) where r = (x, y), k = (k1 , k2 ), and the integral is over the whole plane, and diagonalization of the linear part of the system is achieved by using the transformation ⎛ −ik2 k1√ωk +ik2 f0 −k√ 1 ωk +ik2 f0 ⎞ ωk 2ωk |k| 2ωk |k| uk ⎟ ak ⎜ ik1 −k√ k2√ ωk −ik1 f0 2 ωk −ik1 f0 ⎟ ⎜ (3.2) vk = ⎝ ωk bk , 2ωk |k| 2ωk |k| ⎠ zk ck f0 |k| |k|

√

ωk

2ωk

ωk = + |k|2 , f = f0 = 1. where |k| = From (3.2) for real-valued u, v, z we have the following symmetries k12

∗ ak = a−k ,

+ k22 ,

√

2ωk

∗ ck = b−k ,

f02

(3.3)

where ∗ denotes the complex conjugation, and one can consider only the equation for bk . The equation for ck is obtained by the complex conjugation.

3. Slow manifold in the spectral variables on the f -plane

133

As a result of the diagonalization we have the system ∂ak + − 0 ∗ al am +Uklm al bm + Uklm al bm , = Uklm ∂t

(3.4)

resonant

∂bk − 0 ∗ al am + V + al bm +Vklm al bm + iωk bk = Vklm klm ∂t resonant slow + 0 ∗ + Wklm bl bm + Wklm bl bm

− ∗ ∗ + Wklm bl bm ,

(3.5)

where the matrix elements are known (see next section) and the following compact notation is used

− − ∗ ∗ Uklm al bm = Uklm al bm δ(k − l + m) dl dm. Let us call the variable ak the slow variable, because of no evolution for this variable from the linear part of the system. And let us call the variable bk the fast variable, because it has the high frequency greater than f0 . These fast/slow definitions are different from the conventional definitions for the fast/slow variables from the averaging theory (e.g. Chapter 2). In the system (3.4)–(3.5) we separate the resonant terms and the slow terms, which are important for our further consideration. See the details below. 3.1. Fast and slow formal invariant manifolds Our method of splitting is based on the following definition of the formal invariant manifold (Arnold and Il’yashenko, 1988, p. 76). Consider a formal differential equation x˙ = Λx + f (x, y),

y˙ = My + g(x, y),

f (0, 0) = 0, g(0, 0) = 0, ∂f ∂f ∂g ∂g (0, 0) = (0, 0) = (0, 0) = (0, 0) = 0. ∂x ∂y ∂x ∂x

(3.6)

D EFINITION 1. The subspace y = 0 is called formal invariant manifold of equation (3.6), if g(x, 0) = 0. The equation x˙ = Λx + f (x, 0) is called the restriction of equation (3.6) to this formal invariant manifold. The results on the finite-dimensional formal invariant manifold can be found in Arnold and Il’yashenko (1988).

134

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

The aim of the following consideration is to use this definition for a description of the fast and slow invariant manifolds of the shallow water equations on the f -plane. Let us first define the fast manifold which is an analog of the potential motion for the inviscid fluid without the Coriolis force. Assuming ak = 0 we get the fast manifold F = (ak , bk ): ak = 0 and an equation on the fast manifold ∂bk + − ∗ ∗ 0 ∗ bl bm + Wklm bl bm + Wklm bl bm . + iωk bk = Wklm ∂t

(3.7)

This formula is exact because equation (3.4) for the slow variable ak does not ∗ . And as a consequence, the fast variable b contain terms of the form Uklm bl bm k does not generate any evolution of the slow variable ak . So if the initial state belongs to the fast manifold then the subsequent evolution takes place on this manifold for all times in future. The slow manifold cannot be determined so easy because equation (3.5) con0 a a which plays a role of an external force (a slow force) tains the term Vklm l m and generates the fast variable for an arbitrary non-zero slow variable ak . But an approximate slow manifold can be found in the case of weak nonlinearity. P ROPERTY 1. Slow terms of the form Vk1...i a1 . . . ai can be eliminated from equation (3.5) for the fast variable bk . Actually, any term Vk1...i a1 . . . ai has zero frequency and the frequency of the fast variable bk is positive ωk f0 , therefore the resonance conditions are not satisfied and this term can be eliminated using a transformation of the form bk → bk + Fk1...i a1 . . . ai .

(3.8)

This transformation is invertible and keeps a finite number of terms in the transformed system. (See the next section for a definition of the resonance conditions.) The original slow term from (3.5) is eliminated by the following transformation 0 bk → bk − iωk−1 Vk12 a 1 a 2 = b k + Fk .

(3.9)

The transformed equations take the form ∂ak + − 0 a1 a2 + Uk12 a1 b2 + Uk12 a1 b2∗ = Uk12 ∂t + − + Uk12 a1 F2 + Uk12 a1 F2∗ ,

(3.10)

3. Slow manifold in the spectral variables on the f -plane

135

∂bk + − a1 (b2 + F2 ) + Vk12 a1 (b2 + F2 )∗ + iωk bk = Vk12 ∂t 0 + Wk12 (b1 + F1 )(b2 + F2 )∗ + − + Wk12 (b1 + F1 )(b2 + F2 ) + Wk12 (b1 + F2 )∗ (b2 + F2 )∗ 0 + − 0 U234 a1 a3 a4 + U234 + 2iωk−1 Vk12 a1 a3 b4 + U234 a1 a3 b4∗ + − 0 + 2iωk−1 Vk12 U234 a1 a3 F4 + U234 a1 a3 F4∗ . (3.11)

Now we are able to compare the slow force of the initial system (3.4)–(3.5) 0 a1 a2 fk = Vk12

(3.12)

and the slow force of the obtained system (3.10)–(3.11) + − 0 0 a1 F2 + Vk12 a1 F2∗ + 2iωk−1 Vk12 U234 a1 a3 a4 fk = Vk12 + − 0 + Wk12 F1 F2∗ + Wk12 F1 F2 + Wk12 F2∗ F2∗ (0) + − + 2iωk−1 Vk12 U234 a1 a3 F4 + U234 a1 a3 F4∗ .

(3.13)

The slow force fk0 of the initial system is quadratic in the slow variable ak and the slow force fk1 is cubic in ak , therefore for weak nonlinearity one can say that fk1 is smaller than fk0 . Using this approach we can continue the procedure of reduction of the slow force. But this procedure generates an additional number of the terms in the expression of the slow force, therefore it is difficult to prove convergence of this series. For practical applications one can compare the order at the slow force of each step, and then choose a variable with a minimal slow force. The respective approximate slow manifolds and the slow equations can be found from the condition bk = 0. Therefore the two first approximations are S2 = (ak , bk ): bk = 0 , ∂ak 0 a1 a2 , = Uk12 ∂t 0 S3 = (ak , bk ): bk = −iωk−1 Vk12 a1 a2 , ∂ak + − 0 0 0∗ a1 a2 − iω2−1 Uk12 V234 a1 a3 a4 − iω2−1 Uk12 V234 a1 a3∗ a4∗ . = Uk12 ∂t

(3.14) (3.15) (3.16) (3.17)

From Property 1 it is easy to obtain the next change of variables eliminating the cubic slow terms from (3.11) + − 0 0 a1 F2 + Vk12 a1 F2∗ + 2iωk−1 Vk12 U234 a1 a3 a4 . bk → bk − iωk−1 Vk12

(3.18)

136

Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

3.2. Normal forms As we saw from the previous subsection it is sufficient to eliminate only the terms of one type to split the slow and fast motions. It is remarkable that these transformations are finite and invertible. But we can also make the next step in the splitting procedure and eliminate all non-resonant terms. Our system has two frequencies: zero frequency for the slow motion and the frequency of the inertia–gravity waves

ωk = f02 + |k|2 for the fast motion. The dispersion law of the inertia–gravity waves is of non-decay type. It means that we cannot satisfy the three wave resonance conditions ω k = ω k1 + ω k2 ,

k = k1 + k2 .

A dispersion law is of decay type if this resonance condition can be satisfied. Therefore it is easy to write the main resonant terms of the shallow water equations ∂ak 0 (3.19) a1 a2 , = Uk12 ∂t ∂bk + + iωk bk = Vk12 (3.20) a1 b2 . ∂t It is well known that the slow motion is independent of the fast motion in the leading order. Let us study the connection of the fast and slow motions at the next orders. After a corresponding transformation the second order normal form takes the form ∂ak 0 1 2 = Uk12 a1 a2 + Uk123 a1 a2 a3 + Uk123 a1 b2 b3∗ , (3.21) ∂t ∂bk + 1 + iωk bk = Vk12 a1 b2 + Vk123 a1 a2 b3 + Tk123 b1∗ b2 b3 ∂t + 0 (3.22) + Sk123 a1 b2 b3∗ + Sk123 a1 b2 b3 . 1 a a a for the slow variIt is clear that we get cubic self nonlinearity Uk123 1 2 3 2 ∗ able and its modulation Uk123 a1 b2 b3 by the fast variables. The fast motion also acquires the modulation and the cubic self nonlinearity due to the four-wave interaction. If we continue to eliminate non-resonant terms in the higher orders then the following property of the our system becomes obvious.

P ROPERTY 2. The equation for the slow variable does not contain terms of the form Uk1...n (b1 b2∗ ) . . . (bn−1 bn∗ ). Actually, the right-hand side of the equation for ak contains only terms with the slow variable ak as a factor. Therefore all transformations, eliminating non-

4. Poincaré normal forms

137

resonant terms, has the same structure. As a result, the use of these transformations does not generate terms of the form Uk1...n (b1 b2∗ ) . . . (bn−1 bn∗ ). Thus the transformations eliminating non-resonant terms from the equation for bk do not change the structure of the slow equation. In this way we show that the equations for the slow variable ak contain only self nonlinearity and the modulation by the fast variable bk . The high-frequency force, which can excite the slow motion, is absent at any order. In the next section it will be shown that the high-frequency force does appears at the second order due to the (small) y-dependent β-term in the Coriolis parameter.

4. Poincaré normal forms In the two previous sections we considered the rotating shallow water equations on the f -plane. In this approximation the frequency of the Rossby wave is equal to zero and as a result the fast and slow motions can be separated at any order. To remove this degeneracy we introduce a variability of the Coriolis parameter. But in this situation we should understand how to eliminate variable non-resonant terms. Therefore in this section an extension of the Poincaré normal form theory for systems of partial differential equations which have linear ordinary equations as a main part, is obtained. Poincaré normal forms for the rotating shallow water equations on the β-plane and for large scales (with respect to Rossby radius) for the variable Coriolis parameter are found. The normal forms of the shallow water equations contain the Charney–Obukhov equation and higher approximation equations for Rossby waves and describe the nonlinear interaction of Rossby and inertia–gravity waves. A successful technique for dealing with many differential equations consists of the transformation of them to a simpler (normal, canonical) form. Poincaré suggested a theory of the normal forms for ODEs (see Arnold, 1983; Arnold and Il’yashenko, 1988). This theory produces these simple forms by means of changes of variables using power series in deviations from the equilibrium or periodic motion. This procedure is very effective in the theory of ODEs (see, for example, Arnold, 1983). Of course, the series are not always convergent (see Bruno, 1989), but already the first few terms of the Poincaré normal form can be useful for qualitative studies of the features of the original differential equation. The method of normal forms for PDEs is also useful (see, for example, Shatah, 1985; Nikolenko, 1986; McKean and Shatah, 1991; Ozawa, Tsutaya and Tsutsumi, 1995; Bruno, 2000), but in this case the method is exploited more effectively after a Fourier transformation when the initial PDEs become an infinite system of ODEs and the method is applicable directly. Homogeneous PDEs in a canonical Hamiltonian form give an excellent example of the

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Chapter 3. The Method of Normal Forms and Fast–Slow Splitting

application of the normal forms method (see Zakharov and Schulman, 1993; Zakharov, Lvov and Falkovich, 1992). A general method analogous to the method of Poincaré normal forms for ODEs does not exist for inhomogeneous PDEs. In the present section the method of Poincaré normal forms is extended to PDEs which have a linear ODE as the main part, depending on independent spatial variables as parameters. In this case the definitions of the resonance conditions and the resonant monomials are extended directly and solutions of a homological equation can be presented in an explicit form. This homological equation (the equation for unknown change of variables) contains partial derivatives. Three examples of the application of the method are given for the rotating shallow water equations. These examples are chosen because the dispersion law has a gap between high- and low-frequency waves. This gap allows to separate easily the fast and slow wave motions. The first example is a long wave approximation to the shallow water equations with a variable Coriolis force. The long wave approximation means that motion with a typical scale that is greater than the Rossby radius is considered. The obtained normal form contains the simplest equations for Rossby and inertia–gravity waves. The variability of the Coriolis parameter is crucial for the influence of the inertia–gravity waves upon the Rossby waves. The second example is the shallow water equations on the β-plane. The main points of this consideration are that the β-term gives a partial derivative after the Fourier transformation, and a smallness of the β-term allows diagonalizing the main linear part of the system. Thus we place the rotating shallow water equations under a hypothesis of the general Poincaré theorem. The β-term plays the crucial role for the interaction of the Rossby and inertia–gravity waves too and generates an additional nonlinearity, a so-called scalar nonlinearity, for the Rossby waves. The presentation of the theory and the examples follow Medvedev (1996, 1999b). The third example is the separation of the equation for short equatorial inertia– gravity waves (Medvedev and Zeitlin, 2005). This example is similar the previous one but the main linear part does not contain the Coriolis term. 4.1. Basic theorems Consider a system of PDEs ∂ui (x, t)/∂t + λi (x)ui (x, t) = Fi (x, u),

u ∈ Rl , x ∈ Rm,

(4.1)

with the right-hand side of the form: Fi (x, u) =

∞

n=1 0 0, x > a,

(1.30)

with initial data U (0, x) = U 0 (x) for x > a. Then as in Example 1.1 one can look for a solution of the form U (t, x) = (1.31) αj0 (x − λj t)rj , j

where rj , λj are the eigenvectors and the eigenvalues of A. We would like to define this for t > 0, x > a. We can define αj0 (x) for x > a by decomposition of U 0 (x), U 0 (x) = j αj0 (x)rj . But then, αj0 (x − λj t) is well-defined for t > 0, x > a only if λj 0. Otherwise, for the j ’s such that λj > 0, formula (1.31) gives a solution if we impose arbitrarily the value of αj0 (x) for x < a. Thus we conclude that the following problem is well-posed: ⎧ ∂ U + A∂x U = 0 for t > 0, x > a, ⎪ ⎨ t U (0, x) = U 0 (x) for x > a, (1.32) ⎪ ⎩ lj U (t, a) = βj (t) for t > 0, λj > 0, where βj (t) are given boundary data. Indeed this defines in a unique way αj0 (y) = βj ( a−y λj ) for y < a. We conclude that in the linear case we have to give boundary conditions only for the incoming characteristics. Notice that one can also put

2. Finite volume schemes for conservative systems

197

boundary data βj that depend on the value of the outgoing characteristics, i.e. of the αj (t, a) for j such that λj 0. In the nonlinear case, one expects that a problem of the type ⎧ ⎪ ⎨ ∂t U + A(U )∂x U = 0 for t > 0, x > a, (1.33) for x > a, U (0, x) = U 0 (x) ⎪ ⎩ U (t, a) ∈ B(t) for t > 0, is well-posed, where B(t) is a submanifold of Rp of codimension the number of positive eigenvalues of A(U (t, a)). Not all such submanifold would give a wellposed problem, but indeed the main restriction is that we do not know in advance the number of incoming characteristics, it depends on the solution itself (and it can change with respect to time). Unfortunately there is no good theory that says that such problem is well-posed. In practice one merely uses physical arguments in order to ensure that (1.33) is well-posed. E XAMPLE 1.5. Consider the system of isentropic gas dynamics (1.14), that we set for t > 0, x > 0, with initial data and with wall boundary condition u(t, 0) = 0.

(1.34)

It involves one scalar boundary condition, thus it is a priori compatible with the case of a single incoming characteristics. However, since whenever u = 0, the eigenvalues of the system are ± p (ρ) (one positive and one negative eigenvalue), we are automatically in this situation, and this problem should be wellposed.

2. Finite volume schemes for conservative systems This section is devoted to the introduction of very basic tools for the numerical resolution of systems of conservation laws (without source term) by finite volume methods. The notions introduced here can be found in Godlewski and Raviart (1991, 1996), Bouchut (2004), Toro (1999), LeVeque (2002), and are not at all specific to shallow water systems. Let us consider a system of conservation laws ∂t U + ∂x F (U ) = 0, (2.1) where U (t, x) ∈ Rp , x ∈ R, t 0, is the unknown and F (U ) ∈ Rp is a given nonlinearity. We would like to approximate its solution by discrete values Uin , i ∈ Z, n ∈ N. In order to do so we consider a grid of points xi+1/2 , i ∈ Z, · · · < x−1/2 < x1/2 < x3/2 < · · · ,

(2.2)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

and we define the cells (or finite volumes) and their lengths Ci = (xi−1/2 , xi+1/2 ),

xi = xi+1/2 − xi−1/2 > 0.

(2.3)

We shall denote also xi = (xi−1/2 +xi+1/2 )/2 the centers of the cells. We consider a constant timestep t > 0 and define the discrete times by tn = n t,

n ∈ N.

(2.4)

Uin

intend to be approximations of the averages of the exact The discrete values solutions over the cells,

1 U (tn , x) dx. Uin (2.5)

xi Ci

A finite volume conservative scheme for solving (2.1) is a formula of the form Uin+1 − Uin +

t (Fi+1/2 − Fi−1/2 ) = 0,

xi

(2.6)

telling how to compute the values Uin+1 at the next time level, knowing the values Uin at time tn . In this section we consider only first-order explicit three points schemes (“three points” means that Uin+1 in (2.6) depends only on three values n , U n , U n ) where Ui−1 i i+1 n Fi+1/2 = F Uin , Ui+1 (2.7) . The function F (Ul , Ur ) ∈ Rp is called the numerical flux, and determines the scheme. We shall often denote Ui instead of Uin , whenever there is no ambiguity. The formula (2.6) is very natural with respect to (2.1), because the exact solution U (t, x) to (2.1) satisfies

1 1 U (tn+1 , x) dx − U (tn , x) dx

xi

xi Ci

Ci

t + (F − F i−1/2 ) = 0,

xi i+1/2 where F i+1/2 is the exact flux F i+1/2

1 =

t

tn+1 F U (t, xi+1/2 ) dt.

(2.8)

(2.9)

tn

This is obtained by integrating equation (2.1) with respect to t and x over ]tn , tn+1 [ × Ci , and dividing the result by xi . Notice also that in (2.6), the interface flux Fi+1/2 represents the exchange of conserved quantities between cells Ci and Ci+1 .

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199

2.1. Consistency The numerical flux F(Ul , Ur ) is called consistent with (2.1) if F (U, U ) = F (U )

for all U.

(2.10)

We can see that this condition guarantees obviously that if for all i, Uin = U a constant, then also Uin+1 = U . A deeper motivation for this definition is indeed the Lax–Wendroff theorem, that states that if the numerical flux is consistent and if Uin defined by (2.6)–(2.7) converges to some bounded function U (t, x), then it is a weak solution to (2.1), see Godlewski and Raviart (1991). 2.2. CFL condition It is always necessary to impose what is called a CFL condition (for Courant, Friedrichs, Levy) on the timestep to prevent the blow up of the numerical values. It comes usually under the form

t ai+1/2 min( xi , xi+1 ),

i ∈ Z,

(2.11)

where ai+1/2 is an approximation of the speed of propagation, computed in terms of Ui and Ui+1 . A typical computation can be ai+1/2 = max λj (Ui ), λj (Ui+1 ) , (2.12) where λj (U ) are the eigenvalues of F (U ). The restriction (2.11) enables in practice to compute the timestep at each time level tn , in order to determine the new time level tn+1 = tn + t (within this view,

t is not constant, it is computed in an adaptive fashion). 2.3. Discrete entropy inequalities The stability of a numerical scheme can be analyzed in different ways. There are merely two classes of analysis: the linearized approach, which is always possible to perform, see for example Godlewski and Raviart (1991), and the nonlinear approach, that puts much more restrictions on the scheme, but which is much more reliable. In the latter approach, exposed in Bouchut (2004), two notions are available: the preservation of invariant domains (some natural bounds of the system must be respected, typically the nonnegativity of density), and the occurrence of discrete entropy inequalities. We shall only introduce here the latter property. Consider a convex entropy η associated to the system (2.1), as defined in Section 1.3. We say that the scheme (2.6)–(2.7) satisfies a discrete entropy inequality associated to η, if there exists a numerical entropy flux function G(Ul , Ur ) which is consistent with the exact entropy flux (in the sense that G(U, U ) = G(U )),

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

such that, under some CFL condition, the discrete values computed by (2.6)–(2.7) automatically satisfy

t (Gi+1/2 − Gi−1/2 ) 0, η Uin+1 − η Uin +

xi

(2.13)

n Gi+1/2 = G Uin , Ui+1 .

(2.14)

with

This definition comes very naturally as a local entropy balance, similarly as the conservative balance (2.6). The interest of such property is twofold. First, it ensures the computation of physically relevant discontinuous solutions, because again by the Lax–Wendroff theorem, if the scheme satisfies a discrete entropy inequality and if Uin converges to some function U (t, x), then U satisfies the entropy inequality (1.20). Second, (2.13) ensures a global a priori bound on the solution. Namely, under some suitable boundary conditions, summing up (2.13) with respect to i, one gets that i xi η(Uin ) does not increase with time. One can define also discrete entropy inequalities by interface, involving only two values Ui and Ui+1 instead of three values Ui−1 , Ui and Ui+1 in (2.13). These are useful in the analysis of the numerical flux F (Ui , Ui+1 ). We shall here only deal with the case of approximate Riemann solvers, and for the general case we refer to Bouchut (2004). 2.4. Godunov approach, approximate Riemann solvers Many methods exist to determine a numerical flux. The two main criteria that enter in its choice are its stability properties, and the precision qualities it has, which can be measured by the amount of viscosity it produces and by the property of exact computation of particular solutions. A general tool that enables to construct numerical fluxes is the notion of approximate Riemann solver in the sense of Harten, Lax and Van Leer (1983), that comes naturally within the Godunov approach. This section is devoted to introducing this concept, which is very intuitively related to the physical problem considered. We define the Riemann problem for (2.1) to be the problem of finding the solution to (2.1) with Riemann initial data Ul if x < 0, U 0 (x) = (2.15) Ur if x > 0, for two given constants Ul and Ur . By a simple scaling argument, this solution is indeed a function only of x/t. D EFINITION 2.1. An approximate Riemann solver for (2.1) is a vector function R(x/t, Ul , Ur ) that is an approximation of the solution to the Riemann problem,

2. Finite volume schemes for conservative systems

201

in the sense that it must satisfy the consistency relation R(x/t, U, U ) = U,

(2.16)

and the conservativity identity Fl (Ul , Ur ) = Fr (Ul , Ur ),

(2.17)

where the left and right numerical fluxes are defined by

0

R(v, Ul , Ur ) − Ul dv,

Fl (Ul , Ur ) = F (Ul ) − −∞

∞

Fr (Ul , Ur ) = F (Ur ) +

R(v, Ul , Ur ) − Ur dv.

(2.18)

0

It is called dissipative with respect to a convex entropy η for (2.1) if Gr (Ul , Ur ) − Gl (Ul , Ur ) 0,

(2.19)

where

0

η R(v, Ul , Ur ) − η(Ul ) dv,

Gl (Ul , Ur ) = G(Ul ) − −∞

∞

η R(v, Ul , Ur ) − η(Ur ) dv,

Gr (Ul , Ur ) = G(Ur ) +

(2.20)

0

and G is the entropy flux associated to η, G = η F . It is possible to prove that the exact solution to the Riemann problem satisfies these properties, but however the above definition is rather motivated by numerical schemes. Let us explain why. Consider a discrete sequence Uin , i ∈ Z, and define the function U n (x) that is piecewise constant over the mesh with value Uin in each cell Ci . Therefore, Uin is nothing but the cell average of the function U n (x). Then, the Godunov method consists in computing the solution U (t, x) to (2.1) for tn < t < tn+1 with U n (x) as initial data, and finally in defining Uin+1 as the average over Ci of U (tn+1 , x). According to the computation (2.8)–(2.9), this leads indeed to a numerical flux given by (2.9). Now, let us generalize the previous construction by replacing the exact solution to (2.1) by an approximate one. We still consider the initial data U n (x) that is piecewise constant, but we now consider the approximate solution Uapp (t, x) for

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 2. Construction of an approximate solution Uapp .

tn t < tn+1 (as illustrated in Figure 2) by x − xi+1/2 n n , Ui , Ui+1 Uapp (t, x) = R t − tn

if xi < x < xi+1 .

(2.21)

This formula is obtained by considering that close to each interface point xi+1/2 , we have to solve a translated Riemann problem. Since (2.1) is invariant under translation in time and space, we can think of sticking together the local approximate Riemann solutions, which leads to (2.21), at least for times such that these solutions do not interact. This is possible until time tn+1 under a CFL condition 1/2, in the sense that

xi 2 t

xi+1 x/t > 2 t

x/t < −

⇒

R(x/t, Ui , Ui+1 ) = Ui ,

⇒

R(x/t, Ui , Ui+1 ) = Ui+1 .

(2.22)

Then, we define Uin+1 to be the average over Ci of this approximate solution Uapp (t, x) at time tn+1 − 0. According to the definition (2.18) of Fl and Fr and by using (2.22), we get Uin+1

1 =

xi 1 =

xi

x i+1/2

Uapp (tn+1 − 0, x) dx xi−1/2

x

i /2

n R x/ t, Ui−1 , Uin dx

0

1 +

xi

0 − xi /2

n dx R x/ t, Uin , Ui+1

2. Finite volume schemes for conservative systems

=

Uin

1 +

xi

203

x

i /2

n R x/ t, Ui−1 , Uin − Uin dx

0

+

1

xi

= Uin −

0

n R x/ t, Uin , Ui+1 − Uin dx

− xi /2

n

t n n Fl Ui , Ui+1 − Fr Ui−1 , Uin .

xi

(2.23)

Therefore we see that with the conservativity assumption (2.17), this is a conservative scheme, with numerical flux F (Ul , Ur ) = Fl (Ul , Ur ) = Fr (Ul , Ur ).

(2.24)

The consistency assumption (2.16) ensures that this numerical flux is consistent, in the sense of Section 2.1. Then let us examine the discrete entropy inequality. Since η is convex, we can use Jensen’s inequality in (2.23), and we get

η

Uin+1

1

xi

x

i /2

n η R x/ t, Ui−1 , Uin dx

0

1 +

xi

n

0

n dx η R x/ t, Uin , Ui+1

− xi /2

= η Ui −

n

t n n Gl Ui , Ui+1 − Gr Ui−1 , Uin .

xi

(2.25)

Under assumption (2.19), we conclude that n

t n n G Ui , Ui+1 − G Ui−1 , Uin 0, η Uin+1 − η Uin +

xi

(2.26)

for any numerical entropy flux function G(Ul , Ur ) such that Gr (Ul , Ur ) G(Ul , Ur ) Gl (Ul , Ur ),

(2.27)

thus we recover the definition of Section 2.3, since (2.16) ensures that this numerical entropy flux is consistent. In this way, to any approximate Riemann solver R we can associate a conservative numerical scheme. If we use the exact Riemann solver, the scheme we get is called the (exact) Godunov scheme.

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

R EMARK 2.1. The approximate Riemann solver framework works as well with interface dependent solvers Ri+1/2 . This is used in practice to choose a solver adapted to the data Ui , Ui+1 , so as to produce a viscosity which is as small as possible. In practice, the exact resolution to the Riemann problem is too complicate and too expensive, especially for systems with large dimension. Thus we rather use approximate solvers. The most simple choice is the following. 2.5. Simple solvers We shall call simple solver an approximate Riemann solver consisting in a set of finitely many simple discontinuities. This means that there exists a finite number m 1 of speeds σ0 = −∞ < σ1 < · · · < σm < σm+1 = +∞,

(2.28)

and intermediate states U0 = Ul , U1 , . . . , Um−1 , Um = Ur

(2.29)

(depending on Ul and Ur ), such that as illustrated in Figure 3, R(x/t, Ul , Ur ) = Uk

if σk < x/t < σk+1 .

(2.30)

Then the conservativity identity (2.17) becomes m

σk (Uk − Uk−1 ) = F (Ur ) − F (Ul ),

(2.31)

k=1

and the entropy inequality (2.19) becomes m

σk η(Uk ) − η(Uk−1 ) G(Ur ) − G(Ul ).

k=1

Figure 3.

A simple solver.

(2.32)

2. Finite volume schemes for conservative systems

205

Conservativity thus enables to define the intermediate fluxes Fk , k = 0, . . . , m, by Fk − Fk−1 = σk (Uk − Uk−1 ),

F0 = F (Ul ),

Fm = F (Ur ),

(2.33)

which is a kind of generalization of the Rankine–Hugoniot relation. The numerical flux is then given by F(Ul , Ur ) = Fk ,

where k is such that σk 0 σk+1 .

(2.34)

We can observe that if it happens that σk = 0 for some k, there is no ambiguity in this definition since (2.33) gives in this case Fk = Fk−1 . An explicit formula for the numerical flux is indeed σk (Uk − Uk−1 ) F (Ul , Ur ) = F (Ul ) + σk 0

2.6. CFL condition for simple solvers For a simple solver we can define the local speed by ai+1/2 = a(Ui , Ui+1 ),

a(Ul , Ur ) = sup |σk |.

(2.36)

1km

Then the CFL condition (2.22) reads 1 (2.37) min( xi , xi+1 ). 2 This is called a CFL condition 1/2, and as the above computations show, it ensures the stability of the scheme (entropy inequalities). However, in practice, one uses rather the CFL 1 condition (as written in Section 2.2),

t ai+1/2

t ai+1/2 min( xi , xi+1 ).

(2.38)

It enables larger timesteps (and thus is less CPU consuming), and it gives sufficiently stable results. 2.7. Nonnegativity and vacuum Either in gas dynamics or shallow water models, the nonnegativity of density or water height is a key requirement for the numerical stability. This nonnegativity property is a particular case of what is called an invariant domain for a system of conservation laws. The corresponding analysis for approximate Riemann solvers

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

206

is performed in Bouchut (2004), but let us here say shortly how it works, in the case of nonnegativity of density. The key remark is that according to the definition (2.23) of Uin+1 as an average of values of Uapp and to the definition (2.21) of Uapp , it is enough that the approximate Riemann solver R(x/t, Ul , Ur ) always has a nonnegative density, whenever the arguments Ul and Ur have nonnegative densities. For simple solvers, it means that the only requirement is that the intermediate states Uk , k = 0, . . . , m, have nonnegative densities. Then, the next question is the possibility to treat data having vacuum (dry states for shallow water). In the computation of an approximate Riemann solver, if the two values Ul , Ur are vacuum data Ul = Ur = 0, there is no difficulty, we can simply set R = 0. The real problem occurs when one of the two values is zero and the other not. We shall say that an approximate Riemann solver can resolve the vacuum if in this case of two values Ul , Ur which are zero and nonzero, it gives a solution R(x/t, Ul , Ur ) with nonnegative density and with finite speeds of propagation σ1 , . . . , σm . This is necessary, otherwise the CFL condition (2.38) would give a zero timestep. The selection of solvers that are able to resolve vacuum is a main point for applications to flows in rivers with shallow water type equations. 2.8. Roe solver The Roe solver, introduced by Roe (1981), is an example of simple solver. It is obtained as follows. We need first to find a p ×p diagonalizable matrix A(Ul , Ur ) (called a Roe matrix), such that F (Ur ) − F (Ul ) = A(Ul , Ur )(Ur − Ul ),

A(U, U ) = F (U ).

(2.39)

Then we define R(x/t, Ul , Ur ) to be the solution to the linear problem ∂t U + A(Ul , Ur )∂x U = 0,

(2.40)

with initial Riemann data (2.15). Denoting by σ1 , . . . , σm the distinct eigenvalues of A(Ul , Ur ), we can decompose Ur − Ul along the eigenspaces Ur − Ul =

m

δUk ,

A(Ul , Ur )δUk = σk δUk ,

(2.41)

k=1

and the solution is given by R(x/t, Ul , Ur ) = Ul +

k0

δUk

if σk0 < x/t < σk0 +1 .

(2.42)

k=1

This defines a simple solver, the assumption (2.39) gives indeed the conservativity (2.31), since σk δUk = A(Ul , Ur )(Ur − Ul ) = F (Ur ) − F (Ul ). (2.43) k

2. Finite volume schemes for conservative systems

207

E XAMPLE 2.2. For the isentropic system (1.14), the classical Roe matrix is given by 0 1 A(Ul , Ur ) = p(ρr )−p(ρl ) (2.44) , − uˆ 2 2uˆ ρr −ρl with

√ uˆ =

√ ρl ul + ρr ur . √ √ ρl + ρr

(2.45)

The Roe method is in practice very accurate, but does suffer from incomplete stability properties. In particular, it is not entropy satisfying, entropy fixes have to be designed. We refer the reader to the literature (Godlewski and Raviart, 1996; Toro, 1999) for this class of schemes. For our purpose of designing good schemes for shallow water models, this method is a priori less interesting than the HLL or HLLC solvers because it can produce negative densities. 2.9. HLL solver A basic simple approximate Riemann solver is the HLL solver, that was introduced by Harten, Lax and Van Leer (1983), and was indeed the first example of approximate Riemann solver. It works for general system of conservation laws (2.1). We take two parameters σ1 < σ2 , and define the two speeds simple approximate Riemann solver by ⎧ if x/t < σ1 , ⎨ Ul R(x/t, Ul , Ur ) =

σ2 Ur −F (Ur ) σ2 −σ1

+

F (Ul )−σ1 Ul σ2 −σ1

if σ1 < x/t < σ2 , (2.46) if σ2 < x/t. Ur The consistency and conservativity conditions (2.16), (2.17) or (2.31) are easily checked for this solver. According to (2.33)–(2.34), the HLL numerical flux is given by ⎧ if 0 < σ1 , ⎨ F (Ul ) σ1 F (Ur ) σ1 σ2 F(Ul , Ur ) = σ2 F (Uσl )2− + (U − U ) if σ1 < 0 < σ2 , (2.47) r l −σ1 σ2 − σ1 ⎩ if σ2 < 0. F (Ur ) The invariant domains and entropy conditions can be analyzed and justified, under the subcharacteristic conditions ⎩

σ1 λj (U ) σ2 ,

(2.48)

where λj (U ) are the eigenvalues of F (u). According to Remark 2.1, a local optimization of (2.48) leads to the choice σ1 =

inf

inf λj (U ),

U =Ul ,Ur j

σ2 =

sup

sup λj (U ),

U =Ul ,Ur j

(2.49)

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

208

even if theoretically one should involve also intermediate values of U . In the case of isentropic gas dynamics, the solver preserves nonnegativity of density and can treat the vacuum. Indeed, according to Section 2.7, we only have to check the nonnegativity of the intermediate density in the approximate Riemann solver (2.46), which is given by σ2 − ur ul − σ1 (2.50) ρr + ρl . σ 2 − σ1 σ 2 − σ1 But (2.49) implies that σ1 ul − p (ρl ) ul and σ2 ur + p (ρr ) ur , which proves that ρ ∗ 0. In practice the HLL solver works quite well except an excessive numerical diffusion of waves associated to eigenvalues other than the lowest and largest ones (but this does not occur for isentropic gas dynamics since there are only two eigenvalues). ρ∗ =

2.10. HLLC solver The HLLC solver (also called acoustic solver) has been introduced for gas dynamics equations in order to restore the central wave traveling at velocity u, and better resolve the associated discontinuities. For the isentropic system (1.14), there is no such wave, and therefore there is no special interest in this solver, except that it is naturally expected to be used in systems that look like gas dynamics systems. The solver is defined as a three speeds simple solver ⎧ if x/t < σ1 , U ⎪ ⎪ l ⎪ ⎨ U ∗ if σ < x/t < σ , 1 2 l R(x/t, Ul , Ur ) = (2.51) ⎪ Ur∗ if σ2 < x/t < σ3 , ⎪ ⎪ ⎩ Ur if σ3 < x/t, where the speeds are defined by σ1 = ul − cl /ρl ,

σ2 = u∗ ,

σ3 = ur + cr /ρr ,

(2.52)

with Ul = (ρl , ρl ul ), Ur = (ρr , ρr ur ), Ul∗ = (ρl∗ , ρl∗ u∗ ), Ur∗ = (ρr∗ , ρr∗ u∗ ), and cl ul + cr ur + πl − πr , c l + cr 1 cr (ur − ul ) + πl − πr = + , ρl cl (cl + cr ) 1 cl (ur − ul ) + πr − πl = + , ρr cr (cl + cr )

u∗ = 1 ρl∗ 1 ρr∗

(2.53)

with πl = p(ρl ),

πr = p(ρr ).

(2.54)

2. Finite volume schemes for conservative systems

209

The consistency and conservativity conditions (2.16), (2.31) are satisfied, and the intermediate fluxes of (2.33) are Fr∗ = ρr∗ u∗r , ρr∗ (u∗r )2 + π ∗ , Fl∗ = ρl∗ u∗l , ρl∗ (u∗l )2 + π ∗ , (2.55) with π∗ =

cr πl + cl πr − cl cr (ur − ul ) . c l + cr

(2.56)

The numerical flux F (Ul , Ur ) is defined by (2.34). As in the HLL solver, one has two parameters cl , cr that need to be chosen satisfying the subcharacteristic conditions ρ 2 p (ρ) cl2 for any ρ between ρl and ρl∗ , and ρ 2 p (ρ) cr2 for any ρ between ρr and ρr∗ . It is proved in Bouchut (2004) via the interpretation by the Suliciu relaxation system that this subcharacteristic condition implies discrete entropy inequalities for the entropy η defined in (1.22), with numerical entropy fluxes Gl∗ = ρl∗ (u∗ )2 /2 + ρl∗ el∗ + π ∗ u∗ , Gr∗ = ρr∗ (u∗ )2 /2 + ρr∗ er∗ + π ∗ u∗ , (2.57) with el∗ = el − πl2 /2cl2 + (π ∗ )2 /2cl2 , er∗ = er − πr2 /2cr2 + (π ∗ )2 /2cr2 ,

(2.58)

and el = e(ρl ), er = e(ρr ). According to Bouchut (2004), the following choice of cl , cr enables to fully satisfy the nonnegativity of ρl∗ , ρr∗ and the subcharacteristic conditions, ⎧c πr −πl + ul − ur + , ⎨ ρll = p (ρl ) + α √ ρr p (ρr ) if πr − πl 0, (2.59) ⎩ cr = p (ρ ) + α πl −πr + u − u , r l r ρ cl + ⎧ cr πl −πr r √ + ul − ur + , ⎨ ρr = p (ρr ) + α ρl p (ρl ) if πr − πl 0, (2.60) ⎩ cl = p (ρ ) + α πr −πl + u − u , l l r + ρl cr with the notation x+ = max(0, x). For this to be valid, we have to assume that d ∀ρ > 0, (2.61) ρ p (ρ) > 0, dρ ρ p (ρ) → ∞ as ρ → ∞, (2.62) d ρ p (ρ) α p (ρ), for some constant α 1. (2.63) dρ In the case of shallow water p(ρ) = gρ 2 /2, this is satisfied with α = 3/2.

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

210

In conclusion, with the choice (2.59)–(2.60) and under assumptions (2.61)–(2.63), the solver preserves the nonnegativity of density, handles data with vacuum, and satisfies a discrete entropy inequality. 2.11. Passive transport If often occurs in fluid dynamics problems that one has to solve a transport equation for a material quantity φ, that follows the particle paths. In other words, φ satisfies ∂t (ρφ) + ∂x (ρuφ) = 0,

(2.64)

where ρ 0, u are assumed to be given by the resolution of a closed system of equations that include the continuity equation ∂t ρ + ∂x (ρu) = 0.

(2.65)

For smooth solutions, the two equations can be combined to give ∂t φ + u∂x φ = 0.

(2.66)

This equation means that φ is simply passively transported with the flow. The functions ρ, u can be obtained by solving a system of equations that can involve other quantities, but we need not specify how they are obtained for what we explain here. An important property of (2.64)–(2.65) is that it gives directly a family of entropy inequalities, because if we multiply (2.66) by S (φ) for any function S, we get ∂t (S(φ)) + u∂x (S(φ)) = 0, thus combining it with (2.65) we get ∂t (ρS(φ)) + ∂x (ρuS(φ)) = 0. Now, if S is convex, one can easily check that ρS(φ) is a convex function of (ρ, ρφ). Therefore, one expects for weak solutions a family of inequalities, ∂t ρS(φ) + ∂x ρuS(φ) 0, S convex. (2.67) In particular, taking S(φ) = (φ − k)+ ≡ max(0, φ − k) or S(φ) = (k − φ)+ ≡ max(0, k − φ), we deduce the maximum principle inf φ 0 (y) φ(t, x) sup φ 0 (y). y

(2.68)

y

Numerically, there is a very natural way to solve (2.64), being given a numerical flux for (2.65). This is done as follows. Assume that we have the discrete conservative formula

t 0 0 F = 0, − Fi−1/2 ρin+1 − ρi + (2.69)

xi i+1/2

2. Finite volume schemes for conservative systems

211

0 for some numerical flux Fi+1/2 that is computed in a way we do not need to specify. Then the natural scheme for solving (2.64) is

ρin+1 φin+1 − ρi φi +

t φ φ Fi+1/2 − Fi−1/2 = 0,

xi

(2.70)

with the by now classical passive transport flux 0 0 Fi+1/2 φi if Fi+1/2 0, φ Fi+1/2 = 0 0 Fi+1/2 φi+1 if Fi+1/2 0.

(2.71)

Denoting x+ = max(0, x) and x− = min(0, x), another way to write (2.71) is 0 0 φ φ + Fi+1/2 φ . Fi+1/2 = Fi+1/2 (2.72) + i − i+1 One can check (Bouchut, 2004) that the passive transport scheme (2.70)–(2.71) is consistent with (2.64), and under a natural CFL condition it satisfies the discrete maximum principle min(φi−1 , φi , φi+1 ) φin+1 max(φi−1 , φi , φi+1 )

(2.73)

and the discrete entropy inequalities

t S S F 0, − Fi−1/2 ρin+1 S φin+1 − ρi S(φi ) +

xi i+1/2 with

S Fi+1/2

=

0 Fi+1/2 S(φi )

0 if Fi+1/2 0,

0 Fi+1/2 S(φi+1 )

0 if Fi+1/2 0.

S convex, (2.74)

(2.75)

E XAMPLE 2.3. Consider the isentropic system in one and a half dimension, or isentropic system with transverse velocity ⎧ ⎪ ⎨ ∂t ρ + ∂x (ρu) = 0, (2.76) ∂t (ρu) + ∂x (ρu2 + p(ρ)) = 0, ⎪ ⎩ ∂t (ρv) + ∂x (ρuv) = 0. This conservative system is hyperbolic with eigenvalues λ1 = u − p (ρ), λ2 = u, λ3 = u + p (ρ), and it has an entropy (the physical total energy) η = ρ u2 + v 2 /2 + ρe(ρ), (2.77) with entropy flux G = ρ u2 + v 2 /2 + ρe(ρ) + p(ρ) u.

(2.78)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

The system (2.76) can be considered as the coupling of the usual one-dimensional isentropic system with a passive transport equation on v. Therefore, a numerical flux for (2.76) can be obtained as l , U r , F v (Ul , Ur ) , F (Ul , Ur ) = F is U (2.79) l = (ρl , ρl ul ), U r = with Ul = (ρl , ρl ul , ρl vl ), Ur = (ρr , ρr ur , ρr vr ), U is (ρr , ρr ur ), F is any numerical flux for the usual isentropic system, and 0 F vl if F 0 0, F v (Ul , Ur ) = (2.80) F 0 vr if F 0 0, l , U r ) is the density component of F is (U l , U r ). The numerical with F 0 = F 0 (U flux (2.79) satisfies the discrete maximum principle on v, and if F is satisfies discrete entropy inequalities, it also satisfies discrete entropy inequalities, associated to η, obtained by addition with (2.74) with S(v) = v 2 /2. R EMARK 2.4. In the previous example, if F is is the numerical flux of the HLLC solver, then the numerical flux (2.79) is associated to a simple approximate Riemann solver obtained by completing the intermediate states by vl∗ = vl , vr∗ = vr . This is due to the fact that in the HLLC solver, F0 0

if and only if

σ2 = u∗ 0.

(2.81)

This property underlines the fact that the HLLC solver is particularly adapted to gas dynamics problems. 2.12. Numerical treatment of boundary conditions The numerical treatment of boundary conditions is usually done quite easily within the finite volume framework, even if we do not really know which solution we select, knowing that the mathematical theory is quite poor on the subject. Consider a system of conservation laws in a bounded interval (all we are discussing here would be valid also with source term or for a quasilinear system) t > 0, a < x < b, ∂t U + ∂x F (U ) = 0, (2.82) together with boundary conditions. To fix ideas, let us first assume that we wish to impose for t > 0 U (t, a) = Ua (t)

and U (t, b) = Ub (t),

(2.83)

even if in general only a part of the boundary values can be retained in the solution, see Section 1.4 and in particular (1.33). Assume that the interval (a, b) is divided into nx cells Ci = (xi−1/2 , xi+1/2 ), i = 1, . . . , nx , on which we compute discrete values Uin , i = 1, . . . , nx . The

2. Finite volume schemes for conservative systems

213

boundary is thus a = x1/2 , b = xnx +1/2 . Then the simple and classical way to discretize (2.82)–(2.83) is to set ghost values U0n and Unnx +1 by U0n = Ua (tn ),

Unnx +1 = Ub (tn ),

(2.84)

and then to apply the finite volume update Uin+1 − Uin +

t (Fi+1/2 − Fi−1/2 ) = 0,

xi

i = 1, . . . , nx ,

(2.85)

with as usual

n Fi+1/2 = F Uin , Ui+1 ,

i = 0, . . . , nx ,

(2.86)

for some given numerical flux. Then (2.84) is naturally taken into account in the computation of F1/2 and Fnx +1/2 . A more accurate choice would be indeed to take U0n and Unnx +1 so that (U0n + U1n )/2 = Ua (tn ), (Unnx + Unnx +1 )/2 = Ub (tn ), which means that we extend the values of U by odd symmetry. However, when doing such extrapolation, one has to care to get admissible values for U0n and Unnx +1 (nonnegative density, or other requirement depending on the system). Thus, formulas for doing this would be U0n = proj(2Ua (tn ) − U1n ), Unnx +1 = proj(2Ub (tn ) − Unnx ), where proj is a suitable projection onto admissible states. Assume now that we would like to solve (1.33) at the boundary x = a. Then one just has to set (2.84), where Ua (t) is chosen in such a way that Ua (t) ∈ B(t).

(2.87)

However, it is not sure at all that the numerical solution will converge to the expected one. It could converge to the solution to another boundary problem, for example U (t, a) ∈ C(t) for another set C(t). We refer to Godlewski and Raviart (1996) for a discussion. E XAMPLE 2.5. Consider the isentropic gas dynamics system (1.14), with the wall boundary condition u(t, a) = 0, as discussed in Example 1.5. Then, the most simple way to define the ghost value U0n is to symmetrize U1n by taking ρ0n = ρ1n , un0 = −un1 . E XAMPLE 2.6. For periodic boundary conditions, the natural choice is U0n = Unnx , Unnx +1 = U1n . E XAMPLE 2.7. For Neumann boundary conditions, i.e. ∂x U (t, a) = 0, one can take U0n = U1n . E XAMPLE 2.8. Assume the we would like to solve the isentropic system (1.14), with set incoming mass flux on the boundary ρ(t, a)u(t, a) = q(t) 0. Then

214

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

since the numerical update (2.85)–(2.86) involves a flux, an accurate way of proceeding is to look for a value U0n such that F 0 (U0n , U1n ) = q(tn ). A priori there is no unique solution to this problem, and of course it depends on the particular form of the numerical flux.

3. Finite volumes for systems with source terms A lot of progress has been made in the past years on the numerical treatment of source terms in systems of conservation laws, and in particular for the shallow water system. To fix ideas, in this section we consider systems of the form ∂t U + ∂x F (U, Z) + B(U, Z)Zx = 0, (3.1) where U (t, x) ∈ Rp is the unknown, Z(x) ∈ Rr is a given vector valued function, and Zx = ∂x Z. The nonlinearities are supposed to be smooth, F (U, Z) ∈ Rp , and B(U, Z) is a p × r matrix. The case when Z is scalar (r = 1) in (3.1) is already interesting, and it covers the special choice Z(x) = x, which writes if we take F = F (U ), B = B(U ), ∂t U + ∂x F (U ) = −B(U ). (3.2) The numerical challenge that occurs in the resolution of (3.1) is the accurate treatment of the balance between the differential term and the source during the time evolution. In this respect, the steady states play a crucial role. They are the solutions U (x) which are independent of time, and hence solve ∂x (F (U, Z)) + B(U, Z)Zx = 0. These solutions play an important role because they are usually obtained as limits when time tends to infinity of the general solutions to (3.1). In order to well resolve (3.1), in particular concerning the source term, main ideas were developed in Greenberg and LeRoux (1996), Gosse and LeRoux (1996), Greenberg, LeRoux, Baraille and Noussair (1997), Bermúdez and Vásquez (1994), LeRoux (1999), Gosse (2000, 2001), Jin (2001), Botchorishvili, Perthame and Vasseur (2003). In these works was introduced the notion of wellbalanced schemes, which are schemes that preserve exactly at the discrete level some discrete steady states solutions. Many schemes were afterwards developed (Vázquez-Cendón, 1999; García-Navarro and Vázquez-Cendón, 2000; Kurganov and Levy, 2002; Perthame and Simeoni, 2001; Chacón Rebollo, Domínguez Delgado and Fernández Nieto, 2003; Chinnayya, LeRoux and Seguin, 2004; Gallouët, Hérard and Seguin, 2003; Katsaounis, Perthame and Simeoni, 2004; Bale, LeVeque, Mitran and Rossmanith, 2002; Audusse, Bouchut, Bristeau, Klein and Perthame, 2004; Parés and Castro, 2004). Nonconservative systems with sources have been considered by Parés (2006) with systems of the form ∂t U + ∂x F (U, Z) + A(U, Z)∂x U + B(U, Z)Zx = 0. (3.3)

3. Finite volumes for systems with source terms

215

Note that the systems (3.1) or (3.3) can be interpreted as quasilinear systems in = (U, Z), by completing them with the trivial equation ∂t Z = 0. the variable U At this level, we have to explain a bit how (3.1) can be well posed, even if the mathematical theory is not very well settled. We shall always assume that the system is hyperbolic with respect to U , which means that FU (U, Z) is diagonalizable. The first case when (3.1) is well posed is when Z(x) is smooth (say Lipschitz continuous). Then this is just a conservative system with nonsingular source, and physically relevant discontinuous weak solutions are selected with the Rankine– Hugoniot conditions (1.18) (which do not involve the source), and with an entropy inequality ∂t η(U, Z) + ∂x G(U, Z) + ηU (FZ + B) − GZ Zx 0, (3.4) where indices U or Z denote partial differentiation, and η(U, Z) is an entropy with entropy flux G(U, Z) (i.e. GU = ηU FU ). Here η needs only be convex with respect to U . The second case when (3.1) is well posed is when Z can have discontinuities, but the eigenvalues of FU do not vanish (and we call resonant a point (U, Z) where 0 is an eigenvalue of FU ). Indeed, if we consider (3.1) as a quasilinear system (with the equation ∂t Z = 0), then the matrix of the system (in the sense of Section 1.1) is FU FZ + B . A(U, Z) = (3.5) 0 0 The eigenvalues of A(U, Z) are those of FU , to which we adjoin the value 0. At a noncritical point, FU is diagonalizable and does not have 0 as eigenvalue, thus obviously A(U, Z) is diagonalizable, and the system is hyperbolic. Then, the good property is that weak solutions to (3.1) are well-defined out of the resonant points, because the nonconservative product BZx could cause difficulties only when Z has discontinuities, and in this location the eigenvalue involved is 0 which is linearly degenerate. We recall that discontinuities associated with linearly degenerate eigenvalues are well-defined, without need of any Rankine– Hugoniot condition (Bouchut, 2004). Therefore, the only Rankine–Hugoniot condition needed is (1.18) when ξ˙ = 0. To complete the system we have to provide an entropy inequality for the quasilinear system in (U, Z), ∂t η(U, Z) + ∂x G(U, Z) 0, (3.6) with η(U, Z) convex with respect to U , GU = ηU FU , and GZ = ηU (FZ + B). In the general case of discontinuous Z with resonant points (which is the case for the shallow water system), there are some data for which several solutions exist, containing shocks or rarefactions attached to stationary contact discontinuities. All these solutions can be stable (limits of sequences of “good” unique

216

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

solutions). At the numerical level, this is even worse, one can have convergence to something that is even not a weak solution (but fortunately this happens rather rarely). Notice that contact discontinuities associated to the vanishing eigenvalue are in particular steady states, which explains the critical role played by these solutions. 3.1. The one-dimensional shallow water system The main example of system with source, resonance and nontrivial steady states is the Saint-Venant system for shallow water with topography. This system is naturally under the form (3.1). Denoting by h(t, x) 0 the water height, and u(t, x) the velocity, the system reads ∂t h + ∂x (hu) = 0, (3.7) ∂t (hu) + ∂x (hu2 + gh2 /2) + hgzx = 0, where g > 0 is the gravitational constant and z(x) is the topography. The system has the form (3.1) with U = (h, hu), Z = z, F (U, z) = F (U ) = F 0 (U ), F 1 (U ) = hu, hu2 + gh2 /2 , (3.8) B(U, z) = B(U ) = (0, gh).

(3.9)

The conservative part is nothing else than the isentropic gas dynamics sys2 tem (1.14) √ with ρ ≡ h and p(h) = gh /2. The resonant points are defined by u = ± gh. In order to obtain the steady states, we subtract u times the first equation in (3.7) to the second, and divide the result by h. We get ∂t u + ∂x u2 /2 + gh + gz = 0. (3.10) Therefore, the steady states are exactly the functions h(x), u(x) satisfying hu = cst, u2 /2 + gh + gz = cst.

(3.11)

Between these steady states, some play an important role, the steady state at rest for which the first constant is 0, or equivalently u = 0, (3.12) h + z = cst. An entropy can be obtained as follows. We multiply the first equation in (3.7) by u2 /2 + gh, we multiply (3.10) by hu, and add the results. This gives ∂t η + ∂x G + hugzx 0,

(3.13)

3. Finite volumes for systems with source terms

where η and G are defined according to (1.22) and (1.24), η(U ) = hu2 /2 + gh2 /2, G(U ) = hu2 /2 + gh2 u.

217

(3.14)

The inequality stands here just because of discontinuous solutions, as usual. Next, we add to (3.13) the first equation in (3.7) multiplied by gz, and since ∂t z = 0, this yields ∂t (η + hgz) + ∂x G + hugz 0. (3.15) Thus we have the entropy (the physical energy) and entropy flux η = η + hgz,

G = G + hugz.

(3.16)

The shallow water system has other specific properties that are worthwhile to state. The first is that, as in gas dynamics, the water height h needs to remain nonnegative. Also the total amount of water need to be preserved, which means that the first equation is conservative. Another property is that the system (3.7) is invariant under translations in z, in the sense that adding a constant to z does not modify the equations. The inequality (3.13) has the advantage to have this property also, in contrast with (3.15). A variant of the system is the one and a half dimensional shallow water system, which is (3.7) completed with the transverse velocity equation ∂t (hv) + ∂x (hvu) = 0.

(3.17)

As commented in Section 2.11, this extended system does not introduce any further difficulty. 3.2. Upwind interface schemes As explained above, the numerical challenge for (3.1) is to well resolve the balance between the conservative term ∂x (F (U, Z)) and the source B(U, Z)Zx . In order to do so, a nice way of formulating the schemes is to discretize these two terms at the same location, the interface xi+1/2 . This is the upwind interface scheme (UIS) approach, formulated in its generality by Katsaounis, Perthame and Simeoni (2004). We shall retain here only this formulation, which is rather general and flexible. It handles data Zi attached to each cell, instead of interface values Zi+1/2 used for example by Jin (2001). Similarly as in Section 2, let us consider a one-dimensional grid of points xi+1/2 , i ∈ Z, as in (2.2)–(2.3). In order to solve (3.1), we consider discrete data n = (U n , Zi ) over the mesh. The data Zi do not evolve with time, as expected, U i i while

t (Fi+1/2− − Fi−1/2+ ) = 0, Uin+1 − Uin + (3.18)

xi

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with n Fi+1/2− = Fl Uin , Ui+1 , Zi , Zi+1 , n , Zi , Zi+1 . Fi+1/2+ = Fr Uin , Ui+1

(3.19)

The functions l , U r , Fl (Ul , Ur , Zl , Zr ) ≡ Fl U l , U r Fr (Ul , Ur , Zl , Zr ) ≡ Fr U

(3.20)

are the left and right numerical fluxes. In this formulation we only consider firstorder three point nonconservative schemes. A more particular form occurs when solving (3.1) with F (U, Z) = F (U ) and B(U, Z) = B(U ) (which is the case for shallow water), because then there is invariance of the system by addition of a constant to Z. Therefore, in this case, Fl and Fr should depend only on Zr − Zl , and not on Zl and Zr independently. 3.3. Well-balancing A main feature that is desirable for the scheme (3.18)–(3.19) is that it preserves some discrete steady states, approximating the exact ones defined as smooth functions (U (x), Z(x)) satisfying ∂x F (U, Z) + B(U, Z)Zx = 0. (3.21) These discrete steady states are discrete sequences (Ui , Zi )i∈Z that satisfy an approximation of (3.21), under the form of a nonlinear relation at each interface, linking Ui , Ui+1 , Zi , Zi+1 , D(Ui , Ui+1 , Zi , Zi+1 ) = 0.

(3.22)

We shall often write this relation only locally, as D(Ul , Ur , Zl , Zr ) = 0. These discrete steady states can be defined in various ways. E XAMPLE 3.1. For a scalar law U ∈ R, and if F (U, Z) = F (U ), B(U, Z) = B(U ) > 0, we can define D(U ) by D (U ) =

F (U ) . B(U )

Then (3.21) becomes ∂x D(U ) + Z = 0.

(3.23)

(3.24)

Therefore, we can take for discrete steady states the relation D(Ul ) + Zl = D(Ur ) + Zr .

(3.25)

3. Finite volumes for systems with source terms

219

E XAMPLE 3.2. For the shallow water system (3.7), since the continuous steady states are those solving (3.11), we can take for discrete steady states the relations hl ul = hr ur , (3.26) u2l /2 + ghl + gzl = u2r /2 + ghr + gzr . In particular, the discrete steady states at rest are those for which ul = ur = 0, hl + zl = hr + zr .

(3.27)

The relations (3.27) will be chosen in our applications. Once some discrete steady states are selected, we define the well-balanced schemes as follows. D EFINITION 3.1. The scheme (3.18)–(3.19) is well-balanced relatively to some discrete steady states defined by a relation D if one has for any Ul , Ur , Zl , Zr satisfying D(Ul , Ur , Zl , Zr ) = 0 the identities Fl (Ul , Ur , Zl , Zr ) = F (Ul , Zl ), Fr (Ul , Ur , Zl , Zr ) = F (Ur , Zr ).

(3.28)

According to (3.18)–(3.19), this property guarantees obviously that if at time tn we start with a steady state sequence (Ui ), then it remains unchanged at the next time level. 3.4. Consistency The consistency of (3.18)–(3.19) with (3.1) comes naturally after the discussion of well-posedness in the beginning of Section 3 in the case of discontinuous Z. D EFINITION 3.2. We say that the scheme (3.18)–(3.19) is consistent with (3.1) if the numerical fluxes satisfy the consistency with the exact flux Fl (U, U, Z, Z) = Fr (U, U, Z, Z) = F (U, Z) for any (U, Z) ∈ Rp × Rr ,

(3.29)

and the asymptotic conservativity/consistency with the source Fr (Ul , Ur , Zl , Zr ) − Fl (Ul , Ur , Zl , Zr ) = −B(U, Z)(Zr − Zl ) + o(Zr − Zl ), as Ul , Ur → U and Zl , Zr → Z.

(3.30)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

We recall here that the notation φ = o(Zr −Zl ) means that φ = |Zr −Zl |φ, with φ → 0. We have to mention that condition (3.30) only requires a property when Zl and Zr are asymptotically close, thus it only concerns continuous functions Z(x). This is coherent with our discussion above: one should impose consistency in the weak sense for Z = cst, and consistency for smooth solutions, which is exactly what (3.29)–(3.30) means (notice that taking Zl = Zr in (3.30) gives in that case the conservativity Fl = Fr ). Further justifications can be found in Bouchut (2004), Perthame and Simeoni (2003). 3.5. Discrete entropy inequalities Stability can be analyzed, as in the conservative case, via invariant domains and entropy. Let us only write explicitly the definition concerning entropy inequalities. D EFINITION 3.3. We say that the scheme (3.18)–(3.19) satisfies a discrete entropy inequality associated to the convex entropy η for (3.1) (in the sense of (3.6)), if there exists a numerical entropy flux function G(Ul , Ur , Zl , Zr ) which is consistent with the exact entropy flux (in the sense that G(U, U, Z, Z) = G(U, Z)), such that, under some CFL condition, the discrete values computed by (3.18)– (3.19) automatically satisfy

t η Uin+1 , Zi − η Uin , Zi + (Gi+1/2 − Gi−1/2 ) 0,

xi

(3.31)

n , Zi , Zi+1 . Gi+1/2 = G Uin , Ui+1

(3.32)

with

3.6. Roe and F-wave methods The most general efficient method that enables to treat source terms is the Roe method, even if it has its usual weaknesses which are the need of entropy fix and the failure of nonnegativity. The use of the Roe method in the context of source and with eventually nonconservative terms, as in (3.3), has been studied in Bermúdez and Vásquez (1994), Vázquez-Cendón (1999), García-Navarro and Vázquez-Cendón (2000), Parés and Castro (2004), see also the references therein. The approach there is to apply the Roe method to the quasilinear system in (U, Z) obtained by completing (3.1) or (3.3) with the equation ∂t Z = 0. A treatment of wet/dry fronts is proposed in Castro, Ferreiro, García-Rodríguez, González-Vida, Macías, Parés and Vázquez-Cendón (2005). The two layer shallow water is also treated with this approach in Castro, Macías and Parés (2001), Castro, GarcíaRodríguez, González-Vida, Macías, Parés and Vázquez-Cendón (2004). A related method is Gallouët, Hérard and Seguin (2003). Independently, the F-wave decom-

3. Finite volumes for systems with source terms

221

position method has been proposed by Bale, LeVeque, Mitran and Rossmanith (2002). It appears that indeed both methods give the same basic scheme, that we describe here. The solver can be put under the form of a simple approximate Riemann solver, as formulated in Section 2.5, generalized to the case of source. Knowing the form (3.5) of the matrix of our system (3.1), and assuming no resonance, one has to take one speed σm0 = 0, and the other speeds nonzero. An approximation of the jump in the flux across the stationary discontinuity can be taken as r − Zl ), δFm0 = −B(Z

(3.33)

of B(U, Z) that needs to be chosen. Then, noticing for some approximation B that for the waves with nonzero eigenvalues, Z has no jump, we can write for the flux jump δFk = σk δUk

for k = m0 .

Now we write F (Ur , Zr ) − F (Ul , Zl ) =

(3.34)

m

k=1 δFk ,

r − Zl ) = F (Ur , Zr ) − F (Ul , Zl ) + B(Z

so that with (3.33) it gives

δFk .

(3.35)

k=m0

It is natural to take δUk eigenvector of Aˆ for k = m0 , where Aˆ is a diagonalizable ˆ k = σk δUk . matrix (with nonzero eigenvalues) approximating FU (U, Z), i.e. AδU Therefore, by (3.34), δFk is also an eigenvector, ˆ k = σk δFk AδF

for k = m0 .

(3.36)

The relations (3.35)–(3.36) define in a unique way the values σk for k = m0 , ˆ and the δFk for k = m0 . The δUk are also which are the distinct eigenvalues of A, determined by (3.34), thus we recover the intermediate states by Uk − Uk−1 = δUk ,

k = m0 .

(3.37)

Finally, the numerical fluxes are given by δFk , Fl (Ul , Ur , Zl , Zr ) = F (Ul , Zl ) + σk 0

The name F-wave decomposition method comes from the decomposition (3.35) ˆ that directly of the jump in fluxes into components δFk in the eigenspaces of A, come into the definition of the numerical fluxes (3.38). The consistency condition (3.29) is obvious since if Zl = Zr and Ul = Ur , all δFk for k = m0 vanish by (3.35), thus the δUk also vanish by (3.34), and

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Uk = Ul = Ur by (3.37). The consistency with the source (3.30) is also easy to get since by (3.38) and (3.35), Fr −Fl = F (Ur , Zr )−F (Ul , Zl )− k=m0 δFk = r − Zl ), which reduces therefore to the consistency of B. −B(Z We notice that in the case without source Zl = Zr , the method reduces m0 −1 to the Roe method only if Um0 = Um0 −1 , i.e. Ul + k=1 δUk = Ur − m ˆ ˆ r− δU , and applying A, which is supposed invertible, this gives A(U k k=m0 +1 ˆ Ul ) = F (Ur , Zr ) − F (Ul , Zl ) i.e. the condition on A to be a Roe matrix. There a Roe matrix obtained by freezing Z fore, a possible choice is Aˆ = A(Ul , Ur , Z), to a value Z. The well-balancing property is also easy to obtain from (3.38), it means that r −Zl ) = 0. δFk = 0 for all k = m0 , or equivalently F (Ur , Zr )−F (Ul , Zl )+ B(Z In the case of the shallow water system (3.7), one can take hl + hr B = 0, g (3.39) . 2 This gives the well-balanced property with respect to the steady states at being 0, this gives the conservrest (3.27). Moreover the first component of B ativity of the water height. However, as in the Roe method, it is not possible to analyze the nonnegativity of water height and the entropy inequality. 3.7. Hydrostatic reconstruction scheme The hydrostatic reconstruction scheme is especially designed to solve shallow water type problems. It has been introduced by Audusse, Bouchut, Bristeau, Klein and Perthame (2004), and widely used in Audusse and Bristeau (2005), Bouchut, Le Sommer and Zeitlin (2004), Le Sommer, Reznik and Zeitlin (2004), Bouchut, Le Sommer and Zeitlin (2005), Mangeney-Castelnau, Bouchut, Vilotte, Lajeunesse, Aubertin and Pirulli (2005), Noelle, Pankratz, Puppo and Natvig (2006), Marche (2005). Its strengths are its simplicity and its robustness. Additionally, the hydrostatic reconstruction scheme has the property to be easily adaptable to systems of the same type as the shallow water problem. The construction directly involves the steady states are rest (3.27). Denote as before U = (h, hu). Then the numerical fluxes are defined by 0 Fl (Ul , Ur , zl , zr ) = F(Ul∗ , Ur∗ ) + , p(hl ) − p(h∗l ) 0 Fr (Ul , Ur , zl , zr ) = F (Ul∗ , Ur∗ ) + (3.40) , p(hr ) − p(h∗r ) where p(h) = gh2 /2 and F(Ul , Ur ) is a given consistent numerical flux for the shallow water problem without source. Good choices are for example the HLLC

3. Finite volumes for systems with source terms

223

numerical flux of Section 2.10 (our favorite choice), or the HLL numerical flux of Section 2.9. The name “hydrostatic” comes from the fact that in (3.40) and (3.42), only the pressure part of the system is really involved, the advection part being neglected. The reconstructed states Ul∗ , Ur∗ are defined by Ul∗ = (h∗l , h∗l ul ), h∗l

Ur∗ = (h∗r , h∗r ur ), ∗

= (hl + zl − z )+ ,

h∗r

(3.41) ∗

= (hr + zr − z )+ .

(3.42)

The positive parts (recall that x+ = max(0, x)) in the right-hand sides of (3.42) are just to ensure that we get some nonnegative heights h∗l , h∗r . The value z∗ is defined by z∗ = max(zl , zr ), and another way of writing (3.42)–(3.43) is, with z = zr − zl , h∗r = hr − (− z)+ + , h∗l = hl − ( z)+ + ,

(3.43)

(3.44)

which shows that Fl , Fr do indeed depend only on the difference z, as expected from the invariance z → z + c. P ROPOSITION 3.4. Consider a consistent numerical flux F for the homogeneous shallow water problem (i.e. with z = cst), that preserves nonnegativity of the water height and satisfies a semi-discrete entropy inequality corresponding to the entropy η in (3.14). Then the scheme defined by the numerical fluxes (3.40), (3.41), (3.44) (0) (i) (ii) (iii) (iv)

is conservative in water height, preserves the nonnegativity of h, is well-balanced, i.e. it preserves the discrete steady states at rest (3.27), is consistent with the shallow water system (3.7), satisfies a semi-discrete entropy inequality associated to the entropy η in (3.16).

P ROOF. These results have been proved in Audusse, Bouchut, Bristeau, Klein and Perthame (2004), but for completeness we rewrite here the arguments, except about the semi-discrete entropy inequality (iv). Here, semi-discrete refers to the limit scheme obtained as t → 0. Denote the components of the numerical fluxes by upper indices 0 and 1, Fl = (Fl0 , Fl1 ), Fr = (Fr0 , Fr1 ). Then the conservativity property (0) is obvious since (3.40) gives that Fl0 = Fr0 . For property (ii) of well-balancing, consider discrete steady states at rest, i.e. ul = ur = 0, hl + zl = hr + zr . Then, (3.42) gives h∗l = h∗r , and by (3.41),

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Ul∗ = Ur∗ . Therefore, by consistency of F and since ul = ur = 0, (3.40) gives 0 Fl = F (Ul∗ ) + (3.45) = F (Ul ), p(hl ) − p(h∗l ) 0 ∗ Fr = F (Ur ) + (3.46) = F (Ur ), p(hr ) − p(h∗r ) which gives (3.28). The consistency (iii) has to be looked at carefully, it is the less obvious property. We have to check the two properties of Definition 3.2. The consistency with the exact flux Fl (U, U, z, z) = Fr (U, U, z, z) = F (U ) is obvious since Ul∗ = Ul and Ur∗ = Ur whenever zr = zl . For consistency with the source (3.30), we write 0 . Fr − Fl = (3.47) p(h∗l ) − p(hl ) + p(hr ) − p(h∗r ) ∗∗ Now, we can write p(h∗l ) − p(hl ) = g(h∗l − hl )h∗∗ l for some hl between hl and ∗ ∗ ∗ ∗∗ ∗∗ hl , and p(hr ) − p(hr ) = g(hr − hr )hr for some hr between hr and h∗r . Then, assuming h > 0, the positive parts in (3.44) play no role if hl − h, hr − h and z ∗ are small enough. Thus we have p(h∗l )−p(hl ) = −h∗∗ l (g z)+ , p(hr )−p(hr ) = ∗∗ −hr (−g z)+ , which gives (3.30). In the special case h = 0, the positive parts in (3.44) can play a role only when hl = O( z), or respectively hr = O( z), and we conclude that (3.30) always holds, proving (iii). Let us now prove the nonnegativity statement (i). It is a consequence of the property

h∗l hl ,

h∗r hr ,

(3.48)

that comes directly from the definition (3.44). By assumption the numerical flux F preserves the nonnegativity or h, and here it is meant it holds interface by interface. According to Bouchut (2004), this means that there exists some σl (Ul , Ur ) < 0 < σr (Ul , Ur ) such that hl +

F 0 (Ul , Ur ) − hl ul 0, σl (Ul , Ur )

hr +

F 0 (Ul , Ur ) − hr ur 0, σr (Ul , Ur )

(3.49)

for any Ul and Ur (with nonnegative densities hl , hr ). This implies in particular that h∗l +

F 0 (Ul∗ , Ur∗ ) − h∗l ul 0, σl (Ul∗ , Ur∗ )

h∗r +

F 0 (Ul∗ , Ur∗ ) − h∗r ur 0. σr (Ul∗ , Ur∗ )

(3.50)

3. Finite volumes for systems with source terms

225

But since necessarily 1−ul /σl (Ul∗ , Ur∗ ) 0, 1−ur /σr (Ul∗ , Ur∗ ) 0, we deduce with (3.48) that hl +

Fl0 (Ul , Ur , zl , zr ) − hl ul 0, σl (Ul∗ , Ur∗ )

hr +

Fr0 (Ul , Ur , zl , zr ) − hr ur 0, σr (Ul∗ , Ur∗ )

(3.51)

which means that the scheme preserves the nonnegativity of h by interface. Note in particular that the speeds are σl (Ul∗ , Ur∗ ) and σr (Ul∗ , Ur∗ ), thus the CFL condition associated to the scheme is the one of F corresponding to the data Ul∗ and Ur∗ ,

t max −σl Ul∗ , Ur∗ , σr Ul∗ , Ur∗ min( xl , xr ). (3.52) In particular, if the numerical flux F is able to treat the vacuum (see Section 2.7), the hydrostatic scheme can also handle the vacuum since there is no blow up in (3.52) in the limit of vanishing Ul or Ur . We have to remark that the entropy inequality is only semi-discrete, thus does not give any a priori bound on the total energy (see Section 2.3) In practice we do not observe instabilities, as long as hi remains nonnegative, which is the case under the CFL condition (3.52). 3.8. Other efficient first-order schemes for shallow water Several other well-balanced methods were proposed to solve the shallow water system (3.7), like Jin (2001), Kurganov and Levy (2002). However, very few were able to treat correctly the vacuum, and at the same time to work for all type of data, including resonance. The only methods that enjoy similar properties as the hydrostatic reconstruction scheme are the kinetic method (Perthame and Simeoni, 2001) and the Suliciu relaxation method (Bouchut, 2004). Both overall satisfy a fully discrete entropy inequality, but on the counterpart they are quite expensive in terms of CPU and complexity. 3.9. Additional source terms: the apparent topography method In order to treat additional zero-order source terms with the well-balanced property, a general approach is as follows, explained here for the shallow water system. The particular case of Coriolis force can be treated within this method, see next section. Consider the shallow water system with topography and external force E, ∂t h + ∂x (hu) = 0, (3.53) ∂t (hu) + ∂x (hu2 + gh2 /2) + ghzx = hE,

226

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

where z = z(x), E = E(t, x). One can think in particular of a nonlinear coupling like E(t, x) = ϕ(h, u). Now the velocity equation writes ∂t u + ∂x u2 /2 + gh + gz = E, (3.54) and the steady states at rest are given by u = 0,

∂x (gh + gz) = E.

(3.55)

The idea to solve (3.53) is to use an apparent topography, which means to identify the system as the usual shallow water problem with a new topography z+b, where gbx = −E. Now, b depends also on time while it should be time independent, but when using discrete times tn , we can freeze its value on a time interval, thus we take gbxn = −E n and solve the shallow water system on the time interval (tn , tn+1 ) with topography z + bn . Note that for stationary solutions, this approximation is exact, thus the stationary solutions are preserved by this procedure. At the fully discrete level, this is done as follows. We define n n = −Ei+1/2

xi+1/2 , g bi+1/2

(3.56)

where xi+1/2 = xi+1 − xi , and update U = (h, hu) via Uin+1 − Uin +

t (Fi+1/2− − Fi−1/2+ ) = 0,

xi

(3.57)

with n , Fi+1/2− = Fl1d Ui , Ui+1 , zi+1/2 + bi+1/2 n Fi+1/2+ = Fr1d Ui , Ui+1 , zi+1/2 + bi+1/2 ,

(3.58)

where zi+1/2 = zi+1 − zi and the numerical fluxes Fl1d (Ul , Ur , z), Fr1d (Ul , Ur , z) are associated to the usual shallow water problem. If the numerical fluxes Fl1d , Fr1d are consistent with the shallow water system (3.7), then the new scheme (3.56)–(3.58) is consistent with (3.53) in the sense of Definition 3.2, writing (3.53) as hE = hEφx , φ(x) = x, and taking Z = (z, φ). Moreover, if the numerical fluxes Fl1d , Fr1d are well-balanced with respect to steady states at rest, then the new scheme (3.56)–(3.58) automatically lets invariant the data satisfying ui = 0,

ghi+1 + gzi+1 = ghi + gzi + Ei+1/2 xi+1/2 ,

(3.59)

which can be considered as discrete steady states approximating the relations (3.55). Concerning discrete entropy inequalities, there is no reason in general to have a special compatibility with this method, but however if E is bounded, this should not be a problem because the right-hand side is a lower-order term that should not influence the global stability.

3. Finite volumes for systems with source terms

227

3.10. The shallow water system with Coriolis force and transverse velocity Consider now the shallow water system with Coriolis force and transverse velocity ⎧ ⎪ ⎨ ∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x (hu2 + gh2 /2) + ghzx − f hv = 0, (3.60) ⎪ ⎩ ∂t (hv) + ∂x (huv) + f hu = 0, where z = z(x), f = f (x). Since the two first equations of (3.60) identify with (3.53) with E = f v, according to (3.55) the steady states at rest are given by u = 0,

∂x (gh + gz) = f v.

(3.61)

Since the Coriolis force is orthogonal to the velocity (u, v), the system has the same entropy inequality as the shallow water system with transverse velocity, ∂t η(h, u, v, z) + ∂x G(h, u, v, z) 0, (3.62) with η(h, u, v, z) = h u2 + v 2 /2 + gh2 /2 + hgz, G(h, u, v, z) = h u2 + v 2 /2 + gh2 + hgz u.

(3.63)

The particular structure of (3.60) enables to derive an additional conservative quantity, which is obtained as follows. Combining the third and first equations in (3.60) gives ∂t v + u∂x v + f u = 0. Then, differentiating with respect to x gives ∂t (∂x v + f ) + ∂x (∂x v + f )u = 0,

(3.64)

(3.65)

thus the potential vorticity (∂x v + f )/ h is transported at velocity u. A related equation comes by defining a potential Ω(x) by ∂x Ω = f,

∂t Ω = 0.

(3.66)

Then the last equation in (3.60) writes ∂t (hv) + ∂x (hvu) + huΩx = 0,

(3.67)

or in conservative form ∂t h(v + Ω) + ∂x h(v + Ω)u = 0.

(3.68)

Notice that the definition (3.66) of Ω allows to write down the system (3.60) under the form (3.1) with U = (h, hu, hv), Z = (z, Ω). Compared to the usual shallow

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

water system, we have here an extra eigenvalue u associated to the equation on v, and thus an extra resonance occurs when u = 0. This resonance does not imply further difficulty since the eigenvalue u is linearly degenerate. In order to discretize the system (3.60), a natural thing to do is to use the apparent topography method of Section 3.9 for the two first equations. When doing n so, we need a value for the interface force Ei+1/2 in (3.56), and it can be taken simply n Ei+1/2 = fi+1/2

n vin + vi+1

(3.69) , 2 where fi+1/2 is a discretization of f at the point xi+1/2 , that can be taken for example fi+1/2 = f (xi+1/2 ). Then, according to (3.68), in order to get the update for v, a natural thing to do is to apply the passive transport scheme of Section 2.11 to the variable v + Ω. Rewriting it in terms of the physical variable hv and denoting Ωi+1/2 = Ωi+1 − Ωi , this gives finally the scheme

t h h Fi+1/2 − Fi−1/2 = 0,

xi

t hu hu hn+1 Fi+1/2− − Fi−1/2+ = 0, un+1 − hni uni + i i

xi

t hv hv hn+1 F = 0, vin+1 − hni vin + − Fi−1/2+ i

xi i+1/2− − hni + hn+1 i

(3.70)

with h hu n Fi+1/2 , Fi+1/2− = Fl1d hi , ui , hi+1 , ui+1 , zi+1/2 + bi+1/2 , h hu 1d n Fi+1/2 , Fi+1/2+ = Fr hi , ui , hi+1 , ui+1 , zi+1/2 + bi+1/2 , h h Fi+1/2 vi if Fi+1/2 0, hv Fi+1/2− = h h (vi+1 + Ωi+1/2 ) if Fi+1/2 0, Fi+1/2 h h Fi+1/2 (vi − Ωi+1/2 ) if Fi+1/2 0, hv = Fi+1/2+ h h vi+1 if Fi+1/2 0, Fi+1/2 n

bi+1/2 = −fi+1/2

n vin + vi+1

2

xi+1/2 /g.

(3.71) (3.72)

(3.73) (3.74)

Here, Fl1d and Fr1d denote left and right numerical fluxes associated to the onedimensional shallow water equations, as the hydrostatic reconstruction fluxes of Section 3.7. We recall that zi+1/2 = zi+1 − zi and xi+1/2 = xi+1 − xi . We complete (3.72)–(3.73) by setting

Ωi+1/2 = fi+1/2 xi+1/2 .

(3.75)

3. Finite volumes for systems with source terms

229

It is easy to check that whenever Fl1d and Fr1d have the suitable properties (similarly as in Proposition 3.4), the scheme (3.70)–(3.75) is consistent with (3.60), is conservative in water height, preserves nonnegativity of h and is able to compute dry states, and is well-balanced, in the sense that it lets invariants the data satisfying ui = 0,

ghi+1 + gzi+1 = ghi + gzi + fi+1/2

vi + vi+1

xi+1/2 , (3.76) 2

which are obviously a discrete version of (3.61). There is no special property concerning the transport of the potential vorticity (3.65), but numerical experiments show very low numerical diffusion in that equation (Bouchut, Le Sommer and Zeitlin, 2004; Le Sommer, Reznik and Zeitlin, 2004; Bouchut, Le Sommer and Zeitlin, 2005). Concerning entropy inequalities, as commented in the previous section it is enough that the scheme is entropy satisfying when f = 0, since the Coriolis terms are nonsingular zero-order terms. This is the case if Fl1d and Fr1d satisfy a discrete entropy inequality, according to the analysis of Section 2.11. Nevertheless, the structure of the scheme (3.70)–(3.75) enables to prove that an entropy inequality holds, under some condition of sign on the potential vorticity (∂x v + f )/ h. P ROPOSITION 3.5. Assume that the numerical fluxes Fl1d and Fr1d satisfy a discrete entropy inequality, and consider data for which for all i one has fi+1/2 (vi+1 − vi + fi+1/2 xi+1/2 ) 0.

(3.77)

Then there holds a discrete entropy inequality associated to (3.62)–(3.63). n P ROOF. Denote zi+1/2 = zi+1/2 + bi+1/2 . According to Bouchut (2004), 1d the discrete entropy inequality associated with Fl , Fr1d can be written 2

2

hn+1 un+1 /2 + ghn+1 /2 − hi u2i /2 − gh2i /2 i i i

zi+1/2 h

t Fi+1/2 ϑ 1d hi , ui , hi+1 , ui+1 , zi+1/2 + g +

xi 2

zi−1/2 h − ϑ 1d hi−1 , ui−1 , hi , ui , zi−1/2 + g Fi−1/2 0, 2

(3.78)

where ϑ 1d is a numerical entropy flux satisfying ϑ 1d (h, u, h, u, 0) = (hu2 /2 + gh2 )u. But the passive transport scheme applied to v + Ω satisfies the entropy inequality (2.74)–(2.75) with S(φ) = φ 2 /2, thus rewriting it in terms of v gives 2

vin+1 /2 − hi vi2 /2 + hn+1 i

t v v ϑ 0, − ϑi−1/2+

xi i+1/2−

(3.79)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with

v ϑi+1/2−

=

v ϑi+1/2+

=

h if Fi+1/2 0,

h Fi+1/2 vi2 /2

h h (vi+1 + Ωi+1/2 )2 /2 if Fi+1/2 0, Fi+1/2 h Fi+1/2 (vi − Ωi+1/2 )2 /2

h if Fi+1/2 0,

h 2 /2 vi+1 Fi+1/2

h if Fi+1/2 0.

(3.80)

(3.81)

Adding (3.78) and (3.79) yields n+1 2 2 2 hn+1 ui + vin+1 /2 + ghn+1 /2 − hi u2i + vi2 /2 − gh2i /2 i i

t (ϑi+1/2− − ϑi−1/2+ ) 0, +

xi

(3.82)

with

zi+1/2 h v , Fi+1/2 + ϑi+1/2− 2

zi+1/2 h 1d v ϑi+1/2+ = ϑi+1/2 −g , Fi+1/2 + ϑi+1/2+ 2 1d ϑi+1/2 = ϑ 1d hi , ui , hi+1 , ui+1 , zi+1/2 .

1d ϑi+1/2− = ϑi+1/2 +g

(3.83) (3.84)

We claim that this implies a discrete entropy inequality as soon as v v h h ϑi+1/2+ − ϑi+1/2− − g zi+1/2 Fi+1/2 + g zi+1/2 Fi+1/2 0.

Indeed if (3.85) is satisfied, then one can find

v ϑi+1/2

(3.85)

such that

zi+1/2 h

zi+1/2 h v +g Fi+1/2 + ϑi+1/2+ Fi+1/2 2 2

zi+1/2 h

zi+1/2 h v v ϑi+1/2 g −g Fi+1/2 + ϑi+1/2− Fi+1/2 , 2 2 and with (3.83) it implies that −g

(3.86)

zi+1/2 h Fi+1/2 , 2

zi+1/2 h 1d v + ϑi+1/2 +g Fi+1/2 , ϑi+1/2− ϑi+1/2 (3.87) 2 and thus adding to (3.82) gzi times the first equation of (3.70) if gives a discrete entropy inequality associated to (3.62)–(3.63) with numerical entropy flux 1d v ϑi+1/2+ ϑi+1/2 + ϑi+1/2 −g

1d v + ϑi+1/2 +g Gi+1/2 = ϑi+1/2

zi + zi+1 h Fi+1/2 . 2

(3.88)

4. Second-order well-balanced schemes

231

Now let us examine condition (3.85). The left-hand side can be computed using (3.80), (3.81), (3.74) and (3.75), v v n h ϑi+1/2+ − ϑi+1/2− − g bi+1/2 Fi+1/2 h 2 n Fi+1/2 + = Ωi+1/2 /2 − vi Ωi+1/2 − g bi+1/2 h 2 n Fi+1/2 − + − Ωi+1/2 /2 − vi+1 Ωi+1/2 − g bi+1/2 vi + vi+1 h = Ωi+1/2 Ωi+1/2 /2 − vi + Fi+1/2 + 2 vi + vi+1 h + Ωi+1/2 − Ωi+1/2 /2 − vi+1 + Fi+1/2 − 2 h 1 = Ωi+1/2 ( Ωi+1/2 + vi+1 − vi ) Fi+1/2 + 2 h 1 + Ωi+1/2 (− Ωi+1/2 + vi − vi+1 ) Fi+1/2 − 2 h 1 . = Ωi+1/2 ( Ωi+1/2 + vi+1 − vi )Fi+1/2 (3.89) 2 Recalling that Ωi+1/2 = fi+1/2 xi+1/2 , we conclude that this is nonpositive whenever condition (3.77) holds.

4. Second-order well-balanced schemes There are many methods to go to higher-order in the context of finite volume schemes. In the case of source terms with the well-balanced property, several works have been performed in the past few years. A classical second-order method with slope reconstruction has been developed in Gallouët, Hérard and Seguin (2003), Katsaounis and Simeoni (2003), Audusse, Bouchut, Bristeau, Klein and Perthame (2004), while second-order Roe schemes are proposed in Bale, LeVeque, Mitran and Rossmanith (2002), Parés and Castro (2004), Chacón Rebollo, Domínguez Delgado and Fernández Nieto (2003). Higher-order methods are developed in Noelle, Pankratz, Puppo and Natvig (2006), Vukovic and Sopta (2002), Xing and Shu (2005), and the nonconservative context is studied in Castro, Gallardo and Parés (2006). Related works are a method for unstructured mesh in Audusse and Bristeau (2005), a residual distribution scheme in Ricchiuto, Abgrall and Deconinck (2006), and discontinuous Galerkin schemes in Xing and Shu (2006), Lukacova-Medvidova, Noelle and Kraft (2006). We shall describe here the most classical method derived from Katsaounis and Simeoni (2003), Audusse, Bouchut, Bristeau, Klein and Perthame (2004), Bouchut (2004). Since second-order accuracy in time can be obtained as usual by the Heun method, see below, it is enough to build a scheme that is second-order in

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space only. Starting from a first-order method for solving (3.1), the new ingredient to go to second-order in space is a reconstruction operator, which is attached to the mesh defined as previously (2.2)–(2.3). We shall denote by δ = sup xi

(4.1)

i

the size of the mesh. D EFINITION 4.1. A second-order reconstruction is an operator which to a sequence (Ui , Zi ) associates values Ui+1/2− , Zi+1/2− , Ui+1/2+ , Zi+1/2+ for i ∈ Z, in such a way that it is conservative in U , Ui−1/2+ + Ui+1/2− = Ui , 2 and it is second-order in the sense that whenever for all i,

1 1 Ui = U (x) dx, Zi = Z(x) dx,

xi

xi Ci

(4.2)

(4.3)

Ci

for some smooth functions U (x), Z(x), then Ui+1/2+ = U (xi+1/2 ) + O δ 2 , Ui+1/2− = U (xi+1/2 ) + O δ 2 , Zi+1/2+ = Z(xi+1/2 ) + O δ 2 . (4.4) Zi+1/2− = Z(xi+1/2 ) + O δ 2 , We recall that the notation φ = O(δ 2 ) means that φ = δ 2 φ, with φ bounded as δ tends to 0. Given consistent first-order fluxes Fl , Fr in the sense of Definition 3.2, the second-order scheme is defined by Uin+1 − Uin + with

t (Fi+1/2− − Fi−1/2+ − Fi ) = 0,

xi

n n n n Fi+1/2− = Fl Ui+1/2− , , Ui+1/2+ , Zi+1/2− , Zi+1/2+ n n n n , , Zi+1/2− , Zi+1/2+ Fi+1/2+ = Fr Ui+1/2− , Ui+1/2+ n n n n Fi = Fc Ui−1/2+ , Ui+1/2− , Zi−1/2+ , Zi+1/2− ,

(4.5)

(4.6)

where the arguments are obtained from the reconstruction operator applied to the sequence (Uin , Zin ), and the centered flux function Fc needs to be chosen. As for first-order, Zi does not change with time, but however the interface values n can be time dependent if the reconstruction operator is not performed Zi+1/2± componentwise. We define n n n = Zi+1/2+ − Zi+1/2− ,

Zi+1/2

n n

Zin = Zi+1/2− − Zi−1/2+ .

(4.7)

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233

4.1. Achievement of second-order accuracy The scheme (4.5)–(4.6) generalizes the first-order scheme (3.18)–(3.19), the latter being a formal particular case of the former if the reconstruction operator is trivial, Ui+1/2− = Ui , Ui+1/2+ = Ui+1 , Zi+1/2− = Zi , Zi+1/2+ = Zi+1 , and if Fc (U, U, Z, Z) = 0. The idea in (4.5)–(4.6) is to distribute the variation of the solution between interfaces and centers of the cells, depending on the smoothness of the computed solution. One hand, if the solution has large discontinuities, the reconstruction step should give (U, Z)ni+1/2− = (U, Z)ni−1/2+ = (Uin , Zi ), thus reducing to the first-order scheme, with source discretized at the interfaces. On the other hand, if the solution is smooth, the interface jumps (U, Z)i+1/2+ − (U, Z)i+1/2− should be O(δ 2 ), and the source cannot appear at the interface, because (3.30) gives then O(δ 2 ). It has to be handled by the centered term Fi . In order that this term gives a second-order resolution, we require that Ul + Ur Zl + Zr Fc (Ul , Ur , Zl , Zr ) = − B , 2 2 2 2 + O |Ur − Ul | + |Zr − Zl | (4.8) (Zr − Zl ), as Ur − Ul → 0 and Zr − Zl → 0. This condition implies that Fc (Ul , Ur , Z, Z) = 0. A slight restriction on the reconstruction is also necessary, which is that whenever the sequence (Ui , Zi ) is realized as the cell averages of smooth functions (U (x), Z(x)), as in (4.3), then Zi+1/2+ − Zi+1/2− = O ( xi + xi+1 )δ , Zi+1/2− − Zi−1/2+ = O( xi ).

(4.9)

These last assumptions (4.9) are very weak and of local nature, since in general one would have according to the definition of a second-order reconstruction operator errors in δ 2 and δ respectively on the right-hand sides of (4.9) (thus (4.9) is true for a mesh of uniform size x). It is proved in Bouchut (2004) that if the numerical fluxes Fl , Fr are consistent and sufficiently smooth, and if (4.8) and (4.9) are satisfied, then the scheme is second-order accurate in space in the weak sense, for smooth solutions. 4.2. Well-balanced property The well-balanced property can be achieved for the second-order scheme if all the ingredients involved in the formulas (4.5)–(4.6) satisfy a natural condition. We assume here that a family of discrete steady states is chosen, as described in Section 3.3.

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

D EFINITION 4.2. We say that the second-order reconstruction operator is well-balanced if to a sequence (Ui , Zi ) that is a discrete steady state (i.e. D(Ui , Ui+1 , Zi , Zi+1 ) = 0 for all i), it associates a sequence of interface values . . . , (U, Z)i−1/2− , (U, Z)i−1/2+ , (U, Z)i+1/2− , (U, Z)i+1/2+ , . . . (4.10) that is again a discrete steady state (i.e. D(Ui−1/2+ , Ui+1/2− , Zi−1/2+ , Zi+1/2− ) = 0 and D(Ui+1/2− , Ui+1/2+ , Zi+1/2− , Zi+1/2+ ) = 0 for all i). We remark that here it is important to consider global steady states sequences, since the reconstruction operator can have a large dependency stencil. P ROPOSITION 4.3. Assume that the reconstruction operator is well-balanced, that the numerical fluxes Fl , Fr are well-balanced in the sense of Definition 3.1, and that the centered flux Fc satisfies that whenever (Ul , Ur , Zl , Zr ) is a local steady state (i.e. D(Ul , Ur , Zl , Zr ) = 0), one has Fc (Ul , Ur , Zl , Zr ) = F (Ur , Zr ) − F (Ul , Zl ).

(4.11)

Then the second-order scheme (4.5)–(4.6) is well-balanced, in the sense that steady states sequences are let invariant. The proof of this result is obvious by the formulas (4.5)–(4.6). 4.3. Choice of centered flux The centered flux Fc has to satisfy (4.8) for second-order accuracy, and (4.11) for well-balancing. In practice we define Fc directly by an algebraic formula. For Example 3.1 and if F (U ) > 0, we can take Fc (Ul , Ur , Z) = −

F (Ur ) − F (Ul )

Z. D(Ur ) − D(Ul )

(4.12)

For the shallow water system (3.7) and if we retain only steady states at rest, we have the simple choice hl + hr g z . Fc (Ul , Ur , z) = 0, − (4.13) 2 An important point is that the first component in (4.13) vanishes, thus the water height remains conservative.

4. Second-order well-balanced schemes

235

4.4. Choice of reconstruction operator We shall not give general considerations on second-order reconstructions here, we rather refer the reader to Godlewski and Raviart (1991) for a general introduction. We would like here only to give the most simple and well known reconstructions for scalar conservative equations, in order to be able afterwards to discuss the shallow water system. E XAMPLE 4.1. In the case of a scalar function U ∈ R, the second-order minmod reconstruction is defined as follows, Ui−1/2+ = Ui − with

xi DUi , 2

DUi = minmod

Ui+1/2− = Ui +

xi DUi , 2

Ui − Ui−1 Ui+1 − Ui , , ( xi−1 + xi )/2 ( xi + xi+1 )/2

(4.14)

(4.15)

and

⎧ ⎨ min(x, y) if x, y 0, minmod(x, y) = max(x, y) if x, y 0, ⎩ 0 otherwise. A good property of this reconstruction is the maximum principle, inf Uj Ui−1/2+ , Ui+1/2− sup Uj . j

(4.16)

(4.17)

j

E XAMPLE 4.2. In the previous example, a loss of accuracy in the derivative can occur. This is corrected by the second-order ENO (essentially non oscillatory) reconstruction. For a uniform mesh, it is defined as follows for a scalar sequence Ui , Ui−1/2+ = Ui −

x Deno Ui , 2

Ui+1/2− = Ui +

x Deno Ui , 2

(4.18)

where

x 2 Ui − Ui−1 + θeno D Ui−1/2 , Deno Ui = minmod

x 2

x 2 Ui+1 − Ui − θeno D Ui+1/2 , (4.19)

x 2 Ui+1 − 2Ui + Ui−1 Ui+2 − 2Ui+1 + Ui D 2 Ui+1/2 = minmod , , (4.20)

x 2

x 2

with θeno = 1.

(4.21)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

This reconstruction does however somehow produce some instabilities (oscillations), thus one can prefer a value 0 θeno < 1, even if it kills the second-order accuracy of the slopes. A value θeno = 0.25 is a good value in practice. Notice that θeno = 0 gives the minmod reconstruction (4.15). A counterpart of the increased accuracy in the ENO reconstruction is the loss of the maximum principle, which is often needed, for example for positiveness of density or water height. A possible way to obtain this is to consider the modified ENO reconstruction, defined still by (4.18) but with slopes Denom Ui = minmod(Deno Ui , 2θenom Dmm Ui ),

(4.22)

where Dmm Ui is the minmod slope of (4.15), and 0 θenom 1. The value θenom = 1 is enough to recover the maximum principle, but in order to avoid arithmetic exceptions we prefer to use a slightly stronger limitation θenom = 0.9.

(4.23)

The maximum principle holds for this ENOm reconstruction, at the price of loosing the second-order accuracy of the slope close to local extrema. E XAMPLE 4.3. A second-order reconstruction for the shallow water system (3.7) can be performed on the vector (Ui , zi ) = (hi , hi ui , zi ) as follows. Let us first describe the reconstruction for U , ignoring the variable z. Let us denote Ui+1/2± ≡ (hi+1/2± , hi+1/2± ui+1/2± ) the reconstructed values. Then the conservation constraint (4.2) reads hi−1/2+ + hi+1/2− = hi , 2 hi−1/2+ ui−1/2+ + hi+1/2− ui+1/2− = hi ui . 2 It is easily seen to be equivalent to the representation

xi

xi hi+1/2− = hi + Dhi , Dhi , 2 2 hi+1/2− xi = ui − Dui , hi 2 hi−1/2+ xi = ui + Dui , hi 2

(4.24)

hi−1/2+ = hi − ui−1/2+ ui+1/2−

(4.25)

for some slopes Dhi , Dui . Moreover, the second-order accuracy (4.4) means that Dhi , Dui have to be consistent with dh/dx and du/dx respectively. Thus we can take for Dhi and Dui the minmod reconstruction (4.15), where we put the values hi or ui respectively in the right-hand side. A more accurate choice is to use the ENOm slope (4.22) for Dhi and the ENO slope (4.19)–(4.20) for Dui . This last

4. Second-order well-balanced schemes

237

choice is the one we select as our favorite reconstruction. It keeps the water height h nonnegative Concerning the variable z, we perform the ENOm reconstruction on ζ = h + z. This means that we set ζi = hi + zi , from these values we compute ζi+1/2± by the ENOm reconstruction, and we finally set zi+1/2± = ζi+1/2± − hi+1/2± .

(4.26)

This reconstruction is obviously well-balanced with respect to steady states at rest. Consequently, according to Proposition 4.3, with this choice of second-order reconstruction and with Fc defined by (4.13), the whole second-order scheme is well-balanced. This choice of reconstructing ζ to get z has the advantage to treat correctly interfaces between dry and wet cells, contrarily to other choices (like reconstructing ζ and z and getting h as h = ζ − z, or reconstructing h and z independently), see Audusse, Bouchut, Bristeau, Klein and Perthame (2004), Marche (2005). 4.5. Stability and CFL condition Entropy inequalities are not available at second order, only very special schemes can verify such criterion. In practice this is not really a problem since the schemes usually degenerate at first order whenever the computed solution has sharp gradients. Therefore, the stability of second-order schemes can be analyzed only by two main approaches. The first is the study of the non oscillatory behavior, which can be really justified only for scalar equations (see Godlewski and Raviart (1991) for example). The second is the study of invariant domains, like nonnegativity of density or water height. With the approach presented here (4.5)–(4.6), the invariant domains are preserved if one assumes that the first-order numerical fluxes Fl , Fr and the reconstruction operator do, and if the centered flux Fc satisfies a suitable requirement (it has to “look like” Fr − Fl ). This result uses strongly the conservativity of the reconstruction (4.2), see Bouchut (2004). The analysis shows that at secondorder, the CFL condition has to be divided by 2 with respect to the one used at first-order. In the case of shallow water with the reconstruction proposed above in Example 4.3, with the centered flux (4.13), and with the first-order fluxes given by the hydrostatic reconstruction method, we have conservativity and nonnegativity of water height, the interface between dry and wet is well resolved under half the CFL condition (3.52). 4.6. Second-order boundary conditions Boundary conditions are treated at second-order very similarly to the first-order case discussed in Section 2.12: one has to define suitable ghost values. Assume

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

that the interval of computation (a, b) is divided into cells Ci = (xi−1/2 , xi+1/2 ), i = 1, . . . , nx , on which we have discrete values Uin , Zi , i = 1, . . . , nx . The boundary is a = x1/2 , b = xnx +1/2 . The first step is to define ghost values U0n , Z0 , Unnx +1 , Znx +1 , and if necessary also ghost values for i = −1, i = nx + 2 . . . Then, n n one applies the second-order reconstruction to get the values of Ui+1/2+ , Zi+1/2+ , n n i = 0, . . . , nx − 1, and of Ui+1/2− , Zi+1/2− , i = 1, . . . , nx . The next step is to n n , Z1/2− , Unnx +1/2+ , Znnx +1/2+ . Finally, one applies define ghost values for U1/2− the update (4.5)–(4.6) for i = 1, . . . , nx . E XAMPLE 4.4. Consider the isentropic gas dynamics system (1.14), with the wall boundary condition u(t, a) = 0, as discussed in Example 1.5. Then, one can n = ρn, define the ghost values by even/odd symmetry, ρ0n = ρ1n , un0 = −un1 , ρ−1 2 n n u−1 = −u2 , . . . (as many values as needed in order to be able to get the reconn n structed values at 1/2+). Finally, we define ρ1/2− = ρ1/2+ , un1/2− = −un1/2+ . E XAMPLE 4.5. For periodic boundary conditions, one takes U0n = Unnx , Unnx +1 = n = Un n n n n n U1n , U−1 nx −1 , Unx +2 = U2 , . . . , and U1/2− = Unx +1/2− , Unx +1/2+ = n U1/2+ . E XAMPLE 4.6. For Neumann boundary conditions ∂x U (t, a) = 0, we proceed by even symmetry. 4.7. Second-order accuracy in time The second-order accuracy in time can be classically obtained by the Heun method, which reads as follows. The second-order method in x can be written as U n+1 = U n + tΦ U n , (4.27) where U = (Ui )i∈Z , and Φ is a nonlinear operator depending on the mesh. Then the second-order scheme in time is n+1 = U n + tΦ U n , U n+1 n+2 = U n+1 + tΦ U U , n+2 Un + U (4.28) . 2 It is easy to see that if the operator Φ does not depend on t, this procedure gives a fully second-order scheme in space and time. The convex average in (4.28) enables to ensure the stability without any further limitation on the CFL. Indeed, because of this average, the above second-order algorithm is more stable than the first-order one (4.27). U n+1 =

4. Second-order well-balanced schemes

239

4.8. The shallow water system with Coriolis force and transverse velocity An extension to second-order of the scheme proposed in Section 3.10 for the system (3.60) is performed according to the principle above: one needs to define a centered numerical flux, and a reconstruction operator. Denote U = (h, hu, hv). The first-order scheme (3.70)–(3.75) can be written

t n n n Uin+1 − Uin + Fl Ui , Ui+1 , zi+1/2 + bi+1/2 , Ωi+1/2

xi n n , Ωi−1/2 = 0, − Fr Ui−1 , Uin , zi−1/2 + bi−1/2

(4.29)

with n

bi+1/2 = −fi+1/2

n vin + vi+1

2

xi+1/2 /g,

Ωi+1/2 = fi+1/2 xi+1/2 ,

(4.30) (4.31)

and zi+1/2 = zi+1 − zi , xi+1/2 = xi+1 − xi . The second-order scheme then takes the form

t n n n n n Fl Ui+1/2− , Ui+1/2+ , zi+1/2 + bi+1/2 , Ωi+1/2 Uin+1 − Uin +

xi n n n n n , zi−1/2 + bi−1/2 , Ωi−1/2 − Fr Ui−1/2− , Ui−1/2+ n n , Ui+1/2− , zin + bin , Ωin = 0, − Fc Ui−1/2+ (4.32) with now n n n

zi+1/2 = zi+1/2+ − zi+1/2− ,

n n

zin = zi+1/2− − zi−1/2+ .

(4.33)

For the centered flux Fc , we take similarly as in (4.13) Fc (Ul , Ur , z + b, Ω) hl + hr hl ul + hr ur = 0, − g( z + b), −

Ω , 2 2

(4.34)

and it remains to explain how we perform the reconstruction of U , z, and how we n n define bi+1/2 , Ωi+1/2 , bin , Ωin . The reconstruction of h, u, z is performed exactly as in the shallow water case of Example 4.3. The transverse velocity v is reconstructed similarly as u, by setting hi+1/2− xi Dvi , hi 2 hi−1/2+ xi = vi + Dvi , hi 2

vi−1/2+ = vi − vi+1/2−

(4.35)

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Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

with slope Dvi computed by the ENO reconstruction involving the values vi . Concerning Ω, we indeed make a time independent reconstruction, using directly the relation (3.66), and the formulation (3.67). Given the values fi+1/2 , i ∈ Z, we can define up to a constant the values of Ωi , i ∈ Z, by Ωi+1 − Ωi = fi+1/2 xi+1/2 ,

(4.36)

with as before xi+1/2 = xi+1 −xi . From these values Ωi , we perform the ENOm reconstruction to get interface values Ωi+1/2− , Ωi+1/2+ . We finally set

Ωi+1/2 = Ωi+1/2+ − Ωi+1/2− ,

Ωi = Ωi+1/2− − Ωi−1/2+ . (4.37)

The variable b is treated similarly, defining values bin by n bi+1 − bin = −fi+1/2

n vin + vi+1

(4.38)

xi+1/2 /g. 2 From these values bin , we perform the ENOm reconstruction to get interface values n n bi+1/2− , bi+1/2+ , and we finally set n n n = bi+1/2+ − bi+1/2− ,

bi+1/2

n n

bin = bi+1/2− − bi−1/2+ .

(4.39)

bin

Notice that in this procedure, an arbitrary constant added to does not modify the differences (4.39). Moreover, according to the formulas defining the ENOm n − bn , the result can be computed reconstruction that involve only differences bi+1 i n locally without really computing bi . The scheme we obtain is second-order, conservative and nonnegative in water height, and computes dry states. It is well-balanced with respect to the steady states at rest (3.76), because this relation can be written hi+1 + zi+1 + bi+1 = hi + zi + bi , and it is easily seen that for such data the reconstruction also satisfies h + z + b = cst.

5. Two-dimensional finite volumes on a rectangular grid This section is devoted to the extension of the previously developed finite volume schemes with source terms to two dimensions on a rectangular grid. The case of unstructured mesh is more involved, and we refer to Audusse and Bristeau (2005). The generalization of (3.1) that we are going to consider is a two-dimensional system of the form ∂t U + ∂x F1 (U, Z) + ∂y F2 (U, Z) + B1 (U, Z)∂x Z + B2 (U, Z)∂y Z = 0,

(5.1)

where the space variable is x = (x, y) ∈ R2 , and the partial derivatives ∂x , ∂y refer to these variables. The unknown is U (t, x) ∈ Rp , Z = Z(x) ∈ Rr is given, and

5. Two-dimensional finite volumes on a rectangular grid

241

the nonlinearities F1 , F2 , B1 , B2 are smooth. As in the one-dimensional case, this system can be written as a quasilinear system in two dimensions for the variable = (U, Z), by completing the system with the trivial equation ∂t Z = 0. U By two-dimensional quasilinear system we mean a system of the form ∂x U ∂y U + A1 U + A2 U = 0. ∂t U (5.2) For such a system we can consider planar solutions, i.e. solutions of the form (t, x) = U (t, ζ ) with ζ = x · n and n = (n1 , n2 ) is any unit vector in R2 , which U leads to ∂ζ U + An U = 0, ∂t U (5.3) with = n1 A1 U + n2 A2 U . An U

(5.4)

Via this reduction, the notions introduced in Section 1 can be applied to the one-dimensional quasilinear system (5.3), and thus one defines hyperbolicity, entropies, and other notions for (5.2) by requiring the one-dimensional properties for all directions n. In particular, the system (5.2) is hyperbolic if for any unit , An (U ) is diagonalizable. The reader is referred to the vector n in R2 and any U literature for more details, for example Serre (1999). 5.1. Resolution interface by interface One of the basic principles of finite volume methods is the very simple way to solve a multidimensional quasilinear system like (5.2) by solving onedimensional problems (5.3)–(5.4) at each interface between two cells, n being the normal to the interface. In the case of a rectangular grid, the formulation comes very simply, as follows. Consider a two-dimensional mesh made of rectangles Cij = (xi−1/2 , xi+1/2 ) × (yj −1/2 , yj +1/2 ),

i ∈ Z, j ∈ Z,

(5.5)

and denote their lengths by

xi = xi+1/2 − xi−1/2 > 0,

yj = yj +1/2 − yj −1/2 > 0.

(5.6)

The centers of the cells are denoted by xij = (xi , yj ),

(5.7)

with xi =

xi−1/2 + xi+1/2 , 2

yj =

yj −1/2 + yj +1/2 . 2

(5.8)

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We would like to approximate a solution U (t, x) to (5.1) by discrete values Uijn that are approximations of the mean value of U over the cell Cij at time tn = n t, 1 Uijn

xi yj

x i+1/2 y j +1/2

U (tn , x, y) dx dy.

(5.9)

xi−1/2 yj −1/2

A finite volume method for solving (5.1) takes the form Uijn+1 − Uij + +

t (Fi+1/2−,j − Fi−1/2+,j )

xi

t (Fi,j +1/2− − Fi,j −1/2+ ) = 0,

yj

(5.10)

where as before, Uij stands for Uijn . In this formula, the terms Fi+1/2±,j are exchange terms between the cells Cij and Ci+1,j , and the terms Fi,j +1/2± are exchange terms between the cells Cij and Ci,j +1 . As in the one-dimensional case (3.18)–(3.19), these exchange terms are computed by Fi+1/2−,j = Fl1 (Uij , Ui+1,j , Zij , Zi+1,j ), Fi+1/2+,j = Fr1 (Uij , Ui+1,j , Zij , Zi+1,j ), Fi,j +1/2− = Fl2 (Uij , Ui,j +1 , Zij , Zi,j +1 ), Fi,j +1/2+ = Fr2 (Uij , Ui,j +1 , Zij , Zi,j +1 ), Fl1 ,

(5.11)

Fr1 , Fl2 ,

for some numerical fluxes Fr2 . With the formulation (5.10)–(5.11), the consistency with (5.1) can be defined obviously, by saying that the x difference in (5.10) must be consistent with terms with x derivatives in (5.1) in the sense of (3.29)–(3.30), and similarly for the y difference and the y derivatives. Entropy inequalities and nonnegativity properties can be formulated very similarly to the one-dimensional case for the scheme (5.10)–(5.11), and incell properties are deduced from interface properties at the price of diminishing the CFL condition. In practice one has only to use the CFL condition attached to each of the two directions,

tai+1/2,j min( xi , xi+1 ),

tai,j +1/2 min( yj , yj +1 ),

(5.12)

where ai+1/2,j and ai,j +1/2 are estimates of the speeds in x and y directions respectively. We shall not give the details here. 5.2. Well-balancing Since the above formulation means that we are solving (5.1) independently in the directions x and y, it follows that the steady states that can be preserved by such a

5. Two-dimensional finite volumes on a rectangular grid

243

method are those that can be seen locally as one-dimensional steady states in the directions x or y. In particular, more general steady states where the dependence in x and y are nontrivially balanced cannot be exactly resolved. Nevertheless, the generalization of equations (3.28) that enable to preserve a discrete steady state is that the fluxes computed by (5.11) must give the value Fi+1/2−,j = F1 (Uij , Zij ),

Fi+1/2+,j = F1 (Ui+1,j , Zi+1,j ),

Fi,j +1/2− = F2 (Uij , Zij ),

Fi,j +1/2+ = F2 (Ui,j +1 , Zi,j +1 ).

(5.13)

It is obvious from (5.10)–(5.11) that if at time tn we start with a sequence (Uij ) that satisfies (5.13), then it remains unchanged at the next time level. 5.3. Second-order accuracy For rectangular grids, the extension to second-order is straightforward, it follows directly the one-dimensional case developed in Section 4. Given first-order numerical fluxes Fl1 , Fr1 , Fl2 , Fr2 , the second-order version of (5.10)–(5.11) writes

t 1 Fi+1/2−,j − Fi−1/2+,j − Fi,j

xi

t 2 Fi,j +1/2− − Fi,j −1/2+ − Fi,j = 0, +

yj

Uijn+1 − Uij +

(5.14)

with Fi+1/2−,j = Fl1 (Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j ), Fi+1/2+,j = Fr1 (Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j ), Fi,j +1/2− = Fl2 (Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ ), Fi,j +1/2+ = Fr2 (Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ ), 1 = Fc1 (Ui−1/2+,j , Ui+1/2−,j , Zi−1/2+,j , Zi+1/2−,j ), Fi,j 2 Fi,j = Fc2 (Ui,j −1/2+ , Ui,j +1/2− , Zi,j −1/2+ , Zi,j +1/2− ).

(5.15)

The centered numerical fluxes Fc1 , Fc2 correspond respectively to the x and y parts of (5.1). The reconstructed values can be obtained from the one-dimensional case. Indeed, for fixed j , from Uij , Zij , i ∈ Z, we can get the values Ui+1/2−,j , Ui+1/2+,j , Zi+1/2−,j , Zi+1/2+,j , and for fixed i, from Uij , Zij , j ∈ Z, we can get the values Ui,j +1/2− , Ui,j +1/2+ , Zi,j +1/2− , Zi,j +1/2+ . The CFL condition has to be restricted by a factor 1/2 with respect to (5.12).

244

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

5.4. Two-dimensional shallow water system with Coriolis force Consider now the two-dimensional shallow water system with Coriolis force ⎧ ∂ h + ∂x (hu) + ∂y (hv) = 0, ⎪ ⎨ t ∂t (hu) + ∂x (hu2 + gh2 /2) + ∂y (huv) + hg∂x z − f hv = 0, (5.16) ⎪ ⎩ 2 2 ∂t (hv) + ∂x (huv) + ∂y (hv + gh /2) + hg∂y z + f hu = 0, where h(t, x) 0 is the water height, u(t, x), v(t, x) are the two components of the velocity field, g > 0 is the gravity constant, z(x) is the topography, and f (x) is the Coriolis force. Apart from the Coriolis terms, the system can be written as (5.1) with Z = z, and ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ hu hv h ⎠, huv U = ⎝hu⎠ , F1 = ⎝hu2 + gh2 /2⎠ , F2 = ⎝ 2 2 hv huv hv + gh /2 (5.17) ⎛ ⎞ ⎛ ⎞ 0 0 B1 = ⎝gh⎠ , (5.18) B2 = ⎝ 0 ⎠ . 0 gh Denoting u = (u, v) the vector velocity, the equations of (5.16) can be combined to get, for smooth solutions, ∂t u + u · ∇u + ∇(gh + gz) + f u⊥ = 0

(5.19)

with u⊥ = (−v, u), or, using the identity u · ∇u = ∇(|u|2 /2) + u⊥ curl u with curl u = ∂x v − ∂y u, ∂t u + ∇ |u|2 /2 + gh + gz + (curl u + f )u⊥ = 0. (5.20) From this we deduce the entropy inequality ∂t h|u|2 /2 + gh2 /2 + hgz + div h|u|2 /2 + gh2 + hgz u 0.

(5.21)

Taking the curl of (5.20) we also get the potential vorticity equation ∂t (curl u + f ) + div (curl u + f )u = 0.

(5.22)

The steady states at rest are characterized by u = v = 0,

h + z = cst.

(5.23)

A property of the system (5.16) is that it is invariant under rotation. The consequence is that x and y play the same role, one has only to exchange the role of u and v. This invariance enables to define numerical fluxes by applying the same one-dimensional procedure in x and y directions. This is done as follows, by using the same approach as in Section 3.10. Consider one-dimensional numerical

5. Two-dimensional finite volumes on a rectangular grid

fluxes Fl (Ul , Ur , z), Fr (Ul , Ur , z) associated to the problem ⎧ ⎪ ⎨ ∂t h + ∂x (hu) = 0, ∂t (hu) + ∂x (hu2 + gh2 /2) + hg∂x z = 0, ⎪ ⎩ ∂t (hv) + ∂x (huv) = 0.

245

(5.24)

Our choice is indeed to take the hydrostatic reconstruction fluxes for the two first equations, and the passive transport fluxes for the last equation, according to Sections 3.7 and 2.11. Then, denoting the exchange of x and y components by = (h, hv, hu) U

if U = (h, hu, hv),

the two-dimensional scheme is defined by (5.10) with numerical fluxes 1 , Fi+1/2−,j = Fl Uij , Ui+1,j , zi+1/2,j + bi+1/2,j 1 Fi+1/2+,j = Fr Uij , Ui+1,j , zi+1/2,j + bi+1/2,j , 2 l U ij , U i,j +1 , zi,j +1/2 + bi,j Fi,j +1/2− = F +1/2 , 2 r U ij , U i,j +1 , zi,j +1/2 + bi,j Fi,j +1/2+ = F +1/2 ,

(5.25)

(5.26)

with

zi+1/2,j = zi+1,j − zij ,

zi,j +1/2 = zi,j +1 − zij ,

(5.27)

vij + vi+1,j

xi+1/2 /g, 2 uij + ui,j +1 2

bi,j (5.28)

yj +1/2 /g, +1/2 = fi,j +1/2 2 and xi+1/2 = xi+1 − xi , yj +1/2 = yj +1 − yj , where fi+1/2,j and fi,j +1/2 are discretized values for f (x) at the locations (xi+1/2 , yj ), and (xi , yj +1/2 ) respectively. As in the one-dimensional case, whenever Fl and Fr have the suitable properties (which is the case for the hydrostatic reconstruction fluxes with passive transport), the two-dimensional scheme is consistent with (5.16), is conservative in water height, preserves nonnegativity of h and is able to compute dry states, and is well-balanced, in the sense that it lets invariants the data satisfying 1

bi+1/2,j = −fi+1/2,j

uij = 0,

vij = 0,

hij + zij = cst.

(5.29)

There is no special property concerning the transport of the potential vorticity (5.22), but numerical experiments show very low numerical diffusion in that equation (Le Sommer, Reznik and Zeitlin, 2004; Bouchut, Le Sommer and Zeitlin, 2005). The comments of Section 3.10 are still valid concerning entropy inequalities.

246

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

The second-order version works similarly as the one-dimensional case explained in Section 4.8. The scheme writes as (5.14), with 1 Fi+1/2−,j = Fl Ui+1/2−,j , Ui+1/2+,j , zi+1/2,j + bi+1/2,j , 1 , Fi+1/2+,j = Fr Ui+1/2−,j , Ui+1/2+,j , zi+1/2,j + bi+1/2,j 2 l U i,j +1/2− , U i,j +1/2+ , zi,j +1/2 + bi,j +1/2 , Fi,j +1/2− = F 2 r U i,j +1/2− , U i,j +1/2+ , zi,j +1/2 + bi,j Fi,j +1/2+ = F +1/2 , 1 1 1 , = Fc1 Ui−1/2+,j , Ui+1/2−,j , zi,j + bi,j Fi,j 2 2 2 2 Fi,j = Fc Ui,j −1/2+ , Ui,j +1/2− , zi,j + bi,j , (5.30) hl + hr g( z + b), 0 , Fc1 (Ul , Ur , z + b) = 0, − 2 hl + hr 2 Fc (Ul , Ur , z + b) = 0, 0, − (5.31) g( z + b) . 2 The variations of z, b1 and b2 are now defined by

zi+1/2,j = zi+1/2+,j − zi+1/2−,j , 1

zij

= zi+1/2−,j − zi−1/2+,j ,

1 1 1 = bi+1/2+,j − bi+1/2−,j ,

bi+1/2,j 1 1 1 = bi+1/2−,j − bi−1/2+,j ,

bij

zi,j +1/2 = zi,j +1/2+ − zi,j +1/2− , 2

zij

= zi,j +1/2− − zi,j −1/2+ ,

(5.32)

2 2 2

bi,j +1/2 = bi,j +1/2+ − bi,j +1/2− , 2 2 2

bij = bi,j +1/2− − bi,j −1/2+ .

(5.33)

The reconstruction of U , z, b1 , b2 is performed as follows. A reconstruction in x is performed on U , z, b1 exactly as it was done in the one-dimensional case, see Section 4.8. The same is done in y for U , z, b2 . The values of b1 and b2 are defined by vij + vi+1,j

xi+1/2 /g, 2 uij + ui,j +1 = fi,j +1/2

yj +1/2 /g, 2 = xi+1 − xi , yj +1/2 = yj +1 − yj .

1 1 bi+1,j − bi,j = −fi+1/2,j 2 2 bi,j +1 − bi,j

with still xi+1/2

(5.34)

6. Numerical tests The usual shallow water system (3.7) has already been intensively tested with the hydrostatic reconstruction method (Audusse, Bouchut, Bristeau, Klein and Perthame, 2004; Marche, 2005). In this section we provide numerical results for the one-dimensional shallow water system with Coriolis force (3.60). We consider

6. Numerical tests

247

a few test cases that are representative of the difficulties that can be encountered for this system. We use the scheme proposed in Section 3.10, with Fl1d , Fr1d the hydrostatic reconstruction fluxes (with HLLC homogeneous flux F ). The second-order version described in Section 4.8 is also tested. 6.1. Test 1: Stability of steady states In this test we consider a steady state which is partly at rest, i.e. of the form (3.61), and partly with h = 0. The numerical stability of this steady state is tested, by observing the discrepancy with the exact solution for large times. The interface dry/wet is also tested concerning its stability. The space variable x lies in (0, 25), the topography is defined by 0.2 − 0.05(x − 10)2 if 8 < x < 12, z(x) = (6.1) 0 otherwise, the initial data are ⎧ ⎪ ⎨ max(0, 0.1 − z(x) + 0.1(25 − x)/g) h(x) = max(0, 0.1 − z(x) + 0.1(x − 5)/g) ⎪ ⎩ max(0, 0.1 − z(x))

if x 15, if 5 x < 15,

(6.2)

otherwise,

u(x) = 0, and ⎧ ⎨ −0.1/2π if x 15, v(x) = 0.1/2π (6.3) if 5 x < 15, ⎩ 0 otherwise. The constant g is taken g = 9.81, and f (x) is a constant, f = 2π. The boundary conditions are Neumann on both sides, and the final time is tmax = 100. We use 200 points in space. The first run is at first-order in space. Figure 4 shows the water level at final time. The computation for first-order in time at CFL = 1 is unstable, while if either we take second-order in time or CFL = 0.5 we recover the exact initial data. Notice that the instability seems to be generated by the point x = 15 where f (∂x v + f ) < 0. This situation allows to apply Proposition 3.5, ensuring the stability. But in order for this to apply rigorously one needs the half CFL condition, which corresponds to the numerical experiment. We then run the second-order scheme, with also second-order in time (firstorder in time is unstable, even for CFL = 0.25), at CFL = 0.5. The steady state is well preserved. Figure 5 shows a zoom around the bump of the water level, to compare with the projection on the grid of the exact solution. Figure 6 shows a zoom of the transverse velocity. The oscillations in the center are meaningless

248

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

since they correspond to the dry region where h = 0, and there only hv is meaningful and not v itself. The computation takes 2285 timesteps.

Figure 4.

Figure 5.

Water level h + z for Test 1, first-order.

Water level h + z for Test 1, second-order (zoom).

6. Numerical tests

Figure 6.

249

Transverse velocity v for Test 1, second-order (zoom).

6.2. Test 2: Accuracy This test is devoted to a numerical measurement of the accuracy of the first-order and second-order schemes. The space variable x lies in (0, 40), and the topography is given as z(x) =

2 if |x − 20| 4, 0.48 1 − x−20 4 0

(6.4)

otherwise.

The initial data are constant, h0 = 4, u0 = 10/4, v 0 = 0, the constant g is taken g = 9.81, and f (x) is chosen as f (x) =

2 π 1 − x−22 if |x − 22| 10, 10 0

(6.5)

otherwise.

The final time is tmax = 1. We use 50 points in space. We run both the first-order scheme in space and time at CFL = 1, and the second-order scheme in space and time at CFL = 0.5. We compare the results to a reference solution computed with 5000 cells. Figure 7 shows of the water level, Figure 8 shows the velocity, and Figure 9 shows the transverse velocity. The second-order computation takes 22 timesteps.

250

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 7. Water level h + z for Test 2.

Figure 8.

Velocity u for Test 2.

Then, we compute the L1 error for several grid sizes. The results are shown on Table 1, together with the numerical order of accuracy. The accuracy corresponds to what can be expected taking into account the regularity of the solution.

6. Numerical tests

Figure 9.

Table 1

251

Transverse velocity v for Test 2.

L1 error and numerical order of accuracy for Test 2

Cells

First-order

50 100 200 400 800

48.0 24.0 12.0 6.07 3.07

Second-order / 1.00 1.00 0.98 0.98

5.13 1.95 0.670 0.243 0.0866

/ 1.40 1.54 1.46 1.49

6.3. Test 3: Stability of steady states with nonzero velocity In this test we would like to show the stability of the method and convergence to a steady state with u = 0. Such a steady state is characterized by the relations hu = cst, ∂x u2 /2 + g(h + z) − f v = 0, ∂x v + f = 0. (6.6) The two last equations can indeed be combined to get u2 /2 + v 2 /2 + g(h + z) = cst.

(6.7)

As in Test 1, the space variable x lies in (0, 25), and the topography is as in (6.1). The initial data are constant, h = 0.33, u = 0.18/0.33, v = 0. The constant g is

252

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

taken g = 9.81, and f (x) is a constant, f = 2π/50. The boundary conditions are hu(x = 0) = 0.18,

v(x = 0) = 0,

h(x = 25) = 0.33,

∂x v(x = 25) = 0.

The final time is tmax = 200. We use 200 points in space.

Figure 10.

Water level h + z for Test 3.

Figure 11.

Velocity u for Test 3.

(6.8)

6. Numerical tests

253

We use the second-order scheme in space and time, at CFL = 0.5. Figures 10– 14 show respectively the water level h+z, the velocity u, the transverse velocity v, the discharge hu, and the invariant u2 /2 + v 2 /2 + g(h + z). We observe that

Figure 12.

Transverse velocity v for Test 3.

Figure 13.

Discharge hu for Test 3.

254

Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models

Figure 14.

Invariant u2 /2 + v 2 /2 + g(h + z) for Test 3.

the solution has converged to a steady state. The computation takes 9574 timesteps.

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Chapter 5

Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment V. Zeitlin LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. Introduction 2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW 2.1. General features of the model 2.2. Lagrangian approach to 1dRSW 2.3. Theoretical predictions for the geostrophic adjustment in the 1dRSW model 2.4. Wave breaking and shock formation in the 1dRSW model

258 259 259 260 264 268

2.5. High-resolution numerical simulations of fully nonlinear geostrophic adjustment

270 274

in 1dRSW 2.6. Finite-amplitude wave solutions of the 1dRSW equations

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW 3.1. General features of the model 3.2. The existence and uniqueness of the adjusted state 3.3. The role of baroclinicity: obstacles to the standard adjustment scenario 3.4. Lagrangian approach to 2L1dRSW and symmetric instability 3.5. Stationary waves of finite amplitude in the 2-layer model

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide 4.1. Equatorial waves and geostrophic adjustment in the equatorial waveguide 4.2. Wave-breaking and shock formation by equatorial waves 4.3. Solitons and modons formed by equatorial Rossby waves 4.4. Transport and mixing phenomena during the equatorial geostrophic adjustment

5. Summary and conclusions Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02005-3 257

281 281 285 286 291 292 297 298 300 305 314 317

© 2007 Elsevier B.V. All rights reserved

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Acknowledgements References

319 319

1. Introduction The inspection of the dispersion curve for inertia–gravity waves (IGW) in the RSW model on the f -plane, see Chapter 1 of this volume, shows that IGW are dispersionless in the short-wave limit and strongly dispersive in the long-wave limit. So, at least at the heuristic level, the short IGW behave like acoustic waves which is not surprising, as in the absence of rotation the model is equivalent to the acoustics of a two-dimensional barotropic gas. One can therefore anticipate that at finite amplitude IGW can break and form shocks. The question then arises, whether rotation could prevent breaking, and if not, what is the role of this phenomenon in the fundamental process of geostrophic adjustment, and what are its consequences for the dynamical separation of fast and slow motions. On the other hand, if the dispersion due to rotation does prevent breaking at least for some configurations, one can expect existence of finite-amplitude nonlinear stationary waves in the model. In what follows, we will study essentially nonlinear wave phenomena in the RSW model, most often within the adjustment context. Weakly nonlinear phenomena in RSW involving resonant interactions of wave triads or quartets, or their ensembles (wave, or weak turbulence) will be not addressed below (see Falkovich and Medvedev, 1992, and Medvedev and Zeitlin, 2005, in this relation). We start our study (Section 2) within the framework of the simplified “one-and-a-half” dimensional version of the RSW model on the f -plane (1dRSW). The dependence of any field on one of the spatial coordinates will be prohibited in such model, and the dynamics will be reduced to that of straight fronts/jets and plane-parallel waves. The advantage of this reduced model is that the slow manifold is here unambiguously defined, as it is just an “infinitely slow” steady state. Some exact results on the fully nonlinear geostrophic adjustment may be obtained due to this simplification. The use of Lagrangian variables turns to be advantageous because of the quasi one-dimensional character of the model, and allows to treat the fully nonlinear adjustment process and to prove that rotation cannot prevent breaking for a large class of initial conditions. On the other hand, we show that for periodic boundary conditions, exact solutions in the form of finite-amplitude propagating waves of special form appear in the model thus proving that rotation does equilibrate nonlinearity for some sets of initial conditions. Similar analysis will be performed in the framework of the “one-and-a-half” dimensional two-layer RSW model on the f -plane (Section 3) in order to study the role of the baroclinic effects. We will show that new phenomena appear due

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to baroclinicity, such as wave-trapping, symmetric and Kelvin–Helmholtz instabilities, the latter being related to the loss of hyperbolicity of the system. New peculiar types of finite-amplitude propagating waves become possible due to baroclinicity. The differential rotation effects arising beyond the f -plane approximation in the RSW model introduce new dispersive phenomena, which manifest themselves in a specific form on the equatorial β-plane, where the leading constant-rotation term in the Coriolis parameter vanishes. Some of the equatorial waves are strictly dispersionless (Kelvin waves), some of them strongly dispersive in the shortwave limit, and weakly dispersive in the long-wave limit (Rossby waves), some of them have weak or strong dispersion depending of the propagation direction (Yanai waves), and some of them are close to the f -plane waves (inertia–gravity waves)—see e.g. Gill (1982) and Chapter 1. In addition, waves in the RSW model on the equatorial β-plane are quasi-one-dimensional, especially the long ones, as they are strongly localized in the vicinity of the equator and have a discrete spectrum in the meridional direction (cf. Chapters 1 and 2). One can therefore anticipate nontrivial nonlinear phenomena both in the sector of non-dispersive waves (breaking and shock formation) and in the sector of weakly dispersive waves (formation of solitons) in the equatorial wave-guide. We will display these phenomena in Section 4.

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW 2.1. General features of the model The RSW equations on the f -plane with no dependence on one of the coordinates (y) are: ∂t u + u∂x u − f v + g∂x h = 0, ∂t v + u∂x v + f u = 0, ∂t h + u∂x h + h∂x u = 0.

(2.1)

Here u, v are the two components of the horizontal velocity, h is the total fluid’s depth, f is the constant Coriolis parameter, and g is the acceleration due to gravity. Here and below ∂t , ∂x , . . . denote the partial derivatives with respect to the corresponding arguments. As seen from these equations, the model is “one-and-a-half” dimensional, because although the y-dependence is forbidden, flows in y-direction are allowed. In what follows we often consider localized jets (localized distributions of v(x)) and/or fronts (localized distributions of ∂x h(x)). A configuration is front-like if

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u, v, ∂x h have a common compact support in x. It is worth noting that the 1dRSW model was first considered in the pioneering work of Rossby (1938) where the very concept of geostrophic adjustment was introduced in the context of the relaxation of the oceanic jet to the state of geostrophic equilibrium. The model possesses two Lagrangian invariants: the generalized (sometimes called geostrophic) momentum M = v +fx

(2.2)

and the potential vorticity (PV) Q = (∂x v + f )/ h, (∂t + u∂x )M = 0,

(2.3) (∂t + u∂x )Q = 0.

(2.4)

Linearization around the rest state h = H = const gives a zero-frequency mode and surface inertia–gravity waves with the standard dispersion law: 1/2 ω = ± c02 k 2 + f 2 (2.5) . √ Here c0 = gH , ω is the wave-frequency and k is the wavenumber. As usual, one may evoke a parallel between the shallow-water dynamics and the gas dynamics (modified by rotation), so c0 is analogous to the sound speed. The conceptual advantage of this reduced model with respect to the full twodimensional one is that the geostrophic equilibria (cf. Chapter 1): f v = g∂x h,

u = 0,

(2.6)

are the exact stationary solutions of equations (2.1). Hence, in this model the slow manifold (cf. Chapter 2) is just a steady state, i.e. it is “infinitely slow”. In the state of geostrophic equilibrium velocity is entirely determined by the height perturbation (and vice versa) and the geostrophic PV is given by Q(g) =

f + g ∂2 h f xx h

.

(2.7)

2.2. Lagrangian approach to 1dRSW The system (2.1) may be reduced to a single nonlinear partial differential equation by using Lagrangian variables following Zeitlin, Medvedev and Plougonven (2003). Although physical Lagrangian particles are moving both in x and in y directions in the model, we introduce the coordinates of Lagrangian ‘quasiparticles’ X(x, t) moving in x direction via the mapping x → X(x, t), where x is position of a quasi-particle at t = 0 and X—its position at time t. From the Eulerian point of view X are just the Eulerian coordinates (we use the notation

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x for the Lagrangian labels because the standard notation a is reserved for another purpose—see below). Therefore X˙ = u(X, t) and the over-dot will denote the material time-derivative from now on, while the prime notation will be used for differentiation with respect to x for compactness, e.g. X = ∂x X. The two momentum equations in (2.1) are then rewritten as X¨ − f v + g∂X h = 0, v˙ + f X˙ = 0,

(2.8)

where v is considered as a function of x and t. By virtue of mass conservation the variable h in Eulerian variables (X, t) may be obtained from its initial distribution hI (x) via the Jacobi transformation: h(X, t) = hI (x)∂X x.

(2.9)

By time differentiating the corresponding equation for h−1 it is easy to see that it is equivalent to the continuity equation h˙ + h∂X X˙ = 0.

(2.10)

The second of equations (2.8) may be immediately integrated giving v(x, t) + f X(x, t) = M(x),

(2.11)

where the conserved generalized momentum M(x) is to be determined from the initial conditions. If vI (x) is an initial distribution of the transverse velocity then, as X(x, 0) = x, M(x) = f x + vI (x).

(2.12)

By applying the chain differentiation rule to (2.9) we may express the partial derivative ∂X as follows ∂X h = ∂X hI (x)∂x X = hI (X )−2 − hI (x)X (X )−3 , (2.13) and, thus, get a closed equation for X: ghI −2 X¨ + f 2 X + ghI (X )−2 + (2.14) = f M. (X ) 2 This equation may be rewritten in terms of the deviations φ(x, t) of quasi-particles from their initial positions X(x, t) = x + φ(x, t): ghI (2.15) (1 + φ )−2 = f vI . 2 ˙ It is to be solved with initial conditions φ(x, 0) = 0; φ(x, 0) = uI (x), where uI is the initial distribution of the velocity in the x-direction. φ¨ + f 2 φ + ghI (1 + φ )−2 +

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It should be emphasized that in the absence of rotation, f = 0, the nonlinear equation (2.14) contains only derivatives of X and not X itself. It may be reduced to a linear problem and completely integrated in this case by the hodograph ˙ X or, equivalently, u and h as new variables, and swapping demethod using X, pendent and independent variables: x = x(u, h), t = t (u, h). A linear PDE for x then results—cf. e.g. Landau and Lifshits (1975). The presence of rotation makes the hodograph transformation ineffective. Equation (2.14) may be obtained from an action principle. Indeed, by multiplying the l.h.s. by hI we get 2 ghI 2 ¨ hI X + f X − f M + (2.16) = 0. 2X 2 Equation (2.16) is an Euler–Lagrange equation following from the variational principle with the action S = dt dx L, (2.17) where the Lagrangian density is given by ˙2 gh2 1 X2 X − f2 + f MX − I . L = hI 2 2 2 X

(2.18)

˙ X we get the corresponding HamilIntroducing canonical variables P = hI X, tonian 2 gh2 1 X2 P + hI f 2 − f MX + I . H = dx (2.19) 2hI 2 2 X As usual, vanishing the first variation of the Hamiltonian δH gives the equation for the steady state: 2 ghI hI f 2 X − f M + (2.20) = 0; P = 0. 2X 2 This is the state of geostrophic equilibrium (2.6) expressed in Lagrangian coordinates. A straightforward calculation of the second variation δ 2 H shows that it is always positive-definite and, thus, the geostrophic equilibria are formally stable. It is important to note that the equation for the adjusted geostrophic equilibrium solution may be obtained by simply omitting the term with time derivatives in (2.14). Equations (2.14), (2.15) have advantage of being physically transparent and of incorporating explicitly arbitrary initial conditions. The latter fact leads, however, to the appearance of variable coefficients in the equations. Further simplification may be achieved by additional change of variables x = x(a), which “straightens” the initial elevation profile hI (x). We then get a following relation between J and h (we remind that h is everywhere positive): J = ∂X/∂a = H / h(X, t), where

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the uniform mean height H is introduced. By the chain differentiation rule we get g ∂X h = ∂a P , where P = gH /(2J 2 ) is the pressure variable. The Lagrangian equations of motion in a-coordinates take the form: u˙ − f v + ∂a P = 0,

(2.21)

v˙ + f u = 0, J˙ − ∂a u = 0.

(2.22) (2.23)

This system is equivalent to a single equation, which may be as well obtained by differentiation and change of variables in (2.14): 2 P = f H Q. J¨ + f 2 J + ∂aa

(2.24)

Here Q(a) is the potential vorticity in a-coordinates: Q(a) = H1 (∂a v(a, t) + ˙ = 0. Note, that equations (2.21)–(2.23) f J (a, t)) = H1 (∂a vI (a) + f JI (a)), Q may be deduced, as well, from the original Eulerian system (2.1) by applying the transformation of independent variables mixing the independent and the dependent variables which is known in the theory of partial differential equations (see for details Rozhdestvenskii and Janenko, 1983) da = h(x, t) dx − h(x, t)u(x, t) dt,

dτ = dt,

(2.25)

where a is the so-called Lagrangian mass variable. For positive h one gets ∂x = h∂a ,

∂t = ∂τ − hu∂a = ∂τ − u∂x

(2.26)

and the system (2.21)–(2.23) results. Again, the nonperturbative slow manifold is a stationary solution Js of (2.24). The advantage of the present formulation with respect to equation (2.14) is that it highlights the role of PV: for a given set of (localized) initial conditions the adjusted state is a stationary state with the same potential vorticity as the initial one. It is given by the equation: gH d2 1 (2.27) + f Js (a) = H Q(a). f da 2 2Js2 (a) This equation is nonlinear and can be rewritten as a non-autonomous ODE describing the motion of a material point in a given potential under the action of a “time”-dependent force, where a is “time”. Using the non-dimensional pressure variable p = P /(gH ) and introducing the Rossby deformation radius Rd2 = gH /f 2 one gets: f 1 1 d2 p = Q. + 2√ 2 g da Rd 2p

(2.28)

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264

The corresponding boundary conditions are: the “external force” in the r.h.s. exactly equilibrates the “potential force” (the second term in the l.h.s.) at a = ±∞. Hence, a stationary solution, if it exists, is a separatrix relating two states of unstable equilibrium in the phase-space of this simple one-dimensional system, and thus resembles the soliton or instanton solutions in a number of physical models. 2.3. Theoretical predictions for the geostrophic adjustment in the 1dRSW model 2.3.1. Existence and uniqueness of the adjusted state The first fundamental question which arises in the adjustment context is whether the adjusted state, i.e. the steady state with the same values of Lagrangian invariants as the initial state, exists, and whether it is unique. The simplest proof of the existence and uniqueness theorem for the adjusted state follows from the reformulation of the problem in terms of the variables X: dX = J da. This gives the following equation for the adjusted state: −

g d2 h(X) + h(X)Q(X) = f. f dX 2

(2.29)

Here PV is considered as a function of X given by the inverse mapping x = 1 (f + ∂vI (x(X)) ). x(X, t) (or a = a(X)): Q(X) = hI (x(X)) ∂x The following theorem giving sufficient conditions for existence and uniqueness of the adjusted state was proved by Zeitlin, Medvedev and Plougonven (2003) using the standard tools of the theory of ordinary differential equations: T HEOREM 1. For positive Q(X) with compact support derivatives and arbitrary constant asymptotics (frontal case) equation (2.29) has unique bounded and everywhere positive solution h(X) on R1 . The proof of Theorem 1 is not constructive in the sense that it uses the mapping x = x(X, t) which is not known explicitly. However, the Lagrangian conservation of PV guarantees that this mapping preserves the positiveness of PV and its asymptotics at infinity (provided the infinity is a fixed point). It should be noted that positiveness of Q is a sufficient condition for the absence of inertial/symmetric instability in the full 3d primitive equations models (cf. e.g. Holton, 1979). As shown below, this instability is absent in 1d RSW. 2.3.2. Analysis of small perturbations about the adjusted state The second question in the adjustment context, once the existence and uniqueness of the adjusted state is proved, is whether this state is attainable. Indeed, in the absence of dissipation, the 1dRSW model is energy-preserving and, although the adjusted state is a minimal energy state, it cannot be an attractor in the usual

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sense. However, the evacuation of energy can be provided by outgoing IGW if the radiation boundary conditions are imposed. In order to understand this process, we consider the small perturbations around the adjusted state (Zeitlin, Medvedev and Plougonven, 2003). If the initial conditions vI and hI are in geostrophic balance ghI = f vI ,

(2.30)

then φ = 0 is an exact solution of (2.15). Therefore, for small deviations from the exact geostrophy equation (2.15) may be linearized and solved perturbatively. A necessary condition for this is smallness of the initial imbalance AI = f vI − ghI . It is easy to see that this condition is equivalent to the smallness of the Rossby number: Ro = fUL 1, where U and L are characteristic velocity and scale in the x-direction. An advantage of the Lagrangian approach is that contrary to the Eulerian one, where it is necessary to attribute the initial imbalance either to vI or to hI , the imbalance AI enters the perturbative equations as a whole. Note that another source of imbalance is a non-zero uI . The linearized equation (2.15) is φ¨ + f 2 φ − 2ghI φ − ghI φ = AI .

(2.31)

We represent the solution as a combination of the slow and the fast components: φ = φ¯ + φ˜ where the over-bar denotes the time mean and the tilde corresponds to fluctuations around it. For the time-mean (slow part), corresponding to the shift of Lagrangian particles which is necessary to arrive to the new equilibrium positions, we get f 2 φ¯ − 2ghI φ¯ − ghI φ¯ = AI ,

(2.32)

a linear second-order inhomogeneous differential equation. Introducing a new variable Φ¯ = ghI φ¯ and dividing by ghI we get: 2 f + ghI ¯ Φ¯ = AI , −Φ + (2.33) ghI (g)

where the geostrophic PV constructed from the initial height perturbation QI , cf. (2.7), explicitly enters. Equation (2.33) is a linear inhomogeneous ODE with variable coefficients and its solution may be obtained by the method of variation of constants once solutions of the homogeneous equation are known. We are mostly interested in the case of localized fronts where AI is a compact support function and hI is monotonous and has constant asymptotics. Solutions of the homogeneous equation are then exponentially growing/decaying at both spatial infinities. However, it is not obvious that a decaying at both spatial infinities so(g) lution of the inhomogeneous equation may be found for arbitrary QI (x). By the same method which is used in the full nonperturbative proof above it may be

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shown that unique solution φs (x) of (2.32) exists for non-negative geostrophic (g) PV: QI 0. Hence, at least for non-negative initial PVs one can always find a corrected adjusted state. For the variable in time fast part of the solution a homogeneous equation φ¨˜ + f 2 φ˜ − 2ghI φ˜ − ghI φ˜ = 0

(2.34)

results. By introducing a new variable Φ˜ = ghI φ˜ this equation may be rewritten as (2.35) Φ¨˜ + f 2 + ghI Φ˜ − ghI Φ˜ = 0. Solution of the Cauchy problem for arbitrary initial conditions φ˜ I , uI = φ˙˜ I can ∞ +iω ˜ be obtained via e.g. the Fourier transform Ψ (x, ω) = −∞ e t Φ(x, t) dt for which we get:

f + ghI f ω2 f − Ψ = 0. −Ψ + g hI ghI

(2.36)

(g)

Note that QI plays a rôle of potential in this Schrödinger-type equation. As φI = φ˜ I + φ¯ I = 0 the initial condition for φ˜ and, hence, Φ˜ follows once φ¯ ¯ is found: φ˜ I = −φ. The solution of (2.35) represents a packet of inertia–gravity waves propagating out of the initial localized perturbation plus, possibly, some bound states (trapped modes). The presence of the trapped modes would mean the non-attainability of the adjusted state, as part of the energy of the initial perturbation which is projected onto these modes cannot be evacuated due to radiation. It may be proved however (Zeitlin, Medvedev and Plougonven, 2003) that trapping by an isolated front is impossible The following argument shows that the trapped modes should be sub-inertial, i.e. having a frequency below f . Consider the Fourier-transform ˜ φ˜ = dω(ψ(ω, x)e−iωt + c.c.). Then for each Fourier-component ψ(ω, x) of φ: we get from (2.34): ghI ψ + 2ghI ψ + ω2 − f 2 ψ = 0, (2.37) which is equivalent to (2.36). In the case of a front-like initial configuration the asymptotics of φ˜ I and uI at infinity are zero and those of hI and Q(g) are constant: hI |±∞ = h± ,

f Q(g) ±∞ = . h±

(2.38)

Hence, at x → ±∞ equation (2.36) becomes −Ψ +

f 2 − ω2 Ψ =0 gh±

(2.39)

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and in order to have decaying at spatial infinity bound states we should have ω < f . However, this is impossible, as seen from the following simple consideration. By multiplying (2.37) by ghI ψ ∗ , where the asterisk denotes the complex conjugation, we get 2 2 ∗ g hI ψ ψ − g 2 h2I ψ ∗ ψ + ω2 − f 2 ghI ψ ∗ ψ = 0 (2.40) and for decaying at ±∞ states an estimate +∞ dx g 2 h2I |ψ |2 2 2 f2 ω = f + −∞ +∞ 2 dx gh |ψ| I −∞

(2.41)

follows by integration. This contradicts the initial hypothesis and, hence, subinertial trapped modes do not exist, and the frequency spectrum is continuous. ˜ Therefore, all of the initial φ-perturbation will be dispersed leaving only the stationary part φs in the vicinity of the initial perturbation. As shown in Reznik, Zeitlin and Ben Jelloul (2001) the outgoing waves do not exert any drag upon the stationary state in lowest orders in Ro and, thus, slow and fast variables are split in the perturbation theory, at least for non-negative PVs. The speed of the relaxation toward the adjusted state will depend on further details of the potential Q(g) in the Schrödinger equation (2.36). If quasi-stationary states, i.e. those which decay only by a sub-barrier tunneling, are present the decay rate will be exponential, as is well-known from quantum mechanics (cf. e.g. Migdal (1977)). Otherwise the decay will be dispersive according to the t −1/2 law. Here and below we mean by decay the time-decrease of the amplitude of a spatially localized perturbation. It should be also emphasized that equation (2.41) also shows that negative values of ω2 are impossible in the 1dRSW model. This will not be the case in the two-layer 1dRSW model treated in the next section. There both sub-inertial and negative values of the frequency square may exist, the latter case corresponding to the appearance of the so-called symmetric instability. 2.3.3. Low-resolution numerical simulations of the geostrophic adjustment in Lagrangian variables Equation (2.15) is ready for studies of the Cauchy problem which is equivalent to the fully nonlinear geostrophic adjustment problem. The advantage of having a single PDE with two independent variables is that it is readily solvable by the PDE solvers available in standard software packages. An example of low-resolution calculation using the standard MATHEMATICA routine NDSolve (Wolfram, 2004) is presented in Figure 1 giving a qualitative picture of adjustment. This example is taken from Zeitlin, Medvedev and Plougonven (2003). The 2 Gaussian profile of initial elevation: hI (x) = 1 + e−x is chosen. The along-jet 2 velocity is slightly imbalanced vI (x) = −2(x + 0.2 sin(x))e−x , and non-zero

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Figure 1. Fully nonlinear geostrophic adjustment of the double-jet configuration, see the main text for explanation.

cross-front initial velocity is imposed uI (x) = 0.1e−x . The horizontal axes correspond to time and initial positions of Lagrangian particles, respectively, and the particle displacement X − x is plotted on the vertical axis. Emitted fast gravity waves and slowly dispersing quasi-inertial oscillations concentrated in the vicinity of the jet are clearly seen, as well as systematic displacements of fluid particles necessary to reach the final adjusted state. A zone of steep gradients forms and propagates toward the left. It indicates a shock formation but such process cannot be properly resolved by a generic solver and results in the fuzzy region on the left. The shock formation thus needs attention in the model. If confirmed, being an essentially dissipative process the shock formation would provide an alternative sink of energy due to waves. 2

2.4. Wave breaking and shock formation in the 1dRSW model The formulation of the 1dRSW equations using the independent variable a allows to understand the shock formation, which manifests itself in the low-resolution simulations presented above. For this purpose the Lax method of analysis of the quasi-linear hyperbolic systems (Lax, 1973; Engelberg, 1996) is applied.

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Following Zeitlin, Medvedev and Plougonven (2003), let us rewrite equations (2.21)–(2.23) as a system of two equations u˙ + ∂a p = v, J˙ − ∂a u = 0,

(2.42) (2.43)

where v is not an independent variable and should be determined from the relation ∂a v = Q(a) − J . To avoid cumbersome formulae, all dimensional parameters are taken to be equal to unity in this subsection; the correct dimensions are easy to recover. Equations (2.42), (2.43) represent a quasi-linear system in a standard form (cf. Whitham, 1974): u v u˙ 0 −J −3 ∂a = . (2.44) + −1 0 J 0 J˙ The left eigenvectors of the advection matrix in the l.h.s. are (1, ±J −3/2 ) and the corresponding eigenvalues are μ± = ±J −3/2 . Hence, the Riemann invariants are w± = u ± 2J −1/2 and obey the following equations: w˙ ± + μ± ∂a w± = v.

(2.45)

Expressions of original variables in terms of w± are given by 1 (w+ + w− ), (2.46) 2 16 > 0, J = (2.47) (w+ − w− )2 w + − w− 3 μ± = ± (2.48) . 4 The following equations for the derivatives of the Riemann invariants r± = ∂a w± readily follow: u=

∂μ± ∂μ± r+ r± + r− r± = ∂a v = Q(a) − J. (2.49) ∂w+ ∂w− These equations may be rewritten, using the “double” Lagrangian derivatives along the characteristics dtdr± = r˙ + μ± ∂a r, as r˙± + μ± ∂a r± +

dr± ∂μ± ∂μ± (2.50) + r+ r± + r− r± = Q(a) − J. dt± ∂w+ ∂w− Wave-breaking and shock formation correspond to the loss of smoothness by the Riemann invariants in finite time, i.e. to r± going to ±∞ in finite time. Equations (2.50) are the generalized Ricatti equations on the characteristics. The qualitative analysis of such equations (see Lemmas 1 and 2 in Engelberg, 1996) gives both conditions of shock appearance, and of their absence:

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(1) If the initial relative vorticity Q − J = ∂a v is sufficiently negative breaking always takes place whatever the initial conditions are. (2) If the relative vorticity is positive, as well as the derivatives of the Riemann invariants at the initial moment, breaking never takes place. In the adjustment context this result means that shocks should be more easy to produce on the anticyclonic (negative relative vorticity) side of the jet. As one of the Riemann invariants has always a negative derivative for the stepwise height profiles, shocks should be always produced by the pure height (no vI ) adjustment. The shock production in course of adjustment of front-like perturbations is likely, because these latter never satisfy the second condition. 2.5. High-resolution numerical simulations of fully nonlinear geostrophic adjustment in 1dRSW High-resolution numerical simulations of fully nonlinear geostrophic adjustment in 1dRSW were undertaken by Bouchut, Le Sommer and Zeitlin (2004) in order to check the theoretical predictions and to test the finite-volume numerical method by Bouchut (Chapter 4 in this volume). We first remind the general framework for the shock description in the RSW model in Eulerian variables, and then display some of the results of the numerical simulations following this paper. 2.5.1. Hydraulic theory as applied to the 1dRSW model Equations (2.1) may be reformulated in the form of conservation laws for the two components of momentum and the mass (although momentum is not conserved due to the presence of the Coriolis acceleration in (2.1) we use the term “conservation laws” for the equations below, as is frequently done in the literature, cf. e.g. (Rozhdestvenskii and Janenko, 1983)): 1 2 2 (hu)t + hu + gh − f hv = 0, 2 x (hv)t + (huv)x + f hu = 0, ht + (hu)x = 0.

(2.51)

The analogy with the rotating compressible gas with a specific √ heat ratio γ = 2, h as a density variable, and the equivalent sound speed c0 = gH , gives a mathematically rigorous framework for a description of the hydraulic jumps as gas dynamics shock-waves, see Lighthill (1978), and allows to use powerful computational methods developed in gas dynamics. The basic properties required in order to define the weak solutions to (2.51) are the conservation of mass and momentum across a supposed discontinuity, see e.g. Lighthill (1978). The weak solutions may be also obtained as solutions of

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a dissipative set of equations corresponding to (2.51) in the limit of vanishing viscosity, cf. e.g. Schär and Smith (1993). The standard discontinuity calculus holds for shallow water. (For non-rotating case see Whitham, 1974, for rotating case see Pratt, 1983, 1984; and Nof, 1984.) Weak solutions are known to be completely determined if one specifies the Rankine–Hugoniot (RH) conditions together with the entropy condition. In our case, the RH conditions are: −U [hu] + hu2 + gh2 /2 = 0, −U [hv] + [huv] = 0, −U [h] + [hu] = 0,

(2.52)

where U is the speed of the discontinuity and [A] is the jump, following a fluid particle, of any quantity A across the discontinuity. These conditions do not depend on the Coriolis parameter f and express, respectively, the conservation of momentum in the x and y directions and mass conservation across the discontinuity. Yet, physically relevant solutions are not specified unless the dissipation of the total energy E across the discontinuity is ensured (cf. Whitham, 1974): h E + gh2 /2 u 0 with E = gh + u2 + v 2 , (2.53) 2 which is an analog of the gas dynamics entropy condition. As a direct consequence, this implies that [h] > 0 across a shock. Note that the conditions (2.52) are satisfied by any finite-volume numerical scheme whereas the fulfillment of the condition (2.53), which is not straightforward, requires special care. From the RH and entropy conditions it follows that the rate of the energy dissipation in every material volume V (t) which contains a discontinuity depends on the amplitude of the jump only g d E dx = [h]2 [u]. (2.54) dt 4 −U [E] +

V (t)

However, the PV field is not affected by dissipation due to the passage of shocks. Following Pratt (1983, 1984) and Peregrine (1998), we consider the jump in the PV equation −U [hq] + [huq] = 0 and continuity equation [(U − u)h] = 0 in the absence of contact discontinuities i.e. for [v] = 0, and obtain (U −u)h[q] = 0. Hence [q] = 0 in one dimension. Only transverse variations in shock strength can affect the PV in this framework, cf. Pratt (1983). Hence in the y-independent reduction of the shallow water equations no PV deposit by shocks is possible. As shown by Houghton (1969), shocks are modified by rotation. This implies a decrease in shock strength with time, but, as just explained, shocks do not affect the vortex part of the flow.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

272

2.5.2. The results of high-resolution numerical simulations of the adjustment process The high-resolution finite-volume numerical method by Bouchut (2004) (see also Chapter 4 of the present volume) applied to the system (2.51) perfectly resolves shocks, automatically satisfies the RH conditions including the entropy condition (2.53), and allows for the efficient treatment of the Coriolis source terms. It was extensively tested in Bouchut, Le Sommer and Zeitlin (2004) and shown to perform better than other comparable methods (see Appendix A of this paper). The shock formation was studied in a number of cases, and found to be ubiquitous. For example for the classical Rossby problem of the adjustment of the unbalanced jet with initial conditions of the form:

h(x, 0) = H = const, (2.55) u(x, 0) = 0, v(x, 0) = V NL (x), where (1 + tanh(4x/L + 2)) · (1 − tanh(4x/L − 2)) NL (x) = (2.56) (1 + tanh(2))2 and the relevant parameters are Rossby and Burger numbers, respectively: gH V , Bu = 2 2 , (2.57) fL f L the shocks are formed in the wide range of parameters, as shown in Figure 2. A typical evolution of the height field h(x, t) in the Rossby adjustment is shown in Figure 3 for Ro = 1 and Bu = 0.25. (The figures here and below are taken from Bouchut, Le Sommer and Zeitlin, 2004.) As expected, the height field adjusts to the momentum imbalance by emitting IGW which propagate out from the jet. As seen in Figure 3, and in agreement with the results of the previous subsection and Zeitlin, Medvedev and Plougonven (2003), two discontinuities form rapidly at the wave front. The formation of shocks in RSW, i.e. the fact that rotation does not inhibit wave-breaking, was first predicted by Houghton (1969) and observed by Kuo and Polvani (1997) in their numerical study of the rotating dam-break problem. As shown in Bouchut, Le Sommer and Zeitlin (2004), the energy is conserved with high accuracy before the appearance of shocks, and drops up to 10% when a shock forms. Thus the energy evacuation in the adjustment process due to shocks, which is important for understanding the scenario of adjustment, is substantial. In spite of such sink of energy, the adjusted state is not rapidly attained due to the quasi-inertial oscillations remaining for the long time at the location of the initial jet, as shown in Figure 4. Although persistent quasi-inertial (i.e. having frequencies close to the inertial one) oscillations were observed in all of the numerical simulations of Bouchut, Le Sommer and Zeitlin (2004), confirming the earlier results by Kuo and Polvani (1997, 1999), no significant deviations of their decay Ro =

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

273

Figure 2. Statistics of the shock formation in the Rossby adjustment problem on the Bu–Ro plane: each point corresponds to a run with the corresponding parameters Bu and Ro. Breaking happens in t < π/f : shaded circles; breaking event in π/f < t < 2π/f : open circles; breaking in t > 2π/f : crosses. Appearance of transonic shocks, i.e. those with propagation velocity changing sign in course of evolution, are marked by the superscript t. Drying was observed for large Ro and small Bu: squares.

law from the t 1/2 were found. These numerical simulations allowed to complete the theoretical results on existence of the adjusted state. Thus, it was observed, cf. Figure 5, that negative PV-states adjust as well. The wave-breaking criteria exposed in the previous subsection were also tested and confirmed numerically. The adjustment of a PV-less wave perturbation super-imposed onto a simple balanced jet was performed in order to check how the vorticity in the jet region influences shock formation. The parameters are the Rossby and the Burger numbers of the jet, Rojet , Bujet and the Rossby and the Froude numbers which specify the perturbation, Rop , Frp . The typical evolution of the perturbation is shown in Figure 6. The perturbation splits into a left and a right propagating parts but, in this case, only the former breaks inside the jet. Let us remind that, as shown above, two factors lead to breaking: strong enough gradients of the Riemann invariants and strong enough anticyclonic shear. The perturbation is small enough in Figure 6, and the derivatives R± of the Riemann invariants rewritten in Eulerian variables

274

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 3. The Rossby adjustment of a jet. Two shocks are formed at t = 0.3 and propagate to the left and to the right from the jet, respectively. One of the shocks is formed immediately within the jet core.

(these expressions are obtained by a change of variables from corresponding expressions in Lagrangian coordinates calculated before): 3/4 1/4 g h H ux ± hx , R± = 8 (2.58) h H H are dominated by the derivative of the height field (the second term). Since the sign of this term is fixed by the geostrophic balance of the basic state, the part of the perturbation propagating toward the small h region is more likely to break. Hence, as observed, breaking happens at the cyclonic part of the balanced jet as the influence of anticyclonicity is overcome by the strength of pressure gradients in this case. 2.6. Finite-amplitude wave solutions of the 1dRSW equations In the previous subsections we explored the previously theoretically established fact that rotation does not prevent shock formation for a large class of initial con-

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

275

Figure 4. Check of balance between f v and ghx in the Rossby adjustment: although a close to the geostrophic balance mean state (second panel) is rapidly achieved, oscillations persist in the jet core. The amplitude of oscillations is decreasing with time and depends on the parameters Ro and Bu. The period of oscillations is close to Tf . The scale of the graphs is c02 /L.

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 4.

(Continued.)

Figure 5. PV-shift during the Rossby adjustment process. The initial distribution of PV and the mean distribution at t = 34.2Tf are shown. Note that the initial PV distribution contains a negative-PV part.

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

277

Figure 6. Wave breaking in a simple jet: Rojet = 1, Bujet = 1 and Rop = 0.8. Left column: h, u in the case up > 0. Right column: h, u in the case up < 0. In both cases, the fields are plotted for successive times 0, 0.1, 0.2, 0.3Tf (from top to bottom). The arrows indicate the shock locations.

ditions. We now turn to the opposite case of initial conditions when it does. It is known (Ostrovsky, 1978; Shrira, 1981, 1986; Grimshaw, Ostrovsky, Shrira and Stepanyants, 1998) that the dispersion due to rotation is sufficient to warrant the existence of periodic steady-propagating finite-amplitude plane-parallel waves in RSW. Their particular feature is that amplitudes are bound from above by some limiting value, and that solution tends to form cusps at it crests while approaching the limiting amplitude. These solutions may be obtained using Lagrangian variables as well (Bühler, 1993; Zeitlin, Medvedev and Plougonven, 2003). However, in view of subsequent generalization to the two-layer RSW model below, we present here the derivation in Eulerian coordinates following Plougonven and Zeitlin (2003). We start from the system (2.1) and look for stationary propagating solutions depending on ξ = x − ct only; the prime will denote the derivative with respect to ξ . Therefore, −cu + uu − f v + gh = 0,

(2.59a)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

278

−cv + u(f + v ) = 0,

−ch + (uh) = 0.

(2.59b) (2.59c)

Integration of (2.59c) gives u = c + K/ h, where K is an integration constant. We look for spatially periodic solutions with a given (unknown) wavelength λ and suppose that there is no overall mass-flux in the x direction. Then integration of uh over one wavelength λ gives zero: λ (K + ch) dξ = 0,

(2.60)

0

and the value of the integration constant is K = −cH , where H is the mean (rest) height of the fluid. The expression for u results: H u=c 1− (2.61) . h Note that solutions in question do not carry PV anomaly. Indeed, the PV (2.3) by virtue of (2.59b) and (2.61) is equal to q=

f + v f = , h H

(2.62)

on the solutions, i.e. is equal to the background PV of the unperturbed fluid layer. This is consistent with the result of Falkovich and Medvedev (1992) showing that the invariant definition of the wave component of motion in RSW is that of the absence of the PV anomaly. Hence, the solutions we are looking for are finiteamplitude generalizations of the linear inertia–gravity waves. By introducing the dimensionless elevation χ: h = H χ and the Mach number √ M = c/ gH the following equation for χ follows from (2.61) and (2.59a)– (2.59c):

2 f2 M + χ − (χ − 1) = 0. (2.63) 2χ 2 c02 Considering this equation at the wave-crest which is supposed to be regular, χ = 0, we get 1−

M2 f2 χ − 1 = . χ3 c02 χ

(2.64)

At the wave-crest χ − 1 > 0, χ < 0, and hence the r.h.s. of this expression is negative. Therefore, 1 < χ 3 < M 2 . Hence, the wave solutions exist only for M > 1.

2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW

279

2

By multiplication by [ M2 χ12 + χ] equation (2.63) may be integrated and transformed into a “particle-in-a-well” equation at zero energy level of the “particle”: χ2 +

f2 V(χ; M, A) = 0. c02

(2.65)

Note that only non-negative values of χ are physically acceptable. Zero values are excluded as well, as they lead to infinite u, cf. (2.61). The expression for the “potential” is: 2

− 1) − A] [(χ − 1)2 ( M N (χ; M, A) χ2 V(χ; M, A) = = , 2 (D(χ; M, A))2 (M − 1)2 χ3

(2.66)

where A is an integration constant. It can be easily seen that A is necessarily nonnegative (it is zero for the fluid at rest, strictly positive otherwise, and increases with the amplitude of the wave solutions to be obtained). Stationary waves of finite amplitude correspond to the oscillations of the particle around the rest position χ = 1. They exist if there is a potential well which is situated between two strictly positive zeros, χl and χr , 0 < χl < 1 < χr , where the potential is negative and bounded from below. It is easy to show (Plougonven and Zeitlin, 2003) that: The denominator (D)2 is always positive; thus the sign of the potential is determined by the numerator; D has a unique zero for χ0 = M 2/3 > 1 The numerator N tends to +∞ as χ → 0; for χ = 1, it is equal to −A and, hence, is strictly negative for positive A. Therefore, there is a zero of N in the interval ]0, 1[. For χ → +∞, N → −∞, and N has a single maximum in ]1, +∞[. This maximum occurs for the same value of χ as the zero of the denominator: χ0 = M 2/3 > 1. Therefore, if N (χ0 ; M, A) is strictly positive, the numerator has one zero in the interval ]1, χ0 [ : the potential then has a well and stationary waves exist. This situation is illustrated in Figure 7. For A close to zero, it is easy to check that the numerator has two zeros close to 1, and waves of small amplitude always exist. For small deviations from the rest-state, (2.66) becomes the

Figure 7.

The numerator N (dashed), the denominator (D)2 (gray) of the potential V, and the potential itself (plain) for M = 2 and A = 0.18.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

280

harmonic oscillator equation with “frequency” (= wavenumber) 1 . (2.67) −1 This expression is equivalent to the dispersion relation for linear surface inertia– gravity waves (2.5). Thus there exist a smooth transition from infinitesimal IGW to finite-amplitude non-harmonic waves with increasing amplitude, so that the solution we found may be called finite-amplitude IGW. As A increases, the graph of N descends and there exists a critical value Ac for which the maximum of the numerator in ]1, +∞[ is zero: N (χ0 ; M, A) = 0. Asymptotic analysis in the vicinity of χ0 = M 2/3 shows that the potential itself is continuous and has no well: hence, waves exist only for 0 < A < Ac . Note that as A increases, the potential becomes more and more asymmetric. Correspondingly, the wave-crests sharpen, and the waves tend to develop cusps as A → Ac . However, as it was shown by Zeitlin, Medvedev and Plougonven (2003) a true cusp is an asymptotic envelope of the solutions and is not attainable. Summarizing, stationary periodic waves exist if the equation 2 2 M −1 −A=0 N (χ; M, A) = (χ − 1) (2.68) χ2 k2 =

M2

has two distinct roots in the interval ]1, +∞[ or, equivalently, if N (M 2/3 ; M, A) > 0. The height of the crests of the waves is bounded from above by the limiting value M 2/3 . The difference between the roots of the numerator gives the amplitude of the wave; its wavelength depends on the value of the deformation radius f 2 /c02 . Thus obtained nonlinear waves are therefore the plane-parallel surface inertia–gravity waves of finite amplitude. Note that they are not strictly one-dimensional as the cross-propagation velocity v is non-zero and varies periodically (cf. equation (2.59b)). A limiting-amplitude nonlinear wave solution is presented in Figure 8 (Bouchut, Le Sommer and Zeitlin, 2004). The stability analysis of the finite-amplitude waves is a nontrivial technical problem. Some idea of their robustness, and their emergence from the arbitrary periodic initial conditions may be made on the basis of direct numerical simulations by Bouchut, Le Sommer and Zeitlin (2004). Figure 9 illustrates stability of the limiting-amplitude wave with respect to localized perturbations. No destabilization of the finite-amplitude wave were found within a large class of superimposed perturbations. In Figure 10 spontaneous emergence of nonlinear periodic waves during the evolution of a finite amplitude harmonic wave is displayed (Bouchut, Le Sommer and Zeitlin, 2004). An exact harmonic solution to the linearized equations (2.1) with Ro = 0.1 and k = π/2 is taken as initial condition in this simulation. As seen from Figure 10, the initial wave breaks down and, after a stage of geostrophic adjustment, a quasi-steady propagative state is reached which resembles closely

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

Figure 8.

281

Exact nonlinear-wave solution; the limiting-amplitude configuration with the phase-speed c = 2c0 .

the cusp pattern of the exact propagative nonlinear solution. The adjustment is achieved through shock formation and related energy dissipation. The energy decay due to dissipative processes in the shock is displayed in Figure 11 and shows that in this case the energy tends to a nearly steady value after a rapid stage of adjustment. These simulations allow to conjecture that finite-amplitude wave solutions are attracting in the constant PV sector of periodic initial data.

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW 3.1. General features of the model The 2-layer 1dRSW (2L1dRSW) is obtained by superposition of shallow-water layers of different densities (see Chapter 1 of this volume), with a constraint of no variations in one of the spatial directions. In order to reduce the number of the degrees of freedom, the rigid-lid vertical boundary conditions are used. Although

282

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 9. Stability of the nonlinear wave solution of limiting amplitude: localized perturbation is added to the nonlinear wave solution—left panel; the resulting profile after ten inertial periods—right panel. Between these two snapshots, the wave propagated at the speed c = 2c0 so that 12 recurrences in the computation domain were observed.

some features of the model are close to the 1dRSW, with the displacements of the interface between the layers replacing the free-surface elevation, it will be shown below that new essentially baroclinic effects due to the difference in motions of the layers appear in the 2L1dRSW dynamics (Le Sommer, Medvedev, Plougonven and Zeitlin, 2003). The equations of the 2L1dRSW model are: ∂t u1 + u1 ∂x u1 − f v1 + ρ1−1 ∂x π = 0,

(3.1a)

∂t v1 + u1 (f + ∂x v1 ) = 0,

(3.1b)

∂t u2 + u2 ∂x u2 − f v2 + ρ2−1 ∂x π

+ g ∂x η = 0,

∂t v2 + u2 (f + ∂x v2 ) = 0, ∂t (H1 − η) + ∂x (H1 − η)u1 = 0, ∂t (H2 + η) + ∂x (H2 + η)u2 = 0,

(3.1c) (3.1d) (3.1e) (3.1f)

where (u1 , v1 ) are two components of velocity in the upper layer, (u2 , v2 ) are two components of velocity in the lower layer; π is the barotropic pressure imposed by the rigid lid and equal to the pressure field in the upper layer π1 ; η is the

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

Figure 10.

283

Time evolution of a finite-amplitude harmonic wave.

interface displacement, H1 and H2 are the heights of the two fluid layers at rest; H = H1 + H2 = const is the total height, finally, g is the reduced gravity: g = g(ρ2 − ρ1 )/ρ2 . If the full layers’ heights h1,2 = H1,2 ∓ η are used as dynamical variables, the pressure field in the lower layer is expressed as π2 = π1 + g(ρ1 h1 + ρ2 h2 ); The approximation of the rigid lid leads to a dynamical constraint following from combining equations (3.1e) and (3.1f): (H1 − η)u1 + (H2 + η)u2 = H Ub (t). (3.2) Here Ub is the barotropic velocity in the x-direction. In what follows we will impose Ub = 0 which corresponds to vanishing of the overall x-momentum of the system. The potential vorticities in each layer are: Q1 =

f + ∂x v1 h1

and Q2 =

f + ∂x v2 . h2

(3.3)

They are Lagrangian invariants of (3.1a)–(3.1f). Linearizing the equations of motion about the rest-state gives: ∂t u1 − f v1 + ρ1−1 ∂x π = 0,

(3.4a)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

284

Figure 11. Energy decay during the evolution of a high-amplitude harmonic wave. The evolution of the non-dimensional total energy e/e(0), of the non-dimensional available potential energy ep /e(0) and of the non-dimensional kinetic energy ec /e(0), with ep (t) = dx g(h − H )2 /2, ec (t) = dx h(u2 + v 2 )/2 and e = ep + ec computed in the whole periodic domain.

∂t v1 + f u1 = 0, ∂t u2 − f v2 + ρ2−1 ∂x π

(3.4b)

+ g ∂x η = 0,

(3.4c)

∂t v2 + f u2 = 0,

(3.4d)

−∂t η + H1 ∂x u1 = 0,

(3.4e)

∂t η + H2 ∂x u2 = 0.

(3.4f)

Two kinds of solutions of these equations are easily found. They are stationary flows in geostrophic equilibrium and linear inertia–gravity waves. The stationary solutions corresponding to the geostrophic equilibria are, as in the 1dRSW case, the exact solutions of the full nonlinear equations and are given by: 1 ∂x π, fρ1 1 g v2 = ∂x π + ∂x η. fρ2 f

v1 =

(3.5a) (3.5b)

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

285

The stationary (balanced) solutions are, therefore, baroclinic in the general case, i.e. v1 = v2 . The fast motions in the linear approximation are internal inertia–gravity waves propagating along the interface between the layers; their form can be deduced from the equation for the variable H U = H2 u2 = −H1 u1 following from (3.4a)– (3.14f): 2 ∂tt + f 2 U − ce2 ∂xx U = 0. (3.6) Here ce2 = gHe is the phase speed of the waves, He =

H1 H2 ρ H1 + ρ1 H2

is the equivalent

2

height for the baroclinic modes of the model. The dispersion relation for the waves with frequency ω and wavenumber k reads: ω2 = f 2 + ce2 k 2 .

(3.7)

It is important to note that it is possible to eliminate the barotropic pressure in the full system (3.1a)–(3.1f) or, which is equivalent, to eliminate both pressures π1 , π2 , by expressing them in terms of other variables. From (3.1a), (3.1c) and (3.2) the following expressions result:

h2 −1 h1 + f (h1 v1 + h2 v2 ) − ρ1 ρ2 gh2 ∂ (ρ1 h1 + ρ2 h2 ) , − ρ2 ∂x ∂π2 h2 −1 h1 + = f (h1 v1 + h2 v2 ) − ∂x ρ1 ρ2 gh1 ∂ (ρ1 h1 + ρ2 h2 ) . + ρ1 ∂x

∂π1 = ∂x

∂ h1 u21 + h2 u22 ∂x (3.8) ∂ h1 u21 + h2 u22 ∂x (3.9)

To summarize, the structure of the stationary (balanced) states in the 2L1dRSW model is more complex than in the 1dRSW one. The dynamics is more complex, too, as the elimination of the barotropic pressure via (3.8), (3.9) generates new nonlinearities, see below. The geostrophic adjustment process will be more complex, as well, and will display qualitative differences as compared to the 1dRSW adjustment studied in the previous section. 3.2. The existence and uniqueness of the adjusted state As in the one-layer model, conditions for existence and uniqueness of the adjusted state can be obtained directly in the Eulerian framework as conditions for existence and uniqueness of solutions to the PV equations (Le Sommer, Medvedev,

286

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Plougonven and Zeitlin, 2003): f + ∂x v1 = Q1 (x ), h1 f + ∂x v2 = Q2 (x ). h2

(3.10a) (3.10b)

Here x is an Eulerian coordinate, which is related to the x-coordinate in the original equations via some (unknown explicitly) change of variables (see Section 2 and Zeitlin, Medvedev and Plougonven, 2003). Q1 and Q2 are the potential vorticity distributions in the layers, and v1 and v2 are given by (3.5a) and (3.5b). These equations can be combined to give two ordinary differential equations for the depths of the layers: g h1 − (Q2 + rQ1 )h1 = − −f (1 − r) + H Q2 , f g h2 − (Q2 + rQ1 )h2 = − f (1 − r) + rH Q1 , f

(3.11a) (3.11b)

where notation r = ρ1 /ρ2 for the density ratio of the layers has been introduced and the prime denotes the x -differentiation. An essential difference of these equations from their one-layer counterpart is that the forcing terms in the r.h.s. are not constant. They may be, nevertheless, analyzed by the same method (cf. Appendix A in Zeitlin, Medvedev and Plougonven, 2003). For an equation of the form h − R(x)h = −S(x), the existence and uniqueness of solutions are guaranteed for R and S with constant asymptotics at ±∞. Furthermore, the solution is positive if R and S are positive. Hence, for the initial states with localized PV anomalies such that Q1 0 and Q2 (1 − r)f/H,

(3.12)

the above equations have unique solutions h1 and h2 that are everywhere positive. Therefore, positiveness of the PV subject to the additional constraint (3.12) is sufficient for existence and uniqueness of the adjusted state in the baroclinic two-layer model. 3.3. The role of baroclinicity: obstacles to the standard adjustment scenario Although positiveness of the PV was also found in 1dRSW to be a formal criterion of existence of the adjusted state, the numerical experiments carried by Bouchut, Le Sommer and Zeitlin (2004) and presented in the previous section showed that negative-PV initial configuration were also adjustable. The constraint of positiveness of the PV is more drastic in 2L1dRSW than in the 1dRSW, and has a clear physical meaning, because negative PVs correspond to the appearance of the trapped modes and so-called symmetric instability in the baroclinic

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287

systems. Both phenomena are absent in the barotropic one-layer 1dRSW model. On the other hand, the baroclinicity gives rise to the classical Kelvin–Helmholtz instability, if the initial velocity shear is strong enough. These two typically baroclinic instabilities constitute obstacles to the standard scenario of adjustment in the two-layer system with strong enough initial shears. We briefly describe them below. Note that essentially two-dimensional (“non-symmetric”) Rossby–Kelvin instability of the two-layer system discussed in Chapter 1 does not appear in the 1.5-dimensional model we are considering here. 3.3.1. Trapped modes and symmetric instability An entirely new phenomenon with respect to the 1dRSW model arises in the 2LdRSW one due to the baroclinic effects. Namely, some part of the initial perturbation can be trapped inside the jet in the form of fast oscillations, thus rendering adjustment incomplete. Consider a background state consisting of a balanced jet with nonzero velocities in both layers. The jet (cf. (3.5a)–(3.5b)) is characterized by its velocity Vig in each layer, the barotropic pressure Π and the variable interface height: V1g = ∂x Π,

(3.13a)

V2g = r∂x Π + Bu∂x h2g .

(3.13b)

Here we use the non-dimensional form of equations and introduce the Burger 1 ,H2 ) . Note that h1g and h2g are not independent as h1g + number Bu = g Min(H f 2 L2 h2g = 1. Linearizing around the jet gives: ∂t u1 − v1 + ∂x Π = 0,

(3.14a)

∂t v1 + u1 (1 + ∂x V1g ) = 0,

(3.14b)

∂t u2 − v2 + r∂x Π + Bu∂x η = 0,

(3.14c)

∂t v2 + u2 (1 + ∂x V2g ) = 0,

(3.14d)

∂t η − ∂x (h1g u1 ) = 0,

(3.14e)

∂t η + ∂x (h2g u2 ) = 0.

(3.14f)

The above equations can be combined into a single one for the variable U = h2g u2 = −h1g u1 :

2 Bu∂xx U−

= 0.

rh2g + h1g 2 ∂ 2 h2g 1 2 Π ∂tt + 1 + r∂xx + Bu xx U h1g h2g h1g h2g h2g (3.15)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

288

It is easy to check that for h1g → ∞, we recover from this equation the corresponding equation for the one-layer model and that in the absence of the jet the equation is reduced to that of free inertia–gravity waves at the interface (3.6). The existence of trapped modes may be analyzed following Le Sommer, Medvedev, Plougonven and Zeitlin (2003). As in the one-layer model the localized modes necessarily have frequencies below the inertial frequency f (which is equal to 1 in non-dimensional terms). By introducing the following auxiliary functions rh2g + h1g F (x) = (3.16) , h1g h2g ∂xx h2g 1 2 + Bu G(x) = r∂xx Π (3.17) h1g h2g h2g (ω, x)e−iωt + c.c. we and considering the Fourier-transform of U (x, t) = dω U get for each of the Fourier-components U (ω, x): 2 = 0. Bu∂xx (3.18) U − F 1 − ω2 + G U ∗ and integration over the whole domain the following After multiplication by U estimate for the frequencies ω of the localized modes is obtained: |2 dx + G|U |2 dx Bu |∂x U 2 ω =1+ (3.19) . |2 dx F |U Here F is, by definition, positive while G may be negative, particularly in the 2 Π < 0, cf. (3.16), (3.17). Thus, contrary to the anticyclonic regions where ∂xx situation in the one-layer model, sub-inertial frequencies are possible if the anticyclonic shear is strong enough. Note that anticyclonic shear means negative 2 Π is the relative vorticity in relative vorticity. According to (3.13a), (3.13b), ∂xx the upper layer, while h2g G(x) is the relative vorticity in the lower layer. When their values become lower than minus one (lower than −f in dimensionful units), the respective PVs become negative. To demonstrate the existence of the trapped modes we choose the simplest case of the barotropic jet with no displacement of the interface (η = 0). Equation (3.15) is then reduced to: 2 2 Bu∂xx (3.20) U − ∂tt2 + 1 He−1 + r∂xx Π(H1 H2 )−1 U = 0. eiωt + c.c. we get: Seeking solutions in the form U 1 2 −1 −1 2 = 0. Π U ω He − He + (H1 H2 )−1 r∂xx Bu This is the quantum-mechanical Schrödinger equation: 2 ψ + E − V (x) ψ = 0 ∂xx 2 ∂xx U+

(3.21)

(3.22)

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289

for a particle having the energy E = ω2 (He Bu)−1 and moving in the potential 2 Π). It is known (cf. e.g. Migdal, 1977) V (x) = Bu−1 (He−1 + (H1 H2 )−1 r∂xx that in the case of potential well this equation has both propagating solutions corresponding to the continuous spectrum ω2 1 and localized, trapped in the well, solutions corresponding to the discrete spectrum Min(V (x))He < ω2 < 1. The potential well corresponds to the region of anticyclonic shear and the related trapped modes will be localized there, oscillating at sub-inertial frequencies. It is worth noting that small Burger numbers makes trapping easier, cf. Chapter 2 of this volume. Furthermore, if the potential is deep enough (i.e. the anticyclonic shear is strong enough) modes with ω2 < 0, i.e. non-oscillatory unstable modes, are possible thus giving rise to the symmetric instability, which bears this name precisely due to its one-dimensional, independent of the along-jet direction character. Whether this instability is saturated by interactions with e.g. continuous spectrum (in this case a secondary source of wave emissions during adjustment would appear), or leads to some sort of breaking inside the jet is an open problem. Some hints to its solution may be provided by the Lagrangian approach to the 2L1dRSW presented below. 3.3.2. The loss of hyperbolicity and Kelvin–Helmholtz instability In order to study wave-breaking and shock formation in the 2L1dRSW model it is useful, following Le Sommer, Medvedev, Plougonven and Zeitlin (2003), to rewrite the system (3.1a)–(3.1f) in the canonical hyperbolic form. In doing this we use (3.8) and (3.9) and reduce the system to four equations for four independent variables u2 , h2 , v2 and v1 , i.e. lower(heavier)-layer variables plus upper-layer jet velocity: ∂u2 ∂u2 ρ1 + u2 − f v2 + f (h1 v1 + h2 v2 ) ∂t ∂x ρ2 h1 + ρ1 h2 gρ ∂h2 ∂ − (3.23) h1 h1 u21 + h2 u22 + = 0, ∂x ρ1 ∂x ∂h2 ∂u2 ∂h2 + u2 + h2 = 0, (3.24) ∂t ∂x ∂x ∂v2 ∂v2 + u2 + f u2 = 0, (3.25) ∂t ∂x ∂v1 ∂v1 ∂v1 + u2 + (u1 − u2 ) + f u1 = 0, (3.26) ∂t ∂x ∂x where u1 =

h2 u2 , h2 − H

h1 = H − h2 .

(3.27)

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Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

As in the one-layer case, useful insights may be gained rewriting these equations in terms of the mass Lagrangian variable a (h2 = ∂a/∂x): ∂ ρ1 du2 − f v2 + h1 u21 + h2 u22 f (h1 v1 + h2 v2 ) − h2 dt ρ2 h1 + ρ1 h2 ∂a ∂h2 gρ h1 h2 = 0, + (3.28) ρ1 ∂a dh2 ∂u2 (3.29) + h22 = 0, dt ∂a dv2 (3.30) + f u2 = 0, dt dv1 ∂v1 (3.31) + (u1 − u2 )h2 + f u1 = 0. dt ∂a This system may be rewritten in the matrix form ⎛ ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ 0 0 0 0 −f u2 v2 v2 d ⎜ v1 ⎟ ⎜ 0 T 0 0 ⎟ ∂ ⎜ v1 ⎟ ⎜ −f u1 ⎟ ⎝ ⎠+⎝ ⎝ ⎠=⎝ ⎠, 2 ⎠ 0 0 0 h h h 0 dt ∂a 2 2 2 ρ2 v2 −ρ1 v1 u2 0 0 M 2N u2 f h1 ρ2 h1 +ρ1 h2 (3.32) where we introduce, for compactness, the following auxiliary variables: H u2 T = (u1 − u2 )h2 = h2 , H − h2 H 2 u22 ρ1 h2 ρ M= (H − h2 ) − g , ρ2 h1 + ρ1 h2 ρ1 (H − h2 )2 H h2 ρ1 h2 u2 . N =− ρ2 h1 + ρ1 h2 H − h2 In order to reduce (3.32) to the canonical for the hyperbolic systems form (cf. e.g. Whitham, 1974), the matrix in the l.h.s. of (3.32) should be diagonalized and the left eigenvectors found. The equation for the nontrivial eigenvalues (the trivial ones are 0 and T ) is −λ h22 det (3.33) = λ2 − 2N λ − h22 M = 0 M 2N − λ and its solution is λ± = N ± N 2 + Mh22 . The discriminant of this equation is H 2 u22 ρ1 ρ2 h1 h32 h2 h1 + gρ − . D= ρ1 ρ2 (ρ2 h1 + ρ1 h2 )2 h21

(3.34)

(3.35)

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The eigenvalues (3.33) are, thus, real (and, hence, the system is hyperbolic) only when D is positive. Therefore, if h1 h2 2 (u2 − u1 ) > gρ (3.36) + ρ1 ρ2 then D < 0 and the system loses hyperbolicity. We, thus, see that, unlike its onelayer counterpart, the 2L1dRSW model changes type if the vertical shear of the velocity transverse to the front is sufficiently strong. One may recognize in (3.36) the condition for Kelvin–Helmholtz (KH) instability. The KH instability is known to produce breaking of the growing interface wave with the subsequent formation of the KH billows (cf. e.g. Gossard and Hooke, 1975). This expected singularity is of a different nature as compared to shock formation which should be also present in the hyperbolic system (3.32). The left eigenvectors l(α) , α = 1, 2, 3, 4, corresponding to the eigenvalues λ(α) = (0, T , λ± ) may be easily found. They are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, M, λ± ) and (3.32) may be rewritten in the canonical form, required for the application of numerical Riemann solvers: ∂ ∂ (3.37) U + λ(α) l(α) · U = l(α) · B, ∂t ∂x where U, B denote the 4-vector of dynamical variables and the 4-vector in the r.h.s. of (3.32), respectively. However, finding explicit expressions for Riemann invariants r(α) ; dr(α) = l(α) · U is a non-trivial technical task. Its complexity prohibits the analytical study of the shock formation along the lines of the previous section, although we have all reasons to expect the existence of initial configurations leading to shock formation in the 2-layer model. l(α) ·

3.4. Lagrangian approach to 2L1dRSW and symmetric instability Thus, the 2L1dRSW model displays a number of qualitatively new phenomena. As in the 1dRSW one, sufficient conditions for existence and uniqueness of the adjusted state are established in terms of the initial distributions of PV in both layers. They prove to be more restrictive than in 1dRSW and the question of adjustment in situations not covered by these criteria remains open. The configurations with strong anticyclonic shears corresponding to negative PV are of particular interest, as we showed that they may support trapped states which render adjustment incomplete. For strong enough anticyclonic anomalies unstable localized states are possible, whose evolution will complicate considerably the adjustment scenario. For strong enough transverse velocity shears the developing KH instability will make the adjustment deviate from the standard scenario, too. In the prospect of investigating the fully nonlinear adjustment in the model including the nonlinear stage of the instabilities, especially the symmetric one

292

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

which is rather poorly understood, the Lagrangian approach similar to that developed for 1dRSW is useful. We start from the system (3.23)–(3.27), taken for simplicity in the frequently used in oceanography limit r → 1, and introduce the Lagrangian coordinate X(x, t) corresponding to the positions of the fluid particles in the lower layer. In terms of displacements φ with respect to initial positions ∂ X(x, t) = x+φ(x, t). The corresponding Lagrangian derivative is dtd = ∂t∂ +u2 ∂x and will be denoted by the overdot, as usual. The dependence of the height variable h2 on the Lagrangian labels, and transformation of its derivatives are obtained via the mass-conservation in the lower layer: h2I dx = h(X(x, t), t) dX, where the index I means initial value, as usual. Again, as in the one-layer case, equation (3.25) expresses the conservation of the geostrophic momentum in the lower layer, and allows to eliminate v2 in terms of its initial value and φ: v2 (x, t) + f φ(x, t) = v2I (x).

(3.38)

For simplicity, we will consider the particular case of the barotropic initial flow, which was already treated above to illustrate the mechanism of symmetric instability. Hence h2I = H2 = const, v2I = v1I = vI (x) and, by introducing the H2 1 shorthand notation α1 = H H0 , α2 = H0 , α1 + α2 = 1 we obtain: φ˙ 2 α2 α2 α1 + φ 2 ¨ (v1 − vI ) − · φ+f 1− φ+f 1+φ 1+φ α1 + φ 1 + φ α 1 − g H2 φ = 0, (3.39) (1 + φ )4 φ˙ α2 v1 − v˙1 − (3.40) φ˙ = 0, α1 + φ α1 + φ where prime denotes x-differentiation, as usual, and g is the reduced gravity. ˙ The equations are to be solved with initial conditions φ(x, 0) = 0, φ(x, 0) = u2I , v1 (x, 0) = vI . These equations are, certainly, more complicated than the corresponding onelayer equation for φ, but still may be solved by standard package solvers. For instance, the nonlinear stage of symmetric instability of the barotropic jet may be studied with the help of (3.39), (3.40) by imposing a profile of vI (x) having strong enough anticyclonic vorticity. It is easy to see that for vI (x) satisfying (3.13a) the system (3.39), (3.40) in the linear approximation reproduces equation (3.20) at r → 1. 3.5. Stationary waves of finite amplitude in the 2-layer model In spite of various singularities which lead to wave breaking in the 2L1dRSW model, as in the one-layer 1dRSW model there still exist exact solutions in the form of propagating finite-amplitude waves (Plougonven and Zeitlin, 2003),

3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW

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where nonlinearity and dispersion due to rotation are compensated. These waves, again, are fully-nonlinear counterparts of the linear inertia–gravity waves on the interface between the layers. They also carry no anomaly of potential vorticity, and give standard linear waves in the limit of small amplitudes. However, their variety is richer due to additional parameters (the density ratio and the depth ratio of the layers) present in the model. Three families of finite-amplitude waves thus appear. 3.5.1. General equation for the stationary waves in the two-layer model The stationary wave solutions of (3.1a)–(3.1f) which are functions of ξ = x − ct should verify the following system of ordinary differential equations (the prime denotes the ξ -derivative, as usual): −cu1 + u1 u1 − f v1 + ρ1−1 π = 0,

(3.41a)

−cv1 + u1 (f + v1 ) = 0,

(3.41b)

− f v2 + ρ2−1 π + g h + v2 ) = 0, + u1 (H − h) = 0,

−cu2 + u2 u2 −cv2 + u2 (f −c(H − h)

−ch + (u2 h) = 0.

= 0,

(3.41c) (3.41d) (3.41e) (3.41f)

We use here the height of the lower layer h (the absolute interface position) as a dependent variable. The last two equations can be directly integrated: u1 = c +

C1 , H −h

u2 = c +

C2 , h

(3.42)

where C1 and C2 are the integration constants. The zero-flux condition u1 (H − h) + u2 h = 0 yields: C1 + C2 = −cH.

(3.43)

The expressions (3.42) are then inserted into (3.41a) and (3.41c), and v1 and v2 are eliminated using (3.41b) and (3.41d). The following single equation for h follows: 2

2 r r C1 1 1 C2 2 − + g h + cf + h 2 h 2 H −h C2 C1 cH +f 2 1 − r − r (3.44) = 0, C1 where r = ρ1 /ρ2 is the ratio of the densities of the layers. By integrating (3.44) over one period of the supposed periodic solution, and by using the fact that the

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

294

integral of the interface position h over the period is H2 by definition, the following constraint results: cH2 cH1 1+ (3.45) −r 1+ =0 C2 C1 which, together with (3.43), allows to fix the values of C1 and C2 : C1 = −cH1 , C2 = −cH2 , and to determine u1 and u2 : u1 = c

−(h − H2 ) , H −h

u2 = c

h − H2 . h

Equation (3.44) then becomes:

2 2 2 c H1 f2 H2 −r (h − H2 ) = 0, + gh − 2 h H −h He

(3.46)

(3.47)

1 H2 is introduced. As in the one-layer case, where the equivalent depth He = HH1 +rH 2 the PV-anomaly in each layer is zero, as may be easily seen by using (3.41b), (3.41d) and (3.46):

q1 =

f , H1

q2 =

f . H2

(3.48)

Introducing the non-dimensional variable χ = h/H , and the notation γ = H2 /H , M = c/ g He , ϕ(r, γ ) = He /H = (1 − γ )γ /(rγ + 1 − γ ) equation (3.47) may be rewritten in the form:

2 M ϕ(r, γ ) γ 2 (1 − γ )2 f2 (3.49) − r − (χ − γ ) = 0. + χ 2 χ2 (1 − χ)2 ce2 This equation is similar to (2.63), but the first term here have singularities both at χ → 0 and χ → 1, i.e. when the interface approaches each of the boundaries. It is easy to check that in the limit H1 → ∞, equation (3.47) takes the form of the equation for nonlinear waves in the one-layer model with the replacements H → H2 , g → g . It can also be seen that the linearization of (3.47) for χ close to γ , η = χ − γ → 0 gives g He − c2 η − f 2 η = 0, (3.50) with the dispersion relation for linear internal inertia–gravity waves at the interface (3.7) which readily follows. After multiplication by the first derivative of the expression in the square brackets, (3.49) may be integrated once and gives: χ2 +

f2 V2 (χ; M, r, γ , A) = 0 ce2

(3.51)

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where 1−γ (χ − γ )2 [ϕM 2 ( χγ2 + r (1−χ) 2 ) − 1] − A N2 = , V2 (χ; M, r, γ , A) = 2 2 2 γ (1−γ ) (D2 ) (ϕM 2 ( χ 3 + r (1−χ)3 ) − 1)2

(3.52) is the effective potential, playing the same role as V in the one-layer case. Only the values of χ between 0 and 1 are physically meaningful. As in the one-layer case A > 0 and M > 1. 3.5.2. Analysis of the effective potential The structure of the effective potential is similar to that of the potential V in the one-layer case. However, there are two additional parameters, the ratios of the heights and densities of the two layers, respectively γ and r, and the numerator and denominator now diverge at two points: χ → 0 and χ → 1. In order to have stationary waves, a well in the potential is required about the rest state χ = γ . The following properties of the effective potential may be established (Plougonven and Zeitlin, 2003): The denominator (D2 )2 is always positive; the sign of the potential is given by the sign of the numerator. The numerator N2 tends to +∞ for χ → 0 and for χ → 1. Furthermore, for χ → γ N2 (χ, M, A, r, γ ) ∼ (χ − γ )2 M 2 − 1 − A, (3.53) with A > 0. Hence, N2 has at least one zero in each interval ]0, γ [ and ]γ , 1[. Only the zeros closest to γ , χl and χr , respectively, matter. The possibility of having waves in the system depends on the behavior of the denominator: in order to have wave solutions it is sufficient that the denominator does not have zeros in the interval [χl , χr ]. Note that the derivative of N2 with respect to χ is equal to 2(χ − γ )D2 . Therefore, as in the one-layer case, local extrema of the numerator other than χ = γ coincide with the zeros of D2 . The square root of the denominator, D2 , tends to +∞ at χ → 0 and χ → 1, and has a single minimum in between. Elementary analysis shows that the minimum corresponds to χm (r, γ ) =

1 1/2 1 + r 1/4 ( 1−γ γ )

.

(3.54)

Note that the location of this minimum is a function of parameters r and γ only. If D2 (χm ; M, r, γ ) > 0, the denominator never vanishes in ]0, 1[, and hence wave solutions always exist, whatever the value of A. If D2 (χm ; M, r, γ ) = 0, the denominator vanishes only at the point χm , while if D2 (χm ; M, r, γ ) < 0 it has two zeros, one on each side of χm . It is then necessary to investigate the location of roots of the numerator relative to those of the denominator.

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The existence (non-existence) of wave solutions follows from the existence (non-existence) of zeros in the denominator. The fact that the denominator at the rest position χ = γ is equal to (M 2 −1)2 > 0 is crucial. Two important properties of the denominator are: (1) For any given value of r, there always exists a value of γ such that the denominator never vanishes: this value γ0 (r) is given by χm (r, γ0 ) = γ0 . In this case stationary wave solutions exist for any A. For example, for layers of close densities waves exist for any A at least for the values of γ close to 0.5. (2) If D2 has two zeros, both are situated either in the interval ]0, γ [ or in the interval ]γ , 1[. More precisely, if γ < γ0 (r), both are in ]γ , 1[, and if γ > γ0 (r), both are in ]0, γ [. The location of zeros of D2 determines the orientation of the cusps at the wave extrema. For γ greater than γ0 (r) (or equivalently χm (r, γ ) < γ ), cusps are downward-oriented, for γ below γ0 (r), cusps are upward-oriented. 3.5.3. Wave families and transitions between them with the change of parameters The investigation of different configurations of zeros of the numerator and denominator in (3.52) is given in Plougonven and Zeitlin (2003). Three families of finite-amplitude wave solution are possible: 3.5.3.1. Family A: D2 (χm (r, γ ); M, r, γ ) 0 and γ < γ0 (r). The corresponding waves behave qualitatively in the same manner as waves of the one-layer RSW: there exists a limiting amplitude, the wave profiles tend to develop cusps at their crests as the amplitude approaches the critical one. The typical form and the response of the waves of this family to the increase of amplitude is shown in Figure 12. The cusp is an asymptotic envelope and the solutions are continuous with their derivatives. Qualitatively, waves of the family A arise when the depth of the upper level is considerably larger than the depth of the lower layer (what “considerably” precisely means depends on the values of other parameters). 3.5.3.2. Family B: D2 (χm (r, γ ); M, r, γ ) > 0. In the absence of zeros in the denominator, the potential always has a well around χ = γ , whatever the value of A. Hence, formally, there is no limiting amplitude for the waves other than that imposed by the boundaries. Profiles of the waves of this family for given values of M and r, and for various amplitudes are shown in Figure 13. Note the sharpening of the gradients of the interface displacement with increasing amplitude (a kinklike structure). 3.5.3.3. Family C: D2 (χm (r, γ ); M, r, γ ) 0 and γ > γ0 (r). Qualitatively, these waves are the upside-down version of the waves of the family A. They

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Figure 12. Waves of the family A and their response to the increase of amplitude (from top to bottom). Panel d) shows a wave approaching the limiting amplitude. The values of parameters are H1 = 1, H2 = 2.8, r = 0.1 and M = 2 (Plougonven and Zeitlin, 2003).

are limited by a critical amplitude and tend to form cusps at the troughs as their amplitude increases. They correspond to the situation when the depth of the lower layer is considerably larger than the depth of the upper one. For a given value of r, the domains corresponding to the families A, B, and C in the parameter space (M, γ ) may be obtained by plotting the curve corresponding to the zero level of D2 (χm ; M, r, γ ) as shown in Figure 14. Crossing this curve corresponds to a bifurcation. For example, for given values of γ and r, a transition from the family A to the family B is produced by increasing the phase speed, i.e. the value of M, cf. Figures 15 and 12.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide Due to the absence of the constant part of the Coriolis parameter, the only true geostrophically balanced states at the equator are mean zonal flows. Hence, the geostrophic adjustment of any perturbation localized in the zonal (x-) direction will be wave-dominated, the net result of it being the transformation of the initial perturbation into packets of waves of different nature. Increasing the amplitude of initial perturbation will result in essentially nonlinear wave phenomena. The equatorial waves exhibit a variety of dispersion properties, and nonlinearity will manifest itself differently for, e.g. strongly dispersive and weakly dispersive

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Figure 13. Waves of the family B and their response to the increase of amplitude (from top to bottom): these waves ‘feel’ the presence of the rigid lid and are bounded only by the lower and upper boundaries. Parameters are: H1 = 1, H2 = 1, r = 0.9 and M = 2 (Plougonven and Zeitlin, 2003).

waves. We show below that different kinds of waves display different patterns of breaking behavior, where breaking is understood as irreversible changes in the flow due to propagation of a finite-amplitude nonlinear wave. As in the previous sections, in spite of the fact that resonant triadic interactions of weakly nonlinear equatorial waves produce a number of interesting phenomena (Majda, Rosales, Tabak and Turner, 1999; Majda and Biello, 2003; Medvedev and Zeitlin, 2005; Reznik and Zeitlin, 2006) we will limit ourselves below only by strongly nonlinear phenomena. 4.1. Equatorial waves and geostrophic adjustment in the equatorial waveguide As shown in Chapter 1 of the present volume, the equatorial Kelvin waves are dispersionless, the long Rossby waves are weakly dispersive, while the short Rossby waves are strongly dispersive, the short inertia–gravity are almost dispersionless, and eastward-propagating and long westward-propagating Yanai waves are weakly dispersive (cf. Figure 11 of Chapter 1). This variety of dispersion properties indicates that different patterns of behavior may be expected for non-

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Figure 14. Parameter regimes on the (M, γ )-plane corresponding to the waves of family A (bottom region), B (center) and C (top region), for a) r = 0.9 and b) r = 0.1. As the difference between the densities of the two layers increases, the domain in parameter space corresponding to family C shrinks.

linear equatorial waves of different nature. For example, breaking and shock formation can be expected for nonlinear Kelvin waves (Boyd, 1980a; Ripa, 1982). Soliton formation can be expected for long Rossby waves (Boyd, 1980b; Ripa, 1982). Breaking can be expected for short enough inertia–gravity waves, and for short eastward-propagating Yanai waves. Already this first guess indicates that shock formation should be rather common in the dynamics of nonlinear waves in the equatorial RSW model. Moreover, the argument of Section 2 showing that PV distribution cannot be changed by one-dimensional shock is no more applicable at the equator, as the waves are not one-dimensional anymore. So one can expect nontrivial phenomena produced by the PV “deposits” by shocks. Illustrations of nonlinear wave phenomena in the equatorial waveguide are given in the next section. To give a synthetic idea of various nonlinear phenomena in the equatorial waveguide, an example of fully nonlinear geostrophic adjustment of a long-wave initial perturbation is given in Figure 16. The spectral gap in the equatorial wave spectrum, which was discussed in Chapter 1 means that slow (Kelvin and Rossby waves) and fast (inertia–gravity) parts of motion are dynamically well-separated (cf. Le Sommer, Reznik and Zeitlin, 2004) and (almost,

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Figure 15. Nonlinear waves of family B for the same values of the model parameters as in Figure 12, except for a slightly higher phase speed (M = 2.1 instead of M = 2), which corresponds to a longer wavelength. The amplitude of the wave on panel a) is deliberately chosen close to that of Figure 12c). The response of the wave to the increase of amplitude, panels b)–e), differs significantly from that on Figure 12. In particular, there is no cusp formation anymore (Plougonven and Zeitlin, 2003).

see below) non-interacting. The antisymmetric Yanai wave signal does not appear for symmetric initial perturbations. In addition, Rossby and Kelvin wave-packets travelling in the opposite directions do not interact (see also below). 4.2. Wave-breaking and shock formation by equatorial waves 4.2.1. Kelvin-wave breaking Shock formation is straightforward in the Kelvin-wave sector, as nothing can prevent breaking in the absence of dispersion. This phenomenon was pinpointed in the early works by Boyd (1980a), and Ripa (1982), and studied numerically by Fedorov and Melville (2000). We give here results of high-resolution numerical simulations following Le Sommer, Reznik and Zeitlin (2004) which used the method exposed in Chapter 4 of the present volume. The evolution of the Kelvin-wave packets corresponding to positive and negative height anomalies, respectively, is presented in Figures 17 and 18. Time (T ) and length (Re ) units in all

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Figure 16. Adjustment of a localized large-scale height anomaly symmetric with respect to equator. Rossby number is 0.3. The Rossby-wave part of initial perturbation is moving west forming a dipolar soliton, the Kelvin-wave part is moving east and breaks down in finite time forming a shock. A packet of inertia–gravity waves centered at the location of the initial perturbation’s maximum (x = 30) is slowly dispersed.

simulation were chosen according to the equatorial scaling explained in Chapter 1, Section 6. The mechanism of the wake formation, observed in Figure 17 was explained in Fedorov and Melville (2000) by a direct resonance of the nonlinear Kelvin wave and inertia–gravity waves. Indeed, due to nonlinearity, the phasespeed of positive-anomaly Kelvin wave is larger than that of linear waves. Hence the effective slope of the dispersive curve for nonlinear waves is greater than for linear waves, and it can intersect the dispersion curves of inertia–gravity waves (see the dispersion curves in Figure 11 of Chapter 1). On the contrary, negativeanomaly nonlinear waves are slower than linear ones, and the effect cannot take place. As seen from these figures, the Kelvin shock-front is curved, and hence one may expect a redistribution of PV due to the shock formation. The idea of a dissipative vorticity change due to discontinuities in gas dynamics has been studied by Hayes (1957) and Berndt (1966). More recently this analysis was generalized by Kevlahan (1997). For the shallow-water equations, a potential vorticity jump

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Figure 17. Evolution of a strongly nonlinear Kelvin wave with positive height anomaly (initial Rossby number 0.3). The wave breaks down in finite time (∼10T ) forming a shock. Inertia–gravity wave emission starts at this moment behind the front and continues on.

formula was derived by Pratt (1983) in the case of straight shocks moving with a constant speed. In all of these works the along-shock momentum equation was written in a local reference frame in order to calculate the vorticity jump. Following this prescription we consider a shock (S) propagating through a material volume V (t) and we introduce a local reference frame with normal and tangential unit vectors ( n, s) with respect to the shock. This frame is moving at the local speed of the shock Cn n. The velocity and Bernoulli function in the moving frame are: (n) (n) u − Cn u¯ = gh + u¯ 2 /2. = , B u¯ = (4.1) u¯ (s) loc u(s) loc With a notation [A] = Afront − Arear the Rankine–Hugoniot conditions are: −Cn [h] + hu(n) = 0,

(n) 1 2 (n)2 −Cn hu + gh = 0, + hu 2

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Figure 18. Evolution of a strongly nonlinear Kelvin wave with negative height anomaly (initial Rossby number −0.3). The wave breaks down in finite time (∼10T ) forming a shock. No inertia–gravity wave emission is observed.

−Cn hu(s) + hu(n) u(s) = 0,

(4.2)

where u(n,s) are the velocity components normal and tangential with respect to the is obtained from the first and the second of the Rankine– shock. The jump in B Hugoniot conditions (4.2); see e.g. Lighthill (1978), where the subscript f (r) denotes the front (the rear) state: 3 = − g [h] . B 4 hf hr

(4.3)

Following the aforementioned works, we obtain the potential vorticity jump across the shock from the along-front momentum equation: . hu¯ (n) [q] = −∂s B (4.4) For moving shocks of any shape: [q] =

g [h]3 ∂ . s 4hu¯ (n) hf hr

(4.5)

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

304

Figure 19.

A sketch of a shock front which propagates in a material volume V at local speed Cn n, and the local reference frame ( n, s).

Note that for shocks with ∂s [h] no change of potential vorticity is possible. The rate of change of the amount of vorticity contained in any material volume V may be, thus, calculated: d − B . hq dV = − hu[q] ¯ dS = B (4.6) A B dt V

(S)

Figure 20 taken from Le Sommer, Reznik and Zeitlin (2004) shows the breaking of a Kelvin wave of depression and the subsequent PV-deposit. Choosing V to be the whole southern half-plane, the formula (4.6) suggests that the overall negative potential vorticity will arise there. The right panel of the figure confirms this result. It should be emphasized that such dipolar deposit of PV results in the corresponding zonal jet behind the shock. A theory of vorticity transport due to breaking waves in rotating shallow water on the f -plane was developed by Bühler (2000). It was based on generalized Lagrangian-mean theory (GLM) and confirmed by numerical simulations. However, its generalization to fully nonlinear processes on the (equatorial) β-plane is difficult due to the problem of extending the standard GLM pseudomomentum definition beyond the f -plane (cf. Bühler and McIntyre, 1998). 4.2.2. Yanai and inertia–gravity waves breaking Direct numerical simulations with finite-amplitude sinusoidal waves of a given family taken as initial conditions confirm the qualitative predictions made at the beginning of this subsection. Thus, eastward propagating Yanai wave, as well as eastward and westward propagating inertia–gravity waves break and form specific

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 20.

305

Potential vorticity deposit in the wake of a broken Kelvin wave at Ro = 0.6. Left panel: v −u +βy

the height field. Right panel represents the PV-anomaly q = x Hy the interval (−0.2, 0.2) are shown.

βy − H . The isolines of q in 0

patterns of shocks displayed in Figures 21, 22, 23 taken from Bouchut, Le Sommer and Zeitlin (2005). The shock zone is defined by the enhanced dissipation and grey-shadowed in the figures. In these and subsequent figures the height anomaly isolines (solid for positive and dashed for negative anomalies), and velocity field (arrows) are shown. The measure of nonlinearity is the Rossby number ε. It should be emphasized that breaking and shock formation are not limited to spatially periodic waves. They are ubiquitous at already moderate nonlinearities and appear during the evolution of localized wave-packets as well, as shown in Figure 24 where a snapshot of the evolution of a packet of eastward-propagating inertia–gravity waves with a non-dimensional amplitude ε = 0.3 is presented. Shocks are formed and propagate across the whole wave pattern. 4.3. Solitons and modons formed by equatorial Rossby waves The soliton formation by long equatorial Rossby waves was predicted by Boyd (1980b), and Ripa (1982). Boyd (1985) has also predicted the existence of equa-

306

Figure 21.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Finite-amplitude eastward-propagating Yanai wave—upper panel, which breaks and forms an asymmetric chevron-type shock pattern—lower panel.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 22.

307

Finite-amplitude eastward-propagating inertia–gravity wave—upper panel, which breaks and forms a symmetric chevron-type shock pattern—lower panel.

308

Figure 23.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Finite-amplitude westward-propagating inertia–gravity wave—upper panel, which breaks and forms a horseshoe-type shock pattern—lower panel.

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 24.

309

Snapshot of an eastward-propagating inertia–gravity wave packet with k0 = 1.0/Rd at time t = 4.0. The shock-front area is shaded.

torial modons, the finite-amplitude Rossby solitons. The analytic description of adjustment of the long-wave equatorial perturbations given in Chapter 2 of this volume (see also Le Sommer, Reznik and Zeitlin, 2004) shows that the Rossby-wave part of the response to a localized perturbation is described by the Korteweg–de Vries (KdV) equation and, thus, leads to appearance of the solitons. High-resolution numerical simulations confirm that a dipolar initial perturbation forms a typical soliton sequence consistent with the predictions of the KdV theory, Figure 25. Note that, in full agreement with the KdV theory results, the soliton tail is formed only for positive initial height anomalies. Negative height anomalies disperse and do not form coherent structures (Bouchut, Le Sommer and Zeitlin, 2005). It should be emphasized that such typically KdV behaviour holds for strong nonlinearities here, in spite of the fact that the KdV theory is formally valid for weak nonlinearities only. What is more surprising, a build-up in amplitude for the leading soliton with respect to the initial perturbation is observed. As shown below, at such amplitudes the solitons become modons, i.e. closed streamlines are formed in the dipole core and, therefore, a trapping of material within the vor-

310

Figure 25.

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Long-time evolution of a nonlinear dipolar perturbation. Initial perturbation forms a sequence of modons of diminishing amplitude and phase-speed.

tex core, and subsequent anomalous transport are taking place—see also the next subsection. The transition between solitons and modons may be demonstrated by considering nonlinear equatorial Rossby waves in a fluid layer of varying depth. Let us consider, following Le Sommer and Zeitlin (2005), and Bouchut, Le Sommer and Zeitlin (2005) the evolution of an equatorial Rossby soliton in a fluid layer with for x > 0 smoothly decreasing thickness. We take the linear slope between h = H for x −20. The initial condition is a zeroth order Boyd (1980b) and h = 0.4H soliton centered at x = 15 with B = 0.25. In order to capture eventual fluid particle trapping in the soliton core, the evolution of the passive tracer field is calculated during numerical simulation together with the fluid velocity and elevation. The tracer field is taken to be complex (i.e. there are, in fact, two real tracer fields ar and ai ), and initialized in order that initial tracer contours form a rectangular grid easy to follow in the subsequent evolution: a(x, y, 0) = ar + iai = x + iy. The evolution of the height (solid contours), velocity (arrows) and passive tracer (gray contours) is shown in Figure 26. The soliton is initially propagating westward, as expected. At this stage, the tracer field is advected reversibly. Then the

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Propagation of a solitary Rossby wave on a sloping topography and related Lagrangian tracer transport.

311

Figure 26.

312

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

soliton enters the sloping topography region. Both its phase speed and meridional extent are weakening as may be expected from the WKB analysis (Yang and Yu, 1992). During this phase a partial reflection in a form of low-amplitude Kelvin wave takes place. (It is below the threshold of the displayed h-perturbation and not visible in the figure.) The tracer contours are getting more elongated as the wave propagates. Finally the wave enters the left flat region and keeps a quasisteady shape shown at time t = 150. The soliton is now enclosing a recirculation region. It has been transformed into a modon. The details of the mass-trapping process are visible. Contrary to the intuitive expectations, mass (passive tracer) trapping does not occur at the maxima of h around which velocity vectors are circling due to the geostrophic balance, but rather in the vicinity of the saddle point at the equator where h has a local minimum. Thus, when a solitary Rossby wave propagates over a sloping topography, it “breaks” and forms a modon which encloses a recirculation region. This recirculation region is trapped within the wave pattern so that the underlying particle distribution is irreversibly modified. This allows to qualify such process as breaking, where breaking is understood as irreversible changes of the flow due to nonlinearity. Such breaking is of a totally different nature than shock formation observed above for Kelvin, inertia–gravity and Yanai waves. The difference between the Rossby waves and the other types of equatorial waves is that Rossby waves are essentially balanced, i.e. their meridional velocity is in geostrophic balance with the height field, while other types of waves (except for short westward-moving Yanai waves) are essentially unbalanced, as shown in Bouchut, Le Sommer and Zeitlin (2005). Hence, balanced and unbalanced waves display qualitatively different patterns of breaking. 4.3.1. Elastic scattering of Rossby solitons and Kelvin fronts Perhaps the most striking phenomenon observed in the numerical simulations of Bouchut, Le Sommer and Zeitlin (2005) was quasi-elastic scattering of Rossby solitons and Kelvin fronts showing the robustness of the patterns emerging from breaking equatorial waves. The initial condition is taken as a superposition of two distinct patterns. A Kelvin wave-packet with Gaussian zonal structure is centered at x = 10.0. Its zonal extent is Lx = 3Rd and its non-dimensional amplitude is ε = −0.3. A Kelvin wave-packet with negative height anomaly was deliberately chosen in order to have no inertia–gravity waves tail appearing for positive height anomalies (cf. Figure 17). A solitary Rossby wave was initialized at x = 45.0 with amplitude ε ∼ 0.18. Snapshots of the evolution of this configuration are shown on Figure 27. At time t = 15.0, a shock front has formed at the rear of the Kelvin wave packet as in Figure 18. Dissipation in the shock induces a slow decrease of the Kelvin wave amplitude as it is propagating eastward. On the other hand, the solitary Rossby wave is moving westward with no change of shape. At time t = 30.0, the two wave patterns collide but the interaction is quasi-elastic, as seen

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

Figure 27.

313

Interaction between a Kelvin wave and a Rossby solitary wave. See the text for explanation.

at time t = 45.0. The elastic character of collision can be checked by comparing the results at time t = 45.0 of two different runs respectively with and without the initial Kelvin wave (or with and without initial Rossby wave). The comparisons show that results are practically identical. Such almost-non-interaction between Kelvin and Rossby wave was observed for a wide range of parameters. The only observed effect was a slight phase-lag in the solitary Rossby wave propagation for high enough amplitudes ε > 0.5. 4.3.2. Interaction of Rossby solitons with inertia–gravity waves In order to test the interactions of Rossby solitons and inertia–gravity waves the evolution of a “soliton” with non-dimensional amplitude ∼0.3 arising in the adjustment process of the type shown in Figure 16 was followed in Le Sommer, Reznik and Zeitlin (2004). A typical evolution of the height field of this part of the perturbation is presented in Figure 28. A field of inertia–gravity waves of weak intensity is seen in front of the soliton, while a (mostly Rossby-wave)

314

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 28. Inertia–gravity waves and a Rossby-wave soliton with Ro ∼ 0.3. Upper panel—the height field with isolines h = 0.99, 1.01, 1.1, 1.2, 1.3 shown. Lower panel: a section of the height field along the equator. Inertia–gravity waves in front of the westward-propagating soliton are clearly seen. Behind the soliton there is a wake of weak Rossby waves

wake is formed behind. Although the modons or “solitons” like the one presented on Figure 28 are quite stationary, this simulation suggests that they emit or capture inertia–gravity waves. The precise origin of the observed IGW cannot be determined at this stage. Several mechanisms may be at work, and to distinguish among them further investigations are necessary. As possible explanations we should mention a direct resonance of the soliton with the west-moving inertia– gravity waves (like in the Kelvin front case; however the time-evolution of the field presented in Figure 28 suggests that it is the group velocity of the inertia– gravity wave packet which coincides with the speed of the Rossby solitons, while phase velocity of the inertia–gravity waves is higher), a non-local resonant triad interaction of the type described by Ripa (1982), and a synchronization of IGW produced by adjustment of the initial perturbation while they pass through the modon. 4.4. Transport and mixing phenomena during the equatorial geostrophic adjustment The mixing and transport properties by nonlinear equatorial waves formed during the adjustment process may be studied by the same method which led

4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide

315

to the identification of mass-trapping by the equatorial modons (Le Sommer and Zeitlin, 2005). The passive tracer behaviour during the adjustment of localized mass anomalies and current anomalies at the equator may be thus obtained. 4.4.1. Transport and mixing during adjustment of a symmetric mass anomaly A Gaussian mass anomaly centered at the equator at x = 0 with the amplitude ε = 0.5 is taken as initial condition. Its zonal extent is Lx = 3 and the meridional extent is Ly = 1. The subsequent evolution of this anomaly is illustrated in Figure 29. The initial field is immediately split up into a Kelvin wave, a Rossby wave and a gravity wave packet, as seen at time t = 15. These three components then evolve quasi-independently. The Kelvin wave is propagating eastward. During this propagation its gradients are steepening and a Kelvin front is formed. This wave is rapidly leaving the computation domain and the resulting effect on the tracer field consists in a reversible Stokes drift with the same meridional structure as predicted by Li, Chang and Pacanowski (1996). The Rossby wave is propagating westward. As expected in this parameter regime, it encloses a recirculation region so that the tracer isolines are modified irreversibly during the adjustment, see the snapshot at t = 30. Furthermore, the tracer field never attains a steady structure, which is significantly different from the behavior predicted by the small amplitude Stokes drift analysis. Finally, a gravity wave packet is left at the location of the initial perturbation. Since it is mainly composed of inertia–gravity waves propagating eastward, its effect should be to induce an eastward drift at the equator. This effect is not visible in the figure due to its small amplitude It is worth noting that the Kelvin-wave induced drift and the Rossby wave trapping are acting in the opposite directions so that the tracer at the location of the perturbation is stretched in the x-direction as seen at time t = 45. Another interesting fact is the complicated spatial structure of the isolines of the tracer in the core of the Rossby wave. We can therefore expect a strong mixing to occur in this region if diffusion is added. Hence, the three main transport phenomena due to adjustment are (i) the ballistic transport of the tracer toward the west, (ii) tracer stretching at the perturbation location and (iii) an intense stirring in the Rossby wave pattern. 4.4.2. Transport and mixing during adjustment of a zonal current anomaly A localized zonal current anomaly centered at the equator (produced e.g. by a sudden wind blast) is considered. This situation is typical for the ocean which is submitted to intense zonal winds in the tropics. The initial condition is √ chosen to be a Gaussian zonal velocity patch with maximum velocity u = −0.5 gH . As

316

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 29.

Transport and mixing during adjustment of a mass anomaly centered on the equator. See the text for details.

5. Summary and conclusions

317

in the previous simulations the zonal and meridional extents are set to be Lx = 3 and Ly = 1, respectively. Figure 30 shows the evolution of the dynamical variables and the advection of the tracer field. As previously, the perturbation is split up into a Kelvin wave, a Rossby wave and a gravity wave packet. The Kelvin wave is now an upwelling wave, which signature is a reduction of the depth. Therefore, the gradients are concentrated at its rear front (cf. Le Sommer, Reznik and Zeitlin, 2004, and above). At time t = 15, it induces a reversible westward drift. This drift was not predicted by previous studies which considered only the drift induced by harmonic waves. This fact emphasizes the importance of considering finite amplitude localized wave patterns while studying the transport properties. Again, the Rossby wave is trapping fluid which is then advected westward. As in the previous simulations, the tracer field is stretched along the equator and stirred at the Rossby wave location.

5. Summary and conclusions Two archetype phenomena characterize the essentially nonlinear wave dynamics in the RSW model on the f -plane, where only a single type of waves (IGW) is present. The first is shock formation due to overturning. We have rigorously demonstrated that rotation does not prevent shock formation for large classes of initial data. Shock formation is ubiquitous in the geostrophic adjustment, i.e. initial value problems. It leads to efficient energy evacuation, allowing to reach the adjusted states, whose existence and uniqueness was proved for non-negative PV configurations. Shock formation does not qualitatively change the classical adjustment scenario in the simple 1.5-dimensional configurations. The second is existence of exact nonlinear stationary wave solutions of specific form, and their emergence from periodic signals. These waves were shown to be the finite-amplitude counterparts of the infinitesimal harmonic IGW. Numerical evidence was presented in favor of the hypothesis that they are attractors in the constant PV-sector of the model. New effects are introduced by baroclinicity in the two-layer RSW model. Although existence and uniqueness of the adjusted states may be proved for nonnegative PV configurations with some minor additional constraints in the 1.5d version of the model, the attainability of such states is compromised by appearance of the trapped modes for strong enough horizontal anticyclonic shears. These modes become unstable for negative PV and such symmetric instability may give rise to new kind of breaking. In any case, one cannot expect that negative PVconfigurations in the two-layer case follow the same scenario of adjustment as the positive PV ones, unlike the one-layer case, where this fact was demonstrated by direct numerical simulations. The Kelvin–Helmholtz instability which appears

318

Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water

Figure 30.

Adjustment of a zonal current anomaly centered on the equator. See the text for details.

References

319

for strong enough vertical shears (at positive PV) leads to the loss of hyperbolicity of the system and a new kind of breaking, too. The exact nonlinear wave solutions in the two-layer models appear in form of three different families, two of them (up to reflection) similar to the one-layer ones, and a new one which has a kink-like form with narrow zones of strong gradients of the interface displacement. Beyond the f -plane approximation, the differential rotation effects, appearing in their most spectacular form in the equatorial region, introduce new type of waves, Rossby waves, with specific dispersion which allows to compensate weak nonlinearities for long enough waves through the KdV mechanism, and leads to the soliton formation. Such solitons, however, survive even at severe nonlinearities by becoming modons through the process of “dispersive breaking”, which leads to reconnection of the streamlines and formation of closed recirculation zones moving with the wave. At the same time, weakly dispersive equatorial waves exhibit “standard” shock formation by overturning. This process is most pronounced for specifically equatorial Kelvin waves which are strictly dispersionless. The geometry of the shock pattern depends on the nature of the waves. Shock fronts are, generally, curved and produce recirculation zones, i.e. the PV deposit, behind them. Nonlinear wave phenomena at the equator are accompanied by nontrivial transport and mixing processes: ballistic transport by modons and specific mixing patterns related to the wave-breaking.

Acknowledgements This review is based on the work done in collaboration with F. Bouchut, J. Le Sommer, S. Medvedev, R. Plougonven, and G. Reznik, which is gratefully acknowledged. Special thanks are to J. Le Sommer and R. Plougonven for their help in preparation of the figures.

References Berndt, S.B., 1966. The vorticity jump across flow discontinuity. J. Fluid Mech. 26, 433–436. Bouchut, F., 2004. Nonlinear Stability of Finite Volume Methods for Hyperbolic Systems of Conservation Laws, and Well Balanced Schemes for Sources. Frontiers in Mathematics. Birkhäuser, Basel. Bouchut, F., Le Sommer, J., Zeitlin, V., 2004. Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. Numerical simulations. J. Fluid Mech. 514, 35–63. Bouchut, F., Le Sommer, J., Zeitlin, V., 2005. Breaking of balanced and unbalanced equatorial waves. Chaos 15, 013503-1–013503-19. Boyd, J.P., 1980a. Equatorial solitary waves. Part 1: Rossby solitons. J. Phys. Oceanogr. 10, 1699– 1717. Boyd, J.P., 1980b. The nonlinear equatorial Kelvin wave. J. Phys. Oceanogr. 10, 1–11.

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Boyd, J.P., 1985. Equatorial solitary waves. Part 3: Westward travelling modons. J. Phys. Oceanogr. 15, 46–54. Bühler, O., 1993. A nonlinear wave in rotating shallow water, GFD Summer School preprint (Woods Hole), unpublished. Bühler, O., 2000. On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263. Bühler, O., McIntyre, M.E., 1998. On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343. Engelberg, S., 1996. Formation of singularities in the Euler and Euler–Poisson equations. Physica D 98, 67–74. Falkovich, G.E., Medvedev, S.B., 1992. Europhys. Lett. 19, 279–284. Fedorov, A.V., Melville, W.K., 2000. Kelvin fronts on the equatorial thermocline. J. Phys. Oceanogr. 30, 1692–1705. Gill, A.E., 1982. Atmosphere–Ocean Dynamics. Academic Press, New York (Chapter 7.2). Gossard, E.E., Hooke, W.H., 1975. Waves in the Atmosphere. Elsevier, Amsterdam. Grimshaw, R.H.G., Ostrovsky, L.A., Shrira, V.I., Stepanyants, Yu.A., 1998. Long nonlinear surface and internal gravity waves in a rotating ocean. Surveys in Geophys. 19, 289–338. Hayes, W.D., 1957. The vorticity jump across a gas-dynamic discontinuity. J. Fluid Mech. 2, 595–600. Holton, J.R., 1979. An Introduction to Dynamic Meteorology. Academic Press, San Diego, CA. Houghton, D.D., 1969. Effect of rotation on the formation of hydraulic jumps. J. Geoph. Res. 74, 1351–1360. Kevlahan, N.K.-R., 1997. The vorticity jump across a shock in a non-uniform flow. J. Fluid Mech. 341, 371–384. Kuo, A.C., Polvani, L.M., 1997. Time-dependent fully nonlinear geostrophic adjustment. J. Phys. Oceanogr. 27, 1614–1634. Kuo, A.C., Polvani, L.M., 1999. Wave–vortex interactions in rotating shallow water. Part 1. One space dimension. J. Fluid Mech. 394, 1–27. Landau, L.D., Lifshits, E.M., 1975. Hydrodynamics. Academic Press, San Diego, CA. Lax, P.D., 1973. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia, PA. Le Sommer, J., Medvedev, S.B., Plougonven, R., Zeitlin, V., 2003. Singularity formation during the relaxation of jets and fronts towards the state of geostrophic equilibrium. Commun. Nonlinear Sci. Numer. Simul. 8, 415–442. Le Sommer, J., Reznik, G.M., Zeitlin, V., 2004. Nonlinear geostrophic adjustment of long-wave disturbances in the shallow water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170. Le Sommer, J., Zeitlin, V., 2005. Tracer transport during the geostrophic adjustment in the equatorial ocean. In: Collet, P., et al. (Eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems. In: NATO Science Series C. Kluwer, Amsterdam, pp. 413–429. Li, X., Chang, P., Pacanowski, R.C., 1996. A wave-induced stirring mechanism in the mid-depth equatorial ocean. J. Mar. Res. 54, 487–520. Lighthill, J., 1978. Waves in Fluids. Cambridge Univ. Press, Cambridge, UK. Majda, A.J., Biello, J.A., 2003. The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci. 60, 1809–1821. Majda, A.J., Rosales, R.R., Tabak, E.G., Turner, C.V., 1999. Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances. J. Atmos. Sci. 56, 4118–4133. Medvedev, S.B., Zeitlin, V., 2005. Weak turbulence of short equatorial waves. Phys. Lett. A 342, 217–227. Migdal, A.B., 1977. Qualitative Methods in Quantum Theory. Frontiers in Physics, vol. 48. Benjamin, Reading, MA. Nof, D., 1984. J. Phys. Oceanogr. 14, 1683–1702.

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Ostrovsky, L.A., 1978. Nonlinear internal waves in the rotating ocean. Oceanology 18, 119–124. Peregrine, D.H., 1998. Surf zone currents. Theoret. Comput. Fluid Dyn. 10, 295–309. Plougonven, R., Zeitlin, V., 2003. On finite-amplitude stationary inertia–gravity waves propagating without change of form along the sharp density gradients. Phys. Lett. A 314, 140–149. Pratt, L.J., 1983. On inertial flow over topography. Part 1. Semi-geostrophic adjustment to an obstacle. J. Fluid Mech. 131, 195–218. Pratt, L.J., 1984. On inertial flow over topography. Part 2. Rotating channel flow near the critical speed. J. Fluid Mech. 145, 95–110. Reznik, G.M., Zeitlin, V., 2006. Resonant excitation of baroclinic Rossby waves in the equatorial waveguide and their nonlinear evolution. Phys. Rev. Lett. 96 (3), 034502. Reznik, G.M., Zeitlin, V., Ben Jelloul, M., 2001. Nonlinear theory of geostrophic adjustment. Part I. Rotating shallow water. J. Fluid Mech. 445, 93–120. Ripa, P., 1982. Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial β-plane. J. Phys. Oceanogr. 12, 97–111. Rossby, C.-G., 1938. On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res. 1, 239–263. Rozhdestvenskii, B.L., Janenko, N.N., 1983. Systems of Quasilinear Equations and Their Applications to Gas Dynamics. AMS Translation of Mathematical Monographs, vol. 55. Amer. Math. Soc., Providence, RI. Schär, C., Smith, R.B., 1993. Shallow water flow past isolated topography. Part 2: Transition to vortex shedding. J. Atmos. Sci. 50, 1401–1412. Shrira, V.I., 1981. Propagation of long nonlinear waves in the layer of rotating fluid. Sov. Phys.— Izvestija, Atmos. Ocean Phys. 17 (1), 55–59. Shrira, V.I., 1986. On the long strongly nonlinear waves in rotating ocean. Sov. Phys.—Izvestija, Atmos. Ocean Phys. 22 (4), 298–305. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York. Wolfram, S., 2004. The Mathematica Book. Cambridge Univ. Press, Cambridge, UK. Yang, J., Yu, L., 1992. Propagation of equatorially trapped waves on a sloping thermocline. J. Phys. Oceanogr. 22, 573–582. Zeitlin, V., Medvedev, S.B., Plougonven, R., 2003. Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory. J. Fluid Mech. 481, 269–290.

Chapter 6

Experimental Reality of Geostrophic Adjustment A. Stegner Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France Contents 1. The Holy Graal of rotating shallow-water flows

323 324 329 331 334 339 340 343 350 352 352 358 368 375 376 377

1.1. Single layer f -plane configuration 1.2. Influence of the centrifugal force 1.3. Non-hydrostatic wave modes 1.4. Two-layer stratification

2. Potential vorticity measurements: a new challenge 2.1. Particle image velocimetry and vorticity field measurements 2.2. Height field measurements 2.3. Potential vorticity measurements

3. Simple case studies of geostrophic adjustment 3.1. “Warm-core” lens 3.2. Cyclonic and anticyclonic PV patches 3.3. Uniform PV front

4. What do we learn from laboratory experiments? Acknowledgements References

1. The Holy Graal of rotating shallow-water flows The rotating shallow water model (RSW) is probably the most pedagogical and useful model to understand geophysical fluid dynamics. Even if the RSW equations are based on drastic assumptions (hydrostatic balance, quasibidimensionality, weak dissipation) it is a surprisingly good model of many Edited Series on Advances in Nonlinear Science and Complexity Volume 2 ISSN: 1574-6909 DOI: 10.1016/S1574-6909(06)02006-5 323

© 2007 Elsevier B.V. All rights reserved

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phenomena in the atmosphere and the ocean. Nevertheless, as far as laboratory experiments are concerned, one should keep in mind that the dynamics of a rotating and stratified fluid is given by the full three-dimensional Navier–Stokes equations at the final place. As it was shown in Chapter 1, the RSW equations can be derived from the primitive equations according to an asymptotic expansion which indeed remains valid for some restricted range of dynamical parameters. However, this derivation starts from the hydrostatic and non-dissipative primitive equations (Chapter 1), while both hydrostatic and dissipative effects could play a role in the laboratory. Hence, we recall in this section the derivation of the RSW model from the Navier–Stokes equations. The main purpose is to understand here which dynamical processes are filtered out by the RSW model while they occur sometimes in real experiment. Moreover, we will try to specify the values of the dynamical parameters needed to be achieved in laboratory experiments in order to be close to the RSW dynamics. 1.1. Single layer f -plane configuration Let us consider first a single barotropic and incompressible fluid layer in a rotating tank with a flat bottom and a free upper surface, as shown in Figure 1. In order to get a dimensionless set of equations we use: L and H0 as horizontal and vertical scales, T the characteristic time-scale for the flow evolution, U and U (H0 /L) as horizontal and vertical velocity scales, ρgH0 as the characteristic hydrostatic pressure scale and ρf U L the scale of pressure deviation from hydrostatic balance (f = 2Ω0 the Coriolis parameter). Using this dimensionless formulation, the Navier–Stokes equations can be written as follows: ε∂t u + RoDu − v = −∂x π + Ek u,

(1)

ε∂t v + RoDv + u = −∂y π + Ek v,

(2)

Figure 1. Single water layer on a rotating turntable H0 = 10 cm and D = 90 cm.

1. The Holy Graal of rotating shallow-water flows

α 2 [ε∂t w + RoDw] = −∂z π + α 2 Ek w

325

(3)

where D = u∂x + v∂y + w∂z and = ∂z2 + α 2 (∂x 2 + ∂y 2 ). Besides, in this formulation we decouple the hydrostatic pressure PH corresponding to the fluid at rest and the dynamical pressure π (pressure deviation induced by the fluid motion) according to: Ro (4) π(x, y, z, t) Bu where PH (z) = 1−z+P0 and P0 is the dimensionless pressure at the free-surface. In addition one should consider the continuity equation P = PH (z) +

∂x u + ∂y v + ∂z w = 0

(5)

with upper (z1 = 1 + λη) and lower (z0 = 0) boundary conditions: u(z0 ) = v(z0 ) = w(z0 ) = 0, (6) Ro w(z1 ) = λ ε∂t η + Ro(u∂x η + v∂y η) , (7) λBu π(x, y, z1 , t) = (8) η(x, y, t) Ro where η(x, y, t) is the dimensionless deviation of the free-surface. We have introduced in this formulation the following non-dimensional parameters: √ √ ν • The Ekman number Ek = f H Ek H0 = ν/f 2 fixes the vertical scale δE = of the viscous Ekman layer, where ν is the fluid viscosity. According to the standard boundary layer theory, this viscous layer cannot be neglected at the bottom boundary where the no-slip condition (6) must be satisfied (Gill, 1982; Pedlosky, 1987; Vallis, 2006). In the laboratory, the thickness of this boundary layer is fixed only by the rotation rate Ω0 = 2π/T0 . For typical values of Ω0 1–10 rpm we get δE 1–2 mm. Hence, as far as H0 δE , we usually neglect viscous effects in the upper part of the fluid layer (z 2 − 3δE ). Nevertheless, the Ekman layer forces a secondary re-circulation which induces an efficient transfer of angular momentum from the boundary to the whole fluid domain (Greenspan, 1968). For a fluid layer close to the geostrophic balance √ the √ characteristic time of this Ekman pumping is TE = H0 / νf = T0 /(4π Ek ) (Pedlosky, 1987). Therefore, if we want to neglect this dissipative process over at least several rotation periods T0 , the Ekman number should be quite small Ek 10−4 . Such values can be easily reached if the fluid layer is thick enough. • The aspect ratio parameter α = H0 /L. While this parameter is generally small for synoptic atmospheric or oceanic structures (α 10−2 –10−3 ), this is not always the case in laboratory experiments. Indeed, we could hardly work with

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ultra-thin layers. The first limitation is due to the surface tension which acts on a millimeter scale. The second constraint is due to the Ekman pumping described above. Hence, typical layer depths in rotating tank experiments are about few tens of centimeters H0 10–50 cm. Therefore, in order to get a small aspect ratio (at least α 0.1) the characteristic horizontal scales should be about L 1–5 m. To study the dynamics of multiple structures or to avoid the end effects, the experiment should then be done on a very large turntable D 10 m. A unique installation reaches such large scale (D = 13 m), the Coriolis turntable1 in Grenoble, France. However, for medium size experiments (D 1–2 m) the aspect ratio parameter cannot be asymptotically small and is often close to unity α 1. • The classical Rossby number Ro = fUL characterizes the importance of rotation in the fluid layer. In order to be consistent, the horizontal scale L in the Rossby number Ro should correspond to the characteristic scale of the horizontal velocity gradient. In other words, the Rossby number quantifies the ratio of the relative vorticity ζ = ∂x v − ∂y u with respect to the planetary vorticity (Ro ζ /f ). For large-scale flows, the Rossby number is generally small or finite in the atmosphere and the ocean, leading to the geostrophic balance. It will exceed unity only for very intense vortices, such as hurricanes. This parameter is usually well controlled in a rotating experiment and both small and large values could be obtained. Nevertheless, to reach large values which would exceed unity an external forcing is generally needed. Indeed, without external energy source, from any initial state the geostrophic adjustment process will quickly lead to a mean flow in geostrophic balance which implies small or finite Rossby numbers (Ro 1). • We introduce here the time evolution parameter ε = f1T . This parameter quantifies the dynamical evolution of the flow. It depends on the flow response to the initial condition or to the external forcing. Hence, this parameter cannot be fixed by the experimental setup. Classical textbooks (Gill, 1982; Pedlosky, 1987) usually consider the case of large-scale and slow advective motion and therefore the time-evolution parameter and the Rossby number are fixed to be of the same order ε Ro. However, for high frequency linear waves (i.e. short gravity-waves) ε 1 and Ro 1 while for intense cyclones Ro 1 and ε 1. Hence, as far as experiments on geostrophic adjustment are concerned, it is useful to consider both the cases of slow (ε 1) advective motion or fast (ε 1) wave motion independently of the√Rossby number value. • The Burger number Bu = (Rd /L)2 where Rd = gH0 /f is the Rossby deformation radius. As far as we consider a relatively thick layer H0 10–50 cm and a relatively slow rotation rate Ω0 1–10 rpm we get a large deformation radius Rd 50 cm which is usually close to the size of the experimental apparatus. Hence, with a single barotropic layer, we can hardly obtain 1 http://www.coriolis-legi.org.

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small Burger number values. Only few experiments using high rotation speed (Ω0 60 rpm) reached small Burger number value within a single layer configuration. However, in such case a parabolic vessel is needed to compensate the resulting parabolic shape of the free-surface (Nezlin and Snezhkin, 1993; Stegner and Zeitlin, 1998; van de Konijnenberg, Nielsen, Rasmussen and Stenum, 1999). For such setups the strong curvature of the fluid layer induces, as in the spherical planetary geometry, a strong beta effect. Hence, such parabolic configurations are relevant for modelling large-scale planetary flows, as the Jovian atmosphere for instance. • The relative elevation parameter λ = η0 /H0 where η0 is the characteristic amplitude of the free surface deviation. When the flow is close to geostrophic balance, namely when the dynamical pressure gradient ∇π is balanced at the same order by the Coriolis force, the relative geopotential deviation λ depends on both the Rossby and the Burger number λ ∼ Ro/Bu (cf. Chapters 1 and 2). The RSW model is based on three main approximations: weak dissipation, hydrostatic balance and quasi-bidimensionality of the horizontal velocity. We discuss, in what follows, when and in which range of dynamical parameters these approximations could be valid or not. We first assume the hydrostatic balance for the whole pressure field. The vertical acceleration in (3) could be neglected if both α 2 Ro 1 and α 2 ε 1. Note that for rotating flows, the shallow-water constraint (α 1) is not necessary to get the hydrostatic balance. Indeed, a weakly viscous (Ek 1) slow (ε 1) and geostrophic (Ro 1) flow will follow the hydrostatic balance even if the aspect ratio parameter α is finite. Hence, hopefully for the experimentalists, quasi-geostrophic motions can be accurately reproduced in a rotating tank while α 1. Moreover, the shallow-water constraint is not a sufficient condition that guarantees the hydrostatic balance. Indeed, if the system supports high-frequency waves (ε 1) they could be a source of non-hydrostatic motion or instability. Besides, the case of intense (Ro 1) shallow-water (α 1) vortices or jets is also complex. These intense structures could exhibit in anticyclonic vorticity region an inertial or centrifugal instability which generates three-dimensional and nonhydrostatic perturbations within the large scale flow (Teinturier, Stegner, Viboud, Didelle and Ghil, 2006). Such short-wavelength instabilities could amplify smallscale perturbations (having a finite aspect ratio αp 1) with a rapid growth rate (εp 1). In such case, the cyclonic vorticity regions may satisfy the hydrostatic balance, while intense non-hydrostatic motion occurs in the anticyclonic regions (cf. Figure 5 below). However, when the evolution of a shallow-water flow (or its unstable perturbations) is not fast (ε 1) the hydrostatic balance is satisfied at the first order of approximation and equation (3) leads to: ∂z π = 0.

(9)

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Then according to (8) the dynamical pressure becomes directly proportional to the free-surface geopotential deviation: π(x, y, t) = η(x, y, t).

(10)

We then assume that the fluid layer experiences a weak dissipation. The viscous terms in equations (1)–(2) could be neglected at a first order of approximation if the Ekman number is small enough Ek 1. However, we cannot totally suppress the no-slip condition at the bottom and we should introduce an Ekman layer. This layer will then change the bottom boundary conditions for the upper inviscid layer. It will allow a free-slip condition for the horizontal velocities (u(z0 ), v(z0 )) but will induce a non-zero vertical velocity. In the case of hydrostatic and slow geostrophic motions this vertical velocity is proportional to the horizontal flow vorticity (Pedlosky, 1987; Vallis, 2006) and the boundary conditions (6) should be replaced by: Ek Ek (∂x v − ∂y u) = ζ w(z0 ) = (11) 2 2 where ζ is the vertical component of vorticity. Hence, the√ horizontal dissipation and the Ekman pumping mechanism could be neglected if Ek 1. Practically, in laboratory experiments the dissipation will be a second order process when Ek 10−4 at least. The third approximation assumes a quasi-bidimensional horizontal flow, in other words, the vertical derivatives ∂z u and ∂z v are expected to be negligible. This assumption corresponds to the Taylor–Proudman theorem which is valid only in the limit of small Rossby number (geostrophic flows). A similar approximation is made by the closure hypothesis (22) given in the introduction, which decorrelates the vertical averaging of the horizontal velocity field. Then integrating the continuity equation (22) along the vertical and using the boundary conditions (7) and (11) we finally obtain the following dimensionless formulation of the RSW model: ε∂t u + RoDh u − v = −∂x η, ε∂t v + RoDh v + u = −∂y η, λ[ε∂t η + RoDh η] + (1 + λη)Ro[∂x u + ∂y v] = Ro

(12)

(13) Ek ζ 2

(14)

where Dh = u∂x + v∂y and ζ = ∂x v − ∂y u. Strictly speaking, these RSW equations are valid in the asymptotic limit of slow quasi-geostrophic flows even if the aspect ratio parameter is finite α 1. However, this model is often accurate beyond its limit of validity for finite Rossby numbers (Ek 1; ε 1; Ro 1) and could be applied to a wide variety of laboratory experiments. This will be indeed the case if the vertical motions remain

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weak enough (w 1). This latter condition implies both the hydrostatic balance (9) and a quasi-bidimensional horizontal flow. Nevertheless, in such case, highorder terms should be added in (11) to account for non-linear Ekman pumping (Zavala Sanson and van Heijst, 2000; Hart, J.E., 2000). 1.2. Influence of the centrifugal force Since the Newton’s bucket experiment (1689) it is well known that the freesurface of a fluid layer in solid body rotation is deformed under the action of the centrifugal force. The surfaces of constant pressure for a fluid at rest in the rotating frame (i.e. equipotential surfaces Φ = cst) are given by the potential function: 1 Φ(R, Z) = − Ω02 R 2 + gZ. (15) 2 Hence, the free-surface of a rotating fluid layer takes a parabolic shape. Moreover, according to (15), all the equipotential surfaces corresponds to the same paraboloid simply translated along the rotation axis (Figure 2(a)). We use in what follows a dimensionless formulation where H0 is the mean thickness of the layer, D is the tank diameter and Zc = g/Ω02 is the curvature radius at the center of the parabola. In cylindrical coordinates, the equation for the unperturbed free-surface can be written as: 1 1 2 Z 1β 2 1 2 2 =1+ r − d . h(r) = (16) R − D =1+ H0 2H0 Zc 8 2α 8

Figure 2. Parabolic deformation of the equipotential surfaces (thin lines) due to the centrifugal acceleration in rotating laboratory experiments (a). Schematic description of the tangent plane approximation when the corrective terms of order β 2 d could be neglected (b).

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We have introduced here two dimensionless parameters: • The dimensionless tank diameter d = D/L. The experimentalists tend to use a large tank d 1 in order to satisfy the shallow-water constraint and to avoid the boundary effects. However, for such case, the influence of the centrifugal force could become non negligible close to the wall. • The curvature parameter β = L/Zc quantifies the influence of the curved equipotential surfaces on a dynamical structure of horizontal size L. For the atmosphere or the oceans an equivalent parameter is induced by the spherical geopotential where Zc is replaced by the earth radius RE . It is therefore natural to chose a coordinate system so that the unperturbed water surface, or any equipotential surface, is given by z = cst. Hence, paraboloidal coordinates should be used for rotating laboratory experiments (Nycander, 1993) while spherical coordinates are used for planetary flows (Pedlosky, 1987). However, for small values of the curvature parameter (β 1) the tangent plane approximation is generally made. In other words, a Cartesian system of coordinates is used locally and the corrective terms induced by the parabolic curvature will appear only at the order β 2 d (Nycander, 1993; Stegner and Zeitlin, 1998). If we consider a medium scale experiment (D 100 cm) and a typical horizontal scale L 10 cm, these corrective terms could be neglected for moderate rotation rate (Ω0 10 rpm). The main difference between rotating laboratory experiments and planetary flows is that the effective gravity g e is variable in both directions and amplitude in the laboratory. Indeed, when the centrifugal force is not negligible, it induces a tilting of the effective gravity but also a change in its amplitude (Figure 2(a)). The latitudinal dependence of the effective gravity (also called the γ -effect) in paraboloidal coordinates and in the tangent plane approximation is given by: 1 1 ge = = 1 + β 2r 2 1 + β 2r 2. (17) g cos θ 2 For paraboloidal equipotential surfaces this latitudinal dependence of the effective gravity is of the same order as the latitudinal dependence of the Coriolis force (classical β-effect). Ωz 1 f = = cos θ 1 − β 2 r 2 . (18) f0 Ω0 2 Under the tangent plane approximation the equipotential surface are assumed to be locally flat (Figure 2(b)) while the radial variations of the effective gravity g e , the local component of rotation Ωz and the unperturbed layer depth h are expanded at the first order in y = r − a the local latitudinal coordinate centered at the radial position r a. If we consider the center of the rotating tank (a = 0 and y = r), according to (16)–(18) the latitudinal variations of h(y), Ωz (y) or g e (y) are all quadratic

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and when β/α = L2 /(Zc H0 ) 1 the fluid layer respects the f -plane configuration at the first order of approximation. This will be generally the case for moderate rotation rate (Ω0 10 rpm) in a central region of several tens of centimeters (r 10–20 cm). However, out from the center, the latitudinal variations could be linearly expended in y and they reach their extremal values at the tank wall. Therefore, in order to quantify the relative influence of the β-effect, the γ effect or the topography, we estimate (when a = d) the magnitude of the first derivatives as follows: ∂y h/ h ∝ βd/α, ∂y f/f ∝ β 2 d and ∂y g e /g e ∝ β 2 d. Hence, for standard experimental configurations (β 1, α 1, d 1) the y-dependence of the equilibrium layer depth induced by the parabolic free-surface deformation is the dominant effect. Due to this topographic effect, the equilibrium fluid layer can support topographic-Rossby waves. The linear dispersion relation of these low-frequency waves is analogous to planetary Rossby waves and they are strongly coupled to the slow geostrophic motion. This effect may induce, for instance, a significant drift velocity (Vd f L(∂y h/ h) when L Rd ) and the dispersion of localized vortices (Masuda, Marubayashi and Ishibashi, 1990; Carnevale, Kloosterziel and van Heijst, 1991; Flor and Eames, 2002). Hence, the dynamical influence of the topographic variations could be neglected in the whole tank, from the center to the wall, if ∂y h/ h ∝ βd/α ε. The latter criterion will be generally satisfied in a medium scale experiment (D 100 cm) if the rotation rate is weak enough (Ω0 4 rpm). However, if the ratio βd/α becomes too large, the variation of the layer thickness induced by the centrifugal force could be compensated by a parabolic bottom topography or a parabolic vessel adjusted to the rotation rate (Nezlin and Snezhkin, 1993; Stegner and Zeitlin, 1998; van de Konijnenberg, Nielsen, Rasmussen and Stenum, 1999). 1.3. Non-hydrostatic wave modes Focusing on the geostrophic adjustment problem, where both slow geostrophic motion and fast waves are generated, we look here more carefully at the wave motion that may occur in a rotating fluid layer. We linearize the primitives equations (1)–(8) assuming small amplitudes for the velocity Ro 1 and the free surface displacement λ 1. We neglect all dissipative terms (Ek 1) and take the deformation radius as a characteristic horizontal scale L = Rd of the unperturbed rotating fluid layer, therefore Bu = 1 and α = H0 /Rd . Besides, we keep in mind that the aspect ratio α cannot be asymptotically small for laboratory experiment and we keep the vertical acceleration in (3). Hence, we get: ε∂t u − v = −∂x π,

(19)

ε∂t v + u = −∂y π,

(20)

α ε∂t w = −∂z π,

(21)

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Chapter 6. Experimental Reality of Geostrophic Adjustment

332

∂x u + ∂y v + ∂z w = 0

(22)

with upper (z1 = 1) and lower (z0 = 0) boundary conditions w(z0 ) = 0,

(23)

w(z1 ) = ε∂t η,

(24)

π(z1 ) = η.

(25)

According to the space and time shift invariance of the system, we use the following Fourier decomposition A(x, y, z, t) = A0 (z)ei(t−kx−ly) for all variables. In this case the temporal evolution parameter corresponds to a dimensionless wave frequency ε = ω/f . This linear system finally leads to: ∂z2 π0 +

α 2 ε2 K 2 π0 = 0 1 − ε2

(26)

where K 2 = k 2 + l 2 , with the boundary conditions ∂z π0 (0) = 0,

(27)

∂z π0 (1) = α ε π0 (1).

(28)

2 2

For inertia–gravity waves (ε > 1) we obtain the following dispersion relation: α 2 ε2 K 2 (29) ε2 − 1 corresponding to a single vertical mode π0 (z) = ch(γ z). We can see here that we will recover the dispersion relation of Poincaré waves γ tanh(γ ) = α 2 ε 2 ;

ε2 = 1 + K 2

γ2 =

(30)

only if αK 1, in such case short inertia–gravity waves are dispersionless. This condition is more restrictive than the shallow-water constraint α 1, and indeed short enough gravity waves will always deviate from the RSW model. We have plotted in Figure 3 the deviation from the Poincaré dispersion relation for various values of the aspect ratio parameter α that could be found in laboratory experiments. For a finite value of the aspect ratio parameter, α = 0.3 for instance as it is shown in Figure 4, high-frequency waves (ε 1) or, in other words, short-waves (K 1) will satisfy the dispersion relation of non-rotating surface gravity waves (SGW). K (31) tanh(αK). α It can be shown that the same dispersion relation (31) applies for boundary Kelvin waves propagating along a lateral wall of the tank. Hence, unlike the RSW model, ε2 =

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Figure 3. Log–log plot of the dimensionless dispersion relation for inertia–gravity waves. The curves correspond to different values of the aspect ratio parameter: RSW model or α = 0 (thick line), α = 0.1 (thick dashed line), α = 0.3 (thin dashed line), α = 1 (thin dotted line).

Figure 4. Dimensionless dispersion relation of inertia–gravity waves in a non-rotating (thin line), a rotating fluid layer (thick dashed line) and for the RSW model (thick line) for the fixed value of α = H0 /Rd = 0.3.

Chapter 6. Experimental Reality of Geostrophic Adjustment

334

both inertia–gravity waves and Kelvin waves will become dispersive in the shortwave limit if the aspect ratio parameter is not small enough. Nevertheless, these non-hydrostatic effects could be neglected for a wide range of the inertia–gravity wave spectrum if αK 1, which corresponds to λ˜ 2πH0

(32)

where λ˜ is the characteristic wavelength. For inertial waves (ε < 1) we obtain a discrete spectrum of n vertical modes which correspond to the dispersion relations: −γn tan(γn ) = α 2 εn2 ;

γn2 =

α 2 εn2 K 2 1 − εn2

(33)

where (1 + n)π/2 < γn < (1 + n)π. These non-hydrostatic waves exhibit strong variations of pressure and velocity along the vertical axis (Figure 5(b)). When the vertical wavenumber becomes large (n 1) the wave frequency approaches the Coriolis frequency ε 1 for a wide range of horizontal wavenumber components. According to Figure 5(a), for a given horizontal wavelength k, the short wavelength perturbations along the vertical will have the highest horizontal phase speed. Hence, the non-hydrostatic inertial waves play a crucial role in the vertical alignment and the rapid formation of Taylor columns in a rotating fluid layer. If now we add a mean flow component, such type of non-hydrostatic modes will lead to inertial instability in anticyclonic vorticity regions (Johnson, 1963; Yanase, Flores, Metais and Riley, 1993). Such instabilities occur when the Rossby number Ro exceeds unity and the growth rates for these three-dimensional modes exceed significantly the two-dimensional unstable Rossby modes when Ro 2. Hence, for finite Rossby numbers, starting initially from a two-dimensional flow the three-dimensional perturbations could grow exponentially and break both the hydrostatic and the geostrophic balance (Afanasyev and Peltier, 1998; Stegner, Pichon and Beunier, 2005). As it can be seen in Figure 6, such small-scale instability may occur in shallow-water anticyclonic vortices when the Rossby number is large enough. 1.4. Two-layer stratification We have seen previously that with a single barotropic layer experiment the deformation radius is generally close to the tank size. In other words, for a single layer f -plane experiment we are restricted to large Burger number dynamics. Nevertheless, if we use a two-layer stratification we introduce a baroclinic deformation

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335

(a)

(b) Figure 5. Dimensionless dispersion relation of non-hydrostatic inertial waves (a) corresponding to various vertical modes (b). All the wave modes are calculated for the aspect ratio parameter α = H0 /Rd = 1.

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Figure 6. Dye visualization of von Karman wake in a rotating shallow-water layer with α 0.07, Ro 2 and Re 20000. Small-scale instability is visible in anticyclonic vortices (black dye), while cyclonic vortices (red dye) remain stable (Teinturier, Stegner, Viboud, Didelle and Ghil, 2006).

radius which could be much smaller than the barotropic one. Besides, the Ekman pumping affects only the lower layer, and for an appropriate set of parameters the upper layer dissipation could be strongly reduced. To create a density stratification in water, salt or sugar are generally used instead of temperature. Indeed, the thermal diffusivity (κT 10−7 m2 s−1 ) is a hundred times larger than the salt diffusivity (κS 10−9 m2 s−1 ). In a motionless fluid layer, an initial salt perturbation will diffuse over a 1 cm distance in half a day instead of ten minutes for a thermal perturbation. Hence, for typical layer depths about few to tens of centimeters an initial salt or sugar stratification will remain robust for at least several hours. To obtain a sharp density jump corresponding to a well-defined two-layer stratification we generally proceed as follows. The tank is first filled with the deep and dense lower layer. Then, we start to spin up the rotating table and when the solid-rotation is reached we slowly inject the light upper layer ρ1 at the surface of the dense bottom layer ρ2 . To reduce the vertical

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mixing during the injection we could use floating Hele-shaw cells or small tubes to inject the light fluid horizontally at the free-surface. Another method consists in injecting very slowly the upper layer through floating porous plates. Let us consider a two-layer salt stratification. According to the classical dimensional analysis we add to the previous ones at least four new dimensionless parameters: • The thickness ratio parameter δ = H1 /(H1 + H2 ). This parameter controls the dynamical interaction between the two layers. For equivalent depth layers δ 0.5 the two layers are strongly coupled and baroclinic instability may occurs even if the vertical velocity shear is weak. On the other hand, according to the standard two-layer Phillips model (Pedlosky, 1987), for small ratios δ 1, the baroclinic growth rates tend to vanish. Hence, to avoid a strong baroclinic destabilisation of the flow, we will first consider laboratory experiments with a thin upper layer and a deep lower layer having δ 0.1. Besides, in order to keep the Ekman number small enough in the lower layer (Ek(2) = (δE /H2 )2 10−4 Ek ), we generally fix H2 = 10–20 cm, and therefore the upper layer thickness is about H1 2 cm. • The density ratio parameter N = 2(ρ2 − ρ1 )/(ρ2 + ρ1 ). With salt stratification, we can easily obtain N small up to 10−3 . We then introduce the reduced gravity g = Ng g which controls the dynamics of internal gravity waves at the interface between the layers. • The internal Burger number Bu = (Rd /L)2 corresponding to the baroclinic deformation radius Rd = g H1 H2 (H1 + H2 )/f . We can see here that for small N 10−3 and δ 0.1 the baroclinic deformation radius could be two orders of magnitude smaller than the barotropic deformation radius Rd √ g H1 /f 10−2 Rd , where Rd = g(H1 + H2 )/f . Hence, for a thin upper layer with H1 2 cm and a weak density difference ρ2 − ρ1 2–10 g l−1 we can reach deformation radius as small as Rd 1 cm. Therefore, with a two-layer stratification, the internal Burger number could be easily varied from small to large values 0.01 Bu 100. • We introduce the stratification parameter ES = (δE /dS )2 in order to quantify the dissipation induced by the fluid–fluid interface. This parameter is an equivalent Ekman number for a continuously stratified fluid where dS is the characteristic scale of the vertical density gradient. Indeed, for salt stratification, due to the molecular diffusion and the injection process, the density gradient is always continuous between the upper and the lower layer. Even with very slow laminar injection of both layers the characteristic size dS cannot be infinitely small, and we generally get a density gradient thickness of dS = 3–5 mm (Stegner, Bouruet-Aubertot and Pichon, 2004). For geostrophic flows, the vertical gradient of the horizontal velocity will be directly proportional to the vertical density gradient. Hence, the dissipative terms in the

Chapter 6. Experimental Reality of Geostrophic Adjustment

338

right-hand side of the horizontal momentum equations (12) should scale with ES = (δE /dS )2 . This parameter is larger than the Ekman number defined using the upper layer thickness Ek(1) = (δE /H1 )2 . However, due to the absence of the no-slip boundary condition, there is no boundary Ekman layer for the upper layer. The fluid–fluid interface will then induce (if any) a much weaker recirculation than a classical bottom Ekman layer if the vertical stratification is not too sharp ES = (δE /dS )2 1. We get a dimensionless set of equations for the two-layer RSW model with rigid lid and bottom boundary condition using: L as horizontal scale and T the characteristic time-scale for both layers, U (i) as the horizontal velocity, Hi the vertical thickness and ρi f U (i) L the pressure deviation from hydrostatic balance in each layer. (i)

ε∂t u(i) + Ro(i) Dh u(i) − v (i) = −∂x π (i) , ε∂t v

(i)

+ Ro

(i)

(i) Dh v (i)

+u

(i)

= −∂y π

(i)

(34) (35)

where Dh(i) = u(i) ∂x +v (i) ∂y and the superscript i = 1, 2 corresponds respectively to the upper and the lower layer. The pressure continuity at the interface gives: λBu η = (1 − N )Ro(1) π (1) − (1 + N )Ro(2) π (2) 1−δ and the mass conservation in each layer leads to: (1) λ ε∂t η + Ro(1) Dh η + (1 + λη)Ro(1) ∂x u(1) + ∂y v (1) = 0, (2) λδ ε∂t η + Ro(2) Dh η − (1 − δ − λδη)Ro(2) ∂x u(2) + ∂y v (2)

(2) Ek (2) = (1 − δ)Ro(2) ζ 2

(36)

(37)

(38)

where the relative elevation parameter λ corresponds here to the characteristic deviation of the internal interface rescaled by the upper layer thickness H1 . According to the above equations, if the thickness ratio parameter δ and the density ratio parameter N are small enough and if the motion has a strong baroclinic component (intense velocities in the thin upper layer while the deep lower layer remains almost at rest Ro(2) δRo(1) ), the interface deviation η is controlled by the upper layer pressure only η π (1) . In such case, at the first order of approximation, the upper layer motion is not affected by the lower layer which acts as a neutral layer (Cushman-Roisin, Sutyrin and Tang, 1992). Hence, the upper layer dynamics can be described by the shallow-water reduced-gravity model. Namely, a one-layer RSW model where the gravity g is replaced by the reduced gravity g induced by the two-layer stratification.

2. Potential vorticity measurements: a new challenge

339

Table 1 Vortical motion

Wave motion

Single layer

εα 2 1; Ro α 2 1; Ro < 1

kH0 1

Two layers

δ 1; Ro(2) δRo(1) ; Ro(i) (α (i) )2 1; Ro(1) < 1

kH2 1

However, as for the single-layer case, non-hydrostatic wave motions or inertial instability may occur in the two-layer experiment when respectively λ˜ 2πH2 or Ro(1) > 1. Note that the hydrostatic constraint on the wave activity is fixed here by the deep layer thickness H2 and not the thin upper layer H1 . Recent laboratory experiments performed in a two-layer configuration (α 0.66; δ 0.2) exhibit non-hydrostatic wave behaviour for λ˜ 80 cm wavelength, while H1 12.5 cm and H2 50 cm (Thivolle-Cazat, 2003). Taking into account the above mentioned laboratory constraints, the physical modelling of rotating shallow-water flows looks like a Holy Graal for experimentalist. Nevertheless, for a specific range of the dynamical parameters, the motion in rotating fluid layers could be close to the one layer RSW model. We recall in Table 1, both for single-layer and the two-layer configurations, the distinct conditions needed to be satisfied, respectively, for the slow vortical motion and the fast wave motion in order to follow the RSW dynamics.

2. Potential vorticity measurements: a new challenge Both vorticity and potential vorticity play an important role in the dynamics of rotating fluid layers. The application of the Kelvin theorem to a non-dissipative rotating shallow-water flow implies the Lagrangian conservation of potential vorticity (Chapter 2) in each layer: D 1 + Roζ = 0. (39) Dt 1 + λη An elementary fluid parcel (i.e. fluid column) moving within a layer could be stretched or compressed. These changes in the height of the fluid parcel during its motion will be accompanied by a change in its vorticity. In other words, for a purely incompressible two-dimensional flow when the free-surface or the interface deviations are negligible (λ Ro), we recover the Lagrangian conservation of vorticity: Dζ = 0. (40) Dt In this case, vorticity will be generated in the flow only if there is an external source (boundary layer or fluid injection, for instance). For a rotating fluid layer,

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if the layer thickness varies sufficiently, relative vorticity could be generated from an adjustment process without any external source. The potential vorticity conservation is a key concept for adjustment processes even in the presence of dissipative forces. Hence, as far as laboratory experiments on geostrophic adjustment are concerned, it is crucial to perform quantitative measurements of the potential vorticity field. Such measurements in a rotating fluid layer are indeed not simple. Both the vorticity field ζ and the height field η should be measured simultaneously. If such measurements are now possible, it is mainly due to recent progress in computers, lasers and cameras technology. Besides, additional difficulties are encountered on a rotating turntable where sufficiently compact devices (especially lasers) and remote control of the whole setup are needed. Therefore, direct measurement of the potential vorticity field is always challenging for experimentalists. We give below some details on the non-intrusive methods which can be used to achieve such measurements for specific experimental configurations. 2.1. Particle image velocimetry and vorticity field measurements The particle image velocimetry (PIV) was developed since 1994 to perform accurate and quantitative measurements of fluid velocity vectors at a very large number of points simultaneously (Adrian, 2005). Presently, the 2D PIV method consist to add small neutrally buoyant beads to the working fluid and lightened them with a laser sheet. The 2D particle motion along this plane are recorded with a digital video camera. Cross-correlation image processing is then performed to measure the mean particle displacement in small box region between two successive images (Fincham and Spedding, 1997). Standard systems are sold by commercial companies and it is now the most efficient and non-intrusive technique used in fluid mechanics to obtain a vorticity map in a given region of the flow field. Nevertheless, some technical limitations appear which restrict the spatial resolution of such measurements in rotating fluid layers. The energy necessary to illuminate fine particles and produce images of sufficient exposure and clarity is the first limitation of PIV. The maximum size of the measurement window is then fixed by the laser intensity and the camera exposure time. Hopefully, vertical motions are strongly damped in a rotating fluid layer, therefore neutrally buoyant particles could stay for a relatively long time inside the horizontal laser sheet. Besides, the horizontal velocities of geostrophic motions are usually not too large (V 1–10 cm s−1 ) and the camera exposure time could be optimized to 10–20 ms. But nevertheless, if the intensity per unit area is too small the clarity of recorded images may not be sufficient enough with classical digital cameras. On a medium size turntable (D 1 m), high power lasers which require cooling systems are generally excluded. However, the last

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Figure 7. Horizontal velocity field of a cyclone obtained from particle image velocimetry. For clarity, only half of the vectors (47 × 35) are displayed instead of the full (94 × 70) field. The measurement was made in the thin upper layer of a two-layer stratified fluid corresponding to: Bu 0.4, λ 0.5, α 0.75, δ 0.1.

generation of compact high power laser diodes2 can generate an uniform intensity line (non-Gaussian) with an output power up to 1 W. With such system we could easily detect the horizontal particle motion (V < 10 cm s−1 ) from small (10 cm × 10 cm) to large (1 m × 1 m) areas of investigation. The second limitation is induced by the pixel resolution of the camera. Indeed, to obtain a precise cross-correlation between two interrogation windows, a minimum number of particles (∼5) should be present in the interrogation box (Raffel, Willert and Kompenhans, 1998). This constraint induces practically a minimum size of a 12 × 12 pixel box. Hence, with a standard 750 × 550 pixels camera we usually get a velocity field of 95×70 vectors as it is shown in Figure 7. Using digital cameras with higher resolution (3000 × 2000 pixels) we could, for instance, reach a 370 × 250 vector grid field. However, even with very high resolution camera and optimized software, PIV measurements will always give a coarse-grid resolution in comparison to direct dye tracer visualizations (3000 × 2000 pixels) or high-resolution numerical simulations (4096×4096 for two-dimensional flows (Bracco, McWilliams, Murante, Provenzale and Weiss, 2000)). The third limitation comes from the limited precision of the velocity field. Even with hierarchical correlation methods, where correlations deduced from a large interrogation box are used to guide correlation analysis at smaller boxes (Hart, D.P., 2 Lasiris Magnum Laser (www.laser2000.fr).

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Figure 8. Cyclonic (red) and anticyclonic (blue) dimensionless vorticity ζ /f calculated from an instantaneous velocity field (a) or calculated from the time averaged velocity field (b) shown in Figure 7. The Rossby number deduced from the maximum vorticity is about Ro = ζmax /f 0.6. The measurement area is a rectangular window of 280 mm × 220 mm.

2000), the available dynamical velocity range is about 100:1. In other words, the method cannot detect fluctuations in the velocity field below 1%. Besides, experimental noise on recorded images could easily produce 5% error in the velocity field. This can be a serious problem, because a weak noise in the velocity field

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induces a stronger noise in the derivatives and therefore strongly influences the vorticity measurement. Typical errors of order of 20% (or higher) in the vorticity field could be frequent, and strong efforts on the improvement of the image quality and the software used in the PIV process are needed to reach such precision on the vorticity field measurements. However, if the flow evolves slowly in comparison with the frequency of the PIV acquisition system, time averaging of the velocity field is a simple and efficient way to reduce the experimental noise. For the case of geostrophic adjustment, when a quasi-steady motion is reached, time averaging will lead to sufficiently precise vorticity measurements. For instance, the velocity field shown in Figure 8 is an average of ten velocity fields separated by a time interval of 120 ms. Hence, this corresponds to a time averaged velocity measurements over 1.2 s which is smaller √ than the inertial period Tf = 12 s or the characteristic decay time TE = H2 / νf 200 s. Such time averaging reduces the noise in the vorticity field (Figure 8(b)) by a factor 10 in comparison with the instantaneous vorticity measurements (Figure 8(a)). According to the above comments, we should emphasize that even if we can easily obtain a vorticity map from the standard PIV system the accuracy of such measurements should be checked carefully. Let us recall, that if PIV measurements have a coarse grid resolution corresponding to 12 × 12 pixels on the digitized image, this will be even more pronounced for the vorticity field. Indeed, to resolve accurately a velocity gradient, at least 3–5 grid points are needed. Therefore, quantitative vorticity measurements will not be possible if the dynamical structure under consideration is too small. We can roughly estimate a limiting value as 36 × 36 pixels on the digitized image. Thus, small vortices or thin vorticity filaments are generally smoothened by the PIV process. In such case it could be useful to use two cameras with a wide and a zoom angle in order to quantify accurately the large scale flow and smaller vortical structures (Perret, Stegner, Farge and Pichon, 2006). Hence, the standard PIV method is well suited to quantify slow and large-scale structures in rotating fluid layers. However, for fast and small-scale structures such as high-frequency waves, this velocimetry method can hardly provide quantitative measurements, unless an expensive high-speed PIV technology is used. 2.2. Height field measurements Laboratory techniques for measuring the velocity field, such as the PIV method described above, are quite advanced. However, methods for making accurate measurements of the height field of a fluid layer have remained relatively elusive. As far as we know, four non-invasive techniques were used to detect or to measure the height field fluctuations in rotating fluid layers: light absorption, optical altimetry from the parabolic free-surface, optical rotation of the working fluid and laser induced visualization (LIV).

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2.2.1. Light absorption The light absorption technique is based on the optical density of a dyed layer. It consists in measuring the light intensity after absorption through a uniformly dyed fluid layer (Holford and Dalziel, 1996). The fluid layer is usually lightened from below through a transparent vessel while a video camera records the intensity fluctuations from the top (Figure 9(a)). A specific pass-band-filter, which is centered at the maximum absorption of the dye, is put on the video camera to

Figure 9. Light absorption technique for a parabolic vessel experiment (a) (Stegner and Zeitlin, 1998). Intensity fluctuations view from the top of the experiment (b) post-processed calibrated image corresponding to a relative elevation λ 1 of the layer depth (c).

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increase the sensitivity. With this method a local increase of the layer thickness induces a higher absorption and this region will appear darker on the video image (Figure 9(b)). This altimetric measurements was used successfully on small-scale parabolic vessel where small (10%) and large (60–100%) relative free-surface deviations were detected with an accuracy of less than one mm (Stegner and Zeitlin, 1998). Nevertheless, caustic effects induce systematic errors of order 5–10% of the layer thickness which limits the precision on the height measurements. 2.2.2. Optical altimetry of the parabolic free-surface The free-surface fluctuations of a rotating fluid layer can be imaged and analysed using its parabolic free-surface as a Newtonian telescope mirror. Parallel light rays from a source high above the rotating table reflects from the water surface and converges on the parabolic focus Zf = 12 Zc (Figure 10(a)). However, parallel light rays can hardly be obtained on a rotating table. The image of a point-light source located at Zc (the radius of curvature at the center) no longer have sharp focus but converge through a small disk located at the same height Zc (Figure 10(b)). For practical purpose the light source and the camera are symmetrically displayed off the axis. Then, putting a knife-edge barrier in the middle of this singular disk, where all the rays converge, can partially obscure the image giving great sensitivity to slight imperfections of the reflecting surface. This optical altimetry technique is one of the most sensitive methods used so far in geophysical fluid dynamics experiments. Indeed it is potentially able to detect free-surface fluctuations with a one micron precision, independently of the mean thickness

Figure 10. Focusing of parallel light-rays reflected by the parabolic free-surface (a). Sharp convergence of a point light-source located close to Zc the center of curvature of the apex (b). A knife-edge barrier induces a contrasted black and white image of the free-surface. It increases the sensitivity to deviations from a perfect paraboloid.

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Figure 11. Optical altimetry visualization of inertia–gravity waves (50 µm in amplitude) interacting with localized vortices. The wave maker is on the bottom left, while the vortices appear in the center. Courtesy Y. Afanasyev.

of the parabolic layer. Therefore, this method is particularly suited for the investigations of small amplitude waves (less than 0.1% fluctuation) which are often difficult to detect by other methods. For instance, an inertia–gravity wave having an amplitude of 50 µm (0.04% of the mean layer thickness) could be visualized in Figure 11. Qualitative observations of a large variety of dynamical features such as gravity waves, inertial waves, Rossby waves and small-scale convection could then be performed (Rhines, Lindahl and Mendez, 2006; Rhines, 2006). At the same time, a quantitative method of determination of the slope variation using speckle patterns is possible. A reference image of the fluid layer in solid-body rotation is first made. The slope is measured by comparing the original pattern and a reflected image of this pattern distorted by the surface perturbation induced by the relative flow motion. The procedure is analogous to PIV process where correlations are computed between the small areas of the image and the reference. Nevertheless, the speckle method is limited by large-amplitude deformations and have a limited spatial resolution due to the minimum size of the correlation boxes (Rhines, Lindahl and Mendez, 2006; Afanasyev, Rhines and Lindahl, 2006). A different quantitative method based on optical color coding was also developed using every pixel of the image. A color slide is fixed just below the light

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source. For a given rotation rate (the null point) the entire surface of water is illuminated by only one color. Any perturbation of the free surface results in the appearance of color different from the null point. It is then possible to measure from each pixel of the image the x and the y component of the slope with a 0.1% sensitivity (more details are given in (Afanasyev, Rhines and Lindahl, 2006). Hence, by integrating the slope field quantitative height measurements of the parabolic fluid layer could be achieved. Besides, for a rotating fluid layer satisfying the geostrophic balance, the local slope is directly proportional to the geostrophic velocity. Hence, the primary purpose of optical altimetry is to perform an Altimetric Imaging Velocimetry (AIV) having a much higher spatial resolution than PIV. 2.2.3. Optical rotation Another sophisticated remote sensing method for measuring the thickness of a fluid layer relies upon the optical rotation properties of the working fluid. The liquid is chosen to be optically active (limonene and CFC-113 for instance), so that plane-polarized white light propagating vertically through the fluid layer has its plane of polarization rotated by an angle which depends upon both the wavelength and the layer depth. After leaving the fluid, the angulardispersed white light passes through a sheet of polaroid. For a given layer depth, only light of a certain wavelength has its polarization axis rotated into exact alignment with the polaroid. Light of other wavelengths is either partially or fully extinguished by the polaroid, giving a correlation between the interface height and colour registered by the camera (Hart and Kittelman, 1986; Williams, Read and Haine, 2004). A high-sensitivity up to 1–2% of the layer height could be reached with this technique. Both the large-scale geostrophic flow and small-scale waves could be accurately measured with the technique (Williams, Read and Haine, 2003). According to Figure 12 small-scale fluctuations in the two-layer interface having 1 to 5 mm amplitude are quantitatively detected. However, the sensitivity will be optimal when the mean rotation angle is about 90◦ and this implies for the limonene/CFC mixture that the fluid layer should be relatively thick H = 10–15 cm. Besides, specific precautions should be taken to prevent harmful limonene vapours from evaporating into the laboratory. 2.2.4. Laser induced visualization Laser induced visualization (LIV) technique could also be used in rotating experiments to measure with precision the fluid layer thickness along a line. Initially, the working fluid is uniformly mixed with a fluorescent dye. A vertical laser sheet crosses the horizontal fluid layer and induces the fluorescence of the dye within this plane (Figures 13(a) and 14). In order to optimize the fluorescence, the maximum of dye absorption λabs should be close to the laser wavelength. Hence, we choose the fluoresceine (λabs = 490 nm) or the Rodhamine 6G (λabs = 530 nm)

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Figure 12. Color calibrated visualization of the internal interface of a two-layer fluid using optically active CFC-13/limonene for the lower layer. Small-scale waves (5 mm amplitude and 2 cm wavelength) are visible during one cycle of a large-scale unstable baroclinic mode 2. Courtesy P.D. Williams.

Figure 13. Side view visualization (a) of the fluorescent upper layer. The lower layer appears dark because it does not contain any fluorescent dye and remains therefore transparent to the laser sheet. Edge detection processing (b) allows for a precise measurement of the layer thickness corresponding here to a cyclonic depression: Bu 0.4, λ 0.5, α 0.75, δ 0.1.

if we use, respectively, an argon laser (488 nm) or Nd:Yag laser (532 nm). A video camera, fixed on the side of the tank and perpendicular to the laser sheet could then record the fluorescence of the fluid layer. Using an adequate

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Figure 14. Measurement of the vertical salinity profile from the fluorescent light emission (Stegner, Bouruet-Aubertot and Pichon, 2004). Initially, the upper fluid is uniformly mixed with a fluorescent dye. (a) The upper water initially confined in a transparent bottomless cylinder appears white owing to the fluorescent light emission while the dense water is black. (b) Vertical distribution of the light intensity in the central rectangle shown in (a). (c) The salinity profile can be deduced from (b) if we perform a careful calibration of the laser sheet intensity along the vertical plane. The disturbance at z = −3.3 cm is due to light reflection at the bottom of the cylinder.

image processing we then detect the position of the interface between the light fluorescent and the dark transparent fluid (Figure 13(b)). With this non-intrusive technique we were able, for the two-layer configuration, to measure the displacement of the fluid interface between the fresh and the dense water with an accuracy of 2% at fast acquisition rates (Stegner, Bouruet-Aubertot and Pichon, 2004; Perret, Stegner, Farge and Pichon, 2006). The acquisition frequency is limited by the acquisition rate of the camera and the transfer capacity of the video card. A frequency of 100 Hz could be easily reached nowadays with standard firewire cameras. Note in Figure 15 that the LIV camera is not exactly perpendicular to the fluid layers, in order to reduce the image distortion due to the ray diffraction through the stratified interface between the two layers. Unlike the previous techniques which estimate the height field or its fluctuations in the whole layer, the LIV method gives a measurement of the height field only along a line. Hence, the position of the vertical laser sheet should be

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Figure 15. A horizontal red (670 nm) laser sheet with a vertical green (532 nm) laser sheet are used simultaneously in order to couple PIV and LIV measurements in the upper layer.

carefully chosen. However, this limitation is compensated by the possibility to detect small-scale and three-dimensional structures along the vertical. Indeed, this method measures precisely the dye distribution at each point (x, z) of a vertical plane and does not integrate the information along a vertical ray path. Besides, we could also measure the density field from the fluorescent dye emission (Figure 14(a)). Indeed, on short time scale (i.e. few minutes) the mixing of the initial uniform concentration of both dye and salt is expected to be driven mainly by convection (i.e. turbulent mixing). Hence, dye and salt gradient are not affected by relative diffusion and they are therefore proportional. In a first step, we measured the relative fluorescent light emission (Figure 14(b)) which depends mainly on the dye concentration and the laser sheet intensity. Then, taking into account the vertical distribution of the laser sheet intensity, we correlate the light intensity with the local salinity (i.e. density) as shown in Figure 14(c). This could be, in the next future, an efficient non-invasive technique to measure the density field in a continuously stratified fluid. 2.3. Potential vorticity measurements In order to measure the potential vorticity field according to (39) we chose to use both PIV and LIV measurements simultaneously. However, due to the restrictions of a LIV technique, which gives the height field only along a line, the combination of these non-invasive methods is best suited for unidirectional flows. It is then possible from a line measurement to estimate a global potential vorticity field for either circular (Stegner, Bouruet-Aubertot and Pichon, 2004) or parallel flows (Perret, Stegner, Farge and Pichon, 2006). A typical experimental setup for

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Figure 16. Plots of the averaged vorticity profile measured by PIV (a) and the averaged height profile (b) measured by LIV corresponding to the mean steady state at t = 2Tf . The potential vorticity (c) is deduced from these two profiles. These measurements correspond to the same cyclonic PV anomaly as shown in Figures 8(b) and 13.

a circular cyclonic PV anomaly is shown in Figure 15. Two lasers having different wavelengths are used in addition with specific optical filters, fixed on each camera, in order to detect the dye emission only in the vertical plane and the buoyant particles only in the horizontal plane. Time averaged vorticity and height profiles along a diameter are displayed in Figure 18. The profiles correspond to the PIV and LIV measurements shown in Figures 8(b) and 13. The temporal averaging (over one inertial period Tf = 2π/f ) filters out the fast wave motion from both the density interface and the azimuthal horizontal velocity. Hence, these profiles correspond to the mean adjusted state of the system which evolves slowly in comparison with the wave motion.

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From these data we can then easily quantify the potential vorticity (Figure 16(c)) of the cyclonic PV anomaly. The PV profile is rescaled here by Q0 = f/H1 , the intrinsic PV of the unperturbed upper fluid layer (solid line in Figure 16(c)). For this case, the initial circular PV anomaly was a constant Q patch with Q/Q0 = 2. As far as we know, this technique is the first attempt for direct and quantitative measurements of the potential vorticity field in a rotating shallow-water layer experiment. In other laboratory studies, either the height field or the velocity field were measured but not both of them simultaneously. In such case, the “missing” field could be estimated according to the geostrophic or cyclo-geostrophic balance and the potential vorticity field reconstructed. However, these indirect methods could induce significant errors especially when ageostrophic or non-hydrostatic motions become non negligible. A more refined method based on data assimilation was used recently (Thivolle-Cazat, Sommeria and Galmiche, 2005). The experimental results were compared with a two-layer isopycnal model and data assimilation was used to extrapolate from PIV measurements both the interface position and the potential vorticity field. However, such PV extrapolation depends strongly on the underlying assumptions of the numerical model used and on the assimilation scheme. Therefore, we do believe that coupled measurements is the best way to quantify the PV. Nevertheless, data assimilation will fully benefit from these coupled measurements and it could become an optimal method to test the limits of validity of the shallow-water modelisation for real flows.

3. Simple case studies of geostrophic adjustment We describe in what follows few cases of geostrophic adjustment based on lock released experiments performed in rotating fluids. In a two-layer configuration, vertical boundaries (i.e. locks) are used to fix initial height (or density) steps in the upper layer. For such cases when there is no relative motion in the layers the initial PV field is precisely controlled by the layer thickness. If the release of the vertical walls is rapid enough, we could then follow the geostrophic adjustment of a well defined initial condition corresponding to discontinuous profiles of constant PV. The simplicity of the initial condition makes these experiments easily reproducible. 3.1. “Warm-core” lens 3.1.1. Initial state and experimental configuration The term warm-core lens is generally used for mesoscale vortices which contain a finite volume of warm and light water at the ocean surface. A simple experimental configuration leads to similar dynamical structure (Griffiths and Linden, 1981;

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Figure 17.

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Initial configuration of the “warm-core” lens: the setup (a), and the initial profile of the corresponding potential vorticity (b).

Rubino and Brandt, 2003). A fixed volume of buoyant water is initially confined within a bottomless cylinder of radius Rc on the top of a dense rotating fluid (Figure 17(a)). Assuming that the thin upper layer follows the reduced-gravity RSW equations (cf. Chapter 1), the initial PV distribution is constant for r < Rc and exhibit a singularity at r = Rc (Figure 17(b)) due to the vanishing layer thickness. Similar experiments were performed to study the baroclinic instability of a density front leading to meanders and eddies (Griffiths and Linden, 1981; Bouruet-Aubertot and Linden, 2002). The present experiment was made with a smaller thickness ratio parameter δ 0.1 to reduce the growth rate of baroclinic disturbances and focus the study on the adjustment process (Stegner, BouruetAubertot and Pichon, 2004). 3.1.2. Dynamical stages Three stages were observed during the adjustment process. Just after the rapid withdrawal of the transparent cylinder, the fresh water spreads radially as a gravity current. During this initial stage, the flow is fully three-dimensional (Figure 18(a)) and the effects of rotation are expected to be weak. After approximately half of the inertial period, the radial extension of the lens is stopped (Ungarish and Huppert, 1998). The second stage corresponds to a radial contraction of the lens where steep jumps at the interface may appear (Figure 18(b)). Then, after about two in-

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Figure 18. Dynamical evolution of the interface for the initial configuration corresponding to Bu = (Rd /Rc )2 0.4; α 0.76; δ 0.08 (Stegner, Bouruet-Aubertot and Pichon, 2004). The snapshots are taken at t = 0.5Tf (a), t = 0.8Tf (b), t = 3Tf (c) and t = 0.5Tf (d). The dark rays on the right-hand side of the image are experimental shadows produced by the upper fixations of the cylinder. The full movie could be downloaded from http://gershwin.ens.fr/stegner.

ertial periods, the density front reaches an equilibrium characterized by a standing wave mode superimposed on the mean state (Figures 18(c), (d)). In all our experiments, this third stage is rapidly reached after approximately one or two inertial periods. These results agree with previous studies (Mahalov, Pacheco, Voropayev, Fernando and Hunt, 2000) who also found that the inertial period is the characteristic time of the transition from a density current to a geostrophic front. 3.1.3. Rotating shallow-water predictions We present here the approximations and the calculation of classical Rossby adjustment theory for the axisymmetric warm-core lens configuration (Figure 17). According to the small thickness ratio parameter δ 0.1 and the weak motion observed in the lower layer Ro(2) δRo(1) (Stegner, Bouruet-Aubertot and Pichon, 2004) the reduced gravity RSW equations are expected to provide, in first order of approximation, an accurate description of the upper layer dynamics. Besides, we assume that viscosity and dissipative effects are negligible and that the system reaches a final steady state. Using the deformation radius as the characteristic horizontal scale (i.e. Ro = 1 and λ = 1) we get the following dimensionless

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cyclo-geostrophic balance for an axisymmetric steady state: v2 + v = ∂r η. (41) r The Lagrangian conservation of PV implies a constant value for all fluid parcels within the upper lens: Q(r rf ) = 1 =

1 + ∂r v + v/r 1+η

(42)

where rf is the final radius of the density lens. According to (41) and (42) we get: v v2 1 ∂r (r∂r v) − 2 − v + =0 (43) r r r with the boundary conditions v(0) = 0, h(rf ) = 1 + η(rf ) = 0

⇒

v ∂r v + + 1 (rf ) = 0. r

(44) (45)

For a given radius rf , we can solve (43), (44) and (45) numerically with standard shooting methods. Then, the angular momentum conservation or mass conservation or mass conservation both give the same implicit relation between rf and the initial radius of the cylinder rc : rc2 = rf2 + 2rf v(rf )

(46)

The velocity v(r) and the height h(r) profiles of the steady adjusted density lens are fixed by a single parameter rc = Rc /Rd = Bu−1/2 . Examples of velocities and height profiles are given in Figure 19 for the same initial state and two different deformation radii corresponding to Bu = 0.05 and Bu = 5. For small Burger number we expect an axisymmetric jet (or large-scale ring) whereas for large Burgers number an eddy (close to solid rotation) is expected. 3.1.4. Mean adjusted state In the warm-core lens configuration, the interface between the two fluids intersects the free surface. Hence, unlike the standard Rossby adjustment problem (Gill, 1982; Vallis, 2006) inertia–gravity waves cannot propagate away from the region of the initial density anomaly. Therefore, the separation between the adjusted state and the wave motion is not direct. Hence, we used time averaging over one or two Tf , as described in Section 2, in order to extract the slow dynamics of the height profile and the velocity field. We first observe that the averaged height profile, displayed in Figure 20(a), remains almost constant between t 1.5Tf and t 7Tf . During that time, the averaged velocity profile experiences a slow

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Figure 19. Final steady state according to the standard Rossby adjustment. Two adjusted velocity (dashed line) and height (solid line) profiles resulting from the same initial density anomaly (h = H1 , r = Rc ) are plotted for two different deformation radii Bu = 0.05 and Bu = 5.

dissipation (Figure 20(b)). Therefore, even if a strong wave activity is present according to Figure 18, the averaged mean state remains quasi-steady after one or two inertial periods. Moreover, in the central region, this quasi-steady state is relatively close to the cyclo-geostrophic adjusted state predicted by the PV conservation in the RSW framework. According to this inviscid adjustment model the velocity reaches its maximum and is discontinuous at the edge of the lens. This is obviously unrealistic in a physical system where dissipative processes occur. Indeed, according to Figure 20(b) the maximum velocity is almost three times smaller than the predicted one. Hence, both the velocity and the potential vorticity of these anticyclonic lenses are smoothened near the edge front over a characteristic distance equal to the deformation radius (in the present case Rd = 3.2 cm while Rc = 5.25 cm). 3.1.5. Small-scale instabilities Detailed analysis of the velocity field evolution shows that strong and localized dissipation occurs in the very initial stage of adjustment (t 2Tf ) while the flow experiences only a weak dissipation afterwards. This rapid dissipation which occurs at the edge of the anticyclonic lenses induces a significant deficit in the

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Figure 20. Mean height (a), velocity (b) and PV (c) profiles averaged over one inertial period. The initial density anomaly confined within the bottomless cylinder is plotted with a thin line in (a). The thick solid line corresponds to the cyclo-geostrophic adjusted state predicted by the geostrophic adjustment scenario of the inviscid RSW model.

kinetic energy of the adjusted flow up to 50% or 80% (Stegner, Bouruet-Aubertot and Pichon, 2004). Dye visualization reveals that transient and rapid three-dimensional instabilities occur in the very first stage of adjustment (Figure 21). A first unstable perturbation having a short wavelength grows very quickly, then

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Figure 21. Dye visualization of the three-dimensional perturbations at the edge of the anticyclonic lens (a) t = 0.3Tf , (b) t = 0.5Tf , (c) t = 0.7Tf and (d) t = 1.7Tf (Stegner, Bouruet-Aubertot and Pichon, 2004). The movie is available on http://gershwin.ens.fr/stegner.

spiralling arms appear with a larger wavelength. The first instability scales with the viscous diffusion length Lv = νTf 3–4 mm and does not depend on the Burger number while the secondary mode corresponding to the spiralling arms does scale with the deformation radius. These three-dimensional instabilities localized in time (less than one inertial period) and space (the edge of the anticyclonic lens) provide an efficient mechanism of turbulent dissipation which cascades energy toward small scales in the frontal region. However, outside the outcropping region the potential vorticity conservation is well verified. 3.2. Cyclonic and anticyclonic PV patches 3.2.1. Initial state and experimental configuration We used the term “PV patches” for localised positive or negative potential vorticity anomalies of constant values within a uniform PV layer. The “PV patch” model is the generalisation of the Rankine vortex (cylindrical vorticity patch) for a rotating shallow-water layer. It is the simplest description of potential vortic-

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Figure 22.

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Initial configuration of the experimental setup corresponding to an anticyclonic (a) and a cyclonic (b) PV-patch and their respective PV profiles (c) and (d).

ity front with no outcropping. It could be, for instance, a simplified description of the cyclonic polar vortex in the stratosphere. The corresponding experimental configurations for anticyclonic and cyclonic “PV-patches” are shown respectively in Figures 23(a) and (b). A two-layer stratification with a small thickness parameter δ = 0.125 is first realized. Then, a bottomless cylinder is used to produce a height step at the two-layer interface. Assuming that the thin upper layer follows the reduced-gravity RSW equations, the initial PV distribution is uniform inside (r < Rc ) and outside (r > Rc ) the cylinder. Unlike the “warm core” lens configuration (Figure 17) the potential vorticity exhibits a discontinuity (but not a singularity) at r = Rc due to the finite jump in the layer thickness. Besides, both positive and negative circular PV jumps could be obtained (Figures 23(b), (c)). A positive (negative) thickness anomaly in the upper layer corresponds to a negative (positive) PV jump and will generally lead to a localized anticyclonic (cyclonic) circular ring or vortex. 3.2.2. Dynamical stages The very initial stage of adjustment differs from the warm-core lens configuration. Just after the withdrawal of the transparent cylinder, the vertical density jump gets tilted and a local overturning motion is initiated at the initial position of the cylindrical wall. However, due to the rotation, the overturning motion is stopped after

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Figure 23. Dynamical evolution of the interface of an anticyclonic PV-patch corresponding to Bu = (Rd /Rc )2 0.084; λ = 0.5; α (1) 1.6; δ 0.125. The snapshots are taken at (a) t = 0, (b) t = Tf , (c) t = 2Tf , (d) t = 3Tf and (e) t = 10Tf . The movie is available on http://gershwin.ens.fr/stegner.

one inertial period and a localized shock (steep density front) occurs as shown in Figures 23(b) and (b). Due to the absence of outcropping front no gravity current head is visible for the PV-patch configuration. Afterwards, the thickness anomaly reaches an equilibrium. Even if small fluctuations could be detected, this mean adjusted state holds for a relatively long time. According to Figures 23(c)–(e) and 24(c)–(e), for a small Burger number configuration (here Bu = 0.084) the amplitude and size of the mean adjusted state remain close to the initial unbalanced height profile. Besides, the thickness anomaly remains almost unchanged from t = 2Tf to t = 20Tf . Hence, the system reaches a quasi-steady state in a very short time, approximately one or two inertial periods. For higher Burger numbers, the amplitude of the fluctuations is larger and the system seems to be far from an equilibrium. However, using an accurate time averaging to filter out

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Figure 24. Dynamical evolution of the interface of an cyclonic PV-patch corresponding to Bu = (Rd /Rc )2 0.084; λ = −0.5; α (1) 1.6; δ 0.125. The snapshots are taken at (a) t = 0, (b) t = Tf , (c) t = 2Tf , (d) t = 3Tf and (e) t = 10Tf . The movie is available on http://gershwin.ens.fr/stegner.

the fast wave motion (see below), an averaged mean state is reached with the same rapidity. This characteristic time for adjustment (one or two inertial periods) does not depend on the size or the amplitude of the initial PV-patch. 3.2.3. Rotating shallow-water predictions We present here the calculation of classical Rossby adjustment theory for the cylindrical “PV-patch” configuration (Figure 22). As for the “warm-core” lens case, the small thickness ratio and weak motions in the lower layer justify the use of the reduced gravity RSW equations for the upper layer dynamics. Here again we neglect, at the first order of approximation, the dissipation. Therefore, we use the same set of dimensionless equations as the “warm-core” lens configuration, but we need to consider two distinct regions of uniform PV. We will use the index 0 for the inner PV anomaly region (r < rf ) and the index 1 for the outer region (r > rf ) where rf is the radial position of the PV jump in the final adjusted state. For the case of a anticyclonic PV-patch (Figure 22(c)) we expect a radial extension of the PV front (rc < rf ) while for the cyclonic PV-patch (Figure 22(d)) we expect a radial contraction (rf < rc ). The Lagrangian conservation of potential vorticity implies a constant but distinct value of PV for all fluid parcels within each region

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of the upper fluid layer. Hence, for the inner PV anomaly region we have: Q0 (r < rf ) =

1 1 + ∂r v0 + v0 /r = 1+λ h0

(47)

while for the outer region we have: Q1 (r > rf ) = 1 =

1 + ∂r v1 + v1 /r h1

(48)

where the relative PV anomaly is given initially by λ = η1 /H1 . Then, looking for a steady adjusted state, implies the cyclo-geostrophic balance (41) and according to (47)–(48) we get the second order non-linear ordinary differential equations: v2 1 vi (49) ∂r (r∂r vi ) − 2 − Qi vi + i = 0 r r r with the boundary conditions: v0 (0) = 0,

(50)

v0 (rf ) = v1 (rf ),

(51)

h0 (rf ) = h1 (rf ) ⇒ v0 v1 + 1 (rf ) = ∂r v1 + + 1 (rf ). (1 + λ) ∂r v0 + r r

(52)

Besides, far away from the potential vorticity front (r rf ), a localized solution satisfies the geostrophic balance which implies neglecting the non-linear terms in (49). In such case, the general solution of the linearized equation (49) is expressed through Bessel functions. The outer velocity of a localized adjusted state should then decay at infinity as a modified Bessel function of the second kind v1 (r → +∞) ∝ K1 (r).

(53)

For a given radius rf , we can solve equations (49)–(53) numerically by appropriate shooting methods. Then, as for the “warm-core” lens configuration, the angular momentum conservation leads to the same implicit relation (46) between rf and the initial position of the front rc (i.e. the dimensionless cylinder radius). The velocity and the height profiles of the steady adjusted PV-patch are then fixed by two dimensionless parameters: rc = Rc /Rd = Bu−1/2 and λ = η1 /H1 . Examples of velocities and height profiles for both cyclonic and anticyclonic PV patches are given in Figure 25. For large Burger numbers, in other words for a small cylinder radius in comparison with the deformation radius, the adjusted state corresponds to a localized vortex (Figure 25(a)). The velocity profile exhibits a core solid rotation analogous to Rankine vortices. For small Burger numbers (Figure 25(b)), the adjusted state corresponds to a circular jet (i.e. a circular veloc-

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Figure 25. Velocity (dashed lines) and thickness profiles (solid lines) predicted by the standard geostrophic adjustment for two different sizes of the initial PV-patch: rc = Rc /Rd 0.45 (a) and rc = Rc /Rd 4.47 (b). The anticyclonic (thin line) and the cyclonic (thick line) profiles are given respectively for λ = 0.5 and λ = −0.5.

ity ring). For all these cases, the maximum velocity radius corresponds to rf , the final position of the PV jump. Unlike, the “warm-core” lens configuration, the velocity profiles for PV patches are always continuous. Besides, in agreement with previous studies (Kuo and Polvani, 2000), the geostrophic adjustment process

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Figure 26. Profiles of the upper layer thickness for a cyclonic (a) and an anticyclonic (b) vortex resulting from the initial PV-patch rc = Rc /Rd 3.47 (Bu = 0.083) and λ = 0.5 or λ = −0.5. The corresponding velocity profiles are displayed in (c) and (d). All the profiles were time-averaged over one inertial period Tf . These mean profiles are shown at various time: t = 2Tf (filled circle), t = 5Tf (filled triangle), t = 10Tf (open circle) and t = 20Tf (open square).

induces a cyclone–anticyclone asymmetry. For the same amplitude of the initial potential energy fluctuation, the cyclonic structures will be here more intense than the anticyclonic ones. Indeed, according to Figure 25, for the same relative amplitude of the initial thickness anomaly, the maximum velocity of cyclonic vortices (vmax /(f Rd ) = 0.3 for λ = −0.5 and rc = 4.47) will always be higher than of the anticyclonic ones (vmax /(f Rd ) = −0.2 for λ = 0.5 and rc = 4.47). 3.2.4. Mean adjusted state As for the warm-core lens configuration, we used time averaging over one or two inertial periods Tf in order to extract the slow dynamics of the height profile and

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Figure 27. Spatio-temporal diagram of the relative fluctuations of the upper layer thickness for an anticyclonic (a) and a cyclonic (b) PV-patch. The time evolves along the y axis from top (t = 0) to bottom (t = 7Tf ), while the layer thickness is plotted along the x axis corresponding to a full length of 260 mm. The grayscale levels were decalibrated and intensified in order to enhance the contrast. The white rectangular area on both panels corresponds to the initial positive (λ = 0.5) or negative (λ = −0.5) height anomaly.

the velocity field. According to Figure 26, we observe that both the mean height and the velocity profiles remain almost constant for several inertial periods, at least up to 20Tf for PV patches having small Burger number values (Bu = 0.083 in Figure 26). Hence, the time-averaged state have reached an equilibrium even though small wave motions could be detected both in the inner region of the PV anomaly (Figure 27) and the outer region. For the cyclonic structure, the mean adjusted state coincides perfectly with the predictions of standard geostrophic adjustment (Figures 26(a), (c)). However, for the anticyclonic structure a significant discrepancy occurs for the velocity field (Figure 26(d)). The maximum velocity is at least two times smaller than the predicted one. Such anticyclonic

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dissipation, in comparison with the non-dissipative predicted state, was observed in all the experiments from small to large Burger numbers Bu = rc−2 . Hence, the cyclone-anticyclone asymmetry becomes even more pronounced with this unpredicted dissipation. As for the warm-core lens configuration (which corresponds to the asymptotic limit λ → +∞), we could suspect that this rapid dissipation of kinetic energy is due to a transient three-dimensional instability which affects only the anticyclonic PV-fronts. However, dye visualisations appeared to be less efficient for the PV-patch experiments and we could hardly capture small-scale perturbations. We should note that an outcropping PV front (PV singularity) leads to intense velocities (Ro 1) in comparison with a PV-step front (PV discontinuity) which induces continuous velocity field close to the geostrophic balance. Indeed, the Rossby number never exceed Ro = 0.3 in the PV-patch experiment, therefore ageostrophic motions and related instabilities are expected to be weaker than for the outcropping configuration. 3.2.5. Inertial and sub-inertial wave activity The geostrophic adjustment process is expected to transfer a small (large) amount of the initial potential energy to the fast wave motion for PV patches having small (large) Burger number corresponding to rc = Rc /Rd > 1 (rc < 1). Hence, for the small Burger number case described above Bu = 0.083, the amplitude of the wave fluctuations were about few percents of the mean upper layer thickness. The sensitivity of the LIF technique was high enough to quantify this wave activity in the inner region (inside the PV anomaly) and in the outer region. Spatio-temporal diagrams (Hovmöller plots) of the wave oscillations within the cyclonic and the anticyclonic PV patch are rendered in Figure 27. This plot shows qualitatively the temporal variations (y axis) of the upper layer thickness across a diameter (x axis). The grayscale levels were decalibrated and intensified in order to enhance the contrast for a better visualisation. Unlike the outcropping configuration, the two-layer interface extends here to the whole experimental domain and the inertiagravity waves could freely propagate in the outer region outside the PV anomaly. Nevertheless, a significant wave activity remains for a long time (several inertial periods) inside the PV anomaly even if the mean steady state is already adjusted (Figure 26). A similar behavior was found in previous theoretical (Plougonven and Zeitlin, 2005) and numerical (Kuo and Polvani, 2000) studies dealing with sharp PV fronts in the RSW dynamics. The most striking result is a strong cyclone–anticyclone asymmetry in the wave frequency. According to the spatio-temporal plots, the oscillation is faster for the positive PV anomaly (Figure 27(b)) in comparison with the negative PV anomaly (Figure 27(a)). Indeed, if we measure the relative fluctuations of the upper layer thickness at the center (r = 0), the frequency is sub-inertial (ω/f 0.7) in the anticyclonic PV patch while the inertial (ω f ) frequency is found in the cyclonic PV patch (Figure 28).

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Figure 28. Evolution of the relative amplitude of the upper layer thickness at the center (r = 0) of the initial height anomaly. The case of a cyclonic (anticyclonic) PV-patch is displayed in the upper (lower) panel.

In the rotating shallow-water configuration, the appearance of sub-inertial modes (ω f ) corresponds to trapped modes in other words, these modes must have an evanescent structure outside the PV patch. If the relative vorticity is strong enough, a finite number of trapped modes could appear in anticyclonic vorticity region only (Kunze, 1985; Klein and Treguier, 1993; Young and Ben Jelloul, 1997; Llewellyn Smith, 1999; Plougonven and Zeitlin, 2005). The present experiment shows, for the first time in the laboratory, the existence of sub-inertial modes within an anticyclonic PV-patch. However, according to Figure 28, these modes have a finite lifetime. Unlike, the long-lived trapped modes these sub-inertial waves probably radiate their energy to the lower layer. According to Figure 28, there is no cyclone–anticyclone asymmetry in the lifetime of the inner wave modes. Besides, according to the spatio-temporal graph

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Figure 29.

Chapter 6. Experimental Reality of Geostrophic Adjustment

Experimental three-layer setup (a) and initial distribution of the potential vorticity (b).

displayed in Figure 27, the characteristic size of the inner wave structure, both the cyclonic (inertial) and the anticyclonic (sub-inertial) one, decays with time. This is a signature of dispersive effects, which could be induced by the high value of the wave aspect ratio α (2) = H2 /L 10 in the lower layer. 3.3. Uniform PV front 3.3.1. Initial state and experimental configuration The geostrophic adjustment of a motionless horizontal density gradient generally leads to a baroclinic tilted front corresponding to a simplified model of synoptic atmospheric fronts. However, recent studies (Ou, 1984; Blumen and Wu, 1995; Kalashnik, 2004; Plougonven and Zeitlin, 2005) have shown that even if the initial unbalanced state is smooth, a well defined continuous adjusted state may no longer exist. Indeed, for the case of uniform potential vorticity when the horizontal density gradient is sharp enough the steady adjusted solution exhibits discontinuities in both the density and the velocity field when the front outcrop the top or the bottom boundary. We use a three layer setup to study the adjustment of a uniform PV front (Figure 29(a)). Two upper fluid layers having different density ρ1 and ρ2 but the same thickness H1 are initially separated by a bottomless cylinder. A third deep and dense lower layer acts as a neutral layer which separates the thin upper layers from the bottom boundary. According to the small thickness parameter δ = 0.125 and the weak motion in the lower layer we assume that two upper layers follow the reduced-gravity RSW equations. Hence, if the upper layers have exactly the

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369

same thickness, the initial PV distribution is uniform and have the same value inside (r < Rc ) and outside (r > Rc ) the cylinder (Figure 29(b)). However, as for the warm-core lens configuration, the PV distribution exhibits a singularity at r = Rc for both the inner layer ρ2 and the outer layer ρ1 due to the vanishing of the layer thickness. = √ We can define two baroclinic deformation radii namely: Rd ((ρ2 − ρ1 )/ρ2 )gh/f related to the tilted density interface between the two √ upper layers, RD = ((ρ3 − ρ2 )/ρ3 )gh/f related to the horizontal density interface between the upper layers and one barotropic deformation radius correspond√ ing to the dense bottom layer RB = gH /f 1 m. The density differences between the layers (ρ2 − ρ1 = 3–25 g l−1 ; ρ3 − ρ2 100 g l−1 ) were adjusted, in the present experiment, in order to get Rd = 2–3 cm RD 12 cm. Besides, the size of the rotating tank L = 45 cm was large enough (L Rc Rd ) to neglect side wall effects. More details on the experimental procedure are given in (Mitkin, Stegner, Zeitlin and Pichon, 2006). 3.3.2. Dynamical stages Several dynamical stages were observed during the adjustment process. Just after the withdrawal of the separating cylinder, the inner dense fluid spreads radially at the bottom interface. During this very initial stage, the flow exhibits strong three-dimensional motions (Figure 30(b)) identical to those in a gravity current’s head (Paterson, Simpson, Dalziel and van Heijst, 2006). At this stage horizontal vorticity is generated at the interface between the inner and the outer upper fluid layers. After half of the inertial period the radial tilting of the density front is stopped and a reverse flow occurs. Then, in about one inertial period this tilted baroclinic front reaches an equilibrium characterized by an oscillating mean state. Theses oscillations can be seen in the fluctuations of the extremal positions rin and rout of the tilted front (Figures 30(c) and (d)). At longer time (t = 5 − 10Tf ) this tilted front experiences a large-scale baroclinic instability. The initial volume of dense fluid loses its axial symmetry and splits in two vortices which move away from the center of the tank. Hence, the vertical laser sheet does not capture the central cross-section of the density field any more (Figure 30(e)). 3.3.3. Rotating shallow-water predictions We assume here that: viscosity and dissipative effects are negligible (Re 1, Ek 1); each layer follows the rotating shallow-water dynamics (α 1); top and bottom boundary conditions are free-slip and rigid lid (δ 1, Bu∗ 1). Under these assumptions, the geostrophic adjustment of the density front is now controlled by a single parameter, namely the Burger number Bu. We look here for an axisymmetric steady state, solution of the RSW equations in both the light outer layer 1 and the dense inner layer 2. For simplicity, we

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Figure 30. Vertical cross-section of the density front between the inner layer ρ2 (white region) and the outer layer ρ1 (dark region) visualized by LIV. The snapshots were taken at t = 0 (a), t = 0.5Tf (b), t = Tf (c), t = 1.5Tf (d) and t = 5Tf (e) were Tf = 2π/f = 5 s is the inertial period. This experiment corresponds to rc = Rc /Rd = 2.1 (Bu = 0.22; α = 1).

neglect the cyclostrophic terms in the horizontal momentum equations. Such approximation is valid when Rc Rd . In this case, the steady state satisfies an exact geostrophic balance and therefore using the pressure continuity at the interface we get v1 − v2 = ∂r η

(54)

where η is the dimensionless thickness of the inner layer 2. The Lagrangian PV conservation leads to a constant PV value for all fluid parcels in both layers: 1 + ζ1 = 1, 1−η 1 + ζ2 Q2 (r rout ) = =1 η

Q1 (r rin ) =

(55) (56)

where ζi = 1r ∂r (rvi ) is the relative vorticity and rin (rout ) is the position of the upper (lower) intersection of the tilted density front with the top (bottom) boundary (Figure 31(b)). Note that, the boundary conditions η(rin ) = 1 and η(rout ) = 0 im-

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Figure 31. Density front (a), velocity (b) and vorticity (c) fields in the inner and outer layers of the adjusted steady-state according to the Rossby adjustment theory when rc = 3.33 (Bu = 0.09).

ply a singularity in the PV field at the ends of both layers, even if the PV have the same constant value within the layers. Such singularities will be the source of discontinuities in the vorticity and velocity field of the adjusted state (Figures 31(b) and (c)). Then, the angular momentum conservation or the mass conservation gives both the same implicit relations between (rin , rout ) and the initial radius of the density front rc = Rc /Rd = Bu−1/2 : 2 2 + 2rin v1 (rin ) = rout + 2rout v2 (rout ). rc2 = rin

(57)

Besides, outside of the region of the tilted front (r rin and r rout ) there is no radial displacement of fluid parcels. Therefore, the angular momentum conservation implies: v2 (r rin ) = v1 (r rout ) = 0.

(58)

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Chapter 6. Experimental Reality of Geostrophic Adjustment

Then, according to equations (54), (55) and (56) we obtain the following system of linear equations: 1 1 ∂r 2 φ + ∂r φ − 2 + 2 φ = 0, (59) r r ∂r (rψ) = −r (60) where according to (57) and (58) φ(r) = v1 − v2 satisfies the following boundary conditions 1 r2 1 rc2 − rout φ(rin ) = (61) rin − c ; φ(rout ) = 2 rin 2 rout and ψ(r) = v1 (r) + v2 (r) =

1 rc2 −r . 2 r

(62)

For a given initial radius rc = Bu−1/2 we can solve numerically (59) with (61) using a standard shooting method. An example of height, velocity and vorticity profiles in both layers are given in Figure 33 corresponding to rc = 3.33 (Bu = 0.09). Due to the volume conservation in cylindrical geometry, the front displacement in the outer layer (rc − rin ) is not identical to the front displacement in the inner layer (rout − rc ). This leads to higher velocity amplitude in the outer layer (Figure 31(b)) according to (57). Even if the velocity field is strongly baroclinic (of opposite direction in the upper and the lower layer) the vorticity is anticyclonic in both layers (Figure 31(c)) and reaches the extreme value ζ = −f at both ends of the tilted density front. 3.3.4. Mean adjusted state As for the previous cases, we use a time averaging over one inertial period, in order to separate the slow dynamics of the mean front and the fast dynamics of the oscillations. This temporal averaging filters out the fast dynamics from both the density front and the azimuthal velocity field. According to Figure 32 the qualitative structure of the mean adjusted state measured in the experiment is in agreement with the geostrophic adjustment predicted by a simple two-layer RSW model. The averaged velocity profile, measured close to the upper free surface, is displayed in Figure 32(b). According to the standard inviscid adjustment model (solid line) the velocity is expected to be discontinuous in the outer layer at the upper edge of the front. This is obviously unrealistic in a physical system where dissipative processes occur. Hence, during the adjustment process a strong but continuous cyclonic shear is formed instead of the discontinuous velocity jump predicted by the inviscid theory. The width of this cyclonic shear is much smaller than the deformation radius (0.2 − 0.3Rd ).

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Figure 32. Comparison between the mean experimental adjusted state (dots) and the Rossby adjustment model (solid line). The vertical cross-section of the density profile (a) corresponds to Rc /Rd = 2.1 (Bu = 0.22) while the horizontal azimuthal velocity (b) measured close to the upper free surface corresponds to Rc /Rd = 3.33 (Bu = 0.09).

Besides, the vorticity in such thin shear layer exceeds the planetary vorticity (ζ = 3 − 4f in Figure 32(b)) and may induce fast small-scale instabilities. However, the spatio-temporal resolution of the particle image velocimetry we used could hardly capture such small-scale instability patterns. 3.3.5. Small-scale instabilities By means of LIV, we could visualize a horizontal cross-section of the sharp density gradient just below the free surface. The dynamical evolution of this sharp gradient is shown in Figure 33. After the release of the bottomless cylinder the upper front experiences a rapid radial contraction. Due to the angular momentum conservation, this radial contraction generates at the same time a strong azimuthal flow. During this very initial stage, small disturbances appear at the edge of the front. Using an edge detection image processing we could accurately measure the initial wavelength λ of this instability (Figure 33(b)). Due to its rapid growth rate this instability is probably not affected by the rotation and the wavelength of

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Figure 33. Visualization of small-scale disturbances at t = 0 (a), t = 0.25Tf (b), t = 0.5Tf (c) and t = 0.75Tf (d) in a horizontal cross-section of the density front (Bu = 0.09) just bellow the upper free surface. Local image processing of edge front detection are shown in (b) and (c), the black pixels corresponds to high values of the intensity gradient. The movie is available on http://gershwin.ens.fr/stegner.

the small-scale perturbations does not depend on the deformation radius (Mitkin, Stegner, Zeitlin and Pichon, 2006). In the present case, unlike the outcropping lens configuration, no secondary instability occurs and the non-linear saturation of the initial perturbation leads to the formation of strong cusps and small cyclones appear according to Figure 33(c). Here again the size of the intense cyclones could be much smaller that the deformation radius and remain independent from this latter. These cat-eye patterns look like a classical horizontal shear instability. Nevertheless, due to the baroclinic structure of the flow, the vertical extension of these small cyclones is limited and they should be formed preferentially at the top or the bottom edge of the density front. Besides, these vortices are transient features of the adjustment process. Indeed, after half an inertial period the front reaches its maximal contraction and the small cyclones disappear during the reverse oscillation (Figure 33(d)).

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4. What do we learn from laboratory experiments? Laboratory experiments can hardly reproduce the complex thermodynamics (moisture, turbulent boundary layer convection, evaporation, air-sea fluxes. . .) and the wide range of dynamical regimes (Re 1; δ 1) encounters in the atmosphere or the ocean. However, the physical modeling of rotating shallow water flows is very useful especially for the geostrophic adjustment process, where several dynamical features occur at various temporal and spatial scales. Previous laboratory experiments have shown that the geostrophic adjustment is a rapid process (Ungarish, Hallworth and Huppert, 2001; Bouruet-Aubertot and Linden, 2002; Rubino and Brandt, 2003; Thivolle-Cazat, Sommeria and Galmiche, 2005). But very few studies investigate the characteristic time of this process, especially when strong wave activity is present. According to all the cases we studied, a mean adjusted state is reached after approximately one or two inertial periods Tf . The rapidity of the geostrophic adjustment does not depend on the size or the amplitude of the initial unbalance state. The mean state is obtained from a simple time-averaging over Tf , in order to filter out the fast wave motion. We say that this averaged state reaches an equilibrium (i.e. gets adjusted) when the inverse scale of its temporal evolution remain small in comparison with the characteristic wave frequency. Hence, even if a strong wave activity is present in the initial region of unbalance, the mean flow could nevertheless be adjusted. This experimental observation is in good agreement with the standard hypothesis of dynamical splitting between the fast (ε 1) and the slow (ε 1) component of motion (Reznik, Zeitlin and Ben Jelloul, 2001; Zeitlin, Reznik and Ben Jelloul, 2003). In the limit of small Rossby numbers, the asymptotic analysis shows that the slow component of motion does not feel the fast one (Chapter 2). Therefore, the existence of a mean adjusted state does not depend on the presence (or not) of fast wave motion. Besides, the experimental results for both the warm-core lens configuration and the uniform PV-front configurations showed that a mean adjusted state could be extracted from the wave motion with a simple time-averaging even for finite Rossby numbers. Hence, according to the whole set of experiments, the fast component of motion seems to have only a weak influence (if any) on the evolution of the mean adjusted state for a wide range of parameters (Ro < 1, Bu 0.1–10, −0.5 < λ < 0.5). These two-layers or three-layers experiments also show that the PV conservation remain robust even if the initial state does not satisfy the assumptions of the rotating shallow-water model. Indeed, in almost all the cases, three-dimensional and non-hydrostatic motions (shocks or gravity current head) could occur in the early stage of adjustment. Nevertheless, the prediction of the RSW model based on the PV conservation gives a correct estimate of the mean adjusted state. A very

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good agreement is found for the cases of cyclonic PV front when there is no outcropping. However, the PV conservation can be locally broken in the case of outcropping fronts when the initial PV profile exhibits a singularity (i.e. the layer thickness vanish at a given position). In such case, all the experiments exhibit transient and three-dimensional instabilities localized around the PV singularity. These instabilities are an efficient mechanism of turbulent dissipation which rapidly cascades energy toward small scales in the frontal region. For the uniform PV front configuration, small and intense cyclones are formed in a very short time (∼0.5Tf ) during the adjustment of a large-scale anticyclonic front. The rapid formation of these structures, which are much smaller than the deformation radius, were not predicted by the standard scenario of adjustment and they could hardly be captured by standard numerical simulations which have limited spatial resolution. The laboratory experiment shows here a new mechanism of formation of small and intense structures within a large-scale synoptic front. The relaxation of any unbalanced initial state in a rotating shallow-water model will always lead to the emission of Poincaré waves (away from lateral boundaries). In a real laboratory experiment, both hydrostatic inertia–gravity waves and non-hydrostatic inertial waves could be emitted at the same time and the spectral gap between the fast and the slow component of motion could then be filled. However, according to our experiments and previous studies (Bouruet-Aubertot and Linden, 2002; Rubino and Brandt, 2003; Thivolle-Cazat, 2003) the energy released due to the wave modes during the adjustment is mainly concentrated around the inertial frequency. A significant wave activity remain for a long time (several inertial period) inside both the cyclonic and the anticyclonic structures even if the mean steady state is already adjusted. For some specific configurations the anticyclonic structures may exhibit sub-inertial oscillations. Such wave activity is in good agreement with the RSW model predictions (Chapters 2 and 5) and confirm the dynamical splitting between the fast waves and the slow adjusted motion. However, the inertia–gravity waves detected in the experiment have a dispersive behavior due to the finite value of the aspect ratio parameter α. This could explain why we did not see any evidence of the wave breaking events predicted in the RSW framework (Chapter 5). The small-scale shocks we observed in the very initial stage of adjustment seems to be due to a local overturning event rather than a propagating wave leading to breaking.

Acknowledgements The studies on geostrophic adjustment presented in this chapter were done in collaboration with P. Bouruet-Aubertot, T. Pichon, V. Mitkin and V. Zeitlin which are warmly acknowledged.

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List of Contributors

F. Bouchut, Département de Mathématiques et Applications, Ecole Normale Supérieure et CNRS, UMR 8553, 45, rue d’Ulm, 75230 Paris Cedex 05, France; [email protected] (Ch. 4) S.B. Medvedev, Institute of Computational Technologies, 6 Lavrentiev av., 630090 Novosibirsk, Russia; [email protected] (Ch. 3) G.M. Reznik, P.P. Shirshov Institute of Oceanology, Moscow, Russia; [email protected] yahoo.com (Ch. 2) A. Stegner, Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France; [email protected] (Ch. 6) V. Zeitlin, Laboratoire de Météorologie Dynamique, CNRS, ENS, 24, rue Lhomond, 75231 Paris Cedex 05, France; [email protected] (Chs. 1, 2, 5)

381

Author Index

Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages.

Abgrall, R., 231, 256 Abramowitz, M., 15, 31, 35, 43 Adrian, R.J., 340, 377 Afanasyev, Y.D., 346, 347, 377 Afanasyev, Ya.D., 334, 377 Allen, J.S., 9, 10, 43, 51, 64, 118 Arnold, V.I., 124, 133, 137, 157, 163, 184 Aubertin, A., 222, 256 Audusse, E., 214, 222, 223, 231, 237, 240, 246, 254

Boyd, J.P., 108–111, 116, 118, 299, 300, 305, 310, 319, 320 Bracco, A., 341, 377 Brandt, P., 353, 375, 376, 378 Bretherton, C.S., 107, 118 Bretherton, F.P., 4, 44 Bristeau, M.-O., 214, 222, 223, 231, 237, 240, 246, 254 Bruno, A.D., 137, 185 Bühler, O., 33, 43, 277, 304, 320

Babin, A., 65, 118, 154, 184 Baer, F., 41, 43, 52, 118, 124, 131, 184 Bakai, A.S., 148, 184 Bale, D.S., 214, 221, 231, 254 Balkovsky, E., 175, 184 Baraille, R., 214, 255 Ben Jelloul, M., 16, 23, 25, 44, 45, 54, 55, 58, 69, 70, 72, 74, 79, 106, 118, 120, 267, 321, 367, 375, 378, 379 Benilov, E.S., 24, 25, 43, 71, 77, 118, 120 Bermúdez, A., 214, 220, 254 Berndt, S.B., 301, 319 Beunier, M., 334, 378 Bialynicki-Birula, I., 175, 185 Biello, J.A., 298, 320 Birkhoff, G.D., 157, 185 Blumen, W., 19, 43, 48, 118, 368, 377 Bogaevskii, V.N., 154, 185 Bokhove, O., 10, 43, 50–52, 64, 118, 120, 125, 187 Botchorishvili, R., 214, 254 Bouchut, F., 197, 199, 200, 206, 209, 211, 214, 215, 220, 222–225, 229, 231, 233, 237, 245, 246, 254, 256, 270, 272, 280, 286, 305, 309, 310, 312, 319 Bouruet-Aubertot, P., 337, 349, 350, 353, 354, 357, 358, 375, 376, 377, 378

Camassa, A., 42, 43 Carnevale, G.F., 331, 377 Carr, J., 124, 185 Castro, M.J., 214, 220, 231, 255, 256 Chacón Rebollo, T., 214, 231, 255 Chang, P., 315, 320 Charney, J.G., 14, 43, 125, 131, 185 Chinnayya, A., 214, 255 Cho, H.-R., 182, 185 Cicogna, G., 154, 185 Cushman-Roisin, B., 14, 24, 43, 72, 118, 338, 377 Dalziel, S.B., 344, 369, 377, 378 Deconinck, H., 231, 256 Dewar, W.K., 54, 119 Didelle, H., 327, 336, 378 Domínguez Delgado, A., 214, 231, 255 Dorofeyev, V.L., 82, 119 Dritschel, D.G., 15, 43, 50, 51, 119 Dubrovin, B.A., 166, 167, 175, 185 Eames, I., 331, 377 Embid, P.F., 65, 119, 154, 185 Engelberg, S., 268, 269, 320 383

384

Author Index

Falkovich, G.E., 12, 43, 50, 67, 69, 72, 119, 125, 138, 145, 147, 148, 150, 163, 170, 173–175, 185, 187, 258, 278, 320 Farge, M., 343, 349, 350, 378 Fedorov, A.V., 300, 301, 320 Fernández Nieto, E.D., 214, 231, 255 Fernando, H.J.S., 354, 377 Ferreiro, A.M., 220, 255 Fincham, A.M., 340, 377 Flierl, G.R., 14, 26, 43, 44 Flor, J.B., 331, 377 Flores, C., 334, 378 Fomenko, A.T., 158, 185 Ford, R., 31, 32, 43 Gaeta, G., 154, 185 Gallardo, J.M., 231, 255 Gallouët, T., 214, 220, 231, 255 Galmiche, M., 352, 375, 378 García-Navarro, P., 214, 220, 255 García-Rodríguez, J.A., 220, 255 Ge, Z., 175, 185 Gent, P.R., 51, 119 Ghil, M., 327, 336, 378 Giacaglia, G.E.O., 154, 163, 185 Gill, A.E., 2, 19, 21, 25, 34, 35, 41, 43, 82, 109, 119, 150, 185, 259, 320, 325, 326, 355, 377 Godlewski, E., 191, 193, 194, 196, 197, 199, 207, 213, 235, 237, 255 Goncharov, V., 163, 185 González-Vida, J.M., 220, 255 Gossard, E.E., 291, 320 Gosse, L., 214, 255 Greenberg, J.-M., 214, 255 Greenberg, J.M., 214, 255 Greenspan, G.P., 325, 377 Griffiths, R.W., 352, 353, 377 Grimshaw, R.H.G., 55, 70, 71, 81, 82, 84, 86, 94, 96–98, 104, 120, 277, 320 Gryanik, V., 29, 31, 43 Haine, T.W.N., 347, 378, 379 Hallworth, M.A., 375, 378 Hart, D.P., 341, 377 Hart, J.E., 329, 347, 377 Harten, A., 200, 207, 255 Hayes, W.D., 301, 320 Helfrich, K.R., 82, 104, 119 Hérard, J.-M., 214, 220, 231, 255

Hermann, A.J., 82, 119 Holford, J.M., 344, 377 Holm, D.D., 9–11, 15, 42, 43, 44 Holton, J.R., 2–5, 9, 13, 14, 24, 33, 44, 146, 185, 264, 320 Hooke, W.H., 291, 320 Hoskins, B.J., 4, 44, 51, 119 Houghton, D.D., 271, 272, 320 Hunt, J.C.R., 354, 377 Huppert, H.E., 353, 375, 378 Il’yashenko, Yu.S., 124, 133, 137, 184 Ishibashi, M., 331, 377 Janenko, N.N., 263, 270, 321 Jeffreys, H., 5, 44 Jin, S., 214, 217, 225, 255 John, F., 139, 140, 185 Johnson, E.R., 82, 119 Johnson, J.A., 334, 377 Kalashnik, M.V., 368, 377 Kamenkovich, V.M., 70, 89, 105, 119 Karasev, M.V., 174, 175, 185 Kats, E.I., 157, 185 Katsaounis, T., 214, 217, 231, 255 Kevlahan, N.K.-R., 301, 320 Killworth, P.D., 54, 119 Kittelman, S., 347, 377 Klein, P., 367, 377 Klein, R., 107, 110, 119, 214, 222, 223, 231, 237, 246, 254 Kloosterziel, R.C., 331, 377 Kompenhans, J., 341, 378 Koshlyakov, M.N., 70, 119 Kraft, M., 231, 256 Krasitskii, V.P., 173, 185 Kreiss, H.-O., 125, 185 Kruse, H.P., 175, 185 Kunze, E., 367, 377 Kuo, A.C., 82, 104, 119, 272, 320, 363, 366, 377 Kurganov, A., 214, 225, 255 Kuznetsov, E.A., 67, 69, 72, 119, 145, 148, 150, 175, 185, 187 Lajeunesse, E., 222, 256 Lamb, H.H., 28, 44 Landau, L.D., 262, 320

Author Index Larichev, V.D., 18, 44, 82, 119 Lax, P.D., 200, 207, 255, 268, 320 Le Sommer, J., 34, 44, 55, 108, 110, 111, 113–116, 119, 222, 229, 245, 254, 255, 270, 272, 280, 282, 285, 286, 288, 289, 299, 300, 304, 305, 309, 310, 312, 313, 315, 317, 319, 320 Lebedev, V.V., 157, 185 LeBlond, P.H., 21, 44 Legras, B., 51, 52, 120, 124, 131, 187 Leith, C.E., 124, 185 LeRoux, A.-Y., 214, 255 LeVeque, R.J., 197, 214, 221, 231, 254, 256 Levermore, C.D., 42, 43 Levy, D., 214, 225, 255 Li, X., 315, 320 Lifshits, E.M., 262, 320 Lighthill, J., 28, 33, 44, 97, 117, 119, 270, 303, 320 Lindahl, E.G., 346, 347, 377, 378 Linden, P.F., 352, 353, 375, 376, 377 Llewellyn Smith, S., 367, 377 Lorenz, E.N., 52, 119, 124, 185 Lorenz, J., 125, 185 Lukacova-Medvidova, M., 231, 256 Lvov, V.S., 138, 147, 148, 163, 170, 173, 175, 187 Machenhauer, B., 41, 44, 52, 119, 124, 125, 185 Macías, J., 220, 255 Mahalov, A., 65, 118, 154, 184, 354, 377 Majda, A.J., 34, 44, 65, 107, 108, 110, 119, 154, 185, 298, 320 Mangeney-Castelnau, A., 222, 256 Marche, F., 222, 237, 246, 256 Marchuk, G.I., 157, 185 Margolin, L.G., 42, 44 Marsden, J.E., 11, 44, 175, 185 Marubayashi, K., 331, 377 Maslov, V.P., 174, 175, 185 Masuda, A., 331, 377 McIntyre, M.E., 32, 33, 43, 304, 320 McKean, H.P., 137, 186 McKenzie, J.F., 145, 150, 186 McWilliams, J.C., 51, 53, 82, 119, 341, 377 Medvedev, S.B., 12, 43, 50–52, 67, 69, 72, 119, 123, 125, 138, 145, 147, 148, 150, 153, 154, 160, 169, 173, 174, 176, 177, 179, 180, 184, 185–187, 258, 260,

385

264–267, 269, 272, 277, 278, 280, 282, 285, 286, 288, 289, 298, 320, 321 Melville, W.K., 82, 120, 300, 301, 320 Mendez, A.J., 346, 378 Metais, O., 334, 378 Migdal, A.B., 267, 289, 320 Mikhailov, A.V., 175, 185 Miles, J., 42, 44 Milnor, J., 158, 186 Mitkin, V., 369, 374, 377 Mitran, S., 214, 221, 231, 254 Mohebalhojeh, A.R., 50, 51, 119 Mokhov, O.I., 170, 171, 186 Monin, A.S., 56, 70, 119 Morrison, P.J., 175, 185 Murante, G., 341, 377 Mysak, L.A., 21, 44 Nadiga, B.T., 42, 44 Natvig, J.R., 222, 231, 256 Nezlin, M.V., 18, 44, 144, 145, 149, 186, 327, 331, 378 Nicolaenko, B., 65, 118, 154, 184 Nielsen, A.H., 327, 331, 378 Nikolenko, N.V., 137, 186 Noelle, S., 222, 231, 256 Nof, D., 271, 320 Nore, C., 169, 172, 186 Norton, W.A., 32, 43 Noussair, A., 214, 255 Novikov, S.P., 166, 167, 175, 185, 186 Nycander, J., 330, 378 Obukhov, A.M., 5, 44, 56, 119, 120 Olver, P.J., 163, 167, 175, 186 Ostrovsky, L.A., 277, 320, 321 Ou, H.W., 368, 378 Ozawa, T., 137, 186 Pacanowski, R.C., 315, 320 Pacheco, J.R., 354, 377 Pankratz, N., 222, 231, 256 Parés, C., 214, 220, 231, 255, 256 Paterson, M.D., 369, 378 Pavlov, V., 163, 185 Pedlosky, J., 2, 4, 13, 14, 16, 18, 22, 24, 25, 44, 51, 73, 82, 89, 105, 119, 120, 126, 145, 186, 325, 326, 328, 330, 337, 378 Peltier, W.R., 334, 377

386

Author Index

Peregrine, D.H., 271, 321 Perret, G., 343, 349, 350, 378 Perthame, B., 214, 217, 220, 222, 223, 225, 231, 237, 246, 254–256 Petviashvili, V.I., 145, 149, 150, 154, 186 Philander, S.G., 110, 120 Phillips, N.A., 25, 44, 82, 120 Pichon, T., 334, 337, 343, 349, 350, 353, 354, 357, 358, 369, 374, 377, 378 Pirulli, M., 222, 256 Piterbarg, L.I., 169, 170, 187 Plougonven, R., 184, 187, 260, 264–267, 269, 272, 277, 279, 280, 282, 285, 286, 288, 289, 292, 295–298, 300, 320, 321, 366–368, 378 Pokhotelov, O.A., 145, 149, 150, 154, 186 Polvani, L.M., 272, 320, 363, 366, 377 Poulin, F.J., 26, 44 Povzner, A., 154, 185 Pratt, L.J., 82, 104, 119, 271, 302, 321 Provenzale, A., 341, 377 Puppo, G., 222, 231, 256 Raffel, M., 341, 378 Rasmussen, J.J., 327, 331, 378 Ratiu, T., 11, 44 Raviart, P.-A., 191, 193, 194, 196, 197, 199, 207, 213, 235, 237, 255 Read, P.L., 347, 378, 379 Reznik, G.M., 16, 18, 23–25, 34, 43–45, 54, 55, 58, 69–71, 74, 77, 79, 81, 82, 84, 86, 89, 91, 94, 96–98, 103–106, 108, 110, 111, 113–116, 118–120, 222, 229, 245, 255, 267, 298–300, 304, 309, 313, 317, 320, 321, 375, 378, 379 Rhines, P.B., 82, 119, 346, 347, 377, 378 Ricchiuto, M., 231, 256 Riley, J., 334, 378 Ripa, P., 108, 111, 116, 120, 299, 300, 305, 314, 321 Roe, P.L., 206, 256 Romanova, N.N., 14, 44 Rosales, R.R., 298, 320 Rossby, C.-G., 13, 44, 118, 120, 260, 321 Rossmanith, J.A., 214, 221, 231, 254 Rozhdestvenskii, B.L., 263, 270, 321 Rubenchik, A.M., 148, 174, 187 Rubino, A., 353, 375, 376, 378

Sakai, S., 26, 27, 44, 117, 118, 120 Salmon, R., 11, 42, 44 Saujani, S., 32, 44, 51, 120 Schär, C., 271, 321 Schulman, E.I., 138, 155, 187 Scovel, C., 175, 185 Seguin, N., 214, 220, 231, 255 Serre, D., 191, 193, 241, 256 Shatah, J., 137, 186 Shepherd, T.G., 32, 44, 50–52, 64, 118, 120, 125, 151, 169, 172, 176, 180, 182, 185–187 Shrira, V.I., 277, 320, 321 Shu, C.-W., 231, 256 Shukla, P.K., 145, 150, 186 Simeoni, C., 214, 217, 220, 225, 231, 255, 256 Simpson, J.E., 369, 378 Smith, R.B., 271, 321 Smolarkevich, P.K., 42, 44 Snezhkin, E.N., 18, 44, 144, 145, 149, 186, 327, 331, 378 Sobel, A.H., 107, 118 Sommeria, J., 352, 375, 378 Sopta, L., 231, 256 Spedding, G.R., 340, 377 Stegner, A., 14, 44, 72, 120, 327, 330, 331, 334, 336, 337, 343–345, 349, 350, 353, 354, 357, 358, 369, 374, 377, 378 Stegun, I.A., 15, 31, 35, 43 Stenflo, L., 145, 150, 186 Stenum, B., 327, 331, 378 Stepanyants, Yu.A., 277, 320 Sutyrin, G.G., 24, 43, 55, 91, 103, 105, 120, 338, 377 Tabak, E.G., 298, 320 Tang, B., 24, 43, 338, 377 Teinturier, S., 327, 336, 378 Temperton, C., 124, 186 Thivolle-Cazat, E., 339, 352, 375, 376, 378 Tomasson, G.G., 82, 120 Toro, E.F., 197, 207, 256 Treguier, A.M., 367, 377 Tribbia, J.J., 41, 43, 52, 118, 124, 131, 184, 187 Tsutaya, K., 137, 186 Tsutsumi, Y., 137, 186 Turner, C.V., 298, 320

Author Index Ungarish, M., 353, 375, 378 Vallis, B., 325, 328, 355, 378 Vallis, G.K., 50–52, 64, 120, 125, 187 van de Konijnenberg, J.A., 327, 331, 378 van Heijst, G.J.F., 329, 331, 369, 377–379 Van Leer, B., 200, 207, 255 Vásquez, M.E., 214, 220, 254 Vasseur, A., 214, 254 Vautard, R., 51, 52, 120, 124, 131, 187 Vázquez-Cendón, M.E., 214, 220, 255, 256 Viboud, S., 327, 336, 378 Vilotte, J.-P., 222, 256 Vladimirov, V.A., 182, 185, 187 Voropayev, S.I., 354, 377 Vukovic, S., 231, 256 Warn, T., 50–52, 64, 120, 125, 187 Weinstein, A., 11, 44, 158, 163, 176, 187 Weiss, J.B., 341, 377 Whitham, G.B., 3, 33, 44, 97, 120, 269, 271, 290, 321 Willert, C.E., 341, 378 Williams, G.P., 14, 44 Williams, P.D., 347, 378, 379

387

Wolfram, S., 267, 321 Wu, R., 368, 377 Xing, Y., 231, 256 Yakhot, V., 175, 187 Yamagata, T., 14, 44 Yanase, S., 334, 378 Yang, J., 312, 321 Young, W.R., 70, 120, 367, 378 Yu, L., 312, 321 Zakharov, V.E., 138, 147, 148, 150, 155, 163, 169, 170, 173–175, 187 Zavala Sanson, L., 329, 379 Zeitlin, V., 14–16, 23, 25, 29, 31, 34, 44, 45, 54, 55, 58, 69, 70, 72, 74, 79, 106, 108, 110, 111, 113–116, 118–120, 138, 150, 151, 153, 154, 171, 184, 186, 187, 222, 229, 245, 254, 255, 258, 260, 264–267, 269, 270, 272, 277, 279, 280, 282, 285, 286, 288, 289, 292, 295–300, 304, 305, 309, 310, 312, 313, 315, 317, 319–321, 327, 330, 331, 344, 345, 366–369, 374, 375, 377–379

Subject Index

complex coordinates, 29, 66 complex potential, 29 complex velocity, 29, 65, 77 conservation laws, 193 conservative, 193, 198, 203, 217, 232 consistency, 199, 201, 219 Coriolis parameter, 3 critical layers, 33 cusps, 280, 296, 297, 300

1.5d model, 258, 259 2d Euler equations, 15, 28 β-plane, 4, 16, 18, 70, 72, 121, 123, 137, 138, 145, 148, 150, 151, 171, 183 β-plane approximation, 9 A absolute vorticity, 9 acoustic waves, 258 advective time scale, 13, 54 ageostrophic instabilities, 26 ageostrophic velocity, 14, 17

D density-weighted vertical average, 6 diagnostic equation, 14, 51 discontinuity calculus, 271 dispersion equation, 39 dispersion relation, 11, 17, 23, 285 drift waves, 18

B balanced dynamic, 64 balanced manifold, 52 balanced model, 48, 64, 69 baroclinic, 8, 77, 78, 285 baroclinic deformation radius, 23 baroclinic instability, 25, 287 barotropic, 8, 77, 78 barotropic gas, 8, 258 barotropic jet, 288 Bernoulli function, 302 Bessel equation, 30 boundary conditions, 4, 196, 212, 237 boundary Kelvin wave, 20, 21, 27, 80, 87, 94–96, 98, 101, 103–105 Burger number, 13, 18, 23, 24, 31, 54, 69, 272, 287, 289

E entropy, 194 entropy condition, 271, 272 entropy inequalities, 194, 199, 203, 220 equatorial β-plane, 33, 259 equatorial deformation radius, 34 equatorial inertia–gravity waves, 38, 108, 113, 298, 301, 304, 307 equatorial Kelvin wave, 35, 106, 107, 110, 113, 115, 298, 300, 315 equatorial Rossby wave, 38, 107, 110, 113, 115, 298, 305, 312, 315 equatorial waveguide, 35, 297, 299 equivalent depth, 294 equivalent height, 23, 285

C Casimir invariant, 73, 176, 177 Cayley formula, 160, 162 center of vorticity, 29 centered flux, 232, 234 CFL condition, 199, 202, 205, 237, 242 characteristic function, 122, 123, 157–160 characteristics, 269 Charney balance equation, 123, 125, 131, 183 Charney–Obukhov equation, 18 compensating jet, 101–103, 105

F f -plane, 3, 9, 11, 17, 49, 106, 121, 123–126, 131, 134, 137, 183, 258, 259 fast manifold, 123, 134 fast motion, 11, 19, 55, 56, 60, 61, 75, 83, 106, 111, 114, 265, 285 fast–slow splitting, 16, 48, 57, 72, 88, 96, 104, 106, 111, 115, 116, 258 fast-time average, 54, 58, 83 389

390

Subject Index

finite volumes, 198, 242 finite-amplitude wave, 274, 279, 281, 292 finite-volume numerical method, 272 frontal geostrophic (FG) dynamics, 14, 15, 24, 65, 77 Froude number, 26, 29, 34 G gas dynamics, 193, 195 geostrophic adjustment, 12, 13, 19, 48, 54, 74, 104, 106, 258, 260, 268, 299 geostrophic balance, 12, 13, 49, 57, 59, 60, 104, 260, 262, 265, 284, 312 geostrophic mass, 81, 91, 101, 104 geostrophic momentum, 260, 261, 292 geostrophic velocity, 14 Green function, 15 H Hamiltonian, 10, 262 Hasegawa–Mima equation, 18 Helmholtz equation, 56 hodograph method, 262 homological equation, 138, 140, 142, 148, 154 I improved quasi-geostrophic PV equation, 64, 76, 98, 106 inertia–gravity wave (IGW), 11, 48, 64, 74, 76, 104, 124, 125, 127, 136–138, 145, 147–150, 152, 153, 173, 174, 258, 260, 280, 284, 294, 314 inertial instability, 334 inertial oscillations, 12, 65, 67, 78, 79, 268, 272 initialization, 16, 52, 62, 111, 121, 123–125, 129, 131, 132, 183 instanton, 264 internal inertia–gravity waves, 23 invariant domains, 199, 205, 207, 237 invariant manifold, 121, 124, 125, 132–134, 163, 183, 184 J Jacobi transformation, 261 Jacoby identity, 168, 169 K Kelvin circulation theorem, 9

Kelvin front, 312, 314 Kelvin wave, 150, 151 Kelvin–Helmholtz instability, 26, 259, 287, 289, 291 kink, 296 Kircchoff vortex, 29, 31 Klein–Gordon equation, 56, 57 Korteweg–de Vries equation, 108, 115, 309 L Lagrangian conservation, 9 Lagrangian variables, 260, 261 linearized equation, 11, 17, 18, 20, 23, 25, 35, 49, 265, 283 linearized potential vorticity, 49 loss of hyperbolicity, 259, 291 M mass Lagrangian variable, 263, 290 mass trapping, 312, 315 material surfaces, 5 mixing, 315, 316 modons, 18, 309, 310, 312, 314, 315, 319 multipolar expansion, 29 N no-flux boundary condition, 19, 80 nonlinear frequency shift, 114 nonlinearity parameter, 13, 34, 54 numerical flux, 198, 201, 203, 218 P parabolic cylinder functions, 35, 109 plane-parallel wave, 258 planetary vorticity, 9, 17 point vortex, 15, 28 Poisson bracket, 10, 175–177, 179–182, 184 potential vorticity measurements, 339, 340, 350, 352 potential vorticity (PV), 9, 14, 20, 260, 263, 283, 286, 301, 305 primitive equations, 3 prognostic equations, 14, 51 pseudo-height, 4 PV anomaly, 9, 12, 17, 49, 50, 278 PV conservation, 11, 14, 16, 22, 74 Q QG potential vorticity, 24 quasi-geostrophic PV, 14, 22, 260, 265

Subject Index quasi-geostrophic PV equation, 59 quasi-geostrophic (QG) dynamics, 14, 15, 24, 25, 56, 75, 82 quasi-linear systems, 191, 215, 241, 268 quasi-stationary states, 267 R Rankine–Hugoniot conditions, 194, 195, 215, 271, 302, 303 rapid time scale, 55 recirculation region, 312, 315 reconstruction operator, 232, 234, 235 reduced gravity, 23, 283, 292 relative vorticity, 9 resonance condition, 134, 136, 138, 144 resonant interactions of wave triads, 258 Reynolds number, 4 Ricatti equation, 269 Riemann invariants, 269, 273 Riemann solvers, 200, 204 Rossby deformation radius, 13, 51, 75, 123, 125, 126, 128, 129, 131, 132, 137, 138, 144–146, 150, 183, 263 Rossby number, 13, 14, 16, 18, 23, 27, 28, 32, 34, 50, 54, 59, 64, 110, 132, 272 Rossby wave, 17, 18, 27, 122, 123, 125, 137, 138, 145, 147–153, 163, 169, 170, 172–174, 183 Rossby–Kelvin instability, 26, 287 rotating shallow water model, 2 S scalar nonlinearity, 18 Schrödinger equation, 67–69, 109, 266, 288 screening, 15 shock, 32, 258, 268–273, 291, 299, 301, 302, 304, 312 slaved variable, 51 slaving variable, 51 slow manifold, 16, 53, 121, 124, 125, 127–129, 131–136, 183, 258, 260, 263

391

slow motion, 11, 15, 16, 19, 24, 55, 56, 59–61, 70, 75, 83, 104–106, 111, 114, 122, 125, 128, 131–137, 145, 265 solitons, 108–110, 115, 264, 301, 305, 309, 310, 312, 313, 319 sound speed, 260, 270 spectral gap, 11, 30, 42, 104, 299 stratification parameter, 23 sub-barrier tunneling, 267 sub-inertial modes, 366, 367, 376 superbalance condition, 52 symmetric instability, 259, 264, 267, 286, 289, 291 synoptic-scale motions, 2, 13, 16 T tangent plane approximation, 3 three-dimensional instabilities, 357, 358, 366, 376 tracer transport, 311, 314–316 transonic shocks, 273 V variational principle, 262 vertical averaging, 5 vortex pair, 28, 29 vorticity equation, 9, 28 W wave emission, 28, 62 wave-breaking, 97, 108, 110, 116, 258, 269, 270, 289, 291, 298–300, 305, 312, 317 wave-drag, 63, 68, 115 wave-trapping, 259, 266, 286, 289, 317 weak solution, 193, 195, 215, 270 weak turbulence, 258 well-balanced schemes, 214, 218, 233, 242 Y Yanai wave, 36, 108, 113, 150, 151, 298, 304, 306

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