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.
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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 <<
,
D
respectively.
There is some experimental evidence that one can assume
u
- u % a ! T
where
a!
1/2
(21)
is an empirical constant.
Then, if one sets
-
G 6 = M o + n 6
Mo
+
(1
(22
T 3 / 2
and restrict attention to the case of steady fluxes at the air-sea interface,
does not depend on time and may be used successful y
G6
as one of the external characteristic parameters
(G6
61'3
is the ve-
locity scale) of turbulence in the wind mixed layer of the ocean. Along the same line, one may assume that the temperature is uniform in the upper homogeneous layer.
Then, integrating eq.(l) in the
mixed layer and over the turbulent entrainment layer, one gets s
s(2) = s
s-
= s+
-
O
s
D
dD + dt (0 -
-
Z
(23)
0+)
Hence
-
The time scale of turbulent energy dissipation ceed a few minutes whereas the deepening of the has a characteristic time of several hours.
+'
does not ex-
mixed layer
One may thus regard the
turbulence as adapting itself instantaneously to the modifications of the mixed layer and following the "slow" evolution of the latter. At the scale of turbulence, this slow evolution is not noticeable and the turbulent energy may be regarded as quasi steady, i.e.
7
In this case, one can usually assume
where
c
is a constant of order unity
(co
%
2
- 3,
in the atmosphe-
ric boundary layer) or, mcre generally
e = c 6
2/3
(28)
G15
which is valid also in the case of turbulence generated by a turbulent energy flux
Mo
in the absence of momentum flux.
The values of the variables at the lower boundary of the turbulent entrainment layer depend on the characteristics of the layer of fluid below.
These can be affected by different factors
sion (Mellor and Durbin, 1 9 7 5 ) ,
:
molecular diffu-
internal waves generated inside the
turbulent entrainment layer,but also below it,by turbulent disturbances at the bottom of the mixed layer and radiating momentum and energy away, turbulent diffusion caused by the instability and breaking of the internal waves or other mechanisms not directly related withthe local wind mixing process (Garnich, 1 9 7 5 ) . The effect of both molecular and turbulent diffusion below the turbulent entrainment layer is not perceivable at the time scale of the mixed layer deepening
(of the order of a day) and it may accordingly
be neglected. The contribution, to the energy balance, of internal waves excited by the upper turbulent layer is difficult to evaluate (Linden, 1 9 7 5 Thorpe, 1 9 7 3
;
;
Kantha, 1 9 7 7 ) .
In the absence of vertical density stratification below the turbulent entrainment layer
-- 0 az
be regarded as being at rest.
for
z
D
+
h)
, the fluid there may
The internal waves generated by the
upper turbulent layer do not propagate downwards and, in the nonviscous approximation which is actually made here, the fluid motion reduces to weak irrotational fluctuations which are rapidly attenuated as they move deeper from the bottom of the upper homogeneous layer and which contribute very little to and
M+
.
e
and not at all to
'I
+ '
s+
In that case, one may take e+=u+=.r
= M + = s
= o
(29)
This assumption is not quite valid when the fluid below the turbulent entrainment layer is stratified but, in a first approximation, when the density variation accross the turbulent entrainment layer is large, perturbations below it can be regarded as insignificant and
can be used.
eq.(29)
The temperature at the lower boundary of the turbulent entrainment layer can be written quite generally
-
O+ = G o o where
(30)
yD
is a constant and
OOO
the temperature gradient below the
y
turbulent entrainment layer. With the approximations described above (18, 19, 21, 22, 2 4 ,
25,
26, 28, 29, 3 0 ) , the basic equations (9)-(11) can be written
dt
(gt3AOD 2
:I3]
+
= Gg
+ 1 dD
(I:( -
2
- (ED - +
;h)
2 dt
-
DgBSo 2
(34)
*
Similarly the entrainment fluxes reduce to T-
=
dD d t u-
s-
=
dD dt
dD M - = E h + - e dt
-
-
h '
*Attention
is restricted here to the case dD As a result of 0 ("reverse eq. (291, eqs. (36)-(38) cannot be used when 'd:/it entrainment"). As suggested by Krauss and Turner (1967), one should write s-
=
o
,
T-
=
o
for
< dt -
o
( 3 9),( 40 1
When one considers the upper homogeneous layer throughout a long period when strong storms may give way to perfectly still weather, it is important to describe the reverse entrainment process (Garnich, 1975 ; Kosnizev et al, 1976).
9
It is important to note that, for
h
0
+
,
Eh
has a finite but
small value.
Then when the turbulent kinetic energy e - is smaller 1 2 than the mean flow kinetic energy 7 IIyII , the flux of turbulent energy can be directed from the turbulent entrainment layer into the upper homogeneous layer, contributing to the turbulent mixing in that layer.
This is confirmed by experiments in the laboratory (Kato and
Phillips, 1 9 6 9
;
Eqs.(31)-(33)
Kantha et al, 1 9 7 7 ) . can be integrated easily since
known functions of time.
and
s
to
are
In particular, when they are constant, one
gets
1
-
AOD
= AO(o)D(o)
-yD2
2
-
1 -yD2(o)
+
2
sot
(41)
T
DL =
(f2+ ~ ( o ) v ( o ) sin ) ft + ~ ( o ) U ( o ) cos ft
-
DV = -
where D(o)
,
-
sin ft + (2 + ~ ( o ) V ( o ) ) cos ft - 2 f f
D(o)U(O)
-
,
u(o)
(42)
-
v(o)
(43)
are the initial values.
To close the system of equations (31)-(34), one needs an estimate of the integrated viscous dissipation in the turbulent energy balance equation.
A
question arises
:
what portion of
G6
for mixing and what portion dissipates into heat ?
and
llh
is used
This is probably
the most important problem in one-dimensional modelling of mixed layer deepening and the models can be classified according to their particular parameterization o f the integral energy dissipation.
*
Let
ED
+
where
Eh = (1 - Q 1 ) G 6 Q1
O2
and
+
(1
-
Q2)IIh
(44)
are non-dimensional functions depending o n the
external parameters of the problem.
Their dependence will be examined
later, using similarity theory together with experimental data obtained in well controlled external conditions. However, by simple inspection of eq.(34) and (44), one can get some information about the functions
and
O1
Q2
.
If the deepening of the mixed layer i s produced only by the diffusion of turbulent energy down from the surface into the stratified
*fluid
(Mo # 0
,
T~
=
0
, IIh
= O ) ,
Attention is restricted here to
then obviously so
0
.
When the ocean surface
is cooling n = i 0 acts a s a complementary source o f energy q 2 and (44) must generalized to
10 0 ( @ , 2 1
On the other hand, in the case of shear generated turbulence (Mo < <
nD ,
IIh
G6
;
Q
IID
= IIg)
deepening of the upper homogeneous
layer is possible only if
Hence
If one considers, to begin with, a situation where the effects of rotation and surface heating can be neglected, in the asymptotic case
h + D
0
, there
is one non-dimensional number characterizing the problem,
the overall Richardson number defined as
Q1
The functions
and
In the following,
-u
i n s t e a d of
,
-v
,
O2
should then be functions of
one s h a l l w r i t e f o r s i m p l i c i t y
u,
only.
v,
...
...
FUNDAMENTAL RESULTS OF LABORATORY EXPERIMENTS In laboratory studies of the entrainment process in a stratified fluid, two essentially different mechanisms were used to generate the turbulence. In the first type of experiments, (e.g. Turner, 1 9 7 3
;
Thorpe, 1 9 7 3 ;
Long, 1 9 7 4 ) turbulence is produced by an oscillating grid placed at a given depth
Z1
below the free surface in the tank.
The depth of
the mixed layer is D = Z
1 4- D '
where
ZD
is the distance from the grid to the turbulent entrainment
layer. The most detailed studies of entrainment in this case have been performed in a fluid consisting of two homogeneous layers of different density.
The results have been interpreted, as a rule, using a rela-
tionship of the form
11
where
C
is some appropriate dimensional factor independent of the
grid oscillation frequency
Wo
and of the density jump
the turbulent entrainment layer
(po
AP
accross
is the mean density in the
mixed layer). The exponent
n
is a function of the Peclet number which tends
asymptotically to the value 1973
;
1.5
Crapper and Linden, 1974
for large Peclet numbers (Turner, ;
Wolanski, 1972).
Hence, in the asymptotic case u3
1 - = dD _ w dt
(53) 0
Long (1974), using experimental values of
uo
and
-
g Ap
,
sug-
PO
gested, on the basis of dimensional arguments, the following form for C
where
Zo
is the amplitude of the grid oscillations kept constant
during the experiment, Y is some non-dimensional function tending + 0 and z 1 +. 0 . to a constant value Y ( o , o ) for D D The flux of turbulent energy must be proportional to the third
“0
power of
where
c1
u
and
, i.e
x
is a non-dimensional constant
Combining - - = 1
dD 1/3 dt
Z
‘2
.
(51), (53), (54) and (55), one obtains R-3/2 iM
MO
where
C2
is another non-dimensional constant defined by
+ z
3/2
(57) and where the overall Richardson number R.
1M
=
R.
iM
is defined by
gf3AOD 2/3 MO
The results can also be presented using a “turbulent“ Richardson number (Turner, 1973) %
At least, c 1
is a constant for each particular geometry of the grid.
12
RiT
gf3AOP. -
=
(59)
WL
where
L!
is the turbulent integral length scale in the vicinity of
the upper boundary of the turbulent entrainment layer and
w
the
root mean square turbulent velocity, as before. The variations of the non-dimensional entrainment velocity = -1- dD dt plotted a s a function of the Richardson number RiT
w
ac-
cording to Turner’s experimental data (Turner, 1968, 1973) show that, for w
e
= -1 - dD =
w dt
where
RiT
5
‘3
-3/2 RiT
,
one can write with a good approximation (60)
C 3 % 2 .
In all Turner’s experiments however (Turner, 1968, 1973 and Turner, 19751, Z 1 , Z D , Z o Computing
were fixed
(Z1
= Z,
;
Thompson = 1 cm).
= 9 cm ; Z
from the measurements made by Linden (1975),(Z1 = 6 cm,
C3
ZD = 7 cm) , one finds
c3
%
0.1,
0.6.
On the other hand, if the results are interpreted in terms of RiM, both sets of experiments give similar values for the constant C2C11’3
(%
and this is an argument in favor of formula
1.85
(561 where
C2
may be regarded as a universal constant.
Linden (1975) extended its study to the case where there is a constant density gradient below the turbulent entrainment layer and he found an appreciable decrease in the entrainment rate which he interpreted by the l o s s of energy due to radiating internal waves. Wolanski and Brush (1975) presented experimental evidence that the entrainment rate depends mainly on the overall Richardson number RiM and that the influence of the non-dimensional time negligible.
Since the Brunt-VaisZlS frequency
N
( g 8 ~ ) ~ ’ ~ is t =
(gBy)
in the
fluid below the turbulent entrainment layer is one of the main factors determining internal waves radiation, one can conclude that, in this
*
particular experiment, this effect was very small
.
One of the most important mechanisms of turbulence generation by wind mixing is related to the presence of a mean drift current.
* Wolanski and Brush (1975) also showedthat the entrainment rate depended on the Peclet number. This was later explained by Phillips (1977) who determined that molecular transfer processes play a role in the ocean when
Ri*=
gapD po=o
> lo8
-
if
Ap
is due to a salinity gradient and
lo7 if A p is due to a temperature gradient whereas in when €Ii*) laboratory experiments (Kato and Phillips, 1969 ; Kantha et al, 1977) it occurs for R~~ 103
.
13 Laboratory experiments in which shear generated turbulence is respon' sible for entrainment are thus very important and, in this respect, the work of Kato and Phillips can be considered as a classical one. To avoid the influence of the limited length of the channel and the generation of secondary flows, a circular tank was designed where, at the fluid surface, a constant stress was applied by a rotating screen. that
T
In the top layer, the fluid velocity
u
(t)
was set such
remains essentially constant in the course of the experi-
ment. In the case of a linear continuous density stratification, Kato and Phillips (1969) found
ue *
In the case of two superposed layers of fluid, the dependence of on Ri, was more variable but, for a large range of Richardson
numbers
U
Ri,
(Ri,
5
,
2 102)
one can write a similar formula
-1 8.2 Ri,
- =
u* The main difference between eq.(61) and (63) was attributed by Phillips (1977) to internal waves energy radiation and by Garnich an( Kitaigorodskii (1976, 1977) to the different turbulent energy produc tion in the entrainment layer. If the initial density profile is linear, eq.(61) yields
The laboratory experiments discussed above have been performed wi thout heat flux accross the sea surface tion
(f = 0 ) .
( s o = 0)
and without rota-
In this case, eqs. (31)-(34) give, using (44) and zerl
initial conditions for the velocity AflD
- T1
y D 2 = A@(o)D(o)
-
1 - yD2(o) 2
(65)
Krauss and Turner (1967), restricting attention to the particular case
14 T3/2 C6
= 0
;
Q2 = 0
;
Q1 = 0 G6
D(o) = 0
;
derived the formula
This result seems in very close agreement with (64) but one may argue that this is nothing but a fortuitous coincidence.
Indeed, in
the experiments by Kato and Phillips (19691, the fundamental role in the entrainment process was played by the turbulent energy production
-
in the turbulent entrainment layer and not by the term
@lG6 In fact, with the hypotheses of Krauss and Turner one gets, in
the case of two superposed layers - _ = dD 1 / 2 dt
2
0
which should be compared to eq.(63). The hypothesis that the term
01G6 -
T
plays the main role in
the process of mixed layer deepening in the presence of a velocity shear cannot therefore be considered as sustained by laboratory experiments. An alternative approach was proposed by Pollard et a1 (1973) who assumed
@1 = 0
,
C6
= 0
,
$2 = 1
.
In this case, when
D ( o ) = 0,
eqs. ( 6 5 ) - (67) yield
which differs from
Kato
and Phillips' result (eq.64).
The authors argued that eq.(70) was in good agreement with experimental results but their attempt is not very convincing because the final choice between dependence of
D
,
(64) and (70) lies on the verification of the
not only on time
problem which is still to be solved. the deepening process (t
t
but also on
N
and this i s a
Moreover, at the beginning of
10 - 60 sec), experimental results (Kato
and Phillips, 1969) lie far away from the curve represented by eq.(70) For a fluid of two superposed layers, the assumptions of Pollard et a1 (1973) lead to the following formula for the drag coefficient (71) which is often very different from the observed values.
It i s c l e a r t h a t t h e m o s t i m p o r t a n t s i m i Z a r i t y c r i t e r i u m in t h e a n a l y s i s of t h e m i x e d layer d e e p e n i n g i n t h e o c e a n i s t h e buZk
15
R i c h a r d s o n number
(RiG
or
~ i * or
R ~ M J .In s p i t e o f
f e r e n c e s b e t w e e n i n s i t u and l a b o r a t o r y e x p e r i m e n t s ,
enormous d i f -
t h e Richardson
number s i r n u l a r i t y h o l d s a n d t h e t h e o r y of t h e o c e a n i c m i x e d l a y e r c a n and must r e Z y on t h e e x p e r i m e n t a l d a t a o b t a i n e d b y Kato and P h i l l i p s ( 1 9 6 9 ) a n d Kantha e t a 1 ( 1 9 7 7 ) . THEORETICAL MODELLING OF THE DEEPENING OF THE UPPER HOMOGENEOUS LAYER B Y WIND MIXING
(I) Two superposed homogeneous layers of different densities, no ro;
tation, no heat flux and steady momentum and turbulent energy fluxes accross the air-sea interface.
In this case, setting gBA0D = gBAO(o)D(o)
=
y
=
0
in eq.(65),
one has
const.
and the overall Richardson number
(72) RIG
,
and thus
Q1
and
,
Oz
are independent of time. When
y
=
0
,
it is reasonable to neglect momentum and energy
losses due to internal waves radiation.
Then, the functions
O I and
-
Q, can be determined as follows (Garnich and Kitaigorodskii, 1977). Using
G:I3
as a velocity scale and
D(o)
as a Length scale,
one can write eq.(67) in the non-dimensional form
:
(73)
(74) (75) ( 7 6 ) The solution of eq.(73) satisfying the condition
5 = 1
at
n = 0
is readily found to be
(77)
16 and
where
The limiting value
is independent of
@2 The same result is obtained directly in the model where deepening
of the mixed layer is determined by turbulent energy diffusion from the upper portion of the mixed layer, turbulence being generated by a vertically oscillating grid. G6
=
,
Mo
R ~ G= RiM
)
(In this case
T
-
For sufficiently large values of
R ~ G ( R ~ G> >
=
0
,
2C6)
C*=
,
0
,
combining
(56) and (82), one gets
For small values of
R ~ G, the entrainment velocity must tend to
a constant independent of
.
R ~ G
The simplest interpolation formula between these two asymptotic cases is
where
al
and
a2
To determine ment process
are two non-dimensional universal constants.
a 2 , one examines the final period of the entrain-
( 5 > > 1)
and assumes that, in the same time,
p > > 1
.
Then, from eqs.(78) and ( 8 0 ) , one gets
This formula obviously applies in the case of two superposed layers with velocity shear when turbulent energy production in the turbulent entrainment layer is mainly responsible for the entrainment process as in the laboratory experiment of Kantha et a 1 (1977).
In this case,
for sufficiently large Richardson number, the non-dimensional
-1
entrainment velocity is proportional to
*
-1
RiG Thus, for
m2
Rix (eq.63) and thus to
RiG > >
c6
"2 -
-J
RiG
64 n2s-37T
where
*
Mo
In the general case, when the contribution of
be taken into account, the dependence of be considered.
The value of
Cw
n2
on
a
to
G6
must
and
Cw
must
must then be estimated from the mo-
mentum and energy transfers from wind to drift currents and waves (Garnich and Kitaigorodskii, 1 9 7 7 ) . On the model of eq.(84), one can write a more general relation for
which is valid for a wide range of Richardson number
Q2
RiG
,
i.e.
o2
"3
=
(87)
1 -t a 4 RiG
where
a3
and
are non-dimensional constants.
a4
The values of
al, a2, a3
and
a4
can be determined by a care-
ful inspection of the experimental data (Turner, 1 9 7 3
;
Kantha et al,
1977).
The empirical expressions
are found in very good agreement with the laboratory data-(Garnich and Kitaigorodskii, 1 9 7 7 ,
1978
;
Kitaigorodskii, 1 9 7 7 ) .
In the experiments by Kantha et a1 ( 1 9 7 7 1 , bution to
is
G6
TI6
.
Thus, from eqs.(22)
To estimate the relative importance of in real situations, the value of C* , < a-4/3 computations such that C
-
*M o
%
u$ < <
n6
=
a u$
( a I', 1 0 )
the essential contriand (76)
Mo
due to wave breaking
must be varied in numerical
.
.
Hence
G6s a :u
and RiG = Ri*a
-2/3
.
Knowing the ai ' s , it is possible to verify the hypothesis p > > 1. One finds that it is satisfied over most of the range of interesting Richardson numbers.
))c
18 Because of the similarity between laboratory experiments and in situ conditions, one can argue that the empirical expressions (88) can be applied to the analysis of wind mixing in the ocean.
They
will be used in the following.
THEORETICAL MODELLING OF THE UPPER HOMOGENEOUS LAYER BY WIND MIXING
(11) Continuously stratified fluid, no rotation, no heat flux and steady momentum and turbulent energy fluxes accross the air-sea interface
With a constant temperature gradient below the turbulent entrainment layer, eq.(65) shows that the Richardson number RiG = gBAOD G2/3
6
varies with time as follows
The difference
RiG(o)
-
gByD2 ( O )
of the initial buoyancy jump2 G 2 / 3 which can be interpreted as the result
w-
characterizes the deviation from the value
P,v
of mixing a fluid with a
constant density gradient down to a depth
D(o)
.
One shall assum
here
where (94
I t is illuminating to consider two asymptotic cases of eqs.(84) and (i)
(871,
i.e. and
Q2
constant
(95
which correspond respectively to small values of the Richardson number in the initial stage of development of the mixed layer in a
19 continuously stratified fluid and to large values of the Richardson number in the final stage of the deepening process. (i) In the first case, A 1 , bles from
f(S)
to
q
A2
and A 3
are constants.
Changing varia-
defined by
one gets the linear equation 2 + -
fll
f'
5
-
A2
A3
(98)
F1 ( lS 2+ - - ) f = o
The solution of eq.(98) which satisfies the initial condition Q
= 0
at
5
= 1
is readily found to be
(100) (101)
v-1
Kv+l(aJ - a Kv (a1 A =
~ ~ + ~ (+ av-l ) I (a) a v
I2
One notes that for small values of 7 h
,
corresponding to the case
of "purely diffusive" entrainment consiaered by Krauss and Turner (1967), one has a <<
1
,
a(5
-
1)
v - 12 < < 1
<< 1
and
5
(A1n)
'L
113
(103)
On the other hand, for large values of one obtains using the asymptotic
A2
?-of expansions
such that I
and
a5 > > 1
,
Kv
This case corresponds to the model of Pollard et a1 (1973) where the main mixing source is the turbulent energy production in the turbulent entrainment layer
(nh)
(ii) In the second case, X 1
A1
=
5
;
A2
=
A2
-
c2
. and
X2
can be written
(105) (106)
20
x1
where
and
x2
are constants.
Substituting in eq.(93),
one gets
The change of variables
L
leads again to a linear differential equation of which the solution has the form (99) with a defined as before and
For
a5 > > 1
,
the solution takes the asymptotic form
and
i.e.
the non-dimensional entrainment velocity is proportional to the
inverse of the overall Richardson number in agreement with the observations in the laboratory (cf.eq.(61)). In the general case, eq.(73) (84)
and (87) for
a1
Q2
and
can be solved numerically, using eqs.
.
The results can be compared with
Kate and Phillips (1969).
the actual measurements made by Fig.1 shows the depth
D
of
the mixed layer in a linearly strati-
fied fluid as a function of time.
The dots represent the observed
values in two typical situations differing by the magnitudes of the density gradient and of the surface stress. The continuous curves are the predictions of the model in the same conditions.
Although the
trend is well reproduced, the theoretical results seem to give systematically larger values of the depth trainment process.
D
at the beginning of the en-
This may indicate that the turbulent diffusion
part of the entrainment (described by the function
Q ) 1
is overesti-
mated. The calibration of the model
(the determination of the ails)
was
partly made with the results of laboratory experiments where there was no stratification below the turbulent entrainment layer, i.e.
21
6a
30
90
0
30
60
90
120
150
120
150
180
210
240
270
180
Ilmc)
Fig. 1. Depth of the mixed layer in a stratified fluid as a function of time. The dots represent the observed values in two typical situations. The plain curves are drawn from the predictions of the model. y = 0
.
(Kantha et al, 1977).
cable in the case
y
# 0
The fact that the model remains appli-
is essentially due to the similar set-up
of the two types of experiments (Kato and Phillips, 1969
;
Kantha et
al, 1977) where, in particular, the relationship between the surface In other words, the ex-
stress and the drift current was the same. perimental values of
Csc
were quite similar
(Csc
5
a -4’3
%
0.06
-
0.02).
In real ocean situations, as mentioned before, one should expect values of
Csc
slightly less than in the laboratory, as a result of
the existence of a turbulent energy flux breaking.
The determination of
C,
Mo
associated with wave
in situ is however still an open
question. The results of the numerical computation of the non-dimensional entrainment velocity are in good agreement with the experimental data (fig. 2 ) . The models described above (both in the case case
y
y = 0
and in the
# 0) are of course limited to initial Deriods of mixing s u f -
ficiently smaller than
f-l
.
This is actually the case for many in
situ measurements. Fig. 3 s h o w s a comparison between predicted and observed values (Kullenberg, 1977) of the non-dimensional entrainment velocity.
22
u.uo 0.04
-
-
0.03 0.02
-
0.01
-
0.008. I
I
,
I
I
,
I
Ri,
20 30 40 60 80 100 200 300 400 The non-dimensional entrainment rate
10
Fig. 2 .
as a function of the
The overall Richardson number Ri, for a linearly u stratified x fluid. continuous curve corresponds to the solution of the model. TWO short lines in the upper and l o w e r p a r t o f the figure correspond to Kato and Phillips'approximation.
"
f
10-1
a
-
,
A *
\
'
I
"
-1
Fig. 3. The non-dimensional entrainment rate
3
as a function of the
to the theoretical u* Richardson number Riw. The upper curve corresponds predictions in the case y = 0. The lower curve corresponds to the theoretical predictions in the case y # 0 , constant. The experimental points (x, 0, A ) are taken from the measurements made by Kullenberg in the Baltic Sea.
23 The two continuous lines correspond to the two cases
y
,
0
=
# 0,
y
The experimental points suggest that, in natural conditions, the actual stratification is somehow intermediate between the two cases. Indeed, at the initial moment, an important density jump is created accross the turbulent entrainment layer while a continuous density gradient exists below it. The excellent agreement shown i n fig. 3 between in situ observations and theoretical predictions in an argument in favor of using the present model, calibrated with laboratory data, to the study of wind mixing and entrainment in seas and lakes.
THEORETICAL MODELLING OF T H E UPPER HOMOGENEOUS LAYER BY WIND MIXING
(111) The influence of the rotation of the earth
;
no heat flux and
steadv momentum and turbulent enerav fluxes accross the air-sea interface.
Substituting the first integrals (42) and (43) in eq. (341, neglecting small terms and changing variables to
5
and
rl
as before
(eqs. 74, 75, 761, one obtains
where (113)
The functions
and
Q1
Q2
have the same meaning as before.
@I
is associated with the vertical diffusion of turbulent energy and should be rather insensitive to the earth's rotation.
One shall
thus assume that it has the same form as in the non rotating case
O2
(eq. 84).
represents the portion of the turbulent energy
which i s dissipated. rotation
(nh
IIh
nh
With
strongly influenced by the earth's c * dD = - ( 1 - cos f t ) -) the question arises whether @2 dt fES2
can still be parameterized as before. rization of
O2
Conserving the same paramete-
has the advantage that the solutions o f eq. (112) tend
asymptotically to the solutions obtained in the non-rotating case and found in good agreement with the laboratory experiments
(Garnich and
Kitaigorodskii, 1977, 1978).
y
solutions of eq.(112) lutions of eq. (73) for
In both cases
y
=
0
and
# 0
,
the
are found to be practically the same as the soft
: II/3 .
24
Thus, the assumption will be made here that eq.(87)
is applicable
when the effects of the earth’s rotation are no longer negligible.
, TIh
ft > lI/3
For
decreases and this reflects on the deepening
process as shown on figs 4 and 5. In the case
y = 0 (fig.41, the earth’s rotation produces a devia-
tion from the linear law (77) valid for y
ft < < 1
# 0 , the rotation has a radical effect.
.
In the case
The entrainment rate de-
creases and the depth of the mixed layer appears to tend to some maximum value. The maximum depth
IIh
ce of
the mixed layer can reach under the influen
(@l = 0 ) can be estimated using eqsi(92) and
alone
One finds, for
D*
ft =
The non-dimensional depth different values of
(96).
n
6
=
IL is plotted on fig.6 for three
of N (Garnich and , D function f One can see that a relationship D
f t , as a
Kitaigorodskii, 1978).
%
C
u*
N -
most often
encountered.
C
can be slightly larger
than
1
The value of the coefficient
(N2$l)
1’3
is well verified in the range of values of
and this may be attributed to the diffused energy flux (the
influence of
Q1).
The variations, however, are small and the value
can be used as a characteristic depth of penetration of wind mixing in a rotating stratified fluid. Eq.(115)
D
st
a -8ll4 3
can be compared with the results of Pollard et a1 (1973) N
1/6
D*(r)
and by Phillips (1977)
The comparison is shown on fig.7.
The three formulae are found
to give very close values when, typically,
f a
,
N
.
Q
To find a definitive answer to the problem of choosing between the different parameterizations of the turbulent energy balance in the upper homogeneous layer (Phillips, 1977 : Pollard et al, 1973
;
25
4. T h e o r e t i c a l c u r v e s S ( f t ) a c c o r d i n g t o t h e m o d e l i n t h e c a s e y
D D
= 0.
The curves correspond to the values
@
R i x f o ) = 30
@
Ri
( 0 )
=
;
100 ;
@ @
Rix(O) = 50 Ri*(o)
;
140
=
@
R i w ( 0 ) = 70
.
__ = 5 (0)
90 80
70 60 50 40
30
20 10 xi4
JIt2
21I
JI
ft
Fig. 5. T h e o r e t i c a l c u r v e s S ( f t ) a c c o r d i n g t o t h e m o d e l i n t h e c a s e Y # 0 . The curves correspond t o the values
@
Rix
0 )
=
@
Rix
0 )
= 8.8
4.4
lo-’
;
@
~
;
@
R i x ( o ) = 4.4 10-1
~
~ =( 2 .02
11 0 - 2
\
1
b
c
60
40
20
80
>
I
I
100
120
N_ f
140
Fig. 6. T h e n o n - d i m e n s i o n a l t h i c k n e s s o f t h e m i x e d l a y e r as a function of the non-dimensional ratio for three values of ft f
6
=
.
@
f t = E
;
@ f t = n
;
O f t = z n
"';T 2
0.4
-
I
1
I
Fig. 7. T h e n o n - d i m e n s i o n a l t h i c k n e s s o f t h e w i n d m i x e d l a y e r D ( f t = n ) a s a f u n c t i o n of t h e n o n - d i m e n s i o n a l r a t i o N D* D* f 1. F o r m u l a (116) 2. Formula (117) 3. (Dashed curve ---) Formula (115)
D s-t
D*
27 Garnich and Kitaigorodskii, 1977, 1978) and to determine which one provides the best estimate for the characteristic depth of the oceanic wind mixed layer, many careful experiments must be performed both in the laboratory and in nature, for
.
a broad range of values of
It would be interesting to use,as a basis,the stock of oceanic observations as it was done previously by Kitaigorodskii and Filushkin (Kitaigorodskii, 1960
;
Kitaigorodskii and Filushkin, 1 9 6 4 ) .
In this study, a different parameterization of the energy dissipation was used, i.e.
-
ED + Eh = G g Y l (Ri*
fD
, -
1
u* and this is not, in general, equivalent to the parameterization used in this paper. In a subsequent publication (Resnyanskiy, 1975) a simpler form of eq. (118) was used, i.e.
The thickness of the mixed layer was derived from the simple energy -
G g = ED
balance
+
Eh
which of course gives the well-known Rossby-
Montgomery formula (e.g. Kitaigorodskii, 1970) u* Dst
n,
T-
In situ experimental data (Kitaigorodskii, 1960
;
Kitaigorodskii and
Filushkin, 1964) do not rule out eq.(120) but the arbitrariness Of the approximation (119) must be emphasized.
On the other hand, as
shown above, the dependence of the energy dissipation on the Richardson number must be taken into account to reproduce satisfactorily the laboratory experiments and it has a primary importance in modelling the deepening of the mixed layer.
THEORETICAL MODELLING OF THE DEEPENING OF THE MIXED LAYER BY WINDMIXING
(IV) Influence of the earth's rotation and of the heating of the ocean surface
;
steady heat momentum and turbulent energy fluxes
accross the air-sea interface.
L
Eq. (41) can be written R ~ G = R i ~ ( 0 ) + R1(C2-1) + R2V where
6
and
are defined as before
(121) (eqs. 74 and 75)and where
28 BgYD;
Bg S O D O = -
R 1 = -2 G ; / 3
L,
L*
G6
R 2 =
;
f
(122) (123)
L*
- Obukhov l e n g t h s c a l e
d e n o t i n g t h e Monim
G6 -
=
98s0
From e q s . ( 3 4 ) a n d
R i ~ ( 0 )+
( 4 4 ) , o n e g e t s , u s i n g eq. ( 1 2 1 ) ,
1 2 R1(C2-
1)
+
1
+
R q 2 2
Q1
and
Q2
i s g i v e n by e q . ( 1 2 1 ) .
RiG
c o s f*ll)02
5
= Q 1 - -; 2 The f u n c t i o n s
c* (1 -
-
C
a r e d e f i n e d by e q s . ( 8 4 ) a n d
Q1
The f u n c t i o n s
and
are t h u s
O2
assumed i n d e p e n d e n t o f t h e e a r t h r o t a t i o n and f u n c t i o n s o f t i f i c a t i o n a n d of RiG
the stra-
f l u x t h r o u g h t h e e x p r e s s i o n of
*
Since both of
the surface heat
( 8 7 ) where
eq.(125)
5
and
increase with time,
RiG
t h e right-hand
side
t i m e w h i c h may v a n i s h f o r
i s a d e c r e a s i n g f u n c t i o n of
ncr and ‘cr Restricting attention t o small t i m e s
some c r i t i c a l v a l u e s
( 1 9 7 8 ) h a s shown t h a t ,
(f*q
E
i n t h e c a s e where
an a n a l y t i c a l s o l u t i o n of
eq.(125)
f t << 1) $2
and
c o u l d be found.
,
Garnich
are constant,
The c r i t i c a l
d e p t h i s t h e n g i v e n by
and t h e c r i t i c a l e n t r a i n m e n t v e l o c i t y
l i m
5’
zero.
5,,
For times l a r g e r than t h e c r i t i c a l time, t i v a l u e d and t o d e s c r i b e t h e mixing process
*
5%
i s d i f f e r e n t from
dr! t h e s o l u t i o n becomes mula d d i t i o n a l hypotheses
are
required.
The t y p i c a l w i n d - m i x i n g s o l u t i o n of
When t h e
eq.(125)
at
surface heat
length scale
f t E f
q =
*
n ,
flux i s zero,
accross the turbulent entrainment
Dx-
c a n be c o m p u t e d f r o m t h e
one f i n d s
the temperature difference
l a y e r i s c r e a t e d by t h e s t r a t i f i c a -
t i o n below and one c a n w r i t e
*F o r
i n s t a n c e , one can imagine t h a t f o r t > tcr t h e v e r t i c a l s t r u c t u r e o f t h e t e m p e r a t u r e f i e l d i s d e s t r o y e d a n d a new w i n d - m i x e d l a y e r is generated near the surface.
Substituting in eq.(127),
one finds
r
(114')
identical to eq. (114). On the contrary, if one assumes that the temperature jump
A0
is mainly due to the surface flux one can write, according to eq.(41) where
,
y = 0
AO(o) = 0
,
In the fundamental stages o f the deepening process, the mixing of the entrained fluid is mainly due to the turbulent energy production in the turbulent entrainment layer and one has, approximately
For small values of
ft, using eq.(42)
and (431, one gets then
and in the asymptotic case (96)
Comparison of formula (132) with observations in the ocean (Kitaigorodskii, 1960) has confirmed a relationship of the form D
%
C L ,
but the coefficient determined from the experimental data
was found different.
The difference can be attributed partly to the
contribution of the stratification
y
to the temperature jump accross
the turbulent entrainment layer during the spring-summer heating period when the observations were made. The solution of eq.(125)
has been computed numerically for diffe-
rent values of the surface heat flux. figs. 8 and 9.
(The value
so
=
The results are presented on
10-40C m/sec corresponds to a maximum
value o f the heat flux in mid latitudes during a summer day). Fig. 10 shows the computed values of I[
times
ft = ; i
, II
and 2II
=
EL
D, as a function o f the
at three different non-dimensional
30
-_ L, f
80
10
80
50
40
30
20
10
F i g . 8. T h e o r e t i c a l curve S ( f t ) f o r d i f f e r e n t v a l u e s of t h e s u r f a c e h e a t f l u x so i n t h e c a s e y = 0 . T h e c u r v e s I , 2 , 3 w e r e c a l c u l a t e d f o r R i w ( o ) = 5 0 , us = 1 , 5 1 0 - 2 m / s e c , D ( o ) = lm a n d so = 0 ; 2 . 1 0 - 5 a n d 1 0 - 4 0 C m / s e c , r e s p e c t i v e l y . The c u r v e s 4 , 5 , 6 w e r e c a l c u l a t e d f o r R i * ( o ) = 1 0 0 , u* = 1 , 5 10-2m/sec, D(o) = lm a n d so = 0 ; 2 . 1 0 b 5 a n d 1 0 - 4 " C m / s e c , r e s p e c t i v e l y .
T
F i g . 9. T h e o r e t i c a l c u r v e S ( f t ) f o r d i f f e r e n t v a l u e s o f t h e s u r f a c e h e a t f l u x so i n t h e c a s e y # 0 . The c u r v e s 1, 2 , 3 w e r e c a l c u l a t e d f o r N 2 = g 6 y = 5 . 1 C J - 6 s e c T 2 a n d s o = 0 ; 2 . 1 0 - 5 a n d 1 0 - 4 0 C m/sec r e s p e c t i v e l y . The c u r v e s 4 , 5 , 6 w e r e c a l c u l a t e d f o r N 2 = 5 . 1 0 T 5 s e c ' * a n d so = 0 ; 2.10-5 and 10-40C m/sec r e s p e c t i v e l y .
31
:
+ + * +f+ $f $ *; $$q+$fg
4-
+
+
++ ++
$ +++ +++
+
+
++
$
+
4+
tt
t++
++
#
I
Fig. 10. The non-dimensional thickness of the mixed layer as a function of the non-dimensional parameter gBso values of ft(Il/2, Il, 2Il) and five different ~2 = 10-6sec-2 ; ~2 = 2.10-6sec-2 ; ~2 = 5.10-6sec-2 (u* = 1,s 10-2 m/sec , D(O) = Im)
N'uZ
.
parameter
gBs0
7
-
D
=
D(t) D,
for three values of N. ; ~2 = 5.105 sec-2
.
NU* Although there are fairly important variations with the surface heat flux in the representation used in figs. 8 and 9, one can see gBs0 - a condition which is usually reaon fig. 10 that for 2 < N u* lized, in mid latitude, during the spring summer heating period ,
-
a short range forecast of the thickness of the upper homogeneous layer under the effect of wind mixing can be based on eq.(115) provided the initial stratification is known. CONCLUSIONS In the present review, attention has been restricted to a discussions of previous studies contributing to a better understanding of the dynamics of vertical mixing processes and the problem of modelling the upper ocean layer.
One has voluntarily left aside the question of
large scale well-organized motions such as, Langmuir circulation, for instance. Observations clearly show that large scale vertical motions (like
Langmuir v o r t i c e s )
a r e c l o s e l y r e l a t e d t o t h e u p p e r homogeneous l a y e r
and t h e q u e s t i o n r e m a i n s t o be s o l v e d whether t h e t h i c k n e s s o f t h e mixed l a y e r c a n b e d e t e r m i n e d by s u c h o r g a n i z e d p a t t e r n s o r w h e t h e r t h e s e s t r u c t u r e s d e p e n d o n t h e v e r t i c a l d e n s i t y p r o f i l e b u i l t up by t u r b u l e n c e i n t h e u p p e r homogeneous l a y e r .
ACKNOWLEDGMENT
T h i s r e v i e w i s m a i n l y b a s e d on t h e r e s u l t s r e p o r t e d i n and K i t a i g o r o d s k i i ,
1977,
i s much i n d e b t e d t o D r .
N.
1978
;
Kitaigorodskii,
1977).
(Garnich The a u t h o r
Garnich f o r permission t o p r e s e n t h e r e
a l s o some o f t h e g r a p h s a n d r e s u l t s o f
h i s PhD d i s s e r t a t i o n
(Garnich,
1978).
REFERENCES
C r a p p e r , P . F . , L i n d e n , P . F . , 1 9 7 4 . The s t r u c t u r e o f t u r b u l e n t d e n s i t y i n t e r f a c e s . J . F l u i d Mech., 65:45-64. 1973. A t i m e - d e p e n d e n t model o f t h e u u p e r o c e a n . J . Phys. Denvan, K . L . , O c e a n o g r . , 3:173-184. Garnich, N.G., 1975. A model o f t h e c o n t i n u o u s e v o l u t i o n of t h e s e a s o n a l t h e r m o c l i n e . O c e a n o l o g i a , 15:233-238. Garnich, N . G . , 1 9 7 8 . The t h e o r y o f w i n d - m i x e d l a y e r . Ph.D. D i s s e r t a t i o n , L a b o r a t o r y o f P h y s i c s o f Atmosphere-Ocean I n t e r a c t i o n s , I n s t i t u t e o f O c e a n o l o g y , Academy o f S c i e n c e , U S S R . K i t a i g o r o d s k i i , S . A . , 1 9 7 7 . On t h e r a t e o f m i x e d l a y e r Garnich, N.G., d e e p e n i n g . I z v . Acad. o f S c i e n c e s USSQ, P h y s i c s of Atmosphere and Ocean, 13:1287-1296. Garnich, N.G., K i t a i g o r o d s k i i , S . A . , 1 9 7 8 . T o t h e t h e o r y o f wind mixed l a y e r d e e p e n i n g i n t h e O c e a n . I z v . A c a d . o f S c i e n c e s USSR, P h y s i c s O f Atmosphere and Ocean, 14:1287-1296. 1 9 7 7 . N o t e on t h e r o l e o f i n t e r n a l waves i n t h e r m o c l i n e Kantha, L . H . , e r o s i o n . Chap. 1 0 a . I n : E . R . K r a u s ( E d i t o r ) , M o d e l l i n g and u r e d i c t i o n o f t h e u p p e r l a y e r s of t h e o c e a n . Pergamon P r e s s . K a n t h a , L.H., ~ h i l l i p s ,o . M . , ~ z a d ,R . S . , 1 9 7 7 . On t u r b u l e n t e n t r a i n m e n t a t a s t a b l e d e n s i t y i n t e r f a c e . J. F l u i d M e c h . , 7 9 : 7 5 + - 7 6 8 . Kato, H . , P h i l l i p s , O.M., 1 9 6 9 . On t h e p e n e t r a t i o n o f a t u r b u l e n t l a y e r i n t o s t r a t i f i e d f l u i d . J. F l u i d Mech., 37:643-655. K i t a i g o r o d s k i i , S . A . , 1 9 6 0 . On t h e c o m p u t a t i o n o f t h e t h i c k n e s s o f t h e wind-mixing l a y e r i n t h e Ocean. J z v . Acad. o f S c i e n c e s U S S R , G e o u h y s i c a l s e r . , 3:425-431 ( E n g l i s h e d i t i o n up. 284-287). 1970. P h y s i c s of a i r - s e a i n t e r a c t i o n . L e n i n g r a d , Kitaigorodskii, S.A., Gidromet. I z d a t e l ' s t v o ( E n g l i s h e d i t i o n 1973, Jerusalem I s r a e l Progr. Scien. T r a n s l . ) . Kitaigorodskii, S.A., 1 9 7 7 . O c e a n i c s u r f a c e b o u n d a r y l a y e r . R e p o r t of t h e J O C / S C O R J o i n t S t u d y C o n f e r e n c e on g e n e r a l c i r c u l a t i o n models o f t h e o c e a n a n d t h e i r r e l a t i o n t o c l i m a t e ( H e l s i n k i , 23-27 may 1977) 1 , Geneva. 1 9 6 4 . A ~ p l i c a t i o no f t h e s i m i K i t a i g o r o d s k i i , S.A., F i l u s h k i n , B . N . , l a r i t y t h e o r y t o t h e a n a l y s i s o f t h e o b s e r v a t i o n s i n t h e upper ocean. Oceanological S t u d i e s , 13, Izd-vo., " N a u k a " , MOskVa.
33 Kosnizev, V.K., K u f t a r k o v , Yu.M., F e l s e n b a u m , A . I . , 1 9 7 6 . Oned i m e n s i o n a l a s y m p t o t i c model o f t h e upper ocean. Proceedings o f A c a d . o f S c i e n c e s . USSR ( D A N ) , 1 : 7 0 - 7 2 . Kraus, E.B. and Turner, J . B . , 1967. A one-dimensional model of t h e s e a s o n a l t h e r m o c l i n e : 11. T h e g e n e r a l t h e o r y a n d i t s c o n s e q u e n c e s . T e l l u s , 19:98-106. Kullenberg, G . , 1977. Entrainment v e l o c i t y i n n a t u r a l s t r a t i f i e d v e r t i c a l s h e a r flow. E s t u a r i n e and c o a s t a l marine s c i e n c e , 5:329338. Linden, P.F., 1975. The d e e p e n i n g o f a mixed l a y e r i n s t r a t i f i e d f l u i d . J. F l u i d Mech., 71:385-405. Long, R . B . , 1974. T u r b u l e n c e and m i x i n g p r o c e s s e s i n s t r a t i f i e d f l u i d s L e c t u r e s , T e c h . R e p . NO6 ( s e r i e C ) . T h e J o h n s H o p k i n s U n i v e r s i t y . Mellor, G . L . and D u r b i n , P.A., 1975. The s t r u c t u r e a n d d y n a m i c s o f t h e ocean s u r f a c e mixed l a y e r . J. Phys. Cceanogr., 5:718-728. Long, R . U . , 1 9 7 1 . An e x p e r i m e n t a l i n v e s t i g a t i o n o f t u r Moore, M . J . , b u l e n t s t r a t i f i e d s h e a r i n g f l o w . J. F l u i d Mech., 49:635-656. 1977. E n t r a i n m e n t . Chap. 7. I n : E.B. K r a u s ( E d i t o r ) , P h i l l i p s , O.M., M o d e l l i n g and P r e d i c t i o n o f t h e u p p e r l a y e r s i n t h e o c e a n . Pergamon Press. P o l l a r d , R.T., R h i n e s , P . B . , Thompson, R . O . R . V . , 1973. The d e e p e n i n g o f t h e w i n d - m i x e d l a y e r . G e o p h . F l u i d Dyn., 3 : 3 8 1 - 4 0 4 . R e s n y a n s k i y , Yu.D., 1975. P a r a m e t r i z a t i o n o f t h e i n t e g r a l t u r b u l e n t energy d i s s i p a t i o n i n t h e upper quasi-homogeneous l a y e r o f t h e o c e a n . I z v . A t m o s p h e r i c and O c e a n i c P h y s i c s , 11:726-733 (English e d i t i o n pp. 453-457). Thompson, S . M . a n d T u r n e r , J . S . , 1975. Mixing a c r o s s a n i n t e r f a c e d u e t o t u r b u l e n c e g e n e r a t e d by a n o s c i l l a t i n g g r i d . J . F l u i d Mech., 67:349-368. Thorpe, S.A., 1973. Turbulence i n s t a b l y s t r a t i f i e d f l u i d s a review of l a b o r a t o r y e x p e r i m e n t s . B o u n d a r y - l a y e r m e t e o r o l o g y , 5 : 9 5 - 1 1 9 . Turner, J.S., 1968. The i n f l u e n c e o f m o l e c u l a r d i f f u s i v i t y on t u r b u l e n t entrainment across a density interface. J . F l u i d Mech., 33:639-656. T u r n e r , J . S . , 1973. Buoyancy e f f e c t s i n f l u i d s . Cambridge Univ. P r e s s . Wolanski, B . J . , 1972. T u r b u l e n t e n t r a i n m e n t across s t a b l e d e n s i t y s t r a t i f i e d l i q u i d s a n d s u s p e n s i o n s . P h . D . D i s s e r t a t i o n . The Johns Hopkins U n i v e r s i t y . Brush, L.M., 1975. T u r b u l e n t e n t r a i n m e n t a c r o s s s t a b l e Wolanski, B . J . , d e n s i t y s t e p s t r u c t u r e s . T e l l u s , 27:259-268.
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35
LARGE SCALE AIR-SEA INTERACTIONS AND CLIMATE PREDICTABILITY
CLAUDE FRANKIGNOUL
Department of Meteorology, Massachusetts I n s t i t u t e of Technology, Cambridge, MA
ABSTRACT
After reviewing r e c e n t empirical s t u d i e s of short-term climate p r e d i c t a b i l i t y based on observations of t h e sea s u r f a c e temperature (SST), t h e p h y s i c a l processes t h a t govern t h e generation and decay of l a r g e s c a l e SST anomalies a r e discussed. Using a s l a b model of t h e oceanic mixed l a y e r , we f i n d t h a t l a r g e s c a l e midl a t i t u d e SST anomalies can be described a s a f i r s t - o r d e r a u t o r e g r e s s i v e process i n regions of small mean c u r r e n t , a s suggested by Frankignoul and Hasselmann (1977). The SST anomalies a r e continuously generated by t h e n a t u r a l v a r i a b i l i t y of t h e a i r - s e a f l u x e s . Short time s c a l e v a r i a t i o n s i n t h e l o c a l h e a t exchanges seem dominant, although mixed-layer depth v a r i a t i o n s a r e important during c e r t a i n seasons. Temperature advection p l a y s a l a r g e r o l e i n some regions, and mesoscale eddies mainly c o n t r i b u t e a s m a l l s c a l e noise. The decay of t h e SST anomalies can be represented by a l i n e a r negative feedback, and seems l a r g e l y c o n t r o l l e d by t h e i r back-interaction on t h e atmosphere. The importance of t h e feedback processes f o r c l i m a t e p r e d i c t a b i l i t y i s s t r e s s e d , a s w e l l a s t h e seasonal v a r i a b i l i t i e s i n t h e SST anomaly dynamics.
INTRODUCTION
In recent y e a r s , t h e r e has been growing concern about t h e impact of climate changes on man's a g r i c u l t u r a l , economic and s o c i a l a c t i v i t i e s , leading t o increasing climate research e f f o r t s ( s e e GARP r e p o r t No. 16, 1975).
On monthly
t o decadal time s c a l e s , t h e s u r f a c e l a y e r s of t h e ocean a r e believed t o play a prominent r o l e i n climate v a r i a t i o n s , through exchanges of h e a t , moisture and momentum a t t h e a i r - s e a i n t e r f a c e .
Indeed, t h e r e i s i n c r e a s i n g evidence of large-
s c a l e r e l a t i o n s h i p s between monthly o r seasonal anomalies i n t h e sea s u r f a c e temperature (SST) and t h e atmospheric c i r c u l a t i o n (e.g.,
Namias, 1969; Bjerknes,
1969; R a t c l i f f e and Murray, 1970). The dominant SST anomaly p a t t e r n s a r e l a r g e , nearly ocean-wide, persistence t i m e of t h e order of seasons.
and have a
This p e r s i s t e n c e r e f l e c t s t h e l a r g e
thermal and mechanical i n e r t i a of t h e upper ocean, and suggests t h a t t h e ocean
is l i k e l y t o c o n t r i b u t e most t o t h e p r e d i c t a b i l i t y of t h e coupled oceanatmosphere system.
Empirical c l i m a t e f o r e c a s t s based on t h e observed d i s t r i b u t i o n
of SST anomalies have been made, reaching some degree of success (Namias, 1978). However, higher p r e d i c t i v e s k i l l s a r e linked t o our understanding of t h e physics of these l a r g e s c a l e a i r - s e a
interactions.
The e f f e c t of SST anomalies on t h e
atmosphere is difficult to model simply, and there are complex teleconnections between tropical and extratropical systems (Bjerknes, 1969). Much information comes from simulations with general circulation models (GCM) of the atmosphere. In these numerical experiments, a fixed SST anomaly is prescribed, and its effects on the atmosphere circulation analyzed. variability of the atmosphere
In mid-latitudes where the natural
is largest, the effects of SST anomalies seem
mainly local, although some non-local effects have been found (Kutzbach et al., 1977).
In the tropics, the response of atmospheric models is larger (e.g.,
Shukla, 1975). To investigate the influence of SST anomalies on climate changes, the SST field must be allowed to vary.
It has often been suggested that SST anomalies
arise through positive ocean-atmosphere feedback processes, by which the presence of a SST anomaly modifies the atmospheric circulation in such a manner that the anomaly is strengthened (e.g., Namias, 1963).
However, the empirical evidence
suggests that it is the atmosphere that is driving the ocean, rather than viceversa (Davis, 1976; Trenberth, 1975; Haworth, 1978).
This is consistent with
recent theoretical studies by Salmon and Hendershott (1976), and Frankignoul and Hasselmann ( 1 9 7 7 ) , who suggestedthat SST anomalies arise spontaneously in response to the natural variability of the weather fluctuations.
The climate predictability
problem is therefore twofold, since the evolution of the SST anomaly field and its back interaction on the atmosphere must be predicted.
In their numerical
experiment, Salmon and Hendershott found that the atmosphere was too noisy to be much affected by the SST anomalies on the time scales over which the anomalies are themselves predictable. plate ocean).
However, their ocean model was very simple (a copper
The observations suggest
that climate predictability on time
scales of months exists only during certain seasons (Davis, 1978).
Hence, more
refined models that can simulate the seasonal variability of the atmosphere and the ocean are needed.
To avoid excessive computational costs, only the essential
physical mechanisms should be retained in these models. The main emphasis of this paper is on extratropical air-sea interactions. In section 2, we briefly review recent empirical studies of short-term climate predictability based on observations of SST anomalies.
Encouraging results have
been obtained, but it is difficult to distinguish between competing predictive models.
As discussed by Davis (1977) and Hasselmann (1978), the most useful
selection of predictors will be based on theoretical concepts.
This introduces
naturally section 3 , where the physical processes that govern the evolution of large scale SST anomalies are discussed, using a slab model of the oceanic mixed layer.
It is found that in typical central ocean conditions, the rate of
change of the SST anomalies is described by a first-order autoregressive process, as suggested by Hasselmann (1976), and Frankignoul and Hasselmann (1977).
The
upper ocean responds as an integrator of the short-time scales changes in the
37 air-sea
fluxes.
Random walk SST f l u c t u a t i o n s with l i n e a r l y growing variance
develop, u n t i l negative feedback processes lead t o a s t a t i s t i c a l l y steady s t a t e . The need t o i n v e s t i g a t e f u r t h e r t h e feedback processes i s s t r e s s e d i n t h e conclusions, because of t h e i r important r o l e i n c l i m a t e p r e d i c t a b i l i t y .
SEA SURFACE TEMPERATURE AND SHORT-TERM CLIMATE PREDICTABILITY
In t h e absence of s a t i s f a c t o r y dynamical models of t h e l a r g e s c a l e i n t e r a c t i o n s between anomalies i n t h e weather, SST, sea i c e , snow cover, e t c . , observations a r e b e s t s u i t e d f o r attempting c l i m a t e f o r e c a s t s .
historical
If there are
recurrent p a t t e r n s of developments o t h e r than t h e s e occurring by chance, f u t u r e developments may be i n p a r t p r e d i c t a b l e from p a s t and p r e s e n t observations.
For
s u f f i c i e n t l y small excursions about an equilibrium mean c l i m a t i c s t a t e , l i n e a r r e l a t i o n s can be used f o r d e s c r i b i n g c l i m a t i c changes (e.g. Leith, 1975; Hasselmann, 1976).
Then, t o p r e d i c t a v a r i a b l e y from N observations
zi
of t h e
same o r o t h e r v a r i a b l e s a t p r i o r times ( t h e p r e d i c t o r s ) , one can w r i t e
i
N
z ai zi i=1 where denotes t h e p r e d i c t e d value of y. =
one which minimizes t h e mean square e r r o r indicate ensemble average.
The optimum l i n e a r p r e d i c t i o n i s t h e <E>=<
(y-y) '>,
where t h e angle b r a c e s
The s o l u t i o n i s given by
N ai =
c
j=1
zj>-l
The q u a l i t y of t h e p r e d i c t i o n i s g e n e r a l l y c h a r a c t e r i z e d by t h e s k i l l
which r e p r e s e n t s t h e f r a c t i o n of v a r i a n c e t h a t can be p r e d i c t e d .
As first
discussed by Lorenz (1956), t h e performance of s t a t i s t i c a l e s t i m a t o r s i s influenced by t h e f a c t t h a t t h e mean products i n ( 2 ) a r e not known b u t must be estimated from f i n i t e d a t a s e t s .
This introduces a r t i f i c i a l p r e d i c t i v e s k i l l
t h a t i n c r e a s e s with i n c r e a s i n g number of p r e d i c t o r s N , and decreasing length of the d a t a s e t used t o e s t i m a t e t h e mean products ( s e e a l s o Davis, 1976). therefore necessary t o l i m i t N b e f o r e c a r r y i n g out t h e a n a l y s i s .
It is
Davis (1977)
and Hasselmann (1978) d i s c u s s i n d e t a i l why it i s much p r e f e r a b l e t o s e l e c t t h e number of p r e d i c t o r s by an a p r i o r i c r i t e r i a , r a t h e r than by c o e f f i c i e n t screening.
They a l s o p o i n t o u t t h e d i f f i c u l t y i n d i s t i n g u i s h i n g between d i f f e r e n t
competing models on p u r e l y empirical grounds.
The b e s t d a t a s e l e c t i o n techniques
w i l l t h e r e f o r e b e based on t h e o r e t i c a l s t u d i e s , and Hasselmann concludes t h a t
"the c r e d i b i l i t y of a model w i l l have t o rest t o a l a r g e p a r t a l s o on t h e i n t r i n s i c c r e d i b i l i t y of t h e physics of t h e model". A s mentioned i n t h e i n t r o d u c t i o n , numerous case s t u d i e s of t h e co-occurrence
of SST and atmospheric anomaly p a t t e r n s have been reported i n t h e l i t e r a t u r e . T o use SST a s climate p r e d i c t o r , however, some c o r r e l a t i o n m u s t be observed
between SST anomalies and subsequent weather anomaly p a t t e r n s .
Some e a r l y evidence
f o r such c o r r e l a t i o n s i s r e p o r t e d by Namias (1969). L i t t l e a t t e n t i o n had been given t o t h e o t h e r l a g c o r r e l a t i o n s , u n t i l Davis (1976) made t h e f i r s t systematic a n a l y s i s of t h e r e l a t i o n s h i p between SST and s e a l e v e l p r e s s u r e (SLP) over t h e North P a c i f i c .
Using s t a t i s t i c a l e s t i m a t o r s of t h e form (l), Davis found t h a t
t h e only s i g n i f i c a n t connection between SST and SLP anomalies i s one where SLP leads SST i n time by a l a g of z e r o t o 2 o r 3 months, and t h a t t h e r e was no p r e d i c t i v e s k i l l f o r p r e d i c t i n g SLP from SST or SLP.
Subsequently, Namias (1976)
showed convincing evidence t h a t summer SST anomalies i n t h e Aleutian low a r e s i g n i f i c a n t l y c o r r e l a t e d with SLP ( o r mid-tropospheric h e i g h t ) over t h e North P a c i f i c and with a i r temperature and p r e c i p i t a t i o n anomalies i n t h e United S t a t e s (downstream) i n t h e following f a l l .
Davis (1978) showed t h a t t h e d i f f e r e n c e i n
t h e r e s u l t s occurs because he had used year-round d a t a , whereas Namias only considered t h e t r a n s i t i o n from summer t o f a l l .
Using s e a s o n a l l y s t r a t i f i e d
s t a t i s t i c s , Davis found t h a t up t o 20% of t h e variance of autumn and winter SLP anomalies could be p r e d i c t e d from p r i o r observations of e i t h e r SST o r SLP,
while t h e r e was no p r e d i c t i v e s k i l l f o r SLP i n t h e o t h e r seasons. These s t u d i e s suggest t h a t u s e f u l climate f o r e c a s t s based on t h e observed SST anomaly f i e l d can be made, a t l e a s t f o r some seasons.
The h i g h l y abnormal
w i n t e r of 1976-1977 was c h a r a c t e r i z e d i n t h e United S t a t e s by s e v e r e cold over t h e c e n t r a l and e a s t e r n p a r t s of t h e country, and drought over t h e West.
Figure
1 reproduces t h e f o r e c a s t made by N a m i a s (1978) with a very simple p r e d i c t i o n
model; t h e winter SST anomaly p a t t e r n was estimated from t h e f a l l p a t t e r n using p e r s i s t e n c e and advection around t h e North P a c i f i c gyre, and from t h i s t h e US climate was p r e d i c t e d on t h e b a s i s of teleconnections observed i n t h e p a s t . The g e n e r a l p a t t e r n s of a i r temperature and p r e c i p i t a t i o n w a s reasonably a n t i c i p a t e d , b u t t h e extreme s e v e r i t y of t h e w i n t e r was not f o r e t o l d .
Interestingly,
Davis (1978) notes t h e i n a b i l i t y of h i s s t a t i s t i c a l model t o p r e d i c t t h e North P a c i f i c SLP during t h e same winter. S i m i l a r c o r r e l a t i o n s between SST anomalies and subsequent weather p a t t e r n s might be found i n o t h e r oceans.
R a t c l i f f e and Murray (1970) d i s c u s s some
evidence of a r e l a t i o n s h i p between l a r g e s c a l e S S T anomalies south of Newfoundland and weather p a t t e r n s over Europe during t h e following month, mainly during t h e fall.
Near t h e equator, t h e a i r - s e a i n t e r a c t i o n s a r e s t r o n g e r .
For example,
Newel1 and Weare (1976) found t h a t t h e t r o p i c a l SSTIanomalies i n t h e P a c i f i c were leading t h e g l o b a l l y averaged temperature i n t h e t r o p i c a l troposphere by 6 months.
More r e c e n t l y , Hasselmann and B a r n e t t (1978) were a b l e t o p r e d i c t
SST and atmospheric anomalies i n t h e t r o p i c a l P a c i f i c up t o about 7 months i n
advance.
39
Fig. 1. Predicted and observed temperature and p r e c i p i t a t i o n p a t t e r n s f o r Winter 1976-77 made on 1 December 1976. Shaded a r e a s denote t h e t h r e e c a t e g o r i e s below ( B ) , near ( N ) and above normal ( A ) whose range a r e determined a s t e r c i l e s from a 30-year c l i m a t i c record (from Namias, 1978).
MODELLING SEA SURFACE TEMPERATURE ANOMALIES
Because of wind mixing and t u r b u l e n t convection, t h e upper l a y e r of t h e ocean
is nearly homogeneous.
Large seasonal changes a r e observed i n t h e temperature
and depth of t h e mixed l a y e r , b u t t h e r e i s p r a c t i c a l l y no seasonal v a r i a b i l i t y below t h e l a r g e s t depth it reaches i n l a t e winter.
Thus, it seems s u f f i c i e n t
to consider t h e near s u r f a c e l a y e r s f o r understanding t h e SST v a r i a b i l i t y over periods of a few years. The h e a t exchanges and t h e wind s t r e s s a t t h e sea s u r f a c e f l u c t u a t e with a dominant time s c a l e of a few days, mainly on t h e synoptic s c a l e of motion, r e f l e c t i n g t h e r a p i d changes i n t h e weather.
The weather has l i m i t e d p r e d i c t a b i l i t y ,
hence can be represented a s a s t o c h a s t i c process on t h e c l i m a t i c time s c a l e s . Although t h e oceanic mixed-layer
responds only weakly - because of i t s i n e r t i a -
40 t o t h e s e d a i l y f l u c t u a t i o n s , t h e i n t e g r a l e f f e c t of t h e s t o c h a s t i c atmospheric forcing is large.
Frankignoul and Hasselmann (1977) have shown t h a t random walk
SST f l u c t u a t i o n s with l i n e a r l y growing variance develop, u n t i l negative feedback processes lead t o a s t a t i s t i c a l l y s t e a d y s t a t e .
Using a copper p l a t e model of
t h e oceanic mixed l a y e r , they suggested t h a t t h e SST anomalies T'
can be
approximated by a f i r s t - o r d e r a u t o r e g r e s s i v e (Markov) process obeying t h e equation aT' = _ v' -
at
where v
h
'
-
AT'
(4)
r e p r e s e n t s t h e anomalous h e a t exchanges of s h o r t c h a r a c t e r i s t i c time
s c a l e T ~ ,h t h e ( c o n s t a n t ) mixed-layer depth and X t h e negative feedback processes. The SST anomaly spectrum F
where F,
(0)
T
(0)
i s then given by
denotes t h e (approximately) c o n s t a n t power d e n s i t y of t h e forcing
a t low frequencies
( t h e white n o i s e l e v e l ) .
With
A - ( a few month)-',
t h e model
( 4 ) reproduces t h e main s t a t i s t i c a l f e a t u r e s and orders of magnitudes of t h e SST
anomalies observed i n t h e m i d - l a t i t u d e s c e n t r a l oceans ( s e e a l s o Reynolds,
1978;
Frankignoul, 1978). In t h i s s e c t i o n , we consider a more r e a l i s t i c mixed l a y e r model and d i s c u s s t h e d i f f e r e n t mechanisms c o n t r i b u t i n g t o t h e generation and decay of SST anomalies I t i s found t h a t equation ( 4 ) d e s c r i b e s indeed, t o a reasonable approximation,
t h e r a t e of change of SST anomalies i n regions f a r from s t r o n g c u r r e n t s and l a r g e mesoscale eddy a c t i v i t y .
A
s l a b model of t h e mixed-laver
We r e p r e s e n t t h e upper oceanic l a y e r s by a well-mixed
l a y e r of depth h and
uniform temperature 8' overlying a s t r a t i f i e d region of temperature B s ( z ) . s u b s c r i p t t denotes t u r b u l e n t f l u c t u a t i o n s and t h e symbol
A
If the
a Reynolds averaging,
t h e thermodynamics energy equation can be w r i t t e n
where u and w denote t h e h o r i z o n t a l and v e r t i c a l v e l o c i t y , p t h e water d e n s i t y , C i t s s p e c i f i c h e a t , q t h e h e a t sources due t o s u r f a c e h e a t exchanges and
r a d i a t i o n e f f e c t s , and
v
a,,a
= (-
a I n t e g r a t i o n of q).
z = 0 ) t o j u s t below t h e mixed l a y e r y i e l d s
( 6 ) from t h e s u r f a c e ( a t
41
h
a
-T
aT
+
+ A8w
-
hU.VT
=
2 XhV T
-
+
(7)
DC
where T and U denote averaged values over the mixed-layer.
-
incompressibility condition V-u layer. w = e
aw = 0, +aZ
The entrainment velocity w
A
ah at
(-
+
Here we have used the
and assumed zero fluxes below the mixed-
is defined by
-
V-hU)
Following Kraus and Turner (1967), we have introduced the functionA= 1 for > 0 and zero otherwise, because the mixed-layer temperature does not
ah at + V'hU
-
change by detrainment.
In (7), A8 = T
-
8 is the temperature jump at the bottom
of the mixed-layer, and Q the total heat flux across the air-sea interface. have introduced a constant horizontal diffusivity coefficient horizontal mixing.
x
We
to parameterize
Without advection, ( 7 ) and (8) reduce to the mixed-layer
equations used by Kraus and Turner (1967), and others. We decompose each variable into a (seasonally varying) mean denoted by an overbar, and a random fluctuation or anomaly denoted by a prime.
The SST anomaly obeys
the equation
-
Aegie -
A8'w; (m)
(1)
+ Ae'we' + x(V~T* + DIV~T + (n)
(0)
D'V~T'-
(P)
(9)
D'V 2TI) (r)
d a where - = - + U-V is the time derivative following the mean notion, D' = ht/g dt at the ratio of anomaly to mean mixed-layer depth, and We = we/h. Although the
-
magnitude of the different terms in (9) depends on scale, geographical position and season, some simplifications can be made for typical central ocean conditions. -1 We shall consider large scale SST anomalies, say of wavenumber k 5 2~/1000 km . The rate of change of the mixed-layer depth depends on the energy input by the wind, and the rate of turbulent energy production and dissipation in the mixedlayer (e.g., Niiler, 1977).
This equation will not be written here.
It is suf-
ficient to remark that the characteristic time scale of w ' is normally the one T
of the wind forcing.
From ( 8 ) , one sees that the dominant time scale of h'
(or 0 ' ) will be determined by the rate of energy dissipation.
It has not been
calculated, but experiments with one-dimensional mixed-layer models generally suggest that it is only slightly larger than
T
(i.e. dissipation occurs rapidly).
42
Approximate equation for the rate of change of SST anomalies a. SST anomaZy f o r c i n g .
The fluctuations in the surface heat fluxes, term (a),
play an important role in the generation of SST anomalies (e.g., Jacob, 1967; Adem, 1975; Clark, 1972; Frankignoul and Hasselmann, 1977).
The main contribution to
these anomalies arise from the short time scale fluctuations in the flux of latent heat HL and, to a lesser extent, sensible heat H solar radiation H H
B
R
.
Fluctuations in the incoming
seem mainly important during summer, and the back radiation
contributes essentially a small negative feedback (see below).
Frankignoul
(1978) has suggested that stochastic forcing by local heat fluxes was most efficient in generating SST anomalies during summer and fall. Term (b) describes the influence of mixed-laye5 depth anomalies on the forcing dT -). -It should be most important dt dT . during spring and early summer, when D' 0(1), and - 1s large. A case study dt by Clark (1972) suggests that term (b) is comparable to term (a) during summer
by the mean heat fluxes (which mainly determine
-
and twice as large during spring, but negligible otherwise. On the basis of simple numerical experiments Willebrand (personal communication) has suggested that the fluctuations in the mixed-layer depth terms (k), (l), and (m) discussed below
--
--
including contributions from
may be as efficient as the local heat
exchanges in generating SST anomalies. Further investigations are needed. The mean products (c), (f), (h), (1) , (n), and (r) contribute to SST fluctuations at zero frequency (neglecting seasonal variations) and are not of interest here. Terms (d) and (e) represent advection by anomalous currents. Namias (1965) has suggested that temperature advection by anomalous ageostrophic wind-driven currents is the main generating mechanism for SST anomalies, but Clark (1972) found that local heat exchanges were more important during summer and fall. However, these studies are based on an empirical formula relating linearly surface drift and geostrophic wind, which corresponds typically to a very small Ekman depth.
It seems more consistent with observations to assume that the
Ekman transport is (nearly) uniformly distributed over the mixed-layer depth (Gill and Niiler, 1973).
-
This yields ITl/pfh as typical drift current magnitude,
where .T is the surface wind stress and f is the Coriolis parameter. The advection terms can then be conveniently compared with the heat exchange term (a) in the spectral domain, knowing the white noise level F stress and heat flux spectra:
(0)
and F H
(0)
of the wind
43 At
ocean weathership (OWS) P (50°N, 145'W),
(latent plus sensible heat flux only), FT (afterFisse1 et al., 1976). VT = 5 x
one has FH (0)
=
(0)
=
9 2 -4 4 x 10 W m /Hz
2 x lo4 N2 m-'/Hz
(one component)
With f = 1.1 x 10-4s-', C = 4 x lo3 J kg-' "C-l,
OC m-' (in general, one has
v?;
>> VT'), the ratio (10) is about
0.2 and suggests that local heat exchanges are more efficient than temperature advection in the North-east Pacific (note that which explains Clark's result). and F
(0)
vy is minimum
At lower latitudes, FH
(0)
in summer and fall,
generally increases
decreases, and the ratio (10) should be even smaller.
However, in
regions of large temperature gradient (VT 1 10-50C m-' in the vicinity of the Subarctic front, between 40 and 45ON), wind driven currents are likely to become important.
Case studies by Daly (1978) suggest comparable effects of local heat
exchanges and temperature advection in the North Atlantic. have
documented
Pettersen et al. (1962)
the distribution of heat exchange in relation to the winds and
weather pattern of typical oceanic cyclones (see also Simpson, 1969).
It can be
shown that the SST anomaly patterns induced by heat exchange and advection (assuming, for example, a uniform meridional SST gradient) will often be rather similar (see also Daly, 1978).
This may explain why observed SST changes have
been reproduced with some success using advection only (e.g., Namias, 1965). Quasi-geostrophic eddies also contribute to terms (d) and (e). Eddy frequency spectra are red, with maximum variance at periods eddy energy is in the oceanic mesoscale
-
-
O(50 km).
0(100d), and most of the Hence, eddies mainly cause
a low-frequency, small scale noise, whose magnitude can be decreased by spatially averaging the SST observations.
A comparison with the heat exchange term (a)
gives :
if we neglect the eddy modulation of the mixed-layer depth.
Here E(w) is the
near surface eddy kinetic energy spectrum, which varies strongly with the geographical position. has E(W) 2 2 x
lo4
In the MODE region that has rather large eddy activity, one m2 s-'/Hz
at 100 day period (this value corresponds to 500 m
depth, hence the inequality sign), F ( 0 ) = 10" W2 m-4/Hz (a guess), h = 50 m H and VT = 3 x OC m-l. Then the ratio (11) is 2 1 and the eddy signal is large.
Correspondingly, mesoscale eddies should be detectable from their sur-
face signature, as observed by Voorhis et al. (1976).
In the central part of
the oceans, however, the eddy activity is smaller than in the MODE region and the eddy noise should be smaller (see also Gill, 1975). In general, advection effects are smaller than local heat exchanges.
Conse-
quently, terms (9)and (i) are likely to be small, except perhaps during spring and early summer, when D'
r~
O(1).
L4
Terms (k), (l), and (m) represent the effect of anomalies in the entrainment velocity and temperature jump at the bottom of the mixed layer.
During the cool-
ing season (fall and early winter), the entrainment terms can be very large. Camp and Elsberry (1978) have shown that during atmospheric events with large wind increase, the entrainment heat flux is generally more important than the surface heat flux at high latitudes (OWS P), where the wind forcing is large. At lower latitudes, the entrainment heat flux is smaller, but not negligible. The overall influence of these terms is likely to be rather small, except perhaps at high latitudes.
b.
Feedback mechanisms. The back interaction of SST anomalies on the atmosphere
determines the role of the upper ocean in climatic changes.
The observations
suggest a complex interplay between the large scale atmospheric and oceanic circulations, involving possibly a number of ocean-atmosphere feedback mechanisms (Bjerknes, 1969; Reiter, 1978).
Some theoretical support for the existence of
a positive feedback has been presented by White and Barnett (1972), and Pedlosky (1975). However, these studies are based on highly idealized models of the coupled ocean-atmosphere circulation, and their relevance difficult to establish. In the present context, the local interactions are mainly of interest, whereby SST anomalies may modify the atmospheric circulation in their vicinity.
Namias
(1963) has argued that there is a positive feedback between SST anomalies and weather patterns:
warm SST anomalies cause an intensification of the cyclo-
genesis, and the increased cyclonic activity keeps the water warm, mainly by altering the local heat exchanges.
However, Frankignoul and Hasselmann (1977)
have suggested that term (a) causes in fact a negative feedback.
This is seen
by using the bulk formulae for the sensible and latent heat fluxes
H S
+
HL = C
H
(l+B)PaCa (Ta-T)]ya]
where CH is the bulk transfer coefficient for the sensible heat flux, B the Bowen ratio of latent to sensible heat flux, and the superscript indicates atmospheric variables.
If we assume that the atmosphere is not affected by the SST
anomalies, a strong linear negative feedback is found, with
month)-'.
=
-1 8 m s , h = 50 m, we find
= (1.3 HS+HL However, the air temperature adjusts itself to a value fairly close
For CH = 1.5 x
B = 3,
to the local SST, hence the negative feedback is not as strong as indicated by (13).
In mid-latitudes, this adjustment depends on the radiative-convective
45
properties of the atmospheric boundary layer and on the large scale eddy flow causing horizontal advection and vertical heat exchanges. This is difficult to model analytically. However, some information can be found in the response of atmospheric GCM to prescribed (and fixed) SST anomalies. Figure 2 suggests that, on the SST anomaly time scale, Ta' is approximately proportional to T' .
The
turbulent heat flux does not depend linearly on T' because of the exponential dependence of the saturation vapor pressure on T.
In the temperature range of
observed SST anomalies, however, a linear dependence seems appropriate, and a
AH S+HL
better estimate of (13) is
= (4 month)-'.
No firm link has been established between SST anomalies and cloudiness, hence
the contribution of HR to feedback is not known. The net long wave radiation HB into the mixed-layer may be estimated using Efimova's formula (Reed, 1976): HB = - 0.97 U T4 (0.254-0.00495e) (1-yC)
where U is the Stefan-Boltzman constant, e the vapor pressure in millibars, C the fractional cloud cover and y a constant depending on the type of cloud (varying between 0.9 and 0.25).
The temperature dependence of e can be found by
assuming that e is 80% of the saturation water pressure e
and using the Clausius-
Clapeyron equation
de, =
5.4
3
lo3
dT
T2
Introducing T =
7+
T' and keeping only the dominant terms, we find that the
long wave radiation causes a linear negative feedback given by =-(-)
HB
a
-HB'
=
1.4
pCc
10-'~-~
(1 - 0.0156eS
h
;= 50 m and T = 295'K, For I
-
+ 21.10 es) (1 - yC)
(16)
T
the negative feedback in the absence of clouds is
A
= (21 month)-l. Even smaller values are found when taking cloud effects HB into account. A discussion of the negative feedback caused by the atmosphere
can also be found in Gill (1978). Term (m) contributes to feedback during periods of entrainment, since A e ' = T' - O ' s .
The character of this intermittent feedback due to vertical mixing
appears to be complex.
If there is no anomaly below the mixed layer ( 8 '
-
term (1) represents a negative feedback, with Ae = w /h.
mixed layer begins to deepen, this negative feedback is very strong. and Elsberry (1978). we take as characteristic values w 35 m, so that
A
=
(2 month)-'.
Later in the season, w
=
0 ) .
In early fall when the
= 6 x
From Camp
m s-l
,
-
h =
decreases, h increases
0
0.
00.
U
0 0
.
0
<
C
+
+(
)*
xx ++ ++
0
0-0
m
&+':
X
+
0
+t
+ +
0
X f t
:*
0.
0
.
+ **
. c. 0 8
+ O X 00
I ' I ' 0
' I
3
c a
mw-
-
0
" =0
-
ZrLz
m
3 A0
; D I
-
Dcn-
v)
c
I
)
-
0.
I
OD-
I
-
0--
O
D O
0
I
c
n
P
l
u
X
'
. c
.
00
0 0
0 0
0
0
* O
0 0
a
a
8a
B
0
+
. . .
+
0
0
0 0
N
-
0 0 0
0 0
I I
I
G I & -
TURBULENT HEAT FLUX ANOMALY (Wrn-2) AIR TEMPERATURE ANOMALY ("C)
-
"
v
SEA SURFACE TEMPERATURE ANOMALY ("C) Scatter plot of 30-day averages of the a i r temperature anomaly and the sensible plus latent heat flux anomaly into the atmosphere versus imposed SST anomaly at grid p o i n t s in the GCM experiments by Kutzbach et al. ( 1 9 7 7 ) . The d i f f e r e n t symbols correspond to the 4 experiments described in the paper. Fig. 2.
47
and the negative feedback becomes progressively small.
On the other hand,
will generally contribute an intermittent positive feedback.
O'S
This effect is likely
to be most important for large scale SST anomalies (i.e. not advected away) on time scales of several seasons or more.
Indeed, 8 '
will have the same sign as
T' during the preceeding winter, thereby favoring the reapparition (or strengthening) of SST anomalies during the fall. Pacific (White, private communication).
This has been observed in the North Note that the heat fluxes below the
mixed-layer should be taken into consideration for a more detailed study. Finally, horizontal mixing contributes to SST anomaly decay. (0)
For
If we keep term
only (our parameterization is crude anyway) we find
x=
lo3 m2 s - ' , a reasonable value for the MODE region, and wavelength larger
x
used (Csanady, 1973).
< (10 month)-'. At smaller scales, smaller should be mHence, horizontal mixing seems to contribute little to
SST anomaly dynamics.
Our conclusion disagrees with Adem's (1975) suggestion
than 1000 km, one has
that horizontal mixing is important. for
X
c.
An approximate equation.
(=
However, Adem was using too large a value
lo4 m2 s - ' ) in his numerical simulations of SST changes.
Our analysis suggests that, in a first approximation,
the rate of change of SST anomalies can be described by the equation dT' (1 + D ' ) dt = where
AM >
-
Q'-PCT
D' dT
a t -U'.V(?+T') -
o represents mixing processes.
heat flux should perhaps be added.
- A T' M
In high latitudes, the entrainment
Combining the forcing and feedback mechan-
isms, (18) may be written under the form
(1 + D') dT' dt
=
V'
-
!'e.8(?+T')
- AT'
where v' represents the short time scale atmospheric forcing (generally dominated by local heat fluxes), (probably dominated by
the eddy velocity and A the negative feedback processes
AH~+H~).
Statistical features of the SST anomalies
Equation (19) reduced to equation (4) considered by Frankignoul and Hasselmann (1977) in regions far from strong currents and large eddy activity, if the overall influence of D' in the left-hand side is small (see below).
Then, the pre-
48 d i c t e d SST anomaly s p e c t r u m ( 5 ) compares g e n e r a l l y w e l l w i t h m i d - l a t i t u d e s o b s e r v a t i o n s (see i n p a r t i c u l a r Reynolds, 1 9 7 8 ) .
I n F i g . 3, SST o b s e r v a t i o n s a t OWS P
a r e compared t o t h e model f i t ( 5 ) u s i n g
(HS+H~)/PC,k = ( 2 month)-'
25 m.
This value o f
-
k is
V'
=
-
and h =
smaller t h a n t h e o b s e r v e d mean o f 60 m b e c a u s e v ' i s
u n d e r e s t i m a t e d and h v a r i e s s e a s o n a l l y w h i l e e n t e r i n g ( 5 ) a s a q u a d r a t i c q u a n t i t y
(see F r a n k i g n o u l , 1 9 7 8 ) . F i g . 4 r e p r o d u c e s t h e o b s e r v e d c o r r e l a t i o n between l a r g e s c a l e SST and SLP a n o m a l i e s o v e r t h e North P a c i f i c , and t h e c o r r e l a t i o n between SST a n o m a l i e s and a t m o s p h e r i c f o r c i n g p r e d i c t e d from ( 4 ) .
The agreement i s e x c e l l e n t .
was assumed t h a t SLP v a r i a t i o n s a r e d i r e c t l y r e l a t e d t o v ' .
Here it
Note t h a t SLP i s
n o r m a l l y n o t c o r r e l a t e d w i t h a i r t e m p e r a t u r e ( W i l l e b r a n d , p e r s o n a l communication) b u t i s r a t h e r c o h e r e n t w i t h wind s p e e d a t low f r e q u e n c i e s ( F i s s e l e t a l . , 1 9 7 6 ) . A s d i s c u s s e d by F r a n k i g n o u l
(1978), a s e a s o n a l modulation o f t h e s t a t i s t i c a l
-
p r o p e r t i e s o f SST a n o m a l i e s i s e x p e c t e d from ( 1 9 ) , s i n c e Fv (o), h,, e t c . v a r y F i g . 5 shows s e a s o n a l SST s p e c t r a a t OWS P , and t h e model f i t .
seasonally.
The
s e a s o n a l v a r i a b i l i t y i s w e l l r e p r o d u c e d , e x c e p t d u r i n g s p r i n g , where much SST variance is a t rather short period. caused by t h e t e r m D ' during t h a t season,
T h i s h i g h - f r e q u e n c y n o i s e i s presumably
i n t h e l e f t - h a n d s i d e o f e q u a t i o n (19),s i n c e D' i s l a r g e s t
- O(1).
T h i s argument s u g g e s t s t h a t some h i g h - f r e q u e n c y
v a r i a n c e w i l l b e p r e s e n t d u r i n g summer, b u t l i t t l e d u r i n g f a l l and w i n t e r , as observed.
Note t h a t t h e c h a r a c t e r i s t i c t i m e s c a l e o f h ' seems t o b e
The year-round
- O ( 2 0 days).
i n f l u e n c e ' o f t h e D ' term i n (19) s h o u l d , however, b e s m a l l , con-
s i s t e n t w i t h t h e o b s e r v e d h i g h - f r e q u e n c y s l o p e o f t h e SST s p e c t r u m i n F i g . 3 , which i s o n l y s l i g h t l y f l a t t e r t h a n w-2 The SST a n o m a l i e s are a d v e c t e d by t h e mean c i r c u l a t i o n a l o n g t h e o c e a n i c g y r e s , v i a t h e d e r i v a t i v e t e r m i n ( 1 9 ) , as o b s e r v e d by F a v o r i t e and McLain (19731, and Davis ( 1 9 7 6 ) .
5 O(5 c m
s-l)
I n c e n t r a l o c e a n c o n d i t i o n s , t h e mean c u r r e n t s a r e g e n e r a l l y s m a l l and t h e s p e c t r u m ( 5 ) s h o u l d n o t b e s u b s t a n t i a l l y a l t e r e d , as m o s t
o f t h e SST v a r i a n c e i s a t v e r y l a r g e scales. a p p a r e n t i n c r e a s e of t h e feedback f a c t o r k .
However, a d v e c t i o n may l e a d to a n I n d e e d , t h e SST s p e c t r u m a t wave-
-
number k i s g i v e n by
f o r u n i f o r m mean flow.
-
Fv ( k , o ) = F v ( k , o ) / 2 r k ,
Using p o l a r c o o r d i n a t e s and assuming i s o t r o p i c f o r c i n g
w e f i n d a t l o w frequencies
w <<
x
49
(OWS P, 50"N, 145OW) I 9 58 - I 967
-t*
0.2 0
14.612.3
cu^ 0.15 0 0-
a m I-
n
0.10
v)
X
z 0.05
W 3
0 W
a
LL
0.00
- 0.05 I O - ~
I
1
1
I
I
10-3 lo-* lo-' I FREQUENCY (CYCLES / DAY)
10
Fig. 3 . Variance spectrum of t h e SST a t OWS P , a f t e r F i s s e l e t a l . (1976). The v e r t i c a l e r r o r b a r s r e p r e s e n t approximately 95% confidence i n t e r v a l s . A t t h e annual and semi-annual p e r i o d s , t h e s p e c t r a l l e v e l s a r e o f f - s c a l e a s i n d i c a t e d by t h e bold arrows with t h e numerical values w r i t t e n below t h e arrows. The continuous curve i s t h e model p r e d i c t i o n (from Frankignoul, 1978).
so t h a t t h e f i t t e d value of
w i l l i n c r e a s e with k.
For
= ( 4 month)-',
U =
5 cm s - ' ,
t h e s p e c t r a l f l a t t e n i n g a t low frequencies would suggest a feedback
f a c t o r of
( 2 month)-'
a t 1000 km wavelength, a s u b s t a n t i a l i n c r e a s e .
In r e a l
s i t u a t i o n s , advection w i l l obviously c r e a t e more complex e f f e c t s , although they should not be q u a l i t a t i v e l y d i f f e r e n t .
This could explain i n p a r t ( t o g e t h e r
with t h e e f f e c t of entrainment discussed above) why values of
estimated from
observations i n c r e a s e with decreasing SST anomaly s c a l e , and vary with t h e geographical p o s i t i o n (Reynolds, 1978).
Fig. 6 i l l u s t r a t e s t h e s p a t i a l dependence
50
.61
1
.4
.2
0 -.2
'
I
I
1
I
I
1
I
- I2
I
1
1
-6
I
I
I
I
I
I
I
0
I
I
1
I
I
6
I
I
I
I
I
I
I
J
I2
::r---1
.2
0
-12
-6
0
6
I2
months Fig. 4. Upper panel: observed correlation between the amplitude of the dominant empirical orthogonal mode o f SST and SLP anomalies over the North Pacific, after Davis (1976). Lower panel: theoretical correlation curve (from Frankignoul and Hasselmnn, 1977).
of A in the North Pacific, since the SST anomaly scale generally decreases with increasing empirical orthogonal function (EOF) number.
Note that the Nyquist
wavelength in these data is 1000 km, so that our rough estimate (21) seems quantitatively correct.
CONCLUSIONS
Using a slab model of the oceanic mixed-layer, we have found that large scale mid-latitude SST anomalies can be represented as a first-order autoregressive process in regions with small mean current, as suggested by Frankignoul and Hasselmann (1977). The SST anomalies are continuously generated by the natural variability of the atmospheric fluxes at the air-sea interface.
This stochastic
forcing seems dominated by the fluctuations in the local heat fluxes, although mixed-layer depth variations are important during certain seasons.
In regions
with large mean SST gradient, advection by anomalous wind-driven currents also plays a role.
The decay of the SST anomalies can be modelled by a linear nega-
tive feedback, that seems largely controlled by their back interaction on the atmosphere.
The SST anomalies are also advected by the mean oceanic circulation,
(M,GbI 'N,O5
L961- 8561 'd SMO)
t
U31NIM
80'o
9NlUdS Y
N
I
I
I
I
wwns
F
2
80'0
D
z
-I
0
m
:
0'0
J
L
X
I\I
-<
0
80'0
c
0
m
;D
-++
7
11Vd
L
80'0 II
01 1 0
II
0I1 1 I- 0
0I1 1 2-0
E-, E-, 0'0
01
FREQUENCY (CYCLEWDAY) Fig. 5. Observed and predicted variance spectra of SST anomalies at OSW P for each of the seasons, as in Fig. 3 ( f r o m Frankiqnoul, 1978).
52
= I
2.0
L
)r
c .
0
t4
0
0
1.5
0
0,
Y
..c
0
Y
0)
0
1.0
0
CI
i b L
b
0
0
0.5
a.
1 0
I
g o
c
1
1
I
1
I
I
I
I
3
5
7
9
11
13
15
Order of
17
EOF
Fig. 6. Fitted value of A for the EOF of SST anomalies in the North Pacific that can be represented by the model (4). These values should be multiplied by 2T to compare with values discussed here (from Reynolds, 1978).
and mesoscale eddies contribute essentially a low-frequency, small-scale noise. On the climatic time scale, the weather forcing is not predictable, hence the generation of SST anomalies cannot be forecasted. cay and advection is predictable.
On the other hand, their de-
Hasselmann (1976) had discussed the predict-
ability of stochastic climate models.
In the SST anomaly phase space, the sto-
chastic forcing by the atmosphere induced a diffusion, whereas feedback and advection cause a propagation.
Because of the diffusion, SST prediction will
always entail some degree of statistical uncertainty.
For a statistically
stationary SST state, the maximal predictive skill for the SST field may be significant, but limited to skill parameters of order 0.5. A substantial part of the SST anomaly predictability is controlled by the
feedback process, and it is therefore of much importance to investigate these mechanisms further.
The strength of the negative feedback varies with the SST
anomaly scale, the geographical position and the season.
Since the intensity
of the forcing (thempredictablepart) also varies with season, the SST predictive skill is likely to depend on the time of year.
53 For climate prediction purposes, the back interaction of the SST anomalies must be understood globally, rather than locally. The problem is formidable, and progress will have to rely mostly on numerical simulations with GCM of the atmosphere having some kind of interacting oceanic surface layer.
The obser-
vations suggest that the climate predictability varies with the season, and it seems of interest to establish if this is partly caused by the seasonal variability of the SST anomaly field. Perhaps it is not a coincidence that Davis (1978) found some climate predictability for fall and winter, while SST anomalies are largest and most persistent during summer and fall (Fig. 5).
ACKNOWLEDGEMENTS
The GCM data were provided by Dr. R. Chervin who is gratefully acknowledged. Support from the National Science Foundation, office of the International Decade of Oceanic Exploration, under grant OCE 77-28349 is gratefully acknowledged. This is MODE Contribution number 107.
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Pedlosky, J., 1975. The development on thermal anomalies in a coupled oceanatmospheric model. J. Atmos. Sci., 32:1501-1514. Pettersen, S., Bradbury, D.L., and Pederson, K., 1962. The Norwegian cyclone models in relation to heat and cold sources. Geophys. Publn., 24:243-280. Ratcliffe, R.A.S., and Murray, R., 1970. New lag associations between North Atlantic sea temperature and European pressure applied to long-range weather forecasting. Quart. J. Roy. Met. SOC., 96:226-246. Reed,R.K., 1976. On estimation of net long-wave radiation from the oceans. Geophys. Res., 81:5793-5794.
J.
55 Reiter, E.R., 1978. The i n t e r a n n u a l v a r i a b i l i t y o f t h e ocean-atmosphere system. J. A t m o s . S c i . , 35:349-370. Reynolds, R.W., 1978. T e l l u s , 30:97-103.
Sea s u r f a c e t e m p e r a t u r e i n t h e North P a c i f i c o c e a n .
1976. L a r g e - s c a l e air-sea i n t e r a c t i o n s w i t h Salmon, R . , and H e n d e r s h o t t , M.C., a s i m p l e g e n e r a l c i r c u l a t i o n model. T e l l u s , 28:228-242. Simpson, J . , 1969. On s o m e a s p e c t s o f sea-air i n t e r a c t i o n i n m i d d l e l a t i t u d e s . Deep-sea R e s . , S u p p l . t o v o l . 16:233-261. Shukla, J . , 1975. E f f e c t o f a r a b i a n s e a - s u r f a c e t e m p e r a t u r e anomaly on I n d i a n J. A t m o s . S c i . , s u m e r monsoon: a n u m e r i c a l e x p e r i m e n t w i t h t h e GFDL model. 32: 503-511. Tenberth, K.E., 1975. A q u a s i - b i e n n i a l s t a n d i n g wave i n t h e S o u t h e r n Hemisphere and i n t e r r e l a t i o n s w i t h s e a s u r f a c e t e m p e r a t u r e . Q u a r t . J. Roy. M e t . SOC., 101: 55-74. Voorhis, A . D . , S c h r o d e r , E.H., and L e e t m a a , A . , 1976. The i n f l u e n c e o f deep mesoscale e d d i e s on t h e sea s u r f a c e t e m p e r a t u r e i n t h e North A t l a n t i c subt r o p i c a l convergence. J. Phys. Ocean., 6:953-961. White, W.B., and B a r n e t t , T.P., 1972. A servomechanism i n t h e ocean-atmosphere system o f t h e m i d - l a t i t u d e N o r t h P a c i f i c . J. Phys. Ocean, 4:372-381.
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57
LOW
FREQUENCY MOTIONS IN THE NORTH PACIFIC AND THEIR POSSIBLE GENERATION BY
METEOROLOGICAL FORCES L. MAGAARD
Department of Oceanography, University of Hawaii, Honolulu, Hawaii 96822 (U.S.A.) ABSTRACT Various sets of internal temperature data from the North Pacific Ocean (time series up to 20 years) have been analyzed with respect to baroclinic Rossby waves. Fluctuations with periods between one and five years can, to a large extent, be explained as first order internal Rossby waves. A theoretical model for local meteorological generation of these Rossby waves has been developed. An application of this model requires the frequency wave number spectra of wind stress and buoyancy flux (or temperature) at the sea surface as forcing functions. Such spectra are presently being determined from observations in the eastern North Pacific. INTRODUCTION Our present understanding of oceanic processes varies greatly from process to process.
For example, much successful work has been done to observe and describe
the sea state, to understand its generation, and to forecast it (e.g. Barnett and Kenyon, 1975). ocean.
Much less, however, is known about low frequency processes in the
We recognize now, mainly from temperature observations, the existence of
significantly strong oceanic fluctuations with time scales of several months to several years (e.g. White, 1977).
But the identification of processes underlying
such fluctuations has just begun.
The generation mechanisms of such processes are
still largely unknown, and the prediction of such processes is a very distant goal. In this paper we review some of our work, done under the NORPAX project, utilizing temperature data from the North Pacific to learn more about low frequency oceanic motion.
We have analyzed data by fitting models to them in order to identify
low frequency processes, and we have studied one of the many possible generating mechanisms of such processes. We consider this a prerequisite for the final goal of forecasting low frequency processes in the ocean. DATA ANALYSIS AND MODEL FITTING We have at our disposal the following data sets: 1) Monthly mean values of temperature from both hydrographic and XBT casts at
various depths in six 2-degree squares between California and Hawaii.
These
58 five-year records are described in Emery and Magaard (1976). 2) Annual mean values of baroclinic potential energy (a bulk measure for the baroclinic temperature fluctuations in the upper 500 m) in the area 20-50NI 145E-130W. These 20-year series have been prepared by White (1977). 3) Objectively analyzed monthly mean values of temperature between San Francisco
and Hawaii from 1966-1974. These data have been prepared by Dorman and Saur (1978). 4) The TRANSPAC data.
These XBT data have been collected since 1975 under the
"Anomaly Dynamics Study" (ADS) program, a subproject of NORPAX.
Objectively
analyzed monthly maps of temperature in the upper 400 m of the area 30-50N, 160E-130W are currently being prepared by Bernstein and White (Scripps Institution of Oceanography).
The records go back to the beginning of 1975. Maps from the
first 3% years are now available. We hope that the TRANSPAC program can be continued over several more years. We have started our work on data set 1) by developing a model consisting of a random field of free baroclinic Rossby waves.
For this model we assumed
vanishing mean flow, an assumption that is justified for the area of data set 1).
Using a cross spectral fit of the Rossby wave model to the data we have shown
that for wave periods from one to two years, 62 to 78% (average 70%) of the observed variances and covariances of the internal temperature field can be interpreted as first order baroclinic Rossby waves traveling at distinct NW directions (one direction per frequency) with wave lengths between 1200 and 1700 km.
The group
velocities of these waves are directed almost exactly westward and have magnitudes of about 4 . 5 cm s-'.
For smaller wave periods (from the cutoff period of five
months to about nine months) the model can interpret only 38 to 73% (average 52%) of the variances and covariances, and the angular range of propagation directions of these shorter waves widens to isotropic propagation in the western half plane. The results of this study are described in Emery and Magaard (1976). In a subsequent paper, Magaard and Price (1977) fitted a more general model, containing the Rossby waves as a special case, to the same data.
The result was that, for
wave periods from one to two years, the Rossby waves led to the best fit again, which reinforced the idea that baroclinic Rossby waves play a central role in the low frequency fluctuations of the North Pacific. Data set 2) is presently under study by Price and Magaard.
From preliminary
results it appears that Rossby waves may not be limited to the restricted area of data set 1) but also may play a role in larger portions of the North Pacific. Moreover, even fluctuations with periods larger than two years (up to about five years) seem to be dominated by Rossby waves. The analysis of the TRANSPAC data requires a model that considers the North Pacific Current which dominates the area of these data.
Such a model was developed
by Kang and Magaard (1978). Their work shows that the influence of the current on the dispersion features of the Rossby waves is small for the barotropic shear mode, but is significant for the baroclinic shear mode.
A model fit to the TRANSPAC
data is now in preparation. A preliminary study has shown that Rossby waves appear to be of importance in this case, too.
We conclude that even though much
of the nature of the low frequency motions is still unknown, baroclinic Rossby waves do represent a significant portion of such motion. GENERATION OF ROSSBY WAVES After part of the low frequency motion has been identified as Rossby waves the obvious questions are, of course: why are these waves there? Where do they come from? We have studied one of the possible generating mechanisms, local meteorological forcing by low frequency fluctuations of barometric pressure, wind stress, and buoyancy flux at the sea surface (Magaard, 1977). that in this context the barometric pressure is negligible.
We have shown
Wind stress and
buoyancy flux produce divergent horizontal flow (wind-driven and thermohaline circulation, respectively) in surface boundary layers (viscous and diffusive layers, respectively).
The divergence of this flow leads to vertical pumping
at the lower edge of the surface boundary layers: the pumping then generates internal Rossby waves in the continuously stratified interior of our model ocean. A viscous bottom layer results in a finite response even in case of resonance to which we have given most attention. A rough numerical estimation shows that actual fluctuations of wind stress as well as buoyancy flux could generate the waves. Emery (University of British Columbia), Gallegos (Texas A&M University) and Magaard are presently trying to determine actual input functions for the generation model by Magaard (1977). These functions are frequency wave number spectra of wind stress and buoyancy flux.
Since we cannot get enough data for the deter-
mination of the buoyancy flux and since the model provides a connection between the buoyancy flux and the sea surface temperature, we use the temperature spectra instead of the buoyancy flux spectra and calculate the former. This study is based on 11-year time series of wind stress and sea surface temperature from the area 10-40NI 120-160W, which includes the area of data set 1).
We want to test
whether the waves observed by Emery and Magaard (1976) can be generated by means of the model by Magaard (1977) using the above-mentioned wind stress and sea surface temperature data that stem from the Marine Deck of the Environmental Data Service, Asheville, North Carolina, U.S.A.
Preliminary results suggest
that the annual fluctuation of the sea surface temperature should generate the strongest internal Rossby wave response, which contradicts the observational
60 finding that in the actual Rossby wave field the annual signal does not play a preferred role.
To what extent local meteorological forcing actually generates
Rossby waves remains unsolved at this time.
We hope, however, that our further
studies will shed more light on that problem. Another mechanism of internal Rossby wave generation has been proposed: Bryan and Ripa (1978) have done a theoretical study about the generation of Fmssby waves by reflection of a meteorologically generated, eastward-traveling surface fluctuation at the North American Pacific coast. While their highly idealized study shows that such a mechanism is possible in principle, it remains open to what extent this mechanism is actually efficient. DISCUSSION We expect the continued acquisition of data and the development of advanced methods of analysis will increase our insight into the nature of the observed fluctuations. We believe, however, that it will take much more time and effort to successfully tackle the generation problem because of its complexity.
That
leaves the final goal of predicting the low frequency fluctuations far away. ACKNOWLEDGEMENTS This study has been supported by the Office of Naval Research under the North Pacific Experiment of the International Decade of Ocean Exploration; this support is gratefully acknowledged. Hawaii Institute of Geophysics contribution no. 923. REFERENCES
Barnett, T.P. and Kenyon, K.E., 1975. Recent advances in the study of wind waves. Rep. Prog. Phys., 38:667-729. Bryan, K. and Ripa, P., 1978. The vertical structure of North Pacific temperature anomalies. J. Geophys. Res., 83:2419-2429. Dorman, C.E. and Saur, J.F.T., 1978. Temperature anomalies between San Francisco and Honolulu, 1966-74, gridded by an objective analysis. J . Phys. Oceanogr., 8:247-257. Emery, W.J. and Magaard, L., 1976. Baroclinic Rossby waves as inferred from temperature fluctuations in the Eastern Pacific. J. Mar. Res., 34:365-385. Kang, Y.Q. and Magaard, L., 1978. Stable and unstable Rossby waves in the North Pacific Current as inferred from the mean stratification. Dyn. Atmos. Oceans, in press. Magaard, L., 1977. On the generation of baroclinic Rossby waves in the ocean by meteorological forces. J. Phys. Oceanogr., 7:359-364. Magaard, L. and Price, J.M., 1977. Note on the significance of a previous Rossby wave fit to internal temperature fluctuations in the Eastern Pacific. J. Mar. Res., 35:649-651. White, W.B., 1977. Secular variability in the baroclinic structure of the interior North Pacific from 1950-1970. J . Mar. Res., 35:587-607.
61
WIND-INDUCED
LOW-FREQUENCY
Jiirgen W i l l e b r a n d l
OCEANIC VARIABILITY
and George P h i l a n d e r
G e o p h y s i c a l F l u i d Dynamics P r o g r a m , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , N e w J e r s e y 08540 'Present
affiliation:
I n s t i t u t fiir Meereskunde an d e r U n i v e r s i t a t K i e l , 2300 K i e l , Germany
ABSTRACT
The g e n e r a t i o n o f l o w - f r e q u e n c y c u r r e n t f l u c t u a t i o n s i n t h e o c e a n by v a r i a b l e w i n d s i s r e c o n s i d e r e d . O b s e r v e d s p e c t r a l c h a r a c t e r i s t i c s of t h e wind f i e l d o v e r t h e o c e a n a r e d i s c u s s e d . A s i m p l i f i e d a n a l y t i c a l model i s u s e d t o d e r i v e s p e c t r a l p r o p e r t i e s of t h e o c e a n i c r e s p o n s e t o b r o a d b a n d a t m o s p h e r i c f o r c i n g . The r e s u l t s q u a l i t a t i v e l y a g r e e w i t h a m o r e r e a l i s t i c n u m e r i c a l m o d e l , a n d may h e l p t o e x p l a i n some a s p e c t s o f d e e p - s e a c u r r e n t meter o b s e r v a t i o n s . INTRODUCTION
V a r i a b i l i t y i n s p a c e and t i m e o f
a l l f i e l d s which d e s c r i b e t h e
o c e a n i c s t a t e i s an u b i q u i t o u s l y o b s e r v e d phenomenon. remarks w i l l b e concerned w i t h v a r i a b i l i t y
t o s e v e r a l months,
The f o l l o w i n g
on t i m e s c a l e s f r o m a d a y
a n d s p a t i a l s c a l e s f r o m a f e w h u n d r e d t o a few
thcusand kilometers.
(Of
course, oceanic variability
i s not confined
to those scales). A major reason
f o r the current interest
i n these fluctuations is
their potential ability t o affect the large-scale, of
t h e atmosphere-ocean
system.
Variable c u r r e n t s ("eddies")
act with the general circulation, e n e r g y i n t o t h e mean c u r r e n t s .
long-time
behaviour can i n t e r -
e i t h e r e x t r a c t energy from o r feed
Also, t h e y can c o n t r i b u t e t o t h e t o t a l
o c e a n i c h e a t t r a n s p o r t , which i s e s p e c i a l l y important i n r e g i o n s where t h e mean f l o w i s z o n a l ( e . g . ,
a n t a r c t i c c i r c u m p o l a r c u r r e n t ) . Here
any p o l e w a r d h e a t f l u x i s p o s s i b l e o n l y d u e t o f l u c t u a t i n g c u r r e n t s . F l u c t u a t i o n s i n t h e t h e r m a l s t r u c t u r e of ocean a l s o f a l l i n t o t h a t range of temperature
scales.
t h e u p p e r l a y e r s of t h e Anomalies i n s e a - s u r f a c e
(SST) can i n f l u e n c e t h e a t m o s p h e r i c c i r c u l a t i o n and most
62
l i k e l y c o n t a i n t h e c l u e towards any p r o g r e s s i n t h e c l i m a t e p r e d i c t i o n problem; s e e Frankignoul,
1979 ( t h i s i s s u e ) f o r a f u r t h e r discuss-
i o n and r e f e r e n c e s . One o b v i o u s c a n d i d a t e f o r t h e g e n e r a t i o n o f o c e a n i c v a r i a b i l i t y i s d i r e c t f o r c i n g by t h e a t m o s p h e r e ,
a s t h e r e l e v a n t f i e l d s of wind
s t r e s s , a t m o s p h e r i c p r e s s u r e and buoyancy f l u x a t t h e a i r - s e a f a c e are themselves h i g h l y v a r i a b l e i n space and t i m e .
The r e l a t i v e
i m p o r t a n c e of s e v e r a l mechanisms i s d i s c u s s e d i n Magaard, F r a n k i g n o u l and M u l l e r ,
inter-
1 9 7 7 , and
1 9 7 8 . Here w e a r e c o n c e r n e d w i t h some o v e r a l l
aspects of atmospheric generation processes. The d i s c u s s i o n w i l l b e l e d i n terms of a s t o c h a s t i c m o d e l ,
and i t
i s a p p r o p r i a t e t o make a f e w r e m a r k s o n t h e p h i l o s o p h y b e h i n d t h i s approach.
The t e r m s t o c h a s t i c i s t o b e u n d e r s t o o d
ministic,
n o t as o p p o s e d t o d y n a m i c a l .
as o p p o s e d t o d e t e r -
S t o c h a s t i c and d e t e r m i n i s t i c
m o d e l s a r e i d e n t i c a l w i t h r e s p e c t t o t h e g o v e r n i n g e q u a t i o n s o f motion. T h e y d i f f e r i n t h a t s t o c h a s t i c m o d e l s e x p l i c i t l y a c c o u n t f o r t h e broad b a n d o f s c a l e s w h i c h i s p r e s e n t i n a t m o s p h e r i c as w e l l a s o c e a n i c motions.
There s t i l l w i l l b e a dominant scale b u t t h a t can be q u i t e
d i f f e r e n t i n b o t h s y s t e m s e v e n i f t h e d y n a m i c a l m o d e l i s l i n e a r , dep e n d i n g on t h e d e t a i l s of t h e o c e a n i c r e s p o n s e . f o r c i n g model i s
-
at least i n principle
-
Thus, a s t o c h a s t i c
a b l e t o account f o r t h e
o b s e r v e d d i f f e r e n c e s i n s c a l e between o c e a n i c and a t m o s p h e r i c f l u c t u ations,
t h e l a t t e r h a v i n g l a r g e r s p a c e and s m a l l e r t i m e s c a l e s .
A
f u r t h e r c h a r a c t e r i s t i c of s t o c h a s t i c models i s t h a t t h e y a r e normally evaluated i n terms of c e r t a i n s t a t i s t i c a l parameters,
e.g.
probabi-
l i t y d i s t r i b u t i o n s , c o r r e l a t i o n f u n c t i o n s o r energy s p e c t r a , whereas t h e d i r e c t l y observable f l u c t u a t i n g f i e l d s a r e considered t o be real i z a t i o n s o f random f u n c t i o n s . S t o c h a s t i c f o r c i n g m o d e l s a r e b y n o m e a n s new t o o c e a n o g r a p h y . S i n c e t h e o r i g i n a l work o f P h i l l i p s ,
1957, they have been used f r e -
q u e n t l y i n s u r f a c e and i n t e r n a l g r a v i t y wave s t u d i e s .
More r e c e n t l y ,
t h i s a p p r o a c h was a p p l i e d t o p r o c e s s e s w i t h much l a r g e r s p a c e a n d time s c a l e s
( F r a n k i g n o u l and Hasselmann,
1 9 7 7 ; Lemke,
1977; Fran-
k i g n o u l and Miiller, 1 9 7 8 ) . I n t h e f o l l o w i n g , a f t e r a b r i e f
discussion
oZ a t m o s p h e r i c f o r c i n g s p e c t r a , we c o n s i d e r a v e r y i d e a l i z e d m o d e l
f o r t h e g e n e r a t i o n of
f l u c t u a t i n g c u r r e n t s w h i c h may h e l p t o r e l a t e
observed oceanic current s p e c t r a t o t h e meteorological f i e l d s . SPECTRAL CHARACTERISTICS OF ATMOSPHERIC FLUCTUATIONS The r e l e v a n t a t m o s p h e r i c v a r i a b l e s a t t h e sea s u r f a c e a r e w i n d
63 s t r e s s ;t, a t m o s p h e r i c p r e s s u r e p a ,
and buoyancy
flux bf.
Instead
f l u x w h i c h i s p o o r l y known f r o m o b s e r v a t i o n s ,
of t h e buoyancy
u s e t h e a t m o s p h e r i c t e m p e r a t u r e as v a r i a b l e , suming t h a t t h e buoyancy
thereby
we w i l l
i m p l i c i t l y as-
flux is essentially proportional t o the flux
a n d t h a t t h e a i r t e m p e r a t u r e f l u c t u a t e s much f a s t e r
of s e n s i b l e h e a t ,
Within t h e framework of a c o r r e l a t i o n t h e o r y ,
than the sea temperature. the statistical properties
of t h e s e f i e l d s a r e completely determined
by t h e s p e c t r a l t e n s o r
(k,w)
Fij
=
I dE d r < < i ( g , t ) S . ( & + r , t + ~ ) >ie( ~ Z - U T )
(1)
J Here
L
=
(c1,<2,<3,<4)
=
( ~ ~ , r ~ , p ~d e, n T o t e~ s ) t h e v e c t o r o f atmo-
otherwise the notation is standard.
speric variables;
We c o n s i d e r a t y p i c a l m i d - o c e a n ,
mid-latitude
g a r d c o m p l i c a t i o n s due t o i n h o m o g e n e i t i e s atmospheric f i e l d s
,
s i t u a t i o n and d i s r e -
and i n s t a t i o n a r i t i e s i n t h e
thereby excluding d i u r n a l and annual s i g n a l s .
A s t h e network of w e a t h e r s t a t i o n s o v e r t h e oceans i s r a t h e r s p a r s e , (lc,w) i s n o t o v e r l y l a r g e . R e c e n t ij a t t e m p t s t o o b t a i n i n f o r m a t i o n on F i j f r o m a t m o s p h e r i c d a t a h a v e b e e n
o u r e m p i r i c a l knowledge about F
made b y W i l l e b r a n d , Miiller,
1 9 7 8 , a n d Emery e t a l , 1 9 7 8 . F r a n k i g n o u l
and
1978, constructed a simple a n a l y t i c a l expression f o r t h e
a t low f r e q u e n c i e s , w h i c h
wavenumber s t r u c t u r e o f t h e s t r e s s - s p e c t r a
i s c o n s i s t e n t w i t h s e v e r a l known c h a r a c t e r i s t i c s o f t h e s e f i e l d s . F r o m t h e s e i n v e s t i g a t i o n s t h e most i m p o r t a n t p r o p e r t i e s of t h e a t m o s p h e r i c s p e c t r a c a n b e d e s c r i b e d as f o l l o w s : i ) The f r e q u e n c y a u t o s p e c t r a , F i i ( w )
i.e.
frequency
value of w
I
=
Fii(k,w)dk
are w h i t e
i n d e p e n d e n t , below a c e r t a i n f r e q u e n c y wo.
corresponds
t o a p e r i o d of
3
-
The
5 days f o r t h e s t r e s s
spectra,
and i s somewhat l o n g e r f o r p r e s s u r e and t e m p e r a t u r e
spectra.
A t higher frequencies, a l l spectra f a l l off
t e l y as w - ~ , with a=1.5
for the stress spectra,
approxima-
and a Y 2 . 5
for
p r e s s u r e and a i r t e m p e r a t u r e . i i ) A t periods
l a r g e r t h a n 10 d a y s ,
t h e f l u c t u a t i o n s have no p r e -
f e r r e d d i r e c t i o n and a r e s y m m e t r i c Fii(k,w)
= Fii(-lc,w).
Accordingly,
i n wavenumber,
i.e.
t h e r e i s a n e q u a l amount of
e a s t w a r d and westward p r o p a g a t i n g energy.
Only a t h i g h e r f r e -
q u e n c i e s t h e e a s t w a r d p r o p a g a t i n g d i s t u r b a n c e s ( c y c l o n e s ) dominate.
With r e g a r d t o t h e n o r t h - s o u t h
direction,
the fluctuations
are n e a r l y symmetric a t a l l f r e q u e n c i e s . i i i ) F o r z o n a l wave n u m b e r s i n t h e r a n g e 8 - 1 5 , b e h a v e s a s k-8
w i t h f3 = O ( 5 ) .
t h e pressure spectrum
( S t r i c t l y , t h e conclusion from
,
64
a t m o s p h e r i c d a t a i s a kY8-law).
I t i s n o t w e l l known how f a r t h i s
power l a w e x t e n d s t o h i g h e r wavenumbers.
Also,
u r e o f t h e w a v e n u m b e r s p e c t r a i s n o t known.
At
the directional structlow f r e q u e n c i e s o n e may
assume h o r i z o n t a l i s o t r o p y which i s i n a c c o r d a n c e w i t h t h e o b s e r v e d symmetry, b u t o t h e r n o n - i s o t r o p i c
s p e c t r a l d i s t r i b u t i o n s w o u l d a l s o be
consis tent with the data. The v a r i o u s
components of
t h e s p e c t r a l t e n s o r a r e of course not
i n d e p e n d e n t , t h e i r r e l a t i o n d e p e n d i n g o n t h e k i n e m a t i c s o f t h e atmos p h e r i c flow.
The s i m p l e s t m o d e l i s t o r e l a t e t h e w i n d s t r e s s l i n e a r -
l y t o the geostrophic velocity,
leading t o
w i t h a s u i t a b l y chosen c o n s t a n t c. T h i s c r u d e model i g n o r e s t h e d i f f e r e n c e s between s u r f a c e wind and h e n c e d e s c r i b e s a n o n - d i v e r g e n t
g e o s t r o p h i c and
s t r e s s f i e l d , a de-
f i c i e n c y w h i c h c a n b e r e m o v e d e a s i l y o n t h e c o s t o f s l i g h t l y m o r e complex algebra than i n (2). velocity-stress
Furthermore,
t h e n o n l i n e a r c h a r a c t e r of the
r e l a t i o n is i g n o r e d i n ( 2 ) .
a r e t h e r e f o r e more f l a t quency and wavenumber,
The a c t u a l s t r e s s s p e c t r a
( w h i t e ) t h a n t h o s e g i v e n by
( 2 ) , both i n fre-
especially at high frequencies
d period).
(1-10
N e v e r t h e l e s s , t h e model i s s u i t a b l e f o r q u a l i t a t i v e d i s c u s s i o n s . With r e g a r d t o t h e buoyancy f l u x r e s p . a i r t e m p e r a t u r e , a r e l a t i o n t o t h e p r e s s u r e f i e l d i s l e s s obvious.
F r a n k i g n o u l and Hasselmann,
1 9 7 7 , assumed t h a t t e m p e r a t u r e changes a r e due t o a d v e c t i o n o n l y . W i t h a mean n o r t h - s o u t h
t e m p e r a t u r e g r a d i e n t , t h i s l e a d s t o Ta-u2
and analogous r e l a t i o n s f o r Fi4(k,w),
w i t h a n o t h e r c o n s t a n t c*.
1 shows t h e o b s e r v e d c o h e r e n c e b e t w e e n a i r t e m p e r a t u r e and
Fig.
n o r t h w i n d v e l o c i t y a t Ocean W e a t h e r S t a t i o n D . days t o 200 days b o t h v a r i a b l e s are w e l l related, mation
or
A t periods
from 2
(though n o t p e r f e c t l y ) cor-
t h e i r p h a s e d i f f e r e n c e b e i n g v e r y s m a l l . Here t h e a p p r o x i -
(3) works
r e a s o n a b l y w e l l , w h e r e a s i t f a i l s b o t h a t l o w e r and
higher frequencies.
65
L
l r
.8 Lu
U
6 = 0
.6
w
Lu
.4
U
.2 0 FREQUENCY (cpd)
Fig.
1.
Coherence and phase d i f f e r e n c e between n o r t h wind v e l o c i t y a n d a i r t e m p e r a t u r e a t O c e a n W e a t h e r S t a t i o n D (44'N,41°W), b a s e d on o b s e r v a t i o n s b e t w e e n J a n u a r y 1949 a n d A p r i l 1 9 7 2 . Phase p o s i t i v e i f wind v e l o c i t y leads air temperature.
A SIMPLE STOCHASTIC MODEL
The n a t u r e of
t h e oceanic response t o v a r i a b l e winds, a i r pres-
s u r e o r buoyancy f l u x s t r o n g l y depends on t h e s p a c e and t i m e s c a l e s of t h e f l u c t u a t i o n s .
Especially, it is c r u c i a l whether o r not these
s c a l e s c o i n c i d e w i t h t h o s e of resonance can occur.
f r e e waves i n t h e o c e a n , i n which c a s e
I n the s u b i n e r t i a l frequency range,
t h e only
f r e e waves a r e Rossby waves which h a v e a w e s t w a r d p h a s e p r o p a g a t i o n . Because t h e a t m o s p h e r e i s dominated by e a s t w a r d p r o p a g a t i n g c y c l o n e s , i t h a s b e e n a r g u e d t h a t t h e r e s o n a n c e mechanism i s r a t h e r u n l i k e l y
66
t o occur. 0.1
cpd,
However,
t h i s argument o n l y a p p l i e s t o f r e q u e n c i e s above
a s t h e a t m o s p h e r i c a s y m m e t r y f a d e s away a t l o w e r f r e q u e n c i e s .
The s i t u a t i o n i s v i s u a l i z e d i n t h e frequency-wavenumber Fig.
2.
d i a g r a m of
The s h a d e d a r e a i n d i c a t e s f o r e a c h f r e q u e n c y w h e r e most of
t h e atmospheric energy is concentrated.
This d i s t r i b u t i o n is based
o n a n a n a l y s i s o f w e a t h e r maps
( W i l l e b r a n d , 1978) and r e p r e s e n t a t i v e
f o r t h e N o r t h P a c i f i c f r o m 30'
-
I t i s c l e a r t h a t r e s o n a n t ge-
50°N.
n e r a t i o n o f b a r o t r o p i c R o s s b y w a v e s by f l u c t u a t i n g w i n d s i s b y n o means u n l i k e l y .
( B a r o c l i n i c R o s s b y w a v e s h a v e a much l o n g e r t i m e
s c a l e and a r e n o t c o n s i d e r e d h e r e ) .
The r e s o n a n c e m e c h a n i s m s h o u l d
b e most e f f e c t i v e around a p e r i o d of
10 d a y s ,
and t h e lower (upper)
f r e q u e n c y l i m i t w i l l depend m a i n l y on t h e e a s t - w e s t
(north-south)
dimension of t h e oceanic basin.
FREQUENCY-WAVENU MBER DISTRIBUTION OF ATMOSPHERIC FORCING
-.O West
Fig.
2.
-.6 -,A
-.2
0
.2
.A
WAVENUMBER (cp 1000km)
.6
.8 East
Frequency-wavenumber d i a g r a m i n d i c a t i n g e n e r g y d i s t r i b u t i o n i n t h e a t m o s p h e r i c p r e s s u r e f i e l d o v e r t h e N o r t h P a c i f i c . The L e a v y f u l l l i n e i s t h e mean z o n a l w a v e n u m b e r , d e f i n e d a s (k,w)/J d k F 3 3 ( k , w ) . T h e s h a d e d a r e a i s l i m i k , ( w ) = 1 d& k, F t e d by 2 o n e s t a a a a r d d e v i a t i o n . T h e d i s p e r s i o n c u r v e s f o r bar o t r o p i c R o s s b y w a v e s a r e p l o t t e d f o r two d i f f e r e n t v a l u e s o f t h e m e r i d i o n a l wavenumber
67
How w i l l t h e o c e a n r e a c t i f f o r c e d a t o f f - r e s o n a n t number c o m b i n a t i o n s ? P h i l a n d e r ,
frequency-wave-
1 9 7 8 , h a s d e m o n s t r a t e d t h a t t h e re s -
p o n s e c a n b e t r a p p e d n e a r t h e s u r f a c e , t h e v e r t i c a l t r a p p i n g s c a l e dep e n d i n g o n w a n d k,. I t t u r n s o u t t h a t , here,
a critical horizontal
f o r t h e t i m e s c a l e s of i n t e r e s t
s c a l e i s O(100 k m ) .
v e r t i c a l t r a p p i n g scale e x c e e d s t h e d e p t h of motion i s n e a r l y depth-independent,
i.e.
For larger scales,
t h e ocean,
the
and t h e f o r c e d
b a r o t r o p i c , whereas f o r smal-
l e r s c a l e s t h e f o r c e d m o t i o n i s t r a p p e d a n d b a r o c l i n i c . The a c t u a l amount o f a t m o s p h e r i c e n e r g y a t t h e s e s m a l l s c a l e s i s p o o r l y k n o w n , a n d t h u s a n e s t i m a t e o f t h e b a r o c l i n i c r e s p o n s e m u s t b e b a s e d on e x t r a p o l a t i o n of a v a i l a b l e atmospheric s p e c t r a (cf.
F r a n k i g n o u l and Miiller,
19 7 8 ) . The l a r g e s c a l e r e s p o n s e ,
however,
can b e deduced d i r e c t l y from
a t m o s p h e r i c o b s e r v a t i o n s , as e . g . w e a t h e r m a p s .
is a vertically integrated, Neglecting variations
The s i m p l e s t m o d e l
l i n e a r model w i t h q u a s i g e o s t r o p h i c dynamics.
i n bottom topography,
t e r m s o f t h e stream f u n c t i o n $ ( x , t )
the v o r t i c i t y balance i n
t a k e s t h e f a m i l i a r form
where f A i s t h e a t m o s p h e r i c f o r c i n g f u n c t i o n .
It i s s u f f i c i e n t t o con-
s i d e r t h e c a s e o f w i n d s t r e s s f o r c i n g , f A = ( V X ~ ) ~ T. h e c o r r e s p o n d i n g e x p r e s s i o n s f o r p r e s s u r e a n d b u o y a n c y f l u x f o r c i n g a r e g i v e n by Magaard,
1977. P r e s s u r e f o r c i n g is g e n e r a l l y n e g l i g i b l e i n t h e t i m e
scale r a n g e c o n s i d e r e d h e r e ( e x c e p t f o r i t s i n f l u e n c e on s e a l e v e l ) . The i m p o r t a n c e o f b u o y a n c y f l u x f o r c i n g d e p e n d s on t h e d i f f u s i v i t y of t h e u p p e r l a y e r of t h e ocean and h a s y e t t o b e e s t a b l i s h e d .
( 4 ) h a s t w o l i m i t i n g c a s e s , d e p e n d i n g on t h e r a t i o o f t h e t i m e s c a l e T t o t h e c h a r a c t e r i s t i c t i m e scale T R o f Rossby waves. F o r T < < T R , t h e f i r s t t e r m on t h e l e f t h a n d s i d e o f ( 4 ) d o m i n a t e s , Eq.
and p l a n e t a r y e f f e c t s are unimportant.
For T >> 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 << A ) .
Taking i n t o account t h e expres-
(6
s i o n s f o r energy momentum and volume f l u x on deep water Q = ua2/2),
-, M
= pg2a2 40
= I
ma2,
it appears t h a t a frequency change i s necessary a t the b a r r i e r i n order
F i g . 2. Frequency doubling a t a plane v e r t i c a l b a r r i e r
-
schematic
( a d d i t i o n a l t o 1 s t o r d e r upstream r e f l e c t e d wave and r e a c t i o n e f f e c t s )
t o convey t h e momentum of t h e i n c i d e n t wave and provide t h e r e a c t i o n t o t h a t i n t h e r e f l e c t e d wave without t r a n s m i t t i n g more energy than f i r s t order theory permits. The downstream i n c r e a s e i n Q due t o t h e frequency change would hav.e t o be balanced by a n induced c u r r e n t q from t h e upstream s i d e . l i m i t e d t h i s c u r r e n t q could g e n e r a t e a set-up
I f t h e downstream f l o w f i e l d were (Longuet-Higgins,
1967).
Were t h e motion everywhere i r r o t a t i o n a l , t h e b a r r i e r would experience no time mean r e a c t i o n a s it i s below t h e l e v e l of t h e wave f o r c e .
I n t h i s case, assuming
no s e t up, t h e mean momentum f l u x of t h e downstream fundamental and harmonics n C jpga,' would be equal t o t h a t o f t h e i n c i d e n t and r e f l e c t e d waves (approximately 1 I n f a c t , t h e main d e p a r t u r e from %pga2 i f t h e r e f l e c t i o n i s n e a r l y complete). i r r o t a t i o n a l i t y i s wave breaking, which i f it d i d not l e a d t o a s i g n i f i c a n t f o r c e on t h e b a r r i e r , would i n c r e a s e t h e proportion of higher harmonics i n t h e downstream motion, a s l e s s energy f l u x would be a v a i l a b l e f o r t h e same momentum flow r a t e ( s e e Longuet-Higgins,
1977 f o r wave f o r c e s on extended submerged o b j e c t s ) .
The
foregoing p o i n t s a r e confirmed q u i t e w e l l by t h e experimental r e s u l t s of J o l a s (1962) f o r wave transmission over a s i l l .
An a p p l i c a t i o n of t h e s e e f f e c t s i s t h e "harmonic
suppressor" (Hulsbergen, 1976) where a s i l l i s p o s i t i o n e d t o a c t as an "anti-noise" device c a n c e l l i n g t h e emission of p a r a s i t i c secondary waves by l a b o r a t o r y wavemakers. The opposite case of a b a r r i e r extending from above t h e s u r f a c e down t o t h e troughs, shows i n t h e l a b o r a t o r y , a s might be expected from t h e flux balances, the
/ / I / / / / // / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /
F i g . 3 . Frequency doubling a t a f i x e d s u r f a c e b a r r i e r - schematic ( a d d i t i o n a l t o t r a n s m i t t e d 1 s t o r d e r wave on downstream s i d e )
double frequency harmonics on t h e upstream s i d e , as i n t h i s c a s e t h e wave momentum is r e f l e c t e d , b u t t h e energy i s l a r g e l y t r a n s m i t t e d . be tpga'
The f o r c e on t h e b a r r i e r would
per u n i t width, assuming almost complete transmission.
Low frequency s u r f a c e waves can be seen from a p h y s i c a l viewpoint a s a "device" t o c a r r y energy, momentum and a displacement of f l u i d a c r o s s g r e a t d i s t a n c e s while i n c u r r i n g very l i t t l e p e n a l t y by way o f r o t a t i o n a l i t y .
The " s e c r e t " i s t h e locking
of t h e mean E u l e r i a n momentum i n t o t h e s u r f a c e deformation l a y e r .
The unlocking by
f l u i d f r i c t i o n may be enhanced by frequency i n c r e a s e s a t shelves and submerged b a r s . Wave a t t e n t u a t i o n
On deep water assuming laminar flow and zero s u r f a c e s t r e s s t h e waves l o s e energy a s a r e s u l t of t h e v i s c o u s r e s i s t a n c e t o ( a t f i r s t ) i r r o t a t i o n a l s t r a i n i n g i n t h e i n t e r i o r of t h e f l u i d (Stokes, 1845a). cannot b e destroyed by i n t e r n a l f o r c e s .
On t h e o t h e r hand t h e wave momentum
The generation of a time mean second
119 o r d e r v o r t i c i t y a t t h e water s u r f a c e opposite i n d i r e c t i o n t o t h e r o t a t i o n of t h e
F i g . 4 . Lowering of t h e wave momentum i n deep water
p a r t i c l e s i n t h e i r o r b i t s (Longuet Higgins, 1960) induces a v e l o c i t y f i e l d i n t h e d i r e c t i o n of propagation of t h e wave which a f f e c t s t h e water a t g r e a t e r depths a s t h e v o r t i c i t y slowly d i f f u s e s downwards. from t h e s u r f a c e region.
I n t h i s way t h e wave momentum i s removed
when "separated" from t h e waves t h e s e second o r d e r flows
may come under t h e i n f l u e n c e of t h e C o r i o l i s f o r c e and manifest themselves a s i n e r t i a c u r r e n t s ( U r s e l l , 1949). I n f i n i t e depth where t h e r e i s g r e a t l y enhanced a t t e n u a t i o n due t o t h e no s l i p c o n d i t i o n a t t h e bed t h e wave momentum must likewise be "brought down" from above trough l e v e l so a s it can be reduced by bottom f r i c t i o n .
H e r e t h e momentum i s
t r a n s f e r r e d downwards,not by t h e v o r t i c i t y generation a t t h e surface which would be a l t o g e t h e r t o o weak (and t h e d i f f u s i o n process too slow) b u t by p r e s s u r e f o r c e s consequent on a d i f f e r e n c e i n phase between t h e motion i n t h e boundary l a y e r and the irrotational interior.
This phase d i f f e r e n c e , f i r s t e l u c i d a t e d by Stokes
( 1 8 5 1 ) , i n t h e s o l u t i o n of h i s "second problem" r e s u l t s i n a component of t h e
i r r o t a t i o n a l pressure f i e l d being i n quadrature with t h e wave " p r o f i l e " i n the boundary l a y e r and a t r a n s f e r of energy and momentum t o t h e flow near the bed. The convection v e l o c i t i e s i n t h e i r r o t a t i o n a l flow a r e such a s t o r a i s e t h e vortex l i n e s where t h e s p i n i s o p p o s i t e i n hand t o t h e o r b i t a l motion and depress those whose r o t a t i o n i s i n t h e same sense. The r e s u l t i n g v o r t e x " s t r e e t " c r e a t e s a j e t (Longuet Higgins, 1953) c l o s e t o t h e bed which has t h e f u n c t i o n of reducing t h e wave momentum i n s t e p with t h e energy loss. sandy bed.
The p r o c e s s can be followed v i s u a l l y when t h e motion i s over a S l e a t h (1970) d e s c r i b e s v o r t i c e s being thrown o u t from behind the sand
g r a i n s a t t h e end of each h a l f c y c l e .
I t appears from t h e diagrams i n t h i s
paper t h a t t h o s e eddies which a r e opposite i n hand t o t h e o r b i t a l a higher t r a j e c t o r y than t h e i r c o u n t e r p a r t s .
motion a r e on
120
/ / / / / / / / / / / / / / / / / / / / / / / / / / /// / //I / /// / , / / / / / / / / F i g . 5. Lowering and subsequent r a i s i n g of t h e wave momentum i n f i n i t e depth
STABILITY OF THE VERTICAL DESCENT
The s t a b i l i t y of f l o a t i n g bodies a g a i n s t capsizing depends p a r t l y o n t h e d i s t r i b u t i o n of weight which provides a r e s t o r i n g o r overturning moment depending on whether t h e c e n t r e of g r a v i t y i s below or above t h e c e n t r e of buoyancy and p a r t l y o n t h e volume and d i s t a n c e a p a r t a t t h e "wedges" t h a t go i n t o and come out of t h e water as t h e r e s u l t of a heel.
The s t a b i l i t y a t f i n i t e angles of h e e l
i s measured by t h e magnitude of t h e r i g h t i n g arm (GZ) where the r e s t o r i n g moment i s given by t h e product of t h i s term and t h e displacement.
For i n f i n i t e s i m a l
displacements ( i n i t i a l s t a b i l i t y ) t h e slope of t h e curve of r i g h t i n g arm a g a i n s t angle of h e e l i s taken a s t h e c r i t e r i o n .
The metacentric h e i g h t (GM) i s the s l o p e
of t h i s curve a t t h e o r i g i n and is measured i n metres p e r r a d i a n . For North Sea platforms t h e c e r t i f i c a t i o n a u t h o r i t i e s s p e c i f y a minimum metac e n t r i c height of about a metre f o r a l l draughts through which t h e platform may p a s s i n i t s descent.
This s t i p u l a t i o n may r e q u i r e t h e c o n s t r u c t i o n of l a r g e
buoyancy chambers i n o r d e r t o provide "wedges",
p a r t i c u l a r l y a t t h e c r i t i c a l junc-
t u r e when t h e water l i n e p a s s e s through t h e roof of t h e base.
I t i s not practical
t o provide t h i s e x t r a s t a b i l i t y by adjustment of t h e weight d i s t r i b u t i o n , a s t h i s would e n t a i l s a c r i f i c i n g completion o f t h e deck and i t s equipment.
I n f a c t one
of t h e m a i n platform design o b j e c t i v e s i s t o maximise t h e deck l o a d , a s t h i s advances t h e d a t e of commencement of o i l production. S t a b i l i t y , however, may a l s o be acquired i n e r t i a l l y i f t h e f l u i d momentum associat e d w i t h t h e t r a n s l a t i o n of t h e body i n t h e chosen o r i e n t a t i o n i s g r e a t e r than
t h a t f o r any o t h e r p o s s i b l e alignment.
I f t h e l o c u s of t h e end p o i n t s of the
s o l i d body t r a n s l a t i o n v e l o c i t y v e c t o r be p l o t t e d f o r c o n s t a n t a s s o c i a t e d f l u i d k i n e t i c energy ( T ) , t h e axes of t h e e l l i p s o i d obtained g i v e Kirchoff ' s t h r e e permanent d i r e c t i o n s of t r a n s l a t i o n (Lamb, 1 9 3 2 ) .
121
F i g . 6 . The k i n e t i c energy e l l i p s o i d
The f l u i d momentum v e c t o r i s normal t o t h e energy e l l i p s o i d a s t h e l a t t e r i s t h e v e c t o r r a t e of change of f l u i d k i n e t i c energy with s o l i d body v e l o c i t y
aT (-)au .
Unless
t h e f l u i d momentum v e c t o r and t h e s o l i d body momentum v e c t o r a r e p a r a l l e l t h e r e w i l l be a couple exerted on t h e body by t h e f l u i d .
when t h e motion of the s o l i d i s i n
t h e d i r e c t i o n of one of t h e axes of t h e e l l i p s o i d , t h i s couple i s a b s e n t and t h e body once s e t t r a n s l a t i n g , c a n ,
i n theory, continue t o do so without r o t a t i o n .
Of
t h e s e t h r e e d i r e c t i o n s of permanent t r a n s l a t i o n only one i s considered t o be s t a b l e although t h e complete v a l i d i t y of t h i s r u l e i n a l l c a s e s has been disputed ( U r s e l l , H.D.,
1940).
For t h e most g e n e r a l motion of a body of a r b i t r a r y shape t h e k i n e t i c energy of t h e f l u i d depends on 21 c o e f f i c i e n t s of i n e r t i a .
I n t h e r e s t r i c t e d c a s e of t r a n s -
l a t i o n of s o l i d s of r e v o l u t i o n (and of h e l i c o i d a l symmetry), i n t o which c a t e g o r i e s t h e d e s c e n t of some platform designs f a l l , t h e number of c o e f f i c i e n t s reduces to two: 2T
=
Au2
+
B(v2
+ w2),
adopting Lamb's n o t a t i o n .
I n t h e p a r t i c u l a r c a s e where t h e body moves with t h e a x e s always i n one plane (that of x and y ) ,the d e f l e c t i n g couple may be shown t o be (A-B)u.v by a simple argument:
c Bv c BV
/ / - -
I
-
U
--
I
Fig. 7. The change of t h e f i u i d impulse i n u n i t time
122 When t h e body t r a n s l a t e s from 1 to 2 i n u n i t time t h e couple B.v.u
-
A.u.v.
must
be a p p l i e d by t h e body t o t h e f l u i d i n o r d e r t o e f f e c t t h e change i n f l u i d v e l o c i t y field.
The r e a c t i o n of t h e f l u i d on t h e body, t h e d e f l e c t i n g couple, i s therefore
(A-B)u.v about t h e z a x i s .
I n t h e c a s e of a t y p i c a l platform t h e couple
(A-B)u.v
may be e s t a b l i s h e d and compared t o t h e s t a t i c r e s t o r i n g moment.
The hydrodynamic r e s t o r i n g couple
F i g . 9 . Annulus w i t h c i r c u l a t i o n
F i g . 8. Descent of base
I n Fig.
8
,u
= V cosa, v = V
The f l u i d couple = (A-B)u.v = Taking A-B
sina where V i s t h e v e r t i c a l v e l o c i t y of descent.
V2sin2a 2 A-B
t o be of t h e o r d e r of t h e displacement M,
Metacentric h e i g h t corresponding (GM)
V2
sin2a 2g = '('2) at
Maximum equivalent r i g h t i n g arm (GZ) =
da
a = 0
Taking t h e maximum value of V t o be 2 metres per minute:
T h i s very small v a l u e ccmpared t o t h e design ( h y d r o s t a t i c ) GM of 1 metre shows t h a t f l u i d i n e r t i a p l a y s an a l t o g e t h e r n e g l i g i b l e r o l e i n t h e s t a b i l i t y of descent. The couple (A-B)u.v could be augmented i n t h e c a s e of a doubly connected (e.g. annular) body.
I f t h e impulse of t h e c i r c u l a t i o n i s
be augmented by a f a c t o r Sv.
5
t h e r e s t o r i n g moment w i l l
However t h e impulse of a f r i c t i o n a l l y generated
c i r c u l a t i o n would be n e g l i g i b l e a t t h e v e l o c i t i e s enyisaged.
More might be
achieved by a r t i f i c i a l l y c r e a t i n g a j e t through t h e annulus b u t t h e p r a c t i c a l i t i e s Seem doubtful. To develop a metacentric h e i g h t of 1 metre by r a p i d descent would r e q u i r e a v e l o c i t y of
J
which could be i n excess of 10 m sec-'.
123 Falling leaf oscillation
F i g . 10. O s c i l l a t i o n s about t h e l i n e of t h e impulse
The equation governing t h e o s c i l l a t i o n of a body about a s t a b l e d i r e c t i o n of t r a n s l a t i o n w a s f i r s t given by Kelvin (1871).
Using t h e n o t a t i o n of Lamb (1932)
and Gray (1950) :
Qe
=
(A-B)uv
+ 5
where Q i s t h e combined s o l i d body and hydrodynamic
moment of i n e r t i a about a n a x i s perpendicular t o t h e plane of t h e f i g u r e .
I f the
generating impulse i s I then: Icose
=
Au
The period T
+ BE, =
21r/(
Isine
ABQ
=
-BV
((A-B) I +BE) I
)
I n t h e absence of a s t a t i c r i g h t i n g moment t h e period of t h e o s c i l l a t i o n may be reduced and t h e s t a b i l i t y of t h e descent enhanced by designing A t o be g r e a t e r than B, augmenting t h e e q u i v a l e n t I by i n c r e a s i n g t h e r a t e of descent and i f p o s s i b l e
making use of t h e impulse 5 of t h e c i r c u l a t i o n .
I n f a c t however t h e f e a r of t h i s
type of motion developing would appear t o be groundless i n view of the magnitude of t h e s p e c i f i e d metacentric h e i g h t . I t should perhaps be made c l e a r t h a t t h i s i n e r t i a l o s c i l l a t i o n i s only t h e i n i t i a l
p a r t of t h e motions of a s o l i d f a l l i n g f r e e l y under g r a v i t y i n a f l u i d .
Maxwell
(1853) h a s shown i n t h e c a s e of a p l a n e r e c t a n g u l a r o b j e c t ( t h e e f f e c t i s e a s i l y seen with h i s example of a s l i p of paper f a l l i n g i n a i r ) , t h a t t h e descent a f t e r an
i n i t i a l "wavering" t a k e s p l a c e a t a c o n s t a n t angle t o t h e v e r t i c a l and i s accompanied by a r a p i d r o t a t i o n about a h o r i z o n t a l a x i s .
These l a s t two e f f e c t s a r e due t o
f l u i d f r i c t i o n , w h e r e a s t h e r e s t o r i n g f o r c e governing t h e i n i t i a l o s c i l l a t i o n s depends a s above on f l u i d i n e r t i a only.
Gyroscopic s t a b i l i t y
I f t h e platform, assumed a s o l i d of r e v o l u t i o n , i s made t o s p i n about t h e generat i n g a x i s , it w i l l a c q u i r e s t a b i l i t y i n t h e manner of a "sleeping" t o p provided t h e angular v e l o c i t y i s s u f f i c i e n t l y g r e a t .
The p o i n t on t h e a x i s , about which
t h e motion would t a k e p l a c e were precession t o occur, corresponding t o t h e p o i n t of c o n t a c t of a t o p with t h e ground, could i n p r i n c i p l e
be obtained by minimising
t h e k i n e t i c energy o f t h e system a s Wendel (1950) has done i n l o c a t i n g t h e r o l l axis of s h i p s . I f P i s t h e s o l i d body manent of i n e r t i a of t h e platform about i t s long a x i s and Q t h e s o l i d body and hydrodynamic moment of i n e r t i a about a perpendicular a x i s
through t h e f i x e d p o i n t , ignoring f r i c t i o n a l couples, t h e conservation of angular momentum g i v e s : Restoring torque
=
-
Pwnsintl
Qn2sintlcostl
where w i s t h e spin about t h e
long a x i s , tl t h e i n c l i n a t i o n of t h i s a x i s t o t h e v e r t i c a l and Q t h e precession. The l e f t hand s i d e of t h i s equation may be w r i t t e n Mgmsintl, where m i s t h e metacentric h e i g h t (GM). The c o n d i t i o n f o r t h e platform a x i s t o r e t u r n t o t h e v e r t i c a l a f t e r a small displacement, is, a s i n t h e case of t h e top, t h a t t h e r o o t s of the q u a d r a t i c equation i n Q be r e a l , t h a t is: 2.
>
4MgmQ* PL
= p2w2 - g i v e s t h e e q u i v a l e n t metacentric h e i g h t developed by the 4MgQ ' Assuming t h e mass d i s t r i b u t i o n about t h e a x i s of s p i n t o correspond t o that
The e q u a l i t y , m spin.
of a s o l i d c y l i n d e r of diameter 80 metres r e s u l t s i n a v a l u e f o r P of 800 tonnes m2. An e s t i m a t e of t h e i n e r t i a d i s t r i b u t i o n along t h e same a x i s r e l a t i v e t o t h e fixed p o i n t f o r a t y p i c a l design g i v e s Q
2
1200 tonnes m 2 .
S u b s t i t u t i n g t h e v a l u e s i n t h e equation shows t h a t a s p i n of 2.6 r e v o l u t i o n s p e r minute would develop a metacentric h e i g h t of 1 metre.
CONCLUSIONS
The conviction t h a t second order wave p r o p e r t i e s a r e s u i g e n e r i s and not mere outgrowths from f i r s t o r d e r e f f e c t s f o r c e s i t s e l f on one s t r o n g l y .
The wave momentu
when followed from i t s g e n e r a t i o n , through i t s conservation t o i t s decay seems t o e x p l a i n a s many and a s v a r i e d phenomenaasthe wave energy. The b a s i c E u l e r i a n Lagrangian ambiguity m a n i f e s t s i t s e l f n o t i n t h e magnitude but i n t h e l i n e of a c t i o n of t h e wave f o r c e .
* The problem of "Columbus'egg" remains t o be solved h e r e i f t h e r e i s water b a l l a s t i n t h e compartments of t h e base.
125 Wavering motions, t y p i c a l of f a l l i n g leaves, cannot occur on platforms descending a t t h e usual speeds due t o t h e smallness of t h e couples involved.
T o acquire
s t a b i l i t y by shnking r a p i d l y i n a s t a b l e d i r e c t i o n o f t r a n s l a t i o n i s i m p r a c t i c a l . I t i s j u s t conceivable t h a t an a d d i t i o n of r o t a t i o n a l l y acquired s t a b i l i t y might
be used t o p a s s through a s e c t i o n of low m e t a c e n t r i c h e i g h t i n t h e descent.
ACKNOWLEDGEMENTS
I would l i k e t o thank t h e following f o r t h e i r h e l p : Professor M.S.
of Cambridge University,
D r J.M.R.
College, M r J. S i o r i s and D r C.D. r e s e a r c h group
and Associates Ltd., behav iou r
Memos of t h e c i v i l engineering hydrodynamics
a t Imperial College and M r R.W.P.
Research S t a t i o n , Wallingford.
Longuet-Higgins
Graham of t h e Aeronautics Department, Imperial
May of t h i s group and t h e Hydraulics
I am very indebted t o M r R . L .
Jack of Noble Denton
London, marine c o n s u l t a n t s , f o r much information on platform
.
REFERENCES
1945. On t h e r e f l e c t i o n of s u r f a c e waves by a submerged plane b a r r i e r . Dean, W.R., Proc. Camb. P h i l . SOC. 41:231-238. Gray, A . , 1918. A t r e a t i s e on g y r o s t a t i c s and r o t a t i o n a l motion. Dover E d i t i o n s , 1959, 530pp. Havelock, T . H . , 1940. The p r e s s u r e of water waves upon a f i x e d o b s t a c l e . Proc. Roy. SOC. London. A175:409-421. 1976. The o r i g i n e f f e c t and suppression of secondary waves. Hulsbergen, C.H., P u b l i c a t i o n No. 132, D e l f t Hyd. Laboratory. J o l a s , P . , 1962. Contribution a 1 ' 6 t u d e d e s o s c i l l a t i o n s p6riodiques des l i q u i d e s p e s a n t s avec s u r f a c e l i b r e . Houille Blanche, 758-769. Kelvin, Lord, 1871. On t h e motion of f r e e s o l i d s through a l i q u i d . P h i l . Mag. X L I I : 362-377. Kelvin, Lord, 1887. On s h i p waves. Popular Lectures and Addresses I I I : 4 5 9 , Macmillan, London (1891). Lamb, H . , 1932. Hydrodynamics. A r t . 124, Cambridge University P r e s s . Longuet-Higgins, M.S., 1953. Mass t r a n s p o r t i n water waves. P h i l . Trans. Roy. SOC. A245: 535-581. Longuet-Higgins, M.S., 1960. Mass t r a n s p o r t i n t h e boundary l a y e r a t a f r e e o s c i l l a t i n g surface. J . F l u i d Mech. 8:293-306. Longuet-Higgins, M.S., 1967. On t h e wave-induced d i f f e r e n c e s i n mean s e a l e v e l between t h e two s i d e s of a submerged breakwater. J. Mar. Res. 25:148-153. Longuet-Higgins, M.S., 1977. The mean f o r c e s exerted by waves on f l o a t i n g o r submerged bodies with a p p l i c a t i o n s t o sand b a r s and wave p o w e r machines. Proc. Roy. SOC. London. A352: 463-480. 1964. Radiation s t r e s s e s i n water waves; Longuet-Higgins, M.S., and Stewart, R.W., a p h y s i c a l d i s c u s s i o n , with a p p l i c a t i o n s . Deep Sea Res. 11:529-562. Maxwell, J . C . , 1853. On a p a r t i c u l a r c a s e of t h e descent of a heavy body i n a r e s i s t i n g medium. Camb. and Dublin Math. J o u r . I X . , C o l l . Papers:115-118. 1972. Etude du passage de l a houle s u r un 6cran v e r t i c a l mince Navarro Pineda, J . M . , immerg6. Thesis. U n i v e r s i t y of Grenoble. 1977. The dynamics of t h e upper ocean. 2nd Edn. Cambridge University P h i l l i p s , O.M., P r e s s , p. 40. S l e a t h , J.F.A., 1970. Velocity measurements c l o s e t o t h e bed i n a wave tank. J. Fluid Mech 4 2: 1 11-1 23.
.
126 Stokes, G.G., 1845. On t h e theory of t h e i n t e r n a l f r i c t i o n of f l u i d s i n motion. Trans. Camb. P h i l . SOC. 8:287, Papers 1:75. Stokes, G.G., 1847. On t h e theory of o s c i l l a t o r y waves. Trans. Carnb. P h i l . SOC. 8:441-455, Papers 1:314-326. Stokes, G.G. 1851. On t h e e f f e c t of t h e i n t e r n a l f r i c t i o n of f l u i d s on t h e motion of pendulums. Trans. Camb. P h i l . SOC. 9:8, Papers 111:l. U r s e l l , F., 1947. The e f f e c t of a fixed v e r t i c a l b a r r i e r on s u r f a c e waves i n deep water. Proc. Camb. P h i l . SOC. 43:374-382. U r s e l l , F . , 1949. Wind and ocean c u r r e n t s . Nature 163:237-238. U r s e l l , H.D., 1940. Motion of a s o l i d through an i n f i n i t e l i q u i d . Proc. Camb. Phil. SOC. 38:150-167. Wendel, K. 1950. Hydrodynamische Massen und Hydrodynamische Massentragheitsnamente. Jahrb. d . STG 44.
127
A HYBRID P M T R I C A L SURFACE WAVE MODEL APPLIED TO NORTH-SEA SEA STATE PREDICTION
H. GbTHER and W. ROSENTHAL I n s t i t u t f u r Geophysik, U n i v e r s i t a t Hamburg, Hamburg (FRG) and Max-Planck-Institut
f u r Meteorologie, Hamburg (FRG)
ABSTRACT A hybrid parametrical wave model based on t h e i d e a s of Hasselmann e t a l . (1976) i s used t o p r e d i c t waves from s u r f a c e wind f i e l d s on d i f f e r e n t s i z e s of p r e d i c t i o n area.
INTRODUCTION
We p r e s e n t h e r e i n a s h o r t version t h e performance of a new kind of s u r f a c e wave p r e d i c t i o n model.
Full d e t a i l s of t h e model and i t s a p p l i c a t i o n s can be found i n
Giinther e t al. (1979a,b) and Ewing e t a l . (1979).
These t h r e e papers give a
summary of an i n t e r n a t i o n a l e f f o r t t o generate extreme value wave s t a t i s t i c s f o r t h e northern North-Sea.
I n t h e p r e s e n t c o n t r i b u t i o n we describe q u a l i t a t i v e l y
t h e b a s i c p r i n c i p l e s l e a d i n g t o our numerical model which can be found i n more d e t a i l i n Hasselmann e t a l . (1973) and Hasselmann e t a l . (1976).
A s an example
of t h e a p p l i c a t i o n s of t h i s model we d i s c u s s a hindcast on a small p r e d i c t i o n a r e a using a dense computational g r i d , as w e l l a s a l a r g e hindcast a r e a with a r e l a t i v e l y coarse g r i d . DYNAMIC BEHAVIOUR OF GROWING W I N D SEA AND SWELL SPECTRA
During t h e l a s t twenty y e a r s t h e r e have been g r e a t advances i n understanding t h e behaviour of t h e energy d i s t r i b u t i o n i n s u r f a c e g r a v i t y wave s p e c t r a .
Never-
t h e l e s s p r e s e n t day wave p r e d i c t i o n models a l l u s e , along with well-established research r e s u l t s , h e u r i s t i c concepts which need f u r t h e r experimental and t h e o r e t i c a l investigations.
Without going i n t o d e t a i l s we give here an o u t l i n e of our present
knowledge about t h e s t a t i s t i c a l behaviour of s u r f a c e waves i n deep water, which
i s revealed by our model. We describe t h e s e a s t a t e a t a f i x e d l o c a t i o n f o r a c e r t a i n time by t h e two dimensional e n e r a spectrum.
For most a p p l i c a t i o n s t h e angular energy d i s t r i b u t i o n
128 r e l a t i v e t o t h e mean p r o p a g a t i o n d i r e c t i o n i n a wind-sea i s known w e l l enough t o use t h e one dimensional energy spectrum f o l l o w i n g from an i n t e g r a t i o n of t h e two dimensional spectrum over a l l wave d i r e c t i o n s .
The one dimensional spectrum g i v e s
t h e energy d e n s i t y i n frequency s p a c e and h a s a u n i v e r s a l shape which c a n be c h a r a c t e r i z e d by a few parameters ( s e e F i g . 1 ) .
Under atmospheric energy i n p u t
t h e peak frequency of t h e spectrum s h i f t s t o lower v a l u e s by n o n l i n e a r i n t e r a c t i o n
Ex,
00
--*Lo,
if-
MAX
PM EMAX
fm F i g . 1 . D e f i n i t i o n o f t h e f i v e JONSWAP parameters f o r a wind-sea
p r o c e s s e s between d i f f e r e n t wavenumber and frequency bands.
spectrum.
The u n i v e r s a l s p e c t r a l
shape i s t h e r e b y r e t a i n e d a l s o due t o t h e n o n l i n e a r i n t e r a c t i o n .
T h i s concept
may be compared t o t h e e s t a b l i s h m e n t of a u n i v e r s a l energy d i s t r i b u t i o n i n an ensemble o f weakly i n t e r a c t i n g g a s molecules resembling t h e B o l t m a n n - d i s t r i b u t i o n , a l s o d u r i n g q u a s i s t a t i o n a r y e x t e r n a l energy i n p u t .
The s p e c t r a l peak s t o p s moving
when t h e peak frequency i s e q u i v a l e n t t o a phase speed roughly e q u a l t o t h e surface wind speed
'10
*
fm
=
0.13,
g where U,O = wind speed a t 10 m h e i g h t f m = peak frequency
g
= gravitational constant
S u r f a c e waves w i t h l a r g e r phase speeds are n o t i n f l u e n c e d by t h e atmosphere and can b e t r e a t e d as independent wave groups p r o p a g a t i n g w i t h t h e i r group v e l o c i t y . The
d e s c r i b e d h i s t o r y of a growing wind-sea i s c l e a r l y shown i n F i g . 2 ( t a k e n
from Hasselmann e t al. ( 1 9 7 3 ) ) , showing
t h e energy s p e c t r a
with increasing
d i s t a n c e from s h o r e f o r a homogeneous wind blowing p e r p e n d i c u l a r t o t h e s h o r e .
129
a?--
C.6
-
0.5
-
41 0.4
-
0.3
-
0.2
-
0.9
-
x & a
Oi HZ h h F i g . 2 . E v o l u t i o n o f wave spectrum w i t h f e t c h f o r o f f s h o r e winds ( 1 1 -12 , S e p t . 15, 1968). Numbers r e f e r t o s t a t i o n s w i t h i n c r e a s i n g d i s t a n c e t o s h o r e . The b e s t - f i t a n a l y t i c a l s p e c t r a are a l s o shown.
CLASSIFICATIOIJ OF NUMERICAL MODELS
Before showing some h i n d c a s t r e s u l t s of our model it may be a p p r o p r i a t e t o g i v e a s h o r t d e s c r i p t i o n of t h e main f e a t u r e s o f wave models i n u s e t o d a y .
D i a g n o s t i c models
These models cam be u s e d f o r s p e c i a l c a s e s when t h e s u r f a c e wave f i e l d i s a f u n c t i o n o f t h e l o c a l wind f i e l d o n l y , s o t h a t t h e p r e d i c t i o n problem can be s h i f t e d completely t o t h e p r e d i c t i o n of t h e wind f i e l d .
A model o f t h i s k i n d , f o r example,
works remarkably w e l l f o r p r e d i c t i o n of h u r r i c a n e s e a s t a t e s (Cardone e t a l . ,
19’77).
130 Prognostic models
These models a r e based on an i n t e g r a t i o n of t h e energy t r a n s p o r t equation
-dF - dt
s
s.i n
=
+ S n 1 + S . dis
where
-d - at
t o t a l time d e r i v a t i v e
F
=
F(r,f,B)
S
=
source f u n c t i o n
Sin
=
energy i n p u t from t h e atmosphere
Snl
=
energy r e d i s t r i b u t i o n between d i f f e r e n t wavenumbers by nonlinear interaction
Sdis =
=
s p e c t r a l energy d e n s i t y
d i s s i p a t i o n of energy
The f i r s t models of t h i s kind were s p e c t r a l models developed by Pierson and h i s colleagues, f o r example Pierson e t a l . , 1966.
The energy i n p u t and d i s s i p a t i o n
i s a p p l i e d t o each s p e c t r a l component s e p a r a t e l y and s o each component grows
u n t i l i t s s a t u r a t i o n s t a t e independent of t h e o v e r a l l energy d i s t r i b u t i o n . means Snl,
That
t h e nonlinear i n t e r a c t i o n i n t h e source f u n c t i o n , i s neglected.
Since t h e work of P h i l l i p s (1960) and Hasselmann (1963) on t h e nonlinear i n t e r a c t i o n s between d i f f e r e n t s p e c t r a l components and t h e confirmation of t h e i r importance i n t h e f i r s t JONSWAP experiment (Hasselmann e t a l . , 1973), wave prediction models were developed which used t h e nonlinear term Snl i n t h e source f u n c t i o n i n a rough parametrization ( B a r n e t t , 1968; Ewing, 1971). Hasselmann e t a l . (1976) proposed a new technique t o parametrize t h e complete prognostic equation i n s t e a d of only t h e nonlinear source term, which i s used i n
our numerical model f o r t h e wind-sea p a r t .
It does not use t h e energy content
of t h e s i n g l e frequency-direction b i n s of t h e spectrum as q u a n t i t i e s t o be predicted, b u t t h e ensemble parameters c h a r a c t e r i z i n g t h e u n i v e r s a l shape of t h e wind-sea spectrum.
This r e s u l t s a l s o i n much lower demands on computer resources because
of a smaller number of p r e d i c t e d parameters. For s w e l l frequencies c h a r a c t e r i z e d by
"lo
'
<
0.13
g
(3)
we use a technique, f i r s t applied by Barnett e t a l . (1969), i n which t h e paths of i n d i v i d u a l wave t r a i n s a r e s t o r e d and a t each time s t e p t h e wave energy i s propagated along t h e s e p a t h s with t h e a p p r o p r i a t e group v e l o c i t y .
131 H i n d c a s t i n g over a s m a l l area T h i s example compares h i n d c a s t e d wave s p e c t r a w i t h measurements t a k e n d u r i n g t h e JONSWAP 1973 experiment by DHI ( F e d e r a l Republic of Germany), KNMI ( H o l l a n d ) and 10s ( U n i t e d Kingdom) and is g i v e n i n more d e t a i l i n Gunther e t a l . ( 1 9 7 9 , b ) . The wind f i e l d i s s p a t i a l l y homogeneous and d e p i c t e d i n t h e upper p a n e l o f F i g . 3. I
Windspeed and Direction
rl-l
0
10mk
-. . .015
.Oll .007
m
0
1
012h
18h
20. Sept. 1973
2Ah
6h
21. Sept. 1973
12h
JONSWAP 73, Station 9
-
computed
o observed
F i e . 3 - T i m e s e r i e s for sea s t a t e p a r a m e t e r s from S e p t . 2 0 t h , 12 h u n t i l Sept. 2 l s t , 12 h a t S t a t i o n 9 o f t h e JONSWAF a r r a y . s o l i d l i n e : h i n d c a s t , c i r c l e s : measurements. * I n t h e upper p a n e l North i s p o i n t i n g v e r t i c a l l y upward.
132 Incoming wave energy from t h e w e s t e r n , n o r t h e r n , or s o u t h e r n boundary i s f e d i n t o t h e model from w a v e r i d e r and p i t c h - r o l l buoys on t h e b o r d e r s of t h e p r e d i c t i o n area.
The e a s t e r n b o r d e r i s a c o a s t l i n e and boundary v a l u e s , n e c e s s a r y f o r o f f s h o r e
blowing wind, were modelled by t h e f e t c h l a w s o f Hasselmann e t a l . ( 1 9 7 3 ) . The c o m p u t a t i o n a l g r i d o f t h e h i n d c a s t had a s p a c i n g o f 2 km and t h e times t e p was 5 min.
For t h e s w e l l p a r t ( t r a n s p o r t a t i o n of wave energy a l o n g c h a r a c t -
e r i s t i c s ) w e used 24 f r e q u e n c y and 8 d i r e c t i o n b i n s . square o f
The h i n d c a s t a r e a w a s a
2
44 x 44 km and t h e h i n d c a s t covered 24 h o u r s .
done on a CDC
The computation w a s
Cyber 76 and r e q u i r e d 3 s e c CPU t i m e p e r c o m p u t a t i o n a l t i m e s t e p
and a t o t a l f i e l d l e n g t h o f 90 000 words. O u r h i n d c a s t example shows i n F i g . 3 s a t i s f a c t o r y agreement between measured and
The s p e c t r a l s c a l e parameter cx
h i n d c a s t e d peak f r e q u e n c i e s and wave h e i g h t s . and shape parameter y do n o t a g r e e as w e l l .
The h i n d c a s t e d s p e c t r a i n F i g . 4 , 5 a l s o show s a t i s f a c t o r y agreement w i t h t h e measurements a l t h o u g h d i f f e r e n c e s can be r e c o g n i z e d , which can be t r a c e d back t o t h e d i r e c t i o n a l assumptions o f t h e model. The F i g u r e s 3,4,5
improved i n f u t u r e work.
The d i r e c t i o n a l assumptions w i l l be o n l y show t h e comparison a t one
s t a t i o n n e a r t h e c e n t e r of t h e p r e d i c t i o n area.
From comparisons of t h e whole
s e t of measured s p e c t r a throughout t h e area and t h e a p p r o p r i a t e h i n d c a s t e d s p e c t r a and from two similar h i n d c a s t s w i t h r a p i d l y v a r y i n g wind f i e l d s i n s p a c e and time we conclude t h a t t h e model i s a b l e t o p r e d i c t s u r f a c e waves i n an area of t h e p r e s c r i b e d s i z e as l o n g as waves a r e s h o r t enough t o be c o n s i d e r e d as deep water waves.
I n r a p i d l y t u r n i n g wind f i e l d s t h e wind-sea i s o v e r e s t i m a t e d by t h e model.
The p a r a m e t r i z a t i o n of t h i s e f f e c t w i l l be improved by numerical c a l c u l a t i o n of t h e Boltzmann i n t e g r a l s of t h e n o n l i n e a r i n t e r a c t i o n i n t u r n i n g wind s i t u a t i o n s and t h e comparison of t h e r e s u l t s w i t h f i e l d data. H i n d c a s t i n g over a l a r g e area Work d e s c r i b e d i n t h i s c h a p t e r w a s done a t t h e I n s t i t u t e of Oceanographic S c i e n c e and t h e Harbour Research S t a t i o n W a l l i n g f o r d . model w a s u s e d , d i f f e r i n g o n l y i n numerical d e t a i l s .
The same h y b r i d p a r a m e t r i c a l The g o a l w a s t o g e n e r a t e an
extreme v a l u e s t a t i s t i c o f s u r f a c e waves i n t h e n o r t h e r n North S e a . purpose t h e model h a s t o b e checked on a v a i l a b l e measurements.
For t h a t
The s u r f a c e wind
f i e l d f o r 42 storms from t h e decade 1965-1975 and from t h e f i r s t h a l f o f 1976 have been a n a l y s e d as d e s c r i b e d by Harding e t a l . ( 1 9 7 8 ) .
The computational g r i d
and t h e p o i n t s where s u r f a c e winds a r e a v a i l a b l e a r e shown i n F i g . Of
g r i d p o i n t s i s 100 km.
of t h e t e s t storms F i g .
6.
The spacing
The t i m e s t e p used f o r computation w a s 1 hour.
For one
7,8 show a comparison a t weather s h i p Famita between modelled
and measured s i g n i f i c a n t wave h e i g h t H s and mean zero-up-crossing
periods
133
5
..
M.M/Hz
4.0 -
::
:*
ri
3.0 -
5.0
sept.20 13O0
1.
Sept. 20
4.0
I : i t
....A
2.0 -
..
1400
30
1.0
-
0, 0
0
0.1
0.2
0.3
0
01
0.2
0.3
Hz
0.1
0.2
03 HZ 0.4
0.4
M
Hz
0.4
JONSWAP 73, Station 9 -observed Wave spedm -----computed Wave spectra Figure 4. Wave s p e c t r a a t S t a t i o n 9 of t h e JONSIAP a r r a y . c i r c l e s : measurements.
s o l i d l i n e : hindcast
134
0.5
0
0.2
0.1
0.3
Hz
0
0.4
Ql
0.2
1.5
0
03 HZ 0.4 Sept. 21
Q1
Q2
0.3
Hz
OA
1.5 10
0.5
M
S 1.0 M
05 0 0
0.2
0.1
0
/
H
Z
:
I;
0.1
1
*
5
,A
Q2
r
:jjk 0 0
Oh
03
n
-..
03 HZ 0.4
Q5
0.1
0.2
:
0.3
0.4
*.’ ’0.
0
0.1
0.2
a3
Hz
0.4
JONSWAP 73, Station 9 -observed Wave spectra computedWave spedra
-..-
Figure 5 . Wave s p e c t r a a t S t a t i o n 9 of t h e JONSWPF a r r a y . c i r c l e s : measurements.
s o l i d l i n e : hindcast
b.
a Y
Y
136
Figure
7. Modelled and observed s i g n i f i c a n t wave h e i g h t H s a t Famita p o s i t i o n ( r e p r o d u c e d f r o m HRS Report N o . EX775, 1977).
i c Y
4 -
2 28 W 74
27 ID 14
26.10 ?b
I
I
Figare
t
8, Nodelied
1
1
29 10,76
I
&nC 3bserved mean zerc-up-crossing
(reprcdaced fron IIFS Report Ho.
1
I
I
p e r i o d s T2 at Famita piisition
2x775, 1977).
138
Correlation of significant waveheight HS
3Dr 25
I
standard deviation: Q22
rnj
correlation coefficient: 0.88
0
0
c
20
E
U
u)
t
I
T! 1.5
!!!I
/
-
1.0
I 0
I
1
05
1.0
1
1
1.5 20 Model H, (m)
I
1
2s
3.0
Figure 9. C o r r e l a t i o n between measured and hindcasted wave h e i g h t s a t S t a t i o n s 7 and 9 of t h e JONSWAP a r r a y . CONCLUSION
From a number of t e s t runs t h e c o r r e l a t i o n s given f o r t h e small hindcast a r e a i n F i g . 9 show t h a t t h e hybrid p a r a m e t r i c a l model i n t h e p r e s e n t form i s a b l e t o p r e d i c t deep water s u r f a c e waves.
Although t h e o v e r a l l performance confirms t h e
underlying p h y s i c a l concept, t h e r e a r e some p o i n t s which need f u r t h e r research. One i s t h e energy t r a n s i t i o n from wind s e a t o s w e l l , which w i l l be t a c k l e d by c l a r i f y i n g t h e r o l e of nonlinear wave-wave i n t e r a c t i o n i n t h a t case.
Further
139 i n v e s t i g a t i o n of n o n l i n e a r i n t e r a c t i o n and comparison w i t h f i e l d d a t a w i l l a l s o h e l p t o improve t h e model b e h a v i o u r f o r r a p i d l y t u r n i n g wind s i t u a t i o n s .
An exten-
s i o n o f t h e model t o s h a l l o w water wave p r e d i c t i o n i s under development.
REFERENCES B a r n e t t , T . P . , 1968. On t h e g e n e r a t i o n , d i s s i p a t i o n and p r e d i c t i o n of ocean wind waves. J . geophys. Res., 73: 513-530. B a r n e t t , T.P., Holland, C.H. J r . , Yager, P . , 1969. A g e n e r a l t e c h n i q u e f o r wind 2 wave p r e d i c t i o n , w i t h a p p l i c a t i o n t o t h e South China Sea. C o n t r a c t ~ 6 306-68C-0285, U.S. Naval Oceanic O f f i c e , Washington, D . C . Cardone, V . J . , Ross, D.B., Ahrens, M . R . , 1977. An experiment i n f o r e c a s t i n g h u r r i c a n e g e n e r a t e d sea s t a t e s . Proc. 11th T e c h n i c a l Conference on Hurricanes and T r o p i c a l Meteorology, Dec. 13-16, Miami, F l o r i d a . Ewing, J . A . , 1971. A numerical wave p r e d i c t i o n method f o r t h e North A t l a n t i c Ocean. Deut. Hydrogr. 2. 24: 241-261. Ewing, J . A . , Worthington, B.A., Weare, T . J . , 1979. A h i n d c a s t s t u d y of extreme wave c o n d i t i o n s i n t h e North Sea. Accepted by J . geophys. Res. Gunther, H . , R o s e n t h a l , W . , Weare, T . J . , Worthington, B.A., Hasselmann, K . , Ewing, J . A . , 1979a. A h y b r i d p a r a m e t r i c a l wave p r e d i c t i o n model. Accepted by J . geophys . R e s Giinther, H . , R o s e n t h a l , W . , R i c h t e r , K., 1979b. A p p l i c a t i o n o f t h e p a r a m e t r i c a l wave p r e d i c t i o n model t o r a p i d l y v a r y i n g wind f i e l d s d u r i n g JONSWAF 1973. Submitted t o J. geophys. Res. Harding, J . , Binding, A . A . , 1978. Wind f i e l d s d u r i n g g a l e s i n t h e North Sea and t h e g a l e s o f 3 J a n u a r y 1976. Met. Magazine, 107: 164-181. Hasselmann, K . , 1962. On t h e n o n l i n e a r energy t r a n s f e r i n a gravity-wave spectrum, P a r t 1 : General t h e o r y . J. F l u i d Mech. ,12:481-500. Hasselmann, K . , 1963. On t h e n o n l i n e a r energy t r a n s f e r i n a gravity-wave spectrum, P a r t 3: E v a l u a t i o n o f t h e e n e r a f l u x and s w e l l - s e a i n t e r a c t i o n f o r a Neumann Spectrum. J. F l u i d Mech., 15: 385-398. Hasselmann, K . , B a r n e t t , T . P . , Bouws, E . , C a r l s o n , H . , C a r t w r i g h t , D . E . , Enke, K . , Ewing, J . A . , Gienapp, H . , Hasselmann, D . E . , Kruseman, P . , Meerburg, A . , Miiller, P . , O l b e r s , D . J . , R i c h t e r , K . , S e l l , W . , Walden, H . , 1973. Measurements o f windwave growth and s w e l l decay d u r i n g t h e J o i n t North Sea Wave P r o j e c t (JONSWAP), Deut. Hydrogr. Z . , Suppl. A . , 8 ( 1 2 ) . Hasselmann, K . , Ross, D . B . , Miiller, P., S e l l , W . , 1976. A p a r a m e t r i c a l wave p r e d i c t i o n model. J. phys. Oceanogr. 6 : 201-228. 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 . , 1977. Numerical wave c l i m a t e s t u d y for t h e North Sea. Report No. Ex775. Pierson, W.J., T i c k , L . J . , Baer, L . , 1966. Computer-based procedures f o r p r e p a r i n g g l o b a l wave f o r e c a s t s and wind f i e l d a n a l y s e s c a p a b l e of u s i n g wave d a t a o b t a i n e d by a s p a c e c r a f t . Proc. 6 t h Naval Hydrodynamics Symposium, Washington, D.C., 499. P h i l l i p s , O.M., 1960. On t h e dynamics of unsteady g r a v i t y waves o f s m a l l amplitude, P a r t I . J . F l u i d Mech., 9 : 193-217.
.
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Modelled and observed mean zero-up-crossing periods TZ at Famita position (reproduced from H F S Report No. EX775, 1977).
141
ON THE FRACTION O F WIND MOMENTUM RETAINED BY WAVES
M. DONELAN Canada C e n t r e f o r Inland Waters, Burlington, Ontario (Canada)
ABSTRACT The fraction of t h e momentum or energy transfer across t h e air-water interface which remains in t h e wave field is a crucial p a r a m e t e r in many wave prediction models. In this paper, d i r e c t measurements of t h e air-water momentum transfer coupled with wave spectra measured at several f e t c h e s provide e s t i m a t e s of this parameter. Both laboratory and field measurements a r e used to provide a wide range of conditions. I t is found t h a t t h e fraction of momentum remaining in t h e wave field c a n be as much as %of t h e total transferred locally across t h e air-water interface, and is largely dependent on t h e wave age. INTRODUCTION Given t h e s u r f a c e wind over a body of water, t h e problem of wind wave prediction may, in t h e simplest instance of still d e e p water, be reduced to t h r e e distinct steps: estimation of t h e s u r f a c e momentum flux, determination of t h e fraction of t h e momentum transferred which remains in t h e wave field, and finally, t h e solution of an appropriate momentum balance equation of t h e type:
where t h e M., v., T. a r e t h e average wave momentum per unit area, average group velocity '
1
'
and surface wind s t r e s s components respectively; Y is t h e subject of this paper. The f i r s t step, essentially t h a t of t h e determination of t h e surface drag coefficient, has been t h e cause of intense empirical study over t h e past t w o or t h r e e decades. Some of this work is summarized by S t e w a r t (1974) and will not further concern us here. The third step, t h e solution of a n equation like (1) or its energy equivalent, is normally carried o u t through finite difference methods on a digital computer. Its solution, which clearly requires a link between t h e average group velocity and t h e momentum, c a n be approached in several ways, t w o examples of which a r e given in Hasselmann et al. (1976)and Donelan (1979).
142
It is to the second step that this paper is directed. W e wish to establish t h e fraction Y of air-water momentum transfer which is retained by the wave field. seek t h e net momentum transfer to t h e wave field from the wind
Or, in other words, we
- the
difference between
wind input to the waves and local wave dissipation - in relation to t h e total momentum transfer across the air-water interface. The fraction Y might reasonably be expected to be a function of other dimensionless parameters which characterize t h e air-water interface, in particular t h e ratio of peak wave phase speed to wind speed, generally termed 'wave age', and some atmospheric stability index
-
here we use t h e bulk Richardson number.
This paper
describes field and laboratory experiments aimed at exploring t h e dependence of Y on inverse wave a g e U/c. The bulk Richardson number dependence of Y is investigated for a restricted range of U/c only. EXPERIMENTAL DESIGN The ideal experimental arrangement for the direct determination of Y obtains when mean wind and wave directions and t h e line joining measuring positions all coincide. In t h e light of figure I, we may rewrite (1) in terms of t h e directional spectrum of wave energy F b , 0) and t h e dispersion relation d k ) .
where F(w, 0) and the corresfmnding one-dimensional frequency spectrum E(w) a r e defined, in
I E dw = t h e mean square surface displacement. W e have, of course, made use of t h e equality of mean potential and kinetic energies applicable to small amplitude waves and the relation between momentum and energy valid for
t h e usual manner, such t h a t I I F dB dw =
progressive waves of constant shape fi=EC/w (Phillips, 1977). w and k a r e t h e radian frequency and radian wave number; p and ow a r e the air and water densities; g, t h e a acceleration due to gravity and -u' w' is t h e Reynolds stress in the air near t h e surface. In order to precisely evaluate Y from (21, we need measurements of the directional
-
distribution of wave energy F(w, 9), the dispersion relation w(k) and t h e Reynolds stress as a function of x and t.
The satisfaction of such stringent requirements being beyond our
resources, we set out to estimate Y from t h e finite difference equivalent of (2) using twopoint measurements of E(w) with estimates of F(w, 9),w (k) and u?;;r at a single location. The estimates of F(w, 0) and w(k) were non-dimensionalized with respect to the peak frequency and applied to the point measurements of E(w). Time differences were rendered insignificant in t h e laboratory and were estimated in t h e field from consecutive 20 minute averages.
It
behooves us, a t this point, to show the experimental arrangement before attempting to provide further details of t h e analysis procedure. Figures 2 and 3 show t h e field and laboratory experimental arrangements. The field site consists of a tower, 1100 m offshore mounted on the bottom in 12 m of water, and a 'waverider' accelerometer buoy, moored an additional 2000 m offshore along t h e normal to
143
Fig. 1. Schematic of idealized directional spectra F (a, 9) for two values of the radian frequency
.
i--:-:1;-.Lake Ontario
0 4 0 km
1
2 3 4km
+
Elevation 100rn
Fig. 2. Map of the field s i t e showing the location of the tower and the waverider accelerometer buoy at the western end of Lake Ontario.
144
t h e shoreline through t h e tower. The tower supports a n a r r a y of 14 c a p a c i t a n c e wave s t a f f s for estimating directional s p e c t r a and various micro-meteorological equipment for estimating t h e Reynolds s t r e s s and t h e bulk Richardson number. Further details of t h e tower and i t s measuring equipment a r e given by Birch et al. (1976) and Der and Watson (1977). Suffice i t t o say t h a t t h e wind speed, direction and Reynolds s t r e s s measurements were made with a Gill anemometer bivane (Young, 1971) mounted at 11.5 m above t h e water and sampled five times a second per channel. The laboratory tests w e r e carried out in a large closed wind-wave flume at t h e Canada C e n t r e for Inland Waters (CCIW). Figure 3 provides information on t h e flume's dimensions and general characteristics sufficient for our purposes here.
The
directional s p e c t r a w e r e estimated using a 1/28 scaled version of t h e tower array placed as indicated in figure 3. One-dimensional frequency s p e c t r a w e r e measured using capacitance wire staffs at s t a t i o n s 4 and 5 (denoted in t h e figure by 54 and 5 5 1, and wind speed and Reynolds s t r e s s w e r e obtained through x-film anemometry at t h e intermediate position shown, 26.2 c m above t h e mean water level.
80.0m I I I I
5 3 . 5 m 39.lm 49.8m
I
! It" '
Fetch = 0 I
I
I
0 1 2m
Section A-A
\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\: Fig. 3. Plan of t h e wind-wave f l u m e at t h e Canada C e n t r e f o r Inland Waters showing the disposition of wave 5 and wind U, u+ measurements. The location of t h e directional wave array is indicated by a n X labelled "D.S.". The lower p a r t of t h e figure shows a n enlargement of t h e directional array.
145
DISPERSION RELATION AND SPREADING FACTORS
* 1 --+
0
Fig. 4. The wave-number s p e c t r u m f o r a frequency band of width 1.97 r a d / s c e n t r e d on 20.7 rad/s, f r o m laboratory run 3. The full c i r c l e s r e p r e s e n t t h e limits of t h e d e e p w a t e r dispersion relation corresponding t o t h e e d g e s of t h e frequency band analysed, while t h e dashed c i r c l e corresponds to t h e energy-weighted mean frequency of t h a t band. The a r e a of integration is shown shaded. Although i t i s c o n v e n i e n t to w r i t e (2) in t e r m s of t h e directional s p e c t r u m F (w,
e),
the
s ..
d a t a a r e f i r s t analysed in t e r m s of t h e frequency wave-number s p e c t r u m x ( w , k), which is, in f a c t , required to d e t e r m i n e t h e dispersion relation w (k). Figure 4 is a s a m p l e of t h e wavenumber s p e c t r u m X for a frequency band A w . number s p e c t r u m here.
Details of t h e c h a r a c t e r i s t i c s of t h e wave-
x a r e given in Donelan et al. (1979) and only a f e w f e a t u r e s need concern us
The solid c i r c l e s shown (figure 4 ) r e p r e s e n t t h e limits of t h e d e e p w a t e r dispersion
relation ( w = m ) corresponding to t h e l i m i t s of t h e frequency band analysed. wave-number
The a v e r a g e
corresponding to t h e frequency band A U is e v a l u a t e d as indicated in ( 3 ) . I t is
assumed t h a t k is a function of w only a n d n o t of 9. In addition t o t h e dispersion relation, we will need spreading f a c t o r s a s s o c i a t e d with t h e t w o f i r s t t e r m s of (2), which w e will designate A( w) and B( w) and d e f i n e thus:
146
Since noise, which is predominant at large wave-numbers, tends to bias t h e relations (3), t h e integration of (3) includes only a band of thickness 2 b and TI radians in angular width c e n t r e d on t h e peak of t h e directional spectrum. Finally, t h e angle 0 is referenced to the mean momentum direction 8 defined thus: tan8 =
II x b,;) k sin 0 dkx dk II X (w, 2)k cos 0 dkx dkY
(4)
Noting t h a t I F k cosn 0 d0 = /I X k cosn 0 dkx dk and t h a t k andw a r e independent of 0, Y i t c a n b e seen t h a t t h e integrals of (2) c a n be reconstructed from t h e spreading f a c t o r s A(w) and B(w) provided t h e dispersion relation and t h e frequency spectrum E(w) a r e known. Having determined A(w), B(w) and 5; (w) at o n e fetch, we a r e faced with t h e problem of applying t h e results to t h e one-dimensional frequency spectrum E(w) at t h e o t h e r fetch. By analogy with t h e results of Hasselmann et al. (1973) f o r t h e one-dimensional frequency spectra, we would e x p e c t t h e spreading f a c t o r s and t h e dispersion relation to scale with the peak frequency w W e have n o t included any f e t c h dependence of t h e scaled spreading P' f a c t o r s A (w/w ) and B (w/w 1. Although Mitsuyasu et al. (1975) have indicated some fetch P P dependence of their spreading parameter, their results a r e not applicable to t h e high values of U/c encountered in our experiments. For example, at U/c=3.5 Mitsuyasu et al. (1975) would suggest a n angular distribution of t h e form cos(0/2), which is certainly n o t in agreement with our observations (Donelan et al., 1979). Figure 5 shows a n example of t h e spreading f a c t o r s A(w) and B(w). The frequency axis has been scaled by t h e peak frequency w and a second order polynomial in {(cob )'-I} fitted to P P t h e d a t a points. A spreading f a c t o r of 1.0 corresponds to all waves at t h a t frequency travelling in t h e s a m e direction.
I t is seen t h a t t h e spectrum broadens away from t h e peak.
Spreading f a c t o r s corresponding to t h e commonly assumed cos'0 or c0s48distributions of F ( w , 0 ) a r e indicated f o r comparison on figure .5. T h e average wave-number
scaled by t h e theoretical value f o r deep water at t h e peak
f r e q u e n c y 3 /g is plotted in figure 6 against (w/w ? a n d a second order polynomial in b / w )' P P P f i t t e d to t h e d a t a points. In all cases, t h e polynomials were f i t t e d so as to minimize
C (Ayi)'* E(wi). This forces t h e f i t to b e closest near t h e peak of t h e spectrum. In t h e case
147 of k(w), t h e point (0,O) was added and given weight equal to t h a t of t h e spectral peak. In t h e s e coordinates, t h e theoretical dispersion relation is t h e straight line through t h e origin shown (dashed) in figure 6.
296175
l.ol---!
1.01- -1
296175
I
- C O S ~ ~ -ws4
0.8
e
-cos2e
A(w) 0.6 '
0.4 0
I I I
e
1
2
1
3
4
4
1.01--7
3
Fig. 5. Samples of directional spreading f a c t o r s A 6 )and B (w) f o r field (296175) and laboratory (3) measurements. The curves shown a r e f i t t e d second order olynomials in {(w/wp) * 1 }. The equivalent c o n s t a n t spreading f a c t o r s f o r cos2 8 and cos 8 directional
-
distributions a r e also indicated by t h e horizontal lines. reasonable approximations only near t h e peak.
e . .
I t c a n b e seen t h a t these yield
The normalized one-dimensional frequency s p e c t r a a r e shown in figure 7 and, in figure 8, t h e calculated phase speed w /k' is compared with its theoretical small amplitude, deep water In all cases, t h e value g/w both normalized by t h e theoretical value at t h e peak g/w P' abscissa is t h e square of t h e frequency scaled by i t s value at t h e spectral peak. It is clear t h a t t h e small amplitude, d e e p w a t e r dispersion relation is not exactly followed and, in f a c t ,
148
81
296175
81 7-
6 71
-
3
6-
51
kg
kg "P2
Fig. 6. Computed mean wavenumbers in each frequency band normalized by t h e theoretical value at t h e peak of t h e spectrum. The solid curves a r e fitted second order polynomials in (w/wp) 2, while the dashed straight lines a r e the theoretical small amplitude, deep water dispersion relation. Only frequency bands whose wave-number spectra could be successfully resolved a r e included (Donelan et al., 1979).
296175
3
Fig. 7. Normalized one-dimensional frequency spectra corresponding to Figures 5 , 6 and 8.
149 t h e disagreement becomes m o r e pronounced at higher U/c values (Ramamonjiarisoa and Coantic, 1976 and Donelan et al, 1979).
b
3
296175
I I I
0
1
I I
I
I
0
I I I
Fig. 8. Observed (dots) and theoretical (curve) phase speeds. theoretical peak value g/w P'
I
I
I
r
Both a r e normalized by t h e
The polynomial kg/$ = a0 + a1 (w/w )' + a2 (w/w )4 may then be used to determine k and P P dw/dk o n c e t h e peak frequency w is known. Similarly, t h e spreading f a c t o r s A(u) and Bbl) P may b e derived and applied, through (31, to t h e evaluation of t h e integrals of (2) for t h e f e t c h
at which only E(w) is known. ANALYSIS OF FIELD DATA Observations w e r e obtained in runs of one hour under mini-computer control triggered by a change in t h e ten-minute average wind speed. During such runs, all wave s t a f f s and relevant meteorological sensors w e r e sampled at 5 Hz and, at t h e s a m e time, t h e waverider recordings w e r e made. Although t h e r e w e r e a large number of these runs, only six satisfied t h e c r i t e r i a t h a t both wind and wave directions b e within 25 degrees of t h e normal t o t h e beach and t h a t t h e peak frequency at t h e waverider b e less than 3.14 rad/s - t h e upper limit of t h e waverider's reliable resolution. These six a r e summarized in table I. The tower runs,
150
TABLE I Summary of field data (1976)
5
V+
Symbol
U
-
-u'w'
-
w
Rb
0 1100
0 3100
84.4 20.5 40.4 22.3 42.4 28.0
178.9 84.1 115.2 39.2 121.0 96.0
Deg. m/s (m/s)' 11.5 10 11.5 1100 1100 1100
11.5 1100
0 0 1100 3100
29617 30209 30219 30317 31106 31821
261 240 238 236 236 248
-0.005 -0.034 +0.004 +0.019 -0.008 +0.003
3.3 4.3 3.7 4.1 3.7 4.0
+
12.1 8.3 10.9 8.8 11.1 9.2
0.334 0.109 0.344 0.166 0.392 0.261
5'
cm2
Units m/s Height(m) 11.5 Fetch (m)1100 12.4 8.5 11.1 8.9 11.3 9.5
-
w 5' P P rad/s rad/s cm2
2.4 3.1 2.7 3.1 2.8 3.0
U/C
Y
10
*
3.4 3.1 3.5 3.3 3.6 3.2
0.060 0.071 0.028 0.034 0.060 0.052
V is the measured wind speed
*
c i s the average of the theoretical phase speeds of the peak frequencies at tower and waverider.
TABLE I1 Summary of laboratory data U -u'w' Symbol V+
-
Rb
P
W
P rad/s 0 49.8
-
c2
cm2
cm2 0 49.8
10
G2
u/c
Y
Units m/s Height ( m ) 0.26
m/s 10
0.26
0.26
rad/s 0
Fetch ( m ) 49.8
49.8
49.8
49.8
39.1
20.99 18.22 18.54 5.49 10.96 4.08 17.67 14.85
0.121 0.110 0.120 0.130 0.112 0.117 1.034 1.038 0.696 0.749 0.025 0.171 0.020 0.651 0.371
+0.001 +0.001 +0.001 +0.001 +0.001 +0.001 0 0 0 0 +0.002 0 +0.003 0 0
10.9 10.9 10.9 10.4 10.9 10.9 7.2 7.9 7.9 7.9 11.6 9.4 15.3 7.9 8.9
9.0 9.9 9.0 9.4 9.0 9.0 6.6 6.6 7.2 7.2 11.6 8.7 13.9 7.2 7.5
2.01 1.78 1.85 2.12 1.81 1.91 16.16 15.09 10.86 10.31 0.56 3.35 0.22 9.99 7.84
2.32 2.32 2.52 2.71 2.41 2.18 20.89 18.80 15.29 12.82 0.69 4.75 0.24 14.67 10.50
9.7 10.1 9.7 9.9 9.5 9.7 14.7 15.4 14.0 14.3 6.5 10.1 6.1 13.6 12.4
0.108 0.205 0.262 0.191 0.223 0.096 0.183 0.143 0.265 0.140 0.189 0.354 0.043 0.304 0.311
4.58 6.86 11.94 15.40 18.83 4.49 21.88
0.020 0.051 0.208 0.426 0.766 0.019 1.179
+0.011 +0.004 +0.001 +0.001 +0.001 +0.007 +0.001
17.0 13.0 10.5 9.5 8.9 16.7 7.8
15.0 11.5 9.0 8.4 7.7 14.9 7.0
0.07 0.51 2.38 3.93 7.37 0.16 13.53
0.19 0.55 3.37 5.54 9.56 0.19 20.08
7.5 8.5 11.8 14.0 15.9 7.2 16.5
0.160 0.019 0.159 0.121 0.088 0.057 0.164
Phase Run 12 13 14 15 19 20 I 23 24 25 26 28 38 49 51 52
6.54 6.54 6.55 6.58 6.45 6.53 11.69 11.71 10.63 10.67 4.06 7.19 2.80 10.31 9.30
2 3 5 6 7 29 30
3.29 4.80 7.79 9.46 10.87 3.24 12.00
9.70 9.56 9.71 9.86 9.50 9.64
20.94
(m/s)'
W
0 39.1
53.5
*
................................................................................ I1
+
*
V is the measured wind speed c is the average of the theoretical phase speeds of the peak frequencies at both stations In phase I1
-
u'w' was not measured but inferred from figure 10.
151 identified by a composite number consisting of t h e Julian day and hour (GMT) of start, were divided into four consecutive sections of 13.64 minutes each.
The waverider runs were
divided into three consecutive 20 minute sections. Directional spectra and Reynolds stresses were available at t h e tower only. Thus non-dimensionalized characteristics of the directional spectra derived from t h e tower d a t a were applied to t h e waverider d a t a closest in time. Evidently the reliability of this method, based on the difference between waverider and capacitance wave staff measurements, rests on accurate calibrations of both types of instruments.
The capacitance wave staffs were calibrated in situ by immersion to marked
intervals about t h e mean water level.
The one-dimensional frequency spectrum discussed
here is the average of the spectra from each of t h e 14 staffs. The original spectrum of 128 estimates is grouped into 15 equally spaced bands between 0.03 and 7.4 rad/s. The waveriders were calibrated on a rotating a r m and their d a t a analysed by t h e Marine Environmental Data Service, Ottawa (Wilson and Baird, 1972) who subsequently provided us with a tape of 62 equally spaced spectral density estimates from 0.32 to 3.14 rad/s. w-5
These were then grouped into frequency bands corresponding to the tower spectra, and an tail appended to fill t h e eight high frequency bands, which a r e beyond the frequency
resolution of t h e waverider.
The tail was computed from the peak spectral estimate and
frequency in the following form: i=8to15
(5)
In order to compare t h e waverider and tower wave measurements, t h e waverider was moored I50 m to t h e northwest of t h e tower for a period of one month. Figure 9 shows the superimposed spectra of tower and waverider measurements for two cases in which conditions were very steady so that, although t h e tower measurements covered only t h e first 68%of the waverider observations, they could be expected to yield t h e same results.
Evidently the
comparison is very encouraging even when t h e peak frequency approaches 3.14 rad/s - the Nyquist frequency of t h e waverider analysis. The addition of a tail is clearly necessary in such cases. As mentioned before, t h e characteristics of the directional spectrum at the tower a r e
applied to the waverider observations and t h e integrals of (2) are evaluated at both the tower and t h e waverider.
Linear interpolation between pairs of tower results is applied to bring
them into time coincidence with t h e waverider results.
The finite differences of (2) a r e then
evaluated using only t h e final two sections of each run, i.e. t h e last 40 minutes of each hour. In t h e evaluation of (21, the time difference was computed from the difference over these last two (20 minute) d a t a sections of t h e space-average of t h e first integral of (2). Similarly, the space difference was computed from 'the difference between waverider and tower of the time-average of the second integral of (2). Since the wind speed change which triggered the recording was frequently accompanied by a wind direction change, t h e wave field was generally steadier during t h e latter part of t h e hour. In general, t h e second term of (2) was twice as large as t h e first.
152
Fig. 9. Comparison of wave s p e c t r a measured by fixed capacitance wave s t a f f s (TOWER) with those measured by t h e waverider a c c e l e r o m e t e r buoy (WAVERIDER). During the comparison t h e waverider was moored near t h e tower at t h e s a m e fetch. The normalizer, Emax is simply t h e highest value of E @ ) f r o m e i t h e r tower or waverider. The dashed lines indicate t h e
IJJ-’
tails appended to t h e waverider s p e c t r a which extend only to 3.14 rad/s.
The Reynolds s t r e s s and all o t h e r meteorological p a r a m e t e r s w e r e derived from averages of tower measurements over t h e last 40 minutes of t h e hour. The wind speed, measured at
11.5 m, was adjusted to 10 m using t h e measured friction velocity u,=(-Tpand a logarithmic velocity profile. The e r r o r s introduced by this procedure, through diabatic distortions to t h e logarithmic profile, a r e small, since t h e wind gradient is weak at these heights. ANALYSISOFLABORATORYDATA The laboratory d a t a were gathered in t w o phases. In phase I, t h e r e were 15 runs, during which recordings w e r e made of surface elevation at stations 4 and 5 and of wind speed and Reynolds s t r e s s near station 4. The sampling frequency and duration were 60 hz and 4.5 minutes for nine of these runs, and w e r e 30 Hz and 9.1 minutes for t h e o t h e r six. In phase 11, t h e directional spectrum was measured 3.75 m downwind of station 4 (figure 3), t h e onedimensional frequency spectrum at s t a t i o n 5 and t h e wind speed at station 4.
There were
seven of t h e s e runs of duration 13.7 minutes and sampling frequency 20 Hz. The still water depth was 125 c m in t h e f i r s t phase and 110 c m in t h e second. In order to i n t e g r a t e t h e t w o phases, w e will a t t e m p t to r e l a t e t h e vertical flux of wind momentum (phase I) and t h e horizontal flux of wave momentum (phase 11) to t h e wind speed, which was monitored in both phases. The r a t i o of t h e measured friction velocity u,, t o the measured wind speed as a function of wind speed is given in figure 10.
The scatter is
sufficiently small to insure a reasonably a c c u r a t e e s t i m a t e of t h e friction velocity from the
153
measured wind speed in phase I1 and t h e t w o highest wind speed cases of phase I wherein t h e x-film anemometer was too frequently assaulted by w a t e r droplets to permit a reliable e s t i m a t e of t h e friction velocity. A more appropriate p a r a m e t e r for t h e abscissa might be U/c, b u t this figure is m e a n t only as a n interpolation tool and t h e wind speed U.26 is more Furthermore, in these experiments, c is determined almost
accurately measured t h a n U/c.
entirely by t h e wind speed. For t h e s a m e reasons, t h e dimensionless r a t i o of t h e downwind flux of wave momentum to t h e wave energy ( Fk/u du/dk cos2 8 d 8 6, is given in
F ) II
figure 11 as a function of t h e measured wind speed. This is simply t h e r a t i o of t h e integral of t h e second t e r m of ( 2 ) to t h e mean square surface dispacement.
Other conditions being
unchanged, this dimensionless r a t i o might b e expected to be a function of U/c o r U.26 for t h e s e experiments. Here again t h e s c a t t e r is small enough to allow use of t h e sketched curve as a n interpolation tool in t h e analysis of phase I data; i.e. to infer t h e integrals of t h e second t e r m of (2) from t h e mean square surface displacement at t h e t w o f e t c h e s being considered.
y* u26
-I
.02
0
I
I
I
I
I
4
6
8
I
2
10
12
Fig. 10. R a t i o of friction velocity u, to measured near surface (26 c m ) mean wind versus t h e mean wind. The smooth c u r v e is f i t t e d by "eye" to t h e laboratory (phase I) results.
For t h e s a k e of uniformity with t h e field data, t h e measured wind speed has been adjusted to 10 rn assuming a neutral logarithmic profile. Both t h e 10 m and 26 c m wind speeds a r e
reported in t a b l e 11. I t should b e noted t h a t t h e extrapolated 10 m wind speed is employed in this analysis only to construct t h e p a r a m e t e r U/c, which, at low values, is not sensitive to
154 small errors in U. It is only at t h e lowest wind speeds t h a t t h e stability index Rb indicates significant departure f r o m neutrality. The bulk Richardson number Rb is defined in terms of t h e wind speed U and air t e m p e r a t u r e Ta evaluated at height Z and t h e water surface t e m p e r a t u r e Tw: Rb = Zg(Ta-Tw)/(273+Ta) U
'.
Fig. 11. The r a t i o of t h e downwind flux of wave momentum to wave energy versus the measured mean wind. The smooth curve is f i t t e d by "eye" to t h e laboratory (phase 11) results. This r a t i o would b e 0.5 if a l l wave components w e r e travelling in t h e s a m e direction and were in s t r i c t obedience of t h e dispersion relation IAI G. Lower values reflect significant directional spread, while higher values occur because t h e r a t i o of group to phase speeds approaches unity for trapped harmonics. For t h e phase I1 data, (2) was solved by ignoring t h e f i r s t t e r m (steady s t a t e ) and applying (3) to t h e one-dimensional frequency spectrum at s t a t i o n 5 and t h e average of t h e spectra of
t h e 14 s t a f f s of t h e directional array. The relations (3) w e r e determined separately for each of t h e seven phase I1 cases. Here, as in t h e field d a t a , t h e original spectrum of 128 estimates was grouped i n t o 15 equally spaced bands between 0.12 and 29.6 rad/s. In t h e case of the phase I d a t a t h e flux of wave momentum at s t a t i o n s 4 and 5 was e s t i m a t e d using figure 11 and t h e mean square s u r f a c e displacement -ST measured at stations 4 and 5. RESULTS AND DISCUSSION The e s t i m a t e s of Y (figure 121, f r o m both field and laboratory observations, show a definite dependence on U/c, although t h e r e is considerable s c a t t e r . The field observations
155
0
PHASE II .FIELD 0 BRETSCHNEIDER (1973)
0
3-
0
0
.24-
Y 16-
Fig. 12. The fraction of wind momentum retained by waves Y versus the inverse wave age U/c. For the laboratory d a t a t h e large crosses indicate t h e averages of t h e d a t a assembled in groups of width 2.5 in U/c. The vertical bar indicates t h e average value of U/c in the group. The cross representing the field d a t a has been placed at the value of Y corresponding to neutral atmospheric stability (figure 13). The point of full development (U/c=0.83, Y=O) according to Bretschneider (1973) provides another e s t i m a t e of Y. The straight line (solid) at lower values of U/c provides convenient access to Y for wave prediction purposes. The extension of t h e line to higher values of U/c (dashed) serves only to illustrate the trend in Y. fall within a rather narrow band of U/c and t h e variability of t h e field estimates of Y seem to
be largely due to stability differences (figure 13). The smooth curve, fitted by 'eye' to figure 13, suggests that the appropriate value of Y for neutral stability is 0.053. This value is indicated in figure 12 by a large cross centred on t h e mean value of U/c for these six field observations. The laboratory observations a r e rather badly scattered but, nonetheless, indicate a general increase in Y as U/c increases from 6 to about 12, then, beyond that, a general decrease. The scatter in t h e laboratory values seems surprising at first sight in view of t h e steady and controlled conditions of these measurements.
Stability effects can easily
156
y .04.02-
0' -.&
-.d3
I
-.02
1
1
-.01
0
I
.01
I
.02
Rb
Fig. 13. Y versus bulk Richardson number Rb f o r t h e field data. The smooth curve has been f i t t e d by "eye", and its intersection with R b = 0 transposed to Figure 12. b e ruled out by examining t h e set of a l m o s t consecutive runs (12, 13, 14, 15, 19, 20) taken under nearly t h e s a m e conditions. In t h e s e runs, Y is seen to vary f r o m 0.10 to'0.26. The explanation appears to b e t h a t t h e mean value of averaged over several hundred wave
7,
periods, is a function not only of t h e x (down-tank) direction but also of t h e y (cross-tank) direction, and t h e y dependence is itself dependent on f e t c h x and t h e mean flow parameters: y (x, U)]. I t is n o t our purpose h e r e to explore t h e e x a c t functional form f, but -merely=[f tox,demonstrate t h a t t h e failure of a t i m e average of ' 5 along t h e centreline of the 52
tank t o represent t h e cross-tank a v e r a g e value at t h a t f e t c h is much of t h e cause of scatter in figure 12. To d o this we examine (table 111) t h e values of ;?- from four of t h e s t a f f s for t h e five runs (2, 3, 5, 6, 7) at t h e beginning of phase 11. The four s t a f f s a r e located in pairs (figure 3) at t h e ends of t h e long a r m s of t h e array. The s t a f f s in e a c h pair a r e separated by 3.6 c m and t h e pairs a r e separated by 73 c m symmetrically across t h e tank at t h e s a m e fetch. Runs 29 and 30 a r e o m i t t e d f r o m this comparison because t h e e n t i r e array was r o t a t e d by 15 degrees for t h e s e runs. Table I11 shows
52 f o r staff
number 1 and t h e r a t i o of
7from the
o t h e r s t a f f s to this. While t h e r e a r e differences of about 6% between t h e members of the pairs, t h e s e differences a r e nearly c o n s t a n t and r e f l e c t calibration differences of about 3%. Much larger and e r r a t i c differences occur between t h e pairs. Also shown in t a b l e 111 is the f r o m t h e 14 s t a f f s of t h e a r r a y to from station 5. The r a t i o of t h e a v e r a g e
52
7
average of t h i s r a t i o is 1.51. Evidently, t h e cross-tank variability of about 12% contributes significantly to t h e e r r o r in determining y. These cross-tank differences do n o t appear t o be d u e to reflected waves from t h e beach since these would be d e t e c t e d in t h e directional
157
s p e c t r a and they w e r e not. I t s e e m s probable t h a t they a r i s e as a consequence of reflection I t would be interesting to know whether similar cross-tank variations
f r o m t h e side walls.
have been d e t e c t e d in o t h e r tanks. In retrospect, it would have been preferable to have made several cross-tank measurements at e a c h fetch, and thereby reduce t h e s c a t t e r of figure 12. Lacking these, w e have averaged t h e e s t i m a t e s of Y in e a c h of five equal bands of U/c. These are shown as large crosses on figure 12. TABLE I11 Cross-tank variance of
2
Right
Pair
Left
Pair
cm 2
0.18 0.58
3
5 6
1.14 0.85
.994 1.06 I .07 I .05
1.10 0.81 1.17
2.62
1.03 1.02
1.41 1.30
1.51
1.06
1.25 I .05 I .09
Average
1.047
1.076
I .026
Std. Dev.
0.03
0.13
0.12
2.93 5.37 9.17
7
*
1.09
1.42
The subscripts I , 2 , 8 , 9 r e f e r to individual wave s t a f f s in t h e directional a r r a y , D . S . r e f e r s to t h e a v e r a g e of all 14 s t a f f s in t h e a r r a y and 5 denotes t h e upstream s t a t i o n 5. I t is commonly held (Bretschneider, 1973) t h a t full development occurs when t h e wave age
reaches 1.2 (U/c=0.83). wave field (Y=O).
That is, at this s t a g e t h e r e is no n e t change in t h e momentum of the
This point of full development has been added to figure 12. Practical wave
forecasting, on all but t h e smallest ponds, is concerned only with t h e region of U/c less than 6 or so. In this region, Y is approximated by t h e straight line shown: Y = 0.02-
U C
-
0.017
,
0.83
<
< 7
(6)
A t larger values of U/c, applicable to very short f e t c h e s or unusually sharp wind transients, y is seen to increase to about 0.24 and then to d e c r e a s e again as U/c exceeds 12. At these very high values of U/c, whitecapping becomes a n important dissipative process. Evidently t h e non-linear wave-wave interactions a r e n o t sufficiently rapid to transfer all the absorbed momentum to longer waves. The waves at t h e peak of t h e spectrum have become q u i t e steep, showing pronounced harmonic distortion (Donelan et al., 1979), and a r e unable to retain t h e absorbed wind momentum themselves, losing most of i t through whitecapping. Thus, at t h e s e high values of U/c, t h e momentum retained by t h e wave field must c o m e partly f r o m non-linear transfer to frequencies below t h e peak and partly f r o m d i r e c t wind input to t h e s e longer waves. An interesting discussion of t h e momentum balance may b e found in Hasselmann et al. (1973).
158 The value of Y for over-developed waves ( 0 < U/c < 0.83 ) and for waves in an adverse wind (U/c
1
100
10
1000
xg U* Fig. 14. Y versus non-dimensional fetch xg/U2. The crosses correspond to those in figure 12. The freehand curve shown illustrates the general trend. Finally (figure 14), we examine Y as a function of non-dimensional fetch xg/U2 (Kitaigorodskii, 1962). assumed cos
It i s interesting to note that Hasselmann et al. (1973), using an
8 spreading factor for F(U,
and the fetch dependence of of
e),
an assumed overall drag coefficient of I.OxIO-’
7,obtained an estimate for
y of 0.05.
The fetch dependence
52 was based on laboratory measurements i n the non-dimensional fetch range of 3.0~10-~
to 1.0 and field measurements in the range of 60 to lo4.
Our measurements lie largely
between these two ranges and demonstrate that considerably more than 5%, perhaps up t o 25 96, of the locally transferred wind momentum may be retained and advected away by the
wave field. ACKNOWLEDGEMENTS Many members of the management and staff of the Canada Centre for Inland Waters have
159
generously supported this work. I a m g r a t e f u l to T. Nudds, D. Beesley and J. Carew for assistance in t h e collection of field
and laboratory data, to t h e staff of t h e Marine Environmental Data Service, O t t a w a for waverider field support and analysis, and to M. Dick, P. Hamblin, J. Hamilton, W. Hui and M. Skafel f o r valuable c o m m e n t s and criticisms. REFERENCES Birch, K.N., Harrison, E.J. and Beal, S., 1976. A computer based system for d a t a acquisition and control of scientific experiments on r e m o t e platforms. Proc. Ocean'76 Conf., Wash., D.C. Bretschneider, C.L., 1973. Prediction of waves and currents. Look Lab/Hawaii, 3.1:17. Der, C.Y. and Watson, A.S., 1977. A high-resolution wave-sensor array f o r measuring wave directional-spectra in t h e nearshore zone. Proc. O c e a n 7 7 Conf ., Los. Angeles, Calif. Donelan, M.A., 1979. A simple numerical model for wave and wind s t r e s s prediction. (In press). Hamilton, J. and Hui, W.H., 1979. Directional s p e c t r a of wind-generated Donelan, M.A., waves. (In press). Hasselmann, K., Barnett, T.P., BOUWS,E., Carlson, H., Cartwright, D.E., Enke, K., Ewing, J.A., Gienapp, H., Hasselmann, D.E., Kruseman, P., Meerburg, A., Muller, P., Olbers, D.J., Richter, K., Sell, W. and Walden, H., 1973. Measurements of wind-wave growth and swell decay during t h e Joint North Sea Wave Project (JONSWAP). Deut. Hydrogr. Z., Suppl. A, 8, No. 12, 22 pp. Hasselmann, K., Ross, D.B., Muller, P. and Sell, W., 1976. A parametric wave prediction model. J. Phys. Oceanog., 6:200-228. Kitaigorodskii, S.A., 1962. Applications of t h e theory of similarity to t h e analysis of windgenerated wave motion as a stochastic process. Bull. Acad. Sci. USSR Geophys. Ser. No. 1:105-117. Mitsuyasu, H., Tasai, F., Suhara, T., Mizuno, S., Ohkusu, M., Honda, T. and Rikiishi, K., 1975. Observations of t h e directional spectrum of ocean waves using a cloverleaf buoy. J. Phys. Oceanog., 5:750-760. Phillips, O.M., 1977. The dynamics of t h e upper ocean. 2nd edition. Cambridge University Press, Cambridge, 336 pp. Ramamonjiarisoa, A. and Coantic, M., 1976. Loi expkrimentale d e dispersion des vagues produites par le vent sur une faible longueur d'action. C.R. Acad. Sc. Paris, Skrie B. 282111-114. Stewart, R.W., 1974. The air-sea momentum exchange. Boundary-Layer Met., 6:151-167. Wilson, J.R. and Baird, W.F., 1972. A discussion of some measured wave data. Proc. Thirteenth Conf. on Coastal Engineering, Vancouver, B.C., pp. 113-130. Young, R.M., 1971. Precision meteorological instruments. Catalogue of t h e R.M. Young Company, Traverse City, Michigan.
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161
TIE NUSC WINDWAVE AND TURBULENCE OBSERVATION PROGRAM (WAVTOP); A STATUS REPORT
2
DAVID SHONTINGlt2 and PAUL TEMPLE
'Naval
Underwater Systems Center, Newport, R. I. 02840 (USA)
2Graduate School of Oceanography, U n i v e r s i t y of Rhode I s l a n d , Kingston, R . I .
02881 (USA)
ABSTRACT
Wind generated white caps have been i d e n t i f i e d as providing t h e key mechanism by which a f l o a t i n g o i l s l i c k i s fragmented and i n j e c t e d as d r o p l e t s beneath t h e sea s u r f a c e . Ambient motions of t h e wind waves and smaller scale t u r b u l e n c e then mix t h e o i l f u r t h e r downward. After t h e i n i t i a l breaking, t h e degree of downward o i l mixing i s dependent upon t h e k i n e t i c energy and Reynolds stresses of t h e wind wave motions. Our wave and turbulence o b s e r v a t i o n program (WAVTOP) has developed instrumentation and techniques t o g a t h e r f i e l d d a t a of t h e wind generated motions i n t h e upper 5-10 m. The sensor package (BLT) i n c l u d e s s m a l l f a s t response i m p e l l o r s , and a capacitance wave s t a f f system. The impellor systems, which w e r e c a l i b r a t e d i n s h i p wave-tow t a n k , are configured i n a r r a y s t o record v e l o c i t i e s a t s e v e r a l depths simultaneously f o r e s t i m a t e s of k i n e t i c energy c o n t e n t , Reynolds stresses, and mean s h e a r of t h e motions. The system i s b a t t e r y powered and s e l f recording, u t i l i z i n g a microprocessor and d i g i tizer. Preliminary measurements w e r e made i n Narragansett Bay w i t h t h e BLT mounted on a f r e e d r i f t i n g AESOP S t a b l e Spar Buoy and from an extended boom on Gould I s l a n d . Measurements w e r e made t o r e l a t e t h e l o c a l wind with t h e growth of wind waves and t h e i r t u r b u l e n t k i n e t i c energy. Preliminary r e s u l t s p o r t r a y t h e energy a t t e n u a t i n g with depth following c l a s s i c a l theory, where changes a r e highly c o r r e l a t e d w i t h t h e l o c a l wind speed. Also t h e r e appear phase l a g s of energy w i t h wind changes and evidence of a downward propagating rate of energy a t 0.5 cm/sec. F l u c t u a t i o n s i n v a r i a n c e appear t o occur i n c u r i o u s p u l s e s over 3-5 minute p e r i o d s and wave energy appears t o "advect" downwind i n packets. V e r t i c a l l y i n t e g r a t e d energy c o n t e n t appears t o correl a t e with t h e cube of t h e wind speed.
INTRODUCTION
A c a t a s t r o p h i c t a n k e r c o l l i s i o n o r grounding can quickly d e p o s i t l a r g e
volumes of o i l a t t h e s e a s u r f a c e producing an immense s l i c k .
The p r e d i c t i o n
of t h e f a t e of t h i s s l i c k poses t o some, an i n t r a c t a b l e problem.
A t best, the
s u r f a c e o i l i s advected by a mean c u r r e n t and d i s p e r s e d l a t e r a l l y i n t o t h e open sea.
L e s s favorable c o n d i t i o n s o f wind and c u r r e n t can contaminate t h e c o a s t .
162 A t w o r s t , o i l churned up by high sea state is mixed downward, and
i n shallow
r e g i o n s , causes o i l p e n e t r a t i o n i n t o t h e bottom sediments as w a s seen i n t h e AMOCO CADI2 Disaster (ENDECO, 1978).
Horizontal advection can f l u s h away
s u r f a c e and volume d i s t r i b u t e d o i l , b u t , contamination i n sediment may r e s i d e f o r months or y e a r s having d i s a s t r o u s e f f e c t s on sea l i f e near and a t t h e bottom.
I t i s t h u s important t h a t w e understand the mechanisms by which o i l
i s mixed v e r t i c a l l y ; knowledge which i s necessary t o d e v i s e systems and techniques t o canbat t h e spread and v e r t i c a l mixing of s u r f a c e o i l .
It i s
f u r t h e r important t o assess t h e energy and i n t e n s i t y of mixing a s r e l a t e d t o t h e sea state i n order t o judge a t what p o i n t o i l containment and recovery techniques becane i m p r a c t i c a l . The Wave and Turbulence Observation Program ( P r o j e c t WAVTOP) w a s i n i t i -
a t e d by t h e Naval Underwater Systems Center (NUSC) under sponsorship by the U.S. Coast Guard t o d e f i n e and a s s e s s the magnitudes of dynamic parameters which promote t h e v e r t i c a l mixing of s p i l l e d o i l on the open sea, and f u r t h e r , t o develop a system and technique t o d i r e c t l y measure wind wave and t u r b u l e n t motions i n t h e upper boundary l a y e r . The P r o j e c t WAVTOP is designed t o compliment s e v e r a l C o a s t Guard s t u d i e s of o i l s p i l l dynamics which include:
Wave t a n k modeling of breaking waves i n
an o i l s l i c k (Lin e t a l l 1978); T h e o r e t i c a l s t u d i e s of the r e l a t i o n of sea
s t a t e t o t h e s u r v i v a l of o i l spills ( F a j , 1977); and S t u d i e s of o i l mixing p r o p e r t i e s and breaking waves i n a tank (Milgram e t a l , 1978).
It i s t h e
necessary welding of theory, l a b o r a t o r y experiments, and a c t u a l f i e l d measure-
ments from which w e can forge workable p r e d i c t o r models of t u r b u l e n t mixing of o i l s l i c k s .
During t h e WAVTOP s t u d i e s , w e have developed a system t o
measure dynamic p r o p e r t i e s wind generated s u r f a c e motions, and made s e v e r a l preliminary f i e l d observations.
This paper serves as a s t a t u s report
Of
the
WAVTOP e f f o r t .
PARAMETERS ASSOCIATED WITH O I L M I X I N G I N THE UPPER LAYER:
WHAT TO MEASURE?
The calm ocean e x h i b i t s a minimal e f f e c t upon a deposited o i l s l i c k ; a
mean c u r r e n t may advect the o i l , spreading it l a t e r a l l y .
The o n l y waves
p r e s e n t p e r se, may b e swells from a d i s t a n t stonn which produce e s s e n t i a l l y i r r o t a t i o n a l motions, t r a n s m i t t i n g no stress o r o i l v e r t i c a l l y . The o n s e t of local wind, s t r e s s i n g t h e sea s u r f a c e , produces a v a r i e t y of s u r f a c e motions.
The i n i t i a l response, is the g e n e r a t i o n of c a p i l l a r y
waves which i n t u r n appear t o f e e d momentum i n t o the sea s u r f a c e t o form prog r e s s i v e l y larger wind waves which are accompanied by a mean v e r t i c a l shear of h o r i z o n t a l wind-generated c u r r e n t .
The waves increase i n s i z e w i t h wind
163 speed and form white caps when 6-7 m/sec i s exceeded.
I t i s these c o n d i t i o n s
i n which w e observe v e r t i c a l mixing of an o i l s l i c k .
What s p e c i f i c parameters or mechanisms govern v e r t i c a l t r a n s f e r of o i l through the sea s u r f a c e and i n t o t h e s u r f a c e l a y e r ?
Recent s t u d i e s shed l i g h t
on this problem: Raj (1977), i n summarizing o i l mixing parameters (as a f u n c t i o n of
1.
wind and sea s t a t e s ) , s p e c i f i e d t h e depth v a r i a t i o n s of the mean square v e l o c i t y f l u c t u a t i o n s , the i n t e g r a l scale of t u r b u l e n c e , and the energy
.
spectrum
Milgram e t a1 (1978) determined f r a n o b s e r v a t i o n s of breaking waves
2.
i n a t a n k , t h a t most s i g n i f i c a n t i n f l u e n c e on the d i s p e r s i o n of o i l i n t o submerged d r o p l e t s i s that of breaking waves.
Once t h e o i l film i s t r a n s -
p o r t e d by t h e high energy turbulence it is d i s p e r s e d as d r o p l e t s .
The motion
of t h e s e d r o p l e t s i s then determined by t h e balance of t h e buoyancy of t h e o i l d r o p l e t s (upward) and the Reynolds stresses tending t o mix t h e o i l (downward).
The s c a l e s of motions which mix o i l dawnward may be much l a r g e r than
t h e breaking t u r b u l e n c e .
I n f a c t , i t may b e a s s o c i a t e d w i t h the scales of
the s u r f a c e wave motions. 3.
Lin e t a1 (1978) observed from wind and wave flume measurements,
that wave breaking i n i t i a l l y i n t r o d u c e s o i l v e r t i c a l l y as a "strong v e r t i c a l b u r s t motion l i k e a plume plunging downwards below the water s u r f a c e " w i t h p e n e t r a t i o n about one wave h e i g h t .
Spectra of the motions show a dominant
peak a t the wave frequency with a "Kohogorov" (-5/3)fall-off of f r e q u e n c i e s . breaking
f o r two decades
O i l d i s t r i b u t i o n s (concentrations) r e s u l t i n g from the wave
were exponentially d i s t r i b u t e d with depth.
Fran the above r e s u l t s , it appears t h a t the v e r t i c a l t r a n s f e r of o i l through t h e sea s u r f a c e i s a two-step process.
F i r s t , breaking white caps
supply l a r g e "point" concentrations of high wave number t u r b u l e n t stress which can o v e r c m e t h e s u r f a c e t e n s i o n by breaking the o i l i n t o globules and spewing them downward (Fig. 1 A ) .
Second, a f t e r the oil g l o b u l e s are s c a t t e r e d
beneath the sea s u r f a c e they w i l l be mixed downward by t h o s e i n t e r n a l motions which are r o t a t i o n a l and t r a n s f e r stress.
The motions e x i s t i n g i n the upper
l a y e r are b a s i c a l l y o r b i t a l motions of the wind waves themselves, having
scales f r a n a few c e n t i m e t e r s t o t h e o r b i t a l diameters of the largest Wind waves p r e s e n t .
The vertical e x t e n t of this mixing is d i r e c t l y r e l a t e d t o
t h e k i n e t i c energy of t h e l o c a l o r b i t a l motions of the waves; found t o decrease almost exponentially w i t h d e p t h (e.9. Shonting, 1967).
Thus, the
n e t effects of both white capping and the i n t e n s i t y of the subsurface wave motions w i l l create a l o c a l steady state o i l g r a d i e n t d i s t r i b u t i o n (Fig. 1B). O i l mixing i s a s s o c i a t e d w i t h motions which t r a n s f e r stress and k i n e t i c
164
A
B
WlkD STRESS
WHITE CAP
INSTANTANEOUS 01L DROPLET DISTRIBUTION #
#
I
Fig. 1 A. O i l s l i c k broken i n t o globules a s breaking wave o v e r c m e s s l i c k s u r f a c e t e n s i o n . B. Subsurface mixing d i s t r i b u t e s o i l downward and produces an o i l c o n c e n t r a t i o n g r a d i e n t . energies.
F i g . 2 i s a c o n t r i v e d spectrum suggesting the t y p e s of wind gener-
a t e d motions a s s o c i a t e d w i t h stress and o i l mixinq.
Calm conditions s p e c t r m
(heavy l i n e ) h a s l i t t l e energy i n t h e scales or frequencies of our "mixing motions" ( o r b i t a l scales and s m a l l e r ) .
The "windy" spectrum, up t o t h e
breaking wave peak, d e l i n e a t e s t h e energy bands producing s t r e s s and mixing
165 (beyond, t h e s c a l e s a r e so small and i s o t r o p i c t h a t l i t t l e n e t stress and o i l t r a n s f e r should o c c u r ) .
Fran F i g . 2 , i t appears t h a t our measurements
should be of motions whose s c a l e s f a l l i n t h e shaded regions.
zw
SPECTRUM
STRESS OR MOMENTUM TRANSFER REGIONS
F i g . 2. Difference i n auto-spectra w i t h and without wind wave a s s o c i a t e d motions.
Very few measurements have been made i n the open sea f o r v e r i f i c a t i o n of
the e f f e c t s of breaking waves and k i n e t i c energy i n t e n s i t y upon o i l s l i c k mixing.
I d e a l l y w e should make observations f i r s t i n uncontaminated open
s e a c o n d i t i o n s t o o b t a i n b a s e l i n e d a t a and t h e n conduct similar observations i n s i t u i n an
o i l s p i l l environment.
This l a t e r study may be over ambitious
s i n c e o i l s p i l l s are n o t o f t e n provided s p e c i f i c a l l y f o r such s t u d i e s . A t the time of t h i s w r i t i n g , t h e Northern Cherbourg Peninsula of France
i s s t i l l recovering f r a n a m o s t d i s a s t e r o u s o i l s p i l l r e s u l t i n g from the wreck of t h e 200,000 t o n o i l t a n k e r , AMOCO CADIZ.
After t h e s p i l l , p e r s i s -
t a n t high winds, and choppy s e a s served t o mix and spread t h e l i g h t d i e s e l o i l over a wide s t r e t c h of t h e c o a s t a l zone.
166 Measurements were made by workers a t ENDECO (1978) of the v e r t i c a l concentration of o i l a t s t a t i o n s along the coast and i n a brackish estuary. Sections made from 7 km offshore t o the s h o a l beaches, indicated high concent r a t i o n s of o i l dispersed 3-4 m downward offshore, but, near the beaches where waves were breaking, t h e dl w a s t o t a l l y d i s t r i b u t e d throughout t h e water column from 5-12 m.
'Ihese r e s u l t s indicated t h a t the o i l d i s t r i b u t i o n
with the water w a s very dependent upon the presence of breaking and high energy wind waves; a c l e a r v e r i f i c a t i o n of the laboratory r e s u l t s s i t e d above.
ON THE ENERGETICS OF THE WAVE-INDUCED
MOTIONS
m e objective of our study is t o determine t h e energetics associated with t h e wind wave motions and r e l a t e i t t o the character of the wind i t s e l f . It i s therefore useful t o construct a framework t o r e l a t e the variables
measured t o physical concepts of momentum and energy.
W e formulate expres-
s i o n s f o r the time r a t e of change of energy and momentum related t o wind waves i n terms of measurable q u a n t i t i e s within t h e upper layer.
The analysis follows
t h a t of S t a r r (1968) who has u t i l i z e d t h e Reynolds formulation of fluxes from covariance relationships of fluctuating q u a n t i t i e s t o understand a wide variety of transport phenomena
i n the atmosphere, oceans and even s p i r a l galaxies.
Consider a surface layer of the ocean (Fig. 3) within which we wish t o measure motions associated with wind generated surface waves.
The physical
Fig. 3. A two-dimensiOMl ocean f r a n which w e estimate the time v a r i a t i o n of mean and eddy k i n e t i c energy. The layer above z t is where the wind etress is d i r e c t l y applied. Tlelocity measurements are ma8e fran z = zt to z = -D, where t h e motions beccme negligible.
167 measurement of these motions, f o r p r a c t i c a l reasons, must be made a t o r below
T which i s j u s t beneath t h e wave trough l e v e l (i.e. a region
the l e v e l 2
occupied by water a t a l l t i m e s ) .
The x axis, defining the 2
T
l e v e l , points
p o s i t i v e i n the d i r e c t i o n of the progressive wind waves and z points p o s i t i v e upward.
For our formulations we make the following assumptions:
1. We consider motions only i n t h e x and z d i r e c t i o n s , neglecting cross wave v e l o c i t i e s i.e., p a r a l l e l t o wave c r e s t s . 2.
Motions occurring a t speeds producing p a r t i c l e displacements associ-
ated with gravity waves which are v i r t u a l l y unaffected by Coriolis forces. 3.
The mean sea surface i s e s s e n t i a l l y horizontal, and the l o c a l
density i s assumed constant.
Therefore, no mean horizontal pressure gradients
a r e present i n the water other than those associated w i t h a slowly varying baratropic t i d e . 4.
"he s t a t i s t i c a l properties of the velocity and pressure f l u c t u a t i o n
a r e homogeneous i n the horizontal over the l o c a l a r e a a s is the mean horizontal c u r r e n t .
These p r o p e r t i e s m y , however, undergo changes much slower
than t h e periods of the wave motions ( i n f a c t , these changes such as those associated w i t h wind v a r i a t i o n s a r e of p r i n c i p l e i n t e r e s t ) .
5.
Motions a r e associated only with wind generated waves, propagating
i n the +x d i r e c t i o n and the mean t i d a l flow: no standing waves being present. The momentum equations f o r t h e u and w motions i n the x and z d i r e c t i o n s respectively, may be written a s
where p i s density, p i s pressure, t i s time, Fx and FZ a r e f r i c t i o n a l retarding f o r c e s per u n i t volume i n the x and z d i r e c t i o n s and g i s the acceleration o f gravity assumed constant.
with t h e a i d of t h e two dimensional continuity
equation
and by multiplication of equations (1) and ( 2 ) by t h e respective v e l o c i t i e s , expansion of the t o t a l d e r i v a t i v e s , and invoking assumptions 1-5 we f o m t h e k i n e t i c enerqv r e l a t i o n
-gpruFx-wFz
Now we consider equation (4) as being averaged i n time over a region
(4)
168 of Ocean where a g a i n w e may invoke s t a t i s t i c a l homogeneity assumption ( 4 above).
we t h e n may d e f i n e t h e p r o p e r t i e s of the measured v a r i a b l e s i n the fonn u ( t ) = ii+u', w ( t ) =
-= k c
w + w",
p(t) =
is + p '
(5)
where (
)
with (
1 ' = 0, n o t i n g t h a t the p e r i o d of averaging
(
)
dt T i s much g r e a t e r than
t h e p e r i o d s of t h e s u r f a c e wind waves and i n s p i t e of t h e slowly varying t i d e l e v e l , w e may s a f e l y assume ij. =
0.
W e have t h u s , a f t e r t h e a p p r o p r i a t e averaging
This r e l a t i o n i s a balance of mean and wave (eddy) k i n e t i c energy.
To the
mathematician, t h e t i m e d e r i v a t i v e on t h e l e f t hand t e r m may be troublesome. W e n o t e however, t h a t t h e p a r t i a l t i m e d e r i v a t i v e is meant t o a s s e s s the
change over time i n t e r v a l s l a r g e compared t o the wind wave p e r i o d s .
the wave p e r i o d s may range from 0.5-10
Thus,
s e c while w e which t o consider t h e
k i n e t i c energy changes i n response t o wind stress changes which occur frcm t e n s of minutes t o s e v e r a l hours. To o b t a i n the energy balance of h o r i z o n t a l mean motion, w e m u l t i p l y
equation (1) by u and form
-
6
2
6
--
F ~ ( P ~ =) -g-(PU'W'
u)
+
-6U - uFx -
pu'w'
(7)
6z
By s u b t r a c t i n g equation ( 7 ) from equation (6) w e o b t a i n
which i s the balance of wave (eddy) k i n e t i c energy. W e n o t e that a l l terms on t h e r i g h t , i n b o t h equations
(7) and (8)
(except the viscous d i s s i p a t i o n s ) involve v e r t i c a l s p a t i a l d e r i v a t i v e s and covariances, b o t h i n d i c a t i v e of t r a n s p o r t p r o p e r t i e s of a f l u i d . t i o n s i n d i c a t e t h e change occurring a t a p o i n t .
The equa-
I t i s of i n t e r e s t a l s o t o
i n t e g r a t e t h e r e l a t i o n v e r t i c a l l y s i n c e o u r concern i s t h e e n e r g e t i c s of a f i n i t e water column and how s u r f a c e wind stress imparts momentum and energy through t h e l e v e l %p.
W e thus i n t e g r a t e equations ( 7 ) and ( 8 ) from the
trough l e v e l ZT down t o a depth
D.
All i n t e g r a n d s , which a r e p a r t i a l
d e r i v a t i v e s with r e s p e c t t o z, need b e evaluated only a t z = ZT s i n c e a t
169 Z = .D, a l l f l u c t u a t i o n s vanish (Shonting, 1 9 6 7 ) .
The i n t e g r a t e d forms of equations (7) and (8) a r e
and
The q u a n t i t i e s DM and De denote the f r i c t i o n a l d i s s i p a t i o n of t h e mean
flow and of t h e eddy k i n e t i c e n e r g i e s r e s p e c t i v e l y f o r t h e u n i t column between zT and D . The f i r s t i n t e g r a l on t h e r h s , i n equations (9) and (10) r e p r e s e n t s t h e transformation term between t h e two types of k i n e t i c energy o r , i n o t h e r words, energy t r a n s f e r r e d from the mean motion through the t u r b u l e n t s h e a r stresses o r production of t u r b u l e n t energy (Hinze, 1959) over t h e water column of u n i t c r o s s - s e c t i o n and ZT
D meters high.
The second term i n
equation (9) i s t h e boundary t r a n s p o r t of mean flow k i n e t i c energy between t h e l a y e r of d i r e c t f o r c i n g and t h e region below.
This i s what Hinze (1959)
r e f e r s t o a s convective d i f f u s i o n by turbulence of the k i n e t i c energy.
This
e f f e c t i s due t o t h e Reynolds stress a c t i n g i n t h e d i r e c t i o n of the mean flow. The second t e n of equation (10) i s a boundary t r a n s p o r t of eddy k i n e t i c energy a c r o s s t h e s u r f a c e ZT caused by t h e n e t advection a s s o c i a t e d w i t h t h e covariance between w' and t h e eddy k i n e t i c energy components t h a t this i s sum of two t r i p l e product c o r r e l a t i o n s ) .
(note
Finally, the third
term of equation (10) i s t h e covariance of p r e s s u r e and v e r t i c a l v e l o c i t y . I t provides a second boundary f l u x of eddy k i n e t i c energy across t h e surface
which may b e w r i t t e n a s
-
ut2w'
-
and w t 2 w '
The sum of t h e v e r t i c a l terms i s t h e t r a n s p o r t of eddy k i n e t i c energy f o r t h e u ' and w' v e l o c i t y components. From a p r a c t i c a l p o i n t of view, w e should n o t e t h a t wave observations have been made i n Narragansett Bay where the t i d a l c u r r e n t s ( i . e . values of u ) , depending upon t h e phase and l o c a t i o n , vary i n speed from 0-100 cm/sec. C l e a r l y , f o r a vanishing mean c u r r e n t , equation (9) becomes i d e n t i c a l l y zero Equation (10) on t h e o t h e r hand, s i m p l i f i e s t o
Now t h e r e i s no mean eddy energy conversion term.
However, w e still have
170 the advection terms of k i n e t i c energy and pressure.
This s u g g e s t s t h e
importance of a s s e s s i n g the r a t i o
f o r a given set of observations. The above d e r i v a t i o n s allow t h e observed parameters t o be evaluated i n S p e c i f i c a l l y , the v a r i a n c e s and
t h e c o n t e x t of balanced energy equations.
-
Reynolds stresses can be used t o c a l c u l a t e the o r d e r of magnitude of terms i n equations ( 9') and (10).
The Reynolds stress component
T~
= -pu'W'
d e f i n e s a f l u x of h o r i z o n t a l momentum p u ' t r a n s p o r t e d downward by the covariance between
U I
mixing of o i l .
and w ' .
These same concepts can b e applied t o the v e r t i c a l
Then, considering again Fig. 1, we could d e f i n e a v e r t i c a l
o i l t r a n s p o r t term analagous t o the Reynolds stress as an
O i l
Concentration
f l u c t u a t i o n ( o i l ) ' c o r r e l a t i n g with w ' i . e . Fo = ( o i l ) ' w W
(13)
Thus, i n an actual o i l s p i l l , one could measure a t a p o i n t i n the ocean the
time v a r i a t i o n of o i l c o n c e n t r a t i o n ( p o s s i b l y by use of a fluorometer system) and w' simultaneously t o determine t h e f l u x Fo. I t i s clear that our o b s e r v a t i o n a l prcblem i s f i r s t t o measure and
g a t h e r simple statistics of wind wave motions from the l a r g e s t wave o r b i t a l diameters down t o scales o f breaking wave turbulence.
Analysis should
include;
1.
The k i n e t i c energy ( v a r i a n c e s ) of t h e motions.
2.
The dominant energy containing scales from the s p e c t r a l d e n s i t y
versus frequency and t h e spatial c o r r e l a t i o n s . 3.
Covariances between h o r i z o n t a l and v e r t i c a l v e l o c i t y components.
These s t a t i s t i c a l p r o p e r t i e s should b e assessed i n r e l a t i o n t o depth, the l o c a l wind c o n d i t i o n s , sea s t a t e (observed from f r e e s u r f a c e e l e v a t i o n s t a t i s t i c s ) and whitecap
occurrance, s i n c e t h e s e d a t a are shown t o b e
r e l e v a n t t o o i l mixing.
INSTRUMENTATION Our t h e o r e t i c a l d i s c u s s i o n has defined a p a r t i c u l a r measurement problem such that i n s t r u m e n t a t i o n that w e choose m u s t c o n t a i n c e r t a i n s p e c i f i c attributes.
Our sensing systems must r e g i s t e r both t h e mean and f l u c t u a t i n g
v e l o c i t y components i n t h e w a t e r column.
The instruments should have a
f a s t response t o measure t h e smallest scale v e l o c i t y f l u c t u a t i o n s and long
171 enough recording capacity t o evaluate v a r i a b i l i t y and mean values.
Further,
the systems m u s t be arranged geometrically f o r simultaneous measurements a t d i f f e r e n t depths. Based upon the above considerations, an instrument package f o r the WAVTOP observations was developed consisting of t h r e e basic cmponents; the
sensors, e l e c t r o n i c s package and cabling connectors and pressure cases.
The
sensor system (Fig. 4A) is a ducted impellor device designed by Smith (1978) a t the University of Washington.
'Ihe p l a s t i c impellor is mounted with jewel
N
.
t20
T 4cm
:
Ic
a
: I, z
.
8
I?.'
J
4,.
* 0 )
0
. . .. :
20
40
60
80
100
TOW SPEED C N / S E C - Y I T T O W TANK CALIBRATION
I P V o L T SEALED P B A C I D C E L L S
Fig. 4A. Ducted impeller meter. B. Calibration curve made i n MIT towing tank. C. Off angle response c a l i b r a t i o n . D o t s a r e a c t u a l data; broken l i n e c i r c l e i s the cosine r e l a t i o n . D. Solid s t a t e cmponents i n the e l e c t r o n i c s of the Central processing u n i t (CPU) controls system, "Boundary Layer Thing" (BLT) input from meters i s sensed f o r 2 s i g n , d i g i t i z e d and averaged and a t i n t e r v a l s i s stored onto d i g i t a l c a s s e t t s . The read only memory (ROM) i s pre-programmed f o r BLT t o record data a t prescribed i n t e r v a l s .
.
172 bearings.
lWo
small magnets a r e mounted on a diameter of t h e impellor.
As
t h e impellor s p i n s i n response t o t h e f l u i d flow, t h e magnetic f i e l d of the magnets i n t e r a c t w i t h a " H a l l e f f e c t " c r y s t a l mounted i n s i d e the t i p of the s t a i n l e s s s t e e l support rod.
With passage of t h e blade magnets, a charge
s e p a r a t i o n occurs i n t h e Hall e f f e c t device forming a p u l s e which i s amplif i e d , shaped i n t o a square wave by a Schmidt t r i g g e r , and coded by a sequence of short/long,
or long/short p u l s e s t o i n d i c a t e t h e sense of r o t a t i o n .
The
H a l l e f f e c t pickup o f f e r s almost no magnetic f o r c e f i e l d which could slow
t h e i m p e l l o r s through i n t e r a c t i o n w i t h the magnets.
Thus, the combination
of a n e u t r a l l y buoyant impellor and t h e H a l l e f f e c t pickup provides a v e l o c i t y t h r e s h o l d response of 3-4 mm/sec.
The angular v e l o c i t y of t h e i m p e l l o r s i s
l i n e a r enough w i t h flow speed (Fig.4B)
s o t h a t the i d e n t i c a l c a l i b r a t i o n s
curve may be used f o r a l l seven s e n s o r s w i t h a maximum e r r o r of 2-3%.
The
response of t h e impellors v a r i e d approximately as the cosine of t h e angle subtended by t h e m e t e r a x i s of r o t a t i o n and t h e flow v e c t o r (Fig. 4C).
A
summary o f t h e flow meter c h a r a c t e r i s t i c s i s given i n T a b l e 1.
TABLE 1
C h a r a c e t e r i s t i c of Ducted Flow Meters
D i a m e t e r Impellor
4 m
Responce Distance
3 cm
Frequency Response
5-10 H,
Threshold Speed
3-5 mm/sec
Linearity
High (One curve can b e used f o r a l l systems)
Cosine Response
Good 10-15% Max Error
Signal Gutput
Nominal 15 V- coded square wave p u l s e s whose sign changes i n d i c a t e a sense of r o t a t i o n .
Max Depth O f U s e
Tested t o 20 atm (200 m)
S o l i d s t a t e e l e c t r o n i c s f o r s i g n a l p r e p a r a t i o n i s incased i n t h e s t a i n -
less s t e e l c y l i n d r i c a l support and t e s t e d f o r 20 atm o r 200 m water depth. The u n i t i s powered by
+
15 v dc v i a a water t i g h t connector.
The s i g n a l s
from each of the sensor u n i t s are led to the BLT d a t a p r o c e s s o r which c o n t a i n s
a complex of s o l i d s t a t e micro c h i p s f o r complete data processing and s t o r a g e . Fig. 4D shows the types of I.C. c h i p s used:
The c u r r e n t m e t e r p u b e s are
converted i n t o signed d i g i t i z e d v a l u e s of v e l o c i t y and a r e averaged every 0.2 sec.
Each of t h e s i x channels of time series d a t a is recorded i n three
Memcdyne d i g i t a l c a s s e t t e s .
The d a t a i s t r a n s f e r r e d t o 7 channel Mag t a p e
f o r processing. The lower l i m i t of t h e s c a l e s of motions measured i s l i m i t e d by t h e
173 dimensions of the i m p e l l o r s e n s o r .
The 4 c m diameter impellor l i m i t s our
d e t e c t a b l e scale s i z e of a n eddy motion t o about 10 cm.
*e
response t i m e
of the Smithometer impellor sensor is about HZ (see Table 1). e q u i v a l e n t t o our sampling frequency of 0.2 s e c .
This is
This g i v e s a Nyquist
frequency of 2.5 Hz; t h e upper l i m i t frequency from which we could expect The nominal continuous recording p e r i o d i s 3-4
any s p e c t r a l information.
hours of continuous d a t a ; the p e r i o d being roughly t h e l i f e of our b a t t e r y pack.
THE VERTICAL D I S T R I B U T I O N OF K I N E T I C ENERGY AND THE ENERGY INTEGRAL
W e wish t o measure t h e v e r t i c a l d i s t r i b u t i o n of k i n e t i c energy from the
sea s u r f a c e down t o a depth where t h e wave induced motions vanish.
If w e
assume t h a t t h e motions u' and w ' d e f i n e d by equations 1 and 2 a r e s t r i c t l y wave o r b i t a l motions of deep water type (wavelength
less t h a n 4 t h e depth)
and can be represented as f l u c t u a t i o n s a t a f i x e d h o r i z o n t a l p o s i t i o n by
u' = a w e w2z/g c o s ( w t ) and w ' = a w e w2z/g s i n (-wt) where a = amplitude w = frequency
g = gravity
( u * 2 + ~ 7 2p )
t h e n , by e l i m i n a t i n g the trigonometric terms and multiplying by (z) =
=
(a w e
w e obtain
w2z/g)+
(14)
which becomes t h e k i n e t i c energy a t depth z a s s o c i a t e d w i t h a two dimensional p r o g r e s s i v e wave i n the x-z p l a n e . Now i f we i n t e g r a t e i n t h e form
we o b t a i n the t o t a l k i n e t i c energy of the wave motions i n the water c o l w between t h e free s u r f a c e
n(t)
and t h e depth
.D where t h e u' and w ' motions
of their s u r f a c e v a l u e ) . a r e n e g l i g i b l e (say 1%
A problem occurs i n the
i n t e r p r e t a t i o n of t h e energy i n t e g r a l a t the s u r f a c e .
It i s clear, since
w e d e s i r e t h e average k i n e t i c energy, we s h a l l i n t e g r a t e from the upper l i m i t Tl (t) = z = 0 by d e f i n i t i o n .
-
One a i m of our f i e l d measurements is t o e v a l u a t e the energy d e n s i t y
-
term ( p w I 2 ) and v e r t i c a l i n t e g r a l by use of a c t u a l values of u a 2 (t)and ~ ' ~ ( 2 )These .
v a r i a n c e s , however, a r e obtained only a t d i s c r e t e depths of
instrument placement; u s u a l l y , f o r the WAVTOP measurements, the uppermost
174 instrument w a s placed a t 10-20 cm below the estimated mean trough level.
The
uppermost r e g i o n must be measured a s a c c u r a t e l y as p o s s i b l e since the energy
is an exponential f u n c t i o n i n c r e a s i n g t o t h e s u r f a c e . OBSERVATIONS
Preliminary s e r i a l records of v e l o c i t y components w e r e taken a t s e v e r a l depths simultaneously allow comparison of the v e r t i c a l d i s t r i b u t i o n of k i n e t i c energy of t h e f l u c t u a t i o n s with the c h a r a c t e r of the local Wind. of wind wave observations are s t u d i e d .
Two records
Each w a s made over a p e r i o d when
t h e r e occurred pronounced wind speed changes. S e r i e s 002 (14/Mar/78):
S i x Smithometers w e r e supported from the AESOP
S t a b l e Spar Buoy (Shonting and B a r r e t t , 1971) a t depths of 1 0 , 110, 210, 310, 410, and 610 cm beneath t h e mean trough l e v e l
Zt
(Fig. 5 ) .
The time series
Fig. 5 . Configurations of t h e i m p e l l o r meters on AESOP t o measure w ( t ) a t s i x depths simultaneously ( l e f t ) and t o measure u ( t ) and w ( t ) a t three depths and p ( t ) and w ( t ) a t 3 ( r i g h t ) .
175 of the v e r t i c a l v e l o c i t y w ( t ) were recorded a t 0.2 sec i n t e r v a l s and running means and v a r i a n c e s were estimated f o r t i m e i n t e r v a l s of 128 sec.
frequency waves
-
Since t h e
s e c , this s u p p l i e s over 30 c y c l e s of t h e lowest
wave p e r i o d s ranged from 0.5-4
s u f f i c i e n t f o r a reasonable estimate.
Assuming w e are measuring e s s e n t i a l l y two dimensional waves i . e . wind waves whose o r b i t s are i n t h e x-z plane, the t o t a l k i n e t i c energy of these motions i s
The running v a r i a n c e s f o r 002 are shown i n Fig. 6 A t o g e t h e r w i t h wind speed v a l u e s .
The v a r i a n c e curves i n d i c a t e the following:
1. The g e n e r a l t r e n d of energy d e n s i t y follows t h e wind speed and is
seen t o vary by over an o r d e r of magnitude a t a l l depths f o r the wind speed range of 2-9 m/sec.
2.
The energy remains q u i t e exponentially d i s t r i b u t e d w i t h depth a s
i n d i c a t e d by t h e s i m i l a r v e r t i c a l spacing w i t h time. 3.
A
pronounced minimum appears i n a l l records b u t appears time lagged
p r o g r e s s i v e l y w i t h depth whereby t h e 610 cm minimum appears about 20 min a f t e r the 10 c m value. 4.
Pronounced f l u c t u a t i o n s i n energy occur a t a l l depths.
Sometimes
t h e s e p u l s e s appear i n phase w i t h a d j a c e n t records above and bllow, b u t o t h e r t i m e s appear randomly.
Tha amplitudes w i t h r e s p e c t t o the slow v a r i -
a t i o n (i.e. t h e signal-to-noise
r a t i o ) appear independent of depth.
There
was no v i s i b l e i n d i c a t i o n t h a t t h e AESOP buoy was f l u c t u a t i n g v e r t i c a l l y (it has a n a t u r a l heave period of 26 seconds).
I t i s tempting t o s p e c u l a t e that
these f l u c t u a t i o n s are wave t r a i n packets which are generated by i r r e g u l a r wind stress p u l s e s upwind i n t h e bay. D i f f e r e n t wind c o n d i t i o n s (and hence, s e a s t a t e s ) w i l l cause d i f f e r e n t energy d i s t r i b u t i o n t o e x i s t i n t h e water column.
An
i n d i c a t i o n of the
d i f f e r e n t d i s t r i b u t i o n may be seen by p l o t t i n g variance v s depth on a loga-
rithmic scale (Fig. 6B).
For t w o d i f f e r e n t wind c o n d i t i o n s , d i f f e r e n t
k i n e t i c energy d i s t r i b u t i o n s were observed.
With a 2 m/sec wind, the decrease
of k i n e t i c energy w i t h depth follows a n exponential form. wind speed, 8 m / s e c , column.
Also i n d i c a t e d
f a c t t h a t the energy
For the larger
a l a r g e r amount of k i n e t i c energy e x i s t s i n the water
i s a l a r g e r amount of energy near t h e s u r f a c e .
- distributing
‘!?he
process i s n o t i n a s t e a d y s t a t e is shown
by t h e non-exponential shape of the curve n e a r e r the s u r f a c e .
The deeper
values tend t o converge; perhaps an i n d i c a t i o n t h a t s u r f a c e energy has n o t reached t o the deepest s e n s o r s .
F i n a l l y , e x t r a p o l a t i o n above t h e shallowest
-
176
WIND SPEED
o a8
A A
u)
A
A
\ =
E
4
B5
A A
A
.O
RUNNING VARIANCE
50
0
T I M E min
100
F i g . 6A. Running variance of t h e v e r t i c a l v e l o c i t y made from consecutive 128 s e c r e c o r d s .
depth measurement may prove u s e f u l i n i n v e s t i g a t i n g t h e energy i n p u t mechan-
i s m a t t h e surface. The t o t a l estimate of wind w a v e k i n e t i c energy i s obtained by numerical i n t e g r a t i o n of t h e r e l a t i o n
layer-by-layer
over t h e depths of o b s e r v a t i o n .
This is shown i n F i g . 6C ( s o l i d
177
7
V E L O C I T Y VARIANCE
500.
tun*
A AT T= = 66 88 m m ii n n NARRAGANSETT B A Y
. ''
I
1
..h ' \
'. 'A'
..
u
..
'
'4
\ \
\
0 a#
. 0.
w '
E V
\ \ \
'. \
WIND
\
\
'
\
M
8 m/sec
WIND \
.
2m/sec"
\\ \
\
\
'O\, \
'\
<
\
\ \\
\
\
\ \
\
\
\\
I
. \\
'A \O
"
05-4 10
\
110
210
310 410 D E P T H cm
610
F i g - 6B- fie vertical d i s t r i b u t i o n of v a r i a n c e of the vertical v e l o c i t y : F i r s t record w i t h wind at 2 m/sec showing an exponential decrease with depth1 second r e c o r d (68 min l a t e r ) w i t h wind a t 8 m/sec shows i n c r e a s e of energy fran 1 0 310 cm b u t a d e l a y i n energy propagation deeper.
-
curve) which r e p r e s e n t s f o u r averages each over 1280 sec ( 2 1 d n ) .
Clearly,
t h i s energy c o n t e n t i s i n c r e a s i n g throughout t h e record. Denman (1973) s p e c i f i e s t h e t u r b u l e n t energy i n p u t by wind stress a s
PA i s a i r d e n s i t y , CA i s a drag c o e f f i c i e n t and U10 i s t h e wind speed observed a t 1 0 m h e i g h t above s e a l e v e l . power i n p u t ) i s given by
The r a t e of working by t h e wind s t r e s s ( o r
178 002 24 MAR 18
-
500
-400
fz O
u
0
m
-300
-
c)
c 200
0
m 0
K I N E T I C ENERGY
Fig. 6C. cubed.
P l o t of v e r t i c a l l y i n t e g r a t e d k i n e t i c energy w i t h t h e wind speed
The cube of t h e wind speed (dashed curve i n F i g . 6C) shows a p o s i t i v e c o r r e l a t i o n with t h e i n c r e a s e i n t h e i n t e g r a t e d energy c o n t e n t .
S e r i e s 006 (9/Aug/78) : This w a s made from t h e n o r t h end of Gould Is, Narr a g a n s e t t Bay where a 3-10 km f e t c h occurs t o the northwest.
The BLT system
w a s p o s i t i o n e d over 17 m water depth using a l a r g e stiffboom (Fig. 7 ) .
Five
S m i t h m e t e r s w e r e suspended from the boom support t o r e g i s t e r w ( t ) a t depths of 1 0 , 50, 100, 150, and 200 c m below ZT (Fig. 7 ) .
The wind was l i g h t and
v a r i a b l e w i t h calms occurring during and a t t h e end of the record (Fig. 8 A ) . The wind waves were small
-
l e n g t h s 2-3 m and h e i g h t s 10-15 c m c m e n s u r a t e
with t h e s m a l l wind stresses.
The running v a r i a n c e s ( c a l c u l a t e d f o r a
sequence of 1 2 8 sec i n t e r v a l s ) show energy p e r t u r b a t i o n s which exponentially decay w i t h depth (Fig. 8A lower c u r v e s ) . The energy d i s t r i b u t i o n a l s o d i s p l a y s s e n s i t i v i t y t o the f l u c t u a t i n g wind f i e l d .
The drop i n wind a t approximately 1 5 min, is r e f l e c t e d i n the
v a r i a n c e s (even down t o 200 cm) w i t h i n s e v e r a l minutes.
The wind i n c r e a s e of
wind peaking a t 3 m/sec i s followed, b u t a t a 15-20 min time l a g , by t h e
179
a ELECTRONICS
The experimental arrangement f o r wave energy observations a t Gould Island i n Narragansett Bay.
Fig. 7.
3% 0
2.
\
€
p'
I
\
0)
VI
/--A.
'A \
I
\
0-
\
*
/
\I' 'A*
001
.\ ..
I
\ I-
I
....-..
\
\
\
W I N D SPEED
9 AUQ 7 0
\
.
\
.A
F i g . 8 A . The running variances of the v e r t i c a l v e l o c i t y observed a t the i n.ndidicated depths p l o t t e d w i t h t h e changing wind speeds (upper c u r v e ) .
180 v a r i a n c e s a t 1 0 , 50, and 100 c m depths.
The p l o t s of the wind speed cubed
and t h e v e r t i c a l l y i n t e g r a t e d energy demonstrates t h e t i m e l a g (Fig. 8B); t h e k i n e t i c energy c e r t a i n l y f a l l s behind both t h e wind peak and t h e t a i l - o f f a t
the end o f the r e c o r d .
Fig. 8B. V e r t i c a l l y i n t e g r a t e d kinetic energy and the wind speed cubed. Note t h e apparent phase s h i f t between the curves whereby the wind stress work l e a d s t h e k i n e t i c energy of t h e wave motions by 15-20 minutes.
CONCLUSIONS AND PLANS FOR FURTHER STUDIES
From t h e p r e s e n t WAVTOP program, preliminary conclusions a r e made:
1.
Review s t u d i e s i n d i c a t e o i l s l i c k s a r e mixed downward by breaking
waves coupled with t h e subsurface wind wave and t u r b u l e n t motions; t h e degree of mixing being dependent upon t h e k i n e t i c energy of t h e f l u c t u a t i n g motions. 2.
Equations have been derived which d e f i n e t h e balance of t h e t i m e
r a t e of change, mean and f l u c t u a t i n g k i n e t i c energy, r e l a t i v e t o dynamic s t r e s s and p r e s s u r e terms.
These equations when i n t e g r a t e d v e r t i c a l l y , pro-
v i d e a p h y s i c a l framework from which observed values of wave motions and dynamic p r e s s u r e may be evaluated i n terms of energy and momentum f l u x e s from t h e wind.
181 3.
The Smithometer ducted impellor systems developed a t the University
of Washington, used i n conjunction w i t h t h e NUSC developed microprocessor d a t a l o g g e r , provide a t o o l t o make a v a r i e t y of observations of wave motion veloci t y , free s u r f a c e and dynamic p r e s s u r e . 4.
The b a s i c s e n s o r s a r e capable of r e g i s t e r i n g wave motions from
4 m / s e c t o 100 cm/sec and f l u c t u a t i o n s up t o a frequency of 5 H ,.
The sensors
can r e s o l v e o r b i t a l motion and t u r b u l e n t components having scales down t o a
f e w c e n t i m e t e r s ; and can r e s o l v e orthogonal v e l o c i t y components t o estimate Reynolds stresses i n t h e wave f i e l d . 5.
Preliminary o b s e r v a t i o n s of wave motions show the r e l a t i o n of wind
s t r e s s v a r i a t i o n t o t h e v e r t i c a l d i s t r i b u t i o n of wave k i n e t i c energy and t h e t o t a l energy i n t e g r a l .
F u r t h e r , t h e a c t u a l phase s h i f t of energy flow from
the s u r f a c e which i n d i c a t e s a downward speed of propagation of roughly
0.5
-
2 cm/sec.
The work of t h e s t r e s s (varying a s t h e wind speed cube) cor-
r e l a t e s w i t h t h e v e r t i c a l l y i n t e g r a t e d energy. These preliminary r e s u l t s of t h e study of the wind generated k i n e t i c energy c o n t e n t i n the upper l a y e r , o f f e r incouraging r e s u l t s .
F u r t h e r anal-
y s e s and more observations w i l l provide information on s p e c t r a l composition, and Reynolds stresses as a f u n c t i o n of t h e l o c a l wind f i e l d and the sea state. These measurements w i l l explore the r e l a t i o n s h i p of t h e wave o r b i t a l motions t o the momentum and energy t r a n s f e r of t h e wind downward through t h e water column.
These observations w i l l be similar t o those made by Shonting (1971)
and by C a v e l a r i e t a 1 (1977).
Added sensor systems w i l l include wave s t a f f
( f r e e s u r f a c e ) records and a dynamic p r e s s u r e s e n s o r . systems a step-by-step mence.
W i t h this a r r a y of
e v a l u a t i o n of t h e terms of energy equations w i l l com-
Once the v a l i d i t y of t h e energy and stress r e l a t i o n s i s demonstrated
these parameters can be used t o v e r i f y o i l mixing m o d e l s and theory. ACKNOWLEDGEMENTS The design and c o n s t r u c t i o n of t h e microprocessor d a t a logger was done by John Roklan and a s s i s t e d by W i l l i a m Ryan of t h e Naval Underwater Systems Center.
The f i e l d a s s i s t a n c e and d a t a a n a l y s i s of Tony P e t r i l l o , Dale L i c a t a ,
Robin Robertson and Capt. Angelo M a z a r r e l l i of t h e Department of O c e a n Fngineering of t h e University of %ode I s l a n d , i s a p p r e c i a t e d . p r e s i d e n t of Shielco Corp. of D a v i s v i l l e , R . I . fabrication.
Mike Kinane,
a s s i s t e d i n t h e designs and
S p e c i a l thanks is owed t o M s . E i l e e n Domingos, our s e c r e t a r y
f o r h e r typing e f f o r t s and helping u s t o organize our chaos.
Guidance f o r
the o v e r a l l program w a s recieved from R. G r i f f i t h s of t h e U.S. Coast Guard R & D Office Washington, who sponsored this study (MIPR NO.
2-70099-7-71825-A).
182 REFERENCES
C a v a l e r i , L., Ewing, J., and Smith, N . , 1977. Measurements o f the p r e s s u r e and v e l o c i t y f i e l d below s u r f a c e waves. NATO Syrqposium: Turbulent f l u x e s , through sea s u r f a c e . Wave dynamics and p r e d i c t i o n . I l e de Bendor, France 12-16/Sep/77. Denman, K.L., 1973. A time-dependent model of the upper motion. J. Phys. Oceanogr., 3/3 :173-184. ENDECO, 1978. Measurement of dynamics o f o i l - i n - w a t e r c o n c e n t r a t i o n s during t h e "AMCCO-CADIZ" o i l S p i l l . Data Report Environmental Devices Corporation, Marion, Massachusetts, 14 pp. Kinze, J.O., 1959. Turbulence an i n t r o d u c t i o n t o i t s mechanism and theory. McGraw-Hill Book Co., New York, 586 pp. Lin, J . T . , Gad-el-Ha, M . , and Ta-Liu, H . , 1978. A s t u d y t o conduct experiments concerning t u r b u l e n t d i s p e r s i o n of o i l s l i c k s . Flow Research Corpora t i o n For t h e Department of T r a n s p o r t a t i o n ( U . S . Coast Guard) N o . 65 (DOT-CG-61688-A) Donnelly, R.G., Van Houten, R.G., and Camperman, J . M . , 1978. Milgram, J . H . , Affects of o i l s l i c k p r o p e r t i e s on t h e d i s p e r s i o n of f l o a t i n g o i l i n t o t h e sea. Massachusetts I n s t i t u e of Technology For U . S . Coast Guard Report NO. CG-D-64-78, 374 pp F a j , P.K., 1977. T h e o r e t i c a l study t o determine the s e a s t a t e l i m i t f o r the s u r v i v a l of o i l s l i c k s i n t h e ocean. Arthur D. L i t t l e Corporation, F i n a l Report N o . CG-D-90-77, Task N o . 4714.21, For the Department of Transport a t i o n ( U . S . Coast Guard), 276 pp. Shonting, D., 1967, Measurement of p a r t i c l e motions i n ocean waves. Journal of Marine Research., 25/2 :162-181. Shonting, D., 1971. Observations of Reynolds stresses i n wind waves. P u r e and Applied Geophysics., 81/4:202. 1971. A stable spar-buoy platform f o r mounting Shonting, D. and B a r r e t t , A.H., i n s t r u m e n t a t i o n . Journal of Marine Research., 29/2:191-196. Smith, J . D . , 1978. Measurement of t u r b u l e n c e i n ocean boundary l a y e r s . Proceedings of a working conference on c u r r e n t measures. University of & l a ware, Newark, Delaware, pp 95-128. S t a r r , V.P., 1968. Physics of negative v i s c o s i t y phenomena. M c G r a w - H i l l Book 256 pp. Company, N.Y.,
.
.
.
183
A NUMERICAL MODEL OF LONGSHORE CURRENTS
M. SABATON, A. HAUGUEL E.D.F., Direction des Etudes et Recherches, Laboratoire National d'Hydraulique, Chatou (France)
ABSTRACT
We introduce here a numerical model which permits to compute the longshore currents for any shape of shoreline and any bathymetry. The integration over the depth and the average over a wave period of the momentum and continuity equations leads to a system of equations identic to those of SaintVenant in which supplementary stresses due to waves appear. The model first includes the wave refraction computation which permits to estimate the driving forces and then the computation of longshore currents themselves. This model has been calibrated in the case of a rectilinear shoreline for which the influence of the various parameters has been studied. At last, the model has been applied to a semi-circurlar bay.
INTRODUCTION The longshore currents are important as they can be as strong as the tidal currents and wind induced currents. Neverthelers, they are very different because from one hand, they are generally restricted to the surf area and because on the other hand, the velocity of the other currents is weak there. This let appear that the longshore currents action is essential for littoral drift and dispersion of outfalls near the coast. Consequently, they must be precisely defined. Thanks to some assumptions particularly concerning the wave breaking, it is possible to compute the current pattern with a two dimensional numerical model.
THE MODEL Assumptiomand equations The assumptions of calculation are as follows H1 Incompressible fluid
132 Irrotational Stokes waves
H3 The vertical component of the current is neglected H4 For the calculation of wave currents, the varations of sea surface elevation are neglected compared to the total depth H5 The mass current is assumed to be constant over the depth H6 Wind action is neglected.
184 With t h e s e assumptions, t h e Navier-Stokes equations can be i n t e g r a t e d over the t o t a l depth and t i m e averaged over a wave period. Then t h e following equations are obtained
+ -aa; +t
-
avij += -
aY
-aUu + ax
a% ay
g(d
- g(d
-=
-
+ 5)
a: +
+ -5 ) -a: + ay
Tx
+
Tx(- d)
+
VAV
TX
+
nx(- d)
+
VAG-
--
+
xv 16
wave bottom induced f r i c t i o n driving force with
E)
6=
(d
V =
(d+E)
+
d Iepth
'\
lass c u r r e n t components
:( 5
mean s e a s u r f a c e e l e v a t i o n
1 C o r i o l i s parameter. The parameter v mainly t a k e s i n t o account t h e eddy v i s c o s i t y of mass current. This one i s c e r t a i n l y due t o wave breaking mainly.
Wave d r i v i n g f o r c e s
These s t r e s s e s can be defined a s follows
Longuet-Higgins has f i r s t obtained t h e t e n s o r S c a l l e d t e n s o r of t h e f l u x Of eXCU momentum i n t h e waves. Under some assumptions ( p a r t i , c u l a r l y H 4 ) , it can be writen as follows.
1 where E = - pgH2 i s t h e wave energy 8
n
i s t h e shoaling number
9
i s t h e wave incidence.
185 Consequently the stresses Tx and Ty are functions of three quantities H (wave height), 0 (wave direction) and n (ratio between group velocity and phase velocity). These three quantities depend on the mass current which modifies the wave propogation here is the difficult problem of the interaction between waves and current. It has been neglected here.
Thanks to this important assumption, the quantities H, 0 and n only depend on the wave characteristics and bathymetry. They can be obtained from a numerical model of refraction.
Generally, this kind of model neglects the breaking of waves. The method we have used is to follow a wave orthogonal and to study in each point if there is or not breaking.
The Battjes criterion has been used. The waves break in a place of depth d if the wave height verifies
where y is an experimental coefficient,function of bottom slope 8 and local wave steepness
:
This coefficient determines the breaking shape. The wave height in the surf area is taken as
H = yd
The relation obtained by Bowen, Inman and Simmons for a rectilinear shoreline and a constant bottom slope was in fact H = y(d +
5).
The expression wehaveused involves two supplementary assumptions,from one hand the relation obtained by Bowen is exact even for a vsriable bottom slope,and on the other hand, the mean sea surface elevation can be neglected.
;
186 Bottom friction The bottom friction has been assumed as proportional to the square velocity
In a first step, the orbital velocity due to waves has been neglected. Then, it has been taken into account in the following form
2
Y
orb
:
-
orb
cS
Scheme of model The scheme of the model is as follow
Refraction programme Computation of wave orthogonales and
Interpolation on a regular mesh and computation of wave height in the surf area Computation of driving stresses
t
1
Numerical resolution of Saint-Venant
elevation
I
187 RECTILINEAR SHORELINE *
Most of actual longshore currents theories have been applied to rectilinear and
constant slope shore. In other respects, for this particular case, many measurements were made o n scale models and few on full scale. In order to compare the results of the numerical model with the experimental data, it was of a great interest to calibrate the model for a rectilinear shoreline.
Boundary conditions
Longshore currents along a rectilinear infinite shoreline must be parallel to the coast and in the direction of waves propaqation. To simulate an infinite beach, it is possible to set as up stream boundary condition the downstream velocities computed in the previous time step : so the lateral boundaries are cancelled (see fig. 1) MAX
IMW 1 I I I I
I I
u=v:o
I I
I
I I I
I
I
I
I
I
I
I
I
I I
I
I
I I
I
I
I
I
c
C
+
-
0 vorlo pping Figure 1 .
Results and influence of parameters
The stresses computed in the refraction programm are applied to the fluid at rest.
188 When t h e f l u i d b e g i n s t o move, t h e c u r r e n t p a t t e r n k e e p s p a r a l l e l t o t h e wave
stresses, b u t v e r y soon, owing t o t h e boundary c o n d i t i o n s t h e c u r r e n t becames parallel
t o t h e s h o r e l i n e . I n t h i s p e r i o d , some water comes from d e e p s e a t h r o u g h t h e breaking l i n e , b u t as soon as t h e e q u i l i b r i u m i s r e a c h e d f o r t h e sea s u r f a c e e l e v a t i o n , t h e incoming flow d i s a p p e a r s and t h e boundary c o n d i t i o n i s v e r i f i e d
:
no v e l o c i t y dnd
i n i t i a l water depth.
See f i g u r e s 2 and 3 f o r an example of r e s u l t s
I
b
i
Uistancr f r o m shorrlinr
F i g . 2 . V e l o c i t y p r o f i l e : ( 1 ) computed ( 2 ) computed w i t h r o r r e c t e d s t r e s s e s ( t a k i n g i n t o account sea surf.ii e elevation) ( 3 ) a n a l y t i r a l c a l c u l a t l o n (eddy v l s c o s l t y n e g l e c t e d ) .
189
\,
profile
,-Analytical
\ I -------
Breaking point
I
0
I
200
100
1
Y
360 Distance from shoreline
Fig. 3. Mean sea surface elevation.
The influence of some parameters has systematically been studied. We can distinguish three kinds of parameters
-
friction parameters fig. 4,
-
:
:
Chezy coefficient C s and eddy-viscosity coefficient w (see
and 6)
waves parameters
- shore parameter
:
:
period T, height H, and incidence fJ (see fig. 7 and 8 )
bottom slope, the influence of which has not been studied (It
has been chosen equal to 3 V
t
%).
mA
k
t
Breaking point
Distance from shoreline Fig. 4. Variation of velocity profile with
W
2
(m / s ) .
190
Distance from shoreiinr Fig. 5. Influence of bottom friction upon velocity profile ( 1 ) Bottom friction
-
5A
11
1I
cS
( 2 ) Bottom friction
-
5 ;I I
+
orb
11
cs
Distance from shoreline Fig. 6. Comparison with Longuet-Higgins results ( 1 ) Computation with a bottom friction
-
5 C
( 2 ) Longuet-Higgins velocity profile.
S
1I
II
191
2
U
-
Tt10r H,- 3 m
cs = LOrn’t2 Is Y = 10m’/r
1
-
(m)
i
loo
0
200
300
*
D i s t a n r e from s h o r e 1 ine
Fig. 7. Influence of wave incidence upon velocity profile.
&t
Breaking points,
.
/ I V
A
\
\
1
\\ Im
1 .
0‘ 100
200
Fig. 8. Influence of wave height upon velocity profile.
30 0
(m) LOO Distanre from shoreline
192 Comparison with measurements
Full scale measurements are very rare. Furthermore, the few experiments are very dissimilar and have sometimes been influenced byanother phenomenaaswind for example.
The measured currents can be compared with the computed ones, but we had to choose quite different friction coefficients in order to obtain suite similar velocities.
c
The comparison between the full scale measurementsand the computed results is hereunder discribed
I
:
I
Full scale measurements Conditions
Torrey Pines Beach (1950) T = 12 s , €Id = 6O, Hd = 1,55 m, m = 0,027 Putnam measurement T = 10 s , = 120, Hd = 1,52 m, = 0,31
measured velocity (ds)
surface bottom
: :
ed
Putnam measurement T = 8 s , ed = 120, Hd = 2,59 m, m = 0,02
1'1°
0,76
Computed results
meters
(m / s )
0,11 0,06
' I '
10 -
0,08 -
40
0,43
I
40 60
0,58
I
25 -
0,69 -
40
I
Computed velocity (m/s)
0,73
1,05
To measure the longshore currents, some experiments have been done in laboratory. They are mainly due to Putman (1949), Savile (1950), Brelner and Kamplanis (19631, Galoin and Eagleson (1965).
You will find hereunder some examples of the measurements made in laboratory compared with the computed results (The laboratory results are transposed to a real case thanks to Froude similitude : geometric scale 1 0 0 ) .
193 Laboratory measurements
Comnuted results velocity
I H = ?,O m T = 12,O s e = 11,5
I
I
m = 0,l
I 3,87
1
I
I 10
1
40
With a constant eddy viscosity coefficient
I V =
4,39
10 m2/s the value of C, to calibrate
the model with the measurements varies from 10 to 60.
Conclusion
Several cases are to be considered depending on the information w have. If any measurements have been made, it is only necessary to take them to calibrate the model. In the contrary, if the model is a predictive one, it is necessary to make a further study to point out the influence of each parameter so that the friction coefficient value could be obtained from the bottom characteristics.
SEMI CIRCULAR BAY
The advantage of a rectilinear shoreline was the possibility of calibrating the model with the measurementsaready made. On the contrary, all the possibilities of thistwo dimensional model have not been used. It is in a case without symetry that it will be the most usefull, because it will give the current direction.
Description of the bay
The studied case is idealized. The shoreline is circular (radius = 260 rn) and the bottom slope is constant (m = 3% ) . The bay is opened on a rectilinear shore where the bottom slope is also 3
%.
Results
The results obtained in the case of a bay are quiete different from those obtained along a infinite shoreline
:
- the current pattern shows big gires - the strong current area is larger than the surf area.
194 T h e s e r e m a r q s makes a p p e a r t h e i m p o r t a n c e of l o n g s h o r e c u r r e n t s i n n o l l u t i o n problems.
T h e r e s u l t s o b t a i n e d f o r t w o d i r e c t i o n s o f wave ( i n t h e d i r e c t i o n o f t h e b a y a x i s and e d g e d ) a r e p r e s e n t e d o n f i g u r e s 9 a n d 10. I n t h e s e c o n d case, t h e r e c t i l i n e a r b e a c h v e l o c i t y p r o f i l e h a s b e e n imposed on t h e l a t e r a l b o u n d a r i e s .
Shoreline
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ...................................................... . . . . . . ..................................... ........................................
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F i g . 9 . L o n g s h o r e c u r r e n t s i n a c i r c u l a r bay (wave p r o p a g a t i o n i n t h e d i r e c t i o n of bay a x i s ) . Cs = 40 m ' / 2 / s Hm = 3m Rm = 90" v = 50 m2/s T = 1 O s V e l o c i t y : u 1 m/s S c a l e 1/45000
Shoreline
I
Breaking l
i
*
-
-
_
.
,.
..-.
~
- / / f ~ l l f l l * . - / ’ f / / f l l , * - * , / / / t 1 , , . - - , , / , I ,
I
.......... ......... -
*
,
#
#
.
#
.
#
.
.
.
..........
........ . . . . . .
. . .
.
............... ------”””‘--..............................
, . ~ , . , , , , , \ \ . - _ - - - - r c ~ ” ” ” ~ - -
..............................
.............................. .............................
........................... ....................... ....................... . . . . . . . . . . . . . . . . . . .
The n u m e r i c a l t w o d m e n s l o n a l model q i v e s a n e s t i m a t i o n o f t h e l o n q s h o r e c u r r e n t s i n d u c e d by waves f o r e a n y s h a p e of s h o r e l i n e .
This Page Intentionally Left Blank
F i g . 8 A . The running variances of the v e r t i c a l v e l o c i t y observed a t the i n d i cated depths p l o t t e d w i t h t h e changing wind speeds (upper c u r v e ) .
197
TIME SERIES MODELLING OF STORM SURGES ON A MEDIUM-SIZED LAKE W.P. BUDGELL' and A. EL-SHAARAWI2 'Ocean and Aquatic Sciences Canada Centre for Inland Waters, Burlington, Ontario, Canada 2Applied Research Division, National Water Research Institute Canada Centre for Inland Waters, Burlington, Ontario, Canada
INTRODUCTION The time series modelling approach of Box and Jenkins (1970) has been extensively applied to model time series data in a variety of fields. This approach has been used by econometricians for modelling and forecasting economic data.
Examples of such studies are given by Feige et al. (19741, Granger
(1969) and Pierce (1977).
In hydrology and water management this procedure
has been used by McKerchar and Delleur (1974) and O'Connell (1971).
Recently,
El-Shaarawi and Whitney (1978) have adopted this approach to develop a sampling plan for the phosphorus loadings from the Niagara River into Lake Ontario. In the present paper we describe the steps that may be followed in developing an empirical time series model within the framework of Box and Jenkins. These steps will be illustrated by modelling the dynamical behaviour of water level fluctuations of a medium-sized lake during storm events. THE DATA SET The data used in this study are water level measurements and wind stress estimates at Lake St. Clair, situated in the Great Lakes of North America (Fig. 1). Hourly water level measurements were obtained from a float-type gauge at Belle River. Hourly wind stress values were estimated from meteorological observations recorded at nearly Windsor Airport (Fig. 2 ) . The wind stress estimates were obtained from the standard formula
198
F i g . 1.
Location of Lake St. C l a i r
199
~
Fig. 2 .
~~
Location of sampling s t a t i o n s
200
-
where T is the wind stress, Cd, is the drag coefficient, pa is the air density
-
and V is the observed wind vector.
The drag coefficient was made dependent
upon the atmospheric stability, as suggested by McClure (19701, and upon the wind speed, as suggested by Smith and Banke (1975).
The exact form of the
equation used for the computation of Lake St. Clair drag coefficients was that proposed by Hamblin (1978): C = O.OOl(1.0 d
-
(T -T ))(1.0
0.09
a
w
+
-
0.0010481VI)
(2)
where T -T is the air-water temperature difference and CGS units are employed. a w The wind stress, T , is resolved into Tx and T components along the x Y and y axes shown in Fig. 2 .
-
THE MODEL The water level fluctuations can be viewed as the response or output of a dynamic system which is under the influence of a set of physical factors such as the x and y components of wind stress. Suppose that the observed data on the system take the form of time series, i.e. the data were collected at N equidistant points in time. These data can be written as
nt, T~~ and T Yt are the water level measurement and the x and y components of wind stress at time t, (t = l,Z, N). The observed data can be regarded as a tri-variate realization or a sample from an infinite population
where
...,
of such realizations generated by some stochastic process.
This process is
assumed to be included in the general time series transfer function model proposed by Box and Jenkins (1970).
This model contains two major components.
The first is given by the class of the discrete transfer function model
-
rlt
61nt-1
S)
... -
,...,N.
for t = 1,2
...,
-
and
\
6,nt-=
= 00 T x,t-bl
+
WITx,t-bl-l
...
,...,r),
(j = 1,2, 1 v) are unknown constants which must be estimated.
The parameters bi, bz, 6; (i = 1 , 2
(k = 1 , 2 ,
+
...,
W.
Writing B for the backward shift operator, where, Bmrlt = 'Itm, then equation ( 3 ) can be rewritten as
201 where 6(B) = 1 - 61B w(B) = w
+
w B 1
+
X(b) = A
+
AIB
+
0
and
-
0
-
. - --
+
wsBS
... +
hvBv
62B2
*.a
6rBr
F i n a l l y , e q u a t i o n ( 4 ) maybe r e a r r a n g e d a s follows
The second p a r t of t h e model a r i s e s by r e a l i z i n g t h a t t h e dynamic system w i l l b e a f f e c t e d by d i s t u r b a n c e s , o r n o i s e , whose n e t e f f e c t i s t o c o r r u p t t h e o u t p u t p r e d i c t e d by t h e t r a n s f e r f u n c t i o n by an amount Nt.
The combined
t r a n s f e r f u n c t i o n - n o i s e model may then be w r i t t e n a s
When t h e d a t a a r e a v a i l a b l e , t h e problem i s t o f i t t h e model d e s c r i b e d by e q u a t i o n ( 6 ) t o t h e d a t a .
The approach we s h a l l employ f o r i d e n t i f y i n g
and f i t t i n g t h e t r a n s f e r f u n c t i o n model d i f f e r s from t h a t given i n Box and J e n k i n s (19701, b u t i t can be regarded a s g e n e r a l i z a t i o n t o t h e approach given by Haugh and Box (1977) f o r f i t t i n g a t r a n s f e r f u n c t i o n when only one independent v a r i a b l e i s a v a i l a b l e .
This approach can be d e s c r i b e d a s follows:
( i ) f i t a u n i v a r i a t e Box and J e n k i n s model t o each of t h e t h r e e s e r i e s
( i i ) e s t i m a t e t h e r e s i d u a l f o r each u n i v a r i a t e model; ( i i i ) f i t a Box and J e n k i n s t r a n s f e r f u n c t i o n t o t h e r e s i d u a l s , t a k i n g t h e r e s i d u a l of t h e r l t s e r i e s a s t h e dependent v a r i a b l e and t h e r e s i d u a l s and -r s e r i e s a s t h e independent v a r i a b l e s ; Yt ( i v ) s u b s t i t u t e t h e e x p r e s s i o n s f o r t h e Tit, TXt and
of t h e T x t
r e s i d u a l s obtained Yt i n s t e p ( i ) i n t o the t r a n s f e r function obtained i n s t e p ( i i i ) t o obtain a T
n t , TXt and T y t ; ( v ) check t h e adequacy of f i t ; r e s p e c i f y and f i t t h e models, i f necessary.
t r a n s f e r f u n c t i o n i n terms of t h e v a r i a b l e s
I n t h e n e x t two s e c t i o n s t h e s t e p s l i s t e d above w i l l b e d e s c r i b e d i n more de t a i 1.
THE UNIVARIATE MODEL
The s t e p s t h a t may be followed i n i d e n t i f y i n g , f i t t i n g and checking t h e
202 Suppose
adequacy o f a u n i v a r i a t e t i m e s e r i e s model w i l l b e d e s c r i b e d h e r e .
we have a t i m e series of t h e N s u c c e s s i v e o b s e r v a t i o n s X 1, X2, a t equidistant points i n time.
..., XN
taken
T h i s series i s assumed t o be a r e a l i z a t i o n
from a n i n f i n i t e p o p u l a t i o n o f such r e a l i z a t i o n s g e n e r a t e d by some s t o c h a s t i c This p r o c e s s i s assumed t o be s t a t i o n a r y .
process.
S t a t i o n a r i t y r e q u i r e s t h e absence o f a t r e n d i n t h e mean v a l u e as w e l l
as t h e absence o f any t r e n d i n a l l h i g h e r moments o f t h e process.
I f the
p r o c e s s i s n o n s t a t i o n a r y , a p r i o r t r a n s f o r m a t i o n i s performed on t h e d a t a t o achieve s t a t i o n a r i t y .
D i f f e r e n c i n g o p e r a t i o n s a r e c o w o n l y used t o o b t a i n A d i f f e r e n c e o f o r d e r d i s d e f i n e d as
the desired transformation. Ht = (1-B)
d
Xt
where Ht i s assumed t o be a s t a t i o n a r y series.
An a l t e r n a t i v e t r a n s f o r m a t i o n
t o d i f f e r e n c i n g i s known as t h e d e t r e n d i n g procedure.
This l a t t e r technique
i s o f t e n used t o e l i m i n a t e d e t e r m i n i s t i c t r e n d s and s e a s o n a l e f f e c t s . Assuming t h a t t h e sequence X1, X2,
..., XN
i s s t a t i o n a r y , t h e Box and
J e n k i n s model i s g i v e n as
Bi ( i
where
= 1,2, . . . , p ) , 8.
3
(j = 1,2, . . . , q ) , p and q are unknown parameters
t o be e s t i m a t e d and at ( t = O,l,Z,
...,N)
v a r i a b l e s w i t h mean 0 and v a r i a n c e U
’.
i s a sequence o f independent random The model d e s c r i b e d by e q u a t i o n ( 7 )
i s o f t e n c a l l e d t h e A u t o r e g r e s s i v e Moving Average Model of o r d e r p and q and i s u s u a l l y denoted by ARMA(p,q).
The form of t h e model i n d i c a t e s t h a t p a s t
v a l u e s o f t h e series c a n be used i n f o r e c a s t i n g i t s f u t u r e v a l u e s . I n o r d e r t o f i t t h e above model t o an observed t i m e s e r i e s , t h e f o l l o w i n g s t e p s are r e q u i r e d : ( i ) t r a n s f o r m t h e series t o a c h i e v e s t a t i o n a r i t y and determine t h e v a l u e s o f p and q; ( i i ) g i v e n t h e v a l u e s of p and q estimate B,, 2 and Ga ; and
8,
,..., Bp;
el,
O2
,..., 84
( i i i ) check t h e adequacy of t h e model.
The e s t i m a t i o n of t h e model parameters can be accomplished by u s i n g comp u t e r r o u t i n e s developed by McLeod (1977) a t t h e U n i v e r s i t y o f Waterloo. S t e p s ( i ) and ( i i i ) may be c a r r i e d o u t by t h e examination of t h e sample autoc o r r e l a t i o n f u n c t i o n (ACF) and t h e sample p a r t i a l a u t o c o r r e l a t i o n f u n c t i o n (PACF). N-k rk = C
t=1
The e s t i m a t e d a u t o c o r r e l a t i o n c o e f f i c i e n t a t l a g k i s
-
-
(x, - x) ( x , + ~ - x ) /
N
C
- 2
(x, - x)
t =1
203
where
-
N
c Xt/N t=l The value of rk specifies how much information is contained in an observation x
=
made at time t about an observation made at time t+k.
The other type of cor-
relation which may be used in model identification is the estimated lag k h
partial autocorrelation,
akk.
h
The values of @kk can be estimated by solving
j = 1,2, ...,k h
The value, ,@
measures the dependence of Xt+k on Xt after eliminating the
influence of X t + l 7 Xt+’”*’’l ‘t+k-l upon ‘t+k’ The first step in identifying the model is to determine if the stochastic process is stationary. This is done by examining the ACF and PACF.
If the
ACF and PACF neither damp out with increasing k nor truncate, but instead
remain large, then the process is nonstationary.
In this case the data have
to be transformed either by taking differences or by detrending.
The ACF
and PACF of the transformed data are then calculated and examined for stationarity.
If the process is still nonstationary another transformation is attempt-
ed until stationarity is achieved. Once the data, or transformed data, satisfy the stationarity requirement, the behaviour of the ACF and PACF determines an approximate estimate for the p and q values defined in equation ( 7 ) .
In the case where q=O, the ACF dies
out slowly but the PACF is zero after lag p.
On the other hand, when p=O,
the ACF is zero after lag q but the PACF damps out slowly. When neither p nor q is equal to zero, both the ACF and PACF decay slowly.
To determine
when the population autocorrelations and partial autocorrelations are effectively zero, the sample estimates must be tested for significance. A test of significance of the sample autocorrelations was given by Bartlett (1935). The estimated variance of the autocorrelation, rk, is
k-1 (10)
i=l
which is calculated under the assumption that the population autocorrelation of lag greater than or equal to k is zero.
The estimate of the variance for
h
the partial autocorrelation of lag k, @kk, was given by Quenouille ( 1 9 4 9 ) , as
G2 ‘kk
2
1/N
204
A f t e r t h i s p r e l i m i n a r y i d e n t i f i c a t i o n of t h e model, t h e parameters of t h e The adequacy
model may be e s t i m a t e d by u s i n g t h e method of maximum l i k e l i h o o d .
o f t h e f i t t e d model i s then checked by examining t h e a u t o c o r r e l a t i o n f u n c t i o n
{st}, where at
of t h e e s t i m a t e d r e s i d u a l s
st
- Blxt-l - ... - 8,xt-, h
= Xt
h
h
h
where B. ( i = 1 , 2 , . . . , p and 8
elct-l A
+
and
ej
( j = 1 , 2 , ...,q
... + eqct-q h
a r e e s t i m a t e s of 6. ( i = 1 , 2 , . . . , p
I f t h e model i s adequate t h e a u t o c o r r e l a -
( j = 1 , 2 , ...,q ) , r e s p e c t i v e l y .
j
t i o n f u n c t i o n of {^a
+
i s e s t i m a t e d from e q u a t i o n ( 7 ) a s
w i l l show t h e p a t t e r n a s s o c i a t e d with a s e t of indepen-
t
dent o b s e r v a t i o n s ( i . e . ,
a l l a u t o c o r r e l a t i o n s a t l a g s g r e a t e r than one a r e
zeros).
THE TRANSFER FUNCTION MODEL
Suppose t h a t u n i v a r i a t e models have been f i t t e d t o t h e t h r e e s e r i e s rl TXt
and
and l e t
T
Yt’ t h e s e f i t t e d models.
arl,
ux and
aY
t’ r e p r e s e n t t h e r e s i d u a l s e r i e s obtained from
I f i t i s assumed t h a t t h e r e i s no feedback, then t h e
r e s i d u a l of q t may be expressed i n terms of t h a t of
T~~
and
T
Yt’
An appropri-
a t e model may be w r i t t e n a s
where )l(B)= Po
V(B)= V
$(B)= 1
+
NIB
+
... +
)lLB
+
V B
+
... +
VmB
+
$1~ +
... +
L
,
m
,
Q ~ B ” , and
*.
The { a 1 r e p r e s e n t s a s e t of white n o i s e with z e r o mean and v a r i a n c e U t problem i s then t o determine t h e v a l u e s o f L , m and n. The c r o s s c o r r e l a t i o n f u n c t i o n (CCF) is t h e a p p r o p r i a t e t o o l f o r t h e i d e n t i f i c a t i o n of t h e s e constants. The c r o s s c o r r e l a t i o n between two given t i m e s e r i e s , Xt and Y t , e s t i m a t e d from t h e e q u a t i o n
where r
(k) is t h e e s t i m a t e d c r o s s c o r r e l a t i o n a t l a g k , XY
can be
205
Cx(0)
=
1.
;
(X,
-
X) - 2
-
- 2 Y)
t=l
c (0) =
Y
1
:
(Y,
t=l
Under t h e assumption t h a t Xt and Y t are independent white-noise s e r i e s , we have
The v a l u e s o f L , m and n can be determined from an examination of t h e estimated cross correlations r
u u
(k), ru
( k ) and ru
(k).
Clearly, the
comparison of r
(k) and ru u x ( h w i t h r h $ i r s t a n d a r a g r r o r s w i l l determine u u 11 xrl t h e a p p r o p r i a t e v a l u e s o f Lana m. Once 1and m have been determined, t h e n
t h e f a c t t h a t urlt is a white-noise series imposes a set of r e s t r i c t i o n s on t h e model, t h e number of t h e s e r e s t r i c t i o n s w i l l be equal t o n. A f t e r t h e o r d e r (L,m,n) of t h e model h a s been e s t a b l i s h e d , it i s p o s s i b l e t o estimate t h e unknown p a r a m e t e r s Uo, U 1
$,.
,..., PI,
vo, v1
,..., vm, $1, dJ2 ,...,
One approach f o r o b t a i n i n g approximate estimates f o r t h e s e parameters
i s t h e method o f moments.
This method might be a p p r o p r i a t e i f t h e number
of observations is larg e.
The o t h e r approach i s t o use an approximate maximum
l i k e l i h o o d e s t i m a t i o n technique.
An a l g o r i t h m f o r computing t h e e s t i m a t e s
i n t h i s manner w a s made a v a i l a b l e t o us through t h e U n i v e r s i t y of Waterloo Mathematics F a c u l t y Computer F a c i l i t y . The complete model i s o b t a i n e d by combining e q u a t i o n (7), which is used
t o d e f i n e t h e r e s i d u a l series uqt,
uXt and u
t i o n model d e s c r i b e d by e q u a t i o n (13).
Yt’
w i t h t h e n o i s e t r a n s f e r func-
The f i n a l form of t h e model becomes
A f t e r t h e model parameters have been e s t i m a t e d , t h e n t h e e s t i m a t e d r e s i d u a l
206
s e r i e s {a
t
1
can be c a l c u l a t e d .
The a u t o c o r r e l a t i o n f u n c t i o n of t h e s e r i e s
i s computed and compared w i t h i t s s t a n d a r d e r r o r t o e n s u r e t h a t t h e model h a s accounted f o r t h e r e l a t i o n s h i p between s u c c e s s i v e o b s e r v a t i o n s . i t i o n , the cross-correlation
f u n c t i o n s between
a"t
h
and uxt,
I n addh
and between a
t a r e c a l c u l a t e d and compared t o t h e i r s t a n d a r d errors t o make s u r e Yt t h a t t h e model h a s adequately d e s c r i b e d t h e r e l a t i o n s h i p between t h e dependent
and
series,
nt,
and t h e two independent s e r i e s ,
TXt
and
Y t*
APPLICATIONS The previous a n a l y s i s h a s been a p p l i e d t o develop a t r a n s f e r f u n c t i o n model for a storm event which took p l a c e J u l y 10-14,
1964.
The storm, which
w i l l b e used t o i l l u s t r a t e t h e nt, TXt and -c Yt' procedures d e s c r i b e d i n t h e p r e v i o u s s e c t i o n s .
h a s 91 h o u r l y o b s e r v a t i o n s i n
Fig. 3 shows t h e time h i s t o r i e s of t h e water l e v e l f l u c t u a t i o n s , Q t ,
and
The p l o t s of t h e ACF ( F i g s . and T xt Yt' 4 , 6 and 8 ) and PACF ( F i g s . 5 , 7 and 9) of Q t , Txt and T suggest t h a t t h e s e Yt v a r i a b l e s behave i n a s i m i l a r manner. It can b e s e e n from t h e s e f i g u r e s t h a t t h e normal wind s t r e s s components,
T
t h e ACF does n o t d i e o u t q u i c k l y , s u g g e s t i n g n o n s t a t i o n a r i t y .
The o r i g i n a l The r e s u l t i n g
s e r i e s a r e t h e n transformed by t a k i n g t h e f i r s t d i f f e r e n c e .
a u t o c o r r e l a t i o n f u n c t i o n s , ( F i g s . 10, 12 and 141, and p a r t i a l a u t o c o r r e l a t i o n f u n c t i o n s ( F i g s . 11, 13 and 15) become v e r y s m a l l , i n d i c a t i n g t h a t t h e d i f f e r e n c i n g o p e r a t i o n h a s induced s t a t i o n a r i t y . Since t h e a u t o c o r r e l a t i o n a t l a g one is h i g h i n a l l t h r e e c a s e s , i t i s assumed t h a t a moving average model of o r d e r one might be a p p r o p r i a t e i n each
The s t r u c t u r e of t h e models i s a s follows:
case.
(l-B)Xt
= (1-8 B ) a
1
t
-
elat-l
or X t
-
Xt-1
= at
The e s t i m a t e s of
el,
t h e s t a n d a r d e r r o r of t h e e s t i m a t e and
of t h e t h r e e u n i v a r i a t e models i s given i n Table 1.
0
for each
m
P)
c,
x
m
m
Lc
3
T i m e s e r i e s p l o t s of water l e v e l s and x and y components of wind stress
207
a
'D
4J
m
Fig. 3 .
208
L:
-7
-1
2u
LL
-
4
' 0
Fig. 4 .
10
LAG
-2a
-
20
7
10
20
LAG F i g . 5.
ACF of qt
-1
PACF of qt
-1 LL
uo
a
I"',
...,I
(,..,',I
[L
7
10
20
' 6
+
-
'0
LAG Fig. 6.
7
ACF of T , ~
Fig. 7 .
PACF of Txt
-1
10
20
'
+
-
'0
10
LAG F i g . 8.
20
LAG
-1
-
10
ACF of T
Yt
LAG F i g . 9.
PACF
Of
T
Yt
20
2a * -2a
209
-1
-1
A
h
LL 00
u
.!
, I
.-I
.
20
I
I
1
I
'
h
-20
20
LL
go a_
.!
I . I . 1
I I I " I I
[
II'I.I
e I
I
I
Fig. 11. PACF of ( l - B ) \
Fig. 10.
ACF Of ( l - B ) \
Fig. 12.
ACF of ( ~ - B ) T
Fig. 13.
PACF Of (l-B)TXt
Fig. 14.
ACF of ( 1 - B ) T
Fig. 15.
PACF of ( ~ - E ) T Yt
xt
Yt
h
-20
I
I
210
TABLE I
Estimated parameters i n t h e u n i v a r i a t e models
h
Variable ‘It TXt
TYt
el
A
s.E.(~])
- 2
oa
0.227
0.103
6.65
0.354
0.099
0.12
0.275
0.101
0.09
Residual s e r i e s were computed f o r each of the t h r e e v a r i a b l e s through t h e use of t h e following equation:
^u were estimated f o r the v a r i a b l e s rl t ’ Yt r e s p e c t i v e l y , and a r e shown i n Figs. 1 6 , 17 and 18.
The r e s i d u a l s e r i e s u,.,~, uXt and T~~
and T
Yt’ To i d e n t i f y a r e s i d u a l t r a n s f e r function model, t h e c r o s s c o r r e l a t i o n s
of ^u and ^unt, and ^unt, and ^u and G were computed. These c r o s s correlaxt Yt xt Yt t i o n functions a r e shown i n Figs. 1 9 , 20 and 2 1 , r e s p e c t i v e l y . From these
Gxt and ^u a r e s i g n i f ‘It i c a n t a t l a g s 0 , 1 and 2 , the cross c o r r e l a t i o n s between ^u and ^uqt a r e s i g n i f Yt i c a n t a t l a g s 0 and 1, and t h e cross c o r r e l a t i o n s between 6 and ^u are xt Yt s i g n i f i c a n t a t l a g s - 2 , 0 and 2 . Thus t h e t r a n s f e r function model described
p l o t s i t can be seen t h a t t h e c r o s s c o r r e l a t i o n s between
by equation ( 1 3 ) w i l l be of t h e form:
The estimated c o e f f i c i e n t s a r e l i s t e d i n the following t a b l e :
211 TABLE I1 E s t i m a t e d c o e f f i c i e n t s f o r n o i s e t r a n s f e r f u n c t i o n model Coefficient
Estimate
h
1.087
I-lO A
1.447
I-ll h
-1.736
I-l2 A
vo v1
1.892
h
3.585
$1
0.072
e2
-0.047
Ga2,
The r e s i d u a l v a r i a n c e ,
was 4.47 cm
2
.
The e q u a t i o n s d e s c r i b i n g t h e u n i v a r i a t e models f o r
nt,
TXt
and
T
Yt
are
s u b s t i t u t e d i n t o t h e n o i s e t r a n s f e r f u n c t i o n model g i v e n by e q u a t i o n (19) t o p r o v i d e t h e f i n a l t r a n s f e r f u n c t i o n model d e s c r i b e d below
nt
-
0.097Qt-3
1.628nt-l
+
1 . 0 8 7 ~- ~0 ~. 1 8 5 ~ ~ ,- ~3.296T - ~
- 1.069Tx,t-4
+
1.892~
-
0.288T
+
at
Yt
-
+
0.725n
+
=
+
t-2
x,t-2
+
3. 355Tx ,t-3
+
2.217~
0.108Tx,t-5
0.595~
Yrt-1
-
4.416T
Y I t-2
Y ,t-3
Y ,t-4
0.927at-l
+
0.349at-2
- 0.080at-3
-
0.013a t -4
To e n s u r e t h a t t h i s model a c c o u n t s f o r t h e a u t o c o r r e l a t i o n of
nt
as w e l l
t h e a u t o c o r r e l a t i o n f u n c t i o n of and T T Yt' xt and t h e c r o s s c o r r e l a t i o n f u n c t i o n s o f y,t w i t h r e s p e c t t o ^Uxt and of 6
as t h e dependence o f rlt upon
^at
with respect t o
^u
nt
were c a l c u l a t e d and p l o t t e d ( F i g s . 22, 23 and 24).
Yt o f t h e r e s i d u a l a u t o o r c r o s s c o r r e l a t i o n s a r e more than m a r g i n a l l y significant.
T h i s i n d i c a t e s t h a t t h e model i s adequate.
A p l o t of t h e
p r e d i c t e d and observed water l e v e l f l u c t u a t i o n s i s shown i n Fig. 25.
None
212
n
. 4
'b
I
I
10
20
LRG
F i g . 16.
ACF of
ii
n
d I '0 10 20 LflG
F i g . 17.
LL
oo
=
/ \""I
I'
T I
I .
2
'
ACF of
ii
G h
-2c
d '0 10 20 LAG F i g . 18.
ACF of
Y
213
. I
1
I
-20
I
I
-10
0
I
1
10
20
1
20
LAG Fig. 19.
r (CCF of u xu n
u^
on
n
u^
)
x
-1
.-I1 I
I
-20
I
I
-10
0
10
I
LAG Fig. 20.
r
(CCF of
uYu O
u^ on 2
I I Y
2;
I
a0 I , , ' , , , , LL 0
',.l..y,l,'~~
Fig. 21.
r (CCF of ti on 6 ) uX uY Y X
I . I ' , I '
,l'.-2;
214
-1 F i g . 22.
ACF of
-1 F i g . 23.
r
uX a
on d
(CCF of
2,
(CCF of
2 on u^
xt
-1
F i g . 24.
r
u a
Y
t
yt)
c,
- 20
I
---- Predicted
-
Observed
I
11
I
I
13
12
July, 1964 Fig. 2 5 .
One s t e p ahead predictions and observed water levels (dependent storm)
I
14
216 I n o r d e r t o t e s t t h e model under an independent set of c o n d i t i o n s i t was a p p l i e d t o an a d d i t i o n a l 18 storm e v e n t s .
On t h e s e 18 s t o r m s , t h e model
accounted f o r 72 per c e n t of t h e t o t a l v a r i a t i o n i n t h e water l e v e l (R' .72, where R i s t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t ) . w a s 16.6 cm'.
=
The r e s i d u a l v a r i a n c e
The l a r g e s t storm i n t h e d a t a s e t occurred November 1-4,
1966.
The p r e d i c t e d and observed h o u r l y water l e v e l s f o r t h i s storm a r e p r e s e n t e d i n Fig. 26. Improved e s t i m a t e s f o r t h e model c o e f f i c i e n t s could be obtained i f t h e e s t i m a t e s were based on d a t a c o l l e c t e d from s e v e r a l storm, e v e n t s . be accomplished by "lumping" s t a t i s t i c a l l y s i m i l a r storm d a t a .
This could
I n addition,
accuracy could be enhanced i f over-lake a s opposed t o over-land winds were used i n t h e modelling study.
U n f o r t u n a t e l y , very few over-lake wind measure-
ments a r e a v a i l a b l e f o r Lake S t . C l a i r .
To d a t e , a t t e m p t s t o u s e lake-land
wind r a t i o s d e r i v e d from o t h e r l a k e s i n Lake S t . C l a i r a p p l i c a t i o n s have been disappointing.
Hamblin (1978) found t h a t c o r r e c t i n g t h e land wind f o r t h e
over-lake e f f e c t , a f t e r P h i l l i p s and I r b e (19781, f a i l e d t o improve t h e accuracy of Lake S t . C l a i r water l e v e l p r e d i c t i o n s o b t a i n e d from a s p e c t r a l dynamica 1 model.
CONCLUSIONS
I n t h i s paper an e m p i r i c a l time s e r i e s model h a s been developed t o r e l a t e water l e v e l f l u c t u a t i o n s t o normal wind s t r e s s components f o r storm s u r g e e v e n t s on a medium-sized
lake.
The model produced a c c e p t a b l e r e s u l t s when
used t o p r e d i c t water l e v e l f l u c t u a t i o n s on Lake S t . C l a i r .
Improved accuracy
could be achieved by combining s t a t i s t i c a l l y s i m i l a r storm d a t a and by u s i n g over-lake a s opposed t o over-land wind measurements.
ACKNOWLEDGMENTS
The a u t h o r s would l i k e t o thank D r . P.F.
Hamblin of t h e N a t i o n a l Water
Research I n s t i t u t e i n B u r l i n g t o n , O n t a r i o f o r many s t i m u l a t i n g d i s c u s s i o n s and h e l p f u l s u g g e s t i o n s .
217
I I
Observed ,
I
I
I
I
2
3
4
5
November, 1966 Fig. 2 6 .
One s t e p ahead p r e d i c t i o n s and observed w a t e r l e v e l s (independent storm)
218 REFERENCES B a r t l e t t , M.S., 1935. Some a s p e c t s of t h e t i m e - c o r r e l a t i o n problem i n regard t o t e s t s of s i g n i f i c a n c e . J o u r n a l of t h e Royal S t a t i s t i c a l S o c i e t y , 98:536-543. 1970. Time S e r i e s Analysis: F o r e c a s t i n g and Box, G.E.P. and J e n k i n s , G.M., Control. Holden-Day, San Francisco. El-Shaarawi, A. and Whitney, J . , 1978. On determining the number of samples r e q u i r e d t o e s t i m a t e t h e phosphorous i n p u t c o n t r i b u t e d by Niagara River t o Lake Ontario. Canada Centre f o r Inland Waters Report S e r i e s , i n p r e s s . 1974. The c a u s a l i t y r e l a t i o n s h i p between money Feige, E.L. and Pearce, D.K., and income: a time s e r i e s approach. Presented a t t h e Annual Meeting of Midwest Economic A s s o c i a t i o n , Chicago. 1969. I n v e s t i g a t i n g c a u s a l r e l a t i o n s h i p s by econometric Granger, C.W.J., models and c r o s s s p e c t r a l methods. Econometrica, 37:424-438. Hamblin, P.F., 1978. Storm s u r g e f o r e c a s t i n g i n enclosed s e a s . Proc. of t h e 1 6 t h Conference on Coastal Engineering, ASCE, Hamburg. 1977. I d e n t i f i c a t i o n of dynamic r e g r e s s i o n ( d i s Haugh, L.D. and Box, G.E.P., t r i b u t e d l a g ) models connecting two time s e r i e s . J o u r n a l of t h e American S t a t i s t i c a l S o c i e t y , 72:121-130. McClure, D . J . , 1970. Dynamic f o r e c a s t i n g of Lake E r i e water l e v e l s . Report No. 70-250 H. Hydro-Electric Power Commission of O n t a r i o , Research Div. Report, Toronto. 1974. A p p l i c a t i o n of s e a s o n a l parametric McKerchar, A . I . and D e l l e u r , J . W . , l i n e a r s t o c h a s t i c models t o monthly flow d a t a . Water Resources Research, 10: 246-254. McLeod, A . I . , 1977. Improved Box-Jenkins e s t i m a t o r s . Biometrika, 64:531-534. O'Connell, P.E., 1971. A simple s t o c h a s t i c modelling of H u r s t ' s Law. Proc. of I.A.S.H. I n t e r n a t i o n a l Symposium on Mathematical Models i n Hydrology, Warsaw. 1978. Lake t o land comparison of wind, temperaP h i l l i p s , D.W. and I r b e , J . G . , t u r e and humidity on Lake O n t a r i o d u r i n g t h e I n t e r n a t i o n a l F i e l d Year f o r t h e Great Lakes. Report No. CL1-2-77. Atmospheric Environment, F i s h e r ies and Environment Canada, Toronto. 1977. R e l a t i o n s h i p s and t h e l a c k t h e r e o f - between economic P i e r c e , D.A., t i m e s e r i e s , w i t h s p e c i a l r e f e r e n c e t o money and i n t e r e s t r a t e s . J o u r n a l of t h e American S t a t i s t i c a l A s s o c i a t i o n , 72:ll-22. Quenouille, M.H., 1949. Approximate t e s t s o f c o r r e l a t i o n i n t i m e - s e r i e s . J o u r n a l of t h e Royal S t a t i s t i c a l S o c i e t y (B), 11:68-74. Smith, S.D. and Banke, E.G., 1975. V a r i a t i o n of t h e s e a s u r f a c e drag c o e f f i c i e n t with wind speed. Q u a r t e r l y J o u r n a l of t h e Royal Meteorological Soci e t y , 101:665-673.
219
W I N D INDUCED WATER CIRCULATION OF LAKE GENEVA
S.W. BAUER and W.H.
GRAF
Laboratoire d'Hydraulique ( L H Y D R E P ) , Ecole Polytechnique FBdBrale, Lausanne Switzerland
(Received 1 7 August 1978; accepted 24 August 1978)
ABSTRACT A numerical modeling technique i s used t o simulate flow p a t t e r n s a t various depths i n t h e Lake of Geneva (Le LBman) f o r a homogeneous s i t u a t i o n encountered u s u a l l y during w i n t e r months. Subsequently, a v e r t i c a l l y i n t e g r a t e d flow p a t t e r n i s obtained. A measuring campaign i s under way which provides d a t a on t h e v e l o c i t y , d i r e c t i o n and temperature of t h e atmosphere and t h e water. Using t h e s e d a t a , two quasi-steady s t a t e s i t u a t i o n s a r e compared w i t h model simulations for winter months of 1977 and 1978.
DESCRIPTION OF MATHEMATICAL MODEL
A system of c u r r e n t s
i n a l a k e may be considered as water movement on a l a r g e
s c a l e . I t can be described by t h e t h r e e components of t h e momentum eauation and t h e c o n t i n u i t y equation f o r a homogeneous (non s t r a t i f i e d ) and incompressible f l u i d ( L i g g e t t , 1970) :
P[-
a U+ a 2 a a (UV) + aY aZ ( U W ) a t ax (U ) + -
av + aw = aU + -
ax
ay
aZ
- fV1
=
-
a aU a ap + -(v) ax aZ, ax
4- -(E-)+
aZ
0
aU ax
a au ay ay
-(E-)
(1)
(4)
The boundary c o n d i t i o n s a p p l i c a b l e on s o l i d boundaries a r e u = v = w = o and a t the f r e e s u r f a c e with z = 0
(5)
220
The symbols i n equation 1-6 a r e as follows:
u , v and w a r e t h e v e l o c i t y components i n t h e x, y and z d i r e c t i o n s r e s p e c t i v e l y where x i s p o s i t i v e towards east, y i s p o s i t i v e towards n o r t h and z i s p o s i t i v e upwards w i t h zero a t t h e water s u r f a c e ,
t
is the t i m e ,
f
i s t h e C o r i o l i s parameter,
p
is the f l u i d density,
p
is the local pressure,
17 and
T
&
a r e t h e v e r t i c a l and h o r i z o n t a l compnents of t h e eddy v i s c o s i t y ,
i s t h e a c c e l e r a t i o n of g r a v i t y
g X
and
and T
a r e t h e wind s h e a r s t r e s s e s on t h e water s u r f a c e i n t h e x and y d i r e c t i o n s Y respectively. Equation 3 expresses t h e h y d r o s t a t i c equilibrium, which i s a v a l i d assumption
f o r shallow l a k e s , i.e.: D/L << 1 where L and D a r e c h a r a c t e r i s t i c v e r t i c a l and h o r i z o n t a l dimensions ( f o r the LQman: D/L
2 0,03).
Equations 1-3 may be f u r t h e r s i m p l i f i e d according t o t h e following assumptions:
aU at --
a) s t a t i o n a r y flow:
0,
av at --
0;
such a s i t u a t i o n might be imagined f o r a wind which blows long enough t o establish a stationary circulation; b ) t h e i n e r t i a f o r c e s are small when compared with t h e C o r i o l i s f o r c e s , i . e . ,
the
Rossby number i s small. I n t h i s c a s e equations 1-3 may be l i n e a r i z e d :
( f o r t h e Ldman t h e Rossby number
2 0 , l )i
c ) h o r i z o n t a l d i f f u s i o n i s s m a l l compared t o v e r t i c a l d i f f u s i o n , which can be accepted f o r shallow l a k e s :
a E -)av a x ( ax
=
0,
a a~
-(&
( f o r t h e LQman: D/L
au -) aY
= 0, etc.
'J 0 , 0 3 ) ;
d) t h e v e r t i c a l component of t h e eddy v i s c o s i t y 17 which i s notknown i s assumed t o be c o n s t a n t over the e n t i r e lake: 2 a v -(n -) = 17, etc. a Z aZ 2
a
av
aZ
e ) t h e i n f l u e n c e of e x t e r n a l i n - and outflows i n t h e Leman on wind induced currents on t h e l a k e i s n e g l i g i b l e . Thus, i n t h e immediate v i c i n i t y of r i v e r mouths, t h e model i s not a p p l i c a b l e .
221 W i t h t h e above s i m p l i f i c a t i o n s and conditions t h e system of equations describing
wind induced c u r r e n t s i n a homogeneous shallow l a k e i s thus:
+pfu=
aU ax
ap
--+qaY
aw av + +ay aZ =
a2v a2
0
The numerical s o l u t i o n of t h e model described thus f a r i s obtained by Gallagher
et al.
(1973) and based on a mathematical formulation by L i g g e t t e t a l .
(1969).
ADAPTATION OF MATHEMATICAL MODEL FOR THE LEMAN
The mathematical model as described above has been coded and provided t o LHYDREP by J . A .
L i g g e t t of Cornell University. Subsequently a s u i t e cf r o u t i n e s
allowing g r a p h i c a l r e p r e s e n t a t i o n i n 2 and 3 dimensions of t h e numerical r e s u l t s
as well as of t h e geometrical r e p r e s e n t a t i o n i n f i n i t e elements
-
s e e Figure 1 -
of t h e l a k e geometry have been developed a t LHYDREP. A f t e r s e v e r a l t r i a l geometries 2 a f i n i t e element g r i d c o n s i s t i n g of 579 nodes (about one node per km 1 , 1025 triangular elements and 131 boundary p o i n t s
w a s found t o g i v e reasonable r e s u l t s
( s e e Bauer e t a l . , 1977).
GRILLE DES ELEMENTS FINIS F) TRBIS DIMENSlBNS RVEC UNE DISTBRTIBN DE X : Y : 2 = 1 : 2.00 : 0.020 LE LEMRN
t l
2 "Y
=m
mm
iii : w
'I:dl .I.-'
'*>.' '1 kil
'*,.I '&m
)n:-
'm'm
Fig. 1. "Three dimensional" Lake of Geneva ( F i n i t e element g r i d , nodes a and b showing p o s i t i o n of i n s t r u m e n t s ) .
I t should be noted t h a t compared t o i t s width,the Leman i s very shallow
(0,Ol < D/L < 0 , 0 3 ) . Due t o numerical reasons of t h e f i n i t e element approximation it i s necessary t h a t t h e minimum depth allowable is about 3 % of t h e maximum depth
222 Thus t h e r e a r e no zero depths a t t h e boundaries ( s e e Figure 1) which however i s n o t a severe l i m i t a t i o n f o r t h e geometrical r e p r e s e n t a t i o n . To execute t h e model a s e r i e s of parameters ( s e e equations 7-9) must be entered:
t h e d e n s i t y of water, p, the C o r i o l i s parameter, f , t h e a c c e l e r a t i o n of g r a v i t y , g, and t h e eddy v i s c o s i t y , q . Also, r e p l a c i n g i n euuation 6 t h e shear s t r e s s , T, by equation 11, v i z . T = c
f
where
.
.
'a
C
(11)
i s a wind shear s t r e s s c o e f f i c i e n t ,
f
P
U
2 "wind
i s t h e d e n s i t y of a i r
.
wind
and
i s t h e f r e e stream wind v e l o c i t y .
Two f u r t h e r parameters, C and p a r e t o be supplied. The following parameters f a have been taken a s c o n s t a n t s i n t h e model simulations: 3 : Pa = 1 , 2 kg/m d e n s i t y of a i r 3 : P = 999,9 kg/m d e n s i t y of water 2 a c c e l e r a t i o n of g r a v i t y : g = 9,80 m / s
C o r i o l i s parameter
:
f
= 0,000105 s
-1
( f o r a mean l a t i t u d e of t h e Leman of
46O 25 )
The parameters rl and C f , which a r e unknown a r e thus s u b j e c t t o determination by a model c a l i b r a t i o n . A t r i a l simulation f o r a s e t of f i x e d and chosen uarameters h a s been done and
i s reported by Bauer e t a l .
(1977).
MEASURING CAKZAIGNS
To allow f o r comparison between t h e c u r r e n t s occurring i n nature and t h e i r simulation by a model it i s necessary t o study ?he d i s t r i b u t i o n of t h e water v e l o c i t i e s and t h e winds which generate them. Therefore a measuring campaign has been s t a r t e d t o measure simultaneously and i n s i t u v e l o c i t i e s and d i r e c t i o n s of t h e wind and c u r r e n t s . Furthermore, temperature p r o f i l e s of atmosDhere and water were measured. I d e a l l y , measurements should be taken by a l a r g e number of recording instruments placed according t o some g r i d system over t h e e n t i r e domaine. Such measurements, h o r i z o n t a l l y and v e r t i c a l l y d i s t r i b u t e d , would allow t o o b t a i n a synoptic view of t h e phenomenon. Such a s o l u t i o n however i s a t p r e s e n t , for f i n a n c i a l and operational reasons, impossible and it was thus necessary t o a r r i v e a t a much more modest solution. Keeping t h e concept of a v e r t i c a l scheme, a s i n g l e s t a t i o n i s nlaced i n t h e l a k e . Measured and recorded a r e :
223 - v e l o c i t y o f t h e atmosphere a t t h r e e ( 3 ) d i f f e r e n t a l t i t u d e s as well a s i t s d i r e c t i o n s and t h e temperature;
- v e l o c i t y , d i r e c t i o n , temperature and p r e s s u r e of t h e c u r r e n t s a t f i v e ( 5 ) d i f f e r e n t depths. A d e s c r i p t i o n of this i n s t a l l a t i o n i s given by P r o s t e t a l .
(1977) and a schematic
view of t h e s t a t i o n i s shown i n Figure 2 . This s t a t i o n has been k e p t f o r a d u r a t i o n of weeks a t one place. Since f o r t h e s e r v i c i n g of t h e c u r r e n t meters it i s necessary t o l i f t t h e e n t i r e i n s t a l l a t i o n o u t of t h e w a t e r , t h i s o p e r a t i o n i s a l s o used t o change i t s p o s i t i o n i n t h e l a k e , thus allowing f o r a coverage of d i f f e r e n t a r e a s . During the winter months, t h e i n s t a l l a t i o n w a s l o c a t e d c l o s e t o t h e two nodes a and b a s i n d i c a t e d i n Figure 1. Measurements:
,
,
. .. .. .. ,,
I .
~
+...
Fig. 2 . Schematic view of measuring s t a t i o n For measurements i n t h e atmosphere, conventional cup anemometers of t h e type Aanderaa (WSS 219) a r e being used t o measure t h e v e l o c i t i e s . The number of r o t o r r e v o l u t i o n s i s counted e l e c t r o n i c a l l y and t r a n s m i t t e d i n d i g i t a l form t o a d a t a logger. Wind d i r e c t i o n s a r e determined with t h e a i d of a wind vane type Aanderaa (WDS 2053) and a magnetic compass measuring t h e o r i e n t a t i o n of t h e buoy with resp e c t t o n o r t h . The a i r temperature sensors used were Aanderaa (1289 A ) .
224 DATA
-
OBTAINED SINCE FEBRUARY 1977
The periods of t h e d a t a c o l l e c t e d thus f a r by LHYDREP i n t h e Lake o f Geneva
are summarized i n Table 1.
TABLE 1
Sumary of d a t a p e r i o d s c o l l e c t e d by LHYDREP (Dates given by day, month and y e a r ) No
Period
P o s i t i o n (km) X Y
Depth ( m )
-
1 2 3 4 5
1/2/77
-
1/3/77
1/3/77 25/4/77 16/6/77 - 24/8/77 31/8/77 - 8/11/77 9/12/77 - 15/2/78
528,28 531,24 538,39 542,02 542,02
149,18 147,84 148,68 147,48 147,48
73,O 198,9 293,4 289,9 289.9
To allow i n s p e c t i o n of t h e s e d a t a s e v e r a l computer programs p e r m i t t i n g analog r e p r e s e n t a t i o n o f t h e d a t a were developed a t LHYDREP. A f t e r a l l d a t a weleplotted, v i s u a l i n s p e c t i o n showed t h a t t h e l a k e has been reasonably homogeneous f o r t h e e n t i r e p e r i o d s 1 and 2 and f o r period 5 a f t e r 14/1/78.
Using average temperatures
from 5,6; 10,O; 20,O; 35,O and 55,O m depth t h e o v e r a l l mean l a k e temperature was 5,725 OC w i t h a s t a n d a r d d e v i a t i o n of t h e means of 0,020 OC f o r p e r i o d 1. Similarly, f o r period 2 , using average temperatures from 5,8; 10,2; 20,2; 84,4 and 148,9 m depth, t h e o v e r a l l mean l a k e temperature was 6,143 OC with a standard d e v i a t i o n of 0 , 4 1 2
OC.
For period 5 a f t e r 14/1/78 t h e o v e r a l l mean temperature obtained from
3,9; 8 , 6 ; 18,5; 92,7 and 195,6 m depth was 5,968 OC with a s t a n d a r d d e v i a t i o n of 0,222 OC. Typical records of t h e d a t a a r e shown i n Figures 3 and 4 f o r t h e periods 2 and 5 r e s p e c t i v e l y . I n Figures 3 and 4, t h e t i m e a x i s is h o r i z o n t a l , whereby every 24 hours a v e r t i c a l l i n e , showing t h e d a t e i n y e a r s , months, days,hours and minutes
i n d i c a t e s t h e s t a r t of a new day. The d a t a represented i n t h e topmost band a r e the temperatures of t h e water i n 5 depths corresponding t o t h e p o s i t i o n s of each current meter. Proceeding downwards, t h e n e x t band shows the wind v e l o c i t i e s and then follow t h e v e l o c i t y observations of t h e 5 currentmeters. The subsequent 6 bands show d i r e c t i o n s , t h e f i r s t being t h e d i r e c t i o n of t h e wind followed by t h e directions of t h e 5 currentmeters. The s c a l e s - shown every f i v e days - have been s e l e c t e d
such t h a t t h e temperature band extends over a range of 5-10 OC, t h e wind speed band over 0-10 m / s , over 0-360°.
t h e water v e l o c i t y bands over 0-10 cm/s and the d i r e c t i o n bands
( I n c a s e t h e s e ranges a r e exceeded by t h e d a t a t o be p l o t t e d , t h e
225
t r a c e o f one band c o n t i n u e s o v e r n e i g h b o u r i n g bands b u t k e e p s i t s o r i g i n a l s c a l e . ) I n F i g u r e s 3 and 4 i t c a n b e s e e n t h a t r e c o r d s of wind were n o t always a v a i l a b l e ( F i g u r e 3: a f t e r 9 / 3 / 7 7 ,
F i g u r e 4 : between 3/1/78 and 2 3 / 1 / 7 8 ) . Thus, wind ob-
s e r v a t i o n s o f t h e S w i s s M e t e o r o l o g i c a l S e r v i c e r e c o r d e d a t 7 , 1 3 and 19 h o u r s a t Lausanne a r e i n s e r t e d i n t h e wind bands assuminq a s i x h o u r s d u r a t i o n ( s t r a i q h t l i n e s e g m e n t s ) f o r each o f t h e s e o b s e r v a t i o n s . Comparison o f t h e LHYDREP and "Lausanne" vind d a t a ( s e e F i g u r e 3, b e f o r e 9/3/77 a f t e r 23/1/78)
and F i q u r e 4 b e f o r e 3/1/78 and
show r e a s o n a b l e agreement. Also, c u r r e n t m e t e r 4 o f F i g u r e 3 r e -
c o r d e d o n l y c u r r e n t d i r e c t i o n s and t h u s no d a t a a r e shown on t h e v e l o c i t y band of c u r r e n t m e t e r 4 of t h i s F i g u r e .
SELECTION OF PERIODS USED FOR SIMULATION
I t was s t a t e d above t h a t t h e m a t h e m a t i c a l model i s c o n c e i v e d f o r t h e f o l l o w i n s
limiting conditions:
(1) homogeneity of t h e w a t e r body, i . e . ,
t h e l a k e must h a v e a l m o s t no t e m n e r a t u r e
stratification; ( 2 ) s t a t i o n a r i t y of t h e v e l o c i t y and d i r e c t i o n of t h e wind and t h e w a t e r . The manner of s a t i s f y i n r j t h e above c o n d i t i o n s i.s d i s c u s s e d i n t h e f o l l o w i n g . S i n c e t h e " s t a t i o n a r i t y " i s more r e s t r i c t i v e we s h a l l s t a r t w i t h i t . ( a d 2 ) A c l i m a t o l o g i c a l p a r t i c u l a r i t y of t h e Leman b a s i n i s t h a t t h e r e e x i s t two i m p o r t a n t and s t r o n g w i n d s : one n o r t h - e a s t e r l y - t h e b i s e
- and one s o u t h -
w e s t e r l y - t h e v e n t ( P r i m a u l t , 1 9 7 2 ) . These winds o f t e n blow f o r u e r i o d s o f days a n d , w i t h t h e p o s s i b l e e x c e p t i o n of t h e e a s t e r n end of t h e l a k e , p r o d u c e p r o b a b l y a c o n s t a n t i n t e n s i t y and d i r e c t i o n of s h e a r on t h e l a k e s u r f a c e . S e a r c h i n g f o r s u c h s i t u a t i o n s i n p e r i o d s 2 and 5 o f T a b l e 1 ( s e e F i q u r e s 3 and 4 ) , we h a v e s e l e c t e d two t i m e p e r i o d s of d a t a , s u b s e q u e n t l y t o b e u s e d f o r a model s i m u l a t i o n . These p e r i o d s a r e : bise : vent
:
29/3/77, 2/2/78,
12h00 - 30/3/77,
24h00
OOh00 -
12h00
3/2/78,
I f ' o n e r e g a r d s t h e d i r e c t i o n of c u r r e n t s a t t h e s e l a y e r s , one f i n d s them t o b e r e a s o n a b l y c o n s t a n t ; however, t h e v e l o c i t y o f c u r r e n t s a t t h e d i f f e r e n t l a y e r s i s n o t a t a l l c o n s t a n t ( I ) . V e c t o r i a l l y a v e r a g e d v a l u e s f o r t h e v e l o c i t i e s of wind and c u r r e n t s w e r e c a l c u l a t e d as i n d i c a t e d
- F i g u r e s 3 and 4 w i t h b o l d l i n e s
and a r e summarized i n T a b l e s 2 (BISE) and 3 (VENT). ( a d 1) For t h e p e r i o d s s e l e c t e d a b o v e , i . e . b i s e and v e n t , d e n s i t y v a r i a t i o n s were c a l c u l a t e d and found t o be weak; r e s u l t s a r e g i v e n i n T a b l e s 2 and 3.
-
227 TABLE 2
BISE: Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 531,24 km and y = 147,84 km between 29/3/77, 1 2 hours and 31/3/77, zero hours 3,5 m/s Average wind v e l o c i t y a t Lausannel) : ('wind'x - (U . ) = - 8 , 4 m / s UWlnd = 9 , l m / s = 32,8 km/h wind Depth ( m )
u (cm/s)
p (g/ml)
t (°C)2)
v (cm/s)
&/Po
(cm/s)
- 14,77 - 12,84
5,6 10,2 20,2 84,4 148,9 Weighted mean with 200 m max. depth
- 1,60
-
14,86 12,89 9,61
6,287 6,293 6,320 5,962
0,999931014 0,999930803 0,999929846 0,999941645
-
9,56
1,16 0,96
-
-
-
1,l
0,02
1,lO
5,664
0,999950001
- 3,64
0.09
3,64
-
0,999942124
-
2,1101221 9r5665536 -1,1799582 -8,3563836
. lo;; .
*
'-5
. 10
-
Po =
Velocity components of wind and c u r r e n t s : u i s p o s i t i v e f o r flow towards e a s t v i s p o s i t i v e f o r flow towards north 2, S u b j e c t t o accuracy of 5 0,Ol OC according t o instrument s p e c i f i c a t i o n
')
TABLE 3
VENT:
Mean (temporal) v e l o c i t i e s and temperatures of c u r r e n t s a t x = 542,02 km and y = 1 4 7 , 4 8 km between 2/2/78 zero hours and 3/2/70, 1 2 hours Average wind v e l o c i t y a t station') : ( U . ) = 4 , 7 m / s
(;wind)y wind x = 4 , 3 m / s = 6 , 4 m / s = 22,9 km/h wind
~~
-
-
-
3,92 - 1,29 - 0,60
4,OO 1,16 1,08 2,66 2,22
0,21
2,20
0 ro
3,9 8r6 18,5 92,7 195,6 Weighted mean with 305 m max. depth
8,40 7,92
See Table 2 2 , See Table 2
9,30 8,OO
6,055 5,930 5,911
0,999938764 0,999942607 0,999943190
4,07 2,96 2,30
5,911 5,978 5,527
0,999943170 0,999941159 0,999953397
2,21
-
Po = 0,999947942
-
-3,8424000
. 10:;
-5r6352933
*
lo
. .
10-5
0
2,0113047 -1,2234736
-6
228 CALIBRATION O F MODEL
A s has been s t a t e d above, t h e two model parameters s u b j e c t t o model c a l i b r a t i o n
( i n this study) a r e t h e wind s h e a r c o e f f i c i e n t , C f ,
and t h e eddy v i s c o s i t y ,
n.
Let u s look f i r s t a t t h e i n f l u e n c e of t h e eddy v i s c o s i t y upon t h e simulated v e l o c i t i e s , keeping t h e s h e a r stress c o e f f i c i e n t constant. This is done i n Figure 5, 2 T l = 1o00, 500 and 100 cm /s f o r t h r e e ( 3 ) p o i n t s
where t h e v e l o c i t y v e c t o r s f o r a r e shown every 2,s m.
I t can be seen t h a t i n each c a s e t h e r e i s a d e v i a t i o n on
-
-
/= :
DIRECTION DU VENT I
Fig. 5. Ekman s p i r a l s a t nodes a , b and c
229
t h e water s u r f a c e of about 45O t o t h e r i g h t between t h e wind d i r e c t i o n and t h e c u r r e n t d i r e c t i o n . Then, proceeding downwards, t h e v e c t o r s diminish and t u r n t o t h e r i g h t forming the &man s p i r a l . A t a c e r t a i n depth, D , t h e vectors have turned 180° r e l a t i v e t o t h e v e c t o r s on t h e w a t e r s u r f a c e . For an i d e a l i z e d ocean of in-
f i n i t e dimensions t h i s depth, Dp,
i s given by ( D i e t r i c h e t a l . , 1975) (12)
Thus, t h e depth, D
, is
p r o p o r t i o n a l t o t h e square r o o t of rl a s seen i n Figure 5.
Furthermore, s i n c e t h e m a s s t r a n s p o r t over t h e depth, D eddy v i s c o s i t y ( D i e t r i c h e t a l . , 1975)
-
i.e.
De
1
i s independent of t h e e' vdz = c o n s t a n t - a change of D
produces a change i n t h e v e r t i c a l d i s t r i b u t i o n of t h e v e l o c i t i e s . This i s a l s o e v i d e n t i n Figure 5 . values on the simulation, it i s t o be noted f t h a t t h i s f a c t o r acts merely l i k e a s c a l e f a c t o r on the v e l o c i t y s c a l e s . A s f o r the i n f l u e n c e of t h e C
Concluding from t h e above remarks t h e following procedure i s proposed f o r c a l i b r a t i o n of t h e model:
(1) t r y i n g d i f f e r e n t values of Q ; s e l e c t Q such t h a t reasonable d i r e c t i o n a l agreement between observations and model s i m u l a t i o n i s obtained ( g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n the t o p l a y e r s ) ; ( 2 ) check t h e thus obtained rl value a g a i n s t values found i n t h e l i t e r a t u r e ;
( 3 ) f o r the chosen value of rl a d j u s t t h e wind shear s t r e s s c o e f f i c i e n t , C f , such t h a t the magnitudes o f simulated and observed v e l o c i t i e s agree reasonably w e l l ( a g a i n , g r e a t e r importance must be a t t a c h e d t o c l o s e agreement i n t h e top layers); ( 4 ) check i f t h e s e l e c t e d value o f C
agrees with t h e ones f observations o r found i n t h e l i t e r a t u r e .
c a l c u l a t e d from d i r e c t
Following t h e above procedure, t h e b i s e ( f o r d e s c r i p t i o n s e e T a b l e 2 ) w a s inv e s t i g a t e d f i r s t and is discussed herewith ( t h e same w a s a l s o done f o r t h e v e n t ) : (ad 1) Trying v a r i o u s values of 50 cm'/s
5 rl 5
2000 cm"/s,
it was found t h a t
2
rl = 500 cm / s gave d i r e c t i o n s of v e c t o r s t h a t agreed w e l l w i t h t h e observations, as it i s t o be seen i n Figure 5 (ad 2) I f t h e rlvalue i s derived from a well-accepted equation (Neumann and Pierson, 1966) : Q
= 0,1825
where
. 10-4
U
5/2
.
wind
/ p
Uwind
i s i n cm/s,
p
i s i n g/cm3 and 2 i s i n cm / s
T)
2 with a wind v e l o c i t y of 32,7 h / h , an eddy v i s c o s i t y of 460 cm / s i s obtained.
230
F i g . 6 . S i m u l a t i o n of c u r r e n t v e c t o r s f o r a B i s e , and o b s e r v a t i o n s .
Fig. 7 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Bise, and observations.
231
Fig. 8. Simulation of c u r r e n t v e c t o r s f o r a Vent, and observations RXE X l K M l
"ISCOSIlE OE TURBULENCE: 187.0 C " Z / S PRRRHETRE OE CORIOLIS: 0.000105 RROIRNSIS
RXE X l K M l
Fig. 9 . Simulation of v e r t i c a l l y i n t e g r a t e d c u r r e n t v e c t o r s f o r a Vent, and observations.
232 2 2 This i s thought t o be reasonably c l o s e t o 500 cm / s , and a value of 460 cm / s was adopted f o r f u r t h e r c a l c u l a t i o n s . (ad 3) I n o r d e r t o o b t a i n agreement of t h e magnitudes of t h e v e c t o r s , t h e wind s h e a r stress c o e f f i c i e n t , C f , w a s taken as C
f
= 0,004.
v a l u e s have been c a l c u l a t e d a t LHYDREP by P r o s t (personal communication, f 1978) f o r our measuring campaign and do indeed corroborate with our chosen values. (ad 4) C
The r e s u l t s o f this model simulation ( c a l c u l a t i o n ) are shown f o r t h e b i s e i n Figure 6 and f o r t h e v e n t i n Figure 8. Drawn a r e t h e computer c a l c u l a t e d c i r c u l a t i o n p a t t e r n s a t d i f f e r e n t l a y e r s and t h e i n s i t u measured v e l o c i t y v e c t o r s a t r e s p e c t i v e depths. Considering t h e assumptions made i n t h e mathematical simulation and t h e d i f f i c u l t i e s encountered i n an i n s i t u measuring campaign, t h e agreement i s cons i d e r e d t o b e reasonably good. Furthermore, i n Figure 7 ( b i s e ) and Figure 9 ( v e n t ) we show a comparison of depth-average v e l o c i t i e s c a l c u l a t e d and measured. Agreement i s extremely encouraging f o r t h e b i s e , b u t c e r t a i n l y less good f o r t h e vent.
(We de n o t exclude t h a t f u r t h e r
c a l i b r a t i o n could b e a p p l i e d t o reach b e t t e r agreement f o r t h e v e n t a s w e l l ) . I n t e r e s t i n g l y enough t h e model c i r c u l a t i o n p a t t e r n l e a d s u s t o agree with a conc l u s i o n r e c e n t l y drawn by Hamblin (1976):"Currents demonstrate t h e g e n e r a l tendency t o follow the wind i n t h e nearshore region, whereas t h e r e t u r n c u r r e n t opposed t o t h e wind d i r e c t i o n i s s i t u a t e d i n t h e c e n t r a l p o r t i o n of t h e l a k e " .
ACKNOWLEDGEMENT
This work was p a r t i a l l y sponsored by t h e Swiss National Science Foundation (FNSFS) under i t s s p e c i a l program "Fundamental problems of t h e water c y c l e i n Switzerland"
REFERENCES
Bauer, S.W., Graf, W.H. and T i s c h e r E . , 1977. L e s courants dans l e L6man en s a i s o n f r o i d e . Une simulation mathematique. B u l l . Techn. S u i s s e Romande, 103 :239-243. D i e t r i c h , G . , K a l l e , K . , Krauss, W . and S i e d l e r , G . , 1975. Allgemeine Meereskunde, Gebnider Borntrager, B e r l i n - S t u t t g a r t . Gallagher, R.H., L i g g e t t , J . A . and Chan, S.K.T., 1973. F i n i t e Element Shallow Lake C i r c u l a t i o n . Proc. Am. SOC. Civ. Engs., Vol. 99, NO HY7 Hamblin, P.F., 1976. Seiches, c i r c u l a t i o n , and storm surges of an i c e - f r e e Lake Winnipeg. J . Fish. R e s . Board Can., 33:2377-2391. L i g g e t t , J . A . , 1970. C e l l Method f o r computing l a k e c i r c u l a t i o n . Proc. Am. SOC. C i v . Engs., V o l . 96, No HY3. L i g g e t t , J . A . and Hadjitheodorou C . , 1969. C i r c u l a t i o n i n shallow homoqeneous l a k e s . Proc. Am. SOC. C i v . Engs., V o l . 95, No H Y 2 .
233 Neumann, G. a n d P i e r s o n , W.H. J r . , 1966. P r i n c i p l e s of p h y s i c a l oceanogranhv. P r e n t i c e H a l l , Englewood C l i f f s , N.J. P r i m a u l t , B . , 1972. E t u d e m d s o c l i m a t i a u e du Canton de Vaud, C a h i e r d e l'amenagement r d g i o n a l 1 4 , O f f i c e c a n t o n a l v a u d o i s d e l ' u r b a n i s m e , Lausanne. P r o s t , J . - P . , B a u e r , S.W., G r a f , W.H. and G i r o d , H., 1977. Campagne d e mesure des c o u r a n t s d a n s l e L h a n . B u l l . Techn. S u i s s e Romande, 103:243-249.
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235
NON-LINEAR THREE-DIMENSIONAL MODELLING OF MESOSCALE CIRCULATION IN SEAS AND LAKES,
Jacques C.J. NIHOUL',
Y. RUNFOLA and B. ROISIN
Mecanique des Fluides Geophysiques, Universite de Liege, Sart Tilman B6, B-4000 LiGge (Belgium) 'Also
at the Institut d'Astronomie et de Geophysique, Universite de
Louvain (Belgium)
ABSTRACT
Two-dimensional and one-dimensional models of mesoscale hydrodynamics are discussed with particular emphasis on the possibility of combining them to obtain a three-dimensional description of the currents in seas and lakes. A particular attention is paid to the variable eddy viscosity multimode model developed by Nihoul (1977) and a numerical generalization of this model is presented in which, by successive iterations at each time step, the non-linear advection terms are taken into account in the one-dimensional Ekman model while the parameterization of the bottom stress is, if necessary, revised in the depth-averaged two-dimensional model according to the vertical structure. An example of application to the North Sea is given in illustration.
INTRODUCTION
Mesoscale phenomena in seas and great lakes are characterized by time scales ranging from hours to days.
They encompass inertial oscil-
lations, tides, wind induced currents, storm surges and diurnal thermally induced fluctuations. Their governing equations are obtained from the general NavierStokes equations by application of the Boussinesq approximation and the quasi-hydrostatic approximation (e.g. Nihoul, 1 9 7 6 ) .
In this con-
text, taking into account the different orders of magnitude of the horizontal and vertical velocity and length scales, one can generally neglect also the components of the Coriolis force where the horizontal component of the Earth's rotation vector appears, and the terms of horizontal turbulent diffusion as compared with the vertical turbulent diffusion. The Boussinesq approximation is tantamount to assuming that the specific mass of sea water is a constant while its specific weight may be
variable; small deviations of specific mass being there multiplied by the acceleration of gravity g
,
much larger than typical accelerations
of the fluid. The variations o f the specific weight appear, in the hydrodynamic equations, as a vertical force, the "buoyancy", the magnitude of which must be regarded as an additional variable for which a supplementary equation is required. In the scope of the Boussinesq approximation, one can relate buoyancy to temperature, salinity and turbidity variations and it is generally assumed that the three governing equations for temperature, salinity and turbidity can be combined in a single equation for buoyancy - Although this is obviously feasible when only one of the three variables (usually temperature) plays a significant role in the density variations, in the general case, it constitutes an additional approximation requiring further assumptions on the turbulent diffusion coefficients and the possibility of expressing volume sources of buoyancy (like the effect of radiation) in terms of buoyancy alone. The system of equations governing mesoscale circulations, even in the simplest case of a single equation for buoyancy, is still a formidable problem. equations.
I t is a system of five non-linear partial differential
In real situations, the boundaries may be very irregular
(coasts, sea floor,
...)
and the boundary conditions are often partly
inadequate (in particular, along open sea boundaries).
The equations
contain eddy diffusion coefficients which are unknown functions
Of
space and time (and, possibly, of the velocity and buoyancy fields) and one of the first problem is to establish an adequate parameterization for them. The solution of the three-dimensional time dependent equations o f the mesoscale circulation does not seem to be possible, at this Stage, without rather severe simplifications. the paper by Freeman et a1 (1972)
-
If one takes, for instance,
one of the very few attempting to
solve the complete three-dimensional model
-,
one finds dangerously
restrictive hypotheses such as constant eddy viscosity, zero bottom stress, uniform depth (when buoyancy is taken into account), zero buoyancy (when depth's variations are included) and a constant wind stress whose relation with the wind velocity, incidently, is not correct. Confronted with the complexity of the three-dimensional model, one naturally tries to reduce its size and one turns to situations which can be described by two-dimensional or one-dimensional models.
237 With t h e i r main i n t e r e s t i n t h e v e r t i c a l s t r u c t u r e o f c u r r e n t s and density,
several authors,
a d v o c a t i n g t h e s m a l l v a l u e o f t h e Rosby
number i n m e s o s c a l e f l o w s ( O ( 1 0 - l ) ) advection terms. gible,
,
have neglected t h e non-linear
Since the horizontal diffusion terms a r e a l s o negli-
t h e r e s u l t i n g e q u a t i o n s known a s t h e Ekman e q u a t i o n s c o n t a i n
,
no d e r i v a t i v e w i t h r e s p e c t t o t h e h o r i z o n t a l c o o r d i n a t e s x1 and x2
e x c e p t f o r t h e p r e s s u r e g r a d i e n t w h i c h a p p e a r s a s a n unknown f o r c i n g
t e r m r e l a t e d e s s e n t i a l l y t o t h e atmospheric p r e s s u r e g r a d i e n t and t h e sea surface slope. Some a u t h o r s
(e.g.
Welander, 1957) have a t t e m p t e d t o f i n d an a n a l y -
t i c a l s o l u t i o n o f t h e Ekman e q u a t i o n s w h e r e f o r c i n g f u n c t i o n s l i k e p r e s s u r e g r a d i e n t and wind s t r e s s a p p e a r a s k e r n e l s o f c o n v o l u t i o n i n tegrals. O t h e r s h a v e t r i e d t o e l i m i n a t e t h e p r e s s u r e g r a d i e n t by c o n s i d e r i n g not the horizontal current, phic current
depth-averaged M o s t of
current.
t h e m o d e l s of
N i i l e r ,
(e.g.
b u t i t s d e v i a t i o n from e i t h e r a g e o s t r o -
( d e f i n e d a s t o b e d r i v e n by t h e p r e s s u r e g r a d i e n t ) o r a
the diurnal thermocline f a l l i n t h i s category
1977 ; P h i l l i p s ,
1977 ; K i t a i g o r o d s k i i ,
1979).
Along a r a t h e r s i m i l a r l i n e , one c a n a l s o d i f f e r e n t i a t e w i t h r e s p e c t t o t h e v e r t i c a l c o o r d i n a t e x3 and d e r i v e , tions,
f r o m t h e Ekman e q u a -
a complete s e t of t h r e e equations f o r t h e v e r t i c a l shear
dU
(where i s t h e h o r i z o n t a l c u r r e n t v e c t o r ) and buoyancy. dx 3 More i n t e r e s t e d i n t h e g e n e r a l c i r c u l a t i o n p a t t e r n o f a c o n t i n e n -
$ I = -
t a l s e a o r a l a k e , many a u t h o r s h a v e r e s t r i c t e d t h e i r a t t e n t i o n t o t h e h o r i z o n t a l d i s t r i b u t i o n of rents. ignored,
s u r f a c e s l o p e and depth-averaged
cur-
When t h e w a t e r c o l u m n i s w e l l m i x e d a n d b u o y a n c y c a n b e i n t e g r a t i o n i s c a r r i e d from t h e bottom t o t h e s u r f a c e .
more c o m p l i c a t e d c a s e s ,
In
s e v e r a l l a y e r s a r e t r e a t e d s e p a r a t e l y and
c h a r a c t e r i z e d by t h e i r d e p t h - a v e r a g e d
properties.
Depth-averaged
models have been e x t e n s i v e l y a p p l i e d i n t h e r e c e n t y e a r s and d e t a i l e d r e f e r e n c e s c a n be found i n numerous r e v i e w s and books 1 9 7 5 ; Cheng e t a l ,
1976 ; N i h o u l a n d Ronday,
The t w o k i n d s o f m o d e l s , averaged two-dimenslonal one-dimensional
l o c a l one-dlmensional
models,
( e . g . Nihoul,
1976). models and depth-
have t h e i r obvious l i m i t a t i o n s .
The
Ekman m o d e l s a r e n o t a p p l i c a 5 l e i n c e r t a i n r e g i o n s
( l i k e t h e v i c i n i t y o f t i d a l amphidromic p o i n t s o r i n c o a s t a l zones) where t h e n o n - l i n e a r
advection terms are n o t n e g l i g i b l e
(e.9.
Ronday,
1976). One c a n a l s o show t h a t t h e s e t e r m s m u s t b e r e t a i n e d e v e r y w h e r e i f
238 the mesoscale circulation model is to be exploited to compute the residual macroscale circulation in tidal seas like the North Sea (Nihoul and Ronday, 1975). The depth-averaged models allow only for a crude representation of stratification and give no information on the vertical profile of the horizontal current which may be rather essential in such fields as sediments transport, off-shore engineering, current meters data interpretation
...
Moreover, neither the Ekman equations nor the depth-averaged equations constitute a closed system.
At one stage or another, one-
dimensional Ekman models cannot be pursued without a knowledge of surface elevation, geostrophic or mean current, bottom stress,
...
to
materialize the results of an analytical solution or to formulate the boundary conditions, at the bottom for instance.
Two-dimensional
depth-averaged models, on the other hand, require a parameterization
of the bottom stress (introduced in the equations by the vertical integration) and classical empirical formulas in terms of the depthaveraged velocity may not be entirely satisfactory, especially in particular situations like the reversal of tides in weak wind conditions (Nihoul, 1977)
.
In fact, it is obvious that the two types of models are complementary and should be run in parallel, following some appropriate iteration procedure.
MESOSCALE HYDRODYNAMIC EQUATIONS
In the scope of the approximations described in section 1 , the equations of mesoscale marine hydrodynamics can be written
aa a t + y.
a5 a t + :.y5
u = o
a
a
+ ( u 3a) = pa + ax ax 3 3
(:a)
=
u3
at
(at ah +
( X- -)aa
(4)
ax3
x3 = 5
u.yh 3
-
(5)
u
) 3
at
x
3
= - h
239 e - a x i s i s v e r t i c a l , p o i n t i n g upwards w i t h i t s o r i g i n a t -3 t h e r e f e r e n c e s e a l e v e l and where
where t h e
el
u = u
+
u2
e2
is the horizontal current velocity vector
u3 i s t h e v e r t i c a l component o f
"'fs! f
1
aa x + 1
t h e three-dimensional
velocity vector
e a i s t h e h o r i z o n t a l Nabla o p e r a t o r - 2 ax2
i s the C o r i o l i s parameter,
t w i c e t h e v e r t i c a l component o f
the
Earth's rotation vector
+
q =
g x3
(where
is the pressure,
p
po
the constant referen-
ce d e n s i t y a n d g t h e a c c e l e r a t i o n o f g r a v i t y )
Y
i s the astronomical t i d a l force
v
i s t h e v e r t i c a l eddy v i s c o s i t y
a
i s t h e buoyancy
(a =
-
g
*
- PO) PO
Pa
i s t h e r a t e o f buoyancy p r o d u c t i o n
A
i s t h e v e r t i c a l eddy d i f f u s i v i t y f o r buoyancy
5
is the surface elevation
h
is the depth
h + 5 = H
i s the water height
DEPTH-INTEGRATED A N D MULTI-LAYER M O D E L S
The d i f f i c u l t y o f
(1) t o
solving the three-dimensional
system of
equations
( 4 ) h a s a l r e a d y been p o i n t e d o u t .
I n t h e case of
s h a l l o w well-mixed
g i b l e buoyancy and r e n o u n c i n g riations,
s e a s and l a k e s ,
t h e d e t e r m i n a t i o n of
assuming n e g l i -
t h e v e r t i c a l va-
one u s u a l l y i n t e g r a t e s t h e e q u a t i o n s o v e r d e p t h and r e s t r i c t
a t t e n t i o n t o t h e c o m p u t a t i o n o f t h e s u r f a c e e l e v a t i o n and of depth-averaged
velocity field
y.
Integration over depth,
the
however,
i n t r o d u c e s t h e bottom s t r e s s i n t o t h e equations. The b o t t o m s t r e s s
*
( p e r u n i t mass o f
sea w a t e r )
is defined a s
I n s e a s and l a k e s , t h e a s t r o n o m i c a l t i d e s a r e u s u a l l y n e g l i g i b l e w i t h r e s p e c t t o wind i n d u c e d c u r r e n t s and i n e r t i a l o s c i l l a t i o n s , o r incoming l o n g waves p r o d u c e d by p e r t u r b a t i o n s of t h e s e a s u r f a c e due t o o c e a n i c t i d e s a n d a t m o s p h e r i c d e p r e s s i o n s ( e . g . Ronday, 1976) i s then neglected. ( I n t h e North Sea, f o r i n s t a n c e , s u r f a c e e l e v a t i o n s d u e t o i n c o m i n g o c e a n t i d e s may r a n g e t o s e v e r a l m e t e r s w h i l e a s t r o n o m i c a l t i d a l e l e v a t i o n s n e v e r e x c e e d a few c e n t i m e t e r s ) .
240
:b
=
%Ix
au
['
--h 3-
and must be parameterized in terms of the mean velocity
-
u although,
from a physical point of view, it should really be expressed in terms of bottom currents. Two-dimensional depth-integrated models can be improved to give some indications of vertical variations by considering different layers.
Multi-layer models determine the depth-mean velocity of each
layer and thus provide a staircase approximation of the velocity profile.
They allow a parameterization of the bottom stress in terms
of the mean current in the bottom layer but they introduce additional approximations such as interfacial friction coefficients in the boundary conditions at the interfaces between layers (e.g. Leendertse et al, 1973). Moreover the volume of computation involved, when the number of layers increases, severely limits this number (or equivalently the number of vertical grid points in a three-dimensional attempt) and the variations in the vertical are very crudely represented. Being limited in the number of layers, it seems reasonable to define them in relation with the vertical buoyancy structure (for instance a well-mixed layer above the diurnal thermocline and a stratified layer below).
Unfortunately, with this definition, the inter-
faces between layers are not fixed and how they vary in time is very often poorly known and, in any case, very difficult to take into account (e.g. Cheng et al, 1 9 7 6 ) .
I n m o s t m e s o s c a l e phenomena, o n e can r e g a r d t h e n o n - l i n e a r a d v e c t i o n t e r m s a s s m a l l e x c e p t , p r o b a b l y , i n ZocaZized r e g i o n s where e x c e p t i o n n a l l y h i g h v e l o c i t i e s o r r a p i d s p a t i a l v a r i a t i o n s s i g n i f i c a n t Z y i n c r e a s e t h e i r order o f magnitude. I f these terms are neglected,
t h e mesoscaZe h y d r o d y n a m i c equa-
t i o n s can be t r a n s f o r m e d i n many d i f f e r e n t ways i n t o a one-dimensional system. Some t y p i c a Z o n e - d i m e n s i o n a l m o d e l s a r e b r i e f t y
sc
discussed i n
t h e f o l l o w i n g w i t h p a r t i c u l a r e m p h a s i s on t h e p o s s i b i Z i t y o f
*The
review has no ambition of being exhaustive and, far from drawing up an inventory, one intends 'to focus one's attention on a limited number of papers which serve to illustrate the discussion. In particular, studies of diurnal thermal fluctuations and the dynamics of the thermocline, which are not the main concern of this paper, will not be considered. An extensive review of them is presented in this volume by Kitaigorodskii.
241
c o m b i n i n g s u c h m o d e l s w i t h t w o - d i m e n s i o n a l o n e s t o o b t a i n t h e full three-dimensional picture. VERTICAL SHEAR MODELS Differentiating eq.(l) tronomical force
* B ,
with respect to
x3
and neglecting the as-
one gets
where
au l & = -
ax
3
is the vertical shear vector.
Eq.(8)
and eq.(4)
(linearized) constitute a closed system for
w
and a. If one excepts estuarine and similar regions where horizontal gradients of buoyancy (related to horizontal salinity gradients, for instance) may play an important part, it is customary to neglect the horizontal gradient of a "
,
ho r i z on t a 11y homo g en eo u s "
regarding locally the marine system as
.
In that case, eq.(8) can usually be solved for
a
,
X
the eddy diffusivity
The velocity field
u
5 and eq.(4) for
being eventually a function of
can then be derived from
tant of integration" (actually a function of x 1
g
.
((*[I
within a'kons-
, x 2 and t) which
depends o n the general circulation in the area. The same result can be obtained by considering a "geostrophic current"
u
where
q'
-g
independent of depth and solution of the equation
, in the hypothesis of horizontal homogeneity and after,
integration of eq. ( 3 ) , is given by (11)
where pa is the atmospheric pressure. The unknown forcing term the velocity difference
u
yq
-
can then be eliminated by studying
sg .
Obviously the geostrophic current plays the same role as the "constant of integration" mentioned above.
*
Vertical variations of (e.g. Ronday, 1 9 7 6 ) .
are, in any case, entirely negligible
242 This type of dels
(e.g.
a p p r o a c h h a s b e e n u s e d e x t e n s i v e l y i n t h e r m o c l i n e mo-
Niiler,
1977
;
Phillips,
1977
Kitaiqorodskii,
;
1979).
The d i f f i c u l t y h e r e r e s i d e s i n t h e e x p r e s s i o n o f t h e b o u n d a r y c o n ditions.
For i n s t a n c e ,
%
t h e v a l u e of
a t t h e bottom i s r e l a t e d t o
t h e b o t t o m s t r e s s w h i c h i s e i t h e r unknown o r p a r a m e t e r i z e d i n t e r m s of t h e depth-averaged
!!
velocity
bottom w i l l r e q u i r e t h a t
5
.
The n o - s l i p
u
from w r i t i n g a f o r m a l a n a l y t i c a l s o l u t i o n , unknown f u n c t i o n o f
ANALYTICAL
xl,
x2 and t
( -u
or y )
=
-
a t the
.
t o determine separately.
9
MODELS
Assuming p s e u d o - h o r i z o n t a l l i n e a r advection t e r m s ,
h o m o g e n e i t y a n d n e g l e c t i n g t h e non-
o n e c a n , w i t h more o r l e s s r e a s o n a b l e
,
t i o n s on t h e e x p r e s s i o n of t h e e d d y v i s c o s i t y v c a l solution of e q . ( l )
Y'*
condition
- u
part u9 -9 i n b o t h c a s e s , t h e r e i s an
be z e r o , i . e .
assump-
d e r i v e an a n a l y t i -
i n t e r m s o f t h e unknown f o r c i n g t e r m
q
(and
i f needed). Y*
If
i s n e g l e c t e d , eq. (1) t a k e s ,
i n these conditions, the s i m -
p l e form
where e q . (11) h a s b e e n u s e d . Eq. ( 1 2 ) i s known a s t h e Ekman e q u a t i o n . A well-known
s o l u t i o n of
t h i s t y p e i s t h e model o f Welander
Neglecting t h e s p a t i a l v a r i a t i o n s of
a
v
and assuming c o n s t a n t v e r t i c a l eddy v i s c o s i t y a n a l y t i c a l s o l u t i o n of e q . ( 1 2 ) ,
t = 0
for
=
where
,
Welander
s e e k s an
with the i n i t i a l conditions
a t t h e bottom "S
= 0
and t h e boundary c o n d i t i o n s
0 T
(1957).
and t h e f o r c i n g f u n c t i o n
i s t h e wind s t r e s s
(per u n i t m a s s of
O b t a i n e d by s u p e r p o s i t i o n o f t o Heaviside
(14) sea water).
elementary solutions corresponding
s t e p f u n c t i o n s f o r c i n g terms
(e.g.
Hidaka,
1933), the
f i n a l s o l u t i o n o f W e l a n d e r a p p e a r s a s t h e sum o f t w o c o n v o l u t i o n i n t e g r a l s with respective kernels
q(t)
and
T "S
(t)
.
The v e l o c i t y p r o f i l e d e t e r m i n e d b y W e l a n d e r d e p e n d s t h u s on t h e time h i s t o r y of
the atmospheric pressure,
surface elevation.
Obviously,
t h e wind s t r e s s and t h e
t h e l a t t e r must be d e t e r m i n e d ,
some
243
way o r a n o t h e r , b e f o r e o n e c a n e x p l o i t t h e r e s u l t i n p r a c t i c a l a p p l i cations. In r e a l i t y ,
t h e a n a l y t i c a l s o l u t i o n o f t h e Eckman e q u a t i o n m u s t
be r e g a r d e d a s a f i r s t s t e p , p a v i n g t h e way t o a c c u r a t e t w o - d i m e n s i o n a l modelling.
Jelesnianski,
for instance,
( J e l e s n i a n s k i , 1970)
o b t a i n s t h e same s o l u t i o n a s W e l a n d e r by a p p l i c a t i o n o f t h e L a p l a c e transform.
From t h e v e l o c i t y p r o f i l e ,
mean v e l o c i t y
-
u
T~
i n terms of
depth-averaged As
and
9
and
T “S
q
,
E l i m i n a t i n g t h e convo-
which can be u s e d i n a s u b s e q u e n t
q
using a s i m p l i f i e d version of J e l e s n i a n s k i ‘ s
1974), t h e two-dimensional
depth-averaged
u(x3)
.
s h o r t c o m i n g o f t h e Welander-Jelesnianski-Foristall approach
i s t h e s e v e r e h y p o t h e s i s made on t h e v e r t i c a l e d d y v i s c o s i t y
c o n s t a n t ) t o o b t a i n an a n a l y t i c a l s o l u t i o n .
(
v =
This hypothesis i s i n
c o n t r a d i c t i o n w i t h t h e o b s e r v a t i o n s and i n d e e d t h e models f a i l reproduce
model
and t h e l a t t e r can be s u b s t i t u t e d i n J e l e s n i a n s k i ’ s
formula t o determine t h e v e l o c i t y p r o f i l e The m a i n
.
one c a n d e r i v e an e x p r e s s i o n
model.
integrals (Foristall, gives
5
shown b y F o r r i s t a l l ,
-
L~
and t h e b o t t o m s t r e s s
l u t i o n i n t e g r a l w i t h unknown k e r n e l for
he d e r i v e s e x p r e s s i o n s f o r t h e
to
c o r r e c t l y t h e b o t t o m b o u n d a r y l a y e r c h a r a c t e r i s t i c s and t h e
b o t t o m s t r e s s t u r n s o u t t o b e a l i n e a r f u n c t i o n o f t h e mean v e l o c i t y and n o t a q u a d r a t i c one,
a s it should.
The same o b j e c t i o n c a n b e made t o a l l m o d e l s c o n s t r u c t e d a l o n g t h e same l i n e
:
t o o s e v e r e h y p o t h e s e s , made a t t h e b e g i n n i n g , h a n d i c a p
t h e whole model an d t h e e x p l o i t a t i o n o f tuations. (1969)
the results i n r e a l i s t i c si-
One c a n r e f e r h e r e t o t h e m o d e l o f L i g g e t t a n d H a d j i t h e o d o r o u
(rectangular lake,
c o n s t a n t eddy v i s c o s i t y ,
a t m o s p h e r i c p r e s s u r e g r a d i e n t ) , Gedney a n d L i c k v e r s i o n of t h e p r e c e d i n g one
s t e a d y s t a t e , no
(1972) an improved
( c o n s t a n t eddy v i s c o s i t y ,
n o a t m o s p h e r i c p r e s s u r e g r a d i e n t ) , W i t t e n a n d Thomas
steady s t a t e ,
(1976)
(steady
s t a t e , no a t m o s p h e r i c p r e s s u r e g r a d i e n t and a n u n r e a l i s t i c and r a t h e r u n f o r t u n a t e e x p o n e n t i a l form f o r t h e eddy v i s c o s i t y ) .
MULTI-MODE
MODELS
Generalizing the concept of v e r t i c a l integration,
Heaps
(1972) sug-
g e s t e d t h a t t h e v e r t i c a l v a r i a t i o n s c o u l d be e l e g a n t l y t a k e n i n t o acc o u n t by e x p a n d i n g t h e v e l o c i t y the turbulent operator d dx (v
3
d -)
dx3
u
i n s e r i e s o f e i g e n f u n c t i o n s of
The p h i l o s o p h y o f t h i s a p p r o a c h c a n b e s u m m a r i z e d a s f o l l o w s s u b s t i t u t i n g t h e s e r i e s expansion
i n t h e Ekman e q u a t i o n ,
:
one g e t s a
system o f e q u a t i o n s f o r t h e c o e f f i c i e n t s , e a c h of which i s a f u n c t i o n of
,
x
x2
and
t
and
s a t i s f i e s a two-dimensional
many ways s i m i l a r t o t h e t w o - d i m e n s i o n a l
equation,
in
equation of a depth-averaged
mode 1. The f i n a l r e s u l t i s o b t a i n e d by s u p e r p o s i n g t h e s o l u t i o n s o f twodimensional models, mode
each o f which c o r r e s p o n d s t o a d i f f e r e n t v e r t i c a l
( a s compared w i t h t h e m u l t i - l a y e r model where s e v e r a l two-dimen-
s i o n a l m o d e l s a r e s o l v e d s i m u l t a n e o u s l y , e a c h o f them c o r r e s p o n d i n g to a different vertical layer). T h i s m e t h o d w a s a p p l i e d t o t h e I r i s h S e a by H e a p s a n d J o n e s A s l i g h t l y more
s o p h i s t i c a t e d v e r s i o n w a s u s e d by D a v i e s
n u m e r i c a l e x p e r i m e n t ( r e c t a n g u l a r sea b a s i n o f
(1975).
(1977) i n a
constant depth) with
t h e o b j e c t of t e s t i n g t h e s e n s i t i v i t y of t h e r e s u l t s t o v a r i o u s parameters. The d i f f i c u l t y h e r e r e s i d e s i n t h e p a r a m e t e r i z a t i o n viscosity tions.
v
of t h e eddy
and t h e d e t e r m i n a t i o n o f t h e c o r r e s p o n d i n g e i g e n f u n c -
Heaps and h i s co-workers
were f o r c e d t o i n t r o d u c e s e v e r a l v
s i m p l i f i c a t i o n s r e g a r d i n g e s p e c i a l l y t h e eddy v i s c o s i t y i n d e p e n d e n t of x3 and a f u n c t i o n of ;/h of
xl,
i s a c o n s t a n t ) and t h e bottom s t r e s s
x2 and
t
(taken a s a l i n e a r function
t h e bottom v e l o c i t y assumed d i f f e r e n t from z e r o , ncondition)
V A R I A B L E E D D Y VISCOSITY
in contradiction
.
with the obvious no-slip
MULTI-MODE
(assumed
such t h a t the ratio
MODEL
Most m o d e l s d e s c r i b e d s o f a r c a n b e l a b e l l e d
"1D
t h a t t h e y s e e k , by a p r e l i m i n a r y o n e - d i m e n s i o n a l
+
2D" i n t h e sense
model,
some a p p r o -
p r i a t e f o r m u l a t i o n of t h e v e r t i c a l dependence o f t h e v e l o c i t y f i e l d t o use s u b s e q u e n t l y i n a complementary two-dimensional
model.
a l m o s t i n e v i t a b l e h y p o t h e s i s o f c o n s t a n t eddy v i s c o s i t y , liminary analytical calculations,
t u r n s o u t t o be a t e r r i b l e embar-
r a s s e m e n t when e x p l o i t i n g t h e r e s u l t s o f t h e o n e - d i m e n s i o n a l two-dimensional
modelling a s ,
The
i n the pre-
model i n
w i t h an i n i t i a l l y i n c o r r e c t r e p r e s e n t a -
t i o n o f t h e b o t t o m b o u n d a r y l a y e r a n d o f t h e b o t t o m s t r e s s , o n e may doubt whether t h e r e w i l l b e , u l t i m a t e l y , r e a l
i m p r o v e m e n t s i n t h e two-
dimensional forecast.
*
. I n h i s n u m e r i c a l e x p e r i m e n t , D a v i e s ( 1 9 7 7 ) was a b l e t o r e d u c e t h e s e v e r i t y o f some o f t h e a s s u m p t i o n s b u t t h e n , e v e n w i t h a s i m p l e p i e c e w i s e l i n e a r e d d y v i s c o s i t y , h i s s o l u t i o n i s e n t i r e l y n u m e r i c a l a n d the p h y s i c a l i n s i g h t o f t h e e i g e n f u n c t i o n e x p a n s i o n s i s somewhat o b s c u r e d .
245 In a recent paper on tides and storm surges in well-mixed seas, Nihoul ( 1 9 7 7 ) approached the problem in a contrary direction and recognizing that two-dimensional depth-integrated models,carefully calibrated,have been successfully applied to many lakes and seas and that the results are available, he suggested a locally one-dimensional multi-mode model using the predictions of the preexisting two-dimensional depth-integrated model to provide the local values of the forcing terms and of the boundary conditions. Indeed, a depth-integrated model provides, at any point where one might desire the vertical current profile, the local surface elevation, mean velocity and associated bottom stress. The only difficulty here is that, in two-dimensional models, the bottom stress is parameterized by a quadratic formula assuming that the stress is (except :or
a small wind stress correction) in the di-
rection of the mean velocity.
This type of parameterization is one
of the things one would like to verify by a three-dimensional model. Although the quadratic law may be generally applicable, one suspects that it could be faulty in certain situations, such as tide reversals, when the mean current becomes very small. However, it is readily seen that, the local wind stress being known, the two-dimensional model provides two additional boundary conditions instead of one for the second order Ekman equation (the velocity and the stress at the bottom).
Nihoul (1977) thus derives an analytical
solution of the local Ekman equation as a series of eigenfunctions of the turbulent operator (15) using the surface stress bottom stress
-
T
as boundary conditions.
'I
- S
and the
The velocity deviation
-b is obtained as a functional of T and T~ and the addi'S tional boundary condition ( 9 = - u at the bottom) is used to deterand to verify the twomine the relationship between T~ and
a -
= y
-
y
dimensional parameterization. "2D
In this model, which could be labelled
+ lD", the necessary matching of the one-dimensional and the two-
dimensional models requires a more realistic parameterization of the eddy viscosity which is taken as a function of
t, x l , x2
and
x
3
and,in particular,respects the observed asymptotic form of the eddy coefficient in the bottom boundary layer. The application of the model to the North Sea shows that,for typical values of tides and storm surges,the classical bottom friction law i s valid over most of the tidal cycle but fails, in magnitude and in direction, during a comparatively short period of time,at tide reversal.
This may be regarded as a validation of the depth-integrated
model, the result o f which can be used to determine the velocity
246 p r o f i l e a t a n y g r i d p o i n t where t h e i n f o r m a t i o n might be r e q u e s t e d (off-shore problem,.
structure,
current-meters
mooring,
sediment
transport
.. ) .
There i s however,
i n p r i n c i p l e , no d i f f i c u l t y i n s e t t i n g , f o r i m -
proved numerical f o r e c a s t i n g , a n
i t e r a t i o n p r o c e s s by which t h e c o r -
r e c t e d r e l a t i o n s h i p between
and
d i m e n s i o n a l model
;
-
etc..
,
5
t o f e e d back b e t t e r v a l u e s of model,
T~
i s i n t r o d u c e d i n t h e two-
t h e two-dimensional
-
model b e i n g r u n a s e c o n d time
and T
i n t o t h e one-dimensional
-b
.
One c a n go a s t e p f u r t h e r a n d i n c l u d e t h e n o n - l i n e a r
advection
I n t h i s way, t h e c o m b i n e d
terms i n the i t e r a t i o n process.
2D
i
1D
model i s a p p l i c a b l e e v e r y w h e r e an d r e s u l t s i n a t r u l y n o n - l i n e a r three-dimensional
model o f m e s o s c a l e c i r c u l a t i o n .
T h i s model i s d e s c r i b e d i n t h e n e x t s e c t i o n .
NON-LINEAR
THREE-DIMENSIONAL
MODEL O F MESOSCALE C I R C U L A T I O N
I N WELL
MIXED SHALLOW SEAS
one s h a l l r e s t r i c t
For s i m p l i c i t y ,
a t t e n t i o n t o well-mixed
s e a s l i k e t h e North Sea and assume t h a t buoyancy i s n e g l i g i b l e
w i l l a l s o be n e g l e c t e d .
astronomical t i d a l force
shallow
*.
The
(The j u s t i f i c a -
t i o n i s given i n section 2 ) . I n t h e i n t e g r a t i o n of
eq.(l)
over depth,
t h e non-linear
advection
t e r m g i v e s t w o c o n t r i b u t i o n s r e l a t e d r e s p e c t i v e l y t o t h e p r o d u c t of t h e mean v a l u e s a n d t h e mean p r o d u c t o f t h e d e v i a t i o n s a r o u n d t h e means.
The s e c o n d c o n t r i b u t i o n i s r e s p o n s i b l e f o r a n e n h a n c e d h o r i -
zontal diffusion cient)
( c o m p a r a b l e t o t u r b u l e n c e b u t g e n e r a l l y more e f f i -
called "shear effect" diffusion
(e.g.
Nihoul,
1975).
Shear e f f e c t d i f f u s i o n i s e s s e n t i a l i n t h e depth-integrated persion equation of
a passive
its turbulent equivalent,
i t r e m a i n s g e n e r a l l y n e g l i g i b l e i n t h e mo-
mentum e q u a t i o n a s c o m p a r e d w i t h t h e e f f e c t s o f and t h e s u r f a c e g r a d i e n t
(e.g.
Ronday,
1976).
t y i n i n c l u d i n g t h e s h e a r e f f e c t i n a 2D i n the following,
*The
dis-
a l t h o u g h it i s l a r g e r than
scalar but,
f o r t h e sake of
+
the Coriolis force There i s no d i f f i c u l -
1D m o d e l .
It
i s excluded
simplicity.
e x t e n s i o n o f t h e m o d e l t o s t r a t i f i e d s e a s i s now u n d e r i n v e s t i gation. I t r e q u i r e s a more s u b t l e p a r a m e t e r i z a t i o n o f t h e e d d y visc o s i t y w h i c h m u s t b e a l l o w e d t o v a r y w i t h t h e R i c h a r d s o n number a n d , a t l e a s t , o n e more i t e r a t i o n l o o p i n t r o d u c i n g t h e e f f e c t o f b u o y a n c y i n t h e p r e s s u r e g r a d i e n t and i n t h e two-dimensional model.
247 S u b s t r a c t i n g t h e depth-averaged
e q u a t i o n from e q . ( l ) , one can de-
.
r i v e an e q u a t i o n f o r t h e v e l o c i t y d e v i a t i o n this stage, x
t o change v a r i a b l e from
It i s convenient,
at
to
x3
+ h
5=- 3
(16)
H The v a r i a b l e
5
i s t a k e n t o v a r y from
0
z
take the rugosity length lower l i m i t a t
50
=
The e q u a t i o n f o r
zo G--
9
<<
to
1 , provided
this
i n w h i c h c a s e one m u s t
does n o t c r e a t e a s i n g u l a r i t y a t t h e bottom
e x p l i c i t l y i n t o a c c o u n t and s e t t h e
7
(In
5
?1
- 10
can t h e n be w r i t t e n
i n t h e North S e a ) .
(Nihoul,
1977)
where
the subscript s denoting surface values.
I n h i s s t u d y o f t i d e s and s t o r m s u r g e s i n t h e North S e a , Nihoul ( 1 9 7 7 ) h a s shown t h a t t h e n o n - l i n e a r n e g l i g i b l e and t h a t ,
terms
a c c o r d i n g t o o b s e r v a t i o n s , t h e eddy v i s c o s i t y
was w e l l r e p r e s e n t e d by a n e q u a t i o n o f
where
5
t i o n of
the
behaving a s y m p t o t i c a l l y a s
form
X
i s t h e Von Karman c o n s t a n t a n d
K
were a l m o s t everywhere
a non-dimensional
5
for s m a l l
5
.
(The eddy
v i s c o s i t y i n t h e b o t t o m l a y e r r e d u c e s t h e n t o t h e well-known layer expression z
= x3
+
v =
K
u*
z
where
uy
t i c a l s o l u t i o n i n t h e form o f a f u n c t i o n a l of
-53 T
and
?b
A*,
Nihoul
a multi-mode
(1977) d e r i v e d an a n a l y expansion giving
t h e North Sea
$(5)
as
*
Using t h e r e s u l t s o f t h e two-dimensional
depth-integrated
model o f
( R o n d a y 1 9 7 6 ) , he c o m p u t e d t h e e v o l u t i o n w i t h t i m e of
the velocity profile
?(t
,
xl,
x 2 , x3)
a t a s e r i e s of
ve h o r i z o n t a l g r i d p o i n t s i n t h e S o u t h e r n B i g h t . u = -
boundary
i s t h e f r i c t i o n v e l o c i t y and
h).
Assuming a p a r a b o l i c l a w f o r
-
func-
$(C0)
representati-
The e x t r a c o n d i t i o n (20)
*I t
was shown b y R o i s i n ( 1 9 7 7 ) t h a t t h e f i n a l r e s u l t s w e r e r a t h e r i n s e n s i t i v e t o t h e e x a c t form o f X ; the cogent f a c t o r being i t s asymptotic behaviour A ‘b 5 for 5 << 1
.
248
was used as a test of consistency
;
a significant discrepancy indica-
ting a local influence of the non-linear terms or a temporary failure of the parameterization of effects for small
-u
in the depth-integrated model (memory
1.
At all the grid points where the study was conducted, Nihoul (1977) found very little difference between the mean velocity calculated by the depth-integrated model and calculated by the local multi-mode solution. However, no point was taken in danger zones near coasts or tidal amphidromic points and the model was not tested for other regions than the Southern Bight and the North Sea or for different types of mesoscale circulations or climatic conditions.
The necessity was
thus felt of a generalization of the model based on an iteration procedure by which the non-linear terms could be included in the onedimensional local Ekman model while the parameterization of the bottom stress could be revised in the two-dimensional model according to the calculated vertical structure.
-
u1
The two-dimensional model gives the components
and
-
u2
of
5 at the points of a
the mean velocity and the surface elevation staggered horizontal grid
U
ul-
5
0
X
2+
The integration of the equation for the velocity deviation made at the velocity (Ax)'
u
<-point
.
is
At that point, the two components of the mean
are calculated by linear interpolation
(the error is in
and consistent with the order of the two-dimensional scheme)
.
To compute the non-linear terms which contain horizontal derivatives of
ii
nine
, the velocity deviation must be calculated simultaneously at S-points of the two-dimensional grid.
[In the application to
the North Sea, these points form a rectangle of At each time step
t
ted numerically setting
+
At f?
,
20
a first approximation
= ~'(t)
km x 26 km) iil
.
is calcula-
.
This approximation, combined with the results of the two-dimensional model, is then used to compute 'i
is then calculated setting
pl(t
5
=
+ At)
&
.
A second approximation
(~'(t + A t )
+ pl(t))
.
With the
second a p p r o x i m a t i o n ,
+
n2(t
i s computed and t h e i t e r a t i o n con-
At)
tinues. A t
the i n i t i a l time,
n = 0
t h e f i r s t approximation
and t h e r e s u l t o f t h e a n a l y t i c a l model can be u s e d .
The n u m e r i c a l m e t h o d i s a n e x t e n s i o n o f thod
developed
b y Adam
t h e c o m p a c t h e r m i t i a n me-
(1976).
A s boundary c o n d i t i o n s f o r t h e determination
t h e s u r f a c e s t r e s s and t h e bottom s t r e s s t h e two-dimensional
-
u = -
i s obtained taking
G
model).
of
13
,
one imposes
( t h e l a t t e r o b t a i n e d from
The c o n s i s t e n c y c o n d i t i o n
a t t h e bottom
indicates i f
t h e model c a n p r o c e e d t o t h e n e x t t i m e s t e p o r i f
l i m i n a r y i t e r a t i o n must be p e r f o r m e d on t h e p a r a m e t e r i z a t i o n
a pre-
of the
bottom stress. As
shown by N i h o u l
( 1 9 7 7 ) , t h e c l a s s i c a l bottom f r i c t i o n law cons-
t i t u t e s t h e a l g e b r a i c p a r t o f a d i f f e r e n t i a l r e l a t i o n where t h e add i t i o n a l terms
(containing derivatives with respect t o the t i m e )
n e g l i g i b l e a s l o n g a s t h e mean v e l o c i t y friction are sufficiently large.
A t
-
small v a l u e s of
t h e mean v e l o c i t y
h o w e v e r t h e c o r r e c t i o n may become c o m p a r a t i v e l y i m p o r t a n t a n d , instance,
for
t h e r e s u l t i n g b o t t o m s t r e s s may h a v e a d i f f e r e n t d i r e c t i o n
f r o m t h e mean v e l o c i t y v e c t o r , A t
are
and t h e a s s o c i a t e d bottom
small velocities,
even i n t h e a b s e n c e of wind.
the correction f o r the non-linear
i s n o t n e c e s s a r y and t h e r e s u l t of
terms
p
t h e a n a l y t i c a l m u l t i m o d e model c a n
be used t o correct f o r t h e bottom stress p a r a m e t e r i z a t i o n .
-g
The r e l a t i o n s h i p b e t w e e n
and
i s r a t h e r complicated but Roisin
zb
g i v e n by t h e m u l t i - m o d e m o d e l
( 1 9 7 7 ) h a s shown t h a t i t c o u l d be
w r i t t e n i n much more c o n v e n i e n t a n d s i m p l e f o r m s a n d t h a t , good a p p r o x i m a t i o n ,
where
D
t i o n law
and (D
m
are t h e c o e f f i c i e n t s of t h e c l a s s i c a l bottom f r i c -
i s t h e d r a g c o e f f i c i e n t ) and where
factor. If
with a
one could t a k e
i s a numerical
-
u = - 0
the consistency condition,
t i s f i e d a t a given t i m e s t e p
t
,
eq.(21)
r a t i o n scheme w h e r e t h e c o r r e c t e d v a l u e o f t e d i n t e r m s of t h e simultaneous value of d i a t e p a s t h i s t o r y of
y
g
()?(t-At)).
a t t h e bottom,
i s not sa-
can be introduced i n a i t e 'I
-b
a t time and
1,
t
i s compu-
a n d t h e imme-
A P P L I C A T I O N TO THREE-DIMENSIONAL M O D E L L I N G OF T I D E S A N D STORM SURGES I N THE NORTH S E A
The t h r e e - d i m e n s i o n a l
model w a s a p p l i e d t o t h e c a l c u l a t i o n o f t i d e s
a n d s t o r m s u r g e s i n t h e N o r t h S e a . The t w o - d i m e n s i o n a l
depth-integrated
m o d e l w a s r u n f o r t h e w h o l e N o r t h S e a w i t h t h e h o r i z o n t a l g r i d u s e d by Ronday
(1976).
The v e r t i c a l p r o f i l e o f
t h e v e l o c i t y v e c t o r was c a l -
c u l a t e d a t a series of p o i n t s i n t h e Southern Bight.
The t e s t p o i n t s
were s e l e c t e d f o r t h e i r r o l e i n t h e e x i s t i n g management p r o g r a m o f t h e N o r t h S e a a n d c o r r e s p o n d e d t o p l a c e s w h e r e c u r r e n t m e t e r s w e r e t o be moored o r w h e r e t h e v e r t i c a l v a r i a t i o n s o f t o study t h e d e p o s i t i o n of etc..
t h e c u r r e n t were d e s i r e d
s i l t , t h e d i s t r i b u t i o n of
e g g s and l a r v a e ,
.
A l t h o u g h some o f t h e s e p o i n t s w e r e c l o s e t o t h e c o a s t o r i n r e l a t i v e l y s h a l l o w i r r e g u l a r d e p t h s , v e r y f e w i t e r a t i o n s t a g e s were f o u n d n e c e s s a r y i n most c a s e s . I n many p l a c e s , non-linear
t h e r e i s no s i g n i f i c a n t im p ro v e m e n t i n t a k i n g t h e
advection t e r m s i n t o account i n t h e c a l c u l a t i o n of
and t h e v a r i a b l e eddy v i s c o s i t y multi-mode
model
it
( N i h o u l 1977) appears
t o give very s a t i s f a c t o r y r e s u l t s . This i s i l l u s t r a t e d ,
i n the following,
t a t i o n a t t h e p o i n t 52O30'N
3"50'E.
Fig.
by t h e r e s u l t s o f
t h e compu-
1 shows t h e n u m e r i c a l g r i d
f o r t h e t w o - d i m e n s i o n a l model o f t h e S o u t h e r n B i g h t o f t h e N o r t h Sea 4 4 0 m ) . The t e s t p o i n t i s i n d i c a t e d by a c i r c l e . ( A x l = 1 0 m ; ~ x ~ = 1 . 3 91 5
v;
sm7. ...............
I.
Fig.
................... .................. ............ . .......... .......... . . . . . . . . . .. . . . . . . .......... ...... ................ .......... .......... ...... .......... . , X X X . .
I
$;
. . . . . . . . . . . . . . . . . .\
1 . N u m e r i c a l g r i d f o r t h e t w o - d i m e n s i o n a l d e p t h i n t e g r a t e d model showing t h e t e s t p o i n t and t h e wind d i r e c t i o n .
251 22 m
The d e p t h t h e r e i s
The w i n d s t r e s s i s o r i e n t e d t o t h e N o r t h - E a s t lower r i g h t - h a n d of
corner of
1.10-3 m
the rugosity length
(C0
5
2,
a s indicated in the
t h e f i g u r e a n d i t h a s a maximum m a g n i t u d e
.
2 10-~m~sec-2
In this exercise, of t h e n o n - l i n e a r
t h e i t e r a t i o n was r e s t r i c t e d t o t h e i n t r o d u c t i o n
advection terms.
No c o r r e c t i o n was made on
emphasize t h e comparison w i t h t h e a n a l y t i c a l multi-mode 1977).
Figs.
2-7
model
T b to (Nihoul,
show t h e r e s u l t s o f t h e m o d e l i n w h i c h t h e n o n -
l i n e a r a d v e c t i o n t e r m s w e r e i n c l u d e d by s u c c e s s i v e i t e r a t i o n s a t e a c h time s t e p .
One n o t i c e s o n f i g s .
6 and 7 t h a t ,
near tide reversal,
b o t t o m c u r r e n t s a n d s u r f a c e c u r r e n t s may f l o w i n o p p o s i t e d i r e c t i o n s a n d t h a t t h e b o t t o m s t r e s s m a i n t a i n e d by t h e c u r r e n t i n t h e b o t t o m l a y e r i s t h e n d i f f i c u l t t o r e l a t e t o t h e mean c u r r e n t w h i c h may become very small.
0.1
0.2
0.3
0.4
0.5
0.6 u (m sec I
Fig.2.
-I\
Evolution with t i m e over t h e f i r s t h a l f t i d a l period of t h e e a s t e r n component of t h e h o r i z o n t a l v e l o c i t y v e c t o r (u,) The c u r v e s f r o m r i g h t t o l e f t a r e v e r t i c a l p r o f i l e s c o m p u t e d a t 54' interval.
.
252 E
0.1
Fig. 3 .
0.2
0.3
0.4
0.5
0.6
Evolution with time, over the second half tidal period, of the eastern component of the horizontal velocity vector (u 1 The curves from right to left are vertical profiles compute& at 54' interval.
0.1
0.2
0.3
0.4
0.5
0.6
u2
(m
s-')
Fig. 4. Evolution with time, over the first half tidal period of the northern component of the horizontal velocity vector ( ~ 2 ) The curves from right to left are vertical profiles computed at 54' interval.
.
253
Fig.
5.
u (m s e c - 1 ) 2 Evolution with time, over t h e second h a l f period of t h e n o r t h e r n component o f t h e h o r i z o n t a l v e l o c i t y v e c t o r ( u 2 ) . The c u r v e s f r o m r i g h t t o l e f t a r e v e r t i c a l p r o f i l e s c o m p u t e d at 54' interval.
0.1
Fig.
6.
0.2
0.3
E v o l u t i o n w i t h t i m e , a t t i d e r e v e r s a l o f t h e e a s t e r n compon e n t of t h e h o r i z o n t a l v e l o c i t y v e c t o r (u,) The c u r v e s from r i g h t t o l e f t a r e v e r t i c a l p r o f i l e s computed a t 18' interval.
.
254
u 2 (m sec 0.1
Fig.
ii(m
7.
-1
E v o l u t i o n w i t h t i m e , a t t i d e r e v e r s a l o f t h e n o r t h e r n component of t h e h o r i z o n t a l v e l o c i t y v e c t o r (u,) The c u r v e s from r i g h t t o l e f t a r e v e r t i c a l p r o f i l e s computed a t 18' interval.
.
s-') T
Fig.
8.
)
0.3
0.2
E v o l u t i o n w i t h t i m e o ve r one t i d a l p e r i o d o f t h e magnitude o f t h e mean v e l o c i t y 5 computed r e s p e c t i v e l y by t h e unc o r r e c t e d t w o - d i m e n s i o n a l d e p t h i n t e g r a t e d model ( f u l l l i n e -1, t h e l i n e a r l o c a l model ( d a s h l i n e ---) and t h e nonl i n e a r l o c a l model ( d o t s . . . )
.
255
'7 10
DEGREES
Fig.
9.
E v o l u t i o n w i t h t i m e o v e r one t i d a l p e r i o d o f t h e d i f f e r e n c e b e t w e e n t h e d i r e c t i o n s o f t h e mean v e l o c i t y c o m p u t e d r e s p e c t i v e l y by t h e u n c o r r e c t e d t w o - d i m e n s i o n a l m o d e l a n d by t h e non l i n e a r m o d e l .
Figs.
1 0 a n d 1 1 show t h e v e e r i n g o f
o v e r t h e w a t e r Column. o c c u r s above there
is
5
%
0.1
about the
and t h e u p p e r l a y e r
.
T h i s i s n o t obvious on t h e
variable
the veering
f i g u r e s because
same number o f p o i n t s i n t h e l o w e r l a y e r
0.1
5 5 5
t h e n u m e r i c a l method which, change of
the horizontal velocity vector
I t must be n o t e d h e r e t h a t most o f
1
.
T h i s i s a c t u a l l y an a r t i f i c e of
for increased accuracy,
y = l n (5c )
5 I< 0 . 1
i n t h e bottom l a y e r
introduces the
( E 0 5 5 5 0.15)
0
where t h e v e r t i c a l g r a d i e n t s a r e l a r g e . fig.
12.
The same r e m a r k a p p l i e s t o
256
+ + + +
+ + + +
+ + + +
++ ++
++
+++I
++++++~+++++
.
*
_
I
UZ(rn Sec-')
Fig.
10. V e r t i c a l v e e r i n g o f t h e h o r i z o n t a l v e l o c i t y v e c t o r at tide
reversal
+ + + +
+
.- -
0.1
+
1
u,<m
Fig. 1 1 .
-
0.3
leC
)
Vertical veering of the horizontal veloclty vector after tide reversal
18'
257
T
0.5
Fig.
-
~
0.6
0.7
1 2 . E v o l u t i o n w i t h t i m e o v e r t h e f i r s t h a l f t i d a l p e r i o d of t h e Ekman d i a g r a m s h o w i n g t h e v e r t i c a l v e e r i n g o f t h e h o r i z o n t a l velocity vector. The s e p a r a t i o n b e t w e e n two s u c c e s s i v e c u r ves i s 18'
.
Fig. large,
1 2 shows t h a t ,
a s l o n g a s t h e mean v e l o c i t y i s s u f f i c i e n t l y
t h e h o r i z o n t a l v e l o c i t y v e c t o r k e e p s t h e same d i r e c t i o n f r o m
t h e bottom t o t h e s u r f a c e .
In that case,
t h e classical bottom f r i c -
t i o n l a w a p p l i e s a n d i t may b e a s s u m e d t h a t t h e b o t t o m s t r e s s i s roughly ty.
( e x c e p t f o r a s m a l l w i n d e f f e c t ) p a r a l l e l t o t h e mean v e l o c i -
On t h e o t h e r h a n d ,
tide reversal,
when t h e mean v e l o c i t y b e c o m e s s m a l l , n e a r
there i s a noticeable rotation of the velocity vector
between i t s d i r e c t i o n s i n t h e bottom and t h e s u r f a c e l a y e r s .
258 During a comparatively short period of time, one may thus expect the classical parametrization of the bottom stress to fail both in magnitude and direction.
As described above, this may be corrected at
each time step in the critical interval by an iteration procedure. The iteration on the non-linear terms does not, on the other hand, appear to bring significant improvement in the example considered. The effect may of course be larger at other grid points but, in general, it seems to constitute only a minor correction. In large-scale practical applications of the three-dimensional model, one would thus be wise to define realistically limiting values of + $ ( C 0 ) and which one is willing to tolerate and instruct the model to proceed without any vainly expensive iteration whenever the calculated values do not exceed the limits of tolerance. REFERENCES Adam, Y., 1977. Highly accurate compact implicit methods and boundary conditions. J. Comput. Phys., 24:lO-22. Cheng, R.T., Powell, T.M. and Dillon, T.M., 1976. Numerical models of wind driven circulation in lakes, Appl. Math. Modelling, 1:141-159. Davies, A.M., 1977. The numerical solution of the three-dimensional hydrodynamic equations, using a B-spline representation of the vertical current profile. In: J.C.J. Nihoul (Editor), Bottom Turbulence. Elsevier Publ. Co, Amsterdam, 1-48. Forristall, G.Z., 1974. Three-dimensional structure of storm generated currents. J. Geophys. Res., 79:2721-2729. Freeman, N.G., Hale, A.M. and Danard, M.B., 1972. A modified sigma equations'approach to the numerical modelling of great lakes hydrodynamics. J. Geophys. Res., 7:1050-1060. Gedney, R.T. and Lick, W., 1972. Wind-driven currents in lake Erie. J. Geophys. Res., 77:2714-2723. Heaps, N.S., 1972. On the numerical solution of the three-dimensional hydrodynamical equations for tides and storm surges. Mem. SOC. R. Sci. Liege, 2:143-180. Heaps, N.S., 1975. Storm surge computations for the Irish Sea using a three-dimensional numerical model. Mem. SOC. El. Sci. Liege, 7: 289-333. Hidaka, K., 1933. Non-stationary ocean currents. Part I. MBm. Imp. Mar. Obs. Kobe, 5:141-266. Jelesnianski, C.P., 1970. Bottom stress time-history in linearized equations of motion for storm surges. Mon. Weather Rev., 98:462478. Kitaigorodskii, S.A., 1979. Review of the theories of wind-mixed layer deepening. In: J.C.J. Nihoul (Editor), Marine Forecasting. Elsevier Publ. Co., Amsterdam, 1-33. Leendertse, J.J., Alexander, R.C. and Liu, S., 1973. Rand Report R-1417 - OWRR. Liggett, J.A. and Hadjitheodorou, C., 1969. Ann. SOC. Civil Eng. J. Hydr. Div., 95:609-617. Niiler, P.P., 1977. One-dimensional models o f the seasonal thermocline. In: E.D. Goldberg, I.N. McCave, J.J. O'Brien, J.M. Steele (Editors)! The Sea. Wiley Interscience Publ., New-York, 6:97-115.
259
Nihoul, J.C.J., 1975. Modelling of marine systems. Elsevier Oceanography Series 10, Elsevier Publ. Co., Amsterdam, 272 pp. Nihoul, J.C.J., 1976. Modeles mathematiques et Dynamique de 1'Environnement. Ele Publ. Co., Liege, 198 pp. Nihoul, J.C.J., 1977. Three-dimensional model of tides and storm surges in a shallow well-mixed continental sea. Dyn. Atmos. Oceans, 2: 29-47. Nihoul, J.C.J. and Ronday, F.C., 1975. The influence of the tidal stress on the residual circulation. Tellus, 27:484-489. Nihoul, J.C.J. and Ronday, F.C., 1976. Hydrodynamic models of the North Sea, Mem. S O C . R. Sc. Liege, 10:61-46. Phillips, O.M., 1977. Entrainment. In: E.B. Kraus (Editor), Modelling and Prediction of the Upper Layer of the Ocean. Pergamon Press, Ch.7. Roisin, B., 1977. Modeles tri-dimensionnels des courants marins. Rep. ACN 3 . Ministry for Science Policy Brussels, 124 pp. Ronday, F.C., 1976. Modeles hydrodynamiques. In: J.C.J. Nihoul (Editor), Publ. Ministry for Science Policy Brussels, 3 , 270 pp. Welander, P., 1957. Wind action on a shallow sea : some generalizations of Ekman's theory, Tellus, 9:45-52. Witten, A.J. and Thomas, J.H., 1976. Steady wind-driven currents in a large lake with depth-dependent eddy viscosity. J. Phys. Oceanogr. 6:85-92.
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261
IRREGULAR-GRID FINITE-DIFFERENCE TECHNIQUES FOR STORM SURGE CALCULATIONS FOR CURVING COASTLINES
W. C. THACKER Atlantic Oceanographic and Meteorological Laboratories, National Oceanic and Atmospheric Administration, Miami, Florida, 33149, U.S.A.
ABSTRACT
Finite-difference computations on irregular grids offer the advantages of resolving the coastal curvature and of allowing for an explicit time step of optimal size. A further advantage is the flexibility associated with editing the grid, which allows for improvements in its design. The techniques are based on the approximation of partial derivatives with slopes of planar surfaces associated with triangular components of the irregular grid. Simulations of several surges associated with various hypothetical hurricanes passing through Mobile Bay are realistic and relatively noise free. Because these techniques are computationally efficient, they provide a practical tool for forecasting storm surges.
INTRODUCTION
Previous approaches to the problem of including curving coastlines into calculations of storm surges have involved either the use of transformed coordinates (e.g., Wanstrath et al., 1976) or the use of finite-element techniques (e.g., Pinder and Gray, 1977).
The method suggested here is to perform finite-difference calculations
directly on an irregular grid that can smoothly represent the coastline.
This offers
the advantages of greater flexibility than the coordinate transformation method and greater economy than the finite-element method. The plan of this presentation is, after first discussing features of the irregular computational grid, the approximation of derivatives, and the equations used for calculating storm surges, to present the results of computations for several hypothetical storms in the forms of sea surface elevation contour maps, flow vector maps, and graphs of water elevations along the coastline, with concluding remarks concerning the use of irregular grids in conjunction with the problem of inundation.
THE IRREGULAR GRID
The computational grid for Mobile Bay, shown in Fig. 1, illustrates some of the features that make irregular-grid finite-difference techniques attractive for use in forecasting storm surges. Most obvious is the excellent representation for the curving coastline by the irregular grid; this is a significant improvement over the
262
M E 8 ILE BPY H N C V I
Fiq.
1.
C I Iu I T y
I r r e q u l a r q r i d f o r computiny s t o r m .;urges
I
i n Mobile Bay.
" s t a i r - s t e p " boundary of c o n v c n t i o i i a l uniform g r i d s because i t i s a t the c o a s t l i n e t h a t t h e f o r e c a s t v a l u e o f tlie w a t e r s u r f a c e e l e v a t i o n is most n e c d e d .
S i n c e one
conCinuous g r i d s e r v e s f o r b o t h thc i n l a n d w a t e r s arid f o r t h e open . l i o n s can be madc s i m u l t a n e o u s l y f o r b o t h r e g i o n s , t h c , r e b y a v o i d i n g t h e question of how properly t~o s p e c i f y boundary c o n d i t i . o n s whcre t h e i n l a r i d w a t e r s meet t h e open sea.
The f a c t t h a t t h e g r i d s p a c i n g i s r o u g h l y p r o p o r t i o n a l to t h e s q u a r e r o o t o f
t h e b a s i n d e p t h (E'iq. 2 ) r e f l e c t s t h e f a c t t h d t t h e s t o r m s u r q e i s e s s e n t i a l l y a s h a l l o w w a t e r wave w i t h phase v e l o c i t y y r o p o r t i o i i a l t o the square r o o t of the w a t e r depth.
It
is t h e r a t i o of 1-he g r i d s p a c i n g to t h e basin depth whii:h
l i m i t s the size
of tiic e x p l i c i t c o m p u t a t i o n a l t i m e s t e p i n o r d e r t o ijuardritcc n u m e r i c a l s t a b i l i t y .
Unlike t h e u n i f o r m g r i d , f o r which t h e t i m e s t e p i s d e t e r m i n e d i n tlie d e e p e s t w a t e r
( w h e r c t h i s r a t i o i s minimum and which i s t h e l e a s t i m p o r t a n t reijiort f o r t h e s t o r m
263
DEPTH CONTOURS (Meters)
Pig.
2.
C o n t o u r s of s t i l l w a t e r d e p t h v a l u e s u s e d i n s t o r m surge c a 1 c : u l a t i o n s .
s u r i j e f o r e c a s t ) , t h e i r r c ? y u L d r g r i d h a s a t i m e s t e p which i s n o s m a l l e r tharl rlecess a r y f o r c a l c u l a t i o n s i r i i n 1 and w a t e r s .
T h i s d i s t r i b u t i o n o f g r i d p o ~ n t sh a s t h e
a d d i t i o n a l advantaqt. t h a t i t q i v e s t h c h i q h c s t d e n s i t y o f p o i n t s a l o n g t h e s h o r e l i n e , a i d i n g i n t h e r e s o l u t i o n o f coastal c u r v a t u r e . w i s e u n i f o r m s p l i c e d q r i d , such as the one f o r the Elbe E s t u a r y
1 9 7 6 ) shown i n F'iq. 3 , h a s t o some e x t e n t the same a t t r a c t i v e f e a t u r e s .
(Ramillq,
I t allows
for c a l c u l a t i o n s s i m u l t a n e o u s l y f o r i n l a n d w a t e r s a n d f o r t h e oywn sea, arld t h e l a r q e r g r i d spdciriq in d e e p e r water h a s a b e n e f i c i a l e f f e c t
011
t h e t i m e step; however, t h e
y r i d r e f i n i ? m e r r t scheme d i c t a t e s t h e derisit-y of g r i d p o i n t s i n eac:h l ' o r t i o n c > 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
<,
the values of the wind velocity and pressure gradient can be calculated in order to evaluate the storm forcing,
at the na
time step.
The computations are initial-
ized with a flat motionless sea, and the storm remains at the initial position for one hour ( 2 4 time steps) as it grows linearly to full strength. For numerical stability the Coriolis term is evaluated at the time level, n+%. The bottom friction and advection terms involve both levels, n+4 and n-li.
However,
the equations can easily be rewritten with the values at step n+i given explicitly in terms of values at steps n and n-li.
Also in order to guarantee numerical stability,
the advection terms are omitted for points on the open boundaries.
271
Fig. 9.
Points c, d, e, f, and a are neighboring points for boundary point i = b. The direction of flow at this p i n t is parallel to the unit vector along a line through the neighbors a and c which are also on the boundary.
COMPUTATIONAL RESULTS
Storm surges which might result from various hypothetical storms have been simulated in order to ascertain that the irregular-grid finite-difference techniques are indeed capable of producing reasonable results for realistic circumstances. Four cases are considered, corresponding to the series of Figs., 10, 11, 12, and 13. The first case corresponds to a hurricane with R = 56 km and AP = 100 mb, moving from (2g030'N, 88O3O'W) to (30°40'N, 88OW) in three hours and continuing inland on the same course for two additional hours.
Although this pressure drop might be rea-
sonable for extremely strong hurricanes in this area, the radius and forward speed are both larger than would be expected. Calculations for this case neglected the Coriolis and advection terms and used the stillwater rather than the total depths. Fig. 10 shows a contour map of sea surface elevation after three hours of simulation with the storm just north of Mobile Bay.
Since the line segments composing these
contours represent linear interpolations of the computed values at the grid points with no smoothing, the smoothness of the contour lines accurately reflects the low level of noise in the computations.
The corners of the islands, which correspond to
curvature that is too great to be resolved by the boundary points of the grid, might be responsible for some of this computational noise.
This level of noise is typical
of these calculations and does not seem to increase on the time scale of these simulations. The second case corresponds to a large hurricane which might be expected to hit Mobile Bay.
It has R = 24 km and AP = 100 mb, and it moves due north from (29°30'N,
88OW) to (30°30'N, 88OW) in five hours and continues north for two more hours. this case, the Coriolis term is included and the total depths are used.
For
The Coriolis
term has little effect and can be dropped for computational efficiency. The total depth does give somewhat diferent results than the stillwater depth, especially for
T I M E S l E P 72 H-VALUES
01;
150
T I M E OF D R Y 3
HOURIS1
AP-IOOnb
Fig. 10.
The smoothness of these surface elevation contours is indicative of the low level of noise in these computations.
extreme water levels.
NO attempt has been made to guarantee that the total depth is
always positive, but this is usually the case, even though the calculations can lead to negative elevations greater in magnitude than the stillwater depth of the basin. -t
Figs. lla through lld show vector maps of depth-averaged velocity (Ui/Di) for times corresponding to 2, 4, 5, and 7 hours of simulations for case two.
The symbol
§ marks the position of the hurricane on the map, but it is absent from Fig. lld be-
cause the hurricane is too far north to be represented.
The smooth variation of the
vectors from grid point to grid point are another indication of the low noise level. in spite of the high curvature of the islands and the constraint that the flow vectors be parallel to the computational boundary, the flow through the inlets is well represented.
The flow through the open computational boundary also seems to be quite
273 reasonable. The fact that the corner points of the grid, where land and sea boundaries meet, are taken to be points in the sea with no flow restriction accounts for the unusual behavior of the velocity vectors at these points.
This is easily cor-
rected but should have little influence on the rest of the calculation. Figs. lle through llh show the variation in computed water levels for points within Mobile Bay and Mississippi Sound and along the Gulf of Mexico coast. cated by letters, A-X,
Points on the map indi-
are also indicated on the horizontal axes of the graphs.
It is interesting to note in Fig. lla that the hurricane has moved ahead of the oceanic gyre.
A
corresponding gyre is not evident in subsequent maps because of the
strong effect of the land boundaries on the flow.
Another interesting point (Fig.
119) is that when the storm is directly over the bay, the water levels are negative in most of the bay, because the storm had pushed the water into the Gulf and Mississippi Sound (Figs. lla and l l b ) . Case three corresponds to exactly the same storm as for case two.
For this case
the Coriolis term is neglected and the stillwater depth is used, just as for case one, but the advection terms are also included. The advection terms seem to introduce some noise into the computation, but they are small and have little other effect, so it seems best to neglect these terms.
For this case the water elevations at the
coastline are shown in Figs. 12a through 12d, which can be compared with Figs. lle through llh. Case four corresponds to a storm of the same size and strength as for case two and to the same terms used in the computations. For this case the storm moves towards the east from (30°15'N, 88O45'W) to (30°15'N, 87O45'W) in four hours (Fig. 13a). Figs. 13a-13c show that as much as 5 m of storm surge might be expected within Mobile Bay and Mississippi Sound for such a storm.
CONCLUSION
The irregular-grid finite-difference techniques presented here provide a simple and economical means for forecasting storm surges in bays, estuaries, and lakes, so long as the elevation of the adjacent land is sufficiently high that inundation is unimportant.
It might be further anticipated that an irregular grid might also be
advantageous if inundation is included in the calculations. Rather than for flooding to proceed square by square as for a uniform grid, it would proceed triangle by triangle.
Since the grid points can be positioned according to the elevation Of the land, the triangles should more nearly approximate areas being flooded. For situations in which roads through low-lying areas are built up, forming barriers to the intruding water, the triangular grid might be especially appealing. How to Proceed
with such calculations, including inundation, remains to be studied.
Fig. l l a .
Flow v e c t o r s after two hours for a l a r g e hurricane which might be expected t o move from south t o n o r t h through Mobile Bay. These r e s u l t s were obtained n e g l e c t i n g t h e advection terms i n t h e m o m e n t u m equations.
Fig. l l d .
A f t e r seven hours.
Y
Fig. l l d .
A f t e r seven hours.
Fig. l l d .
A f t e r seven hours.
PRESSURE OROP-
RADIUS= 29.1KH
TIHE STEP 168 UV-VALUES DT= 150 TlHL OF O R I 7
lOO.ns
HOURfSI
-3
N 4
278
-5 ii
J
§
Fig.
lle.
Fig. Ilf.
Shoreline water e l e v a t i o n s a f t e r two hours.
After f o u r hours
219
MOBILE BAY
swr, IPO
Fig. l l g .
A f t e r f i v e hours.
Fig. l l h .
A f t e r seven hours.
IIIIL, I ~
I
S
I
GULF COAST
SlEP, I20
l l M i 5 HaURl5l
280
-5 II
1
Fig. 12a.
Shoreline water elevations to be compared with those in Fig. lle. Although these results include the advection terms and neglect the Coriolis terms, the principal differences in the two sets of results are due to the fact that these involve the approximation of the total depth by the stillwater depth.
-s " J
Fig. 12b.
-I II J
Compare with Fig. llf.
281
s*l
MOBILE BHY
91 -I
Fig. 12c.
Compare w i t h Fig. llg.
IIIOlUS*2Y.I
KM
PRESSME onor.iw.orn VELOCIlI.S.2 111) M I X UIYO SPEFO. 61.11111
M O B I L E BRY
S l L P t 168
1lMEn I
511
-II
Fig. 12d.
I
Compare w i t h Fig. llh.
ROlMISI
MISSISSIPPI S O U N D 1
GULF CORST
s i m IM
IIIIL~I
norno1
282
-s I
I
Fig. 13a.
Shoreline water elevations after one hour for a large hurricane which might be expected to move from west to east through Mississippi Sound and Mobile Bay.
Fig. 13b.
After three hours.
283
MOBILE BAY 5111
1
-III
1
Fig. 13c.
-III
1
After five hours.
ACKNOWLEDGEMENTS
Thanks are due to Gerald Putland and to Alicia Gonzales for their enthusiastic assistance, without which this presentation would not have been possible, and to Joan Wagner for her expert typing and retyping.
REFERENCES
Jelesnianski, C. P., 1967. Numerical computations of storm surges with bottom stress. Mon. Wea. Rev., 95:740-756. Overland, J. E., 1975. Estimation of hurricane storm surge in Apalachicola Bay, Florida. NOAA Tech. Rep. NWS 17, U. S . Dept. of Commerce, 66 pp. Pinder, G. F. and Gray, W. G., 1977. Finite Element Simulation in Surface and S u b surface Hydrology. Academic Press, London, 295 pp. Ramming, H-G., 1976. A nested north sea model with fine resolution in shallow coastal areas. (Seventh Lisge Colloquium on Ocean Hydrodynamics) Mgmoires de la Soci&; Royale des Sciences de Liege, X:9-26. Thacker, W. C., 1976. A spliced numerical grid having applications to storm surge. NOAA Tech. Memo. ERL AOML-26, U. S . Dept. of Commerce, 19 pp. Thacker, W. C., 1977. Irregular grid finite-difference techniques: simulations of oscillations in shallow circular basins. J. Phys. Oceanogr., 7:284-292. Thacker, W. C., 1978a. Comparison of finite-element and finite-difference schemes. Part 1: One-dimensional gravity wave motion. J. Phys. Oceanogr., 8:676-679. Thacker, W. C., 197833. Comparison of finite-element and finite-difference schemes. Part 2: Two-dimensional gravity wave motion. J. Phys. Oceanogr., 8:680-689. Wanstrath, J. J., Whitaker, R. E., Reid, R. 0. and Vastano, A. C., 1976. Storm surge simulation in transformed coordinates. Vol. I. Theory and Application. U. S . Army, Corps of Engineers Tech. Rept. No. 76-3, 166 pp.
This Page Intentionally Left Blank
285
RECENT STORM SURGES I N THE IRISH SEA
N.S.
HEAPS and J . E .
JONES
I n s t i t u t e of Oceanographic Sciences, Bidston Observatory, Birkenhead, Merseyside, England.'
ABSTRACT The t i d a l and meteorological c o n d i t i o n s a s s o c i a t e d with some r e c e n t very l a r g e Surges generated during a period of storm surges i n t h e I r i s h Sea are described. t e n days i n November 1977 a r e i n v e s t i g a t e d dynamically using a v e r t i c a l l y - i n t e g r a t e d f i n i t e - d i f f e r e n c e model of t h e I r i s h Sea. Deductions a r e made concerning t h e possib i l i t i e s of surge p r e d i c t i o n f o r t h i s a r e a using a numerical model.
INTRODUCTION
Major storm surges occurred i n t h e I r i s h Sea i n January 1976 and, more r e c e n t l y , i n November 1977.
During each p e r i o d , e x c e p t i o n a l l y high water l e v e l s were
experienced a t Liverpool and a t o t h e r c o a s t a l l o c a t i o n s i n t h e north-eastern Sea ( f i g u r e 1 ) . region
-
A s a result,
Irish
s e r i o u s c o a s t a l flooding occurred i n p a r t s of t h a t
p a r t i c u l a r l y so i n November 1977.
The p r e s e n t paper d e s c r i b e s t h e meteoro-
l o g i c a l and t i d a l c o n d i t i o n s under which t h e s e surges were generated.
Emphasis i s
placed on t h e 1977 surge e v e n t s and a two-dimensional n m e r i c a l model of t h e I r i s h Sea i s used t o examine them i n d e t a i l .
A n e a r l i e r study (Heaps and Jones,
1975)
i n v e s t i g a t e d t h e l a r g e I r i s h Sea storm surges of January 1965 using a threedimensional model based on the same g r i d network a s that used here. Lennon (1963) showed t h a t major storm surges on t h e west c o a s t of t h e B r i t i s h
Isles can be a s s o c i a t e d with A t l a n t i c secondary d e p r e s s i o n s which move i n towards t h e c o a s t c r o s s i n g eastwards over t h e B r i t i s h Isles a t a c r i t i c a l speed of about 40 knots.
The depression t r a c k s f o r a number of l a r g e surges a t Avonmouth and
Liverpool were p l o t t e d and shown t o l i e within q u i t e d i s t i n c t i v e approach zones corresponding r e s p e c t i v e l y t o t h e s e p o r t s .
Using an a n a l y t i c c o n t i n e n t a l s h e l f
model, Heaps (1965) found t h a t t h e a r e a of t h e C e l t i c Sea t o t h e south of I r e l a n d
i s a n important a r e a f o r t h e generation of W e s t Coast surges, p a r t i c u l a r l y those which i n f l u e n c e t h e B r i s t o l Channel with Avonmouth a t i t s head ( f i g u r e 1 ) .
The wind
286 f i e l d s of t h e r e s p o n s i b l e secondary d e p r e s s i o n s , sweeping landwards a c r o s s t h e C e l t i c Sea, f o r c e w a t e r towards t h e c o a s t where a consequent rise i n sea l e v e l occurs producing a surge.
There i s evidence f o r a quarter-wave
t i d a l resonance
a c r o s s t h e C e l t i c Sea i n t o t h e B r i s t o l Channel which may e x p l a i n t h e occurrence o f e s p e c i a l l y l a r g e surges t h e r e when t h e meteorological system moves a t t h e c r i t i c a l speed of 40 knots (Fong and Heaps, 1978). The shallow north-eastern area of t h e I r i s h Sea, i n c l u d i n g Liverpool Bay, i s p a r t i c u l a r l y s u s c e p t i b l e t o storm surges.
Heaps and Jones (1975) have shown t h a t
winds over t h e i n t e r i o r of t h e I r i s h Sea, a l s o externally-generated
surge d i s t u r -
bances passing i n t o t h e Sea through S t George's Channel and t h e North Channel, both s i g n i f i c a n t l y a f f e c t surge l e v e l s i n t h i s north-eastern region of Liverpool i s s i t u a t e d .
-
i n which t h e Port
The p r e s e n t paper t a k e s t h e study of I r i s h Sea surges
a s t a g e f u r t h e r by considering t h e r e c e n t l a r g e surges of November 1977, i n v e s t i Comparisons are made
g a t i n g t h e i r p r e d i c t a b i l i t y i n terms of a dynamical model. with t h e surges of January 1976 and January 1965.
TIDES AND SURGES AT LIVERPOOL
A major surge peak o f 1.42
m occurred a t Liverpool a t 0 1 - 0 0 h on 12 November
1977, two hours a f t e r p r e d i c t e d t i d a l high water.
The s u p e r p o s i t i o n of surge and
t i d e t o give t h e e l e v a t i o n of t h e sea s u r f a c e ( t o t a l w a t e r l e v e l ) i s shown f o r s e v e r a l hours covering t h i s e v e n t i n f i g u r e 2 .
A maximum e l e v a t i o n of 6 m above
ODN (Ordnance Datum Newlyn), c o n s i s t i n g of 5 m of t i d e and 1
m of surge, was
a t t a i n e d on t h e t i d a l high water a t 23.09 h on 11 November. An even l a r g e r surge peak of 1.47 m occurred a t 19.00 on 14 November.
However,
a s i s apparent i n f i g u r e 2 , t h i s happened an hour o r so b e f o r e t i d a l low water and t h e r e f o r e t h e e v e n t posed no t h r e a t of c o a s t a l flooding.
Nevertheless, a rathe'r
high water l e v e l of 5.65 m was r e g i s t e r e d on t h e preceding t i d a l high water. For comparison, f i g u r e 3 i n d i c a t e s t h a t a very l a r g e surge peak of 2.14 m occurred a t 23.00 h on 2 January 1976, one hour before t i d a l high water.
But i n
t h i s c a s e t h e t i d e s were smaller and while maximum water l e v e l ( a t 23.20 h) w a s c e r t a i n l y high, 5.73 m, it w a s perhaps n o t a s high a s one might have expected w i t h such a l a r g e surge. Figure 4 shows t h a t a l a r g e surge peak of 1.77 m occurred almost on t i d a l low water a t 02.00 h on 14 January 1965; t h e neighbouring maximum w a t e r l e v e l reached 4 m.
A s i n d i c a t e d i n t h e f i g u r e , t h e r e was another s i g n i f i c a n t surge peak of
1.43 m a t 17.15 on 17 January J965 about one hour b e f o r e t i d a l low water.
On t h i s
occasion, t h e water l e v e l reached 5.11 m on t h e preceding t i d a l high water. A l l t h e water l e v e l s quoted above a r e measured with r e s p e c t t o ODN,
below mean sea l e v e l a t Liverpool.
0.27 m
Concern i s with p o s i t i v e ( r a t h e r than negative)
surges and t h e i r e f f e c t i v e n e s s i n c o n t r i b u t i n g t o t h e r e a l i s a t i o n of abnormally
281
Fig. 1.
Sea areas on the West Coast of t h e B r i t i s h I s l e s .
288
HIGH WATER AND MAXIMUM WATER LEVEL
NOVEMBER 1977
(23-09)
DAY 12
NOVEMBER
-5i
DAY 14
I
1977
DAY 15
Fig. 2. Sea l e v e l s a t Liverpool f o r 11-12 November and 14-15 November 1977 i n d i c a t i n g 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 ; -X-x-xt i d e , -.-m-osurge, t o t a l water l e v e l .
-o-o-o-
289
MAXIMUM WATER LEVEL
JANUARY 1976
( 23.20)
DAY 2
DAY 3
-51
Fig. 3 . Sea l e v e l s a t Liverpool f o r 2-3 January 1976 i n d i c a t i n g t i m e s of occurrence of t h e surge peak and t h e a s s o c i a t e d high water and maximum water level. Notation as i n f i g u r e 2.
high s e a l e v e l s .
It i s clear f r o m t h e above d i s c u s s i o n t h a t t h e 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 time of a surge a r e as important a s surge h e i g h t i t s e l f i n determining an abnormally high water l e v e l .
Figures 5, 6 and 7 p l o t t i d a l high and low water
l e v e l s a t Liverpool throughout November 1977, January 1976 and January 1965, t h e
t i m e s of surge peaks being i n d i c a t e d by arrows.
The surges of November 1977
occurred a t a t i m e of high s p r i n g t i d e s , a f a c t o r c o n t r i b u t i n g t o t h e exceptionally high water l e v e l a t t a i n e d on t h e 11th of t h e month.
On t h e o t h e r hand, t h e surge
of 2 January occurred a t lower s p r i n g t i d e s and near a t i d a l high water reduced by t h e d i u r n a l i n e q u a l i t y so t h a t even though t h e surge peak was l a r g e r than t h a t on 12 November t h e maximum water l e v e l a t t a i n e d w a s not a s g r e a t .
The l a r g e surge
peaks on 14 November 1977 and 17 January 1965 came a t s p r i n g t i d e s , b u t both peaks occurred near t o t i d a l low w a t e r and t h e r e f o r e could i n no way c o n t r i b u t e t o t h e development of an e s p e c i a l l y high sea l e v e l .
METEOROLOGICAL CONDITIONS
Figure 8 shows t h e t r a c k s of t h e c e n t r e s of t h e depressions which gave r i s e t o t h e I r i s h Sea surges described i n t h e preceding s e c t i o n . approach zones d e f i n e d by Lennon (1963)
-
The t r a c k s l i e within
marked o u t by dashed l i n e s i n t h e f i g u r e .
290
m
HIGH
MAXIMUM WATER
SURGE PEAK
I
3
JANUARY 1965
HIGH WATER NEAREST SURGE
(02.001
4
5
.
. 6
. 7
. 8
, 9
HOUR OF DAY
>
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.
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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
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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.
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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+<)
h
undisturbed w a t e r depth
P
d e n s i t y o f s e a water, assumed uniform
R
radius of t h e Earth
g
a c c e l e r a t i o n due t o g r a v i t y
w
angular speed of r o t a t i o n of t h e Earth
-
(1)
( 3 ) a r e depth-averaged e q u a t i o n s w r i t t e n i n s p h e r i c a l p o l a r co-ordinates.
The component d i r e c t i o n s a r e those of i n c r e a s i n g x,$; i . e . t o t h e east and t o the north respectively. Adopting a q u a d r a t i c law, r e l a t i n g bottom s t r e s s t o depth mean c u r r e n t , w e w r i t e
F(B) = kpu(u2
+
v')',
G(B) = kpv(u'
+
4
v2)
,
(4)
where k , t h e c o e f f i c i e n t of bottom s t r e s s , t a k e s t h e value 0.0025. of wind stress, F"),
G"),
The components
and g r a d i e n t s of atmospheric p r e s s u r e supply t h e meteoro-
l o g i c a l f o r c i n g which d r i v e s t h e surge motion.
They are s p e c i f i e d e x t e r n a l l y a s
described i n t h e n e x t s e c t i o n . Equations ( 1 ) - ( 4 ) a r e t o be solved s t a r t i n g from a prescribed i n i t i a l d i s t r i b u t i o n of e l e v a t i o n and motion
5
= c(x,$,t),
u
= u(x,$,t),
(5)
v = v ( X , $ , t ) a t t = t0'
A t a c o a s t l i n e t h e component of t h e depth-mean
c u r r e n t along t h e outward d i r e c t e d
normal t o t h e boundary, q , is r e q u i r e d t o vanish.
On open sea boundaries a
g e n e r a l i s e d r a d i a t i o n c o n d i t i o n i s employed t a k i n g t h e form q =
4M + c4. +; 1 1
where c = (gh)';
(5 - 0,
-
$ti'
(6)
5, and 4, a r e c o n t r i b u t i o n s , a s s o c i a t e d with t h e meteorologically 0. and 4 . are s i m i l a r
d r i v e n motion, t o t h e t o t a l e l e v a t i o n and c u r r e n t 5 and q;
388
60°N
I0"W
0"
IOOE
Figure 1. F i n i t e d i f f e r e n c e mesh of t h e c o n t i n e n t a l s h e l f sea model with g r i d p o i n t s ( x ) of t h e 10-level model of t h e atmosphere.
c o n t r i b u t i o n s a s s o c i a t e d with t h e i t h c o n s t i t u e n t of t h e t i d e .
With
t,,
,$ ,
ti
and $. given, equation (6) r e l a t e s 5 and q, seeking t o prevent t h e a r t i f i c i a l r e f l e c t i o n a t t h e open boundaries of d i s t u r b a n c e s generated within t h e model. The surge i n p u t on the boundary, the n e x t s e c t i o n .
c,, 4M, i s
The t i d a l i n p u t ,
ti, Gi,
s p e c i f i e d e x t e r n a l l y as described i n i s derived from o f f s h o r e measurements
(Cartwright, 1976) and model experiments ( F l a t h e r , 1976a), a s i n d i c a t e d by Davies and F l a t h e r (1977).
Only t h e two l a r g e s t t i d a l c o n s t i t u e n t s , M2 and sg, a r e included.
Equations ( 1 ) - ( 4 ) are now replaced by f i n i t e d i f f e r e n c e approximations which allow t h e numerical s o l u t i o n , d e f i n e d on a staggered s p a t i a l g r i d , t o be developed from t h e p r e s c r i b e d i n i t i a l s t a t e (5) by means o f a simple one-step e x p l i c i t t i m e integration.
The open boundary condition (6) i s used t o determine t h e normal com-
ponent of c u r r e n t q from p r e s c r i b e d v a l u e s of
4,,
4i, OM, 5,
e l e v a t i o n 5 derived from t h e equation of c o n t i n u i t y .
employing t h e t o t a l
A complete d e s c r i p t i o n of
t h e scheme i s given i n Davies and F l a t h e r (1978). The b a s i c sea model used i n t h e p r e s e n t work covers t h e whole of t h e North West European c o n t i n e n t a l s h e l f .
The s p a t i a l g r i d , shown a s an a r r a y of elemental
boxes, appears i n Figure 1.
For each box element, u i s defined a t t h e mid-point
of l o n g i t u d i n a l s i d e s , v a t the mid-point of l a t i t u d i n a l s i d e s and 5 a t t h e box c e n t r e . The g r i d spacing is 11O i n longitude by lo i n l a t i t u d e and t h e t i m e increment used i n t h e i n t e g r a t i o n i s 180 seconds.
Additional sea models covering
t h e North Sea with improved r e s o l u t i o n of shallow c o a s t a l waters (Davies and F l a t h e r , 1977) and t h e Southern Bight o f t h e North Sea with t h e t i d a l p a r t of t h e River Thames (Prandle, 1975) are a l s o being developed f o r use i n t h e surge p r e d i c t i o n procedure.
THE METEOROLOGICAL DATA
I n this s e c t i o n w e consider t h e d e r i v a t i o n of wind s t r e s s and g r a d i e n t s of atmosp h e r i c p r e s s u r e t o g e t h e r with t h e surge i n p u t along t h e open boundaries of t h e sea model.
These d a t a are deduced from t h e o u t p u t of t h e f i n e mesh version of t h e
Bushby-Timpson 10-level numerical model of t h e atmosphere (Benwell e t a l . , 1971) used i n r o u t i n e weather p r e d i c t i o n a t t h e B r i t i s h Meteorological Office. I n e a r l i e r work t h e a v a i l a b l e atmospheric d a t a c o n s i s t e d of hourly values of 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 , i n some cases with t h e a d d i t i o n of 6-hourly values of t h e t h i c k n e s s of t h e l a y e r between t h e 1000 m b and 900 mb pressure s u r f a c e s and t h e components of t h e 1000 mb wind, a l l taken d i r e c t l y from and defined a t g r i d p o i n t s o f t h e 10-level model shown superimposed on t h e s h e l f model mesh i n Figure 1.
D i f f e r e n t methods used t o c a l c u l a t e t h e r e q u i r e d i n p u t t o t h e sea
model from t h e s e v a r i a b l e s a r e described and compared i n F l a t h e r and Davies (1978) using t h e storm of 2nd-4th January 1976.
More r e c e n t l y t h e Meteorological Office
have, themselves, made e s t i m a t e s of hourly g r i d p o i n t v a l u e s of pa,
(J:
the surface
390
interpolated Hassea Wagner (1971)
w
interpolated
I
Ta
I
I
interpolated +ICES Ts data
interpolated
Hasse (1974)
E Heaps(1965)
AT
interpolated
As 3 but with CD € Smith*Banke (1975)
3a.
= information supplied by the Meteorologlcal Office
0=
data input to the sea model
Figure 2. A l t e r n a t i v e methods used t o d e r i v e i n p u t d a t a f o r t h e dynamical storm surge p r e d i c t i o n model.
391 wind, and T
:
the air temperature close to the sea surface, for use in operational The methods used are based on the
forecasts of waves and swell (Golding, 1977).
surface and 900 mb wind relationships due to Findlater, Harrower, Howkins and Wright (1966) and employ information from the two lowest layers of the atmospheric model (see Gadd and Golding (1977) for details).
The new surface data has been
available for our surge experiments since late in 1976 and is used in the present real time predictions. The grid points of the atmospheric model comprise a rectangular array on a stereographic map projection. The relationship between the non-dimensional Cartesian co-ordinates (x,y) of the atmospheric model and latitude and longitude, the co-ordinates of the sea model, is
(x + 35') cos (x + 35O)
x
=
(2R/s) tan ( ~ / 4- $/2) sin
y
=
-(2R/s) tan (7r/4 - $/2)
where
s,
(7)
the grid length at the pole is approximately 100 km. The atmospheric data
are defined at intersections of the lines x Alternative methods used to derive Vp are outlined in Figure 2.
=
and
10(1)33, y
=
-44(1)-19.
from the meteorological data
:(')
Method 1 assumes a knowledge of atmospheric pressure
only. The data are first interpolated from the 10-level model onto the sea model grid where the east and north components of Vp
are calculated using simple centred
finite difference approximations. The geostrophic wind, w
-9'
components in the usual way.
To estimate the surface wind a simple empirical law
w = 0 . 5 6 ~ + 2.4 m/s
(8)
9
is used.
is then derived in
Obtained by Hasse and Wagner (1971) from measurements taken in the German
Bight, this relationship requires no additional information about the atmospheric boundary layer. A constant angle of backing, 6
=
20°, between the directions of
g and w is assumed to approximate the frictional influence. The wind stress is -9 calculated from a quadratic law (S)
(9) CDP,lWlUr where the drag coefficient, CD, varies with wind speed according to the equation 1
=
(Heaps, 1965) for w < 5 m/s
0.565
cD
x
lo3
=
-0.12
+
2.513
0.137~for 5 < IJ < 19.22 for w > 19.22 m/s
The east and north components of Vp
and 5")
I
(10)
constitute the required forcing
terms in the sea model equations. Method 2 assumes a knowledge of both atmospheric pressure and near surface air temperature, which makes it possible to take into account the important influence of stability on the geostrophic to surface wind relationship. The derivation of and $ follows exactly Method 1. In addition T values are interpolated onto 9 the sea model grid where sea surface temperatures, Ts, digitised from the chart
Vp
showing their climatological distribution for the appropriate month in the year
392 (ICES, 1962) a r e a l s o defined.
= Ta - Ts.
S u b t r a c t i n g g i v e s ATa-s
Then using t h e
r e s u l t s of a l a t e r i n v e s t i g a t i o n by Hasse (1974), we t a k e
u=aw + b 9
with a = 0.54 b = 1.68
-
0.012ATa-s 0.105AT
i n m/s.
(11)
1
a-s The angle of backing i s defined as a numerical f u n c t i o n of AT Fig. 3 ) .
(from Hasse (1974), a-s The d e r i v a t i o n of wind stress then follows Method 1. Since Method 2
t a k e s account of t h e e f f e c t of s t a b i l i t y e x p l i c i t l y , it might be expected t o give a b e t t e r e s t i m a t e of t h e s u r f a c e wind and hence t h e wind s t r e s s than does Method 1. Method 3 uses s u r f a c e wind a s w e l l a s atmospheric p r e s s u r e provided by t h e Meteorological Office.
The d e r i v a t i o n of Vp
follows Method 1.
The s u r f a c e winds,
defined a s x and y components a t 10-level model g r i d p o i n t s a r e f i r s t resolved i n t o e a s t and n o r t h components, using r e l a t i o n s h i p s deduced from Equations (71, which
are then i n t e r p o l a t e d on t o t h e sea model g r i d .
The wind stress then follows from
t h e q u a d r a t i c s t r e s s law (9) with drag c o e f f i c i e n t ( 1 0 ) .
I n a v a r i a n t , Method 3a,
t h e d r a g c o e f f i c i e n t obtained by Smith and Banke (1975) from measurements on Sable I s l a n d , Nova S c o t i a , namely
CD x 10’ = 0.63
+
0.066~
replaced t h e one defined i n ( 1 0 ) .
(12) Method 3 d i f f e r s from 1 and 2 i n t h a t t h e surface
winds are derived from 900 mb r a t h e r than geostrophic winds.
The 900 mb wind
d i s t r i b u t i o n i s c a l c u l a t e d d i r e c t l y by t h e 10-level model and so includes dynamical e f f e c t s a s s o c i a t e d with t h e curvature of i s o b a r s , t h e motion and development of d e p r e s s i o n s , f r i c t i o n and a l l o t h e r p h y s i c a l processes represented i n t h e atmosp h e r i c model.
I t should t h e r e f o r e approximate t h e a c t u a l wind a t t h e t o p of t h e
boundary l a y e r more c l o s e l y than does t h e geostrophic wind. s u r f a c e wind r e l a t i o n s h i p s of F i n d l a t e r e t a l .
Also t h e 900 mb and
(1966) a r e defined numerically t o
t a k e account o f t h e i n f l u e n c e of s t a b i l i t y ( o r l a p s e rate) a s does (11).
Method 3
might t h e r e f o r e be expected t o g i v e , o v e r a l l , t h e b e s t e s t i m a t e of s u r f a c e wind and hence wind stress. I n a l l cases t h e surge i n p u t e l e v a t i o n a t open sea boundary p o i n t s i s derived from t h e h y d r o s t a t i c l a w i n t h e form
tM= where
-
(Pa
-
(13)
Pa)/P9,
i s t h e mean atmospheric p r e s s u r e taken t o be 1012 mb.
any o t h e r information, t h e a s s o c i a t e d c u r r e n t ,
4M, i s
In t h e absence of
assumed t o be zero.
THE PREDICTION SCHEME
I t remains t o d e s c r i b e how t h e meteorological d a t a and t h e s e a model a r e used
t o provide storm surge f o r e c a s t s i n real-time on a day-to-day A preliminary scheme w a s described by F l a t h e r (197633).
been modified and t h e r e v i s e d v e r s i o n i s o u t l i n e d here.
o p e r a t i o n a l basis.
This has subsequently The scheme i s i l l u s t r a t e d
:
i
J
J.
sea model doto stored for stort condition of 2
seo model doto stored as back-up
I
.1
i
.1
seo model doto stored as bock-up
seo model doto stored for stort condition of 3 I
I 6
12
12
:
:
real time d
day I
0600
1200
meteorologicol prediction-
---
surge prediction
I800
Figure 3.
0000
0600
I
1200
- -3,6
2?
+
4
seo model doto stored 0s bock-up
seo model doto stored for stort condition ......
day 2 1
0000
:
L
ts
doy 3 I
I
1
1800
0000
0600
Scheme for operational surge forecasting.
I
1200
1800
w (D w
i n Figure 3, t o which t h e r e a d e r should r e f e r i n conjunction with t h e d e s c r i p t i o n which follows. Numerical weather p r e d i c t i o n s using t h e 10-level model a r e c a r r i e d o u t by t h e
Meteorological O f f i c e twice a day. 0
< t
< 36 hours, where t
Each p r e d i c t i o n run covers t h e period
denotes meteorological model t i m e with t
m m ponding t o e i t h e r 0000 GMT o r 1200 GMT on t h e day.
m
=
0 corres-
C o l l e c t i o n of observations
and i n i t i a l i s a t i o n o f t h e model t a k e s about 24 hours of real t i m e , so t h a t t h e model c a l c u l a t i o n , which r e q u i r e s about 14 minutes of c e n t r a l processor t i m e on an IBM 360/195 computer, begins a t about 0230 GMT o r 1430 GMT.
Data required
f o r wave p r e d i c t i o n and f o r t h e surge c a l c u l a t i o n a r e derived from t h e model outp u t and s t o r e d on magnetic d i s c .
x and y components of F) and T
The d a t a comprise values of p a , wx, from a 24
x
(~f (the Y' 26 r e c t a n g u l a r s u b s e t of g r i d p o i n t s
o f t h e atmospheric model covering north-west Europe ( s e e Figure 1 ) .
The d a t a
are provided a t hourly i n t e r v a l s from t = 6 t o t = 36 hours, and become a v a i l a b l e m m a t tm= 3, i . e . a t 0300 GMT o r 1500 GMT. The surge p r e d i c t i o n i t s e l f can now begin. of s e p a r a t e s t e p s c a r r i e d o u t i n t u r n .
Each p r e d i c t i o n c o n s i s t s of a number
F i r s t t h e d a t a a r e processed using one of
t h e methods described i n t h e previous s e c t i o n t o q i v e a p p r o p r i a t e values of t h e components of Vpa and :(')
a t i n t e r n a l g r i d p o i n t s and t h e values of
0, a t
open
boundary p o i n t s of t h e sea model a t hourly i n t e r v a l s , tm= 6 ( 1 ) 3 6 . Because of t h e l i m i t e d d e s c r i p t i o n of t i d e i n t h e sea model, confined t o only t h e M2 and S2 c o n s t i t u e n t s , it i s n o t intended t h a t p r e d i c t i o n s of t o t a l water l e v e l be obtained from t h e model d i r e c t l y .
Rather, t h e sea model i s used t o fore-
c a s t surge r e s i d u a l s , t a k i n g i n t o account t h e important influence of tide-surge interaction.
The f i n a l estimate of t o t a l w a t e r l e v e l a t p o r t s of i n t e r e s t i s
then obtained by combining t h e s e r e s i d u a l s with t h e p r e d i c t e d t i d e c a l c u l a t e d , using 60 c o n s t i t u e n t s o r more, by t h e harmonic method.
This procedure r e q u i r e s
t h a t two s e p a r a t e sea model computations a r e c a r r i e d o u t f o r each f o r e c a s t .
In
t h e f i r s t , t h e meteorological f o r c i n g terms and surge i n p u t a r e s e t t o zero, so t h a t a p r e d i c t i o n of t h e model t i d e i s obtained; i n t h e second, t i d e and surge are c a l c u l a t e d t o g e t h e r .
S u b t r a c t i n g t h e two s o l u t i o n s g i v e s t h e r e q u i r e d
residual. Each sea model run covers a p e r i o d 0 < t model t i m e and t
6 30 hours, where
t
denotes sea
0 corresponds t o t = 6 , being e i t h e r 0600 GMT or 1800 GMT. m The i n i t i a l s t a t e of t h e sea i s taken d i r e c t l y from f i e l d s , r e p r e s e n t i n g t h e d i s =
t r i b u t i o n of t i d e o r t i d e
cast. t
+
surge as a p p r o p r i a t e , computed i n t h e previous fore-
Thus, r e f e r r i n g t o Figure 3 , t h e s t a t e of e l e v a t i o n and motion a t time
= 0 i n f o r e c a s t 3, say, i s i d e n t i f i e d with t h a t a t time tS = 12 i n f o r e c a s t 2 ,
s t o r e d i n t h e preceding run.
I f , because of computer f a i l u r e o r some o t h e r cause,
f o r e c a s t 2 had not been c a r r i e d o u t , then f i e l d s from ts = 24 i n f o r e c a s t 1 would be used as i n i t i a l c o n d i t i o n s for f o r e c a s t 3 .
Thus, by s t o r i n g complete
<,
u, v
data twice, at t = 12 and ts = 24, during each sea model run, a degree of robustness sufficient to survive the loss of one meteorological forecast is embodied in the procedure.
If more than one successive forecast is lost then the sequence can
continue without restarting only if meteorological data from another source is available. within the tide + surge run constant components of Vp applied in the hourly interval ti Surge input
-
+
< ti
+
and T(')
are
% centred on the data time ti.
5, is interpolated in time linearly between the hourly specified values
in order to prevent the introduction of sudden elevation changes on the boundary. Having obtained the tide only and tide
+
surge solutions, the former is subtracted
from the latter to give the residual. Further steps in the procedure produce output in graphical form of time series of tide, tide
+ surge and residual at selected ports
and contours of residual elevation over the shelf at suitable time intervals, showing the temporal and spatial development of the surge. for subsequent analysis.
Some of the results are archived
In particular, hourly elevation and current arrays are
retained for use in extracting statistical extremes. The complete surge calculation starting from the hourly atmospheric data and including the two sea model runs and all associated steps requires about 5 minutes of central processor time on an IBM 370/165 computer at the Science Research Council's Daresbury Nuclear Physics Laboratory. REAL-TIME PREDICTIONS During 1977 it was agreed with the Meteorological Office that experimental realtime predictions using the scheme described above would be undertaken during the winter of 1977-78. The meteorological data was to be transmitted to Bidston in real time and the sea model calculations performed on the computer at the Daresbury Laboratory using existing communication facilities with Bidston. The co-operation of computer experts in both the Institute of Oceanographic Sciences (10.5) and the Meteorological Office was required to establish the necessary data link between Bracknell and Bidston, a link which uses a dial-up connection along telephone lines within the public telephone network.
The link became operational at the end of
January 1978 and the experiments commenced soon after. First real-time predictions were carried out from Monday 13th to Friday 17th February 1978.
Thus ten successive forecasts were run with initial times 0600 GMP
and 1800 GMC on each of the five days.
The procedure described in the last section
was carried out with additional steps required to transmit the essential atmospheric data from the Meteorological Office to 10s Bidston and then on to the computer at Daresbury. To minimise the time taken for data transmission along the telephone lines, Method 1 using only the single variable p
was employed. Results were pro-
duced in the form of line printer output at Bidston.
No special priority was
granted for the experiments so that all computer work was carried out as though
396 by an ordinary computer user.
Consequently the time taken to complete each fore-
cast depended on how heavily the computers were being used.
The morning forecasts,
carried out outside normal working hours, took between 40 and 60 minutes with the exception of the 0600 Gm forecast on Monday 13th February, which was delayed by engineering work on the Daresbury computer and ended at 1307 GMT.
In the after-
noons general computer usage was higher and consequently delays were longer so 1 that the 1800 GMT forecasts required from 3 to 4T hours, ending between 1800 GMF and 1920
GMC.
In a subsequent experiment the scheme was operated again from Tuesday 7th to Wednesday 15th March 1978 to provide predictions over the period of spring tides on the 10th and 11th. The aim here was to extend the system to return some of the output to the Meteorological Office, where it could be examined by members of the Storm Tide Warning Service (STWS) - the organisation responsible for the provision of surge warnings in Britain.
In the event of any interesting surges occurring,
comparisons between predictions from the present dynamical scheme and from the statistical procedure, on which surge warnings are currently based, would have been possible.
Fourteen forecasts were carried out in all, though the sequence was not
completed without interruption. Two of the scheduled forecasts were lost. On Wednesday afternoon 8th March the Daresbury computer broke down during the sea model run and the forecast had to be abandoned. Similarly the IBM 360/195 at the Meteorological Office suffered a power failure on Sunday afternoon 12th March. In this eventuality essential work is transferred to a second computer, an IBM 370/158 on which the time requirement is much greater and consequently non-essential jobs are not processed.
Again the forecast had to be abandoned.
In both instances,
the prediction following the one lost was run successfully using the back-up initial data for the sea model stored by the system.
In practice, mainly because of the
additional time requirement for data transmission, results were returned to the Meteorological Office on only a limited number of occasions. These results were received satisfactorily by STWS. The only storm surge of note occurring on the east coast of England during the periods covered was on 8th March, giving a maximum residual of 0.51 m at 1200 GMT at North Shields increasing to 0.96 m at 2200 GKC at Southend. Unfortunately, the surge occurred within about a day of the start of the sequence of predictions with the 1800 Gm forecast on 7th March.
Switching on the meteorological forcing
suddenly at this time generated a substantial artificial negative surge, -0.82 m at 0900 GMT on 8th March at Southend compared with the true residual of -0.24 m. The influence of the start-up can be expected to persist for perhaps two or three days, reducing the accuracy of predictions during this period. surge peak was badly underestimated.
As a result the
It seems unlikely that the 1800 GMT forecast
on 8th March, lost because of computer breakdown, would have significantly improved on these results from earlier forecasts. Some moderately substantial surges were
0600 GMT 11/1/78
1800 GMT 11/1/78
0600 GMT 12/1/78
I800 GMT 12/1/78 w
ID
Figure 4 .
Weather c h a r t s f o r 11th and 12th January 1978.
4
predicted in other areas towards the end of the period as a complex depression moved east from Scotland over the North Sea. Heysham at 2300
GMT
at Dieppe at 2300
Maximum residuals of 0.87 m at
13th March; 0.96 m at Avonmouth at 2100 GMT on 14th; 1.24 m
GMT
on 14th; 0.79 m at Hilbre Island at 0100 GMT on 15th; and
1.36 m at B l i s m in the German Bight at 1300 GMT on 15th were predicted but have not been checked against observations. The main outcome of these experiments was to demonstrate that surge prediction using the scheme is practicable.
Clearly, the need to transmit atmospheric data
along telephone lines from one computer to another, returning results to their ultimate destination in the same way, increases the complication and the possibility of faults causing failures of the system.
Ideally all the calculations should
be carried out on the computer used to make the meteorological predictions, so that data transmission delays can be eliminated. Then, assuming no delay in obtaining access to this computer, the surge forecast would be completed within a very short time after the atmospheric data becomes available.
It was not the aim of these
experiments to examine the accuracy of the predictions. There is no reason why the requirement for real-time running as opposed to running in a hindcast mode should in any way influence the results achieved by the scheme given identical atmospheric model data.
It is more interesting, then, to choose a period of large
surges, extract the atmospheric data from the 10-level model predictions archived at the Meteorological Office, and compute the associated storm surges at any convenient time.
The results will be exactly the same as would have been obtained
in real time.
This procedure has been followed for the storm surge of 11th and
12th January 1978 as described in the next section.
THE STORM SURGE OF 11-12 JANUARY 1978 In this section computations of the storm surge of 11th and 12th January 1978 are described. These computations were carried out in order to compare the accuracy of the results obtained using the alternative methods 1, 2 and 3 (outlined earlier) to calculate the meteorological forces on the sea.
The results were obtained from
computer programs written to perform real time forecasts described in the last section.
-
used in the experiments
The atmospheric data were extracted from the actual
operational 10-level model predictions. The present results, therefore, are identical to those which would have been obtained by running the scheme in real time to predict the storm surge. The surge was associated with a depression which developed to the west of the British Isles on 10th January.
The depression moved east and intensified reaching
the Wash with a central pressure of about 978 mb on the morning of the following day (see Figure 4). It continued east south east across the Southern Bight of the North Sea before turning north of east into Holland and Germany, slowly filling
399
0.5 0 .o
STORNOWAY
-0.5
WICK
AEERDEEN
NORTH SHIELOS
h
E
Y
c
IMMINGHAM
P
INNER DOWSING
u)
LOWESTOFT
WALTON
1
i*/++'
t J
+++
11/01/78
/?-R SOUTHEND
G+'+
12/01/78 1
I
t
Figure 5. Comparison between surge e l e v a t i o n s computed i n s o l u t i o n 3 -( ) and o b s e r v a t i o n s (+++++I. @ i n d i c a t e s t h e r e s i d u a l c l o s e s t t o t h e time of t i d a l high water.
a s it went.
P r e s s u r e s r o s e q u i c k l y as a following r i d g e of high p r e s s u r e extended
eastward i n t o I r e l a n d , b r i n g i n g a s t r o n g p r e s s u r e g r a d i e n t and n o r t h t o northe a s t e r l y g a l e s over t h e western p a r t of t h e North Sea from midday on t h e 11th t o mid-morning on 12th January. The r e s u l t i n g storm surge coincided with a s p r i n g t i d e t o produce l e v e l s on t h e north-east
c o a s t of England i n excess of those occurring during t h e major surges
of 3 1 s t January 1953 and 3rd January 1976.
Further south t h e l e v e l s were l e s s
extreme, although those experienced on 10th December 1965, when flooding l a s t occurred i n London, were e q u a l l e d i n t h e Thames Estuary.
On t h i s occasion t h e
Thames l e v e l s came w i t h i n 60 cm of t h e t o p of t h e defences i n c e n t r a l London and w i t h i n 30 c m of t h e t o p i n o t h e r p a r t s o f t h e r i v e r .
A t Vlissingen on t h e Dutch
c o a s t t h e p r e d i c t e d high water e a r l y on 12th January was exceeded by more than one metre, b u t f u r t h e r n o r t h l i t t l e e f f e c t was f e l t .
On t h e w e s t s i d e of t h e
B r i t i s h Isles, i n t h e I r i s h Sea and i n t h e English Channel, a f a i r l y s u b s t a n t i a l negative surge was produced by t h e n o r t h - e a s t e r l y winds. Four s e p a r a t e model c a l c u l a t i o n s have been c a r r i e d o u t , using t h e a l t e r n a t i v e methods of d e r i v i n g meteorological i n p u t d a t a i l l u s t r a t e d i n Figure 2.
Each calcu-
l a t i o n s t a r t e d from a s t a t e of rest with no e l e v a t i o n d i s t u r b a n c e a t 0000 GMT on 7 t h January, and a 54 hour run-in p e r i o d with t i d e o n l y e s t a b l i s h e d t h e t i d a l d i s t r i b u t i o n a t 0600 GMT on t h e 9 t h when t h e atmospheric f o r c i n g was introduced.
There-
a f t e r e i g h t complete f o r e c a s t s w e r e c a r r i e d o u t s t a r t i n g a t 0600 GMT and 1800 GMT on each of t h e f o u r days 9 t h t o 12th January.
For t h e purposes of comparison w i t h
observations, t i m e series covering t h e p e r i o d 0000 GMT on t h e 11th t o 0600 GMT on 13th January were c o n s t r u c t e d by t a k i n g a p p r o p r i a t e c a l c u l a t e d r e s i d u a l s from t h e f i r s t 12 hours of t h e s e f o r e c a s t s .
Figure 5 shows a p l o t of t h e r e s u l t i n g
t i m e series a t B r i t i s h p o r t s from s o l u t i o n 3 with a v a i l a b l e hourly values derived from observations.
The r e l a t i o n s h i p of t i d e and surge i s i n d i c a t e d by circles,
which i d e n t i f y t h e observed r e s i d u a l occurring c l o s e s t t o t h e t i m e of p r e d i c t e d high water.
I t can be seen t h a t t h e observed surge p r o f i l e i s reproduced reasonably
w e l l a t most p o r t s .
Perhaps t h e most obvious d e f i c i e n c y occurs a t Southend, where
t h e surge peak a t 0000 GMT on 12th January on t h e r i s i n g t i d e i s omitted.
Since
no e q u i v a l e n t f e a t u r e appears a t o t h e r p o r t s , t h e peak i s probably a s s o c i a t e d with tide-surge i n t e r a c t i o n o r l o c a l wind e f f e c t w i t h i n t h e Thames Estuary. the r e s i d u a l a t high w a t e r some two hours l a t e r i s p r e d i c t e d q u i t e well.
However, The
behaviour of t h e remaining t h r e e s o l u t i o n s was g e n e r a l l y s i m i l a r t o t h a t shown i n the figure. N u m e r i c a l comparisons have been c a r r i e d o u t by c a l c u l a t i n g r o o t mean square (RMS) e r r o r s based on d i f f e r e n c e s between observed hourly r e s i d u a l s and equivalent
model p r e d i c t i o n s .
The r e s u l t s a r e shown i n Table 1.
The t y p i c a l RMS e r r o r of
about 20 cm i s r a t h e r b e t t e r than has been achieved i n some previous experiments ( s e e Davies and F l a t h e r , 1978, F l a t h e r and Davies, 1978).
The improvement might
401
TABLE 1
Root mean square e r r o r s ( c m ) f o r t h e storm surge p e r i o d 0000 GMT 11/1/76 t o 0600 GMT 13/1/78 based on comparisons between observed hourly r e s i d u a l s and values computed i n d i f f e r e n t s o l u t i o n s based on methods 1, 2, 3 and 3a r e s p e c t i v e l y .
1
2
3
3a
Stornoway Wick North S h i e l d s Immingham L o w e st o ft Walton* Southend
23.3 21.8 16.3 19.8 14.5 19.0 23.3
21.5 20.1 16.4 19.3 15.9 18.3 21.1
17.4 16.1 13.3 18.8 17.3 15.2 23.7
19.1 17.9 14.1 18.8 17.9 15.2 22.8
All p o r t s
20.1
19.2
Port
*OOOO GMT 11/1/78
t o 0200 GMT 12/1/78
I
I
I
I
17.9
18.4
only.
TABLE 2
+ Co
Regression c o e f f i c i e n t s C 1, Co ( c m ) where 5
observed = CIScomputed hourly r e s i d u a l s i n t h e p e r i o d 0000 GMT 11/1/78 t o 0600 GMT 13/1/78.
1
I
2
based on
3
3a
c1
cO
I
Wick North S h i e l d s Immingham Lowe s to f t Walton* Southend
I I Mean
C,
*OOOO GMT 11/1/78
1
1.01 1.11 1.02 1.11 0.86 0.93
I
1-00
-21.3 -20.7 -12.0 -12.9 -10.0 - 1.5 - 4.2
t o 0200 G m 12/1/78
0.96 1.07 1.02 1.21 1.03 1.02
1
1.03
only.
0.88 0.96 1.16 1.05 1.10 1.05 1.00
-19.6 -11.7 -13.7 -12.8 - 9.0 - 8.6
I
1
1.03
-15.9 -15.4 - 9.8 - 9.1 -13.0 -10.6 -10.5
I
-16.7 -16.1 - 9.0 - 9.9 -14.2 -11.3 -11.3
0.94 1.03 1.29 1.24 1.29 1.16 1.18
I
1.16
I
I
P
0
N
Figure 6. Successive model f o r e c a s t s f o r Southend from s o l u t i o n 3 s t a r t i n g a t ( a ) 0600 GMC and (b) 1800 GMT on 11th January 1978. E denotes t h e exceedance of high water.
403 be a t t r i b u t e d to the new form of meteorological d a t a used i n the p r e s e n t c a s e . Overall s o l u t i o n 3 g i v e s t h e b e s t r e s u l t s with RMS e r r o r s between 13.3 cm a t North S h i e l d s and 23.7 c m a t Southend.
Using t h e same t i m e s e r i e s , l i n e a r r e g r e s s i o n
a n a l y s e s were c a r r i e d o u t f o r each p o r t seeking c o e f f i c i e n t s C1 and Co i n t h e relationship 4-
co.
'observed = Cl'computed The values obtained a r e given i n Table 2.
(14) A
tendency f o r surge v a r i a t i o n s t o be
overestimated a t Stornoway, where C1 ranges from 0.88 to 0.94, and underestimated between 1.07 and 1.29, i s evident. I n 1 t u r n s o u t t o be negative i n d i c a t i n g t h a t c a l c u l a t e d r e s i d u a l s a r e
a t North S h i e l d s and Lowestoft, w i t h C every c a s e C
c o n s i s t e n t l y higher than observed.
The e f f e c t i s p a r t i c u l a r l y marked a t Stornoway
and Wick and probably c o n t r i b u t e s to t h e r e l a t i v e l y l a r g e RMS e r r o r s a t these p o r t s . This kind o f c o n s i s t e n t e r r o r i n l e v e l may be introduced through i n c o r r e c t surge e l e v a t i o n i n p u t along t h e open boundaries o f t h e model.
Solution 3a, with an
average C1 value of 1.16, underestimates surge v a r i a t i o n s as compared w i t h t h e o t h e r s o l u t i o n s , r e f l e c t i n g t h e f a c t t h a t t h e drag c o e f f i c i e n t from Smith and Banke (19751, Equation ( 1 2 ) , g i v e s a lower e s t i m a t e of wind stress than Equation (10) due t o Heaps (1965) i n t h e important range of wind speeds from 10.6 t o 28.5 m / s . The foregoing d i s c u s s i o n concerns r e s u l t s s t r i c t l y from hours 0 t o 12 only of each sequence of 30 hour sea model p r e d i c t i o n s .
Figure 6 shows two complete fore-
c a s t s f o r Southend, beginning a t 0600 GMT and 1800 GMT, r e s p e c t i v e l y , on 11th January.
Successive f o r e c a s t s overlap, t h e f i r s t giving, i n this i n s t a n c e , an
e a r l y warning more than 20 hours b e f o r e the high water a t 0200 GMT on 12th January most a f f e c t e d by t h e surge.
The second f o r e c a s t , a v a i l a b l e f o r i s s u e 12 hours
l a t e r , updates t h i s p r e d i c t i o n .
Because observations a r e used t o r e d e f i n e the
i n i t i a l s t a t e o f the atmosphere before each 10-level model run, consecutive meteorological p r e d i c t i o n s , and hence a l s o consecutive surge p r e d i c t i o n s , d i f f e r i n t h e p e r i o d o f overlap, t h e l a t e s t f o r e c a s t f o r a given i n s t a n t being i n general most a c c u r a t e .
Thus, although t h e surge p r o f i l e s i n Figure 6 look very s i m i l a r ,
s i g n i f i c a n t d i f f e r e n c e s occur i n t h e p r e d i c t e d r e s i d u a l s a t high water (1.26 m i n (a) and 1.13 m i n (b) compared w i t h t h e observed value of 1.04 m)
.
In addition,
t h e maximum p r e d i c t e d r e s i d u a l s a r e 2.94 m i n ( a ) and 2.48 m i n ( b ) , l a t e r reduced t o 2.10 m i n t h e f o r e c a s t beginning a t 0600 GMT on 12th January, s t i l l 0.34 m higher than a c t u a l l y occurred.
These d i f f e r e n c e s a r e a l s o apparent i n p l o t s of
the s p a t i a l d i s t r i b u t i o n of surge.
Figures 7a and 7b show two such predicted d i s -
t r i b u t i o n s f o r 0300 GMT on 12th January when t h e surge peak was between Immingham and Lowestoft.
Levels i n t h e north-eastern
I r i s h Sea and i n the English Channel
are about 25 cm lower i n Figure 7b (hour 9 o f t h e 1800 GMT 11/1/78 f o r e c a s t ) than i n Figure 7a (hour 21 of t h e 0600 GMT 11/1/78 f o r e c a s t ) .
Similar d i f f e r e n c e s appear
i n o t h e r a r e a s b u t t h e g e n e r a l f e a t u r e s remain t h e same.
The sequence of contour
p l o t s , produced a t 3 hourly i n t e r v a l s throughout each p r e d i c t i o n , should t h e r e f o r e
..*...-,,
. . ,. ..... . .. r
L.
.-r
---
Cole of Vectors
...........* ..,, 1
A
,
Figure 7a. S p a t i a l d i s t r i b u t i o n of surge e l e v a t i o n ( c m ) and c u r r e n t a t 0300 GMT 12/1/78 from hour 2 1 of t h e f o r e c a s t s t a r t i n g a t 0600 GMT 11/1/78 ( s o l u t i o n 3 ) .
25cm/s
50 cm/s 75 cm/s
tl
0.I
Figure 7b. Spatial distribution of surge elevation (cm) and current at 0300 GMT 12/1/78 from hour 9 of the forecast starting at 1800 GMT 11/1/78 (solution 3 ) .
give t o t h e surge f o r e c a s t e r a u s e f u l p i c t u r e of t h e developing s i t u a t i o n . F i n a l l y i n t h i s s e c t i o n we p r e s e n t a f i r s t comparison between p r e d i c t i o n s from t h e model and from t h e e s t a b l i s h e d s t a t i s t i c a l procedure operated by t h e Storm Tide Warning Service (Townsend, 1975).
STWS make p r e d i c t i o n s of surge r e s i d u a l s
a t t h e time of harmonically p r e d i c t e d high water, which a r e then used t o estimate maximum l e v e l s a t p o r t s along t h e e a s t c o a s t of England.
When a maximum l e v e l i n
excess of t h e p r e s c r i b e d danger l e v e l f o r t h e p o r t i s p r e d i c t e d , a p p r o p r i a t e flood warnings a r e issued. The procedure used t o d e r i v e comparable information from t h e model o u t p u t w a s a s follows.
F i r s t , the difference
E
between t h e maximum c a l c u l a t e d t i d e p l u s surge
e l e v a t i o n and the a s s o c i a t e d maximum c a l c u l a t e d t i d a l e l e v a t i o n g i v e s t h e expected amount by which t h e a c t u a l t i d a l high water should be exceeded ( s e e Figure 6 b ) . This q u a n t i t y seems more a p p r o p r i a t e f o r flood p r e d i c t i o n than t h e r e s i d u a l a t a p a r t i c u l a r i n s t a n t s i n c e t i d a l high water and a c t u a l high water w i l l n o t , generally, coincide i n t i m e .
The a p p r o p r i a t e danger l e v e l f o r a p a r t i c u l a r model t i d e i s
assumed t o be t h e same h e i g h t above high w a t e r o f t h e model t i d e as t h e r e a l danger l e v e l i s above t h e high water l e v e l of t h e a c t u a l p r e d i c t e d t i d e .
The maximum height
above t h i s e f f e c t i v e danger l e v e l , t h e times a t which danger l e v e l i s reached and a t which t h e l e v e l f a l l s below danger a r e then read from t a b u l a t e d t i d e p l u s surge l e v e l s produced by t h e model.
F i n a l l y , t h e c r i t i c a l times a r e c o r r e c t e d f o r t h e
e r r o r i n t h e t i m e of occurrence of t h e model t i d a l high water as compared with the = a c t u a l p r e d i c t e d t i d a l high water, thus tcorrected
t a c t u a l Hw - tmodel HW' The r e s u l t s f o r t h e n i g h t t i d e of 11th t o 12th January a t e a s t c o a s t p o r t s a r e +
given i n Table 3 .
I t i s assumed t h a t t h e model p r e d i c t i o n s , obtained from s o l u t i o n
3 , would be a v a i l a b l e one hour b e f o r e t h e i n d i c a t e d f o r e c a s t d a t a t i m e .
STWS pre-
d i c t i o n s a r e those a c t u a l l y produced and i s s u e d a t t h e time of t h e surge. included a r e values derived by STWS from observations.
Also
Thus, r e f e r r i n g t o t h e sec-
t i o n f o r Southend i n Table 3, t h e m o d e l f o r e c a s t s t a r t i n g a t 0600 GMT on 11th January p r e d i c t e d that t h e t i d a l high water (HW) l e v e l a t 0200 GMT on t h e following day would be exceeded by 1.26 m and t h a t l e v e l s would be above t h e danger mark from 0049 GWI? u n t i l 0349 GMT on 12th January. a t about 0500 GMT on t h e 11th tide.
-
This information would have been available
about 21 hours b e f o r e t h e p r e d i c t e d t i m e of high
The s t a t i s t i c a l f o r e c a s t about 7% hours l a t e r was marginally l e s s accurate
with e = 1.32 m. January gave
E
The n e x t model p r e d i c t i o n from the 1800 GMT f o r e c a s t on 11th
= 1.13 m, s l i g h t l y b e t t e r than t h e s h o r t term STWS. f o r e c a s t (1.19 m)
and only 9 cm above t h e observed value.
I n g e n e r a l , t h e e a r l i e s t model p r e d i c t i o n s
24 t o 30 hours before t h e event a r e n o t very a c c u r a t e b u t a r e perhaps good enough t o be u s e f u l a s e a r l y warnings.
Later f o r e c a s t s show considerable improvement and
seem t o compare well with t h e s t a t i s t i c a l p r e d i c t i o n s .
The model a l s o appears t o
show some s k i l l i n p r e d i c t i n g t h e c r i t i c a l t i m e s , e s p e c i a l l y a t Immingham.
Clearly,
many more comparisons of t h i s kind w i l l be r e q u i r e d t o a s s e s s t h e u s e f u l n e s s of the model i n surge f o r e c a s t i n g .
401
Comparison of various predictions of high water for the night tide of 11-12 January 1978. Model predictions are identified by the forecast initial data time and assume that the results would be available one hour before this. Storm Tide Warning Service (STWSl predictions are those actually obtained from the established statistical procedure and issued at the times indicated.
Port and observed HW time
Source and time of forecast
Height Time Time Length Exceedence above danger below of danger of HW reached danger warning level (m) (hoursl GMT GMT (m)
I North Shields 1701 GMT 11/1/78
1800 0600 0625 1315
10/1/78 11/1/78 11/1/78 11/1/78
model model STWS STWS
observed
0.34 0.45 0.75 0.31 0.69
0.05 0.16 0.46 0.02 0.40
1627 1612
1712 1737
1600
1800
-
-
24 12 104 4 0
1 I Immingham 1942 GMT 11/1/78
1800 0600 0625 1318 1800
10/1/78 11/1/78 11/1/78 11/1/78 11/1/78
model model STWS STWS model
observed
0.66 0.85 1.15 0.84 0.88 0.96
0.18 0.37 0.67 0.36 0.40 0.48
1848 1833
2003 2028
1833 1825
2033 2035
-
-
264 14% 134 64 2% 0
I Lowestoft 2259 GMT 11/1/78
1800 0600 1150 1800 1810
10/1/78 11/1/78 11/1/78 11/1/78 11/1/78
model model STWs model STWS
observed
0600 Walton 1320 1800 -0130 GKC 12/1/78*2048
11/1/78 11/1/78 11/1/78 11/1/78
model STWS model STWS
observed
Southend 0200
12/1/78
0600 1320 1800 2048
11/1/78 11/1/78 11/1/78 11/1/78
observed
*tide gauge broken.
model STWS
model STWS
0.74 1.00 1.42 0.94 1.18 1.21
-0.02 0.24 0.66 0.18 0.42 0.45
1.18 1.01 1.07 1.21
0.68 0.51 0.57 0.71
*
1.26 1.32 1.13 1.19 1.04
*
0.72 0.78 0.59 0.65 0.50
2208
0118
2218
0103
2125
0025
-
-
-
30 18 11 6 5 0
I 2357 -
0007
-
0020
0322
-
0312
-
*
0049
0349
0104
0344
0050
0320
-
-
20% 12 8% 41r 0
21 124 9 5 0
408 CONCLUDING REMARKS
Recent developments i n t h e establishment o f a storm surge p r e d i c t i o n scheme f o r B r i t i s h waters based on t h e u5e of dynamical f i n i t e d i f f e r e n c e models of t h e atmosphere and of t h e sea have been described.
I n p a r t i c u l a r , f i r s t real-time predictions
c a r r i e d o u t d u r i n g February and March 1978 have demonstrated t h e p r a c t i c a b i l i t y of t h e method. Using a s a t e s t c a s e t h e storm surge of 11th and 12th January 1978, a severe surge which caused flooding and considerable damage on t h e English c o a s t , four a l t e r n a t i v e methods o f d e r i v i n g t h e meteorological f o r c e s a c t i n g on t h e sea from atmosp h e r i c d a t a have been compared.
A s expected, t h e method ( 3 ) using atmospheric
p r e s s u r e and s u r f a c e winds derived from 900 m b winds i n t h e 10-level model by t h e Meteorological Office gave t h e b e s t r e s u l t s when used with t h e drag c o e f f i c i e n t from Heaps (1965).
The a l t e r n a t i v e d r a g c o e f f i c i e n t , due t o Smith and Banke (19751,
gave a lower e s t i m a t e of t h e wind s t r e s s from t h e same s u r f a c e winds and an i n f e r i o r p r e d i c t i o n of t h e storm surge. A f i r s t comparison h a s been made between p r e d i c t i o n s obtained by t h e s t a t i s t i c a l
procedure operated by t h e Storm Tide Warning Service and those using t h e p r e s e n t dynamical p r e d i c t i o n scheme f o r p o r t s on t h e east c o a s t of England. model p r e d i c t i o n s emerge q u i t e w e l l from t h e comparison.
Generally the
The accuracy of t h e two
methods f o r a given l e n g t h o f warning i s s i m i l a r and t h e model may a l s o give a reasonably good i n d i c a t i o n of t h e times a t which danger l e v e l s a r e passed.
In
a d d i t i o n , t h e new scheme can give p r e d i c t i o n s more than 30 hours i n advance whereas t h e s t a t i s t i c a l method, depending on o b s e r v a t i o n s along t h e e a s t c o a s t i s l i m i t e d by t h e propagation time of t h e t i d e and t r a v e l l i n g component of t h e surge t o a maximum of about 12 hours warning.
The s p a t i a l coverage of t h e model scheme i s
an important advantage o f f e r i n g t h e p o s s i b i l i t y of o f f s h o r e p r e d i c t i o n s . I t i s e v i d e n t t h a t f u r t h e r comparisons of t h i s kind w i l l be required t o gauge
t h e value of t h e new scheme.
To t h i s end it i s hoped t h a t sea model p r e d i c t i o n s
w i l l be c a r r i e d o u t on a r o u t i n e b a s i s a t t h e Meteorological Office during t h e
winter of 1978-79. Service.
The r e s u l t s w i l l be passed d i r e c t l y t o t h e Storm Tide Warning
By t h e end of t h e season, comparisons of dynamical and s t a t i s t i c a l pre-
d i c t i o n s i n a wide v a r i e t y of surge s i t u a t i o n s w i l l have been c a r r i e d o u t . f u l l y t h e r e s u l t s w i l l j u s t i f y t h e continued use of t h e model-based scheme.
HopeFurther
development and improvements can then be a n t i c i p a t e d i n t h e l i g h t of t h e valuable experience gained.
ACKNOWLEDGEMENTS
The author i s indebted t o t h e Meteorological O f f i c e f o r t h e i r continuing coo p e r a t i o n i n t h i s p r o j e c t and t o L t . Cdr. J. Townsend, O f f i c e r i n Charge, Storm
409 Tide Warning Service f o r observations and permission t o include s t a t i s t i c a l predictions.
Thanks a r e a l s o due t o D r N.S.
t o Mr R.A.
Heaps f o r valuable comments and c r i t i c i s m ,
Smith, who prepared t h e diagrams and t o Mrs Young, who typed t h e
manuscript. The work described i n t h i s paper w a s funded by a consortium c o n s i s t i n g of the Natural Environment Research Council, t h e Ministry of Agricul'iure, F i s h e r i e s and Food, and t h e Departments of Energy, t h e Environment and Industry.
REFERENCES
Benwell, G.R.R., Gadd, A . J . , Keers, J.F., Timpson, M.S. and White, P.W., 1971. The Bushby-Timpson l o - l e v e l model on a f i n e mesh. S c i . Pap. Met. Off., London, 32: 23 pp. Cartwright, D.E., 1976. Shelf boundary t i d a l measurements between I r e l a n d and Norway. M6m. SOC. R. S c i . Liege, S e r . 6 , 10: 133-140. Davies, A.M. and F l a t h e r , R.A., 1977. Computation of t h e storm surge of 1 t o 6 A p r i l 1973 using numerical models of t h e north w e s t European c o n t i n e n t a l s h e l f and t h e North Sea. D t . hydrogr. Z . , 30: 139-162. 1978. Application of numerical models of t h e Davies, A.M. and F l a t h e r , R.A., n o r t h w e s t European c o n t i n e n t a l s h e l f and t h e North Sea t o t h e computation of t h e storm s u r g e s o f November-December 1973. D t . hydrogr. Z. Erg.-H. A , N r . 14. Duun-Christensen, J . T . , 1975. The r e p r e s e n t a t i o n of t h e s u r f a c e p r e s s u r e f i e l d i n a two-dimensional hydrodynamic numeric model f o r t h e North Sea, t h e Skaggerak and t h e K a t t e g a t . D t . hydrogr. Z . , 28: 97-116. F i n d l a t e r , J . , Harrower, T.N.S., Howkins, G.A. and Wright, H.L., 1966. Surface S c i . Pap. Met. O f f . , London, 23: 41 pp. and 900 mb wind r e l a t i o n s h i p s . 1976a. A t i d a l model of t h e north west European c o n t i n e n t a l s h e l f . Flather, R.A., M6m. SOC. R. S c i . LiBge, Ser. 6 , 10: 141-164. 197613. P r a c t i c a l a s p e c t s of t h e use of numerical models f o r surge F l a t h e r , R.A., prediction. I n s t i t u t e of Oceanographic Sciences Report No. 30, 18 pp. 1976. Note on a preliminary scheme f o r storm surge F l a t h e r , R.A. and Davies, A.M., p r e d i c t i o n using numerical models. Quart. J . R. met. SOC., 102: 123-132. 1978. On t h e s p e c i f i c a t i o n of meteorological F l a t h e r , R.A. and Davies, A.M., f o r c i n g i n numerical models f o r North Sea storm surge p r e d i c t i o n , with a p p l i c a t i o n t o t h e surge of 2-4 January 1976. D t . hydrogr. Z. Erg.-H. A, N r . 15. 1977. Temperature and wind modelling f o r t h e 10-level Gadd, A . J . and Golding, B.W., model. Meteorological O f f i c e Met. 0 2b Technical Note No. 13. (Unpublished manuscript) Golding, B.W., 1977. Waves and swell can be f o r e c a s t . Offshore Services 10, No. 10: 100- 102. Hasse, L . , 1974. On t h e s u r f a c e t o geostrophic wind r e l a t i o n s h i p a t sea and t h e s t a b i l i t y dependence of t h e r e s i s t a n c e law. B e i t r . Phys. Atmosph., 47: 45-55. Hasse, L. and Wagner, V . , 1971. On t h e r e l a t i o n s h i p between geostrophic and s u r f a c e wind a t sea. Mon. Weath. Rev., Wash., 99: 255-260. Heaps, N.S., 1965. Storm surges on a c o n t i n e n t a l s h e l f . P h i l . Trans. R. SOC. (A) 257: 351-383. ICES, 1962. Mean monthly temperature and s a l i n i t y of t h e s u r f a c e l a y e r of t h e North Sea and a d j a c e n t waters from 1905 t o 1954. I n t e r n a t i o n a l Council f o r t h e Exploration of t h e Sea, Copenhagen. Prandle, D., 1975. Storm surges i n t h e southern North Sea and River Thames. P h i l . Trans. R. SOC. ( A ) 344: 509-539. 1975. V a r i a t i o n of t h e sea s u r f a c e drag c o e f f i c i e n t Smith, S.D. and Banke, E.G., with wind speed. Q u a r t . J. R. m e t . SOC., 101: 665-673. Timmerman, H . , 1975. On t h e importance of atmospheric p r e s s u r e g r a d i e n t s f o r t h e D t . hydrogr. Z . , 28: 62-71. generation of e x t e r n a l surges i n t h e North Sea. Townsend, J . , 1975. Forecasting ' n e g a t i v e ' storm surges i n t h e southern North Sea. The Marine Observer, XLV: 27-35.
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Fig.
14.
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 2 . (in degrees)
411
BELGIAN REAL-TIME SYSTEM FOR THE FORECASTING OF CURRENTS AND ELEVATIONS I N
THE
NORTH SEA
Y.
ADAM
Unite de Gestion du Mod6le Mathematique de l a M e r du Nord e t de l ' E s t u a i r e de 1'Escaut.
Ministere de l a Sant6 Publique e t de 1'Environnement.
ABSTRACT A r e a l time system o f storm surge f o r e c a s t i n g based on a mathematical model o f the North Sea i s presented. Problems concerning t h e r e l i a b i l i t y of r e s u l t s and t i m e l a g s a r e d e a l t with, and sources o f errors a r e analysed.
PRINCIPLES
The a i m of t h i s work i s t o b u i l d a systemwhich has t h e a b i l i t y t o f o r e c a s t t h e response of c u r r e n t s and s u r f a c e e l e v a t i o n s i n t h e North Sea t o t h e influence of t h e t i d e and t h e meteorological f o r c e s .
The system should h e l p warning c i v i l
a u t h o r i t i e s of p o s s i b l e floods along the Belgian c o a s t and i n t h e Scheldt e s t u a r y , and i t should h e l p f o r e c a s t i n g s t r o n g c o a s t a l c u r r e n t s that could cause damage t o harbour c o n s t r u c t i o n s o r o t h e r man-built s t r u c t u r e s . I n t h i s preliminary s t a g e , however, w e are mainly ( i f n o t only) concerned with e l e v a t i o n s , and w i l l r e s t r i c t our a t t e n t i o n t o t h e s p e c i f i c problem o f surges ( p o s i t i v e or negative) induced by stormy weather. As we want t o design a simulation system f o r f o r e c a s t i n g purposes, we r e s t r i c t
o u r s e l v e s t o using only d a t a t h a t can be f o r e c a s t a t l e a s t s e v e r a l hours i n advance, and w e p r o h i b i t t h e use o f real measurements which could n o t be used i n
a t r u e real-time system.
Therefore, as t h e r e s u l t s of our simulations depend
on unascertained d a t a , they w i l l probably be less a c c u r a t e than r e s u l t s obtained by models using real i n p u t s . A s d a t a w e use only t h e following
- t i d a l d a t a (amplitudes and phases of t i d a l harmonics) f o r boundary c o n d i t i o n s , and computation of pure tide-driven
-
circulation
f o r e c a s t s o f wind and atmospheric p r e s s u r e a t the s e a s u r f a c e , f o r boundary
412 conditions and e x t e r n a l f o r c e s computation. The r e s u l t s o f r e a l time simulations a r e compared, a f t e r a while, t o computed and measured e l e v a t i o n s i n s e v e r a l s e a p o r t s along t h e English and Belgian c o a s t s .
METHODS
The method used t o compute the occurence o f storm s u r g e s , i s t h e numerical i n t e g r a t i o n o f a mathematical model of water c i r c u l a t i o n i n t h e North Sea, using a s i n p u t t i d a l and meteorological f o r e c a s t s . The mathematical model w a s developed during t h e "Programme National R-D s u r 1'Environnement-Projet Mer" by Nihoul and Ronday (1976). I t i s i n t e g r a t e d using a numerical algorithm on a f i n i t e d i f f e r e n c e g r i d covering
not only t h e North Sea, l i m i t e d by t h e Dover S t r a i t , as i n t h e o r i g i n a l model, b u t a l s o t h e English Channel i n o r d e r t o be f r e e d from a boundary condition t o o d i f f i c u l t t o deal with i n r e a l time.
Equations and numerical algorithms a r e f u l l y
described i n Nihoul and Ronday (1976). Fig. 1 shows t h e p o r t i o n of ocean which i s being modelled, and t h e l i m i t s of t h e computational g r i d .
2 . Boundary c o n d i t i o n s
The boundary condition e a s i e s t t o impose i s t h e e l e v a t i o n along t h e northern and western boundaries a c r o s s t h e Channel. The t o t a l e l e v a t i o n
5,
is t h e sum of a t i d a l e l e v a t i o n
cM,
and an e l e v a t i o n
generated by a v a r i a t i o n of t h e atmospheric p r e s s u r e
where
n
t,
=
izl
fi
ai cos
( q t + pi)
where p a i s the a i r s p e c i f i c weight, p w t h e water s p e c i f i c weight, p t h e atmospher i c p r e s s u r e , pm t h e mean atmospheric p r e s s u r e , n t h e number of t i d a l harmonics (Darwin t h e o r y ) , a . t h e main amplitude of t h e ith harmonic a t a given p o i n t , i t s wave number, p . i t s i n i t i a l phase (slowly varying with t i m e ) , and
W
i
f . a correc-
t i o n f a c t o r , a l s o very slowly varying with time. 0
For t h e northern boundary, t h e values of a . and 9. a r e computed from Cartwriqht
413 (1976) a t a s e r i e s of p o i n t s p l o t t e d o n F i g .
1 ( w i t h l a b e l s A-M)
and t r a n s l a t e d
t o t h e boundary l i n e using i n t e r p o l a t i o n , e x t r a p o l a t i o n , and phase c o r r e c t i o n . For t h e western boundary, amplitudes and phases are assumed t o be constant a c r o s s t h e Channel, and equal t o t h e s e q u a n t i t i e s a t Devonport. Eq. 3 i s a l s o used t o compute a t h e o r i c a l e l e v a t i o n a t a s e r i e s o f l o c a t i o n s along t h e c o a s t .
interactions
3. A i r - s e a
The i n f l u e n c e of t h e a t m o s p h e r i c c o n d i t i o n s on t h e c i r c u l a t i o n i s modelled by 2 parameters : a i r p r e s s u r e g r a d i e n t and wind stress a t t h e s e a s u r f a c e .
I n t h e equations d e s c r i b i n g the e v o l u t i o n of t h e depth-averaged v e l o c i t y ,
-
,
the p r e s s u r e g r a d i e n t a c t s a s a f o r c e
and the wind stress a s a f o r c e
where
W
i s t h e wind v e l o c i t y , m a parameter (m
c o e f f i c i e n t depending upon t h e wind speed. sen t h e following wind stress model
Cd
=
11 f o r IIW 11
1.26
for
2.4
-
O.l),
and C t h e s u r f a c e drag
In a preliminary s t a g e , we have cho-
:
Q
10 m/sec
>
10 m/sec
b u t t h e following formulation has a l s o been used, w i t h l i t t l e s e n s i b l e improvement ( F l a t h e r and Davies, 1975)
Cd
=
f o r II!
0.565
-
0.12
+
0.137
[IW 11
for 5 f o r I!
2.513
II C 5 m/sec
< [IW 11 < II
19.22 m/sec 19.22 m/sec
4. Available d a t a
Input data a r e
:
- amplitudes and phases of t i d a l harmonics, from Cartwright (1976) and t a b l e s of harmonic c o n s t a n t s (Bureau Hydrographique I n t e r n a t i o n a l , 1953)
-
p r e s s u r e and wind v e l o c i t i e s a t the nodes o f a g r i d covering the North Sea, t h e B r i t i s h Isles and Western Europe (see Fig. 2 ) .
These parameters are t h e r e s u l t s
of numerical weather f o r e c a s t s by the Meteorological O f f i c e , Bracknell, which
414
.I
.I
Pig. 1.
.l
.J
The finite difference grid used for the computations with the mathematical model.
415
i F i g . 2. The m e t e o r o l o g i c a l g r i d c o v e r i n g t h e North Sea.
416 a r e s e n t t o Liege through t h e R6gie des Voies Adriennes by t e l e x . Time i n t e r v a l between 2 d a t a s e t s i s 6 hours; by i n t e r p o l a t i o n , meteorologic a l parameters are t r a n s f e r e d from t h e very coarse atmospheric g r i d t o t h e f i n e r hydrodynamical g r i d .
GENERAL SCHEME OF SURGE WARNING
The f i r s t purpose of t h i s system i s t o h e l p g i v i n g an e a r l y warning of surge and flood danger.
A s i n t h e p r e s e n t s t a g e of computational c o s t s and f a c i l i t i e s
i t i s not p o s s i b l e t o run t h e simulation programs throughout t h e whole w i n t e r
p e r i o d , some kind of warning and alarm system has t o be designed. The proposed scheme i s given on Fig. 3 ; it i s n o t f u l l y o p e r a t i o n a l y e t , i n t h e sense t h a t n o t a l l p a r t s have been t e s t e d i n t r u e stormy conditions. A f i r s t warning o f storm danger comes from t h e I n s t i t u t Royal Meteorologique.
I f t h e r i s k o f an important storm surge e x i s t s (mainly, s t r o n g winds from N , NW, o r very deep depression) t h e simulation system i s s e t i n t o a c t i o n . F i r s t , an i n i t i a l condition i s c a l c u l a t e d ( d i s t r i b u t i o n o f
and
over t h e
North Sea and t h e Channel, using t h e numerical model with no atmospheric i n p u t . In t h e meantime, t h e f i l e o f meteorological information i s updated w i t h t h e l a s t available forecasts.
When t h e pure t i d a l regime is almost reached, a t a time
corresponding t o t h e a v a i l a b l e meteorological d a t a , a f i r s t run of t h e storm surge computational procedure i s done; t h i s run gives a f o r e c a s t of t h e c i r c u l a t i o n i n t h e North Sea f o r more than 1 2 hours ahead. program has computed t h e pure
t i d a l circulation.
I n t h e meantime, another By s u b t r a c t i n g two d i s t r i b u -
t i o n s of v a r i a b l e s ( a t t h e same i n s t a n t ) , one can s e e whether t h e r e i s a surge forming somewhere.
I f t h e r e i s , a n e x t simulation i s run as soon as another
weather f o r e c a s t becomes a v a i l a b l e . l u t i o n i s favourable o r n o t .
I f n o t , one checks whether t h e weather evo-
I f it i s n ' t , t h e s i m u l a t i o n goes on with up-to-date
d a t a ; i f it i s , t h e simulation i s terminated.
SCHEME OF A SIMULATION RUN
A s our system i s designed t o work i n r e a l time, w e must now take care of d a t a
a c q u i s i t i o n , d a t a - f i l e s updating, t r a n s f e r o f d a t a from t h e coarse meteorological g r i d t o t h e hydrodynamical g r i d , d e l a y s , s t a b i l i t y , s a f e t y i n c a s e of lack of data.
417
WARNING ADVICE (I.RM.1 RECEPTION
INITIAL CONDITIONS
I
1
PROGRAM
t
Fig. 3. Warning system flowchart.
NO
418
1. Data a c q u i s i t i o n
Meteorological data come from t h e R.V.A.
through t e l e x .
The messages are s e n t
twice a day ( a t 8 GMT and 20 GMT), contain each d a t e and t i m e of transmission, and two sets of u s e f u l data:
- t i m e o f f o r e c a s t i n g ( t h e f i r s t message provides f o r e c a s t s f o r GMT 18 and
GMT 0
t h e following day, t h e second f o r e c a s t s f o r GMT 6 and GMT 12, t h e following day)
-
f o r each node of t h e coarse g r i d : p r e s s u r e , wind speed, wind d i r e c t i o n . Every day, one r e c e i v e s t h u s f o u r s e t s of meteorological parameters.
Each t e l e x t a p e i s w r i t t e n o n t o a magnetic t a p e ; telexcode i s then t r a n s l a t e d t o EBCDIC code i n a d i s k f i l e ; t h e l a t t e r i s then read, and d a t a a r e converted into
usable d a t a and w r i t t e n on a t h i r d f i l e ; t h i s f i l e i s used t o update the
formerly-built meteorological d a t a f i l e .
2. F i l e updating
The r e s u l t i n g f i l e contains:
- 2 d a t a sets which are o f no use, except f o r s a f e t y purposes (see l a t e r )
-
4 d a t a sets, a t s i x hours i n t e r v a l s , covering t h u s a t i m e p e r i o d of 18 hours
19 d a t a s e t s , a t one hour i n t e r v a l s , covering t h e same p e r i o d , derived from t h e former data by cubic i n t e r p o l a t i o n s .
Let us note t h a t t h e updating program can also:
- f i l l l a c k i n g d a t a by t i m e i n t e r p o l a t i o n using, i f needed, t h e f i r s t two data sets
- f i l l l a c k i n g d a t a by time e x t r a p o l a t i o n using, i f needed, t h e f i r s t two d a t a sets, under automatic programmed procedure o r programmer request. The l a t t e r case happens when longer-term f o r e c a s t i s wanted
These 19 d a t a sets are then used t o provide, by s p a t i a l i n t e r p o l a t i o n , 19 s e t s of d a t a , i . e . p r e s s u r e , and two wind v e l o c i t y components a t each node o f t h e hy-
drodynamical g r i d , which are s t o r e d on a " g r i d f i l e " .
3 . Hydrodynamical computations
Using as i n i t i a l condition t h e s t a t e of the s e a a t t h e time o f t h e f i r s t d a t a s e t of t h e g r i d f i l e , and as i n p u t t h e meteorological d a t a i n t h a t f i l e , the numerical program computes the e v o l u t i o n o f t h e hydrodynamical v a r i a b l e s , and s t o -
res t h e i r d i s t r i b u t i o n a t every hour on a hydrodynamical f i l e . The computation simulates 18 hours of real t i m e ; t h e sea s t a t e corresponding t o t h e 12th hour o f simulation w i l l be used as i n i t i a l condition f o r t h e following
run. The whole procedure i s summarized i n Fig. 4 .
419
Message reception
II
conditions
Fig. 4. R e a l time system flowchart.
420 4. Delays
The problem of t i m e l a g s i n t h e f o r e c a s t i n g and t h e a v a i l a b i l i t y o f r e s u l t s
i s o f course c r u c i a l i n surge f o r e c a s t .
L e t us e s t i m a t e how much i n advance one
can f o r e c a s t t h e e l e v a t i o n o f t h e sea s u r f a c e , and a c r i t i c a l high t i d e . A t 8 GMT, a t e l e x message i s received.
I t t a k e s about 1 hour t o t r a n s f e r t h e
t e l e x tape onto magnetic t a p e ; t h i s o p e r a t i o n i s performed o f f - l i n e and r e q u i r e s a l o t o f manipulations.
The n e x t s t a g e s can be executed completely automatically
by t h e o p e r a t i n g system.
I n emergency p r i o r i t y , b u t t a k i n g i n t o account t h a t the
computer can n o t be t o t a l l y dedicated t o t h i s work ( e x i s t e n c e of a time sharing s y s t e m ) , it takes another hour t o g e t t h e r e s u l t s on l i s t i n g , and a l a s t one t o g e t t h e p l o t s , which are drawn o f f - l i n e ; t h u s , around 11 GMT, a f o r e c a s t i s avail a b l e , which runs t o 24 GMT, i . e .
13 hours i n advance; during t h i s p e r i o d t h e r e
i s one high t i d e .
We come l a t e r t o t h e case where high t i d e (without meteorological i n f l u e n c e ) , t a k e s p l a c e i n t h e f i r s t seven hours. I f high t i d e occurs i n t h e l a s t s i x hours of t h e day, then, i t s a c t u a l h e i g h t can be p r e d i c t e d a t l e a s t 7 hours i n advance, and a warning can be given.
I f t h e high t i d e occurs b e f o r e 18 GMT, it i s too
near f o r a f o r e c a s t t o be u s e f u l . However, as w a s s a i d under Section 3 of t h i s c h a p t e r , meteorological d a t a can be e x t r a p o l a t e d .
I n t h e c a s e h e r e mentioned (high t i d e before 18 GMT) it would
have been wise t o e x t r a p o l a t e t h e d a t a received i n t h e evening o f t h e previous day and t o run a simulation with t h e s e e x t r a p o l a t e d d a t a ; a t 23 GMT ( t h e day b e f o r e ) , one can, i n so doing, g e t an i d e a of t h e high t i d e a t l e a s t 12 hours i n advance.
Of course, e x t r a p o l a t e d parameters a r e probably l e s s r e l i a b l e than
f o r e c a s t parameters, b u t one has t o t a k e i n t o account t h e i n e r t i a of t h e atmosphere and, above a l l , of t h e sea.
Systematic t e s t s have been run t o check t h e
r e l i a b i l i t y of t h e r e s u l t s y i e l d e d by such a procedure.
A p r o v i s i o n a l conclu-
s i o n can be drawn from t h e t e s t s : hydrodynamical computations using extrapolated d a t a a r e almost as good as r e s u l t s computed with t r u e f o r e c a s t s , a s soon a s successive
e x t r a p o l a t i o n s a r e d i s t a n t i n time of a t l e a s t 48 hours.
This r e l a t i v e l y
good f i t might a l s o be due t o t h e poor r e s o l u t i o n of t h e t r u e f o r e c a s t s . The delays estimated h e r e hold when one wants t o f o r e c a s t t h e s i t u a t i o n a t any p o i n t , using t h e numerical output of t h e model a t t h a t p o i n t . However, surge waves t r a v e l a t a f i n i t e speed, and t h e computation of t h e surge a t one p o i n t could be used t o have a good i d e a (through r e g r e s s i o n a n a l y s i s of t y p i c a l cases) of what w i l l happen, some time l a t e r , a t another p l a c e which i s f u r t h e r on t h e way of t h e surge wave.
For i n s t a n c e , t h e surges most dangerous f o r t h e Belgian
c o a s t come from t h e North West. Ostend i n about 1 2 hours. I
Such surge waves t r a v e l from North Shields t o
Therefore, by using a numerical f o r e c a s t of t h e e l e -
421
v a t i o n a t North S h i e l d s
( f o r i n s t a n c e ) , t o i n f e r a f o r e c a s t of t h e e l e v a t i o n
ac Ostend, one can p r e d i c t a surge 18 t o 24 hours i n advance. ANALYSIS OF SOME RESULTS
I t has taken r e l a t i v e l y long t i m e t o develop t h e whole system (decoding of
t e l e x messages, f i l e s t r u c t u r e , updating and i n t e r p o l a t i n g programs, being taken i n t o account t h e number of s a f e t y procedures such programs include) and t o extend t h e model t o t h e English Channel. Moreover, t o check the r e s u l t s , t r u e measurements a r e needed, which a r r i v e with some delays.
However, w e have simulated, i n pseudo-realtime,
a bad weather p e r i o d
i n November 1977, and compared e l e v a t i o n s computed by the model t o e l e v a t i o n s measured i n s i t u (more thorough tests w i l l be run during t h e w i n t e r 78-79). The comparison done was mainly between computed and measured s u r e l e v a t i o n s (surge residuals).
However, such a procedure i s r a t h e r questionable: indeed, what i s a
surelevation ?
I t i s t h e d i f f e r e n c e between something r e a l (measured), and some-
t h i n g t h a t does n o t e x i s t ( t i d a l e l e v a t i o n computed by harmonic series) and i s never observed, even during f i n e weather p e r i o d s .
A b e t t e r method could be t o com-
p a r e t o t a l e l e v a t i o n s (measured and s i m u l a t e d ) , b u t it i s then very hard t o make a d i s c r i m i n a t i o n between good and bad r e s u l t s , because v a r i a t i o n s of t h e amplitudes become very small compared t o t h e t o t a l amplitudes.
Figs. 5 t o 9 show some
r e s u l t s o f comparison f o r some p l a c e s along t h e B r i t i s h Coast. t h e model appears t o give very bad r e s u l t s a t Wick
At
f i r s t sight
(Fig. 5): t h e r e s i d u a l s a r e
p o s i t i v e following t h e model (depression near t h e Northern boundary), while t h e measurements show negative r e s i d u a l s , probably due t o a negative e x t e r n a l surge which i s , of course, completely ignored by t h e model.
Positive residuals are
due t o t h e boundary condition
Results a r e much b e t t e r f o r North S h i e l d s (Fig. 61, where t h e general f e a t u r e s of t h e surges are w e l l simulated by t h e model.
However, p o s i t i v e surges a r e
g e n e r a l l y overestimated, while negative surges are underestimated; this general t r e n d could be due t o t h e i n f l u e n c e of a r a t h e r bad boundary condition as s t a t e d above. The same p a t t e r n shows up a t Immingham (Fig.7) surge i s completely missed.
, where
a s m a l l negative
However, confidence i n measurements i n Immingham
should n o t be t o o h i g h , because of t h e p a r t i c u l a r c o n f i g u r a t i o n of the coastl i n e i n this region.
A t
Lowestoft (Fig. 8 ) , t h e general t r e n d of t h e surge i s
w e l l simulated, b u t t h e value of t h e s u r e l e v a t i o n i s s t r o n g l y overestimated. Apart from t h e i n f l u e n c e of t h e boundary c o n d i t i o n , one could suggest, a s res-
422
.....
Fig. 5. Computed and observed surge at Wick.
-.
.
measured residual elevation
.....
:
computed residual elevation
Fig. 6. Computed and observed surge at North Shields.
-. . measured . - - - :- computed
residual elevation residual elevation
.
.. ......
.
.-2
. .
423
Fig. 7. Computed and observed surge at Immingham. -. . measured residual elevation
.....
:
computed residual elevation
Fig. 8. Computed and observed surge at Lowestoft. _ . . measured residual elevation
.-... : computed
residual elevation
424
::I 2.2
2.0
1
-0.L
v
V
V
-1.2
Fig. 9. Computed and observed surge a t Ostend.
-.
.
measured r e s i d u a l e l e v a t i o n
.....
:
computed r e s i d u a l e l e v a t i o n
ponsible f o r t h i s , t h e accumulation ( i n t h e model) of water masses pushed by t h e wind i n t h e Southern Bight due t o i n s u f f i c i e n t flow through t h e Dover S t r a i t . However, s i n c e t h e o r i g i n a l model (Nihoul and Ronday, 1976) gave r a t h e r b e t t e r r e s u l t s w i t h t r u e meteorological f i e l d s , one can a l s o s u s p e c t , t o some e x t e n t , t h e poor r e s o l u t i o n o f t h e meteorological f o r e c a s t s . A t p r e s e n t , too few numerical experiments are a v a i l a b l e t o d i s t i n g u i s h between
the following sources of e r r o r :
-
t h e hydrodynamical-mathematical model i t s e l f (which, f o r i n s t a n c e , would not reproduce a good water flow through the Dover S t r a i t )
- t h e meteorological i n p u t ( f o r i n s t a n c e , overestimation of s u r f a c e winds) - t h e mathematical model of a i r sea i n t e r a c t i o n (overestimating t h e wind s t r e s s
?).
FUTURE IMPROVEMENTS
I n i t s p r e s e n t s t a t e the real-time f o r e c a s t model of the North Sea cannot give s a t i s f a c t o r y r e s u l t s i n a l l s i t u at i o n s .
One can d i s t i n g u i s h s e v e r a l kinds
of improvements t h a t could be included i n t h e g e n e r a l scheme t o d e a l more c o r r e c t l y with a majority of p o s s i b l e meteorological s i t u a t i o n s ; s h o r t term and mid-long
t e r m modifications a r e suggested.
425 1. Improvements t o t h e mathematical model
-
i n t r o d u c i n g a b e t t e r a i r - s e a i n t e r a c t i o n formula; however, r e s u l t s with Heap's model ( F l a t h e r and Davies, 1975) a r e only s l i g h t l y b e t t e r
-
i n t r o d u c i n g a b e t t e r condition a t t h e northern boundary, i f t h i s makes sense without an e x t e r n a l surge model
- coupling t h e e x i s t i n g model t o
-
an e x t e r n a l surge model, capable of g i v i n g b e t t e r r e a l t i m e e l e v a t i o n s a t t h e boundaries.
This model should cover t h e whole c o n t i n e n t a l s h e l f and,
p o s s i b l y , a p a r t of t h e open ocean.
This model w i l l be u s e l e s s i f t h e
meteorological i n p u t does n o t cover a more extended region than it c u r r e n t l y does
-
a f i n e r mesh model o f t h e Southern Bight-Dover S t r a i t region, t o enhance t h e computation o f t h e flow i n t h i s region.
This i s necessary f o r t h e
reason s t a t e d above, and i f one a l s o wants t o compute t h e storm c u r r e n t s i n t h e region.
2. Improvements t o t h e meteorological i n p u t
-
extension of t h e coarse meteorological g r i d t o cover t h e whole c o n t i n e n t a l s h e l f and a p a r t of t h e a d j a c e n t open ocean.
This is necessary i f one wants
t o develop an e x t e r n a l surge model
-
refinement of t h e mesh of t h e meteorological model, t o g e t h e r w i t h higher f r e quency of meteorological f o r e c a s t s .
3. Improvements t o the system
The above-mentioned improvements a r e suggested t o enhance t h e accuracy of t h e forecasts. computation
One w i l l a l s o t r y t o g e t t h e f o r e c a s t s f a s t e r , by performing t h e whole on-line.
The only l i m i t a t i o n t o such improvements i s t h e c o s t , which i s l i k e l y t o be worthwile when the system i s completely o p e r a t i o n a l .
REFERENCES
Bureau Hydrographique I n t e r n a t i o n a l , 1953. Constantes harmoniques. Publication S p e c i a l e n o 26 Cartwright, D . E . , 1976. Shelf boundary t i d a l measurements between I r e l a n d and Norway. Proc. 7th Liege Colloquium on' Ocean Hydrodynamics, Mem. SOC. Roy. S c i . Liege, 10 : 133-139 1975. The a p p l i c a t i o n of numerical models t o F l a t h e r , R.A. and Davies, A.M., storm surge p r e d i c t i o n , I . O . S . r e p o r t n o 16 and Ronday, F. , 1976. Modeles hydrodynamiques. I n : P r o j e t Mer, Nihoul, J . C . J . Rapport f i n a l , V o l . 3. Programmation de l a P o l i t i q u e S c i e n t i f i q u e , Brussels, 270 p p .
This Page Intentionally Left Blank
427
CYCLOGENESIS A N D FORECAST OF DRAMATIC WATER ELEVATIONS I N VENICE
A.
TOMASIN' a n d R . FRASSETTO
1
' L a b o r a t o r i o p e r l o S t u d i o d e l l a Dinamica d e l l e Grandi Masse, C.N.R.,Venezia
(Italy)
ABSTRACT The o c e a n o g r a p h i c and m e t e o r o l o g i c a l a s p e c t s o f t n ? A d r i a t i c s u r g e s are invest i g a t e d . S i n c e V e n i c e f l o o d s a r e only d u e t o s u c h a n o m a l o u s t i d e s , t h e i r f o r e c a s t is o b v i o u s l y r e q u e s t e d f o r p r a c t i c a l p u r p o s e s . S h o r t term w a r n i n g ( a b o u t s i x h o u r s a h e a d ) i s now p o s s i b l e w i t h good a c c u r a c y , t h a n k s t o p r e s e n t u n d e r s t a n d i n g o f t h e s u r g e g e n e r a t i o n . The s t u d y o f t h e c y c l o g e n e s i s i n t h e a t m o s p h e r e o v e r t h e w e s t e r n Alps c a n e x t e n d t h e time f r e e b o a r d f o r alarm.
TNTRODUCTION A N D HISTORICAL REMARKS
I n t h e h i s t o r y o f Venice, f l o o d s are r e p o r t e d a s a r e c u r r e n t c a l a m i t y f o r t h e c i t y ( U n e s c o , 1 9 6 9 , pp. 3 4 - 6 6 ) .
T h i s i s n o t s u r p r i s i n g when c o n s i d e r i n g t h a t
s t r e e t s a r e n o r m a l l y l e s s t h a n o n e meter o v e r mean w a t e r l e v e l a n d t h e t i d a l range is comparable t o t h i s f i g u r e .
Troubles can c l e a r l y arise even w i t h a s u r g e
h a l f a meter h i g h , w h i c h i s a r a t h e r common o c c u r r e n c e .
F i g u r e 1 shows a n e x -
ample. Even t h o u g h
V e n i c e is l o c a t e d i n a l a g o o n , a n d n o t d i r e c t l y e x p o s e d t o t h e
s e a , f r o m t h e p o i n t o f v i e w o f t i d e s a n d s u r g e s no d i f f e r e n c e i s o b s e r v e d , e x c e p t
f o r a d e l a y o f a b o u t a n h o u r f r o m t h e o p e n sea t o t h e town.
The s i t u a t i o n i n
t h i s c e n t u r y i s c e r t a i n l y w o r s e t h a n i n t h e p a s t , as f a r a s t h e f r e q u e n c y o f f l o o d s 1s c o n c e r n e d .
T h i s is d u e t o v a r i o u s r e a s o n s :
t h e town h a s s u n k a b o u t 2 0 cm
s i n c e t h e l a s t c e n t u r y ( w h i c h is c o m p a r a b l e , p r e s u m a b l y , t o t h e t o t a l s i n k i n g i n the previous millennium).
Also, t h e c o m m u n i c a t i o n b e t w e e n t h e sea a n d t h e l a g o o n
is larger now t h a n Z C O y e a r s a g o , d u e t o e x t e n s i v e d r e d g i n g i n t h e l a g o o n i n l e t s . I n t h o s e d a y s , t h e p e a k l e v e l s o f t h e e x t e r n a l sea were p r e s u m a b l y " c u t " by t h e n a r r o w o p e n i n g s o f t h e l a g o o n , e x c e p t f o r s u r g e s o f long d u r a t i o n ( a n d t h e most t r e m e n d o u s f l o o d , i n 1 9 6 6 , was o f t h i s k i n d ) .
428
.
6
12
18
0
6
12
18
0
6
12
F i g . 1. R e c o r d e d water l e v e l i n V e n i c e , s t a r t i n g A p r i l 20, 1967. r i s e a t t h e e n d o f t h e s e c o n d d a y i s e v i d e n t ( T i m e i s GMT + 1 ) .
18
0
The a n o m a l o u s
ADRIATIC DYNAMICS O n l y t h e A d r i a t i c phenomena ( t i d e s a n d s u r g e s ) w i l l b e c o n s i d e r e d now, s i n c e ,
a s i t was s t a t e d a b o v e , t h e r e is n o s i g n i f i c a n t c h a n g e ( o t h e r t h a n a time l a g ) i n t h e t r a n s i t i o n f r o m t h e o p e n sea t o V e n i c e (Goldmann e t a l . , 1 9 7 5 ) . t a n t f e a t u r e is t h e i n d e p e n d e n c e o f t i d e a n d s u r g e i n t h i s area.
t o h a v e a n y i n t e r a c t i o n , a n d s i m p l y b u i l d up.
An impor-
They seem n o t
A t a first g l a n c e , t h i s c a n be
s u r p r i s i n g , s i n c e t h e n o r t h e r n " h a l f " o f t h e sea i s a p l a t e a u s l o w l y d e c l i n i n g s o u t h w a r d s , u p t o 100 m d e p t h ( s e e F i g . 2 ) . I n s u c h a s h a l l o w a r e a , o c e a n o g r a p h e r s would e x p e c t s u b s t a n t i a l n o n l i n e a r i t i e s i n t h e t i d a l c o n s t i t u e n t s and r e l a t e d i n t e r a c t i o n s between t i d e and s u r g e .
In-
d e e d , t h e r a t i o o f t h e d e p t h t o t h e t i d a l r a n g e i s s t i l l v e r y l a r g e , a n d t h i s accounts f o r t h e observed l i n e a r i t y . S i n c e t i d e a n d s u r g e &re i n d e p e n d e n t , o n e c a n e a s i l y i m a g i n e t h e v a r i o u s rnixt u r e s t h a t can appear.
There can be a c o i n c i d e n c e o f f l o o d t i d e and s u r g e , s o
t h a t t h e l e v e l rises t o d a n g e r o u s v a l u e s , or t h e y c a n o t h e r w i s e c a n c e l e a c h o t h e r . What is i m p o r t a n t f o r n u m e r i c a l a n a l y s i s i s t h a t , g i v e n t h e t i d a l r e c o r d , by s u b t r a c t i n g t h e ordinary astronomical t i d e one obtains a kind of meteorological t i d e
t o be s t u d i e d s e p a r a t e l y .
429
N
40' N
F i g . 2.
The A d r i a t i c S e a .
D e p t h s are i n meters
TIDES A N D SEICHES T h e r a n g e o f t h e a s t r o n o m i c a l t i d e i n c r e a s e s from t h e SE o p e n i n g o f t h e A d r i a t i c t h r o u g h t h e NW e n d w h e r e V e n i c e i s : t h e s p r i n g r a n g e 80 cm ( a b o u t ) ( S t e r n e c k , 1 9 1 9 ) .
2(M2+S )
2
rises from 25 t o
This is consistent with t h e c l a s s i c a l picture
a c c o r d i n g t o w h i c h t h e t i d e , a s o b s e r v e d i n t h e A d r i a t i c , i s f o r c e d by t h e o s c i l l a t i o n o f t h e M e d i t e r r a n e a n by t h e O t r a n t o C h a n n e l , w i t h l i t t l e r e l e v a n c e o f t h e direct lunisolar attraction.
It s h o u l d b e r e m a r k e d t h a t r e c e n t c a l c u l a t i o n s t e n d
t o c h a n g e t h i s p i c t u r e (Tomasin, 1976). 8
A t least for t h e d i u r n a l p a r t of t h e t i d e ,
c o n t r i b u t i o n n o t l e s s t o 30% o f t h e o b s e r v e d r a n g e s h o u l d b e a t t r i b u t e d t o t h e
direct l o c a l f o r c i n g o f the astronomical effects. theoretical interest.
Indeed, t h i s should have only
Much more i m p o r t a n t , i n p r a c t i c e ,
is a n o t h e r r e m a r k :
the
d o m i n a n t t i d a l p e r i o d s , 12 a n d 2 4 h o u r s , a r e c l o s e t o t h e r e s o n a n t p e r i o d s o f t h i s
sea.
The s t u d y o f i t s e i g e n m o d e s i s c r u c i a l , s i n c e f r e e o s c i l l a t i o n s a r e o b s e r v e d
f r e q u e n t l y and w i t h r e l e v a n t amplitude.
T h i s way a common m e c h a n i s m i s f o u n d f o r
a r e g u l a r phenomenon, t h e t i d e , c o n s t a n t t h r o u g h t h e c e n t u r i e s , a n d f o r t h e random b u r s t s o f e n e r g y s t i m u l a t i n g t h e s e a i n t h e storms.
430
The f u n d a m e n t a l mode o f o s c i l l a t i o n ( " s e i c h e " ) o f t h e sea h a s a p e r i o d o f app r o x i m a t e l y 22 h o u r s ( K e s s l i t z , 1 9 1 0 ) .
O r d i n a r y s t o r m s h a v e a much s h o r t e r d u r a -
t i o n , b e i n g s h a r p a t m o s p h e r i c f r o n t s t h a t c r o s s t h e s e a moving e a s t w a r d s .
r e a c t s l i k e a pendulum h i t by a q u i c k p u l s e and s t a r t s o s c i l l a t i n g .
The sea
The r a t h e r
p e c u l i a r s h a p e o f t h e A d r i a t i c makes it as a n o r g a n p i p e , w e l l t u n e d o n i t s t o n e and w i t h l i t t l e d i s s i p a t i o n (Robinson e t a l . ,
1973).
Indeed, one can b e l i e v e
t h a t t h e r e i s l i t t l e i n t e r n a l f r i c t i o n f o r t h e same r e a s o n o f t h e l i n e a r i t y o f T h e r e is a l s o l i t t l e e x t e r n a l r a d i a t i o n o f e n e r g y s i n c e , a f t e r t h e g a t e ,
the tide.
t h e M e d i t e r r a n e a n i m m e d i a t e l y o f f e r s a wi de a n d d e e p b a s i n , v e r y c l o s e t o a n i d e a l i n f i n i t e o c e a n wh ere no e n e r g y c a n b e r a d i a t e d ( T o m a s i n , 1 9 7 1 ) .
A v e r y s l o w damp-
i n g o f t h e f r e e o s c i l l a t i o n s is o b s e r v e d , i f t h e r e i s no f u r t h e r p e r t u r b a t i o n . The 9 - v a l u e ( t h e f i g u r e o f m e r i t ) i s l a r g e r t h a n 1 0 0 , a n d t h e o s c i l l a t i o n s p e r -
s i s t f o r many d a y s ( s e e F i g . 3 ) .
100
-50
-I
I
I
16
17
18
A s i m i l a r b e h a v i o u r i s shown by t h e f i r s t h a r m o n i c ,
a l l y ( b u t n o t a l w a y s ) smaller i n a m p l i t u d e .
*
I
19
20
time (days)
11-hour o s c i l l a t i o n , u s u -
For t h i s f r e q u e n c y a n amphidromy i s
observed i n t h e n o r t h e r n A d r i a t i c , b o t h i n t i d e s and s e i c h e s ( A r t e g i a n i e t a l . , 1972).
I t i s now c l e a r t h a t a s u r g e h a s u s u a l l y a p e c u l i a r s h a p e , s i n c e , d u e t o s e i c h e s , i t is f o l l o w e d by many r e p l i c a s o f s u b s t a n t i a l h e i g h t , s p a c e d l e s s t h a n o n e da y.
It may o c c u r t h a t t h e f i r s t a r r i v a l o f t h e s u r g e d o e s n o t c o i n c i d e w i t h t h e f l o o d t i d e , while t h e second one does: was t h e case i n F e b r u a r y 1972.
p e o p l e f o r g e t t h e s t o r m and t h e f l o o d c o m e s , a s
A s an i n t e r e s t i n g c u r i o s i t y , cases c a n h e found
o f " n e g a t i v e " s u r g e s , c a u s e d by r e v e r s e m e t e o r o l o g i c a l c o n d i t i o n s , r e g u l a r l y f o l lo w e d by a s e i c h e s e q u e n c e (e.g. J u n e 3 0 , 1 9 7 5 ) . t h e problem, c a n b e f a c e d a t t h i s p o i n t .
Storm s u r g e s , t h e r e a l c o r e o f
431 O R I G I N OF SURGES
It t u r n s o u t from t h e r e c o r d s t h a t t h e s e a s o n a l p l o t o f t h e f l o o d s h a s a peak i n November and a c e r t a i n r e l e v a n c e i n December a n d J a n u a r y .
Since t h e r e is
l i t t l e d i f f e r e n c e i n t h e t i d a l p a t t e r n through t h e seasons, s p e c i a l meteorologic a l c o n d i t i o n s must f a v o u r t h e s u r g e s .
By a n a l y z i n g c a s e a f t e r c a s e , o n e s e e s
t h a t p u l s e s o f SE wind a l o n g t h e A d r i a t i c a r e t h e t r u e c a u s e . A more d e t a i l e d a n a l y s i s o f t h e m e t e o r o l o g i c a l d y n a m i c s g i v i n g o r i g i n t o f l o o d s
shows t h a t t h e L i g u r i a n S e a , f a c i n g t h e n o r t h w e s t e r n r e g i o n s o f I t a l y , i s a n i m p o r t a n t r e f e r e n c e p o i n t , s i n c e sometimes t h e st orm i s born t h e r e .
More f r e q u e n t l y ,
m o d e r a t e p e r t u r b a t i o n s from t h e A t l a n t i c r e a c h i n g t h e L i g u r i a n a r e a become much s t r o n g e r , a s i f a l o c a l mechanism were t r i g g e r e d .
One c a n s a y t h a t t h i s p l a c e i s
subject t o cyclogenesis, o r formation o f depressions, t h a t eventually migrate eastward.
There a r e important s t u d i e s concerning t h i s weather f e a t u r e (Speranza,
1975; B u z z i e t a l . , 1 9 7 8 ) .
It can b e mentioned h e r e t h a t t h e s u r g e sket ched i n
f i g u r e 1 was s e l e c t e d t o i l l u s t r a t e a s p e c i f i c case o f c y c l o g e n e s i s a n a l y z e d i n t h e l i t e r a t u r e , whose w e a t h e r maps a r e shown i n f i g . 4 .
The e f f o r t t o f o r e c a s t
t h e f l o o d s w i l l o b v i o u s l y t a k e a d v a n t a g e o f t h i s knowledge.
April 21, 1067, 1x00 GMI' F i g . 4. A t y p i c a l case of cyclogenesis. end o f t h e Alps.
April 22, 1967, 0000 GMT A d e p r e s s i o n is formed a t t h e w e s t e r n
PREDICTION BY STATISTICAL METHODS A b i g e f f o r t f o r t h e f o r e c a s t o f t h e f l o o d s was made i n t h e l a s t t e n y e a r s .
432 The p u r p o s e i s w a r n i n g t h e c i t y i n o r d e r t o r e d u c e t h e d a m a g e s o f t h e f l o o d .
Also, i f t h e p l a n n e d p r o t e c t i o n s f o r t h e l a g o o n w i l l b e b u i l t ( l i k e s l u i c e s a t t h e i n l e t s , t o b e c l o s e d o n l y when a s u r g e i s c o m i n g ) , a w a r n i n g s y s t e m w i l l b e vital. A v a r i e t y o f p r e d i c t i v e s c h e m e s was d e v e l o p e d , u s i n g e i t h e r s t a t i s t i c a l o r
d e t e r m i n i s t i c methods
-
o r both.
A d i s t i n c t i o n c o u l d a l s o b e made b e t w e e n s i m p l e
l o w a c c u r a c y s c h e m e s ( f r e q u e n t l y u s e f u l f o r a l o n g term f o r e c a s t ) a n d more s o p h i s t i c a t e d and p r e c i s e methods. An i d e a f o r t h e f o r m e r o n e s c o m e s from t h e a b o v e c o n s i d e r a t i o n s a b o u t s e i c h e s . They p e r s i s t f o r many d a y s , a n d t h i s means t h a t t h e A d r i a t i c h a s a g o o d memory. T r o u b l e s c o m i n g f r o m t h e s u r g e " r e t u r n s " c a n b e p r e d i c t e d by s i m p l y o b s e r v i n g t h e t i d e of t h e l a s t h o u r s .
On t h e s e a s s u m p t i o n s , a s i m p l e s c h e m e was d e v e l o p e d
( T o m a s i n , 1 9 7 2 ) w h e r e t h e f u t u r e l e v e l o f t h e sea i s e s t i m a t e d by a p r e d i c t i v e l i n e a r filter applied to t h e observed t i d e .
The f i l t e r w e i g h t s were s t a t i s t i c a l l y
o b t a i n e d from t h e r e c o r d i n g s of t h e p a s t . I n m a t h e m a t i c a l s y m b o l s , t h e estimate o f t h e sea l e v e l i n V e n i c e , a t time t , s; , i s o b t a i n e d a s t h e i n n e r p r o d u c t o f t w o v e c t o r s b u i l t by p r o p e r f i l t e r w e i g h t s
(f.
f,
, f, ,
... , f
) a n d sea l e v e l s ( 2 ~t - T s
T h o u r s b e f o r e t h e p r e d i c t e d time.
' s t-T-1 ' ...' t-T-n ) o b s e r v e d u p t o
Then S*
t
r s - f
-
-
The l e n g t h o f t h e f i l t e r , n + 1 , was f o u n d t o b e c o n v e n i e n t a t a b o u t 6 0 h c u r s . S t a t i s t i c s o f f e r another simple idea f o r an approximate expectation:
since the
w e a t h e r b e h a l f i o w i s somehow d e t e r m i n e d i n t h e cases o f i n t e r e s t , t h e s i m p l e s t
l o c a l a t m o s p h e r i c p a r a n e t e r , t h e p r e s s u r e , c o u l d g i v e some i n d i c a t i o n .
A numeri-
cal f i l t e r was b u i l t , l i k e t h e o n e d e s c r i b e d a b o v e , b u t now u s i n g a l s o p r e s s u r e f i g u r e s of t h e last hours.
- L2
Now s* = 2 f, +E , where p is t h e v e c t o r o f t h e observed p r e s s u r e values. t The l e n g t h o f t h e f i l t e r was now r e d u c e d t o 2 4 h o u r s . I n s p i t e of its s i m p l i c i t y ( a n y i n t e r e s t e d V e n e t i a n c o u l d u s e i t by h i m s e l f ) , t h e a d v a n t a g e i s r e m a r k a b l e . F i g u r e 5 s h o w s , f o r t h e c a s e a l r e a d y c o n s i d e r e d ( A p r i l 2 1 , 1 9 6 7 ) how t h e s i m p l e s c h e m e w o u l d h a v e p r e d i c t e d t h e f l o o d s i x h o u r s in a d v a n c e .
(Obviously, t h e
w e i g h t s o f t h e f i l t e r were o b t a i n e d f r o m a s a m p l e w h i c h d i d n o t i n c l u d e t h i s c a s e ) . F u r t h e r a d v a n c e s for t h e l a t t e r method ( w h i c h is s t i l l i n p r o g r e s s ) a r e exp e c t e d by r e l a x i n g t h e l i n e a r i t y of t h e method. A s s t a t e d a b o v e , more c o m p l i c a t e m e t h o d s are r e q u i r e d f o r b e t t e r r e l i a b i l i t y ,
a n d t h e p r e c i s e d y n a m i c s o f t h e sea m u s t b e s t u d i e d .
What r e a l l y g i v e s o r i g i n
t o s u r g e s i s t h e SE w i n d a n d , w i t h smaller e f f e c t , t h e low p r e s s u r e o r more p r e cisely a certain pressure gradient along the Adriatic.
The g o o d c o v e r a g e o f
w e a t h e r s t a t i o n s a l o n g t h e s h o r e s is o b v i o u s l y a f a v o u r a b l e f e a t u r e .
T h e i r re-
p o r t s c a n b e c o r r e l a t e d t o sea l e v e l e i t h e r v i a s t a t i s t i c s o r u s i n g storm s u r g e
433
6
10
12
0
6
12
0
10
F i g . 5. An e x a m p l e o f t h e p r e d i c t i o n o f a f l o o d u s i n g a s i m p l e s t a t i s t i c a l reg r e s s i o n . P r e d i c t . o r s a r e a v a i l a b l e f o r t h e s c h e m e u p t o 1 6 . 0 0 , A p r i l 2 1 , 1967.
equations. The s t a t i s t i c a l a p p r o a c h ( S g u a z z e r o e t a l . , above t o t h e whole A d r i a t i c .
1972) e x t e n d s t h e method d e s c r i b e d
T h e r e are more i n d e p e n d e n t v a r i a b l e s :
not the sim-
p l e a t m o s p h e r i c p r e s s u r e i n V e n i c e , b u t e s t i m a t e s o f t h e p r e s s u r e g r a d i e n t a n d wind
stress i n c e r t a i n i d e a l p o i n t s i n t h e m i d d l e o f t h e A d r i a t i c . s;. ' 2
' f + P l ' F , + P * - E 2+
..... ' + -1w ' G-1 + E 2 - G 2 + ....
Here 2 is a g a i n t h e v e c t o r o f t h e sea l e v e l s i n V e n i c e s t-T
t h e v e c t o r ( P l. , t - T ,
P l,t-T-l .
,...)
In formula
9
s t-T-
, . . - ; pi
is
of t h e p r e s s u r e v d u e s i n t h e i - t h s t a n d a r d
p o i n t . W . i s t h e v e c t o r o f t h e wind s t r e s s f i g u r e s i n t h e j - t h s t a n d a r d p o i n t , . ' -J F G a n d f a r e o b v i o u s " f i l t e r " v e c t o r s . More s p e c i f i c a l l y , t h i s model u s e d -k , -k f i v e p o i n t s for g r a d i e n t and stress estimates ( e q u a l l y s p a c e d a l o n g t h e axis o f t h e A d r i a t i c ) a n d t h i s f i e l d r e c o n s t r u c t i o n was made u s i n g s e v e n c o a s t a l s t a t i s m .
Data were t h e o r d i n a r y s y n o p t i c t h r e e - h o u r l y m e a s u r e m e n t s .
The " p a s t " d a t a en-
t e r i n g t h e f o r m u l a r e f e r r e d t o two d a y s f o r l e v e l a n d o n e d a y f o r w e a t h e r .
It
s h o u l d b e r e m a r k e d t h a t g r a d i e n t a n d s t r e s s were t a k e n o n l y i n t h e c o m p o n e n t al o n g t h e a x i s of t h e A d r i a t i c .
T h i s is a p o i n t o f i n t e r e s t , s i n c e i t emphasizes
t h a t t h e A d r i a t i c d y n a m i c s i s almost c o m p l e t e l y o n e - d i m e n s i o n a l f o r s u r g e s .
In t h e r a n g e o f f o r e c a s t o f s i x h o u r s ( T : 6 i n t h e p r e v i o u s f o r m u l a ) f l o o d s were p r e d i c t e d w i t h i n a few c e n t i m e t e r s i n t h e d o z e n s o f cases when t h e model
was t e s t e d (many o f t h e m i n r e a l t i m e ) .
Fig. 6 g i v e s a n example.
434
F i g . 6. An e x a mpl e o f p r e d i c t i o n u s i n g t h e s t a t i s t i c a l me thod by S g u a z z e r o e t a l . The s u r g e ( i . e . t h e d e v i a t i o n from o r d i n a r y t i d e ) i s shown by t h e s o l i d l i n e . Dashed c u r v e i s t h e f o r e c a s t , e a c h p o i n t b e i n g g i v e n s i x h o u r s i n a d v a n c e . ( s t a r t i n g p o i n t F e b r u a r y 1 2 , 1 9 7 2 , 0300 GMT).
One c a n c o n c l u d e t h a t u s i n g o n l y t h e w e a t h e r r e p o r t s from t h e A d r i a t i c c o a s t a n d t h e o b s e r v e d t i d a l l e v e l s , t h e p r e d i c t i o n i s p o s s i b l e f o r a b o u t s i x h o u r s ahead.
( F u r t h e r f o r e c a s t i n g is p r o g r e s s i v e l y u n r e l i a b l e , due t o t h e a r r i v a l o f
o t h e r a t m o s p h e r i c phenomena i n t h e m e a n t i m e ) .
PREDICTION BY DETERMINISTIC MODELS
B e t t e r s a t i s f a c t i o n comes from a d e t a i l e d i n s p e c t i o n o f w ha t h a p p e n s i n t h e A d r i a t i c when t h e a t m o s p h e r e
forces it.
More o r less t h e same i n f o r m a t i o n ( i . e .
w e a t h e r r e p o r t s from c o a s t a l s t a t i o n s ) c a n be u s e d f o r t h e i n t e g r a t i o n o f t h e s t o r m s u r g e e q u a t i o n s where t h e d r i v i n g f o r c e s a r e , a g a i n , t h e wind stress a nd the pressure gradients.
It i s o b v i o u s l y n o t o n l y a matter o f s a t i s f a c t i o n b u t
a l s o a b e t t e r k n owl edge o f t h e p h y s i c s o f t h e s u r g e a n d a more c o m p l e t e i n f o r m a t i o n f o r t h e various places.
1954)
The s t o r m s u r g e e q u a t i o n s c a n b e w r i t t e n (Proudma n,
435
a (UCOS -
a cos Ip ad
p)
+
av + a< = ax a t
where Q and X are l a t i t u d e and e a s t - l o n g i t u d e ,
0,
t is t i m e , 5 t h e e l e v a t i o n o f t h e
sea s u r f a c e , U a n d V t h e c o m p o n e n t s o f t h e t o t a l s t r e a m , i . e .
v e l o c i t y times
d e p t h , Fs a n d Gs t h e c o m p o n e n t s o f t h e f r i c t i o n o f t h e wind o n t h e sea s u r f a c e , FB a n d G B t h e c o m p o n e n t s o f t h e water f r i c t i o n on t h e s e a b o t t o m , pa t h e a t m o s p h e r i c p r e s s u r e o n t h e s e a , h t h e water d e p t h ,
P
t h e d e n s i t y o f t h e water ( a s-
sumed u n i f o r m ) , a t h e mean r a d i u s o f t h e e a r t h , g t h e a c c e l e r a t i o n o f g r a v i t y , w t h e a n g u i a r s p e e d of t h e e a r t h ’ s r o t a t i o n .
T h e s e e q u a t i o n s are l i n e a r i z e d a n d i n t e g r a t e d w i t h r e s p e c t t o d e p t h . c a n b e s a f e l y assumed after t h e above r e m a r k s .
Linearity
This accounts for t h e suppression
o f a l l non l i n e a r terms i n d e r i v i n g t h e s e e q u a t i o n s a n d a l l o w s t h e a s s u m p t i o n o f a l i n e a r i z e d b o t t o m f r i c t i o n , as i t was d o n e i n p r a c t i c e . A v a r i e t y o f w o r k s were d e v e l o p e d ( T o m a s i n , 1971; S t r a v i s i , 1972;’ A c c e r b o n i e t a l . ,
1973) u s i n g t h e above b a s i c e q u a t i o n s , c o r r e c t l y r e p r o d u c i n g t i d e s , s e i c h e s and surges.
I n t e g r a t i o n t e c h n i q u e s were e s s e n t i a l l y a l r e a d y known f r o m t h e l i t e r a -
t u r e of h y d r o d y n a m i c a l n u m e r i c a l m o d e l s ( H a n s e n , 1 9 5 6 ; H e a p s , 1 9 6 9 ) .
In particu-
l a r , r e d u c t i o n t o o n e d i m e n s i o n was p o s s i b l e d u e t o t h e s h a p e o f t h e A d r i a t i c . And t h i s was r e a l l y t h e s c h e m e more e x t e n s i v e l y t e s t e d i n p r a c t i c e ( F i n i z i o , 1970; Tomasin, 1973).
I n formula
w h e r e t h e x a x i s i s d i r e c t e d a l o n g t h e s e a , w is t h e w i d t h o f t h e c r o s s s e c t i o n a n d Q t h e d i s c h a r g e a c r o s s i t , o r , s a y , t h e a v e r a g e v e l o c i t y times t h e a r e a o f the section. The e q u a t i o n s are c l e a r l y s i m p l i f i e d a l s o b e c a u s e t h e e a r t h c u r v a t u r e i s d i s regarded:
n e e d l e s s t o s a y , a l l t h e s e s i m p l i f i c a t i o n s would n o t b e p o s s i b l e f o r
o t h e r a r e a s , l i k e f o r e x a m p l e t h e M e d i t e r r a n e a n , whose m o d e l l i n g i s m e n t i o n e d below.
436 I n f o r e c a s t i n g , t h e o n e d i m e n s i o n a l s c h e m e was u s e d o n t h e f i e l d , u s i n g s y n o p t i c m e t e o r o l o g i c a l d a t a a n d i n t e g r a t i n g t h e e q u a t i o n s by a n e x p l i c i t f i n i t e - d i f f e r e n c e s scheme. issued:
The w e a t h e r f i e l d i s known u p t o t h e time when t h e f o r e c a s t i s
t h e s i t u a t i o n is s u p p o s e d c o n s t a n t f o r t h e f o l l o w i n g h o u r s and t h e equa-
t i o n s g i v e f u t u r e sea l e v e l u n d e r
this
hypothesis.
Consistent with the statis-
t i c a l a p p r o a c h , i t t u r n s o u t t h a t a t a n y time t h e r e i s e n o u g h i n f o r m a t i o n t o p r e d i c t s a f e l y t h e sea l e v e l f o r a b o u t s i x h o u r s .
A s expected, the
a b o v e method
t u r n e d o u t t o b e s a t i s f a c t o r y as a p r e d i c t o r .
FURTHER DEVELOPMENTS I m p r o v e m e n t s o f t h e p r e d i c t i o n a r e e x p e c t e d by w i d e n i n g t h e l i m i t s i n r e s e a r c h h a s been confined so far. p r e d i c t i o n s was a v o i d e d .
which
It w i l l b e r e m a r k e d t h a t t h e u s e o f w e a t h e r
Over t h e M e d i t e r r a n e a n t h e y a r e made d i f f i c u l t by t h e
i n f l u e n c e o f o r o g r a p h y w h i c h c o m p l i c a t e s t h e f l o w o f a i r masses a n d g e n e r a t e s l o c a l effects:
c y c l o g e n e s i s on t h e w e s t e r n l e e o f t h e A l p s was m e n t i o n e d a b o v e .
The G l o b a l A t m o s p h e r i c R e s e a r c h P r o g r a m (GARPI h a s c r e a t e d a s u b p r o g r a m o n Air Flow Over a n d Around M o u n t a i n s a n d t h e f i r s t e x p e r i m e n t a r o u n d t h e A l p i n e b a r r i e r ( A l p e x ) i s p r o p o s e d f o r 1981.
This e f f o r t should help i n understanding the dy-
namics o f p e r t u r b a t i o n s i n t h e M e d i t e r r a n e a n , w i t h t h e f i n a l o b j e c t i v e o f a n a c c u r a t e p r e d i c t i o n 12 t o 24 h o u r s a h e a d :
Venice, I t a l y and s e v e r a l c o u n t r i e s w i l l
substantially benefit. For t h e s p e c i f i c w a r n i n g o f f l o o d s , a wind f o r e c a s t i n g model is b e i n g d e v e l o p e d
(Palmieri a t a l . , 1 9 7 6 ) .
S t a t i s t i c a l m e t h o d s a l s o are b e i n g i m p l e m e n t e d i n V e n i c e
for the prediction o f the'pressure f i e l d i n the central Mediterranean. From t h e h y d r o d y n a m i c a l p o i n t o f v i e w , i t i s n o t i c e a b l e t h a t n o a t t e m p t was made t o f o r e c a s t e x t e r n a l s u r g e s , i . e . phenomena o r i g i n a t i n g o u t o f t h e A d r i a t i c , b u t g i v i n g s i g n i f i c a n t e f f e c t i n it.
From e x p e r i e n c e , t h e y seem n o t t o e x i s t ; s o
f a r t h e l o w f r e q u e n c y c h a n g e s i n t h e l e v e l o f t h e M e d i t e r r a n e a n h a v e b e e n monit o r e d c l o s e t o t h e O t r a n t o Channel and t r a n s m i t t e d t o Venice f o r a d i r e c t u s e i n t h e p r e d i c t i v e models (Mazzoldi et a l . ,
1973).
For a b e t t e r i n s i g h t i n t h e s e
large s c a l e phenomena a f f e c t i n g t h e M e d i t e r r a n e a n , a s p e c i f i c n u m e r i c a l model i s b e i n g d e v e l o p e d , t o f i l l u p t h e t h e o r e t i c a l g a p t h a t was s o f a r o v e r c o m e by d i -
rect i n s p e c t i o n . Time i s r i p e f o r w i d e r a n a l y s i s , w h e r e t h e M e d i t e r r a n e a n i s f u l l y c o n s i d e r e d f o r its hydrodynamical c o n t r i b u t i o n s and f o r t h e m e t e o r o l o g i c a l p a t t e r n s t h a t develop over it.
The g o a l o f a w a r n i n g f o r V e n i c e f l o o d s w i t h a n a d v a n c e o f
t w e l v e h o u r s o r o n e d a y seems n o t t o b e c h i m e r i c a l .
437 ACKNOWLEDGEMENTS
The h e l p o f R .
D a z z i , G . A l d i g h i e r i , and Mrs. J . Z a n i n , o f t h e CNR l a b o r a t o r y ,
h a s b e e n o f u t t e r i m p o r t a n c e f o r t h i s work.
P r e c i o u s s u p p o r t was g i v e n by t h e
C e n t r o S c i e n t i f i c 0 IBM o f Veni ce and t h e CRIS-ENEL b r a n c h o f V e nic e -Me s tre .
REFERENCES
A c c e r b o n i , E. a n d Manca, B., 1973. S t orm s u r g e s f o r e c a s t i n g i n t h e A d r i a t i c Se a by means o f a t wo-di mensi onal h y d r o d y n a m i c a l n u m e r i c a l mode l. B o l l . G e o f i s . Teor . Appl . , 15: 3-22. A r t e g i a n i , A . , Tomasi n, A . and Goldmann, A . , 1972. Sur l a dyna mique d e l a mer A d r i a t i q u e d u e aux e x c i t a t i o n s m & t & o r o l o g i q u e s . Rapp. Comm. I n t . Mer MGdit., 21:181-183. B u z z i , A . a n d T i b a l d i , S . , 1978. C y c l o g e n e s i s i n t h e l e e o f t h e A l p s : A c a s e s t u d y . Q u a r t . J . R . Met. SOC., 104:271-287. F i n i z i o , C . , Palmieri, S. and R i c c u c c i , A . , 1970. A n u m e r i c a l model o f t h e Adria t i c S e a f o r t h e s t u d y and p r e d i c t i o n o f sea t i d e s a t V e n i c e . 1st. F i s . At mo sf ., STR 12. Goldmann, A . , R a b a g l i a t i , R . and S g u a z z e r o , P . , 1975. P r o p a g a z i o n e d e l l a marea n e l l a l a g u n a d i Venezia: a n a l i s i d e i d a t i r i l e v a t i d a l l a r e t e m a r e o g r a f i c a l a g u n a r e n e g l i a n n i 1972-73. R i v. I t a l . G e o f i s . , 2:119-124. Hansen, W . , 1956. T h e o r i e z u r E r r e c h n u n g d e s W a s s e r s t a n d e s und d e r Stromungen i n Randmeeren n e b s t Anwendungen. T e l l u s , 8:287-300. Heaps, N.S., 1969. A t wo-di mensi onal n u m e r i c a l sea model. P h y l . T r a n s . A 265: 93-137. 1910. Das Gezei t enphanomen i n Hafen von P o l a . Mittl. Geb. K e s s l i t z , W.V., S e e w e s e n s , 38: H . V - V I . M a z z o l d i , A . , D a l l a p o r t a , G . , G a s p a r i , A . , C u r i o t t o , A . a nd P e r u z z o , G . , 1973. S i s t e m a d i t e l e m i s u r e d a s t a z i o n i a u t o m a t i c h e f i s s e . CNR-LSDGM, V e n e z i a , R a p p o r t 0 T e c n i c o n . 37. P a l m i e r i , S., F i n i z i o , C. and C o z z i , R . , 1976. The c o n t r i b u t i o n of m e t e o r o l o g y B o l l . Geofis. t o t h e s t u d y a nd p r e d i c t i o n o f h i g h t i d e s i n t h e A d r i a t i c . T e o r . Ap p l ., 19:191-198. Proudman, J . , 1954. Note on t h e dynami cs o f s t o r m - s u r g e s . Mon. N ot. R . a s t r . S OC . Geophys. S u p p l . , 7:44-48. R o b i n s o n , A . R . , T omasi n, A. a n d A r t e g i a n i , A . , 1973. F l o o d i n g o f V e n i c e : Phenomenology a n d p r e d i c t i o n o f t h e A d r i a t i c s t o r m s u r g e . Q u a r t . J . R . Met. S O C . , 99~688-692. S g u a z z e r o , P . , Giommoni, A . and Goldmann, A . , 1972. An e m p i r i c a l model f o r t h e p r e d i c t i o n o f t h e sea l e v e l i n Veni ce. IBM I t a l i a T e c h . Rep. CSVOO6. S p e r a n z a , A . , 1975. The f o r m a t i o n o f b a r i c d e p r e s s i o n s n e a r t h e A l p s . Ann. d i G e o f i s . , 28:177-217. 1919. Die G e z e i t e n e r s c h e i n u n g e n i n d e r A d r i a . D e ns c hr. Akad. Sterneck, R.V., Wiss. Wien, 96:277-324. S t r a v i s i , F . , 1972. A n u m e r i c a l e x p e r i m e n t on wind e f f e c t s i n t h e A d r i a t i c Se a . Acc. Naz. L i n c e i , Rend. Sc. F i s . , m a t . , n a t . , 52:187-196. To ma sin , A . , 1971. A p p l i c a t i o n o f t h e h y d r o d y n a m i c a l n u m e r i c a l i n t e g r a t i o n method t o t h e A d r i a t i c S e a . P h y s i c s o f t h e S e a , T r i e s t e , 13-16 O c t . , Accad. Naz. L i n c e i , Quad. n . 2 0 6 , 119-121. Tomasin, A . , 1972. A u t o r e g r e s s i v e P r e d i c t i o n o f S e a L e v e l i n t h e N o r t h e r n Adria t i c . R i v . I t a l . G e o f i s . , 21:211-214. To ma si n , A . , 1973. A comput er s i m u l a t i o n o f t h e A d r i a t i c S e a f o r t h e s t u d y o f i t s d y n a mic s and f o r t h e f o r e c a s t i n g o f f l o o d s i n t h e town o f V e n i c e . Comp. Phys. Comm., 5: 51-55.
438 T o m a s i n , A . , 1 9 7 6 . The d y n a m i c s o f t h e d i u r n a l t i d e i n t h e A d r i a t i c S e a : p r e l i m i n a r y r e s u l t o f a r e v i s i t i n g a n a l y s i s . Rapp. Comm. I n t . Mer M d d i t . , 2 3 : 4 9 . U n e s c o , 1969. R a p p o r t 0 s u V e n e z i a . M o n d a d o r i , M i l a n o , 348 p p .
439
THE RESPONSE OF THE COASTAL WATERS OF N.W. ITALY
ALAN J. ELLIOTT SACLANT ASW Research Centre, Viale San Bartolomeo 400, 19026 La Spezia, Italy
ABSTRACT
Two-month long low pass records of the coastal currents and winds have been analysed for two locations off the N.W. coast of Italy. Most of the energy was found to be in the long period (>20 day) motions and there was low coherence between the currents and wind except for time scales around 5 days. This suggests that either the wind was exciting a rotational mode of the entire Western Mediterranean or else that the weather systems were more coherent spatially at the 5 day time scale. A depth-integrated hydrodynamic model is being used to resolve the time scales and the effects of bottom topography. The coastal currents may make a significant contribution to the day-to-day variability in sound speed, especially near frontal zones, due to the alongshore advection of water of differing acoustic properties. Consequently, accurate prediction of the sound speed at a fixed location may not be obtained until the coastal dynamics are clearly understood.
INTRODUCTION
During recent years there has been an increasing interest in problems related to shallow water acoustics. As a result, two distinct problem areas have arisen which need to be addressed by the research.
The first of these involves the
acoustic propagation itself, the second is more oceanographic in nature and concerns the variability of the acoustic properties of the water near a coast.
In the deep
ocean, for many acoustic purposes, the water can be considered as being well-mixed in the horizontal plane and only the vertical variations need to be considered. Thus the majority of sonar models do not incorporate range-dependent temperature fields and variable bottom topography but, instead, use a single vertical temperature profile and a flat bottom to characterise a region of interest. For many purposes this is an acceptable approximation, but it is not generally valid in shallow water.
In the coastal zone, as well as the complicating effect of a
sloping bottom, there can be significant variations in the salinity, temperature and sound speed characteristics of the water over relatively short horizontal distances. Among the mechanisms which can cause the variations are the fresh water input by rivers, upwelling induced by the coastal winds, and the enhanced vertical mixing due to the strong tidal and storm generated currents. As a result the coastal zone is usually a region of high acoustic variability.
The problem
440
Fig. 1. The Ligurian and Tyrrhenian Seas showing the mooring locations. The dashed line represents the boundary of the numerical model.
441
is complicated further since the coastal waters are not static but are constantly being moved by the coastal currents under the influence of the winds. Consequently, measurements made at a selected location on one day may not be valid for the following day due to the combined effect of the high spatial variability in the sound speed and the advective effects of the currents. For the oceanographer there are two distinct problems to be resolved: first, can we obtain insight into the mechanisms which lead to the high variability in the coastal waters, e.g. the processes which generate fronts; second, given that high spatial variability exists in the coastal waters, what are the time scales associated with the coastal currents which would contribute to the temporal acoustic variability at a fixed location, and to what extent can we succeed in modelling the dynamics of the coastal response? In order to answer some of these questions, and to provide an oceanographic input into what is essentially an acoustic problem area, a series of field measurements were made in the coastal waters of N.W. Italy and this has been combined with a numerical study of the region. The purpose of this paper is to present some of the observational results and show that, in general, there was a lack of coherence between the coastal currents measured at different locations. The second part of the paper describes how a depth-integrated numerical model is being used to resolve the role that might be played by the variations in bottom topography: one of the factors which may have contributed to the low coherence in the measurements. THE OBSERVATIONS During April and May of 1977 current measurements were made at two locations off the N.W. coast of Italy in water approximately 100 m deep and 15 km offshore. The two moorings, which were 100 km apart, were located near Elba on opposite sides of the shallow water which extends between the Italian mainland and the island of Corsica (Fig. 1).
Three oceanographic cruises were made at approxi-
mately monthly intervals to survey the temperature/salinity properties of the coastal waters near the mooring positions, and meteorological and sea level data were obtained from established coastal recording stations. Each mooring supported two current meters:
one at a depth of 20 m and the other at 80 m.
The data
series were filtered with a low pass filter to remove the fluctuations with periods less than two days; the data were then resampled at 6-hour intervals. The hydrography There was a uniform warming of the coastal waters during the months of April and May, the surface temperature increasing from about
l4OC
to 18OC. Near-bottom
temperatures remained constant and the warming was confined to the upper layers of the water column. Since there were no significant horizontal temperature gradient$
442
F i g . 2 . I n f r a r e d s a t e l l i t e image of t h e T y r r h e n i a n S e a , A p r i l 2 3 , 1 9 7 7 ; t h e w a r m w a t e r masses show up as d a r k e r p a t c h e s , p a r t s of S i c i l y and C a l a b r i a a r e c l o u d c o v e r e d ( c o u r t e s y of t h e Univ. of Dundee).
443
along the coast and there was no evidence of a warm water mass to the south, it appears that advective effects were not important and that the warming was due to solar heating.
This view was supported by a satellite image of the Tyrrhenian
Sea taken on April 23 (Fig. 2) which showed insignificant large-scale horizontal temperature gradients.
Therefore, there appeared to have been a uniform heating
of the coastal waters, due to seasonal changes, throughout the period of the study. Miller et al. (1970) measured water temperature along a section parallel to the coast and extending from Elba to Calabria.
Their data show an alongshore tempera-
ture change of about 0.5OC over a distance of about 600 km (the warmer water lying to the south). at 5 cm s
-1
If we assume that the mean flow was to the N.W., i.e. up the coast,
and we write
dT then -represents the total temperature change and can be estimated from the data. dt aT Consequently, -which represents the effect of local heating can be computed. at The data showed that: 7.70
=
Total change (100%)
7.28
+
0.42
Local heating
Advection
(95%)
(5%)
which supports the conclusion that the increase in temperature was mainly due to a seasonal warming and not due to the advection of warm water from the south. The increase in temperature caused a reduction in the density of the surface water and at both locations the water column changed from being well-mixed during March to a two-layered system, with a surface mixed-layer about 10 m deep during May.
At both positions, the current meters were moored beneath the depth of the
surface mixed-layer.
The coastal winds
There was significant spatial variability in both the strength and the direction of the low pass coastal winds.
Whereas the mean stress at the three southern
stations (Civitavecchia, Olbia, and Ponza) was directed eastwards with a strength -2
of about 0.5 dynes cm
, the
mean stress at Genova and Pisa was about an order of
magnitude weaker; at Pisa the mean stress vector was directed towards the N.E. while at Genova it was towards the N.W.
The anti-clockwise rotation of the mean
wind may have been caused by cyclogenesis (Cantu, 1 9 7 7 ) , or it may have been due to orographic effects.
To resolve the structure of the wind, empirical orthogonal
function analysis (Wallace and Dickinson, 1972) was applied to the five components of E-W wind stress. Of the five modes isolated (since there were five input series)
444
ALONGSHORE
COMPONENTS
3.01 2.0
5
1.0
‘v)0 0
-
c 1.0 > T3 -2.0
WIND STRESS 0
20-
NORTH MOORING
- 30 301 20-
..
-201
SOUTH MOORING
- 30
APRIL
MAY
1977
I
I
I
I
I
I
1
10
20
1
10
20
Fig. 3. Alongshore components of the low pass currents and coastal wind. The current at 20 m is shown by a solid curve, the 80 m current by a dotted curve
445
the first mode was highly correlated with the components of stress at Pisa, Civitavecchia, Olbia and Ponza but only explained 1% of the variance in the Genova record.
In contrast, the second mode accounted for 95% of the Genova variance
but was uncorrelated with the other locations. A similar result was obtained when the analysis was repeated for the N-S components of stress. As a result of this partition of the wind records it was decided that the Genova wind was not representative and that the record had been unduly influenced by orographic effects; it was therefore excluded from further analysis. The remaining 4 vector series were then averaged to produce a time series of the large-scale
wind.
The geostrophic wind stress, calculated from atmospheric pressure, was not
coherent with the observed wind and was a poor predictor of the current response. The coastal currents In Fig. 3 are shown the low pass alongshore components of the currents and mean wind, positive currents and wind stress being directed up the coast towards the N.W.
The current records showed marked fluctuations at a time scale comparable
to the record length while the wind appeared to contain more energy at the shorter time scales. The flow was northward when the wind was near zero and only reversed its direction during strong southward winds.
Linear regression showed that under
conditions of zero wind the flow would be towards the N.W. at both mooring locations, and that this density-driven flow which appeared to be independent of depth had a strength of about 5 cm s-’. Spectra for current, adjusted sea level and atmospheric pressure at Livorno, and the alongshore wind are shown in Fig. 4 and show that the energy was contained in the longer period motions.
Sea level and wind spectra peaked at around 20 days
while the current and atmospheric pressure had maxima at periods comparable to the record length. The spectra decreased steadily between 20 days and 3 days; in particular, there was no pronounced spectral peak at the 5 day time scale. However, when the coherence was computed between pairs of variables then the coherence was usually greater at 5 days than at other periods.
As an example, Fig. 5 shows the
coherence between the alongshore wind and each of the four current records. At both the north and south mooring the bottom currents were highly coherent with the wind at the 5 day time scale. Comparable results were found when sea level and atmospheric pressure were analysed: the coherent response appeared to be confined to a band around the 5 day period.
Since the forcing variables (e.g. wind and
pressure) did not contain an excess of energy at this time scale it suggests that the coastal waters were responding in an organised manner to the 5 day forcing.
446
.O
.1
.2
.3
.4
.5
FREQUENCY Fig. 4. Normalised spectra of alongshore current and wind, sea level and atmospheric pressure. Frequency is in cycles per day.
0
.1
.2
.3
.4
.5
FR E Q U EN C Y Fig. 5. Coherence squared between the alongshore currents and alongshore wind: N signifies the north mooring, S signifies the south mooring.
447
A MODEL O F THE LIGURIAN AND TYRRHENIAN SEAS
Since the local bottom topography, the geometry of the deep basins and rotational effects were suspected of being factors which could influence the currents, the problem is being approached numerically through the use of a hydrodynamic model. A depth-integrated model, suitable for general application to both deep and shallow water areas, was developed; the model includes density terms as well as a salinity balance and a temperature (or pollutant) balance equation. The full system of equations used in the study was
p
=
P0(1
+
as
+ Bc)
where u1 and uz are the components of the horizontal velocity, 17 is the surface elevation above mean sea level, d is the bottom depth with respect to mean sea level,
s
is the salinity and
c represents the concentration of a pollutant or,
with some modification to the equation, the temperature distribution. N and K represent the horizontal eddy stress and diffusivity. A term representing a fresh
448
Fig. 6. Bottom topography of the Ligurian and Tyrrhenian Basins as used in the numerical study; the mooring locations are circled. The area shown is approximately 550 x 380 km2 and the grid size is 20 km. The dashed lines represent the boundaries used to separate the two basins.
water source was included in the continuity equation to allow for river run-off. The components of surface stress, FS1 and FS2, were calculated using a quadratic drag law for the wind stress, and FBI and FB2 were the components of bottom stress -also quadratic in form. Consequently, the equations form a basically linear system with the exception of the quadratic friction term. The equations were solved explicitly in a standard manner using a leap-frog scheme with centred space differences on a regular grid. spatially staggered
OF.
The variables were
the grid so that surface elevation, density and depth
were specified at the centre of each rectangular element while the two velocity components were specified at the mid-points of adjacent sides of the grid elements (Leendertse, 1970; Heaps, 1969; Tee, 1 9 7 6 ) .
Under many circumstances (including
the present) density effects can be neglected, in which case only the first three equations need to be solved. The time scales of the Ligurian and Tyrrhenian Basins As a first step towards resolving the normal modes of the entire Western Medi-
terranean the model was applied to the more restricted region comprising the Ligurian and Tyrrhenian Seas as shown in Fig. 1. I n addition, a t t e n t i o n Was
directed towards the shorter time scales, i.e. those of about one day and less. The current observations discussed previously provided a problem of a qualitative nature to which the model could be applied:
Before the current records had been
low pass filtered, both progressive vector diagrams and spectral calculations had revealed a marked difference in the currents at the two mooring locations, especially during the first month of measurement. Whereas the currents at the northern mooring were generally of long period (>lo days) and flowed parallel to the coastline (having twice as much energy in the alongshore direction as in the on/offshore), the currents at the southern mooring were mainly inertial with a period of around 17.5
hours.
There was nearly an order of magnitude more inertial energy at the
southern mooring than there was in the north, the tidal energy being insignificant at both locations. Previous observations of inertial motions in the Mediterranean have shown that they are rarely coherent over horizontal scales exceeding 10 km. However, in the present case we are not investigating the coherence between inertial motions -we are, instead, seeking an explanation for the apparent lack of such motions at the north mooring. To determine the spatial variability in the local response to forcing the model
was applied to the region shown in Fig. 6.
For the boundary conditions there was
assumed to be no flow through the Straits of Sicily, and the surface elevations along the left hand open boundary were specified everywhere as being the sum of 250
harmonic terms of equal amplitude but with periods ranging uniformly from 12
450
minutes to 50 hours.
Thus the interior of the model was forced by the periodic
elevations along the left hand boundary, and we shall be looking for evidence of local resonance with the applied forcing. The model was run to simulate 100 hours of currents and elevations within the system and the predicted currents at the two mooring locations and at other selected positions were stored on disc.
Cross-
spectral analysis was then used to isolate the coherent fluctuations. A second numerical experiment was then made by separating the two basins as shown by the dashed lines in Fig. 6 and the calculations were repeated. Similar techniques have been used to compute the periods of oscillation of the English Channel (Flather, 1976), the Gulf of Genova (Papa, 1977a) and the South Sicilian Basin (Colucci and Michelato, 1976). Table I summarises the results for the separate and combined basins.
It shows
that whereas the Tyrrhenian Basin responded locally to oscillations with a nearinertial period (around 17.5 hours) these periods were locally suppressed in the Ligurian Basin. The coherent motions in the Ligurian Basin were found to be at 33 hours, 3.6 hours and between 1 to 2 hours; this agrees with previous calculations (Papa, 1977a,b) which isolated the 3.6 and 1.2 hour response but not that at 33 hours.
The local suppression of near-inertial signals at fhe mooring position
within the Ligurian Basin could explain the current characteristics discussed above. The oscillations at 3.6 and 1 to 2 hours can be identified with the seiche motions that would arise in the northern basin shown in Fig. 6. However, this mechanism cannot explain the coherence at 33 hours.
For the typical shelf widths of the
Ligurian Sea as resolved by the model we can compute the phase speed of the fundamental barotropic shelf wave (Robinson, 1964) to be of the order of 400 km/day; such a wave would take about 30 hours to travel around the Ligurian coastline shown in Fig. 1. Thus if energy were being supplied over a broad spectrum by the deep water lying to the SW of the Ligurian Sea then we might expect to see a local response with a time scale of around 30 hours. TABLE I
Periods of coherent fluctuations (for which coherence squared exceeded the 95% significance level): (a) computed for the combined system, (b) computed for the Ligurian Basin, (c) computed for the Tyrrhenian Basin, (d) from analysis of sea level at Genova and Livorno. Period in hours (a) Combined
33.1
25.0
(b) Ligurian
33.1
-
(c) Tyrrhenian
-
(d) Genova and Livorno sea level
33.4
25.4
-
-
-
3.6
1.2
-
-
-
-
3.6
1.2
19.9
16.6
14.2
12.4
-
-
-
3.6
-
19.9
21.4
-
-
451 An independent check of the calculations was made by analysing hourly unfiltered two-month long sea level records from Genova and Livorno (Fig. 1). The records were coherent at all periods longer than 4 days (due mainly to barometric effects) and also at the periods given in Table I. Good agreement was found between the periods computed by the model and those isolated from the elevation data.
In
particular, evidence for periodicities within the Ligurian Basin at 33 and 3.6 hours was obtained. DISCUSSION The observational data suggest that the coastal waters of N.W. Italy may not respond like a straight open coastline (e.9. like the west coast of the United States) but may instead have characteristics more similar to a large enclosed lake. Thus the dominant response to forcing may involve the dynamics of the entire Western Mediterranean. Consequently work is now in progress to resolve the normal modes of the western basin and to look for a possible 5 day periodicity.
Alternatively,
the weather systems themselves may be more spatially coherent at the 5 day time scale and may thus be more efficient at driving the coastal response:
this possi-
bility is also being investigated further. The data analysis has made use of a 'large-scale wind' which was obtained by computing the vector mean of several coastal wind records.
In view of the poor
coherence found between this wind and the coastal currents it 'suggests that this may not be a meaningful approach. It may be necessary to specify the wind everywhere (as is done in modelling the storm surge effects in the North Sea, e.g. Heaps (1969)) in order to obtain a satisfactory prediction of the coastal response. Returning to the problem of acoustic variability in the coastal waters, we can make some estimate of the effects of the coastal currents. From the observations the alongshore current had an amplitude of around 7.0 cm
at the 20 day time
s-'
scale: this is equivalent to a maximum displacement of the water of about 6.1 km -1
in one day.
Since the maximum observed gradient in sound speed was about 0.1 m s -1
per km the current could produce a change of about 0.60 m s
from one day to the
next at a fixed location. In comparison, seasonal heating caused an increase in -1
sound speed of about 0.26 m s
per day.
Thus, even in this region of extremely
low horizontal gradients, the currents may have contributed significantly to the day-to-day variability in sound speed.
Therefore in other areas, especially near
frontal locations, accurate prediction of acoustic characteristics will not be obtained until we have a better understanding of the dynamics of the coastal zone.
452
ACKNOWLEDGEMENTS
F. De Strobe1 assisted with the collection of data at sea, and P. Giannecchini, E. Nacini and G. Tognarini contributed to the analysis and presentation of the results. REFERENCES
Cantu, V., 1977. The climate of Italy. In: C.C. Wallen (Editor), World Survey of Climatology. Elsevier, pp. 127-183. Colucci, P. and Michelato, A., 1976. An approach to the study of the Marrubbio Phenomenon. Boll. Geof. Teor. Appl., XIX:3-10. Flather, R.A., 1976. A tidal model of the north-west European continental shelf. Mem. SOC. Roy. Sci. Liege, 6:141-164. Heaps, N.S., 1969. A two-dimensional numerical sea model. Phil. Trans., A265:93-137. Leendertse, J.J., 1970. A water-quality model for well mixed estuaries and coastal seas. Rand Corporation, RM-6230-RC. Miller, A.R., Tchernia, P. and Charnock, H., 1970. Mediterranean Sea Atlas. Woods Hole Oceanographic Institute Atlas Series, Vol. 111. Papa, L., 1977a. The free oscillations of the Ligurian Sea computed by the H-N method. Deutsch. Hydrogr. Zeitschr., 30:81-90. Papa, L., 1977b. The free oscillations of the Ligurian Sea. A statistical inves tigation. Boll. Geof. TeOr. Appl., XIX:269-276. Robinson, A.R., 1964. Continental shelf waves and the response of sea level to weather systems. J. Geophys. Res., 69:367-368. Tee, K.T., 1976. Tide-induced residual current, a 2-D nonlinear numerical tidal model. J. Mar. Res., 34:603-628. Wallace, J.M. and Dickinson, R.E., 1972. Empirical orthogonal representation of time series in the frequency domain. J. App. Meteor., 11:887-892.
453
A NUMERICAL MODEL FOR SEDIMENT TRANSPORT
J.P. LEPETIT, A. HAUGUEL E.D.F., Direction des Etudes et Recherches, Laboratoire National d'Hydraulique, Chatou (France)
ABSTRACT
We introduce here a numericaltwodimensional model for sediment transport which permits to compute the impact of a coastal structure on the bottom evolution. The introduction of current disturbance and some assumptions using difference of time scale between current and bottom evolutions permits to obtain a propagation equation driving the bottom evolution. The model has been calibrated in the case of the local scour aroundajetty. A last, it has been applied to the bottom evolution in the vicinity of the new port of Dunkerque. IWRODUCTION One of the impacts of a large coastal structure is its effect on current pattern in the vicinity of the structure. These changes in current conditions will induce changes in the sediment transport pattern and may disturb an existing equilibrum thus causing large changes in bottom topography in the vicinity of the structure. To assess the severity and extend of topographical changes induced by the structure the interaction of the resulting fluid motion with the bottom evolution must be properly reproduced.
The study of sediment drifting and movable bed evolution is a difficult problem from a physical and mechanical point of view. But the sediment transport relationship admitted, the problem is reduced to the study of a conservative phenomena.
An other problem is the difference of time scale between current and bottom evolution. It is impossible (because of cost), to compute simultanously the bottom evolution and the current by the classical way. Nevertheless, the interaction between the two is fundamental for the bottom evolution.
This paper presents a two dimensional mathematical sediment transport model taking into account the influence of the bottom evolution upon the current pattern and shows how this particular aspect of the interaction drives the ripples propagation.
454 THEORETICAL ANALYSIS
Bed c o n t i n u i t y equation and sediment t r a n s p o r t r e l a t i o n s h i p
4 be
Let
t h e sediment t r a n s p o r t v e c t o r and
5
t h e bottom e l e v a t i o n ; t h e bed conti-
n u i t y equation may be expressed a s
at +
div
T
= 0
How express
4
a s a f u n c t i o n of t h e v e l o c i t y ? That i s a r e a l problem. Many r e l a t i o n s
can be found t a k i n g i n t o account waves o r n o t . For o u r s e l v e s w e have used t h e MeyerP e t e r r e l a t i o n s h i p f o r t h e sediment t r a n s p o r t v e c t o r
?
which i s supposed i n t h e
d i r e c t i o n of t h e c u r r e n t bottom shear stress which i s evaluated using Chezy‘s relationship.
So t h e bed c o n t i n u i t y equation can be transformed i n t o :
at
au +
T ~ u
+ T xV z av +
T
aU YU
+
T
yV
av 5 =
0
with TXu =
2
aU
!?
w
T = 8&
Tj=T
v
(T
-
‘IC)3’2
uv
aT
>
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.
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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
K=O
+co Prob [H*(h]=
Prob K=O
[
n=k].(Prob[HCh(H>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.
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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.
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