PREDICTABILITY AND MODELLING IN OCEAN HYDRODYNAMICS
FURTHER TITLES IN THIS SERIES 1 J.L. MERO T H E MINERAL RESOURCES...
58 downloads
832 Views
6MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
PREDICTABILITY AND MODELLING IN OCEAN HYDRODYNAMICS
FURTHER TITLES IN THIS SERIES 1 J.L. MERO T H E MINERAL RESOURCES O F THE SEA 2 L.M.FOMIN THE DYNAMIC METHOD I N OCEANOGRAPHY 3 E.J.F. WOOD MICROBIOLOGY OF OCEANS AND ESTUARIES 4 G.NEUMANN OCEAN CURRENTS 5 N.G. JERLOV OPTICAL OCEANOGRAPHY 6 V.VACQUIER GEOMAGNETISM IN MARINE GEOLOGY 7 W.J. WALLACE THE DEVELOPMENT O F THE CHLORINITY/SALINITY CONCEPT I N OCEANOGRAPHY 8 E.LISITZIN SEA-LEVEL CHANGES 9 R.H.PARKER THE STUDY O F BENTHIC COMMUNITIES 10 J.C.J. NIHOUL (Editor) MODELLING O F MARINE SYSTEMS 11 0.1.MAMAYEV TEMPERATURE-SALINITY ANALYSIS OF WORLD OCEAN WATERS 12 E.J. FERGUSON WOOD and R.E. JOHANNES TROPICAL MARINE POLLUTION 13 E. STEEMANN NIELSEN MARINE PHOTOSYNTHESIS 14 N.G. J E R L O V MARINE OPTICS 16 G.P. GLASBY MARINE MANGANESE DEPOSITS 16 V.M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS 17 R.A.GEYER SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING 18 J.W. CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS 19 J.C.J. NIHOUL (Editor) BOTTOMTURBULENCE 20 P.H. LEBLOND and L.A. MYSAK WAVES I N THE OCEAN 21 C.C. VON DER BORCH (Editor) SYNTHESIS O F D E E P S E A DRILLING RESULTS IN THE INDIAN OCEAN 32 P. DEHLINGER MARINE GRAVITY 23 J.C.J. NIHOUL HYDRODYNAMICS OF ESTUARIES AND FJORDS 24 F.T. BANNER, M.B. COLLINS and K.S. MASSIE (Editors) T H E NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED AND T H E SEA IN MOTION
Elsevier Oceanography Series, 25
FORECASTlNG Predictability and Modelling in Ocean Hydrodynamics PROCEEDINGS OF THE 10th INTERNATIONAL LIkGE COLLOQUIUM ON OCEAN HYDRODYNAMICS
Edited by JACQUES C.J. NIHOUL Professor of Ocean Hydrodynamics, University of Liege, Lihge, Belgium
ELSEVIER SCIENTIFIC PUBLISHING COMPANY 1979 Amsterdam - Oxford - New York
ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 211,1000AE Amsterdam, The Netherlands Distributors for the United States and Canado: ELSEVIER/NORTH-HOLLAND INC. 52,Vanderbilt Avenue New York, N.Y. 10017
Library of Congress Cataloging in Publication D8ta
International Liege Colloquium on Ocean Hydrodynamics, loth, 1978. Marine forecasting. (Elsevier oceanography s e r i e s ; 25) Bibliography: p. Includes index. 1. Oceanography--Mathematical mdels--Congresses. 2. Hydrodynamics--Mathenatice.l models-4onaresses. I. c h o u l i Jacques C. J. 11. T i t l e . 551.4'7'0a184 79-u360
ISBN 0-444-41797-4 (Vol. 26) ISBN 0-444-41 623-4(Series)
0 Elsevier Scientific Publishing Company, 1979
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, P.O. Box 330, 1000 A H Amsterdam, The Netherlands Printed in The Netherlands
V
FOREWORD The International Liege Colloquia on Ocean Hydrodynamics are organized annually.
Their topics differ from one year to another and
try to address, as much as possible, recent problems and incentive new subjects in physical oceanography. Assembling a group of active and eminent scientists from different countries and often different disciplines, they provide a forum for discussion and foster a mutually beneficial exchange of information opening on to a survey of major recent discoveries, essential mechanisms, impelling question-marks and valuable suggestions for future research. Basic studies of atmospheric processes continuously feed a science called Meteorology and a public service called Meteorological Forecasting.
For a long time, ocean sciences have remained more descrip-
tive in nature, more concerned with the understanding of the basic processes and mathematical models were often designed with the main purpose of elucidating particular aspects of the ocean dynamics. However, the rapid advancement, in the recent years, of both the physical sciences of the ocean and the mathematical techniques of marine modelling have made possible the development, in the field of marine hydrodynamics and air-sea interactions, of prognostic models serving a new science and initiating a public service
:
Marine
Forecasting. The papers presented at the Tenth International Liege Colloquium on Ocean Hydrodynamics report fundamental or applied research and they address such different fields as storm surges, mixing in the upper ocean layers, surface waves, cycloqenesis and other air-sea or sea-air interactions.
Their unity resides in a common approach,
seeking a better understanding (by modellers and users) of the scientific maturity and of the incentive new prospects of Marine Forecasting.
Jacques C.J. NIHOUL.
This Page Intentionally Left Blank
VII
The S c i e n t i f i c O r g a n i z i n g Committee
of
the
Tenth
International
L i e g e C o l l o q u i u m on Ocean Hydrodynamics and a l l t h e p a r t i c i p a n t s wish t o e x p r e s s t h e i r g r a t i t u d e t o t h e Belgian M i n i s t e r o f E d u c a t i o n , t h e N a t i o n a l S c i e n c e Foundation LiSge
of and
Belgium,
the University
of
t h e O f f i c e o f Naval Research
f o r t h e i r most v a l u a b l e s u p p o r t .
This Page Intentionally Left Blank
IX LIST OF PARTICIPANTS ADAM,Y., Dr., ~ i n i s t e r ede la Sante Publique et de l'Environnement, Belgium
.
ARANUVACHAPUN,S., Dr., Mekong Project, United Nations, Bangkok, Thai land. BACKHAUS,J.O., Mr., Deutsches Hydrographisches Institut, Hamburg, W. Germany. BAH,A., Ir., Universite de Liege, Belgium. BELHOMME,G., Ir., Universite de Liege, Belgium. BERGER,A., Dr., Universite Catholique de Louvain, Belgium. BERNARD,E., Dr., Institut Royal M&t60rOlOgiqUe, Bruxelles, Belgium. BESSERO,G., Ir., Service Hydrographique et Oceanographique de la Marine, Brest, France. BUDGELL,W.P.,
Mr., Ocean
&
Aquatic Sciences, Burlington, Canada.
CANEILL,J.Y., Ir., ENSTA, Laboratoire de Mecanique des Fluides, Paris, France CAVANIE,A., Dr., CNEXO/COB, Brest, France. CHABERT d'HIERES,G.,
Ir., Institut de Mecanique, Grenoble, France.
D E GREEF,E., Mr., Institut Royal Met60rOlOgiqUe, Bruxelles, Belgium. D E KOK, Mr., Rijkswaterstaat, Rijswijck, The Netherlands. DELECLUSE,P., Melle, M.H.N., Paris, France.
Laboratoire d'Oc6anographie Physique,
DI~TECHE,A., prof., ~ r . ,universite de Liege, Belgium. DONELAN,M.,
Dr., Canada Centre for Inland Waters, Burlington, Canada.
DOWLEY,A., Mr., University College, Dublin, Ireland. DUNN-CHRISTENSEN,J.T., Denmark. ELLIOTT,A.J.,
Dr., Meteorologisk Institut,
Copenhagen,
~ r . ,SACLANT ASW Research Centre, La Spezia, Italy.
EWING,J.A., Mr., I.O.S., Wormley, U.K. FEIN,J., Dr., CDRS, National Science Foundation, Washington D.C., U.S.A. FISCHER,G., Prof., Dr., Meteorologisches Institut, Universitat Hamburg, W. Germany. FRANKIGNOUL,C.J., Dr., Massachusetts Institute of Technology, Cambridge, U.S.A.
X FRASSETTO,E., Prof., Laboratorio per lo Studio della Dinamica delle Grandi Masse, Venezia, Italy. FRITZNER,H.E.,
Mr., Norsk Hydro, Oslo, Norway.
GERRITSEN,H., Ir., Technische Hogeschool Twente, The Netherlands. GRAF,W.H.,
Prof., Ecole Polytechnique Federale, Lausanne, Switzerland.
HAUGUEL,A.,
Ir., E.D.F.,
Chatou, France.
HEAPS,N.S., Dr., IOS, Bidston Observatory, U.K. HECQ,P., Ir., Universite de Liege, Belgium. HENKE,I.M., Mrs., Institut fiir Meereskunde, Universitat Kiel, W. Germany. HUA,B.L., Melle, M.H.N., Laboratoire d'oceanographie Physique, Paris, France. JAUNET,J.P.,
Ir., Bureau VERITAS, Paris, France.
JONES,J.E., Mr., IOS, Bidston Observatory, U.K. JONES,S., Dr., University of Southampton, U.K. KAHMA,K., Mr., Institute of Marine Research, Helsinki, Finland. KITAYGORODSKIY,S.A., Moscow, U.S.S.R., Finland.
Prof., Dr., Academy of Sciences of the U.S.S.R., and Institute of Marine Research, Helsinki,
LEJEUNE,A., Dr., Universite de Liege, Belgium. LOFFET~A., Ir., universite de Liege, Belgium. MAC MAHON,B., Mr., Imperial College, Civil Engineering Dept., London, U.K. MAGAARD,L., Prof., Dr., University of Hawaii, Honolulu, U.S.A. MELSON, L.B., Ir., U.S. Navy Sciences Miinchen, W. Germany.
Technical Group Europe,
de, Prof., University of Sao Paulo, Brazil.
MESQUITA,A.R. MICHAUX,T.,
&
Ir., Universite de Liege, Belgium.
MILLER,B.L.,
Dr., National Maritime Institute, Teddington, U.K.
MIQUEL,J., Ir., E.D.F., Chatou, France. MULLER,P., Dr., Institut fiir Geophysik, Universitdt Hamburg, W. Germany. NAATZ,O.W.,
Mr., Fachbereich See, Fachhochschule Hamburg, W. Germany.
NASMYTH,P.W., NIHOUL,J.C.J.,
Dr., Institute of Ocean Sciences, Sidney, Canada. Prof., Dr., Universitd de Liege, Belgium.
XI NIZET,J.L.,
Mr., Universite de Liege, Belgium.
O'BRIEN,J.J., U.S.A.
Prof., Dr., Florida State University, Tallahassee,
O'KANE,J.P., OZER,J.,
Dr., University College, Dublin, Ireland.
Ir., Universite de Liege, Belgium.
PELLEAU,R., Ir., ELF-AQUITAINE, Pau, France. PICHOT,G., IT., Ministere d e la Sante Publique et de l'Environnement, Belgium. RAMMING,H.G., REID,R.O.,
Dr., Universitdt Hamburg, W. Germany.
Prof.,Dr.,
Texas A&M University, College Station, U.S.A
ROISIN,B., Mr., Universite de Liege, Belgium. RONDAY,F.C., Dr., Universite de Liege, Belgium. ROOVERS,P.,
Ir., Waterbouwkundig Laboratorium, Borgerhout, Belgium.
ROSENTHAL,W., Germany.
Dr., Institut far Geophysik, Universitat Hamburg, W.
RUNFOLA,Y., Mr., Universitd de Liege, Belgium. SCHiFER,P., Mr., K.F.K.I.,
Hamburg, W. Germany.
SCHAYES,G., Dr., Universite Catholique de Louvain, Belgium. SETHURAMAN, S., Dr., Brookhaven National Laboratory, Upton, U.S.A. SHONTING,D.H., U.S.A.
Prof., Naval Underwater Systems Center, Newport,
SMITZ,J., Ir., Universite de LiBge, Belgium. SPLIID,H., Dr., IMSOR, Technical University of Denmark, Lyngby, Denmark. THACKER,W.C.,
Dr., NOAA/AOML Sea-Air Laboratory, Miami, U.S.A.
THOMASSET,F., Ir., IRIA LABORIA, Le Chesnay, France. TIMMERMANN,H., Ir., KNMI, D e Bilt, The Netherlands. TWITCHELL, P.F., Dr., Office o f Naval Research, Boston, U.S.A. VAN HAMME,J.L.,
Dr., Institut Royal MBt6orologique, Bruxelles, Belgium.
VINCENT,C.L., Dr., u.S.A. Engineer waterways Experiment Station, Vicksburg, U.S.A. VOOGT,J.,
Ir., Rijkswaterstaat, ~ ' G r a v e n h a g e ,The Netherlands.
WANG,D.P., Dr., Chesapeake Bay Institute, The Johns Hopkins University Baltimore, U.S.A.
XI1 WILLEBRAND,J.,
Dr.,
WORTHINGTON,B.A., U. K.
Princeton
Dr.,
University,
U.S.A.
H y d r a u l i c s Research S t a t i o n , W a l l i n g f o r d ,
XI11
CONTENTS
........................
FOREWORD..
...................... PARTICIPANTS . . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGMENTS LIST OF
KITAIGORODSKII, S.A. layer deepening FRANKIGNOUL, C.
:
:
1
. . . . . . . . . . . . . . . . .
35
Low frequency motions in the North Pacific and
:
WILLEBRAND, J. and PHILANDER, G. oceanic variability VINCENT, C.L.
:
and RESIO, D.T.
ARANUVACHAPUN, S.
:
:
:
Wind-induced low-freauency
61
A discussion of wave
. . . . . . .
71
. . . . . . . . . . . . . . . . .
91
Correlation between wave slopes and near-
surface ocean currents :
57
Wave height prediction in coastal water
of Southern North Sea S.
...
. . . . . . . . . . . . . . . . . .
prediction in the Northwest Atlantic Ocean
MACMAHON, B.
IX
Large scale air-sea interactions and
their possible generation by meteorological forces
SETHURAMAN,
VII
Review of the theories of wind-mixed
....................
climate predictability MAGAARD, I.
V
. . . . . . . . . . . . . . . . .
101
. . . . . . .
113
T h e tow-out of a large platform
GCNTHER, H. and ROSENTHAL, W. : A hybrid parametrical surface wave model applied to North-Sea sea state prediction DONELAN, M. waves
:
....................... On the fraction of wind momentum retained by
.........................
SHONTING, D. and TEMPLE, P.
:
currents BUDGELL, W.P.
;
A status
........................
SABATON, M. and HAUGUEL, A.
:
141
The NUSC windwave and
turbulence observation program (WAVTOP) report.
127
161
A numerical model of longshore
........................ and EL-SHAARAWI, A.
:
183
Time series modelling of
storm surges o n a medium-sized lake
..........
197
XIV BAUER, S.W. and GRAF, W.H.
Wind induced water
:
circulation of Lake Geneva
. . . . . . . . . . . . . .
RUNFOLA, Y. and ROISIN, B.
NIHOUL, J.C.J.,
:
219
Non-linear
three-dimensional modelling of mesoscale circulation
. . . . . . . . . . . . . . . . . .
in seas and lakes THACKER, W.C.
Irregular-grid finite-difference techniaues
:
for storm surge calculations for curving coastlines HEAPS, N.S.
and JONES, J.E.
FISCHER, G.
:
.
26 1
Recent storm surges in the
......................
Irish Sea
235
285
Results of a 36-hour storm surge prediction
:
of the North-Sea for 3 January 1976 on the basis of numerical models DONG-PING WANG
:
:
....................
:
:
. . . . . . . . . . . . . . . . . . .
:
333
35 1
Recent results from a storm surge
prediction scheme for the North Sea ADAM, Y.
. . . . . . . . .
Tidal and residual circulations in the
English Channel FLATHER, R.A.
323
First results of a three-dimensional model
on the.dynamics in the German Bight RONDAY, F.C.
321
Extratropical storm surges in the
Chesapeake Bay BACKHAUS, J.
. . . . . . . . . . . . . . . . . . .
. . . . . . . . .
385
Belgian real-time system for the forecasting of
currents and elevations in the North Sea
. . . . . . .
411
TOMASIN, A. and FRASSETTO, R. : Cyclogenesis and forecast of dramatic water elevations in Venice ELLIOTT, A.J. Italy
:
. . . . . . . .
The response of the coastal waters of N.W.
........................
LEPETIT, J.P. and HAUGUEL, A. sediment transport
:
439
A numerical model for
. . . . . . . . . . . . . . . . . .
BERNIER, J. and MIQUEL, J.
:
453
Security of coastal nuclear
power stations in relation with the state of the sea SUBJECT INDEX
427
......................
.
465 481
1
REVIEW OF THE THEORIES OF WIND-MIXED LAYER DEEPENING S.A. KITAIGORODSKII PP Shirshov Institute of Oceanology, Academy of Sciences, Moscow (U.S.S.R.). English version prepared from the original manuscript in Russian by Jacques C.J. NIHOUL and A. LOFFET MBcanique des Fluides Geophysiques, Universit6 de Liege, Sart Tilman B6, Liege (Belgium). ABSTRACT One considers here the time evolution of the oceanic surface boundary layer in relation with the synoptic variability of atmospheric processes. Attention is restricted to situations where the major responsability for the short-period variability of the vertical structure of the surface boundary layer lies on the local thermal and dynamic interactions between the atmosphere and the ocean and on the internal thermocline - supported transfer processes. Emphasis is laid on theoretical and experimental results which can be interpreted by means of simple one-dimensional vertical mixing models. INTRODUCTION
The description of the dynamic of wind mixing in oceanic surface layers (e.g. Kitaigorodskii, 1970) is based on the assumption that the main sources of turbulent energy are i) the breaking of wind waves which produces turbulence in a relatively thin surface layer (having a thickness of the order of the amplitude of the breaking waves) which extends into the fluid by turbulent energy diffusion effects (Kitaigorodskii and Miropolskii, 1967
;
Kalatskiy, 1974)
;
ii)the velocity shear associated with drift currents responsible for turbulent energy production throughout the turbulent layer and, primarily, in those parts of it where the velocity shear is large. In oceanic surface layers, the two mechanisms can act simultaneously.
However, in laboratory conditions, it is possible to explore
each of them individually.
n
To study the wind wave breaking effect, the initial stirring of the thin surface layer can be simulated by means of a vertically oscillating grid placed in the vicinity of the fluid surface (Turner, 1973
;
Linden, 1975).
The mixing caused by drift currents can be
modelled by experiments in which a constant stress is applied at the surface of the fluid (Kato and Phillips, 1969 The laboratory experiments (Turner, 1973 and Phillips, 1969
;
Kantha et al, 1977
;
;
;
Kantha et a l l 1977).
Linden, 1975
;
Kato
Moore and Long, 1971) expli-
citly show that all the mechanisms of turbulence production create a thin region of large vertical density gradient in the initially continuously stratified fluid.
This region, referred to as the "turbu-
lent entrainment layer", normally lies below a well-mixed layer, the so-called "upper homogeneous layer".
Beneath the turbulent entrain-
ment layer, lies a relatively unperturbed region of the fluid in which internal waves and irregular irrotational perturbations may exist.
In laboratory test conditions, the intensity of the fluctua-
tions below the turbulent entrainment layer is found rather insignificant and such motions do not appear to contribute to the vertical momentum, heat and energy transfer processes. When a steady stress acts on the free surface, a layer of considerable velocity shear (of thickness mixed layer.
6 ) is formed at the top of the
If one excepts the very beginning of the entrainment
process, the thickness of the shear layer is always much smaller than the depth
D
of the mixed layer
( 6 < < D).
Large mean velocity gra-
dients are also observed in the turbulent entrainment layer (Kato and Phillips, 1969
;
Kantha et al, 1977
;
Moore and Long, 1971) and they
may extend to the lower part of the mixed layer (Moore and Long, 1971). At very large values of the Richardson number (based on the variation of density accross the turbulent entrainment layer) a certain amount of heat and momentum transfer in the core of the entrainment layer can be attributed to molecular diffusion (Kantha et al, 1977 Crapper and Linden, 1974
;
Wolanski and Brush, 1975
;
;
Phillips, 1977).
However, in cases of well-developed turbulence in the mixed layer, the molecular effects in the turbulent entrainment layer are obviously negligible.
(Molecular diffusion can only play a role in the one-
centimeter thick layer of water immediately below the surface). In situ observations show that the thickness
h
of the turbulent
entrainment layer reaches several meters in storm conditions. ratio
-
is then of the order of
10-l.
The
Detailed measurements made
in laboratory test conditions, (Crapper and Linden, 1974 ; Wolanski h and Brush, 1975) show that does not depend on the density
3
variation accross the turbulent entrainment layer (provided the density jump is large enough). increasing Peclet number
Beside, it has become evident that with WD (Pe = - where w is the root mean square
x
of the horizontal fluctuating velocity at the upper boundary of the entrainment layer and X the molecular diffusivity of heat or salt) h decreases and tends to a constant value % 1.5 10-1 Measurements D by Moore and Long (1971), in experiments where turbulence was geneh 0 . 8 10-l. Finally, laboratory rated by a velocity shear, lead to D % O(l0-l) experiments by Wolanski and Bush (1975) also showed that D where g and is independent of the Richardson number (Ri = -), p w2 the density dPfference accross is the acceleration of gravity and A p
.
the entrainment layer. In modelling the deepening process of the upper homogeneous layer,
*
in the ocean as well as in laboratory experiments, one may thus assume
EQUATIONS DESCRIBING THE EFFECT OF WIND MIXING ON THE DEEPENING OF THE UPPER HOMOGENEOUS LAYER IN A STRATIFIED FLUID The basic features of an oceanic wind-mixed layer can be simulated by one-dimensional models, disregarding advection, horizontal diffusion and large scale vertical motions.
It will be assumed here, for
simplicity, that the water density is a function of temperature only (the introduction of variations of salinity or horizontal non-homogeneity is not a major difficulty).
It will be further assumed that
the short-wave radiation is absorbed at the sea surface.
A simple
technique to account for the volume absorption of solar radiation has been described by Kraus and Turner (1967) and Denman (1973).
The
corrections introduced thereby have been found to be not very significant since the thickness of the effective absorption layer is, on the average, about one order of magnitude smaller than
*This
D (Denman,1973).
assumption provides a good approximation in modelling local onedimensional vertical mixing processes but may not be applicable to the study of the evolution of the seasonal thermocline (Kitaigorodskii and Miropolskii, 1970). The analysis of the whole year development of the temperature field in the active layer of the ocean ( 2 0 0 - 400 m) must take into account the universal temperature profiles below the upper homogeneous layer. These profiles were found first by Kitaigorodskii and Miropolskii (1970) and were confirmed later by numerous observations of the vertical distributions of temperature and salinity in many parts of the ocean (Moore and Long, 1971 ; Miropolskii et al, 1970 ; Nesterov and Kalatskiy, 1975 ; Reshetova and Chalikov, 1977).
4
With these assumptions, the equations describing the non-steady, one-dimensional vertical heat, momentum and turbulent energy transfers in a stratified rotating fluid can be written
_ a @ --- at
as az
as ae _ _ - T.-- az at
,
where 0
gBs
and
e
-
E
-
aM az
(3)
denote respectively the mean temperature, the mean
horizontal velocity and the mean turbulent energy and where and
M
s,
are the corresponding fluxes (normalized with respect to the
mean thermal capacity
poCp
and the mean density
respectively).
po
f is equal to twice the vertical component of the earth's rotation vector,
g
is the acceleration of gravity,
coefficient and
E
f3
the thermal expansion
is the rate of turbulent energy dissipation.
The
frame of reference is sinistrorsum and such that the x-axis is in the direction of the surface wind and the z-axis is vertical pointing downwards. (z = O),
At the upper boundary of the' mixed layer cribe the fluxes.
one must pres-
The fluxes depend on the atmospheric conditions and
they are normally parameterized in terms of the meteorological data. In general, they are functions of time.
However, in the following,
the discussions will be restricted to the steady case, for the sake of simplicity. If
-
4
stands for any of the variables 0
,
u,
v, e,
one defines
_
e l e Integrating eqs. 1 - 3 over the upper homogeneous layer and the turbulent entrainment layer, one derives a system of equations for the depth-averaged variables
and
4
.
5
Combining these equations and neglecting small terms of relative (in the hypothesis of a "thin interface" D * < 1 ) D magnitude obtains, after some calculations,
dt
(OD)
dt
(GD) +
=
-
s
+ -dD @
S
dt
f
g
P ~ =D
where
D E D = - /0
,
one
+
+ nD + IIh
- ED
-
Eh - M +
dD +-e dt
+
:a
x. a z dz 3:
D+h
I t h = - /D
Mo
K
-
T.
az
dz
IID
The calculation of
can be most easily done with the assump-
tion that the velocity shear in the upper homogeneous layer is concentrated in the constant stress layer 6
au
n D " J f i 6 = - -l
T . 2 az
where
dz
%
6
.
Then
z o * ( y o - us)
(14)
is the velocity at the lower boundary of the constant stress
layer of thickness 6
.
From eq.(2) and its scalar product by
, one gets, after some re-
arrangement and neglecting small terms involving
h
It can be shown that the turbulent energy production in the upper
homogeneous layer and in the turbulent entrainment layer is not very sensitive to the detailed velocity distribution in the main part of the upper homogeneous layer..
In a first approach, it seems thus
reasonable to make the so-called "slab model approximation" where the vertical velocity distribution is assumed homogeneous for
-
6
5
z
5
D
so that
*Even,
in the hypothesis h < < 1 , such simplification is difficult to D justify because the remainlng terms can partially cancel each other and sum up to be comparatively small. It m u s $ be regarded as a first approximation liable to revision. The term ~h is retained in the absence-of a clear-cut evaluation of the respective orders of magnitude of E and E
.
6
y - = y = 56
(17)
nD
In this particular case, one can write - U) = n6 = To. ( L l o - 5 ) = T o ( U o
h '
1 dD = -2- dt
11% - :+I(
(18)
-
2 +
:+'(!!
-
(19)
:+)
= o
"-6
(20)
Velocity shear layers are thus taken into account as velocity jumps ( g o - g) and ( y - 3 ) in thin layers of thickness 6 < < D and h
ie( ~ Z - U T )
(1)
J Here
L
=
(c1,> T R , t h e p l a n e t a r y t e r m i s d o m i n a t i n g , and t h e s o l u t i o n i s i n form of a S v e r d r u p bal a n c e which a d i a b a t i c a l l y a d j u s t s t o changes i n t h e wind f i e l d . I n e i t h e r case, t h e f r e q u e n c y s p e c t r a of o c e a n i c v a r i a b l e s can b e e x p r e s s e d i n terms o f t h e s p e c t r u m o f c u r l T , F c ( k , w ) . frequency
limit,
I n t h e high-
that relation is
-
w i t h t h e d e f i n i t i o n (...) =
1 dlc(
...)
Fc(lc,w)/l
d k Fc(k_,w).
68
The q u a n t i t y k-4
c u s s i o n of
i n ( 5 ) d e p e n d s o n l y w e a k l y on w .
t h e w i n d s p e c t r a i n t h e p r e v i o u s s e c t i o n , w e t h e r e f o r e ex-
p e c t a p o w e r l a w F (w)-w-'
w i t h q M 3 . 5 a t p e r i o d s between
J,
and q-2
From t h e d i s -
b e t w e e n 3 a n d 10 d a y s .
f o r t h e s p e c t r a of current I n t h e low-frequency
Analogous conclusions
1 a n d 3 days,
can be derived
components.
limit,
t h e corresponding r e l a t i o n s are i n
terms o f t h e c u r r e n t s p e c t r a
The m a g n i t u d e o f k g / k t
d e p e n d s on d e t a i l s of t h e d i r e c t i o n a l d i s -
t r i b u t i o n of t h e wind spectrum. T y p i c a l l y almost independent of current if
frequency
,
,
o n e f i n d s k;/k:dk:/k:>>l
and hence t h e magnitude of zonal
f l u c t u a t i o n s w i l l dominate t h a t of m e r i d i o n a l c u r r e n t s ,even
t h e r e i s no p r e f e r r e d d i r e c t i o n i n t h e atmosphere. Furthermore,
both current
spectra w i l l be white,
a s t h e wind stress c u r l spectrum
is a l s o white i n t h a t frequency range. Fig.
3 shows c u r r e n t s p e c t r a t a k e n f r o m a n u m e r i c a l model which
c a l c u l a t e d t h e o c e a n i c response t o a c t u a l l y observed wind s t r e s s f l u c t u a t i o n s o f t h e North P a c i f i c (Willebrand e t a l . , model r e l a x e s s e v e r a l o f
( 5 ) and ( 6 ) , e . g. absence of
1979).
That
t h e c o n s t r a i n t s which were used t o d e r i v e
linearity,
quasi-geostrophy,
l a t e r a l boundaries.
Nevertheless,
i d e a l i z e d wind f i e l d , t h e r e s u l t s support the
above c o n c l u s i o n s . The s p e c t r a l p e a k s a r o u n d p e r i o d s of c a n b e i d e n t i f i e d as r e s o n a n t b a s i n modes,
10 - 2 0
days
t h e i r a m p l i t u d e and l o c a t -
i o n i s d e t e r m i n e d by b a s i n s i z e and f r i c t i o n a l e f f e c t s .
It i s remarkable t h a t
t h e r a t h e r s t e e p s l o p e s of
current spectra
i m m e d i a t e l y b e l o w t h e i n e r t i a l f r e q u e n c y , which a r e g e n e r a l l y observed i n deep-water
c u r r e n t meter r e c o r d s
( cf.
Thompson,
duced from a s i m p l e a t m o s p h e r i c f o r c i n g model. s t i t u t e s an a l t e r n a t i v e t o n o n l i n e a r well-known
k-3-law
a l s o l e a d t o a w-3 However
,
1971), c a n b e de-
T h u s , t h e model con-
cascade arguments which over the
f o r g e o s t r o p h i c t u r b u l e n c e and a T a y l o r - h y p o t h e s i s s p e c t r a l law.
t h e s h a p e of e n e r g y s p e c t r a of
oceanographic variables
i s notoriously i n s e n s i t i v e t o d i f f e r e n t t h e o r i e s regarding t h e i r orig i n , a n d m o r e s p e c i f i c c o n s e q u e n c e s o f t h e a t m o s p h e r i c f o r c i n g mechanism must be considered.
From ( 4 )
one might e x p e c t t h a t atmosperic
and oceanographic f l u c t u a t i o n s a r e c o r r e l a t e d .
(4) shows, however,
A d e t a i l e d a n a l y s i s of
that the local correlation is generally
low,
69
MODEL OCEAN, HORIZONTAL CURRENT SPECTRA NEAR CENTER BASIN 10y 10' -
loo
-
D
s 2 10-1i -5
95%
10-~
-
10-'1
I
I
.01
.1
a
I
1
FREQUENCY (cpd)
Fig.
3. F r e q u e n c y s p e c t r a o f e a s t (u) a n d n o r t h ( v ) c o m p o n e n t s o f o c e a n c u r r e n t s , c o m p u t e d f r o m a n u m e r i c a l m o d e l of a n i d e a l i z e d ocean b a s i n r e s e m b l i n g t h e North P a c i f i c which w a s d r i v e n by o b s e r v e d w i n d s t r e s s f l u c t u a t i o n s .
p a r t l y due t o t h e broad-band
f o r c i n g s p e c t r u m , a n d p a r t l y due t o t h e
w a v e l i k e n a t u r e o f o c e a n i c r e s p o n s e ( W i l l e b r a n d e t a l , 1 9 7 9 ) . Only a t t h e h i g h e s t f r e q u e n c i e s , t h e s i m p l i f i e d model
( 4 ) p r e d i c t s cohe-
r e n c e between c e r t a i n a t m o s p h e r i c and o c e a n o g r a p h i c v a r i a b l e s . c o h e r e n c e i s r e d u c e d f u r t h e r by i n h o m o g e n e i t i e s and t h e f o r c i n g f i e l d s w h i c h a r e i g n o r e d i n ( 4 ) .
That
i n bottom topography This p i c t u r e agrees
w i t h o b s e r v a t i o n a l e x p e r i e n c e : no l o c a l c o r r e l a t i o n h a s b e e n o b s e r v e d a t low f r e q u e n c i e s , w h e r e a s a t h i g h f r e q u e n c i e s o c c a s i o n a l l y s i g n i f i cant
( i f marginal) c o r r e l a t i o n has been found (e.g.
Baker e t a l ,
1977; Meincke and Kvinge,
Brown e t a l ,
1975;
1978).
I n c o n c l u s i o n , we s t a t e t h a t a s t o c h a s t i c m o d e l f o r t h e a t m o s p h e r i c generation of oceanic v a r i a b i l i t y can, i n c o n t r a s t t o a d e t e r m i n i s t i c one,
a c c o u n t f o r some o b s e r v e d f e a t u r e s o f o c e a n i c c u r r e n t f l u c t u a t i o n s ,
n a m e l y t h e l o n g e r t i m e s c a l e s i n t h e o c e a n ( o c e a n i c s p e c t r a a r e much more r e d t h a n a t m o s p h e r i c s p e c t r a ) a n d t h e l a c k o f s t r o n g l o c a l co-
70
h e r e n c e b e t w e e n a t m o s p h e r e a n d o c e a n . H o w e v e r , o n l y b a r o t r o p i c motions c a n b e e x p l a i n e d i n t h i s w a y , b e c a u s e t h e h o r i z o n t a l s c a l e of wind f l u c t u a t i o n s a s i n f e r r e d f r o m w e a t h e r maps i s t o o l a r g e t o g e n e r a t e b a r o c l i n i c m o t i o n , e x c e p t a t v e r y low f r e q u e n c i e s .
In o r d e r t o e x p l a i n
b a r o c l i n i c e d d i e s , which f r e q u e n t l y dominate upper ocean v a r i a b i l i t y , i n terms o f a t m o s p h e r i c f o r c i n g , w e n e e d much more i n f o r m a t i o n on t h e small-scale
s t r u c t u r e of t h e m e t e o r o l o g i c a l f i e l d s .
ACKNOWLEDGEMENTS We t h a n k M r .
putations.
P a c a n o w s k y f o r a s s i s t a n c e i n t h e n u m e r i c a l com-
R.C.
T h i s work h a s b e e n s u p p o r t e d t h r o u g h t h e G e o p h y s i c a l F l u i d
Dynamics L a b o r a t o r y
-
NOAA G r a n t No. 04
-7-
022
-
44017.
REFERENCES B a k e r , D . J . J r . , W.D. N o w l i n , J r . , R . D . P i l l s b u r y a n d H . L . B r y d e n , 1977. Space and t i m e f l u c t u a t i o n s i n t h e Drake p a s s a g e . N a t u r e , 268:696-699 Brown, W . , W. Munk, F. S n o d g r a s s , H. M o f j e l d a n d B . Z e t l e r , 1 9 7 5 . MODE b o t t o m e x p e r i m e n t . J . P h y s . O c e a n o g r . 5 : 7 5 - 8 5 Emery, W . J . , A. G a l l a g o s a n d L. M a g a a r d , 1 9 7 8 . F r e q u e n c y - w a v e n u m b e r s p e c t r a o f w i n d s t r e s s and sea s u r f a c e t e m p e r a t u r e i n t h e e a s t e r n North P a c i f i c . J . Phys. Oceanogr. ( s u b m i t t e d ) F r a n k i g n o u l , C. , 1 9 7 9 . L a r g e s c a l e a i r s e a i n t e r a c t i o n s and c l i m a t e p r e d i c t a b i l i t y . In: J . C . J . N i h o u l ( E d i t o r ) , M a r i n e F o r e c a s t i n g , Elsevier (this issue) F r a n k i g n o u l , C . a n d K. H a s s e l m a n n , 1 9 7 7 . S t o c h a s t i c c l i m a t e m o d e l s , p a r t 11. T e l l u s , 29:284-305 F r a n k i g n o u l , C. a n d P. M U l l e r , 1 9 7 8 . Q u a s i - g e o s t r o p h i c r e s p o n s e of an i n f i n i t e $-plane ocean t o s t o c h a s t i c f o r c i n g by t h e atmosphere. J. Phys. Oceanogr. ( i n p r e s s ) Lemke, P . , 1 9 7 7 . S t o c h a s t i c c l i m a t e m o d e l s , p a r t 111, a p p l i c a t i o n t o z o n a l l y a v e r a g e d e n e r g y m o d e l s . T e l l u s , 29 :385-392 M a g a a r d , L . , 1 9 7 7 . On t h e g e n e r a t i o n o f b a r o c l i n i c R o s s b y waves i n t h e o c e a n by m e t e o r o l o g i c a l f o r c e s , J . P h y s . O c e a n o g r . , 7:359-364 M e i n c k e , J . a n d T . K v i n g e , 1 9 7 8 . On t h e a t m o s p h e r i c f o r c i n g o f o v e r f l o w e v e n t s . I C E S , C.M. 1 9 7 8 / C : 9 , H y d r o g r a p h i c c o m m i t t e e P h i l a n d e r , S.G.H. , 1 9 7 8 . F o r c e d o c e a n i c w a v e s . R. Geophys. S p a c e P h y s . 16: 15-46 1 9 5 7 . On t h e g e n e r a t i o n of w a v e s b y t u r b u l e n t w i n d . P h i l l i p s , O.M., J . F l u i d Mech. 2 ~ 4 1 7 - 4 4 5 Thompson, R . , 1 9 7 1 . T o p o g r a p h i c R o s s b y w a v e s a t a s i t e n o r t h o f t h e Gulf stream. Deep-sea R e s . 18:l-19 W i l l e b r a n d , J . , 1 9 7 8 . T e m p o r a l a n d s p a t i a l s c a l e s of t h e w i n d f i e l d o v e r N o r t h P a c i f i c a n d N o r t h A t l a n t i c . J . P h y s . O c e a n o g r . ( i n pres8 W i l l e b r a n d , J . , S.G.H. P h i l a n d e r a n d R.C. P a c a n o w s k y , 1 9 7 9 . The oceanic response t o large-scale atmospheric disturbances. In preparation
71
A DISCUSSION OF WAVE PREDICTION IN THE NORTHWEST ATLANTIC OCEAN 1 1 C. L. VINCENT and D. T. RESIO 'Wave Dynamics Division, Hydraulics Laboratory *U. S. Army Engineer Waterways Experiment Station, Vicksburg, Miss. (USA)
ABSTRACT Simulation of a wave climate for the Atlantic Ocean through hindcasting is discussed in terms of methods for obtaining historical wind fields and available numerical models for hindcasting directional spectra. Available, gridded pressure and wind data produced recently by the U. S. Navy Fleet Numerical Weather Central are shown to be distorted near major storms leading to the necessity of redigitizing major storm areas from synoptic weather charts. The optimal method for using most of the available oceanographic data to produce the wind fields is one in which the surface wind field is estimated from the pressure field, and temperature (air and sea) fields through a planetary boundary layer model. Afterwards, the available ships wind-field observations are blended into the estimate. A root-mean-squareerror of less than 3 mps on speed appears obtainable. Several numerical models for directional spectral wave estimation are reviewed. Each model is examined in terms of source mechanisms and propagation schemes. Models with wave-wave interaction source terms appear to perform better in tests of wave growth with fetch as well as in field estimation. Models with a fourthorder or ray propagation schemes appear adequate for oceanic hindcasts while first order propagation schemes appear to disperse.
INTRODUCTION .The prediction of sea state through the use of numerical models has become an '
important method for estimating wave climates. Although the simple wave prediction schemes based on the research of Sverdrup and Munk (1947) as modified are still widely used, several recent studies have used a variety of numerical models that calculate directional spectra for hindcasting wave climates. Included are major studies on the Great Lakes (Resio and Vincent, 1976), the Atlantic and Pacific Oceans (Lazanoff and Stevenson, 1977), and the North Sea (NORSWAM, 1977). The rise in application of numerical methods is in part due to cost-effective computer technology, and in part due to the short time frame required to compute a wave
climate compared with the time and cost of performing an extensive wave-measuring program. Recent studies indicated that sea-state prediction with numerical models can have random error below 1 metre, which leads to the expectation that wave climates constructed numerically will have no more error than those measured, particularly when the difficulties inherent in an observational program are considered. An additional benefit of the hindcast studies is the estimation of the wave direction since this parameter is generally not measured in observational programs. The U. S. Army Corps of Engineers has major responsibilities in the United States in the areas of navigation in coastal waters and shore protection, and, consequently, is one of the primary wave-data consumers in the U. S. As part of its Field Data Collection Program, the Corps of Engineers at the U. S. Army Engineer Waterways Experiment Station (WES) is sponsoring a study to hindcast wave climates for the Atlantic and Pacific Ocean and Gulf of biexico coastal areas based on the concepts developed in the Corps-sponsored studies of the Great Lakes (Resio and Vincent, 1976). The study provides for hindcasts in the open ocean that serve as a boundary for hindcasts, at a finer scale, on the continental shelf and in nearshore waters. The shelf and nearshore studies will include refraction and shoaling effects. Also included will be a hindcast of storm tides so that the joint probabilities of water level and wave height in shallow waters can be estimated. A final part of the project will be the development of a computer-based information system to contain the data base and allow sitespecific calculations based on a combination of detailed refraction studies and the data base by users at the many Corps field offices around the U. S. Further, it is the aim of this project to thoroughly document the procedures used and to provide an estimate of the error involved in each method. This paper presents a discussion of some practical problems encountered in designing and operationalizing a system of numerical routines coupled with a data base to produce a deep-ocean wave climate from historical meteorological information. The sources of meteorological data and models available to compute ocean wind fields at a hindcast wind level and the errors involved in such an effort will be discussed. Several available methods for predicting sea state will be discussed in terms of their capability to reproduce theoretical results as well as their ability to predict observed sets of wave data. Discussion will center upon tests in the Northwest Atlantic Ocean off the Canadian Maritime provinces and some results obtained in wave tests in the Great Lakes. Later phases of the project will treat verification of the wave model in diverse locations.
73
WIND DATA Sources of Meteorological Data Our primary source of historical meteorological data over the ocean is the millions
of observations taken aboard ship and archived on magnetic tape at the National Climatic Center in the U. S. To augment these data are land stations in the world meteorologic network. The U. S. Navy Fleet Numerical Weather Central (FNWC) has recently contracted with Meteorology International Incorporated, Monterey, California, to rework and edit these data into a 63x63 point regular, square grid over the Northern Hemisphere of both winds and surface pressures on synoptic time levels for FNWC's own hindcast efforts.
The WES study initially
accepted these data as a basis for WES wind estimates; but on the basis of a published review of the wind estimates by Lazanoff and Stevenson (1977) and concurrent examination of the data at WES, WES has decided to apply a modified method for obtaining winds. An additional effort jointly funded by WES and FNWC is the addition of several million ship observations available but not in the marine deck to this data base. Review of the 63x63 Grid Pressures Lazanoff and Stevenson (1977) reported that the hindcast winds obtained by Holl's (1976) method had an approximate root-mean-square (RMS) error in wind speed of 7 mps and a bias of 3 mps compared with selected observations. It was further reported that the winds were consistently too low above speeds of 15 mps. Clearly some improvement in the wind speed estimates is desired for accurate predictions of extreme wave conditions. WES review of the 63x63 data began by a comparison (Fig. 1) of contoured 63x63 pressures to correspond to U. S. National Weather Service (NWS) surface weather charts, North American and Northern Hemisphere series, which served as an initial basis of the FNWC charts.
In the 63x63 grid, the averaging or blending
procedures to mix ships observations into the meteorologic data tend to smooth the sharp pressure gradients in storms, particularly the smaller ones.
This is
likely due to the inclusion of Laplacian-type operators in the blending process o f Holl (1976) which requires evaluation of derivations over several of the grid
steps in the 63x63 grid and may result in averaging of values over 1000 miles apart.
For a selection of large storms, WES calculated the maximum geostrophic
winds in each quadrant of each storm for both the NWS and FNWC charts (Fig. 2 ) . The results show an under estimation of the NWS values. A plot of central pressures suggests also that the central pressures in the NWS charts tend to be lower than FNWC-sponsored estimates (Fig. 3).
AVAILABLE 63 NORTH
x
70"N On
0 POLE
70'N 50°W
NORTH
50" N
980 MB
r
12 MPS
50°N 50"W
CP
I*
f
'
1200 GMT 12 SEP 61 Ug
SOON O0
+
7OON 50'W
5OoW %
r
70°N O0
0 POLE
' 50' N
CP
U. S. WEATHER BUREAU
63 GRID
950 MB
' 1
1200 GMT 12 SEP 61
Ug
4
37 MPS
1000 MB
950 MB A
B
950 MB A'
B'
Fig. 1. Comparison of low pressure areas on the 63x63 grid and the NWS charts. The difference in central pressure is 30 mb and in geostrophic wind is 25 mps. The transects AB and A ' B ' show the difference in the storm cross sections. 50
-
0 I
Y)
0-
Y
40-
D-
Y) 0
z B
uI n.
8
30-
I-
Y) 0
W W
Vl ID
a n
Vl
U. S. WEATHER BUREAU GEOSTROPHIC WINDSPEED, MPS
Fig. 2.
Comparison of geostrophic winds analyzed on the 63x63 grid and NWS charts.
75
1020
1010 ---- I-
=
W-
a
2 1000 M
0
W
I I :
n
0
/
/
m
--- I
m W
w
= z!
970
I
U. S. WEATHER BUREAU CENTRAL PRESSURE, MB
Fig. 3. Comparison of central pressures between63~63 grid and NWS charts. These comparisons were primarily drawn for large storms exiting the North American land mass and are, therefore, of primary interest for prediction of waves on the U. S. Atlantic coast. This area has many meteorologic stations and the coastal areas are major shipping lanes. In this area, the observational data are quite good; and the storms have been intensively monitored on their track across the North American continent. Consequently, there is little reason to expect a consistent error in these charts immediately after the storm has departed the mainland. Our results tend to reinforce the conclusion o f Lazanoff and Stevenson (1977). A Method for Improved Pressure Estimates
WES initially attempted to find a simple empirical corrective function to readjust the 63x63 grid data. No function however appeared satisfactory. As a result, the central areas (approximately 720 nautical miles square) of every major storm along the U. S. Atlantic coast for the 25-year period 1952-1977 is
76
being redigitized from the NWS charts and will be inserted into the corresponding 63x63 grid to produce a new pressure field. From the old 63x63 grid, pressures will be interpolated on an approximately 50-mile grid. In the storm center areas where new digitized data are available, the pressure data will be replaced by a new value p
P
= a(x,Y)Pl +8(X>Y)P2 (1) where is the old pressure estimate, p1 p2 is the new value, digitized from the NWS charts, a(x,y) and B(x,y) are blending coefficients chosen so that a + 5 = 1 with 5 = 1 a = 0 in the 200-mile square around the storm center and a = 1, 5 = 0 at the edge of the digitized square.
This preserves the NWS data at the storm center but blends the data smoothly into the FNWC pressure values away from the storm center. Derived Wind Estimates Since the FNWC pressures are being reconstructed, it is feasible to use a planetary boundary layer (PBL) model to derive wind velocities at the 19.5metre level required in the wave method rather than the empirical method of Holl (1976). A PBL is desirable because it is a physical model of the processes relating geostrophic and lower level winds and provides an opportunity to incorporate both the stability and baroclinicity of the lower atmosphere into the wind estimates. The PBL model chosen is a recent upgrade of that of Cardone (1969). This model has been shown to produce results with an RMS error of less than 2 mps (Overland and Street, 1977) f o r cases of prediction from geostrophic conditions. Air-sea temperature differences will be derived from the 5-day mean sea surface temperatures and ships observations of atmospheric temperature. The construction of the air temperature fields from ships data is made difficult by the erratic locations of the ships. A value is constructed at sites where there are no data by an algorithm which accounts for both spatial and temporal gradients. Blending of Ships Wind Data The ships observations of wind speed will be blended into the wind field derived through the PBL model from the pressure estimates because they are observations made independent of pressure. The quality of the wind observations is varied. There are inconsistencies in observation level and method. However, comparison of ships wind speed and direction to instrumented observations at Sable Island indicates reasonable agreement between the two data sets. Accordingly, the ships wind data are a valuable addition to the data base, and it is expected that these data will tend to correct wind fields that are somewhat smoothed because of their pressure grid origin. The method used will be of a restricted type. Ships observations of winds will be allowed to influence only the three
nearest grid values and will result in a smoothing only on the order of 100-200 miles. The blending algorithm is of the form n
where qf
is the blended value at grid location i ,
qi
is the value at grid
i derived from the pressure field,
a. is a weighting based on the position of the ships observations, 1
Aq n
is the difference between the value of q at the ship and the grid location, is the number of ships within the triad of grid points.
The value q may be wind speed or a wind-speed component. The blending is restricted to the nearby grid points to prevent oversmoothing of the wind field. Sources of Error in Wind Estimates The wind velocities that are input to a wave model for a series of hindcasts represent an amalgamation of atmospheric and oceanographic data from different sources, observed at different levels by differing methods, that have been used to estimate spatial-field values from which wind velocities are eventually derived through a series of numerical models and approximations. In this process, there are many sources of error; some can be minimzed, but the rest are inherent to the data sets and cannot be reduced. Since the precise level, placement, and manner in which the millions of ships observations of wind were made is not known, it is impossible to remove these sources of error. Locations at which the data were taken are fixed in history, and it is obviously impossible to ascertain historical data for places and times where it does not exist. Attempts to extrapolate data in time and space into regions of sparse data are subject to a high degree of uncertainty which cannot be removed by objective analysis. It also is impossible to account for observer error in any precise sense, although an editing scheme may catch the larger inconsistencies. The errors that generally can be minimized result from the processing of the basic data into derived quantities usually through the use of numerical models. The interpolation functions used to form gridded data and the numerical difference schemes to compute gradients are based on subjective concepts of what is reasonable. Their accuracy generally increases as the grid system on which they are applied is more closely scaled to the size and magnitude o f variations which they approximate. Thus, error is reduced by proper selection of grid size and appropriate order of derivative approximations. The transformation of the wind velocities from a geostrophic level to a level suitable for input to the wave model is through a numerical model that solves for the wind profile near the water surface. It is appropriate to select a model that is both unbiased and has minimal random error.
78
The objective of this exercise is to provide methods that take the basic data available, massage it, and transform it in such manner that the information is not degraded by the analyses and only a minimal amount of error is introduced by subsequent extension of the data. The ultimate evaluation of the methods is through comparison of diagnosed and observed winds. A small set of comparisons has been completed for Sable Island anemometer (Fig. 4) and shows reasonable agreement. The RMS error in these wind-speed estimates is about 2 . 5 mps. More evaluations involving a wide range of sites are under way.
Ultimately, the
error comparisons will involve all major NOAA data buoys in the Atlantic, Pacific, and Gulf of Mexico and should provide one of the more extensive tests of wind estimates over the open ocean.
I $ 4
w
b
z20
=
-
0 1
SABLE ISLAND
I'9
12
18 21 TIME, HR 29 FEB 64 15
0
3
6
I 1MAR 64
Fig. 4. Comparison of wind speed observed at Sable Island, by ships nearby, and prediction from the geostrophic wind. The geostrophic winds were derived from NWS charts. The Cardone (1969) boundary layer model was used to predict the wind speed, neglecting baroclinicity. WAVE MODELS Directional spectral wave prediction models generally contain two primary sets of computational algorithms excluding input-output and peripheral information handling: source term calculations and energy propagation. Source term calculations are the numerical mechanisms which simulate energy (a) transfer from the atmosphere to the spectrum, (b) transfer within the spectrum, and (c) dissipation through breaking. The energy propagation algorithms are the numerical mechanisms which simulate the propagation of energy across the water body. For purposes of WES evaluakion of wave models, the source and propagation algorithms of a number of published wave models were considered separately under the presumption that the best wave model would be a combination of the best source terms with the best propagation scheme. It would be difficult to conceive of a propagation scheme whose errors counterbalance the errors of a set of source terms over a diverse range of wave-generation conditions.
79
Source Terms The calculation of the energy input and exchange on the spectral calculations can be considered of two types. Parametric source term models are those in which some property of the spectrum such as wave height or period is estimated in terms of wind speed, duration, and fetch. Discrete spectral source term models are models that treat the spectrum in a discrete number of frequency and direction bands. The following discussion will treat only those source terms considered in the WES study and were selected on the basis of widespread use. A wider discussion of wave source terms is given in Resio et al. (1978). Discrete Spectral Methods In these methods, the energy balance equation is solved for a number o f discrete frequency-direction (f, 0 ) bands and the source mechanisms directly contribute to each band. Integral properties of the spectrum are obtained through direct integration over the frequency direction space. In a simple expression, the energy transfer in and out of a spectral component (f, given by aF ( f , e ) at
=
s1 + s2
+
sg
0)
is
(3)
where S1 is the energy exchange with the atmosphere, is the energy exchange within the spectrum by conservative, nonlinear S2 wave-wave interactions, S3 is the irreversible loss of energy due to turbulent interactions and wave breaking. The discrete source term models treated in this study are those of Barnett (1968), Ewing (1971), and Salfi (1974) and a modification of Barnett and Ewing source terms involving a relaxation on the equilibrium range "constant" as described by Resio and Vincent (1977). The source terms of Barnett and Ewing can be represented as in the energy balance equation by aF (f, at
0) =
[a
+
b F(f, e ) ] [l
- LI]+ r
-
TF(t, t)
(4)
In both Ewing and Barnett a parameterizes the Phillips (1957) resmance mechanism,
b parameterizes the Miles (1957) mechanism, r , r parameterize the nonlinear wave-wave interactions o f Hasselmann et al. (1973), 1~ parameterizes the wave-growth limiting term. The details of the formulae of a, b, r and T are not given here, but the differences between the models are relatively minor. In both models, a fully directional spectrum is treated. Salfi (1974) source terms (used in the U . S. Navy Fleet Numerical Weather
Central model) are quite different. Salfi only solves the one-dimensional energy balance equation aE
at
(f) = A[l +
+
BE"1
- A]
(5)
where E
is the one-dimensional (frequency) energy density,
A is a linear mechanism and a function of f, and wind speed, B is a nonlinear mechanism, E' A
is the energy density within 90' of the wind direction, is a ratio E/E_,
Em is a fully developed energy density. It is clear that his model does not treat the details of the directional spread of energy and will clearly only be appropriate where there is not much energy propagating transverse to the wind o r where there are not sudden shifts in wind direction. In comparing, in a one-dimensional sense, these source terms with those of Ewing and Barnett, it is evident that they can be equivalent only if A incorporates Resio et al. (1978) point the effects of a + r and B accounts for b - T out that on the forward face of the spectrum wave growth in the Ewing-Barnett
.
type of terms versus those of Salfi can only be calibrated to give the same source for only a narrow range of energy. It can also be seen that if r is proportional to En (Barnett gives it as E 3 ) then A must be a function o f E to be equivalent to a + r ; however A is not a function of E . Parametric Models The parametric model considered in the study is the model proposed by Hasselmann et al. (1976). In this model a pair of differential equations are written to solve for the nondimensional peak frequency of the spectrum f and the nondimensionk! Phillips equilibrium coefficient a as a function of wind stress. This simplification is achieved by assuming constancy of spectral shape and assuming that the spectrum changes sufficiently fast so that the wave direction rapidly adjusts to the wind direction. This model only treats active wave growth and decay and must be interfaced with a swell propagation routine. Tests of the Source Terms In the field it is very difficult to measure the contribution of the individual source term mechanisms. The detailed field studies of Mitsuyasu (1968) and Hasselmann et al. (1973), however, provide excellent evidence for the growth of wave height with fetch and are in close agreement. Unfortunately, no such welldefined curves are available for growth with time. the growth of wave height
It was decided to examine
81 H
=
u*
8 2
(6)
with dimensionless fetch.
where g is gravitational, is friction velocity of air for the wind condition,
u,
F
is fetch,
E
is total energy in the wave field
predicted by the differing sets of source terms. Two wind velocities were input 15 and 30 mps.
Fig. 5 presents the results of the tests. Barnett and Ewing are closer to the curves of Mitsuyasu and Hasselmann than the curves generated from the FNWC model (Salfi, 1974). 30 mps.
The Barnett and Ewing curves do deviate considerably at
The parametric model of Hasselmann et al. (1973) is not shown because
it was derived to fit his field data. 102
-
N.
w
f 'I
+
I
0
: >
10'
k= YI
G z
0 z
I
1 00
10'
1 o5 DIMENSIONLESS F E T C H I F
1 06
10'
gD/UIl
Fig. 5. Comparison between growth-with-fetch relations from spectral models and empirical studies. Resio and Vincent (1977) show that the difficulties with Barnett's parameterizat are due t o the assumption of a constant Phillips equilibrium value a o f 0.0081. By parameterizing a
as a function of dimensionless wave height, the Barnett
82
curves are brought into agreement with the field data (Fig. 6 ) . This result is confirmed by a range of field data (Fig. 7 ) .
Fig. 6 . Comparison between results from the model of Resio and Vincent (1977 and empirical formulae for relations between nondimensional fetch and wave height.
I
I
LEGEND B K
n
I
0
A
z1
10-3
10'
I
-5
SYMBOL BURLING 119Y)I
]
TAKEN FROM
11960' MITSUYASU 119731 KINSMAN 119601 DELEONIBUS. SIMPSON A N D M A T T I E 119741 L I U 119711 OELEONIBUS AND IIMPSON 119721 COBOURG "ICKS
OUCK E
Z
E
E
E
1
CANADIAN WAVE DATA
103
102
DIMENSIONLESS WAVE ENERGY
IF -
9'
104
E U!l
Fig. 7 . Variation of Phillips equilibrium coefficient as a function of dimensionless wave energy.
83 Propagation Schemes In studies which seek only to describe local sea, the swell propagation problem is not important. However, calculation of a wave climate that will be used for sediment transport requires solution of this problem. The general equation in one dimension for propagation is aE= c aE -at g ax and can be solved by finite differences or through ray methods (sometimes called phonon or method of characteristics (techniques). Finite Differences Finite difference solutions are based on the approximation of the propagation equation based on a Taylor series expansion in time and space. Three types of finite difference solutions are treated here: a first order scheme (used in the FNWC model), a fourth order scheme (used by Ewing, 1971) and a Lax-Wendroff (Lax and Wendroff, 1960) scheme modified by Gadd (Golding, 1977) which involves second order expansions in space and time. The equations of each scheme are presented in Table 1. These propagation schemes were evaluated by examining their abilities to propagate a one-dimensional wave envelope. In a practical sense, the most desirable propagation algorithms are those which conserve wave energy and do not change its spatial distribution. Fig. 8 provides the results of tests to examine
t
2o 01 0
I
I
I
I
10
20
30
40
I
50 TIM4 HOURS
I
I
I
I
60
70
80
90
I
I00
Fig. 8. Comparison of energy conservation in the modified linear and Lax-Wendroff propagation schemes. (Note that the analytical, ray, and fourth order methods all essentially retain 100 percent of the energy and are not shown).
84
TABLE 1 Propagation Schemes A.
FNWC (Salfi, 1974)
n+l
E.
=
n Ei
+
n !J(E~- ~EY)
Unconditionally stable upwind differencing scheme modified by computational logic to reduce diffusion o f swell.
B.
Ewing (1971)
5 2 requires alternate grid/staggered mesh
Stable for
C. Lax-Wendroff (Golding, 1977) Step 1 n+1/2 = 1 (EY + Eq+l) - T1 P ( E ~ + -~ Ei) E.1+1/2 2 Step 2
Stable for
P
5 2 , requires two-step calculation.
i
grid point
n
time level
FC =
C At/hx with C g
g
wave group velocity
85
energy conservation as a function of wave period for the modified Lax-Wendroff and the FNWC schemes. The Lax-Wendroff and fourth order schemes had very similar results and have very slight amplification of energy ( 2 percent).
To reduce confusion in the figure, the fourth order results were deleted. A comparison of the modified Lax-Wendroff and fourth order schemes is given in Golding (1977). The FNWC propagation scheme showed a marked dependence upon wave period, with, a significant loss of energy. F o r a propagation time of 24 hours, the loss was 10 percent for a period, in all cases, of 16.5 sec, and over 50 percent f o r both 6.1and 9.7-sec waves. Fig. 9 provides scheme. Again the Lax-Wendroff and only weakly frequency-dependentand shape. The FNWC scheme is markedly
examples of the diffusive properties of the fourth order schemes characteristics are show excellent preservation of wave envelope frequency-dependentand has the undesirable
property of severely distorting the wave envelope. Ray Methods Propagation of wave energy through so-called ray o r phonon methods is analagous to rays developed through geometrical optics for wave refraction. The water body is covered with a dense network of rays for each propagation direction. Along these rays are a series of storage locations spaced as a function of wave frequency through which the wave energy is jumped in time. Such methods have propagation properties equivalent to the analytical solution for energy features with space scale larger than the spacing of points along a ray. This method tends to be computationally swift, but requires considerable computer storage. Preliminary analysis of this technique of the Atlantic Ocean indicates that 1,800,000 storage locations requiring approximately 230,000 60-bit words on a Cyber 176 computer are needed. Combination of Source Term and Propagation One major reason for splitting the analysis of wave models on the basis of source terms and propagation was to obtain comparison of the wave models against common standards. Table 2 presents a compendium of the major models that embody the various operational combinations of the source terms and propagation schemes and provides estimates of the model errors in published verification studies. Unfortunately, it is virtually impossible to put into operation all these models in the same grid and test them because of the diverse natures of the techniques and in some instances the operationalization of the models in highly sitespecific manners. Based on the theoretical tests discussed in this paper, the following comments can be made concerning the models: (1) The source terms used by Hasselmann et al. (1976) and Barnett (1968) as modified by Resio and Vincent (1977) and Ewing (1971) provide a more accurate growth with fetch than the FNWC source terms of Salfi (1974).
86 W
D
-ANALVTICAL
AND METHOD OF CHARACTERISTICS LAX-WENDROFF . . . . . MOOIFIEO LINEAR
___ T = 6 I SCC I t-24
T= I 6 5 SEC
HR
1-24 HR
1-41) HR
tA72 HR
(-96
HR
t-96
HR
t-W
HR
Fig. 9. Comparison of energy diffusion characteristics of the finite difference propagation schemes. The fourth order, modified Lax-Wendroff and ray methods provided suitable (2) propagation of wave energy over the entire range of the spectrum of interest. ( 3 ) Thk first order, modified scheme of Salfi (1974) should not be used for propagation of intermediate period (6-12 sec) wave energy for periods longer than a few hours because it does not conserve energy and greatly distorts the wave energy envelope. Examination of Table 2 would indicate that the performance of these models in verification studies parallels the theoretical tests. The Hasselmann, Barnett,
TABLE 2 Model Comparisons Model Salfi (1974)
Characteristics Source Terms Growth with Fetch Modified First Order Scheme
L/E; WB
Low by 25%-50%
Model Comparisons Lazanoff & Stevenson (1975) N=ll; u=l.lm Bias=+0.7m Lazanoff & Stevenson (1977) N=76; o=1.5m Ocean Data Systems, Inc. Bias=+O.lm (1975) N=59; a=1.3m A . NOAA-EB-01 Bias=-S% N=48; a=2.5m B. NOAA-EB-03 Bias=+20%
Ewing (1971)
Fourth Order Scheme
L/E; WW; WB
30% LOW to 20% High
Ewing (1971)
N=32; u=l.Om Bias=-0.7m
Barnett (1968)
Method of L/E; WW; WB Characteristics
30% Low to 20% High
Barnett (1968)
N=12; o=0.6m Bias=O.7m
Unbiased
Resio & Vincent (1978)
N=123; u=0.5m Bias=O.lm
Unbiased
NORSWAM Pro ject
u=l .Om Bias=-5%
Resio, Vincent Hybrid Linear(1976) Analytical Hasselmann et al. (1976)
L/E; WW; WB
P Mixed Finite Difference and Method of Characteristics
1 ZL/E - linear and experimental wind input; WW - wave-wave interactions; WB N is number of comparisons; u is rms error in significant wave height.
-
wave breaking; P
-
parametric.
88
and Ewing type models generally perform better than that of Salfi. Part of the increased error in the Salfi model studies may be the difficulties of wind field over the ocean; however, a significant part would appear due to deficiences in the source terms and propagation scheme. The FNWC version of this model is currently having its propagation algorithms changed. Parametric Versus Discrete Models Parametric models such as Hasselmann et al. (1976) are the most recent advance in wave modeling. Because of their computational simplicity, they require less computer time. However, f o r the general open-ocean problem they must be coupled to a swell propagation routine. Two assumptions of the single parameter model which may not be strictly valid in an ocean model are that spectral shape is invariant and that the wave angle so rapidly adjusts to the wind angle that the wind angle and the wave angle are equal. The invariance of spectral shape for duration limited waves has been questioned by Mitsuyasu (1968). The relaxation time for wave angle adjustment may also be sufficiently long that considerable variation in directionality and angular dispersion of an active sea away from the wind angle occurs. Forristall et al. (1978) show a hurricane spectrum in which the direction for forward face of the spectral peak is approximately 90' different from the higher frequency portion of the spectrum. Parametric models using more parameters are being formulated and will likely resolve these difficulties. The discrete spectral models can be adjusted to represent the effects of variable wind angles as demonstrated by Forristall et al. (1978) and do not require assumptions regarding spectral shape, other than in the parametric version of the wave-wave interaction source terms. At present they appear more flexible in terms of handling the many ambiguous wave generation cases that occur. Summary The hindcast of wave conditions requires an accurate resolution of the wind field and an equally accurate solution of the wave generation and propagation equations. Results of this study indicate that great care must be taken to obtain wind data on a grid that does not smooth out the pressure gradients responsible for large wave conditions. These data can be obtained from a detailed analyses of the synoptic weather charts and ships observations with an RMS error of about 3 mps. The prediction of sea state can apparently be made with those source terms incorporating conservative nonlinear wave-wave interactions to a higher accuracy than source terms without them. Wave propagation through a fourth order or Lax-Wendroff finite difference schemes or through a ray propagation technique appears more accurate than linear schemes. ACKNOWLEDGEMENT The authors wish to acknowledge the permission of the U. S. Army Engineer Waterways Experiment Station to publish this paper.
89
REFERENCES Barnett, T. P., "On the Generation, Dissipation, and Prediction of Windwaves," Journal of Geophysical Research, Vol. 73, No. 2, 1968, pp. 513-529. Cardone, V. J., "Specification of the Wind Distribution in the Marine Boundary Layer for Wave Forecasting," Tech. Rept. 69-1, Geophysical Science Laboratory, New York University, 1969, 99 pp. Ewing, J. A., "A Numerical Wave Prediction Model for the Atlantic Ocean," Deutsche Hydrographische Zeitschrift, Vol. 24, 1971, pp. 241-261. Forristall, G. Z., E. G. Ward, L. E. Borgman, and V. J. Cardone, "Storm Wave Kinematics," to be presented at the 1978 Offshore Technology Conference, Houston, Texas, 8-11 May 1978. Golding, Brian, W., "A Depth Dependent Wave Model for Operational Forecasting," Meteorological Office, Berkshire, England, 1977, unpublished manuscript. Hasselmann, K., T. P. Barnett, E. Bonws, H. Carlson, D. C. Cartwright, K. Enke, J. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Muller, D. J. Olbers, ti. Richter, W. Sell, H. Walden, "Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP), Deutshes Hydrographisches Institut, Hamburg, 1973, 95 pp. Hasselmann, K., D. B. Ross, P. Muller, and W. Sell, "A Parametric Wave Prediction Model," Journal Physical Oceanography, Vol. 6, 1976, pp. 200-228. Holl, M. M. "The Upper Air Analysis Capabilities, FIB/UA: Introducing Weighted Spreading, Project M-213, Final Report, Contract No. N-000228-75-CZ374, for Fleet Numerical Weather Central, Monterey, California, 1976. Hydraulic Research Station, NORSWAM Technical Advisory Group Report, Hydraulic Research Station, Wallingford, England, 1977 Lax, P. D. and Wendroff, B., "Systems of Conservation Laws," Communications on Pure and Applied Mathematics, Vol. 13, 1960, pp. 217-237. Lazanoff, S. M. and Stevenson, N. M., "An Evaluation of a Hemispheric Operational Wave Spectral Model," Technical Note No. 75-3, Fleet Numerical Weather Central, 1975. Lazanoff, S. M. and Stevenson, N. M., "A Northern Hemisphere Twenty Year Spectral Climatology," Preprint of Paper presented at NATO Symposium on Turbulent Fluxes through the Sea Surface, Wave Dynamics and Prediction, Ile de Bendon, France, 12-16 September 1977. Miles, J. W., "On the Generation of Surface Waves by Shear Flows," Journal Fluid Mechanics, Vol. 3 , 1957, pp. 185-204. Mitsuyasu, H., "On the Growth of Wind Generated Waves (I)," Rept. Res. Inst. for Appl. Mech., Kyushu University, Fukuoker, Japan, Vol. 16, 1968, pp. 459-482. Overland, J. and R. Street, "Winds in the New York Bight," Journal of Physical Oceanography, 1977, V. 7, pp. 200-228. Phillips, 0. M., "On the Generation of Waves by Turbulent Wind," Journal Fluid Mechanics, Vol. 2, 1957, pp. 417-445.
90
Resio, D. T. and C. L. Vincent, "Design Wave Information for the Great Lakes, Report 1, Lake Erie," U. S. Army Engineer Waterways Experiment Station, CE, TR H-76-1, Vicksburg, Miss. 1976, 54 pp. Resio, D. T. and Vincent, C. L . , "A Numerical Hindcast Model f o r Wave Spectra on Water Bodies with Irregular Shoreline Geometry, Report. 1: Test of Nondimensional Growth Rates," U. S. Army Engineer Waterways Experiment Station, CE, MP-H-77-9, August 1977, 53 pp. Resio, D. T. Vincent, C. L., "A Numerical Hindcast Model for Wave Spectra on Water Bodies with Irregular Shoreline Geometry, Report 2: Model Verification with Observed Wave Data," U. S. Army Engineer Waterways Experiment Station, CE, MP H-77-9, to be published in 1978. Resio, D. T., A. W. Garcia, and C. L. Vincent, Preliminary Investigation of Numerical Wave Models, Coastal Zone 78, ASCE, March 1978, p. 2085-2104. Salfi, Robert E., "Operational Computer Based Spectral Wave Specification and Forecasting Models," City University of New York, University Institute of Oceanography 1974, 130 pp. Sverdrup, H. U. and W. H. Munk, Wind, Sea, and Swell: Theory of Relations for Forecasting," H. B. Pub. No. 601, U. S. Navy Hydrographic Office, Washington, D. C., 1947, 44p.
91
WAVE HEIGHT PREDICTION I N COXTAL WATER OF SOUTHERN NORTH SEA
S.
ARANWACHAPUN
IMekong S e c r e t a r i a t , ESCAP, United Nations, Bangkok (Thailand)
ABSTRACT
The paper d i s c u s s e s l o c a l e f f e c t s such as wave r e f r a c t i o n due t o t h e i r r e g u l a r bottom topography i n t h e nearshore region around t h e E a s t Anglian Coast. I t demonstrates how wave r e f r a c t i o n and s h o a l i n g e f f e c t s could b e introduced i n t o a simple wind-wave r e l a t i o n i n o r d e r t o form a b a s i c model t o e s t i m a t e wave h e i g h t i n the area. R e s u l t s from t h e study suggest t h a t l o c a l i t y i s r a t h e r important t o the p r e d i c t i b i l i t y o f t h e sea s u r f a c e waves i n t h e region being i n v e s t i g a t e d .
INTRODUCTION
Although there are numerous wave p r e d i c t i o n models i n t h e l i t e r a t u r e , very few of t h e s e have taken r e g i o n a l e f f e c t s i n t o account.
The models t h a t allow
f o r l o c a l e f f e c t s can b e very important because they may be more r e l i a b l e than general models,where l o c a l i t y i s a dominant e f f e c t .
To demonstrate l o c a l a f f e c t s ,
wave p r e d i c t i o n s i n the nearshore region around t h e E a s t Anglian c o a s t a r e presented.
The wave c h a r a c t e r i s t i c s i n this p a r t o f t h e Southern North Sea a r e very
complex, due t o the sand bank system (see Figure 1) which makes the bottom topography h i g h l y i r r e g u l a r and causes wave r e f r a c t i o n .
I t has been shown t h a t
wave r e f r a c t i o n i n such an a r e a is rather pronounced and t h a t wave h e i g h t can be a f f e c t e d by the r e f r a c t i o n (Aranuvachapun, 1977a)
.
General wave p r e d i c t i o n models
n e g l e c t i n g the r e f r a c t i o n e f f e c t on wave h e i g h t f o r example, by assuming a f l a t s e a f l o o r (as i n the e a r l y model introduced by Sverdrup and Munk (1947) and improved by Bretschneider (19581, w e r e a p p l i e d to the a r e a .
The r e s u l t s from
t h e s e models a r e no more s a t i s f a c t o r y than the r e s u l t s from a simpler model which allows f o r the e f f e c t of wave r e f r a c t i o n . The r e f r a c t i o n can be simply introduced i n t o any p r e d i c t i o n model by using t h e r e f r a c t i o n c o e f f i c i e n t ( K ) and the s h o a l i n g c o e f f i c i e n t (Ks). R and K i s given by S
H/Ho
= [ ( b o b ) . (C,/CG)
where H
0
i
1 = KR
. KS
The product K
R
(1)
i s t h e wave h e i g h t i n t h e deep w a t e r w i t h d i s t a n c e between two orthogonals
bo, and group v e l o c i t y C
Go
s i m i l a r l y , H , b and C
G
a r e f o r t h e shallow water.
92
Fig. I
2-
Figure 1.
Map of t h e a r e a i n v e s t i g a t e d .
Fig. 2 Values estimated from dlagrams Values estimated from wave dato
I
\
I KR
0.4
0 1 330 Figure 2 .
I
I
340
350
A graph of K
compared w i &
I 360
I
I
10 20 Wave ray a n g l e
I
I
I
30
40
50
I
v a l u e s , from wave r e f r a c t i o n d i a g r a m a g a i n s t ray angles the wave d a t a KR v a l u e s .
93 This d e r i v a t i o n (1) may n o t be r e l i a b l e when b = 0, such a s a t t h e c r o s s i n g of orthogonals, H/Ho
tends t o i n f i n i t y .
Pierson (1951) suggested from h i s experimental
work that a t the c r o s s i n g of orthogonals o r a t t h e c a u s t i c p o i n t , t h e r e might be In h i s experiment, t h e phase v e l o c i t y of the waves seemed
phase s h i f t i n waves.
This phenomenon may
t o vanish a t t h e c a u s t i c p o i n t and reappeared again a f t e r .
n o t b e observed i n the r e a l environment due t o t h e randomness of t h e a c t u a l s e a s u r f a c e (Chao, 1974).
Refraction C o e f f i c i e n t ( K ) R
To i n v e s t i g a t e how a p p l i c a b l e the d e r i v a t i o n (1) i s t o t h e e s t i m a t i o n of wave
h e i g h t i n areas of pronounced r e f r a c t i o n , K
R
values ( i n (1)) were determined
from t h e f i e l d d a t a ( H and H ) i n t h e southern North Sea and t h e shoaling c o e f f i c i e n t from S
=IL1 ' [ 2nd
+
4nd/~ s i n h 4nd/L 32.0 m , t h e a s s o c i a t e d wavelength i s
For f i v e second waves i n w a t e r , depth do L
= 39.62
29.87 m.
S i m i l a r l y i n shallow water ( d = 5.0 m ) , the wavelength i s L =
m.
S u b s t i t u t i n g these values i n ( 2 ) gives Ks equal t o 0.86 and t h e
d e r i v a t i o n (1) becomes KR
=
1 0.86
.
H -
(3)
Ho
H and Ho a t d i f f e r e n t wind d i r e c t i o n s were obtained ( f o r d e t a i l s s e e Aranuvachapun,
1977a), and KR values corresponding t o each wind s e c t o r a r e t a b u l a t e d i n t h e following t a b l e .
K
R
was a l s o evaluated from wave r e f r a c t i o n diagram (Wilson,
1966) constructed by using topographic d a t a of t h e a r e a .
A s e r i e s of t h e s e
r e f r a c t i o n diagrams a t various angles of wave rays gives values of K
w i t h wave r a y d i r e c t i o n s a s shown i n Figure 2 .
(dotted l i n e ) .
d i r e c t i o n i s h i g h l y coherent w i t h t h e wind d i r e c t i o n , K compared t o the values on t h e graph o f Figure 2 . standard d e v i a t i o n o f t h e c a l c u l a t e d K
R
Where wave
values i n Table 1 a r e
I t may be seen t h a t t h e r e i s
suggesting t h e simple i d e a of wave
r e f r a c t i o n (1) can b e a p p l i e d t o t h i s a r e a of southern North Sea. TABLE I
Refraction c o e f f i c i e n t s Wind d i r e c t i o n
100 - 2 0 0 m 20° - 30' NE 300 - 400 r m 40' - 50°
Mean values o f K 0.2350 0.2996 n
R
associated
R The e r r o r b a r s r e p r e s e n t
(Table 1).
R
some agreement between t h e two sets of K
R
S.D.
Of
K
R
0.0902 0.2589 n iq23
94 Wind and Wave Relationship
Kraus (1972, s e c t i o n 4.4) shows by dimensional arguments t h a t t h e mean square o f t h e sea-surface displacement
u
i s r e l a t e d t o t h e shipboard
-
l e v e l wind speed
by
where C1 i s a c o n s t a n t and g i s t h e a c c e l e r a t i o n due t o g r a v i t y .
Longuet-Higgins
(1952) derived a t h e o r e t i c a l expression f o r t h e expected maximum wave h e i g h t i n
t h e form
where Cg = ( l o g N)'
and N i s t h e number of wave c r e s t s i n the record.
study the sample s i z e of N i s approximately 10,000, so C2 = 3.04.
For t h i s
From ( 4 ) and
(5), t h e expected wave h e i g h t i n terms of wind speed should be
Table 2 summarises t h e c o r r e l a t i o n s between wind speed a t Gorleston ( s e e Figure 1) and wave h e i g h t s a t Cromer, Happisburgh and Lowestoft ( d e t a i l s on d a t a c o l l e c t i o n can be found i n Aranuvachapun, 197723).
The h i g h e s t c o r r e l a t i o n
c o e f f i c i e n t i s found a t Lowestoft, f o r which t h e d i s t r i b u t i o n o f wind speed and wave h e i g h t i s r e p l o t t e d onto a log-log s c a l e as shown i n Figure 3. of t h e b e s t f i t l i n e from l i n e a r r e g r e s s i o n is equal t o 1.85. (6) between Hmax
The s l o p e
Hence r e l a t i o n s h i p
and U agrees w i t h f i e l d measurements i n this c o a s t a l area of
southern North Sea, and i s a reasonable f i t i n the region being i n v e s t i g a t e d .
TABLE I1
The c o r r e l a t i o n c o e f f i c i e n t s of wind speed a t Gorleston and wave h e i g h t a t t h r e e d i f f e r e n t s t a t i o n s f o r various wind d i r e c t i o n s . Stations
Wind d i r e c t i o n s
Cromer
a l l directions onshore d i r e c t i o n offshore direction 270' 360° NW, 360°
Happisburgh
a l l directions onshore d i r e c t i o n offshore direction 330' - 360° NW, 360'
Lowestoft
a l l directions onshore d i r e c t i o n offshore direction 360' 70' NE, 150'
-
-
Correlation coefficients
-
-
-
100' NE
-0.034 0.480 -0.139 0.534
1 0 0 O NE
0.277 0.657 0.130 0.702
2 0 0 O SE
0.669 0.690 0.648 0.710
95
Fig. 3
Figure 3 .
A graph of wave height ( H
on a log-log s c a l e .
rnax
)
a t Lowestoft against wind speed a t Gorleston
Wave P r e d i c t i o n Model
To i n c o r p o r a t e sohe e f f e c t s of wave r e f r a c t i o n i n the a r e a , a modified form o f ( 6 ) t h a t includes r e f r a c t i o n and shoaling c o e f f i c i e n t s , i s proposed a s H
max
=
CK K U2/g S R
(7)
where C i s a c o n s t a n t , Ks = 0.86 and K
R
values from Figure 2 .
wave h e i g h t a t the t h r e e s t a t i o n s a r e c a r r i e d o u t using ( 7 ) .
P r e d i c t i o n s of This work concen-
trates o n wind s e c t o r from 3 3 0 O N W t o 5O0NE as i t i s t h e regime of onshore winds for t h i s coastline. Figure 4 shows r e s u l t s from t h e p r e d i c t i o n s compared with t h e a c t u a l measured data.
Other models a r e a l s o used t o p r e d i c t wave h e i g h t a t t h e same t h r e e
s t a t i o n s i n o r d e r t o compare t h e i r accuracy.
The p r e d i c t i o n s made using t h e
model of Darbyshire and Draper (1963) , and t h e Sverdrup-Munk-Bretschneider model, a r e presented i n Figures 5 and 6 r e s p e c t i v e l y ( i n t h e s i m i l a r manner as i n Figure 4 ) .
The c o r r e l a t i o n c o e f f i c i e n t between p r e d i c t e d values and t h e
measured values f o r each model is t a b u l a t e d i n T a b l e 3 .
TABLE I11
The c o r r e l a t i o n between p r e d i c t e d wave h e i g h t and t h e a c t u a l measured values a t t h e three s t a t i o n s . Correlation coefficient
Figure associated
Station
Model of p r e d i c t i o n s
Cromer
Darbyshire and Draper Sverdrup-Munk-Bre ts chneider Simple r e l a t i o n ( 7 )
0.69 0.42 0.62
5a 6a 4a
Happisburgh
Darbyshire and Draper Sverdrup-Munk-Bre tschneider Simple r e l a t i o n ( 7 )
0.63 0.61 0.72
5b 6b 4b
Lowestoft
Darbyshire and Draper Sverdrup- Munk-Bre ts chneider Simple r e l a t i o n ( 7 )
0.66 0.71 0.78
5c 6c 4c
I t i s found t h a t t h e proposed formula ( 7 ) g i v e s , i n g e n e r a l , a b e t t e r
c o r r e l a t i o n than t h e Sverdrup-Munk-Bretschneider model i n the a r e a under i n v e s t i gation h e r e .
This r e s u l t suggests t h a t f o r t h e a r e a l i k e t h e nearshore region
around the E a s t Anglian c o a s t where the wave r e f r a c t i o n i s important, t h e s h o a l i n g and r e f r a c t i o n e f f e c t s represented by Ks and K
the model.
should be included i n R I t a l s o demonstrates t h a t d e s p i t e i t s s i m p l i c i t y , t h e m d e l (7)
gives more s a t i s f a c t o r y r e s u l t s than t h e complex model of Sverdrup-MunkBretschneider f o r t h e a r e a s t u d i e d .
The i n c r e a s e i n accuracy may be due t o t h e
f a c t t h a t model ( 7 ) allows f o r l o c a l e f f e c t ( i - e . , r e f r a c t i o n and shoaling effects)
while t h e o t h e r does n o t by assuming t h e f l a t s e a f l o o r , implying
97
'
s
Figure 4.
A graph of p r e d i c t e d wave h e i g h t using (6) compared w i t h t h e a c t u a l wave height f o r t h e three s t a t i o n s .
98
99
p r e d i c t i o n model.
100 t h e l o c a l i t y can be very important t o t h e p r e d i c t i b i l i t y o f t h e s e a i n this region.
ACKNOWLEDGEMENTS
The author wishes t o thank t h e Mekong S e c r e t a r i a t f o r a i d i n g t h e p r e s e n t a t i o n of this paper a t t h e conference, D r . Phadej Savasdibutr and D r . P. Brimblecombe f o r their h e l p f u l d i s c u s s i o n s .
The i d e a s expressed h e r e a r e of my own and not
n e c e s s a r i l y those of t h e S e c r e t a r i a t .
REFERENCES
Aranuvachapun, S a s i t h o r n , 1977a. Wave r e f r a c t i o n i n t h e southern North Sea. Ocean Engineering, 4:91-99. Aranuvachapun, S a s i t h o r n , 197%. Wave Climate i n t h e southern North Sea and Sediment Transport on t h e E a s t Anglian Coast. PhD Thesis, University of E a s t Anglia, Norwich, U.K. Bretschneider, C.L., 1958. Revisions i n wave f o r e c a s t i n g : deep and shallow water. Proceedings of t h e S i x t h Conference on Coastal Engineering, ASCE, Council of Wave Research. Chao, Yung-Yao, 1974. Wave Refraction Phenomena Over t h e Continental Shelf Near the Chesapeake Bay Entrance. U.S. Army Corps. o f Engineering, Coastal Engineering Research Centre, Tech. Memo. No. 47. Darbyshire, M. and Draper, L . , 1963. Forecasting wind - generated s e a waves. Engineering, 195:482-484. Oxford University P r e s s , Kraus, E . B . , 1972. Atmosphere-Ocean I n t e r a c t i o n . pp. 268. 1952. On t h e s t a t i s t i c a l d i s t r i b u t i o n of t h e h e i g h t s Longuet-Higgins, M . S . , of s e a waves. Journal of Marine Research, 11:245-266. J r . , 1951. The I n t e r p r e t a t i o n of Crossed Orthogonals i n Wave Pierson, W . J . , Refraction Phenomena. U . S . A r m y Corps. of Engineering, Coastal Engineering Research Centre Training, Memo. 2 1 . Sverdrup, H . U . and Munk, W.H., 1947. Wind, s e a and s w e l l ; theory of r e l a t i o n s h i p s f o r f o r e c a s t i n g . U.S. Navy Hydrographic O f f i c e , Washington, P u b l i c a t i o n No. 601. Wilson, W.S., 1966. A Method f o r C a l c u l a t i n g and P l o t t i n g Surface Wave Rays. U . S . Army Corps. of Engineering, Coastal Research Centre, Tech. Memo. No. 17.
101
CORRELATION BETWEEN WAVE SLOPES AND NEAR-SURFACE OCEAN CURRENTS
S. SETHURAMAN Department of Energy and Environment, Brookhaven National Laboratory, Upton, NY, USA.
ABSTRACT
The development of wind generated c u r r e n t s i n t h e ocean was studied with simultaneous observations of mean wind speed, wind d i r e c t i o n , s u r f a c e wave parameters and near-surface ocean c u r r e n t . The measurements were c a r r i e d out during February 23 - March 14, 1976 a s p a r t of a c o a s t a l ocean boundary l a y e r and d i f f u s i o n study off Long I s l a n d , New York i n t h e A t l a n t i c Ocean. The r e s u l t s show a high c o r r e l a t i o n between wave slope and near-surface current i n d i c a t i n g t h e p o s s i b i l i t y of wave age playing a s i g n i f i c a n t r o l e i n t h e generation of c u r r e n t . Wave age is known t o cause v a r i a t i o n s i n momentum t r a n s f e r (Kraus, 1972; SethuRaman, 1978). The wind generated c u r r e n t was found t o have a broad s p e c t r a l peak a s compared with t i d a l c u r r e n t s . This peak was found t o occur a t approximately t h e same frequency a s wind speed s p e c t r a l peak. I n t e g r a l time s c a l e s associated with wind and near-surface c u r r e n t were about t h e same, i n d i c a t i n g t h e dominance of wind f o r c i n g near t h e ocean s u r f a c e f o r t h i s period of observations. INTRODUCTION
A s wind blows over w a t e r , wind-generated c u r r e n t s a r e produced i n t h e water due
t o t h e t r a n s f e r of momentum from a i r t o water a t t h e i n t e r f a c e and by f r i c t i o n between a d j a c e n t l a y e r s w i t h i n t h e water.
The downward, ' h o r i z o n t a l momentum f l u x
from t h e atmosphere i s p a r t l y spent on t h e generation of waves and t h e r e s t on d r i f t c u r r e n t s o r wind generated c u r r e n t s .
The mechanism of momentum t r a n s f e r i s
not y e t f u l l y understood, but t h e magnitude seems t o depend on t h e aerodynamic roughness of t h e sea s u r f a c e (SethuRaman and Raynor, 1975) which i s a f u n c t i o n of sea s t a t e c o n d i t i o n s (Neumann, 1968; K i t a i g o r o d s k i i , 1973; SethuRaman, 1977). V a r i a t i o n s i n wave age caused by t h e changes i n mean wind d i r e c t i o n , d u r a t i o n and f e t c h appear t o influence t h e momentum t r a n s f e r s i g n i f i c a n t l y (SethuRaman, 1978). P a r t i a l l y developed waves have s t e e p e r s l o p e s and move a t a lower speed than t h e low-level winds c o n t r i b u t i n g t o h i g h e r f r i c t i o n a l and form drags.
On t h e o t h e r
hand, f u l l y developed waves have f l a t t e r s l o p e s and move a t s i g n i f i c a n t l y h i g h e r speeds r e l a t i v e t o near-surface winds.
The r e l a t i o n s h i p between wind speed and
d r i f t c u r r e n t h a s been i n v e s t i g a t e d i n t h e p a s t by s e v e r a l i n v e s t i g a t o r s i n t h e
102 f i e l d and i n t h e l a b o r a t o r y (Hughes, 1956; C a r r u t h e r s , 1957; Shemdin, 1972; Wu, 1975).
There seems t o be a g e n e r a l agreement t h a t t h e r a t i o between t h e surface
d r i f t v e l o c i t y and t h e s u r f a c e wind speed assumes a n asymptotic value of about 3 per cent a t long f e t c h e s . The purpose of t h i s study i s t o i n v e s t i g a t e t h e p o s s i b l e mechanism by which the
wind generated c u r r e n t s a r e produced and maintained.
Wave h e i g h t and wave period
measurements and near-surface c u r r e n t and wind observations were used t o study the v a r i a t i o n s i n wind-induced d r i f t .
S p e c t r a l a n a l y s i s of various parameters were
performed t o determine t h e dependence of one on t h e o t h e r .
Wave s l o p e , wind
speed and s u r f a c e c u r r e n t were some of t h e v a r i a b l e s considered important t o determine a s t o whether t h e p a t t e r n of v a r i a b i l i t y of atmospheric momentum is followed i n t h e process of c u r r e n t generation. MEASUREMENTS The oceanographic measurements c o n s i s t e d of a moored instrument a r r a y 5 km off shore i n t h e A t l a n t i c Ocean near Long I s l a n d (Fig. 1 ) .
Observations of c u r r e n t s ,
s a l i n i t y and temperature a t d i f f e r e n t depths were recorded with t h i s spar buoy. A d e s c r i p t i o n of t h e development of t h i s telemetered, moored instrument a r r a y i s
given by Dimmler, e t a 1 (1975).
Wave h e i g h t s and wave periods were observed with
a "waverider" which is e s s e n t i a l l y a buoy t h a t follows t h e movements of t h e water s u r f a c e and measures waves by measuring t h e v e r t i c a l a c c e l e r a t i o n of t h e buoy. The s p h e r i c a l buoy was 0.7 m i n diameter and was provided with an antenna f o r the tranmission of d a t a t o t h e shore.
Mean wind speed and d i r e c t i o n were measured a t
a h e i g h t of 24 m a t t h e c o a s t a l meteorological s t a t i o n a t Tiana Beach (TB).
The
analyses reported h e r e a r e based on measurements made f o r a period of t h r e e weeks from February 23 t o March 14, 1976.
F i g . 1. Map of e a s t e r n Long I s l a n d showing t h e l o c a t i o n of t h e oceanographic spar buoy, and t h e meteorological tower a t Tiana Beach (TB). Wave r i d e r was deployed c l o s e t o t h e buoy.
103 ANALYSIS Passage of synoptic meteorological systems over Long I s l a n d and t h e v i c i n i t y causes v a r i a t i o n s i n near-shote wind speeds and wind d i r e c t i o n s .
A t y p i c a l time
period of mean wind speed measured a t Tiana Beach (TB i n Fig. 1) a t a height of 24 m is shown i n F i g . 2, f o r the d u r a t i o n of t h i s study. Wind speeds v a r i e d from -1 1 t o 18 m s e c . Observations a t t h e beach a r e approximately r e p r e s e n t a t i v e of over-water winds (SethuRaman and Raynor, 1978). Time h i s t o r i e s of t h e wave height and wave periods a r e given i n Fig. 3.
I n c r e a s e i n wind speeds and wave h e i g h t s
w i t h t h e approach of storms can be seen i n Figs. 2 and 3, r e s p e c t i v e l y ,
1.00’ FEB 23
1
I
I
MAR 15
MAR I
Fig. 2. Time h i s t o r y of one-hour mean wind speeds a t Tiana Beach a t a h e i g h t of 24 m f o r t h e d u r a t i o n of t h e experiments. One of t h e o b j e c t i v e s of t h e a n a l y s i s was t o s e p a r a t e t h e wind generated current
from t h e t o t a l c u r r e n t and study i t s v a r i a t i o n .
Separation of t h e t i d a l component
is a d i f f i c u l t procedure due t o i t s dependence on s e v e r a l f a c t o r s .
The t i d a l e l l i p s e seems t o have a n along-shore component of about 17 cm/sec with t h e e l l i p s e inclined t o t h e shore ( S c o t t and Csanady, 1976).
Analysis of near-surface c u r r e n t s during
low wind periods i n d i c a t e d t i d a l amplitudes of comparable magnitude.
A tidal
104
Fig. 3 .
Time history of 20-minute wave heights and wave periods near the buoy (see Fig. 1).
-1 amplitude of 15 cm sec was used to help estimate the wind generated currents from along-shore current observations. Observations of tides at Shinnecock Inlet were used to get tidal cycles. This inlet is in the vicinity of the measurements site.
Any possible nonlinear interactions between the tidal and wind generated
currents were neglected in the present study. Some of the analyses were also performed without separating the tidal current to provide an alternate interpretation. The measurements used here to study the wind generated current were made at an average depth of 3 m below the water surface.
105 Wave s l o p e s The s i g n i f i c a n t wave h e i g h t , H , obtained from t h e waverider i s t h e average h e i g h t of t h e h i g h e s t 1 / 3 of t h e waves. over 20 min. d u r a t i o n a r e used h e r e . same 20 min.
Time p e r i o d s , T , of t h e waves averaged
Mean wind speeds a l s o corresponded t o t h e
Wave l e n g t h , L, was obtained from t h e r e l a t i o n s h i p ,
where g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n .
Mean s l o p e s of t h e waves were t h e n
e s t i m a t e d from s
=H Ll2
A case study A t y p i c a l h i g h wind period h a s been chosen t o s t u d y t h e simultaneous v a r i a t i o n of wind d i r e c t i o n , wave h e i g h t , wave s l o p e and wind generated c u r r e n t .
High winds
w i t h a long f e t c h over t h e w a t e r occurred on February 24 a f t e r a high p r e s s u r e system moved over t h e ocean. s h o r e t o a l o n g shore.
T h i s caused a change i n wind d i r e c t i o n from o f f
Wind speeds i n c r e a s e d from about 3 m/sec t o 15 m/sec.
Maximum wind speeds corresponded w i t h maximum wave h e i g h t o b s e r v a t i o n s shown i n Fig. 4.
A time h i s t o r y of mean wind d i r e c t i o n s and mean wave s l o p e s computed
from Eq. 2 a r e given i n F i g . 5.
The s l o p e s a r e t h e s t e e p e s t immediately a f t e r a The wave s l o p e t h e n reaches a n a s y m p t o t i c a l l y
s i g n i f i c a n t change i n wind d i r e c t i o n .
c o n s t a n t lower v a l u e a s t h e wind d i r e c t i o n becomes more p e r s i s t e n t .
This phen-
omenon i s b e l i e v e d t o be due t o d i f f e r e n t s t a g e s of development of waves o r i n o t h e r words due t o wave age.
An i n c r e a s e i n s u r f a c e d r a g was observed immediately
f o l l o w i n g s i g n i f i c a n t changes i n wind d i r e c t i o n i n previous s t u d i e s (Neumann, 1968; SethuRaman, 1978). a r e shown i n F i g . 6.
Mean wind speeds and t h e e s t i m a t e d wind generated c u r r e n t s The maximum wave s l o p e and t h e h i g h e s t c u r r e n t l a g maximum
wind speed by a few hours.
As t h e wind speed, wind d i r e c t i o n and t h e wave s l o p e
reach approximately c o n s t a n t v a l u e s , wind generated c u r r e n t a l s o tends t o approach a n asymptotic value. 13 cm/sec
This average e q u i l i b r i u m v a l u e can be estimated t o be about
f o r c u r r e n t and about 6 m/sec f o r wind f o r t h i s c a s e .
Assuming f u l l y
rough c o n d i t i o n s , f r i c t i o n v e l o c i t y uj, f o r t h e a i r can be e s t i m a t e d a s 24 cm/sec (SethuRaman and Raynor, 1975) y i e l d i n g a n average r a t i o of wind generated c u r r e n t ,
V, t o f r i c t i o n v e l o c i t y u* a s 0.54.
Values c l o s e t o 0.53 have been found by Wu
(1973) and P h i l l i p s and Banner (1974).
Spectral analysis
Frequencies a s s o c i a t e d w i t h wind generated c u r r e n t s w i l l be r e a d i l y apparent i n a s p e c t r a l a n a l y s i s of t h e time s e r i e s d a t a s i n c e t h e t i d a l f r e q u e n c i e s a r e
106
HOURS
Fig. 4.
Wave h e i g h t v a r i a t i o n s f o r February 23-26. S t a r t i n g d a t e and time a r e a l s o indicated. Increase i n wave h e i g h t due t o i n c r e a s e i n mean wind speed i s seen.
HOURS
Fig. 5. Variance of wave s l o p e s and mean wind d i r e c t i o n s f o r February 23-26. S t a r t i n g d a t e and t i m e a r e a s i n d i c a t e d . S o l i d l i n e s r e p r e s e n t t h e wave slope and dashed l i n e t h e wind d i r e c t i o n . Close c o r r e l a t i o n between wave slope and wind d i r e c t i o n seems t o e x i s t .
FEE 23.1976 1700E
" 15
0
- 5 ~
~~'~o'&'&'&';o';o HOURS
Fig. 6. Wind generated c u r r e n t (estimated) and wind speed f o r February 23-26.
107 d i u r n a l and semi-diurnal.
One advantage of t h i s a n a l y s i s i s t h a t t h e r e i s no need
t o separate the t i d a l currents.
The v a r i a n c e spectrum of t h e one-hour along-shore
mean wind speeds f o r t h e d u r a t i o n of t h e s t u d y is shown i n F i g . 7.
The spectrum
h a s a pronounced peak around .014 c y c l e s p e r hour which corresponds t o a time p e r i o d of about 3 days.
T h i s time p e r i o d r e p r e s e n t s t h e average time elapsed be-
tween two s u c c e s s i v e h i g h wind episodes caused by t h e movement of synoptic systems and i s i n agreement w i t h a s i m i l a r a n a l y s i s made w i t h o b s e r v a t i o n s c o l l e c t e d cont i n u o u s l y over one y e a r (SethuRaman and Brown, 1977).
A small d i u r n a l peak can
Variance s p e c t r a of along-shore c u r r e n t s a t depths of
a l s o be s e e n i n Fig. 7.
3.1 m, 14.3 m, and 24.6 m a r e shown i n Fig. 8.
A pronounced, but narrow peak f o r
a l l d e p t h s w i t h c o n s t a n t amplitude was found a t a frequency corresponding t o semid i u r n a l t i d a l period.
An e s t i m a t e of t h e semi-diurnal along-shore t i d a l c u r r e n t
from F i g . 8 g i v e s about 16 cm/sec.
A v a l u e of 15 cm/sec was assumed h e r e and a
v a l u e of 17 cm/sec was r e p o r t e d by S c o t t andcsanady (1976).
Diurnal t i d a l c u r r e n t s
d i d n o t produce a pronounced peak b u t was found t o be p r e s e n t a t a l l depths with d e c r e a s i n g amplitudes.
Decrease i n s p e c t r a l amplitude between t h e depths of 14.3
and 24.6 mwas more t h a n t h a t between 3.1 and 14.6 m. t h e reason f o r t h i s difference. 30 m.
Bottom f r i c t i o n might be
Depth of w a t e r a t t h e s i t e of t h e buoy was about
The frequency a s s o c i a t e d with'wind-generated c u r r e n t i s a l s o seen i n Fig. 8
which corresponds t o t h e dominant peak of wind speed s p e c t r a i n F i g . 7 .
A com-
p a r i s o n of t h e s p e c t r a l amplitudes a t t h i s frequency would y i e l d a r a t i o of 3 p e r c e n t between t h e wind-generated c u r r e n t and wind speed which h a s been found t o be t h e e q u i l i b r i u m v a l u e by s e v e r a l i n v e s t i g a t o r s OJu, 1973).
S p e c t r a l dens-
i t i e s f o r wind speed and along-shore c u r r e n t a t 3 m have been p l o t t e d a s a funct i o n of frequency i n F i g s . 9 and 10, r e s p e c t i v e l y .
The c u r r e n t s p e c t r a (Fig. 10)
seems t o follow Kolmogorov's i n e r t i a l subrange r e l a t i o n s h i p a t f r e q u e n c i e s more t h a n 0.1 c y c l e p e r hour.
With a mean c u r r e n t of 18 cm s e c
-1
t h i s corresponded
t o a wave l e n g t h of about 2 m which was approximately equal t o t h e depth of measurement.
Atmospheric t u r b u l e n c e was found t o obey Kolmogorov's r e l a t i o n a t
f r e q u e n c i e s above 0.1 Hz (SethuRaman, e t a l . ,
1974).
Time s c a l e s
A u t o c o r r e l a t i o n f u n c t i o n f o r wind speed RU(t) d e f i n e d a s
-2 where u ( t ) u ( t ) i s t h e autocovariance of wind speed and u i s t h e v a r i a n c e ,
1
2
i s a f u n c t i o n of only t h e time d i f f e r e n c e t 2-tl, and d e s c r i b e s t h e memory of u(t).
A s i m i l a r f u n c t i o n , R ( t ) can be d e f i n e d f o r t h e water c u r r e n t .
Varia-
t i o n of R ( t ) and R ( t ) w i t h time l a g s a r e shown i n F i g s . 11 and 12, r e s p e c t i v e l y .
108
5 x lo5
I
I
-
6lo50
% ‘uc
5
v I
c
v
u)
c
n W
g
-
lo4-
fJY
n
5 ?
o
OD0
w
0
I
lO”o00l
F i g . 7.
F i g . 8.
0.01
I
0.I
I
at One-dimensional variance spectrum for one-hour mean wind speeds at T i a M Beach.
One-dimensional v a r i a n c e spectrum f o r one-hour mean along-shore currents a t d i f f e r e n t depths.
109
i
3x10'
' I0
~00.001
Fig. 9.
0.01
0.I n (cycles p e r hour)
Variation of spectral d e n s i t i e s as a function of frequencies for mean wind speeds a t Tiana Beach.
30001
1
I
I
I
0.01 0.I n ( c y c l e s per hour)
Fig. 10.
I
Variation of spectral d e n s i t i e s a s a function of frequencies for onehour mean along-shore near-surface currents (depth: 3 m). /
I
I
I
I
l
-
WIND SPEED
-
-
W
F i g . 11.
,
Autocorrelogram for wind a t Tiana Beach.
-
-
I
!
110
A U N G SHORECURRENT
-
3.1m ---246m
F i g . 12.
Autocorrelogram f o r along-shore c u r r e n t s a t 3 . 1 m and 2 4 . 6 m d e p t h s
The atmospheric a u t o c o r r e l a t i o n f u n c t i o n f a l l s off r a t h e r r a p i d l y , but t h e a u t o c o r r e l a t i o n f o r c u r r e n t h a s s e v e r a l peaks and f a l l s off slowly i n d i c a t i n g longer memories and d i f f e r e n t f o r c i n g f u n c t i o n s . a l s o be seen i n F i g . 12. T~ =
f
Semi-diurnal and d i u r n a l peaks can
An i n t e g r a l time s c a l e ,
(4)
RU(t) d t
0
can be d e f i n e d f o r wind speed and a s i m i l a r one f o r c u r r e n t .
T h i s time s c a l e
was e s t i m a t e d t o be about 10.5 hours f o r wind from F i g . 11 and about 11.5 hours f o r c u r r e n t from F i g . 12.
The c l o s e n e s s of t h e s e two values s u g g e s t s t h e domin-
ance of atmospheric f o r c i n g on t h e ocean.
Coherence
A measure of t h e c o r r e l a t i o n between wave s l o p e , s , and along-shore c u r r e n t , c , a s a f u n c t i o n of frequency can be obtained by computing t h e coherence, Cohscr given by
where Co(n) and Q(n) a r e t h e c o s p e c t r a and q u a d r a t u r e s p e c t r a , r e s p e c t i v e l y , and S i s t h e i n d i v i d u a l spectrum a t d i f f e r e n t f r e q u e n c i e s , n. a r e shown i n Fig. 13 a s a f u n c t i o n of f r e q u e n c i e s .
Values of Coh
A maximum coherence of about
0.55 occurs a t a frequency of 0.16 c y c l e s per hour corresponding t o a time period of about 6 hours.
This i n d i c a t e s t h a t t h e r e i s a good c o r r e l a t i o n between
wave s l o p e and n e a r - s u r f a c e c u r r e n t and t h e maximum c u r r e n t s l a g maximum s l o p e by about s i x hours.
111
0.8
WAVE SLOPE R ALONG SHORE CURRENT
n (cycles per hour)
F i g . 13.
Coherence between wave s l o p e and along-shore c u r r e n t a s a f u n c t i o n of frequency.
CONCLUSIONS
A n a l y s i s of simultaneous o b s e r v a t i o n s of s u r f a c e wave parameters, wind speed and n e a r - s u r f a c e c u r r e n t i n d i c a t e s t h e p o s s i b i l i t y of wave age p l a y i n g an imp o r t a n t r o l e i n t h e g e n e r a t i o n of wind d r i f t c u r r e n t s .
Maximum c u r r e n t s appear
t o l a g maximum wave s l o p e s by about 6 h o u r s ,
ACKNWLEDGEMENTS
Many members of t h e D i v i s i o n of Atmospheric Sciences and t h e D i v i s i o n of Oceanographic Sciences p a r t i c i p a t e d i n t h e experiments.
A s s i s t a n c e i n computer
p r o g r a m i n g was provided by C. Henderson and J. T i c h l e r and i n d a t a a n a l y s i s by
J. Glasmann and K. T i o t i s .
The a u t h o r wishes t o thank T. S . Hopkins and G. S .
Raynor f o r s t u d y i n g t h e manuscript and o f f e r i n g some v a l u a b l e s u g g e s t i o n s . The submitted manuscript h a s been authored under c o n t r a c t EY-76-C-02-0016 w i t h t h e U. S. Department of Energy.
Accordingly, t h e U. S . Government r e t a i n s
a nonexclusive, r o y a l t y - f r e e l i c e n s e t o p u b l i s h o r reproduce t h e published form of t h i s c o n t r i b u t i o n , o r allow o t h e r s t o do s o , f o r U. S. Government purposes.
112 REFERENCES C a r r u t h e r s , J . N . , 1957. A d i s c u s s i o n of "A d e t e r m i n a t i o n of t h e r e l a t i o n between wind and s u r f a c e d r i f t . " Quar. J . Roy. Met. S O C . , 83: 276-277. Dinnnler, D. G . , Greenhouse, N. and Rankowitz, S . , 1975. A c o n t r o l l a b l e a u t o mated environmental d a t a a c q u i s i t i o n and monitoring system. Proc. 1975 Nuclear Science Symposium, San F r a n c i s c o , C a l i f o r n i a , November 1975. Hughes, P., 1956. A d e t e r m i n a t i o n of t h e r e l a t i o n between wind and sea s u r f a c e d r i f t . Quart. J . Roy. Meteor. SOC. 82: 494-502. K i t a i g o r o d s k i i , S . A . , 1973. The physics of a i r - s e a i n t e r a c t i o n . T r a n s l a t e d from Russian by A. Baruch, I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , Jerusalem, pp. 237. Kraus, E. B . , 1972. Atmosphere-Ocean I n t e r a c t i o n . Clarendon P r e s s , Oxford, England, pp. 275. Neumann, G . , 1968. Ocean C u r r e n t s . E l s e v i e r S c i e n t i f i c P u b l i s h i n g Company, New York, N. Y . , pp. 352. P h i l l i p s , 0. M, and Banner, M. L., 1974. Wave breaking i n t h e presence of wind d r i f t and s w e l l . J . F l u i d Mech., 66: 625-640. S c o t t , J. T . and Csanady, G. T . , 1976. Nearshore c u r r e n t s o f f Long I s l a n d . J. Geophys. Res., 81: 5401. Shemdin, 0. H . , 1972. Wind-generated c u r r e n t and phase speed of wind waves. J . Phys. Ocean. 2: 411-419. SethuRaman, S . , 1977. The e f f e c t of c h a r a c t e r i s t i c h e i g h t of sea s u r f a c e on d r a g c o e f f i c i e n t . BNL Report 21668, pp. 33. SethuRaman, S . , 1978. I n f l u e n c e of mean wind d i r e c t i o n on s e a s u r f a c e wave development. J. Phys. Ocean., 8: ( i n p r e s s ) . SethuRaman, S . and Brown, R. M., 1977. Temporal v a r i a t i o n of suspended p a r t i c u l a t e s a t Upton, L. I . , N. Y. AMS Conference on A p p l i c a t i o n s on A i r P o l l u t i o n Meteorology, November 28-December 2, 1977. P r e p r i n t Volume: 16-18. SethuRaman, S. and Raynor, G. S . , 1975. S u r f a c e d r a g c o e f f i c i e n t dependence on t h e aerodynamic roughness of t h e s e a . J . Geophys. Res., 80: 4983-4988. SethuRaman, S. and Raynor, G. S . , 1978. E f f e c t of changes i n upwind s u r f a c e c h a r a c t e r i s t i c s on mean wind speed and t u r b u l e n c e near a c o a s t l i n e . Am. Meteor. SOC. Fourth Symposium on Turbulence, D i f f u s i o n , and A i r P o l l u t i o n , Reno, Nevada, January 15-18, 1977. P r e p r i n t Volume ( i n p r e s s ) . SethuRaman, S . , Brown, R. N. and T i c h l e r , J . , 1974. S p e c t r a of atmospheric t u r b u l e n c e over t h e sea d u r i n g s t a b l y s t r a t i f i e d c o n d i t i o n s . Am. Meteor. SOC. Symposium on Atmospheric D i f f u s i o n and A i r P o l l u t i o n , Santa Barbara, C a l i f o r n i a , September 9-13, 1974. P r e p r i n t Volume: 71-76. Wu, J . , 1973. p r e d i c t i o n of n e a r - s u r f a c e d r i f t c u r r e n t s from wind v e l o c i t y . J . Hyd. D i v i s i o n , ASCE, 99: 1291-1302. Wu, J . , 1975. Wind-induced d r i f t c u r r e n t s . J . F l u i d Mech., 68: 49-70.
113
THE TOW-OUT
OF A LARGE PLATFORM
B. MacMahon
C i v i l Engineering Department, I m p e r i a l College, London
ABSTRACT The time mean r e s i s t a n c e experienced by a f l o a t i n g body being towed a g a i n s t I f these a r e small compared t o t h e waves i s found t o depend on i t s motions. wave amplitude t h e a p p r o p r i a t e frame of r e f e r e n c e f o r c a l c u l a t i o n s i s Eulerian. The magnitude of t h e wave f o r c e i n t h i s system a g r e e s with Lagrangian determinaFrom a consideration t i o n s which however p o i n t t o a d i f f e r e n t l i n e of a c t i o n . of t h e impulsive g e n e r a t i o n of t h e motion t h e l i n e of a c t i o n of t h e Eulerian wave f o r c e i s shown t o be a t a d i s t a n c e ka2/2 below mean water l e v e l . This l e a d s t o some q u a l i t a t i v e conclusions on frequency changes and induced c u r r e n t s a t plane b a r r i e r s and some o b s e r v a t i o n s on t h e transformation of t h e "high l e v e l " Eulerian momentum f l u x when t h e waves a r e a t t e n u a t e d by i n t e r n a l and bottom f r i c t i o n . The study o f t h e s t a b i l i t y of t h e v e r t i c a l motion of l a r g e platforms when sinking on t o a prepared p o s i t i o n on t h e sea bed l e a d s t o a confirmation of t h e usual engineering experience t h a t s t a b i l i t y i s d e a r l y bought. The p o s s i b i l i t y of r e p l a c i n g an expensive s t a t i c s t a b i l i t y by a r e l a t i v e l y cheaply acquired i n e r t i a l s t a b i l i t y i s seen t o be i m p r a c t i c a l .
I NI'RODUCTION
The s t u d i e s which follow arose o u t of a series of c a l c u l a t i o n s t o determine t h e most adverse c o n d i t i o n s of waves and c u r r e n t s compatible with towing and sinking o p e r a t i o n s on a l a r g e concrete o i l production platform. T y p i c a l l y t h i s comprises a c y l i n d r i c a l base about 100-150 m e t r e s i n d i a m e t e r and 50 metres high from which arises a number of towers 150-200 metres long to support t h e production deck which may be about 60-70 metres square.
Depending
on a v a i l a b l e water depth, t h e sequence of c o n s t r u c t i o n o p e r a t i o n s and questions of s t a b i l i t y and towing r e s i s t a n c e , p l a t f o r m s may be towed o u t a t various draughts with t h e w a t e r l i n e passing through e i t h e r t h e base o r the towers.
The f l o a t i n g
phase of t h e l i f e of a platform i s a v u l n e r a b l e one when a very valuable but unshipshape s t r u c t u r e i s being towed t o i t s f i n a l emplacement a t g r e a t expense
114 by a f l e e t of tugs. unlike t h e f i r s t o r d e r o s c i l l a t o r y f o r c e which v a r i e s l i n e a r l y with wave height but a g a i n s t which no n e t t work needs to be done i n towing,the time mean wave force The r a p i d rise of towing r e s i s t a n c e
depends on t h e square of t h e wave amplitude.
with i n c r e a s i n g wave h e i g h t which i s a consequence of t h i s makes t h e p r e d i c t i o n of a s u i t a b l e weather window v i t a l a s beyond a c e r t a i n c r i t i c a l wave height a platform i s no longer towable. The l o c a t i o n on t h e v e r t i c a l of t h e l i n e of a c t i o n of t h e u n i d i r e c t i o n a l wave f o r c e i s a l s o a matter of g r e a t p r a c t i c a l importance as it may govern t h e draught a t which a platform should b e towed i n given wave c o n d i t i o n s .
I f t h e water i s
deep enough it may be p o s s i b l e t o lower t h e p a r t of t h e s t r u c t u r e having a l a r g e a r e a of c r o s s s e c t i o n below t h e l i n e of a c t i o n of t h e wave f o r c e .
Although the
drag due t o t h e t r a n s l a t i o n a l v e l o c i t y through t h e water may i n c r e a s e s u b s t a n t i a l l y a t t h e lower d r a u g h t , i t w i l l be more than counterbalanced by t h e diminution i n t h e wave r e s i s t a n c e when c o n d i t i o n s a r e severe. A s regards t h e sinking operation it i s important t o determine t h e r e l a t i v e
o r d e r of magnitude of t h e f o r c e s c o n t r o l l i n g t h e s t a b i l i t y of the descent.
There
have been a t l e a s t two i n c i d e n t s i n t h e North Sea of t a n g e n t i a l g l i d i n g ("skidding o f f " ) of platforms approaching touchdown on t h e s e a bed and some anxiety expressed over t h e p o s s i b i l i t y of a " f a l l i n g l e a f " o s c i l l a t i o n developing.
I n several
designs l a r g e a d d i t i o n s i n t h e form of buoyancy chambers have been necessary to secure a s a t i s f a c t o r y r i g h t i n g arm a t a l l draughts through which t h e platform may pass.
I t i s a c u r i o u s f a c t t h a t t h e s e chambers,which c o n s t i t u t e a s u b s t a n t i a l
p a r t of t h e s t r u c t u r e , f u n c t i o n during only a few minutes of t h e p l a t f o r m ' s lifetime. I t was of i n t e r e s t t h e r e f o r e t o see i f t h e necessary s t a b i l i t y could be achieved
by o t h e r means.
THE TOWING RESISTANCE
Lagrangian a n a l y s i s
I t h a s been known s i n c e Stokes' 1847 paper t h a t a second order t i m e mean current
i s a s s o c i a t e d with t h e propagation of i r r o t a t i o n a l g r a v i t y waves of f i n i t e ampli-
tude.
Stokes obtained t h i s r e s u l t by a change from a Eulerian t o a Lagrangian
s p e c i f i c a t i o n of t h e motions and t a k i n g account of t h e f i n i t e dimensions of t h e T h i s showed t h e c u r r e n t t o be of magnitude U = ck 2 a 2 e 2kz i. n
p a r t i c l e paths.
L
t h e d i r e c t i o n of propagation of t h e waves when t h e w a t e r i s deep i n r e l a t i o n t o t h e wavelength ( c = phase v e l o c i t y , k = wave number, a = amplitude and z i s t h e v e r t i c a l co-ordina&e of t h e o r b i t c e n t r e s ) . f l u x i s given by
The corresponding time mean momentum
I UL Vgroupdz as wave p r o p e r t i e s 0
such a s energy and momentum are
115 propagated a t t h e group v e l o c i t y (V group) equal t o h a l f t h e phase v e l o c i t y on deep water.
The r e s u l t ,
ipga',
i s i n agreement with t h e Eulerian a n a l y s i s of
Longuet-Higgins (1964). I n f i n i t e depth t h e Lagrangian d r i f t v e l o c i t y i s given ck2a2 cosh 2 k ( h + z ) . When t h i s i s s u b s t i t u t e d i n t h e previous i n t e by UL = 2sinhLkh 2kh gra1,taking account a l s o of t h e new v a l u e of t h e group v e l o c i t y , ;(l + ___ 2kh sinh2kh) ' pga t h e momentum f l u x t u r n s o u t t o be -(1 + 7) , again i n agreement with t h e 4 sinh2kh " r a d i a t i o n stress" approach o f Longuet-Higgins (1977)
.
The l o c a t i o n of t h e mean momentum may be obtained by taking moments about z = 0:
I ULzdz -m
I uLdz -z
0
0
=
whence
-m
-
z=
h 4n
(z6 0 i s t h e "paramstre de repos")
The impulsive generation of t h e motion
Any i r r o t a t i o n a l motion may be considered t o have been generated from r e s t by a system of impulsive p r e s s u r e s p $ a p p l i e d t o t h e f l u i d boundaries.
Following
Lord Kelvin (1887) a r i g i d corrugated s h e e t i s imagined t o cover t h e water surface. I t i s s t r u c k an impulsive blow i n t h e h o r i z o n t a l d i r e c t i o n and immediately with-
drawn ( i n Kelvin's example t h e s h e e t was g r a d u a l l y a c c e l e r a t e d up t o the phase v e l o c i t y when t h e f l u i d p r e s s u r e s on it v a n i s h e d ) .
By a well known theorem t h e
impulse i s equal to t h e t o t a l momentum of t h e f l u i d p l u s t h e impulsive r e a c t i o n s on t h e boundaries a t i n f i n i t y .
These r e a c t i o n s a r e f i n i t e and may be ignored* i f
t h e s h e e t i s envisaged as being many wavelengths long.
This would appear t o be
an example of t h e u n c e r t a i n t y p r i n c i p l e where t h e u n c e r t a i n t y i n the momentum may be reduced by spreading t h e wave motion over a g r e a t l e n g t h .
Fig.
1. Generation of s u r f a c e waves by an impulse
The resolved p a r t of t h e impulse i n t h e h o r i z o n t a l d i r e c t i o n on an element of dn t h e p r o f i l e i s given approximately by p 4 d x -where 17 = a s i n k x i s t h e s u r f a c e o r d i n a t e and ds
2
dx d x i f t h e wave slope i s not too l a r g e .
As
4
=
ekzcos(kx-ot)
*In t h e c a s e of 2 dimensional o r 3 dimensional bodies moving i n an unbounded f l u i d t h e r e i s an i n t e r e s t i n g indeterminacy i n t h e f l u i d momentum d u e t o the f i n i t e reactions a t infinity.
.
116 on deep water where a is t h e frequency, t h e impulse per wavelength i s given by:
A c o s kx ka c o s kx dx
where z has been taken a s zero over t h e p r o f i l e .
0
The r e s u l t , T p a * c , i s
t h e h o r i z o n t a l momentum per wavelength of a g r a v i t y wave t r a i n
on deep water.
The n e t t v e r t i c a l momentum p e r wavelength i s of course zero.
momentum f l u x i s
rrpa2c where ~
The
T i s t h e period, t h e f a c t o r 2 being due t o t h e group
On s u b s t i t u t i n g f r a n t h e d i s p e r s i o n r e l a t i o n w e again o b t a i n
v e l o c i t y a s before.
.
t h e expression
4 I n f i n i t e depth where
+
=
cOshk(h+z) a coshkh
cos(kx-ot) t h e mean momentum per wave-
l e n g t h i s given by t h e same i n t e g r a l a s b e f o r e s i n c e an impulsive motion of t h e plane bottom can make no c o n t r i b u t i o n t o t h e momentum i f t h e f l u i d i s i n v i s c i d . When t h e r e s u l t
v2 2
f l u x i s given a s -(I 4
i s m u l t i p l i e d by t h e a p p r o p r i a t e group v e l o c i t y t h e momentum 2kh
+
a) ..
The l i n e of a c t i o n of t h e wave force
I n t h e absence of v i s c o s i t y t h e r e i s no mechanism by which t h e manentum generated by t h e impulse can be communicated t o o t h e r f l u i d r e g i o n s o u t s i d e t h e "layer of a c t i o n " of t h e h y p o t h e t i c a l corrugated s h e e t .
I n both t h e i n f i n i t e and f i n i t e
depth c a s e s t h e wave momentum i s e n t i r e l y contained i n t h e region of space between t h e b o t t o m s of t h e troughs and t h e t o p s of t h e c r e s t s ( P h i l l i p s , 1 9 7 7 ) . The l i n e of a c t i o n of t h e f o r c e i s a t s t i l l water l e v e l a s i s e v i d e n t from the symmetry of t h e s h e e t . To t h e second o r d e r t h e wave p r o f i l e i s given by ka2 5 = asinkx + - sin2kx r e l a t i v e t o t h e s t i l l water l i n e on i n f i n i t e depth. Both 2 canponents a r e symmetrical about t h i s l e v e l but t h e mean of t h e r e s u l t i n g p r o f i l e ka2 i s r a i s e d by due t o t h e f a c t t h a t t h e f i r s t harmonic l i f t s both the minima of 2 The l i n e of a c t i o n t h e troughs and t h e maxima of t h e crests i n t h e fundamental. ka2 of t h e E u l e r i a n wave f o r c e i s t h e r e f o r e t o t h e second o r d e r of approximation ~
2
below t h e mean water l e v e l .
T h i s l o c a t i o n of the wave f o r c e w a s f i r s t obtained
by Longuet Higgins (personal canmunication, 1 9 7 8 ) . The c o n t r a s t i n t h e l o c a t i o n of t h e wave f o r c e i n t h e two systems i s perhaps only t o be expected.
I t i s w e l l known t h a t t h e fundamental Eulerian and Lagrangian
d e f i n i t i o n s of v e l o c i t y a r e q u i t e d i f f e r e n t .
I n p r a c t i c e however t h e f o r c e on t h e
towrope i s unambiguous and t h e r e s o l u t i o n o f t h e paradox m u s t l i e i n t h e o s c i l l a t i o n s of t h e platform ,as hinted a t by Havelock
(1940) i n t h e c o n t e x t of s h i p r e s i s t a n c e .
I f t h e heaving motions of t h e s t r u c t u r e a r e s m a l l r e l a t i v e t o t h e wave amplitude, t h e a p p r o p r i a t e frame of r e f e r e n c e i s , l i k e t h e body i t s e l f , f i x e d i n space, i . e . Eulerian.
On t h e o t h e r hand i f t h e heaving motions a r e s i g n i f i c a n t and of t h e
order of t h e wave amplitude t h e body w i l l experience a f o r c e even on t h e p a r t of i t which l i e s below t h e wave troughs. I t i s an i n t e r e s t i n g thought t h a t t h e f o r c e could be opposite t o t h e d i r e c t i o n
117 of t h e waves w e r e t h e heaving period a d j u s t e d t o be somewhat g r e a t e r than t h e wave period.
Here, were t h e damping s m a l l enough,the body's motion would be i n
antiphase t o t h a t of t h e wave,so coming under t h e i n f l u e n c e of a p a r t i c l e v e l o c i t y o p p o s i t e t o t h a t of t h e d i r e c t i o n of wave t r a v e l a t t h e t o p l i m i t of i t s v e r t i c a l excursion and a smaller forward v e l o c i t y when i t s downward displacement i s a maximum. This would b e t r u e f o r shapes l i k e t h e platform i n r e l a t i v e l y long waves with slender towers and t h e b a s e submerged.
I t i s well known t h a t s h i p s may d r i f t opposite to
t h e d i r e c t i o n of wave propagation but due t o t h e l a r g e w a t e r l i n e s e c t i o n t h e d i r e c t i o n of t h e e f f e c t s would be reversed.
Wave transmission and r e f l e c t i o n a t plane v e r t i c a l b a r r i e r s
Some consequences of t h e l o c a t i o n of t h e mean E u l e r i a n momentum i n r e l a t i o n t o t h e e f f e c t of f i x e d b a r r i e r s on wave t r a i n s may be of oceanographic i n t e r e s t .
If a
v e r t i c a l p l a t e , normal t o t h e wave d i r e c t i o n , be imagined t o extend from t h e sea bed i n i n f i n i t e depth t o t h e bottom of t h e troughs of t h e r e s u l t a n t motion, t h e e n t i r e mean momentum of t h e wave should be t r a n s m i t t e d p a s t t h i s "Eulerian" o b s t a c l e . I t i s w e l l known however from f i r s t order theory (Dean,1945) t h a t such a b a r r i e r
would r e f l e c t n e a r l y a l l t h e wave energy (a f t h e g r i d , and t h e " s t a i r - s t e p "
b o u n d a r y p r o v l d e s a rouqh r e p r e s p n t a t i o n of
the s i i o r e l i i i e .
S u c h a s p l i c e d g r i d ( T i i a c k e r , 1976) p r o v i d e d t h e m o t i v a t i o r l f o r t h c i r r e q u l a r - q r i d finite-difference
techniqucs.
J u s t ds l i n e a r i n t e r p o l a t i o i i
('a11 bc?
s u c c ~ ~ s s f ~ u~sle ldv
264
to calculate derivatives at the "extra" points along the splices, it should also provide a means for calculating derivatives at points on an irregular grid.
. Fig. 3 .
Piecewise uniform spliced grid for the Elbe Estuary (Ramming, 1975).
The fact that the grid points are connected by line segments to form a mosaic of triangular elements (Fig. 1) is reminiscent of similar grids used in finite-element calculations (see, for example, Pinder and Gray, 1977).
This similarity is due to
the fact that the techniques discussed here as well as those of the finite-element method involve linear interpolation over triangular elements. The fundamental distinction is that the finite-element method is based directly upon approximation of the functions, whereas the finite-difference method is based upon approximation of the derivatives.
The practical distinction is that the finite-difference techniques
provide greater computational economy.
The spatial averages (Thacker, 1978a and
1978b) that result from the finite-element method necessitate a matrix inversion at each time step.
In addition to this computationally expensive matrix inversion,
these averages lead to greater storage requirements, to a greater number of arithmetic operations per time step, and to a smaller value for the length of the time step than required by the corresponding finite-difference calculations. Because the computational grid is irregular, only one index is used to specify the grid points rather than two indices corresponding to distances along coordinate axes as for the conventional uniform grids.
Since the grid point index is neither simply
related to the coordinates of the grid point nor to the indices of neighboring points that provide values necessary for evaluating derivatives, this information must be
tabulated for computation. Also, the differentiation coefficients are not simply the inverse of the grid spacing as for uniform grids.
Since they vary from grid
point to grid point, either they must be tabulated or they must be calculated from the tabulated values of the coordinates and the indices of neighboring points each time they are needed. Because the manner in which the grid points are indexed is unimportant, it is a simple matter to alter the grid in order to add additional points, to remove points, or to respecify neighbors.
After editing the grid, it is also a simple matter to
sort and renumber the grid points for computational efficiency. The scheme used here assigns indices to the interior points first, the lowest for interior points with six neighbors, next for those with five, and then for those with seven, and assigns indices to the boundary points last, also according to the number of neighAdditional editing (Thacker, 1977) guarantees that each interior grid
boring points.
point is situated at the geometric center of the polygon formed by the neighboring grid points.
These editing procedures can also be used for finite-element grids
so long as the matrix inversions are calculated by an iterative technique, but if direct inversion techniques are used, finite-element grids should be numbered so that the differences between the indices of neighboring points be as small as possible.
For storm surge calculations the previous time step provides excellent
values for initializing the iterative techniques, so they should be efficient as well as flexible.
APPROXIMATION OF DERIVATIVES
The slope of the spinnaker-shaped surface in Fig. 4 can be approximated by the slope of the planar surface determined by points a, b, and c. curvature the approximation is better.
Of course, for smaller
The planar surface is a linear interpolating
function, and its derivatives provide approximations of the function specifying the curved surface,
-
-afax
f (y -y ) a b c
+
f (y -Y ) b c a
+
fc(Ya-Yb)
A
In the storm surge calculation the function f can represent the x- and y-components of the vertically integrated horizontal velocity,
+
U,
and the surface elevation, H.
Since the dynamical variables are calculated at the grid points which are vertices of triangles, there is no reason for preferring the approximations corresponding to one adjacent triangle over those corresponding to any other.
For this reason, the
266
Y
Fig. 4. The slope of the plane passing through pints a, b, and c approximates the slope of the curved surface. The plane represents the interpolating function with derivatives that approximate the derivatives of the curved surface. Interior
N - 6
N - 5
N = 7
Boundary
N = 3
Fig. 5.
N-4
N = 5
N = 6
The approximations for derivatives at points on the irregular grid are averages of the approximations obtained from the adjacent triangles. For interior points, the approximations are centered, involving only values associated with the N neighboring points and not the value at the p i n t for which the derivative is evaluated. For points on a boundary, the approximations are "one-sided", with the values at the grid point contributing to the evaluation of the derivative.
261
derivatives at a grid point are approximated by averages of the contributions from all adjacent triangles weighted according to their area, Fig. 5.
The resulting N-point
formulas (Thacker, 1977) are equally as simple as the three-point formulas for the slope of the surface in Fig. 4. For example, if there are five points contributing to the approximation, then the formulas are,
In every case the numerators are given by cyclic sums of products of the values of the function at the grid points with the differences of coordinates at adjacent points, and the denominator is twice the area of the polygon formed by the N points.
For
regular polygons, such as the square and the hexagon shown in Fig. 6, the formulas reduce to the familiar expressions,
- f -f af -- a c ax
af
-=-
aY
ax
-X
a
c
x -x
and f -f b d Yb-Yd
=
aY
1 [fb-ff Yb-Yf
2
Yc-Ye
d
Fig. 6. For uniform grids, with points in square or hexagonal arrays such as these, the N-point formulas for approximating derivatives reduce to simple, recognizable expressions.
When the shallow water wave equations are discretized to obtain equations for the values of
y'
and HY, corresponding to the transport and surface elevation for grid
point i and time level n, the partial derivatives are approximated by the appropriate N-point formulas. Only at the boundary (see Fig. 5 ) is the point at which the derivative is approximated also one of the N points contributing to the approximation.
268 GOVERNING EQUATIONS The hydrodynamic equations governing the storm surge,
+
au+ +v at
.-= D ''
- = -
q.;,
at
-
+ gDVH
. - + + -+T - -+B - fkxU
-+
account for the atmospheric forcing through the term, T, and for the bottom friction through
6.
The term involving the Coriolis parameter, f, and unit vector in the
vertical direction, k, account for the earth's rotation which has a relatively small influence on the storm surge. The term involving the gravitational acceleration, g, and the water depth, D, accounts for flow in response to slope in the sea surface. The flow accelerates in response to these forces and the sea surface rises as the flow converges. The wind velocity and pressure gradient fields for the hurricane forcing are taken to be the same as those used by Overland (1975) for Apalachicola Bay, 2rR
-+
w=-
S
r2+R2
+ vp
=
-
AP
r2+R2
- exp (-f)g. R
r2 The velocity field has two components; one is circularly symmetric with maximum value, WmX, at distance, r
= R,
from the storm's center and with inflow angle speci-
fied by the unit vector, Ip, and the other approximates the assymmetry of the storm
+
The value of W , depending upon max the values of the radius, R, and of the pressure drop, LIP, used to specify the storm, associated with its translational velocity,
S.
is determined (see Fig. 7) as in the SPLASH model (Jelesnianski, 1967) used by the National Weather Service for forecasting storm surges. The symmetric part of the wind speed, the inflow angle with maximum of 22O at 3 R and 17O at large r, and the pressure gradient inward along the radial direction vary as indicated in Fig.
8.
The hurricane forcing associated with these fields is given by
where
p
and
p W
are the densities of air and water and where the drag coefficient
has the value used in the SPLASH model, Cd
= 2.4
-3
x 10
at all wind speeds.
Whereas the SPLASH model uses time-history bottom stress, the more conventional quadratic stress is used here,
with the Chezy coefficient, C
H
4
= 62 m /sec.
The mathematical specification is completed by the boundary conditions requiring
269
RADIUS OF MAX WINDS
Fig. 7.
(MILES)
This nomogram (Jelesnianski, 1967) can be used to obtain the value of the maximum hurricane wind velocity from the values o f the radius to maximum winds and the pressure drop. Tabulated values as used by SPLASH were used for computation.
RADIAL DISTANCE F R O M CENTER OF STORM
Fig. 8 .
Variation of hurricane wind speed, inflow angle, and pressure gradient with radial distance from center of storm (Overland, 1975).
270
that there be no flow normal to the shoreline and that the surface elevation along boundaries separating the portion of the sea included in the computation from that which is excluded be that height of water supported by the atmospheric pressure drop. The finite-difference equations, which govern the values of the dynamical variables at points on the irregular grid, have a "leap-frog''time structure with values for the transport vectors and surface elevation corresponding to different time levels separated by ~ / 2 ,where the length of the time step is
T =
2.5 minutes.
Except for points
on the boundary, which must satisfy the imposed boundary conditions, the values of the dynamic variables at the grid points are obtained from the equations
-(Hi 1 n+l-Hij n =
- n+4 . - (?*?i)i
For those points corresponding to the coastline, the momentum equation must be altered to prevenf flow normal to the coastline. The right-hand side, which represents the forcing, must be projected onto the line tangent to the boundary determined by the unit vector
bi =
+ +
+ +
+
(xa-xc)/lxa-xcl, where x
+
and x
are the coordinates of the
point which are neighbors of point i = b lying on the boundary (see Fig. 9 ) .
This ^
^
is done by taking the inner product of the right-hand side with the dyadic, bibi. For those points on the computational boundary not corresponding to a coastline, the atmospheric pressure determines the value of the surface elevation at each time step. The position of the storm at the na
time step and the velocity of the storm are
calculated from specified coordinates for the center of the storm at two different times, which might correspond to the forecast value for the storm to reach a designated point in the vicinity of the bay and the time that the forecast is issued. From the position of the center, the values of the distances to each grid point and
P
HOUR OF DAY
--431 -4
DAY 17
Fig. 4. Sea l e v e l s a t Liverpool f o r 13-14 January and 17 January 1965 i n d i c a t i n g t h e t i m e s of occurrence of surge peaks and a s s o c i a t e d high waters and maximum water l e v e l s . Notation a s i n f i g u r e 2 .
Fig. 5. Heights of high and low w a t e r a t Liverpool r e l a t i v e t o mean sea l e v e l (MSL), for November 1977, i n d i c a t i n g t i m e s of occurrence of surge peaks. MHW = mean high w a t e r , MLW = mean low water, ODN = ordnance datum Newlyn, CD = c h a r t datum.
SURGE PEAK
SURGE PEAK I HOUR
SURGE PEAK AT MILFORD HAVEN BUT NOT AT LIVERPOOL
MHW
Fig. 6. Heights of high and low water a t Liverpool r e l a t i v e t o MSL, f o r January 1976, i n d i c a t i n g times of occurrence of surge peaks. Notation a s i n f i g u r e 5.
SURGE PEAK I AT LW
SURGE PEAK I HOUR BEFORE LW
Fig. 7. Heights of high and low water a t Liverpool r e l a t i v e t o MSL, f o r January 1965, i n d i c a t i n g t i m e s of occurrence of surge peaks. Notation a s i n f i g u r e 5.
N
294
Fig. 8.
.
Depression t r a c k s f o r f i v e r e c e n t l a r g e surges a t Liverpool; p o s i t i o n a t 0000 h r , 0 p o s i t i o n a t 0600 h r i n t e r v a l s .
However t h e t r a c k a s s o c i a t e d with t h e surye of 14 November 1977 i s an exception and follows a s o u t h - e a s t e r l y course between Iceland and Denmark r a t h e r than an e a s t e r l y t o n o r t h - e a s t e r l y course over t h e B r i t i s h I s l e s . The weather c h a r t s of f i g u r e s 9 , 10 and 11 i l l u s t r a t e t h e developing storm p a t t e r n s a s s o c i a t e d with t h e l a r g e surges recorded i n t h e I r i s h Sea on 1 2 November 1977, 14 November 1977 and 2 January 1976.
The secondary depression which brought
s t r o n g westerly-type winds t o bear on t h e I r i s h Sea during 11 and 12 November 1977 was a poorly-defined f e a t u r e ( f i g u r e 9) b u t n e v e r t h e l e s s a powerful surge-producing agent.
I t c o n t r a s t s with t h e l a r g e r and more c l e a r l y - d e f i n e d cyclone which passed
a c r o s s Scotland i n t o t h e North Sea on 2 and 3 January 1976 ( f i g u r e 11) again b r i n g i n g very s t r o n g westerly-type winds t o t h e I r i s h Sea.
The r a t h e r d i f f e r e n t
synoptic c h a r t s o f 13 and 14 November 1977 ( f i g u r e 10) show a f r o n t a l system and wind f i e l d s sweeping over t h e B r i t i s h I s l e s from t h e north-west,
some of t h e
s t r o n g e s t winds a f f e c t i n g t h e I r i s h Sea. Figures 12, 13 and 14 p l o t recorded wind speed and d i r e c t i o n , along with barom e t r i c p r e s s u r e , a t Ronaldsway i n t h e Isle of Man (a c e n t r a l l o c a t i o n i n t h e northern I r i s h Sea) f o r p e r i o d s which include t h e l a r g e I r i s h Sea surges of November 1977, January 1976 and January 1965.
The times of surge peaks a r e
295
Fig. 9.
1200h 11/11/77
l8OOh ll/ll/77
OOOOh 12/11/77
0600h 12/11/77
Weather c h a r t s f o r t h e storm surge of 11 t o 12 November 1977.
indicated. of 1012 mb.
The barometric p r e s s u r e v a r i a t i o n s a r e shown with r e s p e c t t o a mean Wind angle 8 i n degrees i s measured clockwise from t h e south.
I t is
apparent from t h e f i g u r e s t h a t t h e major surges of 12 November 1977, 2 January 1976 and 14 January 1965 were each preceded by f a l l i n g barometric p r e s s u r e and r a p i d l y s t r e n g t h e n i n g winds v e e r i n g from south-west t o west.
These c h a r a c t e r i s t i c s
r e f l e c t t h e i n f l u e n c e of an i n t e n s e depression moving quickly eastwards a c r o s s t h e northern p a r t of t h e B r i t i s h I s l e s ( f i g u r e s 8 , 9 , 11 h e r e , a l s o f i g u r e 1 given by Heaps and Jones ( 1 9 7 5 ) ) .
Manifestly t h e surge of 14 November 1977 was associated
with s t r o n g west north-west winds maintained f o r over twelve hours a s t h e r e s u l t of a n o r t h e r l y depression e n t e r i n g t h e North Sea ( f i g u r e s 8 , 1 0 ) .
The surge of
17 January 1965 can obviously be l i n k e d t o e x c e p t i o n a l l y s t r o n g west south-west winds again maintained f o r h a l f a day o r so: t h e e f f e c t of a l a r g e depression moving eastwards t o t h e n o r t h of t h e B r i t i s h Isles ( f i g u r e 8 here and f i g u r e 2 given by Heaps and Jones ( 1 9 7 5 ) ) . An o v e r a l l examination of f i g u r e s 12, 13 and 14 shows
296
Fig. 10.
1200h 13/11/77
OOOOh 14/11/77
1200h 14/11/77
OOOOh 15/11/77
Weather c h a r t s f o r t h e storm surge of 14 November 1977.
t h a t t h e winds of January 1965 considerably exceeded those of January 1976 and also those of November 1977.
IRISH SEA MODEL
To s i m u l a t e the storm s u r g e s o f November 1977 a two-dimensional numerical model
of t h e I r i s h Sea was formulated on t h e g r i d network shown i n f i g u r e 15.
The grid
has a square mesh of s i d e 7.5 n a u t i c a l miles and i s constructed with r e f e r e n c e t o a c e n t r a l x-directed l i n e along t h e p a r a l l e l of l a t i t u d e 53O20'N and a c e n t r a l y-directed l i n e along t h e meridian of longitude 4O4O'W. t o t h e e a s t and t h e y coordinate t o t h e north.
The x coordinate increases
Surface e l e v a t i o n 5 i s evaluated
a t t h e c e n t r a l p o i n t of each elemental box, c u r r e n t u ( i n t h e x - d i r e c t i o n ) a t the mid-point of each y-directed box s i d e , and c u r r e n t v ( i n t h e y-direction) mid-point of each x-directed box s i d e .
a t the
Averaging u and v a c r o s s an elemental box
297
Fig. 11.
0600h 2/1/76
1800h 2/1/76
0600h 3/1/76
1800h 3/1/76
Weather c h a r t s f o r t h e storm surge of 2 t o 3 January 1976.
y i e l d s t h e c u r r e n t components a t i t s c e n t r e .
The model has open boundaries across
t h e North Channel i n t h e n o r t h and a c r o s s S t George's Channel i n t h e south. The hydrodynamic equations of t h e model a r e :
g i v i n g t h e v a r i a t i o n s of 5, u, v with r e s p e c t t o time t i n terms of t h e C o r i o l i s e f f e c t ( c o e f f i c i e n t y ) , s e a s u r f a c e g r a d i e n t s ( f a c t o r e d by g t h e a c c e l e r a t i o n of t h e E a r t h ' s g r a v i t y ) , q u a d r a t i c bottom f r i c t i o n ( c o e f f i c i e n t k ) , components wind. stress on t h e s e a s u r f a c e (F
sx' Fs,)' over t h e sea s u r f a c e . Here: y = 1.1667 x -3 A l s o p = 1025 kg m , t h e water d e n s i t y .
Of
and g r a d i e n t s of atmospheric p r e s s u r e pa -2 and k = 0.0026. s-l, g = 9.81 m s
DAY OF MONTH (NOVEMBER 1977) +
1
8
1
9
1 1 0
I
II
1
1
2
1
1
3
1
1
4
1
1
5
1
1
6
1
1
7
I
-
SPEED
WIND SPEED
:I
80
04
t
Fig. 12. Recorded wind speed and direction, and barometric pressure, at Ronaldsway (Isle of Man): 8-17 November 1977.
DAY OF MONTH (DECEMBER 1975- JANUARY 1976)-
1
3
0
1
3
1
I
I
1
2
1
3
1
4
1
5
1
6
1
7
I
s
/ e’ - 240
F
WIND SPEED
I I
.2 0 0 WIND
160DIRECTION
I
I
I
-120
- 80 -40
0
?4 mb
1030 PRESSURE
1020 1010 1000 990 980
Fig. 13. Recorded wind speed and d i r e c t i o n , and barometric p r e s s u r e , a t Ronaldsway: 30 December 1975 - 8 January 1976.
w
0 0
OF
DAY
~
m/r
9
~
1
O
/
I
I
/
MONTH (JANUARY 19651
I
2
/
-t -x -
1
3
~
1
4
~
1
5
SURGE
pEil
SPEED DIRECTION
WINO SPEED
:1
-240 WIND DIRECTION
- 200 - I60 -120
-80 - 40 0
Y
Fig. 14.
Recorded wind speed and direction, and barometric pressure, at Ronaldsway: 9-18 January 1965.
301 In the equations, h denotes the undisturbed depth of water, prescribed realistically over the grid at the mid-points of the box sides. The total water depth at any time is h
+
5, determined at the mid-point of a box side with 5 an average
of the values taken from the centres of the adjacent boxes. An explicit finite difference scheme was used to develop solutions of the dynamica1 equations, yielding elevation 5 and depth-mean currents u , v through time over the Irish Sea.
The scheme is basically similar to that used by Heaps and
Jones (1975) with the frictional term and the total depth h paper by Flather and Heaps (1975).
+
5 treated as in the
In generating solutions through time, starting
from a state of rest with 5 = u = v = 0 everywhere, the 5, u, v are incremented from values at t to values at t
+
At over successive time intervals At.
In this
procedure, elevation 5 is prescribed at the open boundaries as time advances, also and atmospheric pressure gradients 6pa/6x, 6pa/6y wind stress components F F sx' sy at the u and v points of the model. Zero normal flow is postulated at the land boundaries. Having regard to numerical stability, it was found convenient to take At
=
120
S.
TIDAL COMPUTATIONS Tides were generated in the model for the whole of November 1977 in response to specified open boundary tides consisting of the M2 and S2 constituents - the principal harmonic components. Amplitudes and phases of the tidal input, applied at the elevation points adjacent to the northern and southern open boundaries, are given in table 1.
Basically this input comes from cotidal charts and from
a
numerical tidal model of the sea areas on the west coast of the British Isles. The tides generated in the model were analysed to yield the M
2
and S2 components
at Port Patrick, Belfast, Douglas, Workington, Heysham, Liverpool, Hilbre Island, Holyhead, Dublin and Fishguard (see figure 15 for these locations).
In table 2
the results of this analysis are compared with corresponding results derived from the analysis of observations. There is satisfactory agreement, with discrepancies in tidal amplitude for the most part being less than 0.13 m and discrepancies in tidal phase not exceeding 6O for M2 and 16O for
S2.
A similar comparison is also
made in table 2 for tidal flows through the North Channel across section C6C7 in figure 15. The agreement between model and observation is again quite good.
Here,
the observational results come from measurements of voltage across the North Channe by Prandle and Harrison (1975) with conversion from voltage to flow using a calibration factor due to Hughes (1969). The model tide, limited to M
2
and S by our restricted knowledge of the open 2
boundary tides, obviously differs from the predicted tide based on a comprehensive set of harmonic constants.
Some measure of this difference can be gained from the
tidal curves of figures 19 and 20 for Workington and Liverpool: deviations of
302
SCOTLAND
Fig. 15.
-------
Irish Sea model: land boundary; open sea boundary; flow section; 0 tide gauge and 0 corresponding elevation point of the model. Key: PP = Port Patrick, B = Belfast, D = Douglas, R = Ronaldsway, W = Workington, HE = Heysham, L = Liverpool, H = Hilbre Island, HO = Holyhead, DU = Dublin, F = Fishguard, BB = Baginbun.
303
Amplitude H (metres) and phase g (degrees) of t h e i n p u t t i d e s a t t h e e l e v a t i o n p o i n t s a d j a c e n t t o the northern and southern open boundaries
Northern boundary East
West M2
s2
H 9
0.71 32 1
0.76 329
0.84 336
0.90 343
0.94 347
H
0.16 7
0.18 13
0.21 19
0.25 23
0.27 27
9
Southern boundary East
West
M2 s2
H 9
0.92 168
0.94 178
0.98 186
1.06 194
1.14 201
1.22
H
0.39 216
0.40 221
0.41 228
0.43 233
0.44 237
0.46 241
g
208
roughly 10 to 17 p e r c e n t i n range a r e e v i d e n t between t h e model t i d e and t h e t i d e from a f u l l p r e d i c t i o n .
A s it t u r n s o u t , t h e a c c u r a t e reproduction of t i d e i n t h e
model i s n o t e s s e n t i a l f o r our purposes with t h e emphasis on storm surge computation.
Thus, with meteorological as well as t i d a l f o r c i n g included i n t h e model,
t h e t o t a l motion of t i d e and surge i s computed from which t h e regime of t i d e alone
i s subtracted.
The r e s u l t g i v e s t h e computed surge - conditioned by i n t e r a c t i o n
with t h e t i d e .
I t is assumed t h a t an approximate t i d e i s s u f f i c i e n t f o r t h e s a t i s -
f a c t o r y determination of t h i s i n t e r a c t i o n .
STORM SURGE COMPUTATIONS
The model w a s run t o simulate t h e regime of t i d e and surge i n t h e I r i s h Sea f o r the p e r i o d 00.00 h 8 November - 23.00 h 17 November 1977.
w a s added t o t i d a l e l e v a t i o n along t h e open boundaries.
In t h i s , surge e l e v a t i o n Simultaneously, f i e l d s
of wind stress and h o r i z o n t a l atmospheric p r e s s u r e g r a d i e n t were a p p l i e d t o the sea s u r f a c e . A t each e l e v a t i o n p o i n t a d j a c e n t t o the northern open boundary a surge e l e v a t i o n
w a s p r e s c r i b e d ( a t hourly i n t e r v a l s ) equal t o t h a t observed a t P o r t P a t r i c k .
At
each e l e v a t i o n p o i n t a d j a c e n t t o t h e southern open boundary a surge e l e v a t i o n w a s p r e s c r i b e d ( a l s o a t hourly i n t e r v a l s ) from a l i n e a r i n t e r p o l a t i o n , with r e s p e c t t o d i s t a n c e , between t h e observed surge a t Fishguard and t h a t a t Baginbun.
The
s u r g e s observed a t P o r t P a t r i c k , Fishguard and Baginbun ( l o c a t i o n s shown i n f i g u r e 15) a r e p l o t t e d through t i m e i n f i g u r e 16.
The changing f i e l d s of wind stress and atmospheric p r e s s u r e g r a d i e n t were evaluated a t three-hourly i n t e r v a l s over s i x r e c t a n g u l a r sub-areas of t h e I r i s h
304 TABLE 2
Amplitude (in metres) and phase (in degrees) of the M and S surface tides at 2 various Irish Sea ports, comparing results from the numerical model with those derived from ob ervation. A similar comparison is made for the M and S2 tidal 5 3 2 flows (in 10 m / s units) through the North Channel, section C C 6 7'
Amplitude Model M2
Port Patrick Belfast Douglas Workington Heysham Liverpool Hilbre Is. Holyhead Dublin Fishguard '6'7
S2
Port Patrick Be1fast Douglas Workington Heysham Liverpool Hilbre Is. Holyhead Dublin Fishguard '6'7
Phase
Observed
Model
Observed
1.36 1.14 2.43 2.68 3.08 2.95 2.95 1.55 1.42 1.32 28.9
1.34 1.20 2.31 2.72 3.15 3.08 2.92 1.79 1.34 1.36 24.0
329 32 1 326 331 325 3 16 316 287 321 214 42
333 315 327 334 326 322 3 18 292 326 2 08 43
0.35 0.27 0.70 0.77 0.91 0.87 0.87 0.48 0.38 0.48 8.9
0.38 0.29 0.72 0.90 1.01 1.00 0.95 0.59 0.40 0.53 8.2
5 357 359 6 0 350 350 312 346 246 68
16 352 7 14 8 5 0 328 357 247 79
Sea region following a method used by Heaps and Jones ( 1 9 7 5 ) . 6pa/6x, 6pa/6y and geostrophic wind (RG, e
)
Pressure gradients
were evaluated uniformly over each
G rectangle in terms of differences of observed barometric pressures taken over distances of approximately 60 nautical miles.
Surface wind (R, 8 ) was then deduced
from the empirical relations: R = 0 . 5 6G ~
+
0.24,
e
=
eG
- 22.
Here: RG, R denote wind speeds in m/s and ElG,
(4) €Iwind
angles in degrees measured
clockwise from the south. Resultant wind stress, F dynes/cm2 in the direction 8 , was subsequently evaluated using the square law: F = 12.5cR2 with the drag coefficient c given by
(5)
103C = 0.554, R < 4.917 = -0.12 = 2.513,
+
0.137R,
4.917 < R < 19.221
R > 19.221
(6)
Then, components of wind stress were determined from: = F cosf3. (7) FSX = F sine, F SY Subjected to open-boundary elevations of tide and surge, wind stresses and atmos-
pheric pressure gradients, the model yielded the combined motion of tide and surge in the Irish Sea through the period 8-17 November.
The tidal motion alone, deter-
mined separately by the model as prescribed in the preceding section, was subtracted from the combined motion to yield the storm surge.
Surge levels (model) are com-
pared with surge levels (observation) for a number of Irish Sea ports in figures 17 and 18. The locations of all these ports are indicated in figure 15.
On the obser-
vational side, the surge level at a place is obtained by taking the difference between the observed and the tidally-predicted water levels there, hour by hour. There is thus a correct correspondence between this procedure and the modelling one for the computation of surges. An examination of the residuals in figures 17 and 18 shows that the large semidiurnal-type fluctuations observed during 8-11 November at Workington, Heysham and Hilbre Island are quite nicely reproduced by the model.
The fluctuations are some-
what overestimated at Douglas, and at Liverpool their phasing is in error due, no doubt, to the inability o f the model to reproduce the influence of the Mersey Estuary. At Holyhead and Dublin the fluctuations are smaller and reasonably well reproduced. They are present at Belfast but, again as at Liverpool, their phasing comes out incorrect due presumably in this case to the omitted influence of Belfast Lough. The main surge peak which occurred near midnight on 11 November is predicted quite well by the model at Workington, Heysham and Hilbre Island.
A
magnified
diagram of the Workington residuals near the maximum is shown in figure 19.
Note
from this diagram that the peak occurred on the rising tide, a feature common to all the other ports apart from Belfast and Liverpool. The Liverpool residuals near the surge maximum are shown in figure 20.
It can be seen from this figure that,
while the model surge maximum occurs on the rising tide, the observed maximum was higher and occurred five to six hours later. In effect, there is a significant contribution missing from our Liverpool surge prediction on 12 November.
The
source of this error is suggested by figure 21 showing wind speeds recorded at Liverpool on 11 and 12 November. A rapid fall followed by a rapid rise evident in the recorded speed between 18.00 h and 23.00 h on the 11th is clearly not represented in the wind field used for the model computations. Other anemometer observations around the coastline of Liverpool Bay indicate that this fall and rise in speed was fairly local to Liverpool
-
at least in its intensity. We propose, there-
fore, that the surge contribution missing from our Liverpool prediction was generated by local wind variations which could not be accounted for by the barometric pressure differences on which the model winds were based.
Figure 2 2 indicates that observed
surge peaks at Liverpool on the 14th not reproduced by the model might also be
306
DAY OF MONTH (NOVEMBER 1977)I
a
'
10
'
11
'
12
'
13
'
14
'
16
i 16
17
m.
0-8 PORT PATRICK
0.4 0.0
0.4
0.0
Fig. 1 6 . Observed r e s i d u a l e l e v a t i o n s a t P o r t P a t r i c k , Fishguard and Baginbun (derived f o r t h e f i r s t two of t h e s e p o r t s on t h e b a s i s of t i d a l p r e d i c t i o n s and f o r t h e t h i r d on t h e b a s i s of t h e X o - f i l t e r ) .
a t t r i b u t e d t o l o c a l v a r i a t i o n s i n wind speed n o t accounted f o r by t h e larger-scale model winds.
A f i n e r r e s o l u t i o n o f t h e wind s t r u c t u r e over t h e I r i s h Sea i s
c l e a r l y r e q u i r e d f o r i n p u t t o t h e model t o improve i t s performance a t Liverpool
-
and q u i t e p o s s i b l y a t o t h e r p l a c e s . Returning t o c o n s i d e r a t i o n of f i g u r e s 17 and 18, it should be pointed o u t t h a t t h e observed surge p r o f i l e s a t Heysham and H i l b r e I s l a n d terminated prematurely a t t h e end of 11 November due t o t h e f a i l u r e of t h e t i d e gauges a t those l o c a t i o n s under storm c o n d i t i o n s .
Moreover, s h i f t s i n datum i n t h e Workington and Holyhead
t i d e gauges occurred on t h e 11th due t o s l i p p a g e of t h e i r recording mechanisms when high water l e v e l s were a t t a i n e d .
I n t h e surge p r o f i l e s shown f o r Workington and
Holyhead, adjustments i n datum have been made i n an attempt t o minimise t h e s e observational errors. I n a r e p e a t run with t h e model f o r the p e r i o d atmospheric p r e s s u r e g r a d i e n t s were set t o zero.
8-17 November, wind stresses and Motion i n t h e I r i s h Sea w a s thereby
obtained due s o l e l y t o t i d e and surge on t h e open boundaries. t i d e then gave t h e externally-generated
surge i n t h e I r i s h Sea.
S u b t r a c t i n g t h e model Associated residual
307
I
m. 0.8
€
DAY OF MONTH 5
10
PORT PATRICK
11
(NOVEMBER 1977)12
'
13
'
14
'
16
'
18
'
17
I
A
0.4
0.0 08 0.4
0.0
0.8 0.4 0.0
1.2 0.8
WORK INGTON
0.4
0.0 1.6
1-2 0.8 04
0.0
Fig. 17.
Residual elevations at various Irish Sea ports: the numerical model, from observation.
------
from
308
DAY OF MONTH (NOVEMBER 1 9 7 7 1 4 I
m.
S
'
I0
11
12
13
IS
14
16
..
1.2
..
0.8 0.4
0.0
Fig. 18. Residual elevations at various Irish Sea ports: from the numerical model, - - - - - from observation.
- -- - - -
17
I
309
I
1.4 -
1.2
DAY
OF MONTH II
(NOVEMBER I
1977) 4 12
SURGE
-
-0.21
Fig. 19.
X
Tide and surge elevations at Workington,
from observations
----__-
and a full harmonic tidal prediction based on observations, from the model. Hourly values are plotted. Tidal heights are given to mean sea level datum.
310
t
DAY OF MONTH ( NOVEMBER 1977)4 I 12 II TIDE LIVERPOOL
I
SURGE 1.2I -0-
"
0.0
I I1 II 1I II I ' 1I 1I II 1
I l l
I l l I 1 1 I I I I I I
12
0
I 1
I
l
l
1
I I I I II 1
I
I I I I I I I I I 1
12
21
>
HOURS
Fig. 20. Tide and surge elevations at Liverpool, from observations and a full harmOnic tidal prediction based on observations, - - - - - - - - - f r o m the model. Hourly values are plotted. Tidal heights are given to mean sea level datum.
FROM ANEMOGRAPH
0
1 12
: 13
: 14
: 15
: 16
: 17
: 18
: 19
:
20
II NOVEMBER 1977
:
PI
:
22
:
23
:
:
0
I
I
:
2
:
3
:
:
5
4
: 6
:
7
" 8
"
: >HOURS 9
10
II
12
12 NOVEMBER 1977
Fig. 21. Wind speeds recorded by the anemometer at Seaforth, Liverpool, 11-12 November 1977. Limits of Wind speeds used in the the anemograph record are shown together with hourly means ( - ~ - o - ~ - ) . Surae elevations at Liverpool are shown, model computations, for Liverpool Bay, are denoted by @ + + -) and as determined from the model (--x-- -x--- x - - - ) . as observed ( - +
- - -
.
7c-k
J /
\ \
t
06
00 4s WINO 40 SPEED
m
,-1
35 30
25 20 15 ROM ANEMOGRAPH
10
5
0
HOURS
I
14 NOVEMBER 1977
Fig. 22. Wind speeds (and t h e i r hourly means) recorded by t h e anemometer a t S e a f o r t h , Liverpool, 14 November 1977. wind speeds used i n t h e model computations, f o r Liverpool Bay, are a l s o shown. The observed and computed s u r g e s a t Liverpool are p l o t t e d . Notation as i n f i g u r e 21.
313
DAY OF MONTH (NOVEMBER 1 9 7 7 1 4 I
9
'
10
'
11
12
'
IS
'
14
'
15
'
18
17
'
Fig. 23. Residual elevations from the model resolved into a part due to disturbances entering across the open boundaries ( ) and a part due to wind and atmospheric pressure gradients over the model area ( - - - - - - -
).
314
DAY OF MONTH (NOVEMBER 1977)I
9
10
11
12
13
14
16
18
17
I
I
0-4
+
HOLYHEAD
n
0.0
0.8 04
00 08
04
0.0
Fig. 24. Residual elevations from the model resolved into a part due to disturbances entering across the open boundaries ( ) and a part due to wind and atmospheric pressure gradients over the model area ( - - - - - - - ).
315
t
tt
DAY 8
'
OF MONTH (NOVEMBER 1977)10
11
'
12
IS
14
15
18
17
I
HILBRE ISLAND
Fig. 25. Computed tides (M2 + S ) and residuals (the smaller variations shown) 2 for (a) Port Patrick, (b) Fishguard and (c) Hilbre Island.
316
DAY OF MONTH (NOVEMBER 1977)I
s
'
10
11
12
13
14
16
I6
I?
I
8
0
I
5 3 Fig. 26. Residual flows (in 10 m /s units) across sections C1C2, C C7, C4C5 and A B . from the numerical model. Positive flow directions are skown in figur2
4;.
elevations are plotted through time in figures 23 and 24.
A l s o plotted are the
residual elevations due to the direct action of wind and atmospheric pressure over the Irish Sea (elevations obtained by subtracting the external surge from the total surge determined originally).
Thus, figures 23 and 24 show the total surge of
figures 17 and 18 resolved into an external part coming from the open boundaries and an internal part coming from the effects of wind and atmospheric pressure (essentially wind) over the Irish Sea.
It is evident from these figures that the
open-boundary influence generally predominates.
Clearly, however, the wind effect
can be equally important at Workington, Heysham, Liverpool and Hilbre Island along the north-eastern coast.
of special interest is the fact that at Workington and
Heysham on 11 November the two surge components were of similar magnitude and were directly superimposed to produce the high surge peaks observed.
There was a some-
what less effective superposition at Liverpool and Hilbre Island on the same day.
317 Evidently the large surges at Heysham, Liverpool and Hilbre Island on 14 November were mainly generated by winds over the interior of the Irish Sea. Figures 23 and 24 indicate that the large semidiurnal-type surge fluctuations during 8-11 November originated mainly from the open boundaries. Such fluctuations are also evident at Heysham and Workington as the result of meteorological forcing over the Irish Sea.
This suggests that the Irish Sea basin has a natural mode of
oscillation of near-semidiurnal period which may be excited by external surges on the open boundaries and, to a lesser extent, by wind stress and atmospheric pressure acting on the surface of the basin.
The magnification of the tides in the Irish
Sea may well depend on the existence of this mode which would seem to have a maximum amplitude in the neighbourhood of Heysham. Figure 25 compares the tidal and total surge profiles from the model at Port Patrick, Fishguard and Hilbre Island. The semidiurnal fluctuations discussed above are shown to occur with their peaks consistently on the rising tide, which suggests that they are primarily the product of surge-tide interaction on the open boundaries which propagates (with the tide) into the interior of the Irish Sea region.
There
may be further interaction within the region itself but, m r e likely, the main internal modifications come from a magnification due to the existence of a natural basin-mode of approximately semidiurnal period.
The dynamics of surge-tide inter-
action in the Irish Sea requires further detailed study. Surge flows across sections C C2, C6C7, C4C5 and A B of the Irish Sea (figure 4 4 151, as derived from the model, are plotted through time in figure 26. These plots complement the results for surface elevation given in figures 17 and 18. The C1C2 and C6C7 flows show an average transport from south to north through the Irish Sea, 5 3 -1 This must be largely due over the period 8-17 November, of around 8 x 10 m s
.
to a southerly wind component between the 8th and the 11th (figure 12) but subsequently, with west to north-west winds, due to a generally downward gradient of residual sea-surface elevation from south to north between the opposite open
ends of the Irish Sea (compare the surge elevation at Port Patrick with that at Fishguard in figures 17 and 18). When this gradient is small on the 12th and on the 14th, the flow is also small. Comparatively little of the sustained south to north transport appears to pass through the eastern part of the Irish Sea across C4C5 and A4B4. Main features of the transports shown in figure 26 are the semidiurnal-type fluctuations representing, particularly during 8-11 November, a succession of flow pulses directed alternately in and out of the northern Irish Sea.
These pulses may be associated with the similar fluctuations of surface
level already discussed.
In an inward pulse, water passes northwards across C C 1 2 and (simultaneously) southwards across C6c7, turning eastwards across C C 4 5 and A4B4 into the eastern region of the Irish Sea. In the following outward pulse the flow directions are reversed.
Fluctuations in flow of approximately quarter-
diurnal frequency are strongly evident across C4C5.
The flows across A4B4 are
smaller and also exhibit these higher-frequency oscillations.
318 CONCLUDING REMARKS
1.
Recent l a r g e storm surges i n t h e I r i s h Sea (during November 1977, January
1976 and January 1965) may be a s s o c i a t e d with t h e type of weather c o n d i t i o n s i d e n t i f i e d by Lennon (1963) as being r e l e v a n t t o t h e generation of major surges on t h e
w e s t c o a s t of t h e B r i t i s h Isles.
An exception was t h e surge of 14 November 1977,
caused by a depression which followed a t r a c k between Iceland and Denmark r a t h e r than one which passed from west t o e a s t a c r o s s t h e B r i t i s h Isles.
2.
An examination of t i d e , surge and t o t a l water l e v e l a t Liverpool during
t h e r e c e n t surge events has emphasised t h e p o i n t t h a t t i d a l c o n d i t i o n s p r e v a i l i n g a t t h e t i m e of a surge may be j u s t a s important as surge h e i g h t i t s e l f i n d e t e r mining an abnormally high water l e v e l .
Thus, a moderately l a r g e surge on a very
high t i d e might raise sea l e v e l t o a g r e a t e r e x t e n t than a major surge on a somewhat lower t i d e .
3.
A two-dimensional numerical model of t h e I r i s h Sea was a b l e t o reproduce
t h e main f e a t u r e s of t h e surges observed a t a number of I r i s h Sea p o r t s d u r i n g the period 8-17 November 1977.
External surges e n t e r i n g t h e I r i s h Sea through t h e
North Channel and S t George's Channel had a s u b s t a n t i a l e f f e c t on t h e i n t e r i o r surge l e v e l s .
Meteorological f o r c e s a c t i n g over t h e I r i s h Sea i t s e l f were respon-
s i b l e f o r important surge c o n t r i b u t i o n s a t p o r t s such a s Workington, Heysham and Liverpool i n t h e north-eastern
4.
region.
Local v a r i a t i o n s i n wind s t r e n g t h appear t o be a b l e t o generate s i g n i f i c a n t
surges a t Liverpool n o t accounted f o r by t h e model with s u r f a c e winds assessed on t h e b a s i s of barometric p r e s s u r e d i f f e r e n c e s taken over d i s t a n c e s of about 60 nautical m i l e s .
Presumably, t h e r e f o r e , t h e model's performance could be u s e f u l l y
improved by running it with a more d e t a i l e d wind s t r u c t u r e over t h e sea surface.
5.
Large semidiurnal-type f l u c t u a t i o n s were a f e a t u r e of t h e surges i n t h e
I r i s h Sea during t h e p e r i o d 8-11 November 1977.
The model reproduced them q u i t e
well and i n d i c a t e d t h a t they o r i g i n a t e d mainly from v a r i a t i o n s of surge l e v e l on t h e open boundaries, p o s s i b l y e x c i t i n g a n a t u r a l mode of o s c i l l a t i o n of t h e I r i s h Sea b a s i n of near-semidiurnal period.
Semidiurnal-type f l u c t u a t i o n s of surge level
on t h e open boundaries, most l i k e l y a r i s i n g from surge-tide i n t e r a c t i o n , were i n f l u e n t i a l i n producing t h e i n t e r n a l f l u c t u a t i o n s . 6.
A model f o r f o r e c a s t i n g storm surges i n t h e I r i s h Sea needs t o be l a r g e r
i n a r e a than t h e r e s e a r c h model of t h e p r e s e n t paper.
For f o r e c a s t i n g purposes
a model i s required which does n o t depend q u i t e so c r i t i c a l l y as t h e p r e s e n t one on open-boundary surge c o n d i t i o n s .
A new model s a t i s f y i n g t h i s requirement,
covering a l l t h e sea a r e a s on t h e w e s t c o a s t o f t h e B r i t i s h I s l e s , i s under development (Owen and Heaps, 1978).
319 ACKNOWLEDGEMENTS The authors are grateful to a number of colleagues at I.O.S. Bidston for advice and assistance in this study. Members of the Tidal Computation Section determined most of the observed residual elevations shown and those for Baginbun came from work by Dr D.T. Pugh. Thanks are due to M r R.A. Smith for preparing the diagrams and to Miss Barker and Mrs Young for typing the manuscript. The work described in this paper was funded by a Consortium consisting of the Natural Environment Research Council, the Ministry of Agriculture, Fisheries and Food, and the Departments of Industry and Energy.
REFERENCES
Flather, R.A. and Heaps, N.S., 1975. Tidal computations for Morecambe Bay. Geophys. J. R. astr. SOC., 42: 489-517. Fong, S.W. and Heaps, N.S., 1978. Note on quarter-wave tidal resonance in the Bristol Channel. Institute of Oceanographic Sciences Report No. 63. Heaps, N.S., 1965. Storm surges on a continental shelf. Phil. Trans. R. SOC., A,257:
351-383.
Heaps, N.S. and Jones, J.E., 1975. Storm surge computations for the Irish Sea using a three-dimensional numerical model. Mbm. SOC. r. sci. Liege, ser. 6, 7: 289-333.
Hughes, P., 1969. Submarine cable measurements of tidal currents in the Irish Sea. Limnol. Oceanogr., 14: 269-278. Lennon, G.W., 1963. The identification of weather conditions associated with the generation of major storm surges on the west coast of the British Isles. Q. J1. R. met. SOC., 89: 381-394. Owen, A. and Heaps, N.S., 1978. Some recent model results for tidal barrages in the Bristol Channel. Proceedings of the Colston Research Symposium 1978, University of Bristol (in press). Prandle, D. and Harrison, A.J., 1975. Recordings of potential difference across the Port Patrick-Donaghadee submarine cable. Institute of Oceanographic Sciences Report No. 21.
This Page Intentionally Left Blank
321
RESULTS OF A 36-HOUR STORM SURGE PREDICTION OF THE NORTH SEA FOR 3 JANUARY 1976 ON THE BASIS OF NUMERICAL MODELS 1)
G.FISCHER Meteorologisches Institut der Universitat Hamburg
ABSTRACT
Within the "Sonderforschungsbereich 94" of the University of Hamburg and in collaboration with the "Deutsches Hydrographisches Institut" and "Deutscher Wetterdienst", a group has been established a few years ago with the aim to explore the feasibility of forecasting North-Sea storm surges by integrating numerically a combined atmospheric-oceanographic physical model. A first step into this direction is the simulation of the severe storm and the resulting water levels occuring on 3 January 1976. For this purpose, the atmospheric model was run with a resolution of 8 levels in the vertical and a horizontal grid spacing of 1.4O in latitude and 2.8O in longitude on the northern hemisphere. The initial conditions are based upon observations of 2 January 1976, 12h GMT, i.e. about 24 hours before the storm reached its greatest intensity in the southern parts of the North-Sea. The surface geostrophic wind predicted by the atmospheric model was converted into stress values through a bulk formula which then entered the North-Sea model to yield the desired water elevations and currents in a 22 km grid. Besides of taking predicted winds, also the observed values stemming from a careful re-analysis of the storm situation were fed into the North-Sea model to give a "perfect forecast". The water levels obtained in this way were then compared with gauge measurements at a number of coastal stations. Though the meteorological model simulated quite well the track and intensification of the storm cyclone the evolving pressure gradient, i.e. the geostrophic wind at the surface, was on the whole weaker than observed. Therefore, a reasonable correspondence with measured water elevations could only be reached by correcting the predicted geostrophic wind with a factor of 1.55. Then the results computed by the North-Sea model became about as good as those on the basis of observed geostrophic winds and known before they would have been a very valuable information about the surge to be expected. It is questionable, however, whether the factor 1.55, introduced a posteriori, is valid in general. Though one knows from experience that numerical weather predicitions tend to underestimate cyclone development, thus justifying a correction to stronger winds, the value will certainly change from case to case. To clarify this point too, further experiments of this kind are planned.
1) The full article is to appear in "Deutsche Hydrographische Zeitschrift" Heft 1 , 1979
This Page Intentionally Left Blank
323
EXTRATXOPICAL STORM SURGES IN THE CHESAPEAKE BAY DONG-PING WANG Chesapeake Bay Institute, The Johns Hopkins University, Baltimore, MD (U.S.A.)
ABSTRACT
Two major extratropical storm (cyclone) surges in the Chesapeake Bay, in 19741975 are examined. The subtidal sea level was the dominant surge component, and it was induced by the local wind set-up and the nonlocal coupling with coastal sea level. The study suggests that the observational study is essential to the improvement of storm surge forecast. INTRODUCTION Extratropical storms (cyclones) over the U.S. Atlantic coast can cause severe damage.
For example, the coastal storm of early March 1962 caused damage over
$200 million.
While storms causing this much damage are rare, storms of lesser
damage potential do occur several times each winter. Accurate forecasts of flooding and beach erosion caused by these storms are important. There are basically two different approachs to storm surge forecast. The empirical method relates the storm surge to meteorological data from a regression analysis.
The theoretical method determines the storm surge from numerical inte-
gration of the equations of motion and continuity, with appropriate boundary conditions. In the empirical method, physical reasoning is essential in selecting the proper predictors. The theoretical method has less uncertainty in selecting meteorological forcing. However, the numerical model is designed for limited area forecast, and therefore, the choice of model domain and boundary conditions can be critical. A better understanding of the nature of storm surge is thus vital to the improvement of forecast skill. With the advancing of computer technology, the three-dimensional model for semi-enclosed sea, lake and estuary, has been developed (Heaps and Jones (1975), Leenderste et al. (19731, Simons (1973)).
In particular, Heaps has applied the
numerical model to operational surge forecast in the North Sea.
In contrast,
there have been few studies on the storm surge from direct observations. Lack of solid observational evidence, makes it difficult to evaluate model performance.
324
Fig. 1. Map of t h e Chesapeake Bay and its t r i b u t a r i e s ( s e a l e v e l and meteorological s t a t i o n s a r e marked).
325
Recently, Wang (1978a) has examined the subtidal sea level in Chesapeake Bay (Fig. 1) and its relations to atmospheric forcing, from a year-long record. His results indicated that the Bay water response depends on the time scale of atmospheric forcing.
At time scales longer than 7 days, sea levels in the Bay were
driven nonlocally by coastal sea level.
Between 4 and 7 days, both coastal sea
level and local forcing (particularly,lateral wind) were important. At shorter time scales (1 to 3 days), the Bay water response was local, driven by the longitudinal wind.
Wang (1978a) also constructed a response model (empirical method)
which accounts for over 90% of the total subtidal variance. The success in explaining the observed sea level suggests that subtidal sea level is closely related to large-scale atmospheric forcing.
In contrast, super-
tidal sea level was strongly affected by inhomogeneous topography, shoreline and small-scale atmospheric disturbances.
It would be interesting to know if sub-
tidal sea level is the major component of storm surge.
In other words, can the
storm surge be adequately determined from subtidal sea level alone, which is relatively well-understood? We will examine the two major storm surge events in the period of our subtidal sea level study (July 1974 to June 1975). We will describe the atmospheric forcing (extratropical cyclone), the Bay water response, and the relation between subtidal sea level and storm surge.
STORM SURGE A.
Event I (December 1 to 4, 1974)
On December 1, 1974, a low pressure disturbance (cyclone) was centered around 35'N,85OW
(Fig. 2a).
Winds were southwestward along the Mid-Atlantic coast (Cape
Cod to Cape Hatteras), which generated an onshore Ekman transport. Consequently, sea levels increased over the entire Bight.
In particular, the sea level rise
was about 70 cm at the mouth of Chesapeake Bay (Kiptopeake B.) (Fig. 3).
Assoc-
iated with coastal sea level change, sea levels also increased throughout the Bay. The cyclone propagated to the northeast, and its center passed over the Bay area on 0600 December 2 (Fig. 2b), which resulted in a local northward wind (Fig. 3).
The northward wind set-up w a s quite pronounced; this explains the high
sea level at the Bay head (Havre de Grace). The cyclone continued moving northeastward, and it was centered around Nova Scotia on December 3 (Fig. 212).
The intensity of the cyclone also had signifi-
cantly increased; the central pressure on December 3 was 982 m b , compared to 1004 m b on December 1. Winds were northeastward along the Mid-Atlantic coast, which generated an offshore Ekman transport. Consequently, sea levels decreased
326
Fig. 2 .
Surface weather (atmospheric pressure) map on: (a) 1200 December 1, (b) K3XJ December 2 , and (c) 1200 December 3, 1974.
327
9AnA
Kiptopeoke B
.
I
v
I dyne/cm2
D e c e m b e r , 1974
Fig. 3.
The o r i g i n a l ( s o l i d l i n e s ) and lowpass (dashed l i n e s ) s e a l e v e l s , and t h e lowpass windstress a t P a t u e n t . Kiptopeahe
5
I
December.
Fig. 4.
1974
The highpass s e a l e v e l s .
B
over the entire Bight.
The additional sea level drop at Havre de Grace was due
to the local wind set-down (Fig. 3 ) . The storm surge was dominated by subtidal sea level.
In fact, the response
model (Wang, 1978a) which was developed for subtidal sea level, gives a satisfactory account of the surge event.
The Bay and coastal sea levels responded to
the E-W windstress at time scales of 4 to 7 days; the rise/fall of sea level was associated with the westward/eastward windstress.
In addition, the N-S windstress
drove.loca1set-up/down at time scales of 1 to 3 days. The supertidal component was small. Fig. 4 shows the highpass records (difference between the original and subtidal sea levels):
the semidiurnal tide was
dominant, and the diurnal tide was also clearly reflected by the "diurnal inequalities." There were indications of storm influence in the upper Bay (Annapolis and Havre de Grace).
However, they were too small compared to the subtidal com-
ponent, to have practical significance. B.
Event I1
(April 3 to 6 , 1975)
On April 3 , 1975, a low pressure disturbance was centered around 45"N,80°W (Fig. 5a).
Winds were westward along the New England coast, however, they were
northward over the southern Bight and Chesapeake Bay.
Coastal sea levels did
not respond to the northward wind, apparently due to the lack of large-scale (coherent) forcing. On the other hand, significant set-up in the Bay was induced by the local wind (Fig. 6). The cyclone propagated to the east, and it was centered around the Gulf of Maine on April 4 (Fig. 5b), which resulted in a southeastward wind along the MidAtlantic coast. As the cyclone continued moving eastward (Fig. Sc), winds became southward over the Chesapeake Bay. large:
The local southward wind set-down was
the sea level difference was over 100 cm between Kiptopeake B. and Havre
de Grace (Fig. 6).
Coastal sea level also dropped slightly on April 4.
The storm surge was dominated by subtidal sea level. The rise/fall of sea level was mainly due to the northward/southward wind set-up/down.
The eastward
wind was partly responsible for the sea level decrease on April 4. The supertidal component was also significant in the upper Bay (Fig.7).
The regular tidal
oscillation was suppressed during the storm period. DISCUSSION
Our analysis of two strong extratropical storm surges in the Chesapeake Bay suggests that subtidal sea level is the dominant surge component. Our results and Wang (1978a) also indicate that surqes can he induced by local wind set-up,
329
.
d
9
F i g . 5.
Surface weather (atmospheric p r e s s u r e ) map on: (b) 1200 April 4, and (c) 1200 April 5, 1975.
(a)
1200 April 3,
330
I
Klptopeoke B
0
0
u 0
ovre de Grace
v)
\
:
:
:
:
:
:
5
I
6.
:
:
I
9
April,
Fig.
:
1975
The o r i g i n a l ( s o l i d l i n e s ) and lowpass (dashed l i n e s ) sea l e v e l s , and t h e lowpass windstress a t Patuxent.
I
Kiotooeoke 8
-
U
-
U
m
H o v r e de G r o c e
April
Fig. 7 .
1975
The highpass s e a l e v e l s .
331 and nonlocal coastal sea level effect. The nonlocal effect (coastal surge) can be very important under favorable large-scale forcing conditions. For example, the maximum surge height (at Havre de Grace) was comparable between the two events, despite the fact that the local longitudinal windstress was about twice the magnitude in event 11. The compensation was due to the large coastal surge in event I. The local wind set-up is well-known; Wang (1978a) found high coherence between longitudinal windstress and surface slope over a year-long period. The wind set-up can be easily adopted and calibrated in the storm surge model. local effect however, is less well-known.
The non-
In the estuary surge model, the
coastal effect is usually modeled as "observed" surface elevations at the open ocean boundary. Wang (1978a) indicated that the Bay and coastal water response to E-W wind forcing is coupled, Thus, it may not be appropriate to treat the two syqems separately. The present modeling of "open ocean" surge is also rather poor. Wang (1978b) indicated that coastal sea levels along the Mid-Atlantic Bight are driven by:
(a) the local Ekman transport, (b) the local alongshore wind set-up,
and (c) the nonlocal shelf waves.
The "open ocean" surge model however, mainly
considers the effect of cross-shore wind set-up (Pagenkopf and Pearce, 1975). It seems unlikely that the "open ocean" surge model is applicable to extratropical storm surges. In conclusion, our study on the storm surge in Chesapeake Bay suggests that observational study should be emphasized. Recognizing that the model validation procedure is usually rather arbitrary, governing processes must be examined from observations. Only if these processes are clearly identified, can the regional storm surge model be formulated and tested properly.
A continuous feedback be-
tween model prediction and field verification is the only lead to a verified model for surge forecast. ACKNOWLEDGEMENTS We thank Mr. Jose Fernandez-Partagas who kindly made the weather charts available to us.
This study was supported by the National Science Foundation, under
Grant WE74-08463 and OCE77-20254. REFERENCES Heaps, N.S. and Jones, J.E., 1975. Storm surge computations for the Irish sea using a three-dimensional numerical model. Mgmoires Societe Royale des Sciences de Ligge, 6e s6rie;"tome VII, 289-333. Leenderste, J.J., Alexander, R.C. and Lin, S.K., 1973. A three-dimensional model for estuaries and coastal sea. The RAND'Corp., R-1417-OWRR, 57 pp.
332 Pagenkopf, J.R. and Pearce, B.R., 1975. Evaluation of techniques for numerical calculation of storm surges. R.M. Parsons Laboratory, MIT, Report No. 199, 120 pp. Simons, T.J., 1973. Development of three-dimensional numerical models of the Great Lakes. Canada Centre for Inland Waters, Scientific Series No. 12, 26 pp. Wang, D.P., 1978a. Subtidal sea level variations in the Chesapeake Bay and relations to atmospheric forcing. To appear in J. Phys. Oceanogr. Wang, D.P., 197833. Low-frequency sea level variability on the Middle Atlantic Bight. Submitted to J. Mar. Res.
333
FIRST RESULTS OF A THREE-DIMENSIONAL MODEL ON THE DYNAMICS IN THE GERMAN BIGHT J. BACKHAUS
Deutsches Hydrographisches Institut, Hamburg (F.R.G.)
ABSTRACT A three-dimensional barotropic fine mesh model of a shallow coastal sea is described. The tidal dynamics in very shallow water, e.g. wetting and drying of mud flats, are simulated by means of a movable horizontal boundary. A critical examination of the model results, especially of the vertical current structure, is carried out. In particular the influence of the wind on the horizontal and vertical current distribution is studied by simulating the extreme case of a storm surge and some idealized mean wind conditions. INTRODUCTION The threat of oil spills, the increased dumping of industrial waste into the sea, and last
-
but not least - storm surges, are common problems in coastal oceanography.
Taking these problems into account, it is essential to have detailed knowledge about the general circulation of water masses in the area under consideration, which - in the case of this study - is the German Bight. The spatial and temporal distribution of current and water-level data about the German Bight is rather incoherent, because a synoptic survey of the entire area has never been carried out. Therefore, knowledge about the wind and tide generated circulation in the German Bight still need improvement. The vertical distribution of residual currents in particular is rather unknown. This has given rise to the development of a three-dimensional numerical model and to extensive measuring efforts, terminating in a synoptic survey of currents and water levels taken over a period of one year within the framework of the 1979 "Year of the German Bight" experiment. Some locations for permanent moorings (current meters, tide gauges, meteorological buoys) to be deployed in the German Bight, were selected by means of the simple
334 model here presented. A good way to develop a model for a particular sea area, is to improve the model
stepwise; beginning with a very simple version, and always comparing the model results with measurements. In so doing, one can hope to learn a great deal about the behaviour of the model, and the physical processes in the area under consideration. In this study, the model equations and numerical techniques will be described only very briefly, more emphasis is laid upon a critical consideration of the "simulation ability" of the model, in order to find out how it could be further improved.
Fig. la. Map of German Bight, dashed line indicates area covered by the model. THE MODEL, GENERAL DESCRIPTION
A fine, horizontal grid resolution of 3 nautical miles was chosen to approximate
the German Bight's topography, which is rather complex, especially in the near shore regions. The largest system of coastal drying banks, which exists in the entire North Sea region, in combination with small islands, is to be found along the coast of the German Bight (Fig. la, lb). Water depths vary between 45 m below mean sea level and 2 m above mean sea level (drying banks) in coastal waters. For this first three-
dimensional modelling approach on the simulation of dynamics in a well-mixed shallow
335
Fig. lb. Depth (m) contours of discretisized bottom topography. sea, a vertical equidistant discretisation of 15 m was chosen.The simulation of the wetting and drying of tidal flats is carried out in the top layer (area between two adjacent computation levels) by means of a movable model boundary (Backhaus, 1976). As the sea is considered to be well mixed, all three layers have equal homogeneous density; therefore, the model is barotropic. The assumption of well-mixed conditions is not valid during summer; however, as far as could be estimated from measurements, baroclinic effects seem to be at least one order of magnitude smaller than the effects arising from bottom turbulence and non-linear wind/tide interactions.
..
*
.
.
t Fig. 2. Sketch of vertical configuration of the model.
336 The computation levels (Fig. 2, dashed lines) are horizontally fixed and completely permeable, so that the water can move freely in the basin. The internal shear stresses ri are defined at these levels; at the surface and the bottom respectively quadratic stress laws are applied. Turbulence is parameterized by means of a constant 2
vertical eddy viscosity coefficient Av = 40 cm / s and by a depth dependent horizontal exchange
-
coefficient Ah= h
5 m/s. The model could be regarded as quasi-linear,
with respect to the non-linear bottom friction. A vertically integrated flow is computed for each layer; the depth mean flow is obtained simply by integrating over the number of layers. The surface elevation is calculated from the equation of continuity (l), using the horizontal divergence of the depth mean flow. In the equations of motion (21, which are given in momentum form for an arbitrary layer, the non-linear terms are omitted. As
-
for example - proposed by Simons (1973)
the layerwise vertically integrated equations of motion are coupled by the internal shear stresses and by the barotropic pressure gradient, which does not vary with depth. No flux normal to closed boundaries may occur, slip along walls is permitted. Water levels are prescribed at open boundaries, and, for all layers, the gradient of the flux normal to the boundary is assumed to be zero. Together with the stresses given at the sea surface and bottom, this set of boundary conditions closes the probl e m for the barotropic case.
The numerical integration technique used is basing on the well-known explicit difference scheme introduced by Hansen (1956). The scheme was extended for the third dimension in a similar manner to that proposed by Sundermann (1971). The coupled system of partial differential equations (1, 2) are solved approximatively on a temporally and spatially staggered grid.
;+fix+?
Y
= O
U z f V - g h c
,
H = D + C = I h L
+(AhU
+(AhU Y
)
x
x
V = - f U - g h C +(AhV Y x
, v=su L
)
x
+(AhV
v,:
, i=mv L
+ ( A v U ~ ) ~
)
Y
Y
)
Y
+ ( A v V ~ ) ~
= depth mean transport, H = actual water depth, where U,V = horizontal transport, D = undisturbed water depth, = surface elevation, h = layer thickness, f = coriolis (constant), g = acceleration due to gravity, Ah, Av = coefficients of horizontal and vertical eddy viscosity, L = number of layers, x,y,z = coordinate system (east, north and down respectively).
c
337 MODEL RESULTS
Before discussing the results of the model, some remarks about tidal dynamics in the German Bight are given. There is an amphidromic point (Fig. 4b) some 2 0 0 km North-West of the vertex of the right-angle shaped coastline. Therefore, tidal elevations have a wide range, varying from a few centimetres near the amphidromic paint to about 1.5 m near the vertex. The tidal wave, travelling through the German Bight, shows a counter-clockwise sense of rotation, which also applies to the currents. Some examples of measured currents ar: shown by means of their current
Fig. 3 . Current ellipses (M2) for near surface ( f u l l line) and near bottom (dotted line) measurements. The sense of rotation is indicated by arrows. ellipse for the M2 tidal constituent (Fig. 3 ) . The 3 0 rn depth contour in the chartlet of the Figure gives an idea of a special formation in the German Bight's bathymetry:
338 the remains of a post-glacial estuary of the River Elbe. From measurements, as well as from model results, it can be observed that this prehistoric estuary has a strong influence upon the vertical structure of the currents, which becomes obvious from the current ellipses shown. In the vicinity of the underwater estuary, a general narrowing of the near-surface current ellipses can be found, indicating a zone of maximum vertical shear in the German Bight.
REPRODUCTION OF THE TIDE
Since the tide is the dominant signal in the North Sea, it should be reproduced correctly in the model, and with sufficient accuracy, before that model is applied to other cases, for example, to wind and tide-induced residual currents. The propagation of the tidal wave in the German Bight is simulated for the case of the
Fig. 4a. Computed co-tidal and co-range (cm) lines for M dominant semi-diurnal lunar tide (M
2
).
tide.
The boundary values (surface elevations),
prescribed at the open boundaries, were previously computed with a general two-dimensional North Sea model. The computed surface elevations of the North Sea model generally agree with observations, those of the fine-mesh German Bight model are of similar accuracy. Co-tidal and co-range lines for the M2 tide, obtained with the German Bight model (Fig. 4a) are compared with a chart, basing upon observations (Fig. 4b). A few improvements were made, due to a better resolution of the coastal topography, especially in the vicinity of the Jade/Weser/Elbe estuaries. The horizontal resolution of the grid is far too coarse to give a correct simulation of dynamics near the coast. Here, processes of sub-grid scale are parameterised very
339
Fig. 4b. Co-tidal (related to moons transit in Greenwich) and co-range (cm) lines for M tide. (adopted from Hansen, 1952) 2 roughly. However, the focus of this study is related mainly to the circulation in the deeper parts of the German Bight, and there the resolution seems to be sufficient. For a comparison with measured tidal currents, the tidal signal had to be extracted from the current meter data by means of a bandpass filter. Filtered nearsurface and near-bottom measurements for a period between spring tide and neap tide (stations 7 and 9 in Fig. 3 ) are compared with computed results in the corresponding layers of the model (Fig. 5, left of vertical dashed line). The agreement for the near-bottom currents is already quite close; whereas, in the near-surface regions larger deviations occur. For these regions the vertical resolution of the model seems to be insufficient, since we know, from observations, that the vertical current shear is strongest near the surface. Comparisons were carried out for more than the two stations shown, and - in general
-
the same features, as described above, were observed.
Apart from the discrepancies in near-surface currents, the sense of rotation and the phase of the currents seem to have been correctly simulated. This is also valid for the amplitudes in the lower layers. Therefore, it might be justified to apply the model for cases other than the pure tide.
340
A
Ly
>
a
.
Y
.
,
I >
-50
.
-Y .
0:
,I"
I ,
o*
1Ih
Oh
I11
t-------r
51
w z
50
e
0
Oh
IP
0"
t-
IIh
Oh
11"
w z
54
50
e
0
ob
il*
oh
11'
oh
I I ~
t-
Fig. 5. Computed (full line) and observed (dashed line) currents. Direction (true north), speed, and north - and east-components (cm/s) of near surface (top) and near bottom (below) currents for st tion 9 (left) and station 7 (right). vertical dashed line corresponds to January 2n8 1976, 12 noon (see Fig. 6).
341 SIMULATION OF A STORM SURGE Storm surges which, from time to time, cause exceptional damage along the coast, are one of the problems in the German Bight. One question concerning modelling aspects is, whether or not a fine-mesh German Bight model
-
which could be regarded
as a nest of the general North Sea model - will improve the accuracy of s t o m surge simulations in the German Bight. For that purpose, both models were run with the same set of three-hourly wind stress fields, computed from re-analysed weather maps for the storm surge of January 3rd, 1976, (Hecht, SuRebach, personal communication, 1 9 7 7 ) . The North Sea model was run first, in order to obtain a consistent set of boundary values for the German Bight model. Both models are running separately, without interaction, because of a restricted computer memory.
Fig. 6 . Observed (dotted) and computed (dashed = 3 dim. German Bight model, full line 2 dim. North Sea model) residuals (m) of surface elevations during storm surge of January 3rd 1976, starting at January 2nd, 12 noon.
=
For some coastal tide gauges, a comparison of measured and computed water level residuals is shown (Fig. 6 ) . The residuals were obtained by subtracting the tidal surface elevations from those containing tide plus surge. When comparing the residuals obtained with the coarse mesh North Sea model (grid size approx. 20 km) with those of the fine mesh model no significant improvement is to be observed, in general. This could have been expected, because the boundary values, computed with the North Sea model, are the dominant forcing, besides that of the wind. Positive effects arising from a better horizontal resolution are either very small, or not present.
Obviously, a fine mesh model is not necessary, when a simulation of surface elevations only is desired. As concerns the storm surge modelling G. Fischer and other participants at this colloquium agreed, that there are still things which are more important than grid refinements. More weight should be placed upon the meteorological input data and surface stress parameterisations in combination with waveinduced motions and surface elevations. However, the computed vertical current structure during the storm surge (Fig.5, Colid curves) right of vertical dashed line) show
a remarkable amplification of the
current speed, especially near the surface, which is up to four times as large as normal (see dashed curves). A similar factor is known from the very few current measurements taken during storm surges in the German Bight. Note that the near-surface inflow is followed almost instantaneously by an outflow near the bottom. The circulation during the storm surge becomes clearer if current residuals are viewed for the tidal cycle - when the maximum inflow and the peak of the surge occur (Fig. 7a); and, for the subsequent cycle, when the piled-up water masses are rushing back into the North Sea basin (Fig. 7b). Note the persistent outflow in the bottom layer. RESIDUAL CURRENTS
The residual currents in the North Sea are driven mainly by wind and tide. The influence of the wind is much stronger (up to one order of magnitude) than that of the tide. During spring and autumn the residual circulation is rather variable, because rapid changes in meteorological conditions occur. The general vertically integrated mean circulation of the North Sea is known fairly well by now (Maier-Reimer, 1977). From observations, it is known that the circulation can vary considerably with depth, which is of extreme interest for all marine pollution problems. Particularly in the German Bight, large differences in speed as well as in direction between near-surface and near-bottom residual currents are observed (Mittelstaedt, personal communication, 1978). In order to study the influence of the wind on the circulation in the German Bight, some computations were carried out, using homogenous and constant wind fields of different direction for the entire North Sea region. This rather idealized wind forcing is far away from reality; but, nevertheless, some principal knowledge will be gained about the processes which are causing the vertical distribution of the residual currents observed in the German Bight. A (moderate) wind speed of 5 m/s was chosen for all wind fields. The computations
were started from a quasi-steady state tidal cycle; again consistent sets of boundary values f o r each wind situation were previously computed with the North Sea model. A quasi-steady state was reached for all cases after at least five tidal cycles. The model's response time on the wind field is of the order of one tidal
343
I
I
STOQN SURGE 3 . J R N .
\I \ \
1916. GEQMRN BIGHT. CURRENTS IN LRYER 1
STOQN SUQGE 3 . J R N .
1976. GEQNON BIGHT. CURWENTS IN I R Y E R 2
STORfl SURGE 3 , J R N .
1976. GEQNRN BIGHT. DEPTH HERN CUQRENTS
l t
,-
\ \ \
t 1 \ \ \
I
\..., ... . ,,-\ .. I
r ,
\ \\ \ \\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I\\\\
.. \.
;0
\ \ r
\ \ \ \ I
4 N
STORM SUQGE 3 . J R N .
1976. GEQNRN BIGHT. CUQRENTS IN LRYEQ 3
I
Fig. 7a. Residual currents during storm surge of January 3rd 1976, for ‘inflow period‘.
I
344
I
I
I
I
I
\\\\.
\ \ \ \ \ \ \ \ I /
\\\\\\\
I t
0
I
L
STOQM SUQGE 3.JRN.
I
1
1976. GERMQN EIGHT, CUQQENTS IN L W E R
I
STOQM SURGE 3.JRN.
1976. GERMRN EIGHT, CUQRENTS IN L W E Q 2
STOQM SUQGE 3.JRN.
1976. GEQMRN EIGHT. OEPTH MERN CUQQENTS
I
4 - 0
4
n.
STORM SURGE 3.JRN. 1976. GERMRN EIGHT. CURQENTS IN LWER 3
Fig. 7b. Residual currents during storm surge of January 3rd 1976, for 'outflow period'.
345 cycle, which can also be observed in nature. The residual currents
i, also
called mean transport velocity elsewhere, were
obtained by integrating over one tidal cycle, using the following formula ( 3 ) :
;=fUdt T where
T
//hdt
(3)
T is the period of the M2 tide.
This formula is applied for each layer, but for the surface layer only, the second integral needs to be computed, because only there the layer thickness h
is depen-
dent upon time. For the cases of winds blowing from North-West, South-West, and South-East, the quasi-steady state residual circulation computed is shown for all three layers, and for the depth mean flow (Figs. 8, 9, * O ) . A considerable vertical current shear, especially in the area of the underwater estuary, can be observed in the flow patterns. Generally, these results are in good agreement with the residuals, computed and selected for certain wind situations by Mittelstaedt, using current meter data obtained in the German Bight, measured during the past 10 years. Again, the largest discrepancies between computation and measurement occur in the near-surface region. From observations, as well as from model results, there is evidence that the vertical change from the near-surface flow to the currents in deeper regions, occurs in a rather narrow “transition zone“. For comparisons between model results and measurements, it is important to know whether or not a current meter was moored above or below, or possibly right in the transition zone. It should be mentioned here, that, for technical reasons, no current meter was moored closer to the sea surface than 8 metres; all “near-surface‘‘data has been measured in a depth range of about 8 to 12 m below the surface. However, if discrepancies between computations and measurements occur, they might be caused by both the model and/or the data. Some further and careful work is necessary here. The variability of computed near-surface currents, in dependence of the wind direction, is much stronger than for the circulation in the deeper areas of the German Bight. There the circulation is rather persistent and significant changes only occur when the wind direction is veering from westerly winds to easterly or vice-versa. The depth circulation is mainly driven by the slope of the mean sea level. All westerly winds pile up the watermasses in the German Bight, causing a compensating outflow in near-bottom regions, focused by the underwater estuary of the River Elbe. The same mechanism causes an inflow for all easterly winds. The knowledge of these features of the vertical distribution of the residual currents is
-
for example
very important for the selection of dumping areas, and for the depth in which chemical waste should be dumped, in order to prevent it reaching the coast.
-
346
,
.....------I
,
I
*.-.c-.cc
, , I
0
9
CEQMPN RIGHT. QESIGUPI CUQQENTS L W E Q I , WING NU 5 MIS
7P.O
GEQMPN RIGHT. QESIGUOL CUQQENTS L P I E Q 7. UlNG NU 5 M/S
I
I
... ...... ................... ..... .....m
I
~
'..a
l l
I
-,,,,-, ......................... . . . . . . . . . ................ . . . . . . . . . . . . . . . . . I ,
\.
3
............................ ........... ...., Y -l m. . . . ,
9
I
I
,
.
.
I
.t I
GCQnPN RIGHT. QFSIOUPL TUQQCNTS LPVEQ 3 . UiNG NU 5 M,$
Fig. 8. Quasi steady state circulation for NW wind (5 m / s ) .
347 I
I7 I
GEQMPN RIGHT. QESlOUPL CUQQENTS L P I E Q I , UiNO Sbi 5 W S
GEQMPN RIGHT. QE5lOLlPL CUQQENTS LPVEQ
7. UINO
5U 5 W 5
I
..-.,,,,,, .......... -.*,,,,,,, --.,,,,,,, -..,.,,,,I
.............. ................ ................. ................... .....................
--.,,,,,,,
- - - . I , , , , ,
I * \ \ . . \ \ \ \ \ \ \ .
- - - - * , , , , I
...................... ,.............-----.~ ................... , , . . - .- - - - . .---
n
GEQMW RIGHT. QESIOUPL CUQQENTS LPIEQ 3 . UtNO 514 5 W S
CEQUIPN RIGHT, GEPTil MWN QESIGUPl CUQQFNTS, UNC
Fig. 9. Quasi steady state circulation for 8W wind (5 m / s ) .
5u 5 w 5
348
GEQMDN BIGHT. QESIOURL CUQQENTS LOVE9
I,
MNO SE 5 M/S
GEQMDN BIGHT. QESiOURL CUQQENTS LDfEQ '2. UlNO SE 5 W S
f GEQMW BIGHT. QESIOURL CUQQENTS LOVE9 3 . UlNO SE 5 M/5
GEQMRN BIGHT. OEPTH MERN QESlOUR CUQQENTS. U1NO SE 5 M/S
Fig. 10. Quasi steady state circulation for SE wind ( 5 m / s ) .
::.:J
349
CONCLUDING REMARKS
Apart from insufficiencies, caused by a poor vertical resolution near the sea surface, the rather simple model version described is able to already simulate the significant features of the dynamics in the German Bight. The knowledge about the horizontal and vertical distribution of currents was improved by applying the model to some significant cases.
ACKNOWLEDGEMENTS
The author is indebted to Prof. K. Hasselmann, who encouraged him to participate at this colloquium. The assistence of Mrs. Barttels and Mrs. Petersitzke in preparing and typing the manuscript is very much appreciated. Thanks to Mr. Hontzsch for adding the final touch to the diagramms.
REFERENCES
Backhaus, J., 1976. Zur Hydrodynamik im Flachwassergebiet, ein numerisches Modell. Deutsche Hydrogr. Zeitschrift, 29:222-238. Hansen, W., 1952. Gezeiten und Gezeitenstrome der halbtagigen Hauptmondtide M2 in der Nordsee. Erganzungsheft, Deutsche Hydrogr. Zeitschrift, Reihe A, Nr. 1. Hansen, W., 1956. Theorie zur Errechnung des Wasserstandes und der Strdmungen in Randmeeren nebst Anwendungen, Tellus No. 8. Maier-Reimer, E., 1977. Residual circulation in the North Sea due to the M2-tide and mean annual wind stress. Deutsche Hydrogr. Zeitschrift, 30:69-80. Neumann, H., Meier, C., 1964. Die Oberflachenstrome in der Deutschen Bucht. Deutsche Hydrogr. Zeitschrift, 17:l-40. Simons, T.J., 1973. Development of three-dimensional numerical models of the Great Lakes. Environment Canada, Scientific series no. 12. Sfindermann, J., 1971. Die hydrodynamisch-numerische Berechnung der Vertikalstruktur von Bewegungsvorgangen in Kanalen und Becken. Mitt. Inst. f. Meereskunde, XIX. Thorade, H., 1928. Gezeitenuntersuchungen in der Deutschen Bucht. Archiv der Deutschen Seewarte, 46.
This Page Intentionally Left Blank
351
T I D A L AND R E S I D U A L
FranGois C.
CIRCULATIONS I N T H E ENGLISH C H A N N E L
RONDAY
Mecanique d e s F l u i d e s GBophysiques, U n i v e r s i t e de L i e g e , B6,
B-4000 L i e g e
(Belgium).
a t t h e I n s t i t u t de MBcanique,
Also
S a r t Tilman
U n i v e r s i t e de Grenoble,
38
S a i n t Martin d'H&res (France).
ABSTRACT
E r r o r s i n t r o d u c e d by v a r i o u s n u m e r i c a l s c h e m e s f o r h y d r o d y n a m i c models have been a n a l y s e d f o r a r e a l s i t u a t i o n : t h e t i d a l c i r c u l a t i o n i n t h e English Channel. T h i s a n a l y s i s i s b a s e d on t h e p r o d u c t i o n of harmonics of t h e M2 t i d e . T h i s s t u d y shows t h e u n a b i l i t y o f For some s c h e m e s t o g i v e a g o o d r e p r e s e n t a t i o n o f t i d a l h a r m o n i c s . t h i s r e a s o n - i n d e p e n d e n t l y o f d i f f i c u l t i e s t o o b t a i n p r e c i s e boundary c o n d i t i o n s - it i s always hazardous t o c a l c u l a t e t h e r e s i d u a l c i r c u l a t i o n by a v e r a g i n g t h e t r a n s i e n t c i r c u l a t i o n .
INTRODUCTION
I n t h e English Channel, g i v e a non n e g l i g i b l e
t i d a l h a r m o n i c s a r e v e r y s t r o n g and m i g h t
contribution t o the residual
flow.
To c a r r y
o u t t h i s i n v e s t i g a t i o n d i f f e r e n t d e p t h a v e r a g e d hydrodynamic models a r e used. The f i r s t s t e p o f
of
t h i s study i s t o determine
t h e main p a r t i a l t i d e o f
lunar
(M2)
tide.
The
(M2)
t h e semi-diurnal
:
f i r s t s i m u l a t i o n b a s e d on a
a l g o r i t h m shows a n e x c e l l e n t a g r e e m e n t between ved
t h e t i d a l harmonics
t h e E n g l i s h Channel
e l e v a t i o n s and c u r r e n t s .
c a l c u l a t e d and o b s e r -
Unfortunately
g i v e s a poor agreement f o r higher harmonics.
simple numerical
t h i s simulation
A s t h e bottom s t r e s s
and t h e a d v e c t i o n g e n e r a t e n o t o n l y h i g h e r harmonics b u t a l s o a r e s i d u a l component, one c a n n o t e x p e c t t o have a n u m e r i c a l hydrodynamic model g i v i n g a good r e p r e s e n t a t i o n o f representation of In the
litterature,
1967 ; F l a t h e r ,
many a u t h o r s
1976) determine
flow and a poor
Therefore
(e.g.
Durance,
1974 ; B r e t t s c h n e i d e r ,
t h e r e s i d u a l c i r c u l a t i o n by a v e r a g i n g
the t r a n s i e n t c i r c u l a t i o n without harmonics.
the residual
t i d a l harmonics.
considering t h e generation of
tidal
i t seems v e r y i n t e r e s t i n g t o v e r i f y t h e a b i l i t y
of d i f f e r e n t numerical hydrodynamic models t o reproduce t h e harmonics.
352 From this study it will be possible to show that the residual flow calculated by averaging the transient flow is very sensitive to the discretization of the advection.
GENERAL EQUATIONS OF DEPTH-AVERAGED TIDAL MODELS
If
denotes the water transport vector and
U
H
the total depth,
the two-dimensional (depth-integrated) hydrodynamic equations for tides can be written (e.g. Ronday, 1976):
-
in the formalism of the depth-averaged velocity
-
or in the formalism of water transport
a H + V.?
= 0
-
at
(3)
(4)
with
-h H = h + C where
h
is the mean depth, 5
rotation vector, mass, g
5
the surface elevation, f
the Coriolis
the astronomical tide-producing force per unit
the acceleration of gravity and
D
the drag coefficient on
the bottom. In the English Channel (and the Dover Straits) the astronomical tide-producing force gives only a very small contribution to the observed M 2
.
Therefore,
can be neglected in our models, and tidal
motions are induced by external forcing along ouen sea boundaries. To solve these equations of motion initial and boundary conditions must be imposed. Initial -
conditions
As forced hyperbolic systems are not sensitive to initial conditions, the following initial conditions will be taken
353
u = o
and
5
=
0
and
5
=
0
or
(7)
- = o
for all points in the English Channel and in the Dover Straits.
Boundary conditions
-
y.5
along the coasts or where
-
u.5
=
0
= 0
is the normal at the coast
along open sea boundaries
a ) Northern open sea boundary. As the distance between coastal stations is not too large, a linear interpolation between observations at Zeebrugge and Foreland gives boundary conditions along the boundary.
8 ) Western open sea boundary After different numerical simulations, ( M 2 , M 4 ,
M6)
data coming
from the physical model of Grenoble (Chabert d'Hi6res
S .
Leprovosl
1970) are used along the western boundary.
NUMERICAL METHODS FOR THE RESOLUTION OF TIDAL EQUATIONS
As described in the previous section, tidal motion can be studied by means of two kinds of hydrodynamic models
-
:
the first uses the concept of depth-averaged velocity,
- the second the concept of water transport. From a physical point of view, no differences exist between the two sets of partial differential equations ( 1 to 4).
However, equa-
tions (3 and 4) have a conservative form and this is extremely important in numerical analysis. To study the propagation of long waves, hydrodynamicists have the
choice between implicit and explicit algorithms.
Implicit algorithms
Implicit algorithms are often unconditionally stable. the ratio
it
-
However,
has to be taken sufficiently small to reduce the error
between the solution of the partial differential equations and that of the finite difference equations.
Leendertse (1967) and Nihoul
Ronday (1976) have shown that the time step
(At)
E
must remain small
354 when a small phase deformation is imposed.
Moreover, implicit algo-
rithms require the resolution of algebraic equations at each time step. Since the advantage of unconditionally stable schemes cannot be exploited for coastal seas, implicit algorithms are not considered in this study.
Explicit algorithms
All explicit algorithms have a stability condition.
The critical
time step is a function of the maximum depth, of the maximum velocity, and of the spatial step. time step is approximatively At
%
with
For the English Channel, the critical :
200 sec Ax = 10 km.
Only explicit algorithms will be considered in this study.
NUMERICAL MODELS USED TO STUDY THE GENERATION OF TIDAL HARMONICS
To carry out the present investigation, three numerical models based on typical numerical algorithms are used.
These models have
several characteristics in common :
-
the same geographical area,
- the external forces
-
the empirical coefficients the numerical staggered grid
(e.g. Ronday, 1976).
These models differ by the discretization of the equations ( 1 t o 4 ) The quality of the numerical solution is a function of
-
the accuracy of the algorithm the conservative or non conservative form of the equations.
Model 1
is based on the concept of the depth averaged velocity,
and has been described by many authors (e.g. Hansen, 1966
;
Ramming,
1976 and Ronday, 1976). The algorithm of resolution is explicit and its accuracy is only O(At, Ax)
due to a simple discretization of the advection terms :
forward or backward derivatives according to the direction of the current.
There arises from this discretization a numerical viscosity
355 Model 2
is based on the concept of the water transport, and has
been used by Fisher (1959) and Ronday
(1972).
The algorithm of resolution is explicit and its accuracy is O(At, Ax').
The centered discretization o f the advection terms in-
duces a weak instability.
T o eliminate this instability, an artifi-
l o 3 m2/s) and viscous terms
( u AU) are intron An order of magnitude analysis shows that the artificial
cia1 viscosity duced.
(vn
%
viscous terms are small compared to the pressure or Coriolis terms.
Model 3
also uses the water transport formalism.
The long wave
propagation is studied by means of an explicit predictor-corrector procedure. First, a dissipative procedure
(the advection terms are calculated
with forward or backward derivatives) gives an estimate of the solution.
Secondly, a "weakly" instable procedure corrects this first
estimate ves).
(the advection terms are calculated with centered derivati-
The accuracy of this two stens procedure is approximatively
equal to
O(At2, Ax2).
ANALYSIS OF RESULTS
Comparison between observed and calculated elevations
Fig.
(1 to 18) show the amplitudes and phases of
tides calculated with models 1, 2 , and 3 .
M2, M4 and M6 Tables ( 1 to 3 ) give the
comparison between the observations and the numerical results for some coastal stations.
Data are taken from the Deutsches Hydrogra-
phisches Institut - Hamburg Monaco
(1966).
(1962) and the Bureau Hydrographique de
The statistical analysis of the elevations is based
on eighteen stations (Fig. 19).
a) M2 tide i) Fig.
( 1 to 6) and Table 1 show that the differences exis-
ting between the results of the three simulations are very small.
T h e standard deviations are
:
for the phases
u+
%
2" (or 4 minutes)
for the amplitudes
u
%
0.13 m
A
ii) The agreement between the in situ observations (Table 1) and the numerical results (Fig.
(1 to 6)) is excellent.
different models give the following standard deviations
The :
356
.
.
for the phases model 1
u4
%
5.4"
(or 1 1 minutes)
model 2
cr4
Q,
6.5"
(or 1 3 minutes)
model 3
u4
Q,
5.4"
(or 11 minutes)
and for the amplitudes model 1
uA
model 2
uA
model 3
uA
%
%
0.12
m
0.08
m
0.09
m
TABLE 1
M 7 tide
Comparison between the observations and the numerical results for some coastal stations (amplitude in meters ; phases in degrees) STATIONS
OBSERVATIONS 3 .84/180° 1.91/230° 2.68/285' 3.11/312' 2.47/323' 2.27/322" 1.47/317' 1.11/178' 1.48/159'
St. Servan Che rbourg Le Havre Dieppe Hastings New Haven Nab Tower Lyme Regis Salcombe
I
I 4
I 9
I 2
MODEL 1
MODEL 2
MODEL 3
3.74/173' 1.87/226O 2.56/280° 3.01/307' 2.48/321° 2-05/3130 1.41/313O 1.28/170° 1.62/156O
3.86/173' 1.90/225O 2.66/27g0 3.11/305' 2.55/318" 2-15/3110 1.48/310° 1.26/17l0 1.62/155'
3.71/172O 1.92/227' 2.72/281° 3.18/307' 2.63/320° 2.24/3120 1.57/313" 1.22/17l0 1.62/157'
I 1
I
0
I
I
I
I
3
Fig. 1 . Lines of equal phases for the M2 tide calculated with model 1 (in degrees).
367
Fig.
2.
L i n e s o f e q u a l p h a s e s f o r t h e M 2 t i d e c a l c u l a t e d w i t h model 2 . ( i n degrees)
Fig.
3. L i n e s o f e q u a l p h a s e s for t h e M 2 t i d e c a l c u l a t e d w i t h m o d e l . 3 3. . (in degrees)
358
4.
Fig.
-
I
L i n e s of model 1 .
I
I
L'
Fig.
9'
5.
L i n e s of model 2 .
e q u a l a m p l i t u d e s f o r t h e M2 t i d e c a l c u l a t e d w i t h ( i n centimeters)
I
I 1'
I
I
0
I
I I
2'
I
S
equal amplitudes f o r the M2 t i d e calculated with ( i n centimeters)
359
I
*'
I
Fig.
0)
I
I
3
6.
MA
,
2
Lines of model 3.
I
I
I
0
I
I
I
9
2
equal amplitudes f o r the M2 t i d e calculated with (in centimeters).
tide
R e s u l t s from t h e t h r e e models a r e p r e s e n t e d
i n Fig.
( 7 t o 12)
and T a b l e 2 g i v e s t h e comparison between t h e n u m e r i c a l r e s u l t s and the observations a t different stations.
The f e a t u r e s w h i c h d i s -
tinguish the respective solutions are a s follows
:
i) The c a l c u l a t e d p h a s e s a r e i n g e n e r a l i n g o o d a g r e e m e n t w i t h the observations : ( o r 24 minutes)
1
a$
Q
23'
model 2
u$
2,
21'
( o r 22 minutes)
model 3
a$
Q
20'
( o r 20 m i n u t e s )
model
D i f f e r e n c e s between
t h e s e s i m u l a t i o n s a l s o remain
A$max = 6 2 '
( o r 64 m i n u t e s )
a@
( o r 28 m i n u t e s )
ii)Fig.
%
27"
( 1 0 t o 1 2 ) show t h e
s p a t i a l d i s t r i b u t i o n s of
tudes i n t h e English Channel. lar,
Shapes of
these
small
:
t h e Mq ampli-
lines are s i m i -
b u t t h e r e a r e l a r g e d i f f e r e n c e s i n i n t e n s i t y between t h e
different simulations :
360
1
.EoAelUA
%
overestimates the M4 tide (Fig. 10 and Table 2) :
0.06 m
The error is amplified with increasing distance from Cherbourg. For example, at The Havre the calculated M4 is of the
0.25 m
Chabert d'Hi&res
0.34 m
instead
observed. and Le Provost (1970) have shown that the M4
tide is mostly generated near the "Cap d e l a Hague" and the "Cap de Barfleur" where the advection is very strong.
As the
accuracy of the scheme i s poor f o r t h e advection terms
O(At,Ax)
one can expect a radiation of errors from these capes. TABLE 2
M4 tide Comparison between the observations and the numerical results for some coastal stations (amplitudes in meters ; phases in degrees) STATIONS St. Servan Cherbourg Le Havre Dieppe Hasti ngs New Haven Nab Tower Lyme Regis Sa lcombe
I
MODEL 1
MODEL 2
MODEL 3
0.28/286O 0.14/359' 0 . 2 5 / 77' 0.27/187O 0.22/228' 0.09/245" 0.16/354' 0.10/ 75" 0.10/132O
0.26/242' 0.19/ 14" 0.34/ 89' 0.32/174O 0.24/212O 0.06/205° 0.16/ 4' 0.23/ 60° o.11/112°
0.30/304' O.O9/338O 0.20/ 80' 0.20/164' 0.14/208'
0.28/293" o.14/350° 0.25/ 76" 0.27/172' 0.23/207' 0.065/214' O.O9/333O 0.14/ 41° 0.09/1100
,
I
b'
OBSERVATIONS
9
I 2
,
I
I 0
O.O35/20r0
0.08/330° 0.10/ 5 3 O 0.09/1170
,
I 2'
I
I
S
Fig. 7. Lines of equal phases for the M4 tide calculated with model 1. (in degrees).
361
I
I
I
i'
-
Fig.
I 2'
5'
I I
I
I
I
I
0
2
I
I
3
8 . L i n e s o f e q u a l p h a s e s f o r t h e M 4 t i d e c a l c u l a t e d w i t h model 2 . ( i n degrees)
I
I b
Fig.
3
9.
I 7
I I
I 0
I I
I
I
I
3
L i n e s o f e q u a l p h a s e s f o r t h e M q t i d e c a l c u l a t e d w i t h model 3 . ( i n degrees)
362
I
*'
I
I
I
3
\'
Fig.
I 2.
I
I I
I
I
0
I
I
I
3'
2'
10. L i n e s of e q u a l a m p l i t u d e s for t h e M4 t i d e c a l c u l a t e d w i t h m o d e l 1. ( i n c e n t i m e t e r s ) .
Fig.
I
I 9
11.
,
I 1'
I
I
I
0'
1
I 1'
I
I
S
L i n e s of e q u a l a m p l i t u d e s for t h e M4 t i d e c a l c u l a t e d w i t h m o d e l 2. ( i n c e n t i m e t e r s ) .
363
L
I
I b
I
Fig. 1 2 .
I
0'
I
I
L S'
1'
Lines of equal amplitudes for the M4 tide calculated with model 3. (in centimeters). underestimates the M q tide (fig. 11 and Table 2 )
.Eo$eA2
oA
I
I
I
1'
3'
%
:
0.06 m
The damping of M4 comes from the discretization of the advection terms (To maintain a stable procedure with centered derivatives, artificial viscous terms have been introduced). As the advection is very strong near Cherbourg, the additional viscosity must be high scheme.
(vn
%
lo3
m2/sec)
to keep a stable
Therefore this numerical viscosity induces a too
large damping of the solution elsewhere.
It is possible to
improve the solution a little by increasing the drag coefficient
(D
(Pingree
&
2.5
and reducing the viscosity
lo2
m2/sec
Maddock, 1977).
---- -
.Model 3 gives the best reproduction of the M4 tide (Fig. 1 2 and Table 2) :
uA
%
0.03 m
Now, there is no difference between calculated and observed tide at The Havre. at Nab Tower. of this area
However, there remains an error
(0.07 m)
This might be due to the spatial discretization :
the narrow and shallow channel between Nab Tower
and Southampton is not taken into account.
364 y)
M
tide
(13 t o 18) a n d T a b l e 3 l e a d s t o t h e f o l l o -
The a n a l y s i s o f F i g . wing c o n c l u s i o n s
:
i ) T h e r e a r e few d i f f e r e n c e s b e t w e e n t h e t h r e e s i m u l a t i o n s : t h e shape and t h e
-Phsas_e-s,
i n t e n s i t y of
The c o n c e n t r a t i o n o f
solution near St. large
: U+
Amplitudes. ---------
%
31"
are similar. l a c k of
c o t i d a l l i n e s and t h e
re-
Malo e x p l a i n why t h e s t a n d a r d d e v i a t i o n s e e m s (or 22 minutes).
uA
The s t a n d a r d d e v i a t i o n
t h e i n t e n s i t y of
TABLE
the iso-lines
t h e Pq6 i s a l s o s m a l l
%
0.021 m
i s small, but
( o f t h e o r d e r of
0.05 m ) .
3
M A -t i d e
Comparison between t h e o b s e r v a t i o n s and t h e n u m e r i c a l r e s u l t s f o r some c o a s t a l s t a t i o n s ( a m p l i t u d e i n m e t e r s ; p h a s e s i n d e g r e e s ) . STATIONS
OBSERVATIONS
,
I
Pig.
9
1
MODEL
2
MODEL
O.O1/283O 0 . 0 5 / 87O 0.28/264O 0.03/300° 0.06/ 78" O.O3/137O 0.06/ 86 O.ll/ 530 0.02/128°
0.03/289' 0.04/100° 0.26/288" O.O4/307O 0 . 0 5 / 95" O.O4/156O O.O7/146O 0.07/ 970 O.03/15lo
O.O2/352O 0.03/101" 0.16/286' 0.02/298° 0.04/ 173' 0.024/160° O.O4/119O 0.05/103' O.03/17Zo
S t . Servan Cherbourg Le H a v r e Dieppe Has t i n g s New Haven Nab Tower Lyme R e g i s Salcombe
L
MODEL
I
0.01/320° 0.04/ 98" 0.25/269' 0.02/289' 0 . 0 6 / 90" 0.025/142° 0.05/ 87' 0.09/ 730 0.02/149"
I 0
I
1
3
I 3
13. L i n e s o f e q u a l p h a s e s f o r t h e M6 t i d e c a l c u l a t e d w i t h model 1. (in degrees).
365
I
I
I
Ir'
Fig.
14.
Fig.
I
2'
I
I
I
I
0'
3'
2
L i n e s of e q u a l p h a s e s f o r t h e M6 t i d e c a l c u l a t e d w i t h m o d e l 22. . (in degrees)
I L'
I
I
I
3
I S'
I
I 2'
I
I
I
0'
I
I 2
I
I
8
1 5 . L i n e s of e q u a l p h a s e s f o r t h e M g t i d e c a l c u l a t e d w i t h m o d e l 3. (in degrees).
366
1
I 4.
I
3.
I
I
I
I
I
1.
0
1-
2.
3-
16. L i n e s of e q u a l a m p l i t u d e s f o r t h e M 6 m o d e l 1 (in centimeters).
Fig.
I 4-
Fig.
I 2.
17.
I
I
I
3-
2.
1-
I 0
tide calculated with
I
I
1.
2-
I 3-
L i n e s of e q u a l a m p l i t u d e s for t h e M6 t i d e c a l c u l a t e d w i t h m o d e l 2 (in centimeters).
367
-
51
-
50
Fig. 18. Lines of equal amplitudes for the M6 tide calculated with model 3 (in centimeters).
ii) The agreement between the observations and the numerical results is satisfactory for the phases, and poor for the amplitudes if one considers the intensity of M6 in the English Channel : for model 1
:'$ oA
for model 2
:'$ uA
for model 3 : u $ UA
iii)As
(or 17 minutes)
%
26O
%
0.034 m
%
38O
n ,
0.044 m 20° (or 14 minutes)
%
0.032 m
(or 26 minutes)
no serious improvement exists from one model to another,
the origin of discrepancies between the observations and the numerical results has to be found elsewhere. It is well known that a good reproduction of the S 2 tide is impossible without the combination of
S2
and M2 tides.
More-
over, the M6 tide generated by friction depends not only on M 2 , but also on S2, N 2 ,
...
For a station located between the
"Cap de la Hague" and Guernesey, Le Provost ( 1 9 7 6 ) showed that the
3uM2
component of bhe friction term (a source of M6)
is
368 overestimated (about 20
%
at spring tides) if the
S2
tide is
not taken into account. A spectral analysis of the friction term for the three simulations locates the main source of M6 near the "Cap de la Hague". Therefore, the radiation of an error, estimated at about 20
%
near the "Cap de la Hague", can produce much larger errors near Lyme Regis and The Havre.
This error is not affecting regions
located near the open sea boundaries where correct M6 elevations are prescribed. In conclusion, a good reproduction of M6 is impossible with M 2 only.
COMPARISON BETWEEN CALCULATED AND OBSERVED TIDAL CURRENTS
I t is always difficult to compare calculated currents to the observations
:
currents rapidly vary from point to point, and they are
often reduced to their surface values by means o f empirical formulas in atlas of currents (Sager, 1975). a) In order to visualize the differences existing between the three models, fifteen stations (fig. 19) are chosen.
Fig. 1 9 . Stations of comparison for vertical tides + horizontal tides
369 Fig.
(20.1 t o 20.15)
give the amplitude
(in cm/s),
the direction
( i n d e g r e e s ) , and t h e t i d a l e l l i p s e of c u r r e n t s c a l c u l a t e d w i t h t h e models
:
model 2
-----
model 3
.....
model
1
The a n a l y s i s o f t h e f i g u r e s l e a d s t o t h e f o l l o w i n g c o n c l u s i o n s
i) Maxima o f t i d a l c u r r e n t s .
:
The t h r e e n u m e r i c a l s i m u l a t i o n s
a p p r o x i m a t e l y g i v e t h e same r e s u l t s
(u
%
0 . 1 5 m/s).
The c o -
h e r e n c e between models 2 and 3 i s h i g h e r . ii) Minima o f
(u
0.1
%
t i d a l currents.
Here t h e d i f f e r e n c e s a r e s m a l l e r
C u r r e n t s c a l c u l a t e d w i t h models 2 and 3 a r e
m/s).
more s i m i l a r . iii) Phases of
o r d e r of
F4odels 2 a n d 3 a p p r o x i m a t i v e l y
tidal currents.
D i f f e r e n c e s w i t h model 1 a r e of
g i v e t h e same r e s u l t s .
the
30 m i n u t e s .
i v ) D i r e c t i o n of
tidal currents.
I f one e x c e p t s t h e t i m e of
tide
r e v e r s a l , t h e d i r e c t i o n of c u r r e n t s i s n o t very s e n s i t i v e t o t h e scheme u s e d i n t h e m o d e l .
I
Fig.
a
I
20.1.
7
9
11
13
15
17
1s
21
28
2s
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 1. ( i n c m / s )
.
370 960
330 800
270 240 210 I80
I50 I20
90 60 30
0
! a5
90
0
75
so *5
30 15 0 1
Fig.
9
5
7
20.2.
9
II
13
15
17
19
2!
23
25
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 2 ( i n cm/s).
360 330
300 270 240 210 1 00
I50 I20
90 SO
so 0
1 as 90
0 75
so 45 SO
15
0 I
Fig.
3
5
20.3.
7
9
II
13
15
!7
19
21
23
25
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 3 ( i n cm/s).
371
a
t
Fig.
B
5
7
9
I1
13
17
19
21
'2s
26
27
20.4. T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t m o d e l s at station 4
Fig.
IS
20.5.
( i n cm/s).
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t m o d e l s at s t a t i o n 5 ( i n cm/s).
372
1
20.6.
Fig.
0
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 6 ( i n cm/s).
360 330
so0 1?0
240 210
I80
150 I20
so so 30 0
710 I80
0 150 I20
so 60 SO
0 I
Fig.
a
5
20.7.
7
9
II
IS
15
I7
19
21
25
2s
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t station 7 (in cm/s).
373 360
380
so0 270 240 210 I80
ISO I20
90
so
so 0 5
3
1
7
I1
9
13
IS
17
19
21
23
25
27
210 I80
0
Iso I?n
so 60
SO
0 5
3
I
Fig.
I
Fig.
7
20.8.
a
s
20.9.
9
1:
13
15
17
19
21
28
l5
21
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t station 8 (in cm/s).
I
s
11
IS
15
17
is
21
29
2s
27
T i d a l c u r r e n t s c a l c u l a t e d wlth t h e d i f f e r e n t models a t station 9 (in cm/s).
374 360
330
so0 270 240 210
Ia0 I50
I20 90
60
90 0
105 90 0
75
so +5
30 15
0 I
Fig.
3
5
7
20.10.
9
I1
13
IS
17
I9
21
19
25
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 10 ( i n c m / s ) .
360 190
so0 770
240
?I0 180
I50 I20
90
so so 0
10s 90
0 75
so 45
so 15
0 I
Fig.
9
5
20.11.
7
9
11
IS
15
17
19
21
25
25
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 11 ( i n c m / s ) .
375 360
330 300 270 2W 110
I80
I50
I20 SO
60 SO
0
I05
SO
0
75 60
$5
30 15
0 1
Fig.
3
5
7
20.12.
9
II
13
15
17
IS
21
13
15
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 1 2 ( i n cm/s).
360
350
so0 270
240 210 I80 150 120
90 60 SO
0
210 I90
0
I50 I20
SO SO 30 0 I
Fig.
a
s
20.13.
7
s
I!
13
IS
I?
14
21
23
2s
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h t h e d i f f e r e n t models a t s t a t i o n 13 ( i n c m / s ) .
376 360 *SO SO0
270 240
210
I00
I50 I20 90
80 80
0
I05
90 0
75 60
15
so 15 0
Fig.
20.14.
T i d a l c u r r e n t s c a l c u l a t e d with the d i f f e r e n t models a t s t a t i o n 14 ( i n c m / s ) .
350
a30
SO11 270 24 0 210
I80
I50 120
90 80
50
0
105 90
0 76 60
45 SO 15
0 I
Fig.
S
5
20.15.
7
9
I1
I3
15
17
19
11
25
25
27
T i d a l c u r r e n t s c a l c u l a t e d w i t h the d i f f e r e n t models a t s t a t i o n 15 ( i n c m / s ) .
377 8 ) As a purpose of this study i s to show that the residual flow calculated by averaging the transient flow is very sensitive to the discretization of the advection, a Fourier analysis of currents is made.
T o clarify ideas, characteristic stations are chosen (Fig. 2 1 ) .
FRRNCE
I
I L'
Fig. 2 1 .
I
I
3'
2
-
I I
I
I
0
I
I
2'
I
3'
Stations of comparison for Fourier Analysis.
- for the Ma currents TABLE 4 Amplitude of M2 currents
STATION
Amplitude of the eastern (u) and northern (v) currents
Model 1 ( d s )
Model 2 (m/s)
Model 3 (m/s)
3
0.53 0.20
0.54 0.17
0.56 0.21
4
1.22 1.07
1.42 1.08
1.61 1.37
6
1.32 0.05
1.35 0.06
1.41 0.06
0.61
0.62
0.60
0.97
0.95
1.03
8
U V
I
LS
378 The a n a l y s i s o f T a b l e 4 results.
H o w e v e r , Model
shows
1 has
t h a t t h e t h r e e models g i v e s i m i l a r
a weak t e n d e n c y t o u n d e r e s t i m a t e c u r -
r e n t s i f o n e c o n s i d e r s Model 3 a s t h e b e s t
- f o r t h e M q and Mo M4
a n d Mo
tive terms.
(higher numerical
(residua1)currents
t i d e s a r e both generated f o r a For t h i s reason,
l a r g e D a r t by t h e a d v e c -
i t seems i n t e r e s t i n g t o compare s i m u l -
t a n e o u s l y t h e d i f f e r e n c e s between t h e models. b e s t r e p r o d u c t i o n of
accuracy)
M4
elevations,
S i n c e Model 3 g a v e t h e
r e s u l t s o f Model 3 a r e t a k e n a s
reference values. T a b l e 5 shows t h e a m p l i t u d e s o f
t h e e a s t e r n and w e s t e r n components
o f M4 a n d T a b l e 6 t h e e a s t e r n a n d w e s t e r n c o m p o n e n t s o f
the residual
currents. TABLE 5
M4
currents
STATION
Amplitude of (u) the eastern and n o r t h e r n ( v ) currents
Model
1
(m/s)
Model 2 (m/s)
Model 3 (m/s)
3
0.05 0.09
0.02 0.05
0.03 0.07
4
0.06 0.12
0.02 0.12
0.03 0.12
6
0.04 0.02
0.03 0.00
0.03 0
0.10 0.16
0.05 0.09
0.07
U
8
V
0.12
TABLE 6
Residual currents
STAT I ON
Eastern (u) and n o r t h e r n ( v ) currents
Model 1 (m/s)
Model 2 (m/s)
Model 3
(m/s)
0.02 0.01
0.00 -0.04
0.02
4
-0.05 0.22
0.04
0.30
0.01 0.25
6
0.01 -0.06
-0.03
0.26 0.40
0.24
3
U V
-
8
U V
0.00 0.10
0.00
0.04 0.00 0.10 0.21
379 The features which distinguish the respective solutions are
:
-
model 1 has the tendency to overestimate the currents and model
-
According to the results of Table 6 the residual currents are
the intensity of M4 and Mo currents are similar
2 to underestimate them
very sensitive to the discretization of the advective terms. At station 3 ,
M o current goes north-east with model 1 ,
with model 2, east with model 3. the
u
south
Near the "Cap de la Hague",
component of the current is negative with model 1 and
positive with models 2 and 3. y ) Calculated currents have to be compared with the observations. The
quality of current measurements is not sufficient to decide the ability (or unability) of models to reproduce harmonics of M2 currents. Nevertheless, one might expect the same conclusions for M 4 (and M o ) currents than those for M4 elevations.
For this reason, only M2
currents will be considered in this section. An important parameter for the comparison is the intensity of the largest M 2 current.
The analysis of Fig. ( 2 2 to 2 5 )
shows a good
agreement between the observations and the three simulations.
Fig. 2 2 .
Largest M g currents deduced from the observations (in (Sager, 1 9 7 5 ) .
m/S)
380
I
I
I
9
23.
Fig.
, 3
24.
I
2'
I
I
I
0
I
I
I
I
3
2'
L a r g e s t M 2 c u r r e n t s c a l c u l a t e d w i t h model 1 ( i n cm/s)
I
1
Fig.
1
I
'r
Largest M2
I 2
I I
I 0
I I
I
2
c u r r e n t s c a l c u l a t e d w i t h model 2
I 3
( i n cm/s).
I
-
51
-
50
381
I
Fig.
,
I
4'
I
1'
3
I
I
I
I
0
I
,
I
a
1'
25. L a r g e s t M 2 c u r r e n t s c a l c u l a t e d w i t h m o d e l 3 ( i n c m / s ) .
Tables
( 7 t o 10) show t h a t t h e d i f f e r e n c e s b e t w e e n t h e t h r e e m o d e l s
and t h e o b s e r v a t i o n s a r e r e a s o n a b l e
( e r r o r s l e s s t h a n 20 % ) .
However,
t h e p a r a m e t e r R - r a t i o between t h e s m a l l and t h e g r e a t a x i s of
-
M2 t i d a l e l l i p s e ( n e a r The H a v r e ) .
T h a t m i g h t be due t o t h e c l o s u r e o f
the Seine's
estuary.
TABLE 7
Amplitude of t h e M2 ellipse ( i n m/s). S T A T I ON
the
i s much l a r g e r t h a n t h a t o b s e r v e d a t s t a t i o n s 3
c u r r e n t a l o n g t h e g r e a t a x i s of
Observation
Model 1
Model 2
the tidal
Model 3
~~
3
0.51
0.54
0.54
0.57
4
1.71
1.62
1.79
1.93
6
1.20
1.33
1.36
1.42
8
1.20
1.16
1.14
1.19
382 TABLE 8
D i r e c t i o n of the North)
t h e g r e a t a x i s of
STATION
Observation
t h e M2
tidal ellipse
(relative to
-
Model
Model 2
1
Plodel 3
3
270°
282'
282O
260"
4
230'
239'
243O
2200
6
255"
278'
278"
258'
8
215O
2220
223'
2000
TABLE 9
R a t i o between
STATION
3
-
the
s m a l l and t h e g r e a t a x i s of
Observation
Model
t h e M2 t i d a l e l l i p s e
Model 2
1
Model
0.15
0.39
0.32
0.38
4
0.08
0.07
0.07
0.01
6
0.08
0.01
0.03
0.01
8
0.00
0.02
0.00
0.03
3
TABLE 10
D e l a y ( i n h o u r s ) b e t w e e n t h e t i m e o f maximum o f p a s s a g e o f t h e moon a t t h e G r e e n w i c h m e r i d i a n
STATION
-
Observation
3
lh.
4
Ih.
30 m i n
Model
c u r r e n t and t h e
Model 2
1
Model
3
Ih.
1 8 min
lh.
18 m i n
Ih.
34 min
Oh.
42 min
Oh.
42 min
Oh.
5 2 min
6
lh.
25 m i n
Ih.
1 8 min
Ih.
24 min
lh.
2 2 rnin
8
5h.
40 min
5h.
54 min
5h.
3 6 rnin
5h.
46 min
CONCLUSION
Even i f currents,
a model y i e l d s a good r e p r e s e n t a t i o n of
PI2
e l e v a t i o n s and
i t s a b i l i t y t o g i v e correct harmonics and subharmonics
(especially residual
currents)
of
?I2,
s t r o n g l y depends on t h e q u a l i t y
383 o f the discretization of the advection terms. To overcome this difficulty, one must (Nihoul
&
Ronday, 1976)
i) solve the transient motions by means of a simple model (model 1 or 2)
;
ii)average the transient equations ( 1 - 2
or 3-4) over T and s o l v e the
steady state resulting equations for the residual flow. In the averaged equations, the transient motions still appear in the non-linear terms producing the equivalent of an additional stress o n the mean motion.
This stress can be calculated explicitly using
the results o f the preliminary long wave equations, and the question o f numerical stability is obviously ignored in the calculation of this stress.
ACKNOWLEDGEMENTS
The author is indebted to Dr. Ch. Le Provost for his valuable advice during the course of this work. his appreciation to Mr.G.
He also wishes to express
Chabert d'Hieres for his constant encoura-
gement and most appreciated support so vital to a project of this nature.
Thanks are also due to Prof. J.C.J. Nihoul for computer time
facilities.
Support for this research has been provided by the
Centre National de la Recherche Scientifique - A.T.P.
Internationale
1976-1977, NO1563.
REFERENCES
Brettschneider, G., 1467. Anwendung des Hydrodynamisch-numerischen Verfahrens zur Ermittlung der M2-Mitschwingungsgezeit der Nordsee. Mittl. Inst. Meereskunde. Univ. Hamburg, 7:l-65. Bureau Hydrographique International, 1966. Pqarees - Constantes harmoniques. Monaco, Publication specidle, 26. Chabert d'HiS.res, G., & Le Provost, Ch., 1970. Etude des phenomenes non lindaires deriv6s de l'onde lunaire moyenne M2 dans la Manche. Cahiers Oceanographiques, 22:543-570. Durance, A , , 1975. A mathematical model of the residual circulation o f the Southern North Sea. Sixth Liege Coll. On Ocean Hydrodynamics, Mem. S O C . R. Sci., Liege, p p . 261-272. Fisher, G . , 1959. Ein numerisches Verfahrens zur Errechnung von Windstau und eezeiten in Randmeeren. Tellus, 9:60-76. Flather, R.A., 1976. A tidal model of the North-West euroDean continental shelf. Seventh Liege Coll. on Ocean Hydrodynamics, Mem. Soc. R. Sci. Liege, pp. 141-164. Hansen, W., 1966. The reproduction o f the motion in the sea means of hydrodynamical - Numerical methods. NATO Subcommittee on Oceanographics Research, Tech. Rep. 25:l-57. Hyacinthe, J.-L., & Kravtchenko, J., 1967. Modele mathematique des marees littorales. Calcul numerique sur l'exemple de la Manche.
384 La Houille Blanche, 6:639-650. Leendertsee, J.J., 1967. Aspects of a computational model for long period water-wave propagation. Ph. D. Dissertation, Technische Hogeschool Delft, 165 pp. Leprovost, Ch., 1976. Technical analysis of the structure of the tidal wave's spectrum in shallow water areas. Seventh Liege Coll. on Ocean Hydrodynamics, Mem. SOC. R. Sci. Liege, pp. 97-112. Nihoul, J.C.J. & Ronday, F.C., 1976. Hydrodynamic models of the North Sea. Seventh Liege Coll. on Ocean Hydrodynamics, Mem. SOC. R . Sci. Liege, pp. 61-96. Pingree, R.D., F, Maddock, L., 1977. Tidal residual in the English Channel. J. Mar. Biol. Ass. U.K., 57:339-354. Ramming, H.G., 1976. A nested North Sea model with fine resolution in shallow coastal areas. Seventh Liege Coll. On Ocean Hydrodynamics, Mem. SOC. R. Sci. Liege, pp.9-26. Ronday, F.C., 1972. Modele mathdmatique pour l'etude de la circulation de mardes en Mer du Nord. Marine Sciences Branch, Manscp. Rep. Ser. Ottawa, 29:l-42. Ronday, F.C., 1976. Modeles hydrodynamiques de la Pier du Nord. Ph. D. Dissertation, Universitd de Liege, 269 pp. Sager, G., 1975. Die Gezeitenstrdme im Englischen und Bristol-Kanal. Seewirtschaft, 7:247-248.
385
RECENT RESULTS FROM A STORM SURGE PREDICTION SCHEME FOR THE NORTH SEA
R.A. FLATHER Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead, U.K.
ABSTRACT During the last four years a new system for the prediction of storm surges in the North Sea has been under development at 10s Bidston. The scheme is based on the use of dynamical finite-difference models of the atmosphere and of the sea. The atmospheric model, the Bushby-Timpson 10-level model on a fine mesh, used in operational weather prediction at the British Meteorological Office, provides the essential forecasts of meteorological data which are then used in sea model calculations to compute the associated storm surge. The basic sea model, having a coarse mesh, covers the whole of the North West European Continental Shelf. Additional models of the North Sea and its Southern Bight, the eastern English Channel and the Thames Estuary with improved resolution are also under development. First real-time predictions were carried out early in 1978. This paper outlines the prediction scheme and presents some recent results.
INTRODUCTION
This paper deals with some aspects of the implementation of the storm surge prediction scheme, based on the use of dynamical finite difference models, proposed by Flather and Davies (1976). The essence of the scheme is to take data from numerical weather predictions carried out by the British Meteorological Office using a 10-level model of the atmosphere (Benwell, Gadd, Keers, Timpson and White, 1971), then to process the data in order to derive, in advance, the changing distribution of wind stress and gradients of atmospheric pressure over the sea surface. Subsequently a numerical sea model taking the processed data as input is used to compute the associated storm surge. The original scheme has undergone considerable development and improvement as a result of a series of experiments carried out in the last four years. The basic linear sea model covering the continental shelf has been replaced by a much-improved non-linear version capable of reproducing the tidal distribution with good accuracy (Flather, 1976a).
Tide and surge can now be calculated together taking account
386 of t h e important e f f e c t s a s s o c i a t e d with t h e i r i n t e r a c t i o n .
A second component
c o n s i s t i n g of a North Sea model with f i n e r s p a t i a l r e s o l u t i o n has a l s o been e s t a b l i s h e d and incorporated i n t h e scheme (Davies and F l a t h e r , 1977).
A t e s t of t h e
scheme with both sea models covering a continuous p e r i o d of 44 days i n November and December 1973 i s perhaps one of t h e l o n g e s t s u c c e s s f u l surge simulations y e t c a r r i e d o u t (Davies and F l a t h e r , 1978).
Other experiments of a p r a c t i c a l n a t u r e
i n which t h e p r e d i c t a b i l i t y of surges w a s examined l e d t o t h e design of a f i r s t o p e r a t i o n a l scheme g i v i n g p r e d i c t i o n s up t o about 30 hours i n advance ( F l a t h e r , 197633).
The procedure described here i s based on t h i s scheme.
The q u e s t i o n of how t o d e r i v e t h e b e s t p o s s i b l e e s t i m a t e of t h e meteorological f o r c e s on t h e sea from l i m i t e d atmospheric information i s of fundamental importance f o r surge p r e d i c t i o n .
Many a l t e r n a t i v e procedures e x i s t with varying degrees of
dynamical and empirical c o n t e n t (see f o r example Duun-Christensen Timmerman ( 1 9 7 5 ) ) .
(1975),
Some of t h e a l t e r n a t i v e s were compared f o r t h e storm surge
of 2nd t o 4 t h January 1976 ( F l a t h e r and Davies, 1978).
Since then t h e Meteorological
O f f i c e has been a b l e t o provide atmospheric p r e s s u r e , s u r f a c e wind and near-surface
a i r temperature i n s t e a d of t h e b a s i c dependent v a r i a b l e s ( t h e h e i g h t of t h e 1000 mb p r e s s u r e s u r f a c e , t h e 1000 mb wind i n components, and t h e t h i c k n e s s of t h e 1000900 mb l a y e r ) from t h e 10-level model.
These r e q u i r e modified procedures f o r
d e r i v i n g t h e meteorological f o r c e s , which a r e described here. The p l a n of t h e paper i s a s follows.
F i r s t , two s e c t i o n s g i v e an o u t l i n e of
t h e s e a model and t h e meteorological d a t a with a l t e r n a t i v e methods f o r processing
it i n t o t h e r e q u i r e d form.
These two i n g r e d i e n t s make up t h e p r e d i c t i o n scheme
a s described i n t h e s e c t i o n which then follows.
F i r s t real-time p r e d i c t i o n s using
t h e scheme were c a r r i e d o u t from 13th t o 17th February 1978, with a second sequence of f o r e c a s t s from 7 t h t o 15th March covering a period of s p r i n g t i d e s . a r e described i n t h e f o u r t h s e c t i o n .
These t e s t s
Since no s u b s t a n t i a l surges occurred during
t h e p e r i o d of real-time running, t h e accuracy of t h e p r e d i c t i o n scheme i s i l l u s t r a t e d f o r t h e c a s e of t h e storm surge of 11th and 12th January 1978: the most r e c e n t severe surge on t h e e a s t c o a s t of England. made and a r e compared with observations.
Four s e p a r a t e p r e d i c t i o n s were
F i n a l l y , a f i r s t comparison between
p r e d i c t i o n s f o r t h e high t i d e on t h e n i g h t of 11th and 12th January obtained from t h e p r e s e n t dynamical method and corresponding p r e d i c t i o n s obtained from a s t a t i s t i c a l procedure a r e presented.
F u r t h e r comparisons of t h i s kind w i l l be p o s s i b l e
during t h e 1978-79 storm surge season when t h e new dynamical scheme i s t o be operated on a r o u t i n e b a s i s a t t h e Meteorological O f f i c e , Bracknell (U.K.).
THE SEA MODEL
The hydrodynamical equations which c o n s t i t u t e t h e b a s i s of t h e sea model a r e
(v c o s @ ) - 2wsin $v R c o s @ 6+ = - A !1% 6Pa +_1 (F(s)R cOS $ 6x pR cos @ 6x pD
6u +""+
R c o s $ 6x
6t
v _6 __
p))
-
where t h e n o t a t i o n is: Xl$
e a s t - l o n g i t u d e and l a t i t u d e , r e s p e c t i v e l y
t
time
5
e l e v a t i o n of t h e sea s u r f a c e
u,v
components of t h e depth mean c u r r e n t ,G(')
F")
components of t h e wind stress ;(')
on the sea Surface
F ( B ),G(B) components o f t h e b o t t o m stress ;(B)
atmospheric p r e s s u r e on the sea s u r f a c e
'a D
t o t a l depth of water (=h+
if
Tc
sediment t r a n s p o r t
S
i f r < r
T = O
C
‘Ic= A
‘I
=
Vs-
m
W2 2
bottom shear s t r e s s
tT-
DM
I
(0,02 < A < 0 , 0 6 S h i e l d s ) . C r i t i c a l bottom shear stress
C
u, v are t h e two components of t h e depth averaged c u r r e n t
w2 = u2
+
v2
KT, ass p e c i f i c weight of water and sediment M mean diameter of sediment.
D
Influence of bottom e v o l u t i o n upon t h e c u r r e n t p a t t e r n
With t h e i n i t i a l bottom shape
5,
and t h e new geometric c o n d i t i o n s t h e depth
averaged flow p a t t e r n i s (uo, v o ) . T h i s c u r r e n t modifies t h e bottom shape which i n
.
t u r n modifies t h e c u r r e n t by (u, (t), v1 ( t ) )
455
At time t, the current pattern in given by (uo bottom level by E(t)
(5,
=
5 - 5,
+
u1 (t), vo
+
v1 (t)) and the
is the bottom evolution).
The resulting disturbance ( u l , v1) is assumed to be without effect upon the surface elevation zo. This assumption is equivalent to neglect the characteristic respnsetime of the surface wave propagation compared to the characteristic response time of the bottom evolution.
The resolution of the fluid continuity equation shows that the current disturbance (ul, vl) can be written in two different terms
:
- the first one ( G , , 31) comes directly from the bottom elevation 5 and expresses the flow conservation along the stream lines of the undisturbed field of currents (uo vo)
-
%
%
the second one (ul, v1) is a deviation of the flow due to the bottom slope. It is governed by
:
Bottom equation
These two terms are introduced in the bed continuity equation (1) which can be written
with C =
:
1 h
(u
2+
aT
v -)
av
Equation ( 2 ) governs a ripples propagation in the direction of the initial current pattern with the celerity C. This phenomena comes directly from the adaptation of
-
-
current disturbance (ul,v,). By neglecting the disturbance it is impossible to reproduce the ripples propagation.
456 The second member can be divided in two differents parts :
-
contribution of the initial current pattern which is conserved at time t 2
,
%
contribution of the deviation of the flow (ul, v,) which drives a ripple deformation.
Fluid equation
To determine the current disturbance (ul, vl) an other assumption is required irrotational current disturbance pattern
(61 +
2
,
-
u1, v1
2,
+ v1 )
:
an
2,
is assumed. So u1 and v1
are obtained from the three-dimensional stream function $, which yields a Poisson type equation ( 3 ) .
So the actual current pattern is defined by
:
2,
h = zo
- 5
actual depth and $ obtained from
NUMERICAL MODEL
A finite difference scheme is used to solve equations ( 2 ) and ( 3 ) . The computa-
tional grids $ and u , v,
5
are shifted. The initial conditions (uo, vo, z o ,
5,)
are
obtained with an other numerical model or recorded on a scale model.
Each time step involvatwo stages
-
computation of the bottom level
5
:
;
equation ( 2 ) is solved by the characteristic
method. All functions are explicited but the scheme is stable.
- Computation of the new velocities
2 , 2 ,
;
only ul, v1 have to be computed. Equation ( 3 )
is solved by an iterative process.
NUMERICAL EXAMPLES
Local scour around a jetty
Several numerical examples have been computed. In figures 1 and 2, the local scour around a jetty, and the flow pattern evolution are shown. The conditions are : flat
457
Fig.1 -EROSIONS AFTER 1 , 2 AND 3 HOURS
458
~
~
~~
F i g . 2 - CURRENT
P A T T E R N A F T E R 1,2 A N D 3 HOURS
EXPERIMENT
COM P U TAT I ON
Fig.3 ,COMPARISON
BETWEEN MEASURED
AND COMPUTED EROSIONS
460
.. ..-.. . . . ..... .. .. ..
Scale : 1 /25000 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
-41 . . .. .. . .. .. .. .. . .. .. . .
7
Numerical model
0
I
meters
...
F i g . &- EROSIONS
NEAR DUNKERQUE PORT
initial bottom, far field mean velocity = 41 cm/s, water depth = 20 cm, width = 46 cm, ratio jetty lenght over flume width = 1 / 3 and particle diameter 4,5 mm. The initial current pattern has been computed with an other numerical model. In figure 3 , comparison between computed and measured scour is shown.
Study of new port of Dunkerque
The Port Autonome of Dunkerque has built a new port able to receive 2 2 meters draught ships. Many studies have been carried on during ten years. Particularly, a movable bed model have been built to study the bottom evolution due to tidal currents near the new port.
The numerical model has been used in this particular case, but to decrease the cost of computation the second kind of disturbance has been neglected. Only equation (2) was solved. The initial current pattern used for the computation was recorded on the scale model.
The comparison between the computed and mesured erosions and accretions is presented on figures 4 and 5. The main difference takes place near the jetties and it probably comes from the initial current pattern which was not conservative because of the precision of measurements on the scale model.
CONCLUSION
A simple kinematical study of the sediment transport equation has shown how can the ripples propagation be obtained. It has also allowed a numerical integration on a computer. The characteristic response time of the surface wave propagation compared to the characteristic response time of the bottom evolution put a stop to any Sort of computation of the disturbed current in the classical way. The introduction of current disturbance and several assumption permits the computation of the bottom evolution during a long time.
This kinematical and mathematical aspect almost understood, studies are going on a more physical and dynamical point of view to determine the influence of the different parameters in transport relationship and to find a best dynamical approximation on the current disturbance. In the same time, a mean of averaging the tide in tidal problems is investigaded.
462
Fig. 5 - ACCRETIONS N E A R DUNKERQUE PORT
463 REFERENCES Daubert, A . , Lebreton, J.C., Marvaud, P., Ramette, M., 1966. Quelques aspects du Galcul du transport solide par charriage dans les ecoulements graduellement varies. Bulletin du CREG no 18. Zaghoul, N.A., Mc Corquodale, J.A., 1975. A stable numerical model for local scour. Journal of Hydraulic Research. Bonnefille, R. Essai de synthese des lois de debut d'entrainement des sediments sous l'action d'un courant en regime continu. Bulletin du CREG no 5. Lepetit, J.P., 1974. Nouvel avant-port de Dunkerque, etude sur modele rdduit sedimentoloqique d'ensemble de l'evolution des fonds au voisinage de l'avant-port. Rapport Electricit6 de France, Direction des Etudes et Recherches. Gill, M.A., 1972. Erosion of sand beds around spur-dikes. Journal of Hydraulic Division.
This Page Intentionally Left Blank
465
SECURITY OF COASTAL NUCLEAR POWER STATIONS IN RELATION WITH THE STATE OF THE SEA
J. BERNIER
, J.
MIQUEL
Laboratoire National d'Hydraulique, Chatou (France)
ABSTRACT
The safety of a coastal power plant is concerned with two phenomena : the wind waves, and the maximum and minimum tide levels. This paper presents methods of statistical analysis for estimating the probabilities of extreme events to be taken into account by the designer. First are recalled the definitions of these phenomena, in particular the relationships existing between the maximum of N waves and the significant wave. Then the case is approached where, because of little information available, the use of either meteorological data or uncommon events recorded in a far-off past is necessary. The paper concludes with an example of statistical study of storm durations.
INTRODUCTION
The figure below shows a vertical cross section of a power plant bordering on the sea :
Plant
Tranqoilliration
466 The rates of flow required for the power plant cooling is pumped in the tranquillization basin, which is protected against the waves by the dike. The tranquillization basin is communicating with the sea and its level is equal to that
of the tide. A maximum residual agitation of 30 cm in the basin is consistent with the operation of the pumping station. The designer needs following complementary information
:
1. Extreme wind waves probabilities, so that the stability of the dike may be ensured against centennial events at least. 2. Maximum and minimum tide level probabilities, so that protection may be ensured against the flood (maximum level) on the one hand, against failing
of the pumps on the other (minimum level). DEFINITION OF THE WIND WAVE TAKEN INTO ACCOUNT
Among the numerous statistical waves characteristics, the most frequently used for the dike design is the significant wave denoted by Hl13 (Average upper third of the greatest waves). This is the parameter that has been selected for the estimation of the wind waves risks. However, it should be indicated that many other parameters may be directly related to H
1/3
.'
1. Cartwright and Longuet-Higgins have demonstrated that in the case the wind waves follow the Gaussian model the following relation may be used
-
:
H113 = 1,6 H = 0,79 Hlllo. These results have been checked on some recordinqs (Miquel, 1975). 2. Utilizing the same assumption, Longuet-Higgins showed that the maximum of N waves is related to H1l3, and gave the expression of its mean value. Bernier, in an internal paper published at the "Laboratoire National d'Hydraulique", verified this expression. Besides, utilizing the results obtained by Cramer and Leadbetter (Cramer and Leadbetter, 1967) he could demonstrate that Hm(N) follows a law of extreme values, the mean and the standard deviation of which are :
12 log, N It appears then possible to evaluate the probabilities of H-(N)
from those
of Hi131 either directly by combining the probabilities of H113 with those of the extreme value distribution, or through simulation by reconstituting a fictitious sample in the following way
:
467 H
MAX
where
(N) = m (N) - U (N). p
[ 0,45
+ 0 , 7 8 loqe(- loge p) ]
is drawn in an uniform law on
] 0, 1 [ .
It is important to take into consideration not only the mean value but also the variability (figured by the standard deviation)
:
the neglect of this varia-
bility runs counter to safety. An exhaustive study of waves hazards should also take into account the periods. At the present, the couple (wave-period) is being studied in a frequential way (Allen, 1977) in order to assign a "probable" period to a given wave, the waves only being probabilized. Another important point, which is likely to be taken into account soon, is the storm duration
an incipient response is given farther.
:
DEFINITION OF TIDE LEVELS Definition 1 Observed maximum level -------------__-___ :
:
It is the level actually reached by
the sea. It will be denoted by HI. Definition 2 Predicted maximum level -----------------:
:
It is the level that the sea would
reach in the absence of atmospheric perturbation
:
it is determined by the posi-
tion of the stars (astronomic tide). In France, this level is computed by the "Service Hydrographique et Oceanographique de la Marine", by summing up the ampli tudes associated with different periods, the semi-diurnal amplitude being the principal one. This level will be denoted by H
0'
Definition 3 : Tide deviation : It is the positive or negative difference ------------_-_ between H I and Ho, mainly due to meteorological conditions (pressure, wind, temperature, etc
...)
It will be denoted by
S.
Predicted tide Time
468
Generally, S is estimated by the difference between the observed HI and the calculated Ho. It will be shown farther that S can be sometimes estimated from meteorological conditions : S (P, V, etc.). Hg and S being estimated, their values are somewhat uncertain. This uncertainty
-
-
should be allowed for in the probabilization. It can be written : H1 = Ho+ S +
E,
where (Eis the residue, the statistical characteristics of which must be given at the Same time as the estimates Ho and S . Everything said about the maxima levels can be symmetrically extended to the minima levels.
WIND WAVES PROBABILITIES
The sample
:
The sample of daily waves is established, namely by choosing for
each day, the surge H1,3(i)
the highest of the day. It should be made sure that
all periods of the year are equally represented in the sample, otherwise a seasonal study would be necessary. The monthly maxima method : For each month, the highest waves is selected from the sample above. The new sample {Hj}
is successively fitted to the Normal, LOT.
Normal, Extreme values distributions. The best of these fittings is chosen. Example
:
NORMAL
L06. NORMAL
EXTREME VALUES
Max. Monthly Wave
469 The “Renewal“ method
:
the shortcomings of the monthly maximum method lead us
to use a method, inspired by the study of the renewal process, which is used already for about ten years to study the rates of flow of rising rivers. Starting from the sample constituted above, the maximum wave each storm, provided that this wave
is selected in
is higher than a given threshold chosen
beforehand, and that two successive waves belong undeniably to two differing storms (independence)
:
woves
t
H’/3
1
lime
t
*
Let us take the month as a reference period. Then, two samples can be constructed :
{ Hj } is the {nk]is
set of the surges higher than the threshold,
the catalogue of the number nk of storms having exceeded the threshold in the course of the kth month.
The calculation o f the monthly probability of exceeding a value h, namely the probability of the monthly maximum H* exceeding the value h, is carried out as follows Prob Prob
Prob
:
[ H* > h]= [ H* 6 h]= +
[ H* 6
h
1
-
Prob[2
0 storm
Prob[3
1 storm
+
Prob[gr
]
=
[ H* 6 h]
Prob
>/ threshold in the course Of a month] >/ threshold and d h
storms% threshold and
+aJ
6 h]
Prob [ 3 k storms 2 threshold and
thrt?shold])
470 +W
Prob [H*
>h ]= 1 -
1
P(k) .Fk(h)
K=O
f
where
-l P(k)
is the probability of having k storms in the course of the month,
F(h)
is the probability of a storm, higher than the threshold, beino lower than or equal to h.
If h is great enough, F(h) is near to 1 and this result can be simplified to
-
> h]zl
Prob [H*
+oo
1
P(k)
:
[ 1 + k(l - F(h))}
K=O Prob [H*
for
{
> h]”,
+m
1
n
=
-
+OD
P(k) = 1 and
1
P(k). k = n
K=O
K=O
-
(1 - F(h))
monthly average number of storms.
From the practical point of view, the nk catalogue enables P(k) or determined for the utilisation of the simplified formula two laws is used
;
n
to be
one of the following
:
Poisson’s law : P(k) = e
- A & k! k
Negative Binomial law : P(k) = k!
r(Y)
Since P(k) may considerably vary according to the month, it would be preferable, when sufficient information is available, to take as a reference period the year instead of the month. The probability F(h) is determined by the sample of the H to which are fitted the followinq laws :
i‘
471
This method has the advantage of utilizing the maximum amount of information, while warranting its homogeneity. It is possible and desirable to calculate the intervals of confidence.
T I D E LEVEL PROBABILITIES
The Observed Maximum Level
:
the most simple way is, like in the case of waves
to -~ constitute the sample of daily maximum levels of the hiqh water. Then, the san
methods are used as for the wave. The result is presented in the following form
:
472
PROBABILITIES
OF HIGH WATERS IN DIEPPE
-
I
- 0,lO -
LOW WATER
- 0,20 -
- 0,30 - 0,40 - q50,
----I
I
I
I
1
Return
I
I l l
Period
1
1
1
I
1 1 ,
( i n years)
However, the question may arise of whether it will be safe to use only one statistical law for explaining the behaviour of a variable, which is made up of two phenomena entirely different : the astronomic tide and the tide deviation due to meteorological conditions. We decided therefore to study also these two phenomena. The Predicted Maximum Level
:
In fact, it‘s a question of a random pseudova-
riable easy to probabilize either by constitutinq directly a catalogue of predicted heights, or by using the estimates based on the semi-diurnal amplitude. The two methods can be compared in the figure below.
413
FREQUENCIES OF PREDICTED LEVEL IN DIEPPE
% Frequencies of overstepping
700
8,OO
9,m
Z level
l0,OO ( in meters)
-
474 Tide differences
:
First, we constitute the catalogue of daily tide differences
obtained either by means of differences Ho - H1 on a series of observed tides, or by reconstitution from meteorological conditions ( s e e farther). Then, we proceed to the same probabilistic study as for the waves. The Sum of predicted levels and tide differences
:
We have H1 = H
0
+ S. If
Ho
and S are independent, the probability of their sum can be easily calculated by writinq Prob
:
+W
jw
[ H1 >
hl] =
f
[x]=
G
[y]=Prob
where
G
[hl
-
x].
[ x < H,- < x +
Prob
f [x]. dx
dx
]
[S>y]
For our part, we found that if the coefficient of correlation between Ho and S could attain 0,3 during slight or medium storms, this coefficient is practically
zero for heavy storms by which we are particularly concerned. This result is only indicative as it corresponds to a particular case and deserves to be tested on other sites.
If the correlation is no more zero but if there exists a relationship of the kind S =
A
Ho
+ S', where
Ho and S' are independent, we can get again to the pre-
vious case by considering the independent variables
:
h1
(1 +
Ho and S ' .
The figure below enables the two methods for estimating the HI level to be compared by studying directly H 1 or by studying the sum
OF TIDE LEVELS
PROBABILITIES
IN
€3 0
DIEPPE
+
S.
-
Level (in meters )
10,90
10,ao
1420
. 1
2
3
4
5
Return
10
Period
20 30 4050 ( in y e a r s )
100
475
For the design, we take the extreme limits of these estimates to which we add confidence intervals at 7 0 % .
CASE OF POOR INFORMATION
Wave data and tide data are frequently very short, rendering the statistical estimates too uncertain : additional information should then be used. Sometimes, it is fortunate to find another wave or tide series in the vicinity of the studied site. If the two series are closely related, the probability estimates of the long series can be easily transposed to the short one. If this is not the case, it is necessary then to consider other possibilities. Utilization of the meteorology
:
in the case where information, such as pres-
sure, wind, temperature in the vicinity of the site, is
available, it is possible
to establish a relationship between these data and the surqes or the tide fluctuations. As a test we tried multiple linear reqressions of the kind
n s
1/3
=
= g
where
f (P, (P,
2
v, v ,
v, v2,
i
nO,Ap,Av, ...I
no,Ap,
P
. . .)
= temporary pressure variation
T
= temperature
V
= wind speed
A
AV,
= pressure
P
h
T,
T,
:
V
= temporary variation of the wind
Although the results are not yet exploitable for high events, they are incentive for low and medium events in so far as the obtained multiple correlation coefficients reached 0 , 8 to 0,9.
Using these relationships, we reconstructed a
fictitious sample of tide differences over a long period of time and we estimated then the probabilities resultinq from this sample. On the figure below, the obtained results can be compared with respect to the probabilities derived from observations
:
416
PROBABILITIES OF TIDE DEVIATIONS IN LE HAVRE
3000
2000
1000
-
-
Return Period (in Days)
From observations within 10 years
-
500 400
300 200
100
50
40 30
-
From meteorology
--
From observations w i t h i n 1 years
Tide Deviation ( in
20
30
40
50
60
70
80
90
100
cm
)
110
In this figure we can see that there is an acceptable compatibility between the estimates for return periods lower than 10 years. Beyond these periods, it will be necessary either to improve the statistical relationships between the meteorology and the sea states or to use mathematical prediction models. Utilization of exceptional events
:
it happens that there exist recorded data
on one or more exceptional events for which an estimate can be fixed, and which are known to be the highest within a long period of time (for instance, a century). This information is precious and may be utilized, thouqh it greatly differs from a complete catalogue of waves or tides. It allows the statistical uncertainty to be reduced and the representativity of the used sample to be proved. The detailed description of this method can be found in the references (Bernier and Miquel, 1977). It was already applied successfully to flood risk estimations
:
477
FLOOD PROBABILITIES AT HAUCONCOURT (MOSELLE 1
Return Period 1000
500 200 100
50 20
10 5
2 1
STORM DURATIONS
Recent works on random waves showed how the storm duration may affect the lifetime of dikes. Using once more the techniques applied to the study of river flow rates (Miquel and Phien BOU Pha, 1978) we can estimate, for instance, the duration probability of a storm exceeding a given surge threshold. The probabilities of the yearly sums
of storm durations can be read in the figure below
:
478
Durations
EXCEEDING A GIVEN WAVE THRESHOLD IN LE HAVRE : TOTAL ANNUAL SUMS -
Thus, in decennial year, the total
Return
duration, sum
Period
over
100 years
the year of storms exceeding the surge level of 3,5 m in Le Havre
1
about 10 days.
0
1
2
3
4
5
6
7
8
9
Wave Threshold ( in meters) A curve of the same kind can be obtained,to describe the durations of individual storms. Indeed, such curves will be useful to designers when they will be able to take simultaneously into account both, the storm durations and their intensities.
REFERENCES
Allen, H., 1977. Analyse statistique des mesures de houle en differents sites du littoral franqais. Edition no 3, rapport EDF HE 46/77.01. Chatou (France) Bernier, J., Miquel, J., 1977. Exemple d'application de la theorie de la decision statistique au dimensionnement d'ouvraqe hydraulique : prise en compte de l'information het6rogGne. A.I.R.H. Baden. Cramer, Leadbetter, 1967. Stationary and related stochastic processes. Sample function properties and their applications. John Wiley. New York.
479 Miquel, J., 1975. Role et importance d'un modile statistique de la houle en vue du depouillement et du stockaqe des donnees. A.I.R.H. Sao Paulo. Miquel, J., Phien Bou Pha, B., 1977, Tempetiage : un modile d'estimation des risques d'etiage. Xime Journee de 1'Hydraulique. Toulouse.
This Page Intentionally Left Blank
481
SUBJECT INDEX Aberdeen, 3 9 9 . Accretion, 4 6 2 . Acoustic propagation, 4 3 9 ,
441,
451
Adriatic Sea, 4 2 8 - 4 3 6 . A.D.S.
Program (Anomaly Dynamics Study), 5 8 .
Advection, see also Currents, 1 6 9 , 273,
274,
280,
351,
354,
170, 242, 246, 251, 270, 271, 360, 363, 378, 383, 441, 443.
355,
Air-sea interaction, 3 5 , 3 6 , 3 8 , 6 1 , 4 2 4 , 4 2 5 . - Surface heat flux, 1 0 , 2 8 - 3 1 , 3 7 , 4 1 , 4 2 , 4 4 , 5 0 . - Air-sea interface, 6 , 1 5 , 1 8 , 2 3 , 2 7 , 3 5 , 4 1 , 6 2 , 1 0 1 , - Surface stress, 2 0 , 2 1 , 1 1 8 , 2 4 5 , 2 4 9 , 3 3 6 , 3 4 2 , 4 4 9 . - Air-sea temperature difference, 7 6 , 2 0 0 .
141.
Aleutian, 3 8 . Alps, 4 3 1 ,
436.
Amphidromic point, 2 3 7 , Anemometer, 2 2 3 , Annapolis, 3 2 7 ,
305, 328,
248,
311,
337,
430.
312.
330.
Antarctic circumpolar current, 6 1 . Apalachicola Bay, 2 6 8 . Atlantic Ocean, 4 3 ,
71,
72,
78,
85,
102,
285,
431.
Atmosphere - Atmospheric boundary layer, 7 , 4 5 . - Atmospheric circulation, 3 5 , 3 6 , 61. - Atmospheric data, see also Meteorological data, 6 3 , 3 9 8 , 4 0 8 . - Atmospheric frequency wave number spectrum, 6 3 , 6 4 , 6 7 , 6 8 . - Atmospheric pressure gradient, 2 3 7 , 2 4 1 , 3 0 3 . - Atmospheric stability, 1 4 2 , 1 5 4 , 1 5 5 , 2 0 0 . - Air temperature, 4 8 , 6 4 , 6 5 , 7 6 , 1 5 4 , 2 2 3 , 3 9 1 . Autocorrelation function, 2 0 2 , Avonmouth, 2 8 5 , Baginbun, 3 0 2 ,
287, 303,
203,
206,
398. 305.
Baltic Sea, 2 2 . Baroclinic - Baroclinic motion, 7 0 , 3 3 5 . - Baroclinic Rossby waves, 5 7 - 6 0 , - Baroclinic shear modes, 5 9 . Barotropic - Barotropic motion, 7 0 , 1 6 7 , 3 3 5 , - Barotropic Rossby waves, 6 6 . - Barotropic Shear modes, 5 9 . Bathymetry, 1 8 5 ,
337.
Bathythermographic data, 5 7 ,
58.
66.
336.
208,
209,
211,
212,
214.
482
Battjes criterion, 1 8 5 . Belfast, 3 0 1 , 3 0 2 , 3 0 4 , - Belfast Lough, 3 0 5 . Belgian coast, 4 1 1 , Belle River, 1 9 7 , Bise, 2 2 6 ,
227,
305,
412,
307,
313.
420.
199.
229,
230,
232.
Boltzmann distribution, 1 2 8 . Boltzmann integrals, 1 3 2 . Bottom - Bottom characteristics, 1 8 9 , 1 9 3 , 4 4 1 , 4 4 7 , 4 4 8 , 4 5 3 . - Bottom evolution, 4 5 5 , 4 6 1 . - Bottom friction, Bottom stress, Bottom turbulence, 1 8 4 , 236, 435,
238-249, 449.
251,
257,
258,
268,
Boussinesq approximation, 2 3 5 ,
236.
Bristol Channel, 2 8 5 ,
270,
297,
335,
336,
190, 193, 351, 387,
286.
British Isles, see also English Coast, Ireland, Scotland, 2 8 5 , 294,
295,
301,
318,
398,
287,
413.
Bowen ratio, 44. Buoyancy, 1 8 , 2 3 6 , 2 3 9 , 2 4 6 . - Buoyancy balance, 1 6 3 . - Buoyancy flux, 5 9 , 6 2 - 6 7 . - Brunt-Vaisala frequency, 1 2 . Biisum, 3 9 8 . Calabria, 4 4 2 ,
443.
California, 5 7 . Canadian Maritime Provinces, 7 2 . Cap de Barfleur, 3 6 0 . Cap de La Hague, 3 6 0 ,
367,
368,
379.
Cape Cod, 3 2 5 . Cape Hatteras, 3 2 5 . Celtic Sea, 2 8 5 - 2 8 7 . 356,
360,
363,
364.
Chesapeake Bay, 3 2 4 ,
Cherbourg, 1 6 5 ,
325,
328,
331.
Chezy coefficient, 1 8 9 , Civitavecchia, 4 4 3 ,
194,
196,
268,
454.
445.
Climate - Changes, 3 5 , 3 6 , 4 4 , 2 4 8 . - Predictability, Forecast, 3 5 - 3 7 , - Record, 3 9 .
53,
62.
Cloudiness, 4 5 Coriolis parameter, Coriolis acceleration, Coriolis force, see also Earth rotation, 4 2 , 1 1 9 , 1 6 7 , 1 8 4 , 2 2 0 , 2 2 2 , 2 3 5 , 2 3 9 , 2 4 6 , 2 6 8 , 270-273,
280,
297,
336,
352,
355.
Corsica, 4 4 1 . Cross correlation function, 2 0 4 ,
206,
210-214
483
Current, see a l s o Oceanic c u r r e n t - Bottom c u r r e n t , 4 4 5 , 4 5 4 - Current e l l i p s e , 337, 338. - F o u r i e r a n a l y s i s of c u r r e n t s , 377. - Current g e n e r a t i o n , 102. - G e o s t r o p h i c c u r r e n t , G e o s t r o p h i c v e l o c i t y , 63. 237, 230, 241. - ~ n e r t i a lc u r r e n t , 1 1 9 . - I r r o t a t i o n a l c u r r e n t d i s t u r b a n c e , 456. - Long s h o r e c u r r e n t , 1 8 3 , 1 8 6 , 1 8 7 , 1 9 2 , 1 9 4 , 1 9 5 , 4 4 1 , 4 4 4 , 4 4 6 . - D r i f t c u r r e n t , 1, 2 , 1 2 , 1 7 , 1 0 1 , 1 0 2 , 1 1 5 . - Mean c u r r e n t , 3 8 7 . C u r r e n t p r e d i c t i o n , 222, 228. - Current p r o f i l e , 245, 251-254, 321, 333, 338, 411, 458, 461. Residual c u r r e n t , Residual c i r c u l a t i o n , 351. - Current spectrum, 62, 445, 446. - Wind i n d u c e d c u r r e n t , 1 8 3 , 2 2 0 , 2 3 5 , 4 4 1 . T i d a l c u r r e n t , see T i d a l . Wave c u r r e n t , s e e Wave.
-
-
Currentmeter, 68,
172,
223,
224,
431,
436,
443.
238,
246,
250,
333,
340,
345,
441,
443.
Cyclogenesis, Cyclone,
43,
Denmark,
294,
Devon P o r t , Dieppe,
65,
360,
364,
301-307,
310.
352,
141, 449.
Driving s t r e s s , Drying banks, Dublin,
398,
353,
158,
460,
Earth rotation,
473.
412,
424,
425.
200,
249,
268,
304,
352,
363,
391,
392,
14, 403,
308,
314.
462. 91,
96,
386,
400,
s e e a l s o C o r i o l i s , 4,
406, 23,
408. 24,
27,
235,
239,
268,
435.
Eddy, s e e a l s o T u r b u l e n c e , 6 1 , 1 1 9 . - Eddy d i f f u s i o n , 2 3 6 , 2 3 9 , 2 4 1 . - Eddy d i f f u s i v i t y , 4 4 7 . - Mesoscale eddy, 43, 52. - Quasi-geostrophic eddy, 43. - Eddy e n e r g y , 4 3 , 1 6 6 , 1 6 9 . - Eddy n o i s e , 4 3 . - Eddy s t r e s s , 4 4 7 . - Eddy v e l o c i t y , 4 7 . - Eddy v i s c o s i t y , 1 8 4 , 1 8 8 , 1 8 9 , 1 9 3 , 236,
328.
186.
E a s t Anglian Coast, 387,
325,
334.
301-305,
Dunkerque,
323,
see a l s o B o t t o m f r i c t i o n , C h e z y c o e f f i c i e n t ,
Drag c o e f f i c i e n t , 114, 413,
321,
318.
Dover S t r a i t s , 105, 408,
294,
413.
356,
Douglas,
44, 63,
239,
Efimova's
242-247,
250,
f o r m u l a , 45.
Ekman - Ekman d e p t h , 4 2 . - Ekman d i a g r a m , 2 5 7 .
336.
194,
196,
220,
222,
228,
229,
484
Ekman - Ekman equation, 2 3 7 , 2 3 8 , - Ekman spirals, 2 2 8 , 2 2 9 . - Ekman transport, 4 2 , 3 2 5 , Elba, 4 4 1 ,
242-245. 331.
443.
Elbe estuary, 2 6 3 ,
264,
338,
345.
Energy - Energy source, 9 . - Energy balance, 7 , 9, 2 7 , 4 0 , 7 9 , 8 0 , 1 6 8 , - Energy dissipation, 9 , 2 7 . - Energy exchange, 7 9 . - Diffused energy flux, 2 4 , 8 6 , 1 6 6 , 1 8 0 . - Energy radiation, 4 3 0 . - Energy spectrum, 6 8 , 1 2 7 , 1 2 8 , 1 3 5 , 1 6 3 . English Coast, 4 1 2 ,
170,
180.
421.
English Channel, 3 5 1 - 3 5 4 ,
359,
367,
400,
403,
412,
413,
416,
421,
450.
Entrainment, 4 4 , 4 9 . - Entrainment heat flux, 4 4 , 4 7 . - Purely diffusive entrainment, 1 9 . - Entrainment layer, 1 3 . - Entrainment process, 2 , 1 6 , 2 0 . - Entrainment rate, 1 2 , 2 2 , 2 4 . - River entrainment, 8 . - Entrainment velocity, 1 6 , 2 0 , 2 1 . - Critical entrainment velocity, 2 8 . Erosion, 3 2 3 ,
457,
459,460.
Eulerian and Lagrangian reference frames, 1 1 4 , Europe, 3 8 ,
394,
116,
124.
413.
Falling leaf oscillation, 1 2 3 . Feedback
,
36,
37,
40-52,
Fishguard, 3 0 1 - 3 0 8 , Forcing, 4 7 ,
204,
246,
331.
314-317.
59, 391,
60, 69, 110, 169, 241, 242, 245, 270, 328, 331, 341, 352, 429, 445, 449-451. - Atmospheric forcing, 4 8 , 6 2 , 6 7 , 6 8 , 7 0 , 1 1 0 , 2 3 7 , 2 6 0 , 3 1 7 , 3 1 8 , 323, 325, 387, 394, 396, 400, 411. - Isotropic forcing, 4 8 . - Stochastic forcing, 5 2 , 6 2 , 6 9 .
Forecast, 1 9 7 , 2 0 2 , 432,
244,
246,
394-407,
411,
413,
416,
418,
420,
424,
433.
Foreland, 3 5 3 . Free surface, see also Sea surface, ocean surface, Air-sea interface, Water surface, 2 , 1 0 , 1 3 . Froude similitude, 1 9 2 . Genova, 4 4 3 ,
445,
German Bight, 3 3 3 ,
450, 334,
451. 337-342,
345-349,
Germany, 3 9 8 . Gould Island, 1 7 8 ,
179.
Gravity waves, 6 2 ,
114,
116,
127,
167.
391,
398.
485 71,
Great Lakes,
327,
Grey P o i n t ,
330. 116,
72,
Gulf of Mexico,
48,
193,
141,
154,
185.
78,
273,
278-283.
273. 124.
Gyroscopic s t a b i l i t y ,
172.
Hall e f f e c t ,
(Ontario) , 143.
356,
Hastings,
Hautconcourt,
360,
364.
477. 325,
Havre d e G r a c e ,
Hawai,
130,
328.
Gulf of Maine,
Hamilton
128,
367.
Guernesey,
38,
197.
115,
Group v e l o c i t y ,
Gyre,
72,
327,
328,
330.
57.
Heat - H e a t exchange, 43, 45. - H e a t f l u x , 41, 42, 44, 47, 50, - T u r b u l e n t h e a t f l u x , 45, 46. - Heat t r a n s p o r t , 61. Heavyside Heysham,
313,
Hindcast,
71-73,
Holyhead,
301-308,
Holland,
77,
314-317,
88,
127,
398.
129,
131-135,
138,
398.
314.
88,
s e e a l s o Storm,
294,
129,
268-274,
282.
318.
399,
Immigham,
401,
403,
s t a b i l i t y of
406,
235,
Inertial oscillations, Inertial
398.
398.
Hurricane, Iceland,
316-318,
301-308,
Hilbre Island,
63.
242.
step function,
301-301,
61,
421,
422.
239.
sinking bodies,
114,
121-124.
399.
I n n e r Dowsing,
I n t e r n a l waves, 2, 7, 12, 1 3 , 1 5 . I n t e r n a l Rossby waves, 5 9 , 6 0 .
-
Ireland,
285,
I r i s h Sea, Italy,
436,
441,
Jade Estuary, JONSWAP,
302,
244,
400.
285-289,
294,
451.
138.
K i n e t i c energy e l l i p s o i d ,
L a k e LGman,
Bay,
220,
Lake O n t a r i o ,
301-308,
338.
128-134,
Kiptopeake
296,
325, 221,
141,
327, 224,
197.
120,
121.
328,
330.
230,
231.
316-318,
400,
403
486
Lake Saint Clair, 1 9 7 - 2 0 0 ,
216.
Langmuir circulation, Langmuir vortices, 3 1 , Lausanne, 2 2 6 ,
32.
227.
Lax-Wendroff scheme, 8 3 ,
85,
86,
88.
Layer, see Atmospheric boundary layer, Entrainment layer, Mixed layer, Oceanic surface layer, Shear layer, Turbulent entrainment layer, Velocity shear layer, Wind mixed layer. Le Havre, 3 5 6 ,
360,
363,
364,
368,
381,
478.
LiQge, 4 1 6 . Ligurian Sea, 4 3 1 ,
440,
Liverpool, 2 8 5 - 2 9 4 , Livorno, 4 4 5 ,
448-451.
301-318.
450,
451.
London, 4 0 0 . Long Island, 1 0 2 , Lowestoft, 3 9 9 ,
103.
401,
403,
421,
422
Low frequency processes, 5 7 - 6 0 . Lyme Regis, 3 5 6 ,
360,
364,
368.
Markov process, 4 0 . Maximum likelyhood method, 2 0 4 , Mediterranean Sea, 4 2 9 ,
430,
205.
435,
436,
449,
451.
Mersey Estuary, 3 0 5 . Mesoscale phenomena, 2 3 5 - 2 3 8 ,
240,
Mixed layer, 2 - 1 1 , 44-47,
50,
- Mixed layer - Mixed layer - Mixed layer
-
15, 16, 18, 20, 61, 67, 162, 163, deepening, 7 , 9 , 1 4 , depth, 4 0 , 4 1 , 4 3 . depth anomaly, 4 2 . model, 4 0 . temperature, 4 1 .
59,
Mixed layer Mixed layer
Mobile Bay, 2 6 1 ,
262,
271-274,
246,
248.
23-27, 31, 32, 166, 181, 239, 16, 27.
35, 36, 39-41, 240, 443.
278-283.
Metacentric height, 1 2 0 - 1 2 4 . Meteorological data, 4 , 420,
424,
436,
441,
Mississipi Sound, 2 7 3 ,
62, 70, 467, 468,
73, 342, 389, 474, 475.
392,
395,
416,
418,
278-283.
Model - Analytical model, 2 4 2 , 2 4 9 , 2 5 1 . - Combined atmospheric-oceanographic-physical model, 3 2 1 . - Depth averaged tidal model, 3 5 2 . - Diagnostic model, 1 2 9 . - Finite difference calculation model, 1 4 1 , 2 6 1 , 2 6 4 , 2 7 0 ,
301,
352,
Irregular grid finite difference model, 2 6 1 , 2 6 3 , 2 7 0 , 2 7 3 . Finite element technique model, 2 6 1 , 2 6 4 . Gaussian model, 4 6 6 . One-dimensional model, 2 3 7 , 2 4 0 , 2 4 6 , 2 4 8 , 2 5 0 . - Two-dimensional model, 1 8 3 , 2 3 7 , 2 3 8 , 2 4 1 - 2 4 8 , 2 5 0 , 2 5 4 , 2 5 5 ,
296,
-
385,
318,
389,
338,
391,
341,
408,
342,
412,
453.
414,
436,
456.
487
Mode 1 - Three-dimensional time dependent model, 2 3 6 , 250,
-
-
-
258,
334. Multi-mode model, 2 4 3 - 2 5 1 . Multi-layer model, 2 4 0 , 2 4 4 . Prognostic model, 1 3 0 . SPLASH model, 2 6 8 , 2 6 9 . Thermocline model, 2 4 2 . Transfer production model, 2 0 0 , 2 0 1 , 2 0 4 - 2 0 6 , Vertical shear model, 2 4 1 . General circulation model, 3 6 , 4 5 , 4 6 , 5 3 .
MODE, 4 3 ,
239,
241,
245,
246,
333,
210,
211
47.
Molecular diffusion, Molecular transfer, 2 , Momen tum - Momentum balance, 1 4 1 . - Momentum flux, 1 4 1 , 1 5 2 , 1 6 6 , 1 6 8 , - Momentum transfer, 1 4 1 , 1 4 2 , 1 5 8 .
170,
7,
12.
180.
Monin -0bukhov length scale, 2 8 . Nab Tower, 3 5 6 ,
360,
363,
Narragansett Bay, 1 6 9 ,
364.
178,
179.
New England, 3 2 8 . New Foundland, 3 8 . New Haven, 3 5 6 ,
360,
364.
Niagara River, 1 9 7 . NORPAX, 5 7 ,
60.
NORSWAM, 1 3 5 . North America, 3 8 ,
72,
73,
North Atlantic Ocean, 4 3 , North Channel, 2 8 6 ,
287,
75,
197.
72. 297,
301,
318.
North-East Pacific Ocean, 4 3 . North Pacific Current, 5 8 . North Pacific Ocean, 3 8 ,
47,
48,
50,
52,
57-60,
66-69.
North Pacific gyre, 3 8 . North Sea, 7 1 , 321,
323,
91-94, 334, 338,
North Shields, 3 9 6 ,
114, 341,
399,
120, 342,
401,
127, 385,
403,
132, 389,
238, 398,
245-250, 294, 400, 411-416,
420-422.
Nova Scotia, 3 2 5 . Nyquist wavelength or frequency, 50, 1 5 1 ,
173.
Ocean - Ocean current, 1 0 2 , 1 0 3 , 1 0 5 , 1 0 7 , 1 1 0 , 111. - Oceanic circulation, 50. - Oceanic cycle, 4 3 . - Ocean surface heating, see also Surface heat flux, 2 7 . - Oceanic surface layer, 1, 5 3 . - Oceanic variables frequency wave number spectrum, 6 7 - 7 0 . - Ocean Weathership P., 4 3 , 4 4 , 4 8 - 5 1 . - Ocean Weather Station D., 6 4 , 6 5 . Oil mixing, 1 6 2 - 1 6 5 ,
170,
180,
181.
295, 424.
488
Olbia, 4 4 3 ,
445.
Open sea boundary, 2 3 6 . Ostend, 4 2 0 ,
421,
424.
Otranto Channel, 4 2 9 , Pacific Ocean, 3 8 , Patuxent, 3 2 7 ,
436.
43,
47-50,
52,
71,
72,
78.
330.
Peclet number, 3 ,
12.
11,
Permanent directions of translation, 1 2 0 , Phase speed, 1 2 8 , Pisa, 4 4 3 ,
141,
147,
149,
150,
121.
154,
185,
262,
450.
445.
Platform (Oil production), 1 1 3 ,
114,
117,
120,
124.
Poisson's law, 4 7 0 . Poisson's type equation, 4 5 6 . Pollution problems, 1 6 1 - 1 7 0 , Ponza, 4 4 3 ,
194,
342.
445.
Port Patrick, 3 0 1 - 3 0 7 , Power plant, 4 6 5 ,
313,
315,
317.
466.
Quasi-hydrostatic approximation, 2 3 5 . Quasi-geostrophic dynamics, 6 7 , Radiation stress, 1 1 5 ,
235.
Reflexion effect, 1 1 7 ,
118,
Refraction effect, 9 1 - 9 3 ,
156,
96,
68.
157.
185,
187.
Reynolds averaging, 4 0 . Reynolds stress, 1 4 2 ,
144,
Richardson number, 2 ,
3,
Righting arm, 1 1 4 ,
120,
294,
163,
27-29,
169-170,
142,
144,
181. 150,
122.
Ripples propagation, 4 5 5 , Ronaldsway,
150-152,
10-22,
461.
298-302.
Rossby-Montgomery formula, 2 7 . Rossby number, 2 2 0 ,
237.
Rossby waves, 5 8 - 6 0 ,
65,
67.
Rugosity length, see also Bottom friction, 2 4 7 , Sable Island, 7 6 ,
78,
392.
Saint Malo, 3 6 4 . Saint Servan, 3 5 6 ,
360,
364.
Saint-Venant equations, 1 8 6 . Salcombe, 3 5 6 ,
360,
364.
Salinity - Salinity balance, 4 4 7 . - Salinity gradient, 1 2 , 4 3 9 . - Salinity-temperature-depth diagrams, 1 0 2 , San Francisco, 5 8 .
'
441.
251.
154,
156,
246.
489
Scheld Estuary, 4 1 1 . Schmidt trigger, 1 7 2 . Scotland, 2 9 4 ,
302,
398.
Sea state, 5 7 ,
101,
127,
-
129,
162,
170,
175,
181, 394,
418,
Sea state forecast, 5 7 , 7 1 , 7 2 , 8 8 , 1 0 0 . Sea state generation, 5 7 .
Sea surface, 3 ,
-
131,
476.
163, 445.
167,
9, 13, 39, 173, 220, 229,
59, 62, 77, 93, 94, 297, 318, 349, 385,
101, 104, 119, 161387, 391, 413, 435,
Sea surface elevation, Sea surface s l o p e , 1 5 2 ,
170, 183-186, 188, 189, 237, 239, 242, 245-248, 261, 262, 265-271, 278, 280, 282, 285-291, 303, 305, 309-311, 317, 318, 321, 325, 327, 333, 336338, 342, 352, 387, 394, 411, 420, 432, 433, 436, 441, 446, 447, 451. - Sea surface temperature, 3 5 , 3 8 , 4 9 , 5 9 , 7 6 , 1 5 4 , 3 9 1 , 4 4 1 . - Sea surface temperature anomaly, 3 5 - 5 3 , 6 1 . - Sea surface temperature anomaly dynamics, 4 7 . - Sea surface temperature anomaly generation and decay, 4 0 , 4 2 . - Sea surface temperature anomaly rate of change, 3 6 , 4 7 . - S e a surface temperature anomaly spectrum, 4 0 , 4 8 . - Sea surface temperature anomaly time scale, 4 5 . - Sea surface temperature field, 3 6 , 5 2 . - Sea surface temperature fluctuation, 3 7 , 4 0 . - Sea surface temperature frequency wave number spectrum, 5 9 . - Sea surface temperature gradient, 5 0 . - Sea surface temperature prediction, 5 2 . Sea surface temperature variance, 4 8 .
-
Sediment transport, 2 3 8 , Seiches, 4 2 9 - 4 3 1 ,
435,
246,
453,
454,
461.
450.
Seine Estuary, 3 8 1 . Shear effect diffusion, 2 4 6 . Shear layer, 2 . Shear stress, 3 3 6 , 4 5 4 . Shelf (Continental shelf), 7 2 ,
285,
385,
388,
Shinnecock Inlet, 1 0 4 . Shoaling effects, 9 1 , 9 3 , 9 6 ,
184.
Sicily, 4 4 2 . Sicily (Straits of Sicily), 4 4 9 . Saint-Georges Channel, 2 8 6 , Slab model, 5 ,
287,
297,
318.
36.
Smithometer, 1 7 3 ,
174,
178,
181.
Solar radiation, 4 2 . Solomon Island, 3 2 7 ,
330.
Sound speed, 4 3 9 , 4 5 1 . Southampton, 3 6 3 . Southend, 3 9 6 ,
399-403,
Southern Bight, 2 4 7 - 2 5 0 ,
406. 389,
398,
424,
425.
389,
395,
425,
450.
490
Stability (numerical), 2 6 2 ,
270,
301,
355.
Static stability of floating bodies, 1 1 3 ,
120.
Storm, Surge, see also Hurricane, Cyclone, 2 ,
8, 72-76, 103, 132, 162, 197, 206, 215-217, 245, 261, 268-270, 273, 285, 294, 323, 328, 385, 387, 389, 392, 396, 406, 408, 411, 416, 421427-436, 439, 468-470, 474-478. Storm duration, 4 6 7 , 4 7 7 . Storm surge, 2 1 6 , 2 3 5 , 2 4 5 , 2 4 7 , 2 5 0 , 2 6 1 , 2 6 2 , 2 6 8 , 2 7 1 , 2 7 3 , 286, 289, 294-296, 303-306, 316, 318, 321, 325, 328, 331, 333, 342-344, 386, 396, 398, 400, 408, 412, 416, 430, 432, 434. Storm surge forecast, 2 6 1 , 2 6 3 , 2 6 5 , 2 6 8 , 2 7 0 - 2 7 3 , 3 1 8 , 3 2 1 , 3 2 3 , 331, 341, 342, 385, 390, 392, 408. Critical surge level, 4 7 8 . Surge peak, 2 8 6 - 2 9 3 , 3 0 5 , 3 1 6 , 4 0 3 . Surge prediction, 3 8 6 , 3 9 3 , 3 9 4 , 3 9 8 , 4 0 6 , 4 2 0 . Surge profile, 3 0 6 , 3 1 1 , 3 1 2 , 3 1 7 , 3 9 2 , 3 9 9 , 4 0 0 , 4 0 3 - 4 0 6 . Surge residual forecast, 3 9 3 . Surge simulation, 3 8 6 , 4 2 5 . Surge tide interaction, 3 1 7 , 3 9 4 , 3 9 5 , 4 0 0 , 4 2 8 . Surge wave, 4 2 0 .
135, 306, 424,
-
-
-
-
Stornoway, 3 9 9 ,
401,
403.
Stratification, 7 , 2 3 , 2 8 , 2 9 , 2 3 8 , 4 3 9 . - Non-stratified fluid, 2 1 9 , 2 2 6 . - Stratified fluid, 3 , 4 , 9 , 1 0 , 2 1 , 2 4 , 4 0 , 2 4 6 . - Stratification in two layers, 1 0 , 1 4 - 1 6 , 2 4 0 , 4 4 3 . - Stratification with constant density gradient, 1 2 ,
13,
18-23,
59.
Subartic front, 4 3 . Subinertial frequency range, 6 5 . Svendrup balance, 6 7 . Swell prediction, 3 9 1 . Taylor hypothesis, 6 8 . Taylor series expansion, 8 3 . Thermocline, 2 3 7 ,
240.
Thermohaline circulation, 5 9 . Thames River, 3 8 9 , Tiana Beach, 1 0 2 , Tide, 2 4 5 ,
-
-
-
-
400. 103,
108,
109.
247, 250, 285-289, 303-306, 317, 318, 333, 338, 352, 385, 389, 394, 396, 408, 411, 416, 420, 427-432, 435, 439, 461, 466, 474-476. Astronomical tide-producing force, 3 5 2 , 4 6 7 , 4 7 2 . Tidal current, 1 6 9 , 1 8 3 , 3 3 8 , 3 4 0 , 3 4 2 , 3 6 8 - 3 7 8 , 4 6 1 . Tidal cycle, 3 4 2 . Tide deviation, 4 7 2 . Tidal distribution, 4 0 0 . Tidal dynamics, 3 3 7 . Tidal ellipse of current, 3 6 9 , 3 8 1 , 3 8 2 . Tidal energy, 4 4 9 . Tidal flats, 3 3 5 . Tide fluctuation, 4 7 5 . Tidal force, 2 3 9 , 2 4 6 . Tidal forcing, 3 0 3 . Tide gauge, 3 3 3 , 3 4 1 . Tidal harmonics, 3 5 1 , 3 5 4 - 3 6 8 , 3 7 7 - 3 8 2 , 4 1 1 - 4 1 3 .
491 Tide - High spring tide, 2 8 9 , 3 4 0 , 3 6 8 , 4 0 0 . - Tidal high water, 2 8 6 , 2 8 9 . - Tide level, 1 6 8 , 3 0 1 , 3 0 9 , 3 1 0 , 3 2 5 , 3 2 8 - 3 3 1 ,
-
466,
467,
406,
412,
421,
434,
169,
177.
471.
Tidal low water, 2 8 6 , 2 8 9 . - Neap tide, 3 4 0 . - Oceanic tide, 2 3 9 , 3 8 5 . - Tidal period, 2 5 1 - 2 5 7 . - Tidal prediction, 3 0 6 , 3 0 9 , - Tidal sea, 2 3 8 . - Reversal of tide, 2 3 8 , 2 4 5 , - Tidal resonance, 2 8 6 . - Tidal wave, 3 3 7 , 3 3 8 . Time series analysis, 1 9 7 , Topographic effects, 9 1 ,
310,
315,
251-257,
200,
202,
394,
412.
369.
207,
216.
93.
Torrey Pines Beach, 1 9 2 . Towing resistance, 1 1 3 ,
114.
Tranquillization bassin, 4 6 6 . TRANSPAC, 5 8 ,
59.
Trapping scale, 6 7 . Turbulence - Atmospheric turbulence, 1 0 7 . - Turbulence generation, 1 0 , 1 2 , 1 6 . - Geostrophic turbulence, 6 8 . - Turbulence production, 1, 2 . - Shear generated turbulence, 1 0 , 1 3 . - Well-developed turbulence, 2 . - Turbulent convection, 3 9 . - Turbulent diffusion, 7 , 2 0 , 2 3 5 , 2 3 6 , 2 4 6 . - Turbulent disturbance, 7 . - Turbulent energy, 1, 6 , 8 , 2 3 . - Turbulent energy balance, 9 , 2 4 . - Turbulent energy diffusion, 1, 9 , 1 6 , 2 3 . - Turbulent energy flux, 7, 9 , 11, 1 5 , 1 8 , 2 1 , 2 3 , 2 7 . - Turbulent energy production, 5 , 1 3 , 1 4 , 1 6 , 1 9 , 2 9 , 4 1 , - Turbulent energy (rate of dissipation), 4 , 4 1 . - Turbulent energy (time scale of dissipation), 6 . - Turbulent entrainment layer, 2 - 2 0 , 2 3 , 2 8 , 2 9 . - Turbulent fluctuations, 4 0 , 1 8 1 . - Turbulent integral length scale, 1 2 , 1 6 2 . - Turbulent interactions, 7 9 . - Turbulent mixing, 9, 1 6 2 . - Turbulent operator, 2 4 5 . - Turbulent stress, 1 6 3 , 1 6 9 , 3 3 6 . Tyrrhenian Sea, 4 4 0 ,
442,
U.S.
Atlantic Coast, 7 5 ,
U.S.
West Coast, 4 5 1 .
443, 323,
448-450. 325,
328,
Veering of horizontal velocity, 2 5 5 - 2 5 7 . Velocity profile, 1 8 8 - 1 9 1 , Velocity shear, 1, 3 ,
5,
243, 14,
16.
Velocity shear layer, 6 . Venice, 4 2 7 ,
428,
432,
433,
436.
247.
331.
492
Vent, 2 2 6 ,
227,
229,
231,
232.
Vlissingen, 4 0 0 . Von Karman constant, 2 4 7 . Vorticity balance, 6 7 . Vorticity generation, 1 1 9 . Walton, 3 9 9 ,
401.
Wash, 3 9 8 . Water level fluctuations, 1 9 7 ,
200,
Water level prediction, 2 1 5 - 2 1 7 ,
206,
207,
216,
328.
321.
Wave, see also Internal waves, Gravity waves, 1 7 ,
-
-
-
-
72, 81, 93, 132, 469-477. 155, 157. Wave attenuation, 1 1 8 , 1 1 9 . Wave breaking, 1, 2, 2 1 , 7 9 , 8 7 , 1 1 8 , 1 6 2 - 1 6 6 , 1 7 0 , 1 8 0 , 1 8 3 - 1 8 5 . Capillary waves, 1 6 2 . Wave current, 1 8 3 , 1 9 5 . Wave-current interaction, 1 9 6 . Wave energy, 8 5 , 8 6 , 1 1 4 , 1 2 4 , 1 3 0 , 1 3 2 , 1 4 2 , 1 5 3 , 1 5 4 , 1 6 4 , 1 6 6 , 168, 173-181, 184. Wave field, 1 4 1 , 1 4 2 , 1 5 7 , 1 5 8 . Wave force, 1 1 4 , 1 1 6 , 1 1 8 , 1 2 4 . Free wave, 6 5 . Wave generation, 8 8 , 1 0 1 . Wave hazard, 4 6 7 . Wave height, 9 1 - 9 4 , 9 7 - 9 9 , 1 0 2 - 1 0 6 , 1 1 4 , 1 3 2 , 1 3 6 , 1 3 8 , 1 6 3 , 1 8 5 , 186, 189, 191. Critical wave height, 1 1 4 . Wave momentum, 1 1 4 - 1 1 8 , 1 2 0 , 1 2 4 , 1 4 1 . Wave momentum flux, 1 1 4 , 1 5 2 - 1 5 4 . Wave number spectrum, 1 4 5 , 1 4 8 . Wave parameters, 1 3 5 . Wave period, wave frequency, 1 0 2 - 1 0 5 , 1 1 6 , 1 6 3 , 1 7 5 , 1 8 9 , 4 6 7 . Wave prediction, 7 5 , 8 5 , 91, 9 6 , 1 3 9 , 1 5 7 , 3 9 1 , 3 9 4 . Wave prediction model, 1 2 7 , 1 3 0 , 1 3 9 . Wave profile, 1 1 6 , 1 1 9 , 1 2 4 , 1 4 1 . Wave propagation, 8 8 , 1 8 5 , 1 8 7 , 1 9 4 , 3 5 3 , 4 6 1 . Wave slope, 1 0 1 - 1 0 3 , 1 0 6 , 110, 111, 1 1 5 . Wave spectrum, 7 8 - 8 0 , 8 5 , 8 8 , 1 3 1 , 1 3 3 , 1 3 4 . Stokes waves, 1 8 3 . Waves stress, 1 8 8 . Surface waves, 1 1 8 , 1 2 7 , 1 2 9 , 1 3 2 , 1 3 8 . Wave velocity, 1 1 5 - 1 1 7 . Wave-wave interaction, 7 9 , 8 7 , 8 8 , 1 3 8 , 1 5 7 . 181,
262,
267, Wave age, 1 4 2 ,
WAVTOP, 1 6 2 ,
173,
180.
Weather anomaly, 3 8 ,
46.
Weather prediction, 3 8 5 ,
394,
Weathership FATIMA, 1 3 2 ,
136,
137.
Well-mixed shallow seas, 2 3 9 ,
245,
Weser Estuary, 3 3 8 . Wick, 3 9 9 ,
401,
403,
421,
422.
403,
412,
416,
424,
246,
335,
443.
425.
493
Wind - C o a s t a l wind, 439. - Wind d r i f t , 4 2 , 4 3 , 5 0 , 5 9 , 1 0 1 , 1 0 2 , 111, 4 4 1 . - Wind f i e l d , 7 7 , 8 8 , 1 3 1 , 1 3 2 . - Wind f i e l d p r o d u c t i o n , 1 2 9 , 4 3 6 . - Wind-generated c u r r e n t , 101-107, 162, 338, 342. - Wind-mixed l a y e r , 3 , 6 , 2 6 , 2 7 . - W i n d - m i x i n g , 1, 3, 7, 1 5 , 1 8 , 2 3 , 2 4 , 2 7 , 3 9 . - Wind-mixing l e n g h t s c a l e , 28. - Wind s p e c t r u m , 4 4 5 , 4 4 6 . - Wind s t r e s s , 4 2 , 5 9 , 6 2 - 6 4 , 6 7 , 1 4 1 , 1 6 2 , 1 6 6 , 1 6 8 , 1 7 5 - 1 7 8 , 1 9 7 , 200, 206, 207, 216, 220, 222, 228, 229, 232, 236, 237, 242, 245, 251, 297, 301, 303-306, 317, 321, 327, 330, 331, 341, 385, 387, 389-392, 403, 408, 413, 424, 433, 434, 443-445, 449. - Wind s t r e s s f l u c t u a t i o n , 6 8 , 6 9 . - Wind s t r e s s f r e q u e n c y wave number s p e c t r u m , 5 9 , 6 3 , 6 4 . - S u r f a c e wind, 141, 424. - Wind-tide i n t e r a c t i o n , 335. - Wind v e l o c i t y , 6 4 , 6 5 , 7 3 , 7 6 - 8 1 , 9 4 , 9 5 , 1 0 1 - 1 1 1 , 1 2 8 , 1 4 1 , 1 4 4 , 149-154, 163, 174-180, 200, 222-224, 227, 229, 236, 268-270, 278283, 294, 298-300, 304-306, 311, 312, 342, 346-348, 391, 403, 413, 418, 475. - Wind w a v e s , 1 6 2 , 1 6 3 , 1 6 6 - 1 7 0 , 1 7 4 - 1 7 8 , 4 6 6 , 4 6 8 . - Wind wave f l u m e , 1 4 4 . - Wind wave p r e d i c t i o n , 1 4 1 . - Wind-wave r e l a t i o n s h i p , 9 3 , 9 6 , 1 9 2 . - S u b s u r f a c e wind w a v e s , 1 8 0 . - G e o s t r o p h i c wind, 42, 63, 73, 74, 76, 78, 304, 321, 391, 392, 445. Windsor
( O n t a r i o ) , 197,
Workington, Zeebrugge,
301-308, 353.
199.
313,
316-318.
This Page Intentionally Left Blank