THREE-DIMENSIONAL MODELS OF MARINE AND ESTUARINE DYNAMICS
FURTHER TITLES IN THIS SERIES 1 J.L. MERO THE MINERAL RESOU...
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THREE-DIMENSIONAL MODELS OF MARINE AND ESTUARINE DYNAMICS
FURTHER TITLES IN THIS SERIES 1 J.L. MERO THE MINERAL RESOURCES OF THE SEA 2 L.M.FOMIN THE DYNAMIC METHOD I N OCEANOGRAPHY 3 E.J.F.WOO0 MICROBIOLOGY OF OCEANS AND ESTUARIES 4 G.NEUMANN OCEAN CURRENTS 5 N.G.JERLOV OPTICAL OCEANOGRAPHY 6 V.VACQUIER' GEOMAGNETISM I N MARINE GEOLOGY 7 W.J. WALLACE THE DEVELOPMENTS OF THE CHLORINITY/SALINITY CONCEPT I N OCEANOGRAPHY 8 E.LISITZIN SEA-LEVEL CHANGES 9 R.H.PARKER THE STUDY OF BENTHIC COMMUNITIES 10 J.C.J. NIHOUL (Editor) MODELLING OF MARINE SYSTEMS 1 1 0.1.MAMAYEV TEMPERATURESALINITY 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. JERLOV MARINE OPTICS 15 G.P.GLASBY MARINE MANGANESE DEPOSITS 16 V.M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS 17 R.A.GEYER SUBMERSIBLES AND THEIR USE I N OCEANOGRAPHY AND OCEAN ENGIEJEERING 18 J.W. CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS 19 J.C.J. NIHOUL (Editor) BOTTOM TURBULENCE 20 P.H. LEBLOND and L.A. MYSAK WAVES I N THE OCEAN 21 C.C. VON DER BORCH (Editor) SYNTHESIS OF DEEPSEA DRILLING RESULTS I N THE INDIAN OCEAN 22 P. DEHLINGER MARINE GRAVITY 23 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF ESTUARIES AND FJORDS 24 F.T. BANNER, M.B. COLLINS and K.S. MASSIE (Editors) THE NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED AND THE SEA I N MOTION 25 J.C.J. NIHOUL (Editor) MARINE FORECASTING 26 H.G. RAMMING and 2 . KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R.A. GEYER (Editor) MARINE ENVIRONMENTAL POLLUTION 28 J.C.J. NIHOUL (Editor) MARINE TURBULENCE 29 M. WALDICHUK. G.B. KULLENBERG and M.J. ORREN (Editors1 MARINE POLLUTANT TRANSFER PROCESSES 30 A. VOlPlO (Editor) THE BALTIC SEA 31 E.K. DUURSMA and R. DAWSON (Editors) MARINE ORGANIC CHEMISTRY 32 J.C.J. NIHOUL (Editor) ECOHYDRODYNAMICS 33 R. HEKlNlAN PETROLOGY OF THE OCEAN FLOOR 34 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF SEMI-ENCLOSED SEAS 35 B. JOHNS (Editor) PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS 36 J.C.J. NIHOUL (Editor1 HYDRODYNAMICS OF THE EQUATORIALOCEAN 37 W. LANGERAAR SURVEYING AND CHARTING OF THE SEAS 38 J.C.J. NIHOUL (Editor) REMOTE SENSING OF SHELF SEA HYDRODYNAMICS 39 T lCHlYE(Editor) OCEAN HYDRODYNAMICS OF THE JAPAN AND EAST CHINA SEAS 40 J C J NIHOUL IEditor) COUPLED OCEAN-ATMOSPHERE MODELS 41 H KUNZENDORF (Editor) MARINE MINERAL EXPLORATION 42 J C J NIHOUL (Editor) MARINE INTERFACES ECOHYDRODYNAMICS 43 P. LASSERRE and J.M. MARTIN (Editors) BIOGEOCHEMICAL PROCESSES AT THE LAND-SEA BOUNDARY 44 I.P. MARTINI (Editor) CANADIAN INLAND SEAS I
Elsevier Oceanography Series, 45
THREE-DIMENSIONAL MODELS OF MARINE AND ESTUARINE DYNAMICS Edited bv
J.C.J. NIHOUL University of L i k e , B5 Sart Tilman, B-4000 Liige, Belgium and
B.M. JAMART MUMM, Institute of Mathematics, 15 Avenue des Tilleuls, B-4000 Likge, Belgium
E LSEV IER Amsterdam - Oxford
- New York - Tokyo
1987
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands Distributors for the United States and a n a d a :
ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N Y 10017, U.S.A.
ISBN 044442794-5 (Vol. 45) ISBN 0 4 4 4 4 1 6 2 3 4 (Series) 0 Elsevier Science Publishers B.V., 1987
A l l rights reserved. N o part o f this publication may be reproduced, stored in a retrieval system o r transmitted in any form o r by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission o f the publisher, Elsevier Science Publishers B.V./Science 81Technology Division, P.O. Box 330, 1000 A H Amsterdam, The Netherlands. Special regulations for readers in the USA - This publication has been registered w i t h the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies o f parts o f this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred t o the publisher. Printed in The Netherlands
V
FOREWORD
The International Libge Colloquium on Ocean Hydrodynamics is organized annually. The topic differs from one year to another in an attempt to address, as much as possible, recent problems and incentive new subjects in physical oceanography. Assembling a group of active and eminent scientists from various countries and often different disciplines, the Colloquia 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 recommendations for future research. The Scientific Organizing Committee and the participants wish to express their gratitude to the Belgian Minister of Education, the National Science Foundation of Belgium, the University of Libge, the Intergovernmental Oceanographic Commission and the Division of Marine Sciences (UNESCO), and the Office of Naval Research for their most valuable support. In May 1986, we learned with sadness that Dr. Norman S. Heaps would not be able to attend the Libge Colloquium as planned because of illness. Norman passed away on 26 July 1986. The modelling community has lost a pioneer, a guide, and a friend. We dedicate this volume of proceedings, which contain a small part of his large legacy, to the memory of Dr. Norman Stuart Heaps.
Jacques C. J. Nihoul
Bruno M. Jamart
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VII
TABLE OF CONTENTS PERSPECTIVE IN THREE-DIMENSIONAL MODELLING OF THE MARINE SYSTEM Jacques C.J. Nihoul and S. Djenidi
.................
............................................
1
ON MODELING THREE-DIMENSIONAL ESTUARINE AND MARINE HYDRODYNAMICS Y. Peter Sheng
................................................................................................................................
35
CIRCULATION MODELLING USING ORTHOGONAL CURVILINEAR COORDINATES Alan F. Blumberg and H. James Herring .......................................................................................
55
PREDICTING OPEN OCEAN CURRENTS, FRONTS AND EDDIES Allan R. Robinson ...........................................................................................................................
89
PREPARATION OF ESTUARY AND MARINE MODEL EQUATIONS BY GENERALIZED FILTERING METHODS
K. W. Bedford, J. S. Dingman and W. K. Yeo
113
A LIMITED AREA MODEL FOR THE GULF STREAM REGION William R. Holland
.........................................................................................................................
127
STUDY OF TRANSPORT FLUCTUATIONS AND MEANDERING OF THE FLORIDA CURRENT USING AN ISOPYCNIC COORDINATE NUMERICAL MODEL Douglas B. Boudra, Rainer Bleck and Friedrich Schott
....................................
149
DYNAMICS OF AGULHAS RETROFLECT'ION AND RING FORMATION IN A QUASIISOPYCNIC COORDINATE NUMERICAL MODEL E. P. Chassignet and D. B. Boudra
169
MODELLING OF MESOSCALE OCEANIC INSTABILITY PROCESSES Aike Beckmann
...............................................................................................................................
195
AN EDDY-RESOLVING MODEL FOR RIVER PLUME FRONTS J. W. Dippner
..................................................................................................................................
21 1
A FINITE DIFFERENCE GENERAL CIRCULATION MODEL FOR SHELF SEAS AND ITS APPLICATION TO LOW FREQUENCY VARIABILITY ON THE NORTH EUROPEAN SHELF J. 0. Backhaus and D. Hainbucher
.................................................................................................
221
A THREE DIMENSIONAL CIRCULATION MODEL OF THE SOUTH CHINA SEA
T. Pohlmann
....................................................................................................................................
245
VIII THE INFLUENCE OF BOUNDARY CONDITIONS ON THE CIRCULATION IN THE GREENLAND-NORWEGIAN SEA. A NUMERICAL INVESTIGATION
S. Legutke .......................................................................................................................................
269
A THREE DIMENSIONAL BAROCLINIC MODEL OF THE WESTERN BALTIC M. J. Boehlich
......................................................................................................
285
A STUDY OF VARIOUS OPEN BOUNDARY CONDITIONS FOR WIND-FORCED BAROTROPIC NUMERICAL OCEAN MODELS L. P. RQed and C. K. Cooper
.........................................................................................................
305
COASTAL CURRENTS, INTERNAL WAVE COLLAPSES AND TURBULENCE IN THE STRAIT OF MESSINA ZONE E. Salusti and R. Santoleri
..............................................................................................................
337
A THREE-DIMENSIONAL FINITE ELEMENT MODEL FOR THE STUDY OF STEADY AND NON-STEADY NATURAL FLOWS J.-L. Robert and Y. Ouellet
..................................................................................................
359
REAL AND SPURIOUS BOUNDARY LAYER EFFECTS IN THREE-DIMENSIONAL HYDRODYNAMICAL MODELS Bruno M. Jamart and JosC Ozer ......................................................................................................
373
A TROPHIC-DIFFUSION 3D MODEL OF THE VENICE LAGOON C. Dejak and G. Pecenik
..........................................................................................................
391
THREE DIMENSIONAL CONTINENTAL SHELF HYDRODYNAMICS MODEL INCLUDING WAVE CURRENT INTERACTION
M. L. Spaulding and T. Isaji
...........................................................................................................
405
THREE-DIMENSIONAL MODEL OF CURRENTS IN THE BAY OF SEINE J. C. Salomon, B. Thouvenin and P. Le Hir
...................................................................................
427
TIDAL STREAMS IN SHALLOW WATER P. P. G. Dyke
..................................................................................................................................
44 1
MODELLING AND OBSERVATIONS OF THE RESIDUAL CURRENT OFF SOUTHWEST NOVA SCOTIA K. T. Tee, P. C. Smith and D. Lefaivre
..........................................................................................
455
IX A THREE-DIMENSIONAL WEAKLY NONLINEAR MODEL OF TIDE-INDUCED LAGRANGIAN RESIDUAL CURRENT AND MASS-TRANSPORT, WITH AN APPLICATION TO THE BOHAI SEA Shizuo Feng .....................................................................................................................................
47 1
THREE DIMENSIONAL NUMERICAL MODEL FOR THERMAL IMPACT STUDIES M. D a m , P. Donnars and P. Pechon
............................................................................................
489
ESTIMATION OF STORM-GENERATED CURRENTS N. S. Heaps and J. E. Jones
............................................................................................................
505
A COUPLED 2-D/3-D MODELLING SYSTEM FOR COMPUTATION OF TIDAL AND WIND-INDUCED CURRENTS J. M. Usseglio-Polatera and P. Sauvaget
........................................................................................
539
A HIGH RESOLUTION 3D MODEL SYSTEM FOR BAROCLINIC ESTUARINE DYNAMICS AND PASSIVE POLLUTANT DISPERSION J. Krohn, K. Duwe and K. D. Pfeiffer
............................................................................................
A THREE DIMENSIONAL NUMERICAL MODEL OF SEMI-DIURNAL TIDES ON THE EUROPEAN CONTINENTAL SHELF A. M. Davies ...................................................................................................................................
555
573
A GENERAL THREE-DIMENSIONAL EDDY-RESOLVING MODEL FOR STRATIFIED SEAS 59 1 I. D. James ...................................................................................................................................... A 3-D MODEL OF THE SEVERN ESTUARY
J. Wolf .............................................................................................................................................
609
THE VARIATIONAL INVERSE METHOD REVISITED (Abstract only)
c. Provost ........................................................................................................................................
625
THE BRANCHING OF THE GULF STREAM REVISITED USING THE VARIATIONAL INVERSE METHOD (Abstract only) F. Martel and C. Provost
.................................................................................................................
627
ABOUT A DIAGNOSTIC ANALYSIS OF THE HISTORICAL HYDROGRAPHIC DATA IN THE TROPICAL ATLANTIC (Abstract only) C. Provost and M. S. Suk
...............................................................................................................
629
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XI
LIST OF PARTICIPANTS ADAM Y., D r . , Management U n i t o f t he Mathematical Models (MUMM),Liege,Belgium AIRAUDO J.L., Ing., Meteorologie Nationale, Paris, France BACKHAUS J.O., P r o f . Dr., U n i v e r s i t a t Hamburg, Hamburg, Germany BAH A., Dr., U n i v e r s i t e Laval, Quebec, Canada BECKMAiill A., M r . , U n i v e r s i t a t K i e l , K i e l , Germany BECKERS J.M. , Ing. , U n i v e r s i t e de Liege, Liege, Belgium BODE L., D r . , James Cook U n i v e r s i t y o f Nor t h Queensland, Townsville, A u s t r a l i a BOEHLICH M.J., M r . , U n i v e r s i G t Hamburg, Hamburg, Germany BOUDRA D.B., P r o f . D r . , Rosenstiel School o f Marine and Atmospheric Science, Miami, USA CHARTIER M., Dr., I n s t i t u t de P r o t e c t i o n e t de SOret6 Nucleaire, CEA, Fontenay-aux-roses, France CHASSIGNET E., Ing., Rosenstiel School o f Marine and Atmospheric Science, Miami, USA CLEMEiiT F., Mr., U n i v e r s i t e de Liege, Liege, Belgium COMELIAU B., Ing., U n i v e r s i t e de Liege, LiCge, Belgium DAHL F.E., Ing., Det norske Ver it as, Hfivik, Norway DAVIES A.M. , D r . , I n s t i t u t e o f Oceanographic Sciences, Birkenhead, UK DELECLUSE P., D r . , Museum d ' H i s t o i r e N a t u r e l l e , Paris, France DEJAK C. , Pro f. Dr., Enternazionale Energie A l t e r n a t i v e , ENEA, Roma, I t a l y DELEERSINIJDER E., Ing., U n i v e r s i t e de Liege, Liege, Belgium DEMUTH C l . , D r . , Management U n i t o f t he Mathematical Models (MUMM), Liege, Be1g i um DESAUBIES Y., P r o f . Dr., IFREMER, Brest, France DINGMAN J.S., Mr., The Ohio S t a t e U n i v e r s i t y , Columbus, USA DIPPNER J.W. , Dr., U n i v e r s i t a t Hamburg, Hamburg, Germany DISTECHE A., Prof. Dr., U n i v e r s i t e de Liege, Liege, Belgium DJENIDI S., Ing., U n i v e r s i t e de Liege, Liege, Belgium DONNARS Ph., Ing. , Labor at oir e Nat ional d'Hydraulique, Chatou, France DYKE P.P.G., P r o f . D r . , Plymouth Polyt echnic, Plymouth, UK EIFLER W., D r . , Commission o f the European Communities, Ispra, I t a l y ELLIOTT A.J., D r . , U n i v e r s i t y College o f North Wales, Menai Bridge, UK EVERBECQ E., Ing., U n i v e r s i t e de Liege, Liege, Belgium FANDRY C. , D r . , CSIRO, Hobart, A u s t r a l i a FEiiG S., P r o f . Dr., Shandong College o f Oceanology, Shandong, The People's Republic o f China FLEBUS C. , M r . , U n i v e r s i t e de Liege, Liege, Belgium FRAI\IKIGHOUL C l . , P r o f . D r . , U n i v e r s i t e P i e r r e e t Marie Curie, Paris, France GOFFART A., Miss, U n i v e r s i t e de Liege, Liege, Belgium GOFFART P . , Ing., U n i v e r s i t e de Liege, Liege, Belgium GOPALAKRISHNAN T.C. , Dr., Kuwait I n s t i t u t e f o r S c i e n t i f i c Research, Safat, Kuwait GUNST D.R.R. , Ing., M i n i s t e r i e van Openbare Werken, Dostende, Belgium HAINBUCHER D., Mrs. , U n i v e r s i t a t Hamburg, Hamburg, Germany HALMES F. , M r . , U n i v e r s i t e de Liege, Liege, Belgium HAPPEL J.J., Ing., U n i v e r s i t e de Liege, Liege, Belgium HECQ J.H., Dr., U n i v e r s i t e de Liege, Liege, Belgium HIRES R.I., D r . , Stevens I n s t i t u t e o f Technology, Hoboken, USA HOLLAiiD W.R. , Dr., Nat ional Center f o r Atmospheric Research, Boulder, USA HUA B.L. , D r . , IFREMER, Br est , France HUIZIiIGA P. , Ing., N R I O - C S I R , Stellenbosch, South A f r i c a
XI1
JAMART B.M., Dr., Management U n i t o f the Mathematical Models (MUMM), Liege, Be1g i um JAMES I . D . , Dr., I n s t i t u t e o f Oceanographic Sciences, Birkenhead, UK KARAFISTAIJ-OEiJIS A., Dr., U n i v e r s i t e de Liege, Liege, Belgium KROHN J., Dr., GKSS Research Centre, Geesthacht, Germany LAIME A.F., Ing. , U n i v e r s i t e de Liege, Liege, Belgium LEBON G., Prof. Dr., U n i v e r s i t e de Liege, Liege, Belgium LEFAIVRE D. , D r . , Centre Champlain des Sciences de l a Mer, Quebec, Canada LEGUTKE S. , Mrs., U n i v e r s i t x t Hamburg, Hamburg, Germany L I L., D r . , Sta te Oceanic Adm inist r at ion, Dalian City, The People's Republic o f China LOHRMAW A. , Ing., Det norske Ver it as, HBvik, Noway LYNN N.M., Mr., Royal Naval College, London, UK MARTEL F., Miss, U n i v e r s i t e P i e r r e e t Marie Curie, Paris, France MILLET 6. , Mr., ORSTOM, M o n t p e l l i e r , France MONREAL A., Dr., CONACYT, Mexico, Mexico MOUCHET A., Miss, U n i v e r s i t e de Liege, Liege, Belgium NEVES R. , D r . , CTAMFUTL, Lisboa, Portugal NIHOUL J.C.J., Pr of . Dr., U n i v e r s i t e de Liege, Liege, Belgium OZER J., Ing., Management U n i t o f the Mathematical Models (MUMM), Liege, Belgium PECENIK G. , D r . , MONTEDIPE SPA, Venezia, I t a l y PECHON Ph., Ing., Labor at oir e Nat ional d'Hydraulique, Chatou, France PEDERSEN G.K., M r . , U n i v e r s i t y o f Oslo, Oslo, Norway PICHOT G., Dr., U n i t e de Gestion Modele Mathematique Mer du Nord e t Estuaire de l 'Escaut, Br uxelles, Belgium POHLMAiW Th. , M r . , Uni v e r s i t l t Hamburg, Hamburg, Germany PONTRELLI G. , D r . , IRAM-CNR, B a r i , I t a l y POSTMA L., M r . , D e l f t Hydr aulics Laboratory, D e l f t , The Netherlands PROVOST Ch., Dr., U n i v e r s i t e P i e r r e e t Marie Curie, Paris, France RANDLES J., Mr., Commission o f t h e European Communities, Ispra, I t a l y ROBERT J L . , P r o f . D r , Uni v e r s i t e Lava1 , Quebec, Canada ROBINSON A., P r o f . D r . , Harvard U n i v e r s i t y , Cambridge, USA ROCKLIFF N.J. , O r . , Plymouth Polytechnic, Plymouth, UK RODRIGUEZ I.,Ing. , D i r e c t i o n Generale des Cates, Madrid, Spain ROED L.P., Dr., Det norske V e r i t a s , HBvik, Norway R O I S I N M., M r . , U n i v e r s i t e de Liege, Liege, Belgium RONDAY F.C., D r . , U n i v e r s i t e de Liege, Liege, Belgium RYGG O.B., Mr. , U n i v e r s i t y o f Oslo, Oslo, Norway SALAS DE LEON D. , D r . , CONACYT, Mexico, Mexico SALOMON J.Cl., D r . , IFREMER, Br est , France SALUSTI S.E., P r o f . Dr., U n i v e r s i t a La Sapienza, Roma, I t a l y SHENG Y., Dr., Aeronautical Research Associates o f Princeton, Princeton, USA SMETS E . , Ing., Waterbouwkundig Laboratorium, Borgerhout, Belgium SMITZ J., Ing., U n i v e r s i t e de Liege, Liege, Belgium SNYKERS Ph. , Ing., U n i v e r s i t e de Liege, Liege, Belgium SOULAIMANI A., M r . , U n i v e r s i t e de Technologie, Compiegne, France SPAULDING M.L., P r o f . D r . , AppliedScience Associates I n c . , Narragansett, USA SPITZ Y., Miss, Management U n i t o f t h e Mathematical Models (MUMM), Liege, Belgium STANLEY P., Mr., Marine Science Labor at or ies, Menai Bridge, UK STEEOMAN R.K. , D r . , Steedman Lim it ed, Subiaco, Western A u s t r a l i a TEE K.T., M r . , Bedford I n s t i t u t e o f Oceanography, Oartmouth, Canada TREGLOS Y., Mr., UNESCO, Par is, France USSEGLIO-POLATERA J.M., SOGREAH, Grenoble, France VALCKE A., Ing., U n i v e r s i t e de Liege, Liege, Belgium WILLIAMS J., M r . , Branch O f f i c e o f Naval Research, London, UK WOLF J . , D r . , I n s t i t u t e o f Oceanographic Sciences, Birkenhead, UK
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PERSPECTIVE I N THREE-DIMENSIONAL MODELLING OF THE MARINE SYSTEM Jacques C.J. NIHOUL and S. DJENIDI* School o f GeoHydrodynamics and E n v i r o n m e n t a l Research (GHER), U n i v e r s i t y o f Liege, Belgium *Also “ U n i t e de M o d e l i s a t i o n de 1 ‘Environnement M a r i n , (MODEM), A s s o c i a t i o n U n i v e r s i t e de Corse - U n i v e r s i t e de L i e g e , C a l v i , Corse INTRODUCTION There i s a g e n e r a l consensus, a t l e a s t i n developed c o u n t r i e s and i n t e r n a t i o n a l i n s t i t u t i o n s , t h a t o u r m a r i n e environment has been d r a m a t i c a l l y d e t e r i o r a t i n g i n t h e l a s t decades. The m i g r a t i o n o f i n d u s t r i e s t o t h e coasts, t h e development o f new h a r b o r s , t h e growth o f v a s t u r b a n c e n t e r s , t h e use o f i n s e c t i c i d e s and f o n g i c i d e s have l e d t o an i m p o r t a n t p o l l u t i o n o f c o a s t a l areas, d e s t r o y i n g m a r i n e l i f e and c r e a t i n g severe h e a l t h problems f o r t h e human p o p u l a t i o n . O f f s h o r e , t h e dumping o f i n d u s t r i a l wastes has d a n g e r o u s l y i n c r e a s e d s i n c e t h e l a s t war. The development o f m a r i t i m e commercial exchanges, t h e e x p l o i t a t i o n o f o r e s and h y d r o c a r bons and o t h e r uses o f t h e sea f l o o r , such as d r e d g i n g , c o n t i n u o u s l y i n c r e a s e t h e p o l l u t i o n o f t h e sea and more p a r t i c u l a r l y t h e c o a s t a l zones. The problems o f f i s h e r i e s a r e c l o s e l y r e l a t e d . N o t o n l y because p o l l u t i o n d e s t r o y s m a r i n e l i f e o r contaminates m a r i n e p r o d u c t s o r because o v e r f i s h i n g i s a form o f p o l l u t i o n b u t m a i n l y because t h e same p h y s i c a l mechanisms which govern the f a t e o f p o l l u t a n t s a r e o f t e n responsible f o r c r e a t i n g the conditions o f marine f e r t i l i t y . I n t h e same t i m e , t h e c o s t o f raw m a t e r i a l s and energy has i n c r e a s e d enormously, c a l l i n g f o r a more e x t e n s i v e e x p l o i t a t i o n o f t h e sea and l i m i t i n g t h e economical r e s o u r c e s w h i c h can be devoted t o p o l l u t i o n c o n t r o l . S i m u l t a n e o u s l y , t h e problem o f s u p p l y i n g t h e i n c r e a s i n g w o r l d p o p u l a t i o n w i t h f o o d has l e d t o a more s y s t e m a t i c h a r v e s t i n g o f m a r i n e p r o d u c t s . I n t h e n e x t decade, t h e problems w i l l become more a c u t e and w i l l c a l l f o r a more thorough u n d e r s t a n d i n g and a more r a t i o n a l and s t r i c t c o n t r o l o f t h e m a r i n e environment.
2
The marine system, however, i s extremely complex and i t i s a overwhelming task to predict the intricated environmental e f f e c t s of man's a c t i v i t i e s and, a f o r t i o r i , to s e t limits to such a c t i v i t i e s - by internationaZ conventions and t r e a t i e s , f o r instance - which are n o t permissive o r unduly severe. Sofar, the simple collection of data and t h e i r descriptive ordering have appeared such formidable tasks that one has often ignored the need for doing more than t h i s . One realizes t h a t what i s necessary now i s the management of the marine system, the search f o r the necessary compromises between the requirements of increasing industrialization and affluent society and the necessity t o preserve the valuable natural resources. T h i s i s a n optimal control problem which can only be approached through mathematical modelling. Mathematical models are the only a1 ternative t o zero discharge, i f ecologically acceptable solutions to environmental problems are t o be provided. AN INTERTWINING OF MATHEMATICAL MODELS The f i r s t step i n modelling a marine system i s the demarcation of the system. This includes the definition of i t s support, i . e . i t s extension in physical ( p t ) space, and of i t s scope, i . e . i t s deployment i n s t a t e space. The demarcation defines the boundaries of the system and the s t a t e variables of the model and t h u s determines the nature, place and time of the boundary and i n i t i a l conditions which will be required.
The support and the scope may d i f f e r appreciably from one study t o another, depending on i t s particular objectives, and t h i s generates a whole hierarchy of different mathematical models, accorded t o t h e i r particular designs. These models may a l l be regarded as "sub-sets" of some - s t i l l tentative - universal the construction of which must be pursued t o keep track of a l l the model aspects which have been sacrificed t o urge conclusive, t h o u g h p a r t i a l , results. The f i r s t c h a r a c t e r i s t i c of a model i s i t s o b j e c t , i.e. the geographical area, the dates and the specific events o r processes one wishes to investigate. One understands easily the difference between model 1 i n g the Bering Sea o r the Mediterranean, investigating ice formation, tidal fronts i n mid-latitudes o r trapped Kelvin waves a t the equator.
The second characteristic of a model i s i t s span, i . e . i t s dimension in physical space and i n s t a t e space. Ideally, a marine model would be a time dependent 3D-model (four-dimensional support). A reductionofthe dimensions of the support can be achieved,however, by r e s t r i c t i n g attention t o timeand space averages, considering for instance (quasi) steady s t a t e models of low frequency residuals, depth-averaged models of shallow continental seas , cross-section averaged models of estuaries, time- and depth dependent models of surface and bottom boundary layers or primary production and f i n a l l y box-models f o r completely spaceaveragedecological variables. Ideally, also, a completely r e a l i s t i c marine model would have an i n f i n i t e number of s t a t e variables. Computing f a c i l i t i e s , o f course, impose limitations on the number of s t a t e variables b u t , independently of such r e s t r i c t i o n s , there are r e l i a b i l i t y and c l a r i t y constraints on the number of s t a t e variables. A model w i t h many s t a t e variables incorporates as many different processes and interactions and involves a correspondingly large number of parameters and boundary conditions which cannot be evaluated from existing data bases without an inevitable margin of error. The results of such a model, on the other hand, - because of i t s increased sophistication and f a l l i b i l i t y - can become impossible to interpret in terms of s c i e n t i f i c diagnosis o r management recommendations. The essence o f modelling i s the s e l e c t i o n o f a l i m i t e d nwnber o f representa-
There must be sufficiently few of them f o r t h e i r evolution equations t o be amenable to analysis b u t enough of them t o describe adequately the system's behaviour. t i v e s t a t e variables.
The s t a t e space can be divided i n several sectors corresponding t o hydrodynamical, chemical , biological processes . . . and one can conceive separate hydrodynamical, chemical and biological models with the necessary input-output links between them. Of these, the hydrodynamic models are by f a r the most advanced. In a sense, t h i s i s rather fortunate because the understanding of hydrodynamic processes i s prerequisite to any form of chemical o r biological modelling a n d , indeed, constitutes, i n the present s t a t e of development of marine models, the most reliable contribution t o the explanation and anticipation of ecological processes. While chemical and b i o l o g i c a l models are s t i l l frequently limited t o i n t e r a c t i o n box-models describing, by means of d i f f e r e n t i a l equations, concentrations and biomasses i n a hypothetic homogeneous environment or averaged over 1arge regions of space, hydrodynamic models have evolved t o transport-dispersion field-models describing , by means of p a r t i a l d i f f e r e n t i a l equations
4
the s p a t i a l d i s t r i b u t i o n and time e v o l u t i o n o f f i e l d v a r i a b l e s determined a t a l l g r i d points. The development o f hydrodynamic models has been considerably s t i m u l a t e d by t h e i r d i r e c t a p p l i c a t i o n t o c o a s t a l and o f f - s h o r e engineering.
It has a l s o been
made p o s s i b l e by t h e absence o f any s i g n i f i c a n t feed-back from chemical and b i o l o g i c a l processes on hydrodynamic phenomena : t r a n s p o r t and d i s p e r s i o n i n the sea a r e determinant f a c t o r s i n marine chemistry and b i o l o g y b u t chemical and b i o l o g i c a l i n t e r a c t i o n s have no appreciable e f f e c t on advection and m i x i n g i n t h e sea. The s t a t e v a r i a b l e s o f hydrodynamic models are t h e thermo-mechanical v a r i a bles, v e l o c i t y , pressure, buoyancy, temperature, s a l i n i t y , t u r b u l e n t k i n e t i c energy,
... depending on
t h e i r degree of s o p h i s t i c a t i o n .
They can e a s i l y be
extended t o i n c l u d e t h e concentrations of passive and semi-passive c o n s t i t u e n t s , i.e.
c o n s t i t u e n t s which a r e simply c a r r i e d along by t h e sea (passive) o r which,
w h i l e c a r r i e d along, can be produced o r destroyed by l o c a l r e a c t i o n s depending on t h e c o n s t i t u e n t ' s c o n c e n t r a t i o n o n l y such as b a c t e r i a l o r r a d i o a c t i v e decay (semi-passive). The combination o f hydrodynamic models and chemical-ecological i n t e r a c t i o n models leads t o a c t i v e transport-dispersion rnodeZs.
These however a r e s t i l l
i n an e a r l y stage o f development p a r t l y because of t h e i r complexity (many coup l e d p a r t i a l d i f f e r e n t i a l equations) and p a r t l y because o f t h e l a c k o f s u i t a b l e chemical and b i o l o g i c a l data f o r t h e i r c a l i b r a t i o n i n three-dimensions and f o r t h e determination o f a p p r o p r i a t e boundary c o n d i t i o n s . L i m i t i n g t h e scope o f t h e model t o a p a r t i c u l a r s e c t o r o f t h e s t a t e space reduces i t s dimensions.
This can be achieved a l s o by r e s t r i c t i n g a t t e n t i o n t o
aggregate averages ; considering, f o r instance, zooplankton biomass ( w i t h no d i s t i n c t i o n between herbivores, c a r n i v o r e s and omnivores and, a f o r t i o r i , between species) , t o t a l organic m a t t e r ( lumping t o g e t h e r d i s s o l v e d and p a r t i c u l a t e o r g a n i c m a t t e r ) , mercury c o n c e n t r a t i o n i n f i s h ( w i t h no s p e c i f i c a t i o n o f i t s d i s t r i b u t i o n ) etc..
..
The t h i r d c h a r a c t e r i s t i c o f a mathematical model i s i t s purview, i . e .
spread
i n p h y s i c a l space,
(its
"durationn and i t s
"reach")
and
its
its
5
aperture arena").
i n s t a t e space
(its
))frequency gunrut"
and i t s
"wave-nwnber
I n p h y s i c a l space, one thus d i s t i n g u i s h e s between ZocaZ, regional and gZobaZ model s. Although t h e terms a r e suggestive enough
-
a model o f t h e n e a r - f i e l d d i s p e r s i o n
o f a coastal discharge can be described as l o c a l , a model o f the A d r i a t i c Sea as r e g i o n a l , a general c i r c u l a t i o n model o f t h e A t l a n t i c as g l o b a l
-
t h i s dis-
t i n c t i o n i s n o t w i t h o u t some ambiguity (A model o f t h e Mediterranean i s g l o b a l compared t o one o f t h e A d r i a t i c and r e g i o n a l compared t o a general ocean c i r c u l a t i o n model). One can however removethe ambiguity by d e f i n i n g ( i ) ZocaZ models as models of "small" s i z e regions where t h e f l o w f i e l d and o t h e r hydrodynamic c h a r a c t e r i s t i c s are given, ( i i ) regional models as models o f "medium" s i z e regions where the flow f i e l d must be determined t a k i n g i n t o account boundary c o n d i t i o n s impo-
gZobaZ models as models o f " l a r g e " sed by l a r g e r s c a l e c i r c u l a t i o n s and (iii) s i z e regions where t h e f l o w i s m a i n l y d r i v e n by i n t e r n a l f o r c i n g and l i t t l e by open-sea boundary i n f l o w s . Obviously, reach and d u r a t i o n a r e r e l a t e d and l o c a l models are n a t u r a l l y i n t e rested i n s h o r t - t e r m m o d i f i c a t i o n s of the near f i e l d w h i l e g l o b a l models are more concerned w i t h t h e long-term e v o l u t i o n o f t h e whole system. The marine system i s c h a r a c t e r i z e d by f a i r l y w e l l - d e f i n e d "spectral windows" i.e.
domains o f l e n g t h - s c a l e ( i n v e r s e l y , wave numbers) and time scales ( i n v e r -
sely, frequencies) associated w i t h i d e n t i f i e d phenomena.
These windows may
correspond t o eigenmodes o f t h e system ( i n t e r n a l waves, i n e r t i a l o s c i l l a t i o n s , Rossby waves, E l NiRo
...
) o r e x t e r n a l f o r c i n g (annual o r d a i l y v a r i a t i o n s o f
i n s o l a t i o n , t i d e s , storm, atmosphere c l i m a t e changes
...
).
The basic hydrodynamic equations c o n t a i n t h r e e c h a r a c t e r i s t i c frequencies (i)
t h e Brunt-Vaisala frequency
n
i s a measure o f t h e s t r a t i f i c a t i o n
(n2 i s d e f i n e d as the v e r t i c a l g r a d i e n t o f buoyancy ; t h e maximum value o f t h e B r u n t - M i s a l a frequency i n t h e sea i s o f t h e o r d e r o f 10-2 s - 1 )
(ii)
;
t h e C o r i o l i s frequency
f
i s a measure o f t h e E a r t h ' s r o t a t i o n
( f i s d e f i n e d as t w i c e t h e v e r t i c a l component o f t h e E a r t h ' s r o t a t i o n vector ; i n mid-latitudes, (iii)
t h e K i b e l frequency
j
f
2r
lo-'+ s - l ) ;
i s a measure o f t h e E a r t h ' s curvature ( i f B
denotes t h e g r a d i e n t o f f , j can be d e f i n e d as j
%
Br
where
6
r % i s t h e Rossby r a d i u s o f deformation and H a t y p i c a l depth ; the maximum value o f t h e Kibel frequency i n the sea i s o f t h e order o f 10-6 s - 1 ) . Diurnal and seasonal v a r i a t i o n s o f thermal exchanges can be characterized s - l and lo-’ s - l r e s p e c t i v e l y . by t y p i c a l frequencies o f t h e order o f A frequency o f t h e order o f s - l can be associated w i t h v a r i a t i o n s i n t h e wind f i e l d . F i n a l l y , a frequency o f t h e order o f lo-* s - l may be introduced i n connect i o n w i t h the year-to-year v a r i a t i o n s o f t h e s t a t e o f l a r g e areas o f t h e ocean and the e n t i r e atmosphere as, f o r example, t h e s e l f - o s c i l l a t i o n o f the Northern branch o f t h e G u l f Stream and t h e E l NiAo Southern o s c i l l a t i o n . Marine processes can thus be c l a s s i f i e d according t o t h e i r time scales as shown schematically i n Table I. I n general, time scales and l e n g t h scales are r e l a t e d and i t i s customary t o associate h i g h frequencies and high wave numbers, small frequencies and small wave numbers although t h e a s s o c i a t i o n may be d i f f e r e n t f o r eigenmodes and forced o s c i l l a t i o n s . The t r a n s f e r of energy between windows i s e f f e c t e d by non-linear i n t e r actions. Chemical and e c o l o g i c a l i n t e r a c t i o n processes can a l s o be characterized by s p e c i f i c time scales and t h e comparison between these time scales and those o f hydrodynamic phenomena i n d i c a t e s which processes are a c t u a l l y i n competition i n the sea. Obviously, a t hydrodynamic scales much smaller than i n t e r a c t i o n scales, very l i t t l e i n t e r a c t i o n takes place over time o f s i g n i f i c a n t hydrodynamic changes and b a s i c a l l y the c o n s t i t u e n t s are transported and dispersed p a s s i v e l y by t h e sea.
On the o t h e r hand, hydrodynamic processes w i t h time scales much l a r g e r
than i n t e r a c t i o n scales scarcely a f f e c t t h e dynamics o f i n t e r a c t i o n s over any time o f i n t e r e s t . The range o f ( t i m e and l e n g t h ) scales t h e model can reproduce defines i t s aperture.
7
Time scale Frequ ncy (s-
1s
f1
1
S ectral windows T e s s e
s
)
Microscale processes 3 D "eddy" turbulence (+ surface waves)
Mol ecul a r diffusion
Mesialscale processes Internal waves Vertical micros tryctu re I n h i b i ted " b 1 i ny " turbulence
Eddy turbulence
Mesoscale processes Tnertial oscillations Tides, storm surges Diurnal variations
"61 iny turbulence"
Synopticscal e processes Frontal currents Meanders, "rossby"X turbulence
Mesoscale v a r i a b i l i t y
Seasonalscale processes
"Rossby t u r b u 1en ce "
Global scale processes Climatic processes
Seasonal v a r i a b i l i t y
l m lo-*
Smaller scale fluctuations ( f i l t e r e d o u t processes)
l h
Id l w
1 month
1 year
10-8
(Pa1eo)climaticscale processes
Table I : Schematic representation of marine v a r i a b i l i t y
*A "bliny" (from the Russian " b l i n i " ) i s a pancake-shaped eddy contributing t o an energy cascade t o smaller scales via epidemic i n s t a b i l i t i e s and internal waves. A "rossby" (from the s c i e n t i s t Rossby) i s a pseudo-twodimensional eddy column of scale of the order o f the Rossby radius o f
deformation.
8 The f o u r t h c h a r a c t e r i s t i c o f a model i s i t s resoZution. The r e s o l u t i o n i n physical space i s determined by the mesh-size o f the numerical g r i d and the time step o f i n t e g r a t i o n , t h e r e s o l u t i o n i n s t a t e space by the margins o f e r r o r allowed on t h e s t a t e variables. The r e d u c t i o n o f support and scope by averaging and aggregation i s , t o some extent, r e l a t e d t o t h e r e s o l u t i o n o f the model. A depth-integrated model f o r instance i s equivalent t o a 3D-model w i t h a very coarse (one g r i d p o i n t ) v e r t i cal r e s o l u t i o n . There i s an obvious connection between the spread o f a model, i t s aperture and i t s r e s o l u t i o n . Given the complexity o f the model, t h e l i m i t a t i o n s i n computing f a c i l i t i e s o r budgets g e n e r a l l y impose a l i m i t on t h e number o f g r i d p o i n t s and time steps and thus, f o r a chosen reach and duration, a maximum resolution.
The maximum r e s o l u t i o n determines t h e l a r g e s t frequencies and
wave-numbers t h a t can be resolved. On the o t h e r hand, phenomena t h e l e n g t h scales and time scales o f which exceed the reach and d u r a t i o n o f t h e support a r e n o t t r u l y "resolved by t h e model" as the s o l u t i o n i s l a r g e l y determined by i n i t i a l o r boundary c o n d i t i o n s ( f o r instance, i f the d u r a t i o n o f the s i m u l a t i o n i s much smaller than the character i s t i c time o f t h e process, r e s u l t s , a t any s i m u l a t i o n time, are completely s e t b y t h e i r i n i t i a l values).
The f i f t h c h a r a c t e r i s t i c o f a model i s i t s accuracy, i . e .
i t s ability to
reproduce the r e a l i t y .
Obviously, t h e accuracy o f t h e model i s n o t simply a question o f r e s o l u t i o n and p r e c i s i o n o f t h e c a l c u l a t i o n s . I t depends f o r instance t o a l a r g e e x t e n t on i t s degree o f s o p h i s t i c a t i o n and on the r e l i a b i l i t y o f the data used f o r the determination o f parameters and boundary conditions.
A s i m p l i s t i c model can produce very p r e c i s e r e s u l t s e n t i r e l y d i s -
connected from r e a l i t y , an i n t r i c a t e d model may c o n t a i n t o o many assumptions and uncertain f i g u r e s t o provide a s a t i s f a c t o r y representation o f i t . As pointed o u t before, f o r each problem, a compromise i s i n e v i t a b l e . A f i n a l d i s t i n c t i o n between models can be made here between "process-modezs"
which emphasize accuracy i n s t a t e space and "engineering modeZs" which emphas i z e accuracy i n physical space.
9
A process model i s generally devised to investigate, i n details, particular mechanisms, scrutinize the behaviour of specific s t a t e variables and elucidate fundamental questions. Very refined i n i t s representation of, sometimes, rather subtle processes, i t may be content with very crude approximations o f the physical world (constant depths, r e c t i l i n e a r coasts, i n f i n i t e ocean, steady two-dimensional fronts, rigid sea surface . .. ).
An engineering model, on the contrary, i s in general called upon to tackle a practical situation and may not ignore the real f i e l d conditions (depths, coastlines, actual atmospheric forcing .. ). I t s aims however, are to assess the consequences of particular events and to provide the marine forecasts which will a s s i s t planning and management. The model must be sound, expeditious and e f f i c i e n t b u t i s not required to provide detailed information on the delicate machinery subtending i t s parameterization schemes.
.
I t i s easy i f carried t o rule of thumb may be useful progressively
t o imagine the excesses t o which b o t h types of models may lead extremes (ivory-tower intellectual game, on one side, foreman's on the other). Although purely process o r engineering models f o r preliminary investigations, t h e i r vocation i s t o enlarge and acquire the virtues of the other.
The final goal i s a diagnostic-prognostic model providing an accurate description of a l l aspects of the real world. Such a model i s often referred t o as a simulation model.
The basic equations of a l l hydrodynamic and active dispersion models are cast in the same mould and may be regarded as different breeds of the same fundamental equations of Geophysical Fluid Dynamics and the same diffusion equations . The differences between the models are essentially the r e s u l t s of t h e i r d i s t i n c t aims, spans, purviews and resolutions. This diversity findsexpression in the choice of s t a t e variables and related acting phenomena, the parameterization of interactions, boundary conditions and sub-grid scale processes and, to some extent, the numerical schemes. I t i s easy to imagine how many different models can be conceived by considering different objectives, different places o r dates, different time scales and length-scales ... even i f some combinations, as pointed-out before, must be excluded.
10 The problems o f t h e management o f c o a s t a l waters and c o n t i n e n t a l seas, coastal and o f f - s h o r e engineering, p o l l u t i o n , e u t r o p h i c a t i o n , primary product i o n , food chain dynamics and f i s h i n g y i e l d s o - c a l l e d "weather" o f the sea, i . e . , the range o f frequencies
-
...
can be associated w i t h t h e
mesoscale and synoptic scale processes i n
s-l.
The marine weather, although d i s p l a y i n g t h e same f e a t u r e s as t h e atmospheric weather, i s c h a r a c t e r i z e d by time scales and l e n g t h scales which a r e o f t e n one order o f magnitude d i f f e r e n t .
The most i n t e n s e phenomena i n c o n t i n e n t a l seas
are f r e q u e n t l y found i n t h e mesoscale range, i n e r t i a l o s c i l l a t i o n s , t i d e s , storm surges
... and
t h e importance o f s y n o p t i c f e a t u r e s such as f r o n t s and
rossbies and macroscale f l o w f i e l d s which dominate atmospheric weather p a t t e r n s has o n l y been recognized, i n t h e sea, r e c e n t l y , w i t h the development o f remote sensing and l a r g e s c a l e s e a - t r u t h experiments p r o v i d i n g unprecedented s y n o p t i c views o f t h e ocean surface p r o p e r t i e s .
MARINE WEATHER EQUATIONS The equations d e s c r i b i n g t h e weather o f t h e sea can be obtained from t h e general S t r a t i f i e d F l u i d Dynamics equations by averaging over a time o f a few hours (say 104s).
The average e l i m i n a t e s mesialscale and microscale processes
from a l l the l i n e a r terms and o n l y t h e e f f e c t s , i n the mean, o f t h e i r nonl i n e a r i n t e r a c t i o n s remain i n t h e equations, i n t h e form o f t u r b u l e n t o r "pseudo-turbulent" d i f f u s i o n terms which can be parameterized w i t h t h e he1 p o f a p p r o p r i a t e eddy d i f f u s i v i t i e s . These d i f f u s i v i t i e s a r e r e l a t e d t o g l o b a l c h a r a c t e r i s t i c s o f the bliny-eddy turbulent f i e l d , v i z (i)
the mean t u r b u l e n t k i n e t i c energy
e = < 71y . y > ( i i ) the t u r b u l e n t k i n e t i c energy d i s s i p a t i o n r a t e
where y :
represents t h e f l u c t u a t i n g v e l o c i t y ,
a double s c a l a r product,
v
the vector operator
t h e kinematic v i s c o s i t y ,
11
0
= el
a a a El + e2 ax2 + e3 ax3
and where angular brackets
Y
denote an average.
Mesoscale and synoptic marine processes s a t i s f y ( i ) the Boussinesq approximation (according t o which the density of sea water may be assumed constant except in the gravity term where density deviations are multiplied by the acceleration o f gravity, several orders of magnitude larger t h a n typical flow accelerations) ; ( i i ) the quasi-hydrostatic approximation (according to which the vertical momentum equation reduces to a balance between vertical pressure gradient and gravity). If p o i s the constant reference ("Boussinesq") density and i f one defines "buoyancy" b and "reduced pressure" q by
q = E + g x 3 + 5 PO
(4)
where g i s the acceleration o f gravity, p the pressure, x3 the vertical coordinate ( t h e vertical axis pointing upwards) and 6 the tidal potential, the equation of continuity and the vertical component of the equation of momentum reduce to
where
-v = y
iv 3 g 3
i s the velocity vector (y i s the horizontal velocity vector). Eqs ( 5 ) and ( 6 ) may be regarded as defining equations f o r v 3 and q. The "marine weather" s t a t e variables are then (i) the two components of the horizontal velocity vector y , ( i i ) the buoyancy b ,
12 t h e t u r b u l e n t k i n e t i c energy e,
(iii) (iv)
the turbulent dissipation r a t e
E
.
I f y stands f o r any o f t h e s t a t e v a r i a b l e s ul, e v o l u t i o n equation can be w r i t t e n
where QY
u2, b y e,
E,
i s the r a t e o f production (destruction i f negative) o f
t h e general
y
and
xy
t h e b l iny-eddy d i f f u s i v i t y . The p r o d u c t i o n o f buoyancy i s e s s e n t i a l l y due t o r a d i a t i o n and i t i s gener a l l y p o s s i b l e t o assume t h a t r a d i a t i o n , absorbed i n t h e upper few meters o f t h e sea, can be represented by a surface source t o be taken i n t o account i n the boundary c o n d i t i o n s a t t h e a i r - s e a i n t e r f a c e . ( I n expressing these boundary c o n d i t i o n s f o r b, one must determine t h e buoyancy f l u x i n terms o f t h e f l u x e s o f s e n s i b l e and l a t e n t heat, evaporation and p r e c i p i t a t i o n , t a k i n g i n t o account t h e f l u x d i s c o n t i n u i t y due t o absorbed or e m i t t e d r a d i a t i o n , e.g.
N i houl , 1984). With t h i s approximation, one can w r i t e Q
b
= O
(9)
and (e.g.
Nihoul 1984, Rodi 1985)
for
y = u. J
for
y = e
for
y =
where
f
E
i s t h e C o r i o l i s frequency ( t w i c e t h e v e r t i c a l component o f t h e
earth's r o t a t i o n vector),
3 = h1 = ?
tum o r " t u r b u l e n t v i s c o s i t y " ,
yl,
yz
i s t h e t u r b u l e n t d i f f u s i v i t y o f momenand y 3 are e m p i r i c a l constants.
13 Parameterization o f b l i n y - e d d y t u r b u l e n t d i f f u s i o n I t can be shown (e.g.
N i h o u l , 1980) t h a t b l i n y and eddy t u r b u l e n c e c o n t r i -
butes t o a pseudo Kolmogorov cascade t r a n s f e r r i n g energy from t h e mean f l o w t o h i g h wave numbers (small s c a l e s ) where viscous energy d i s s i p a t i o n takes place.
The viscous s i n k i s c h a r a c t e r i z e d by
(i)
the length scale
(ii)
t h e time s c a l e
(iii)
the velocity scale t-1
uvn,lv
n,
E114 v 1 / 4
v
and (iv.)
t h e Reynolds number
Averaging and i n t r o d u c i n g an eddy v i s c o s i t y t o account f o r t h e e f f e c t , i n the mean,of m e s i a l s c a l e and m i c r o s c a l e f l u c t u a t i o n s , amounts t o r e p l a c i n g t h e energy t r a n s f e r through t h e cascade and i t s u l t i m a t e d i s s i p a t i o n a t h i g h wave numbers by a s i n g l e s i n k a t t h e s c a l e o f t h e energy c o n t a i n i n g eddies. and 1,
Ifum
a r e c h a r a c t e r i s t i c v e l o c i t y and l e n g t h scales o f these eddies and
t h e associated t i m e scale, one may argue t h a t , f o r t h e concept of eddy lm urn1 v i s c o s i t y t o be c o n s i s t e n t , one must r e q u i r e
Because t h e t u r b u l e n t energy spectrum f a l l s o f f very r a p i d l y from i t s peak
1
l,, ui i s a very s u b s t a n t i a l f r a c t i o n o f t h e t u r b u l e n t k i n e t i c energy e and one
value a t s c a l e l,, may assume
t h e k i n e t i c energy o f t h e eddies a t s c a l e
14 u m
Q
a ell2
Combining eqs (17), (18) and (19), one g e t s
The o t h e r t u r b u l e n t d i f f u s i v i t i e s a r e g e n e r a l l y expressed i n t h e form ;s-
- B
s-
v
where t h e
6"s
a r e new e m p i r i c a l f u n c t i o n s o r constants
One g e n e r a l l y considers t h a t
order 1 b u t t h a t
gb
B~ and
may be t a k e n as constants o f
i s a f u n c t i o n o f t h e s t r a t i f i c a t i o n measured by t h e
Richardson number
o r t h e F l u x Richardson number
where m
n
i s t h e B r u n t - V a i s a l a frequency as b e f o r e
(n2 =
lax,ab I
)
and
i s t h e " P r a n d t l frequency" g i v e n by
T h i s i s e a s i l y understood. I n a s t r a t i f i e d f l u i d , work has t o be done t o r a i s e an i s o l a t e d b l o b o f f l u i d above i t s e q u i l i b r i u m l e v e l .
I n z e r o shear ( i . e .
Ri =
m),
the blob o f
f l u i d w i l l f a l l back t o i t s e q u i l i b r i u m l e v e l a t a r a t e determined by t h e Brunt-VZisala frequency n .
As t h e shear increases ( i . e .
as
R i decreases),
t h e tendency f o r a d i s p l a c e d p a r c e l o f f l u i d t o r e t u r n t o i t s e q u i l i b r i u m l e v e l w i l l decrease, b u t t h e r e w i l l s t i l l be a buoyancy f o r c e a c t i n g on i t t o make i t r e t u r n .
As t h e b l o b o f f l u i d i s t e m p o r a r i l y d i s p l a c e d from i t s
e q u i l i b r i u m p o s i t i o n i t w i l l exchange i t s p r o p e r t i e s w i t h t h e surrounding f l u i d a t t h e new l e v e l .
I n t h e case o f temperature, s a l i n i t y , buoyancy and
o t h e r s c a l a r p r o p e r t i e s o f t h e f l u i d , complete exchange can o n l y be e f f e c t e d
15 by small s c a l e t u r b u l e n t m i x i n g and u l t i m a t e l y b y m o l e c u l a r a c t i o n .
This takes
a c o n s i d e r a b l e t i m e and u s u a l l y t h e p a r c e l o f f l u i d w i l l be dragged back t o i t s e q u i l i b r i u m l e v e l b e f o r e i t can exchange more t h a n a t i n y f r a c t i o n o f i t s heat, s a l t , buoyancy w i t h i t s new and d i s s i m i l a r s u r r o u n d i n g s d u r i n g i t s temporary residence there. F o r momentum, however, t h e s i t u a t i o n i s d i f f e r e n t .
The b l o b o f d s p l aced
f l u i d has a d i f f e r e n t h o r i z o n t a l v e l o c i t y t h a t i t s new s u r r o u n d i n g s t h e r e i s a s h e a r ) , and t h e r e i s a d r a g on it.
i.e.
T h i s i s a b u l k f o r c e which
r e q u i r e s no m o l e c u l a r m i x i n g - i n : t h e momentum i s t r a n s f e r r e d i m m e d i a t e l y by pressure. Thus momentum exchange i s l i k e l y t o r e t a i n i t s e f f i c i e n c y a t h i g h R i c h a r d s o n number, even though t h e buoyancy t r a n s f e r i s reduced as t h e s t r a t i f i c a t i o n increases. 8
b
One s h o u l d t h u s e x p e c t
= f ( R i o r Rf)
< 1
(25)
Parameterization o f sub-grid scale d i f f u s i o n The second t e r m i n t h e r i g h t - h a n d s i d e o f eq. ( 8 ) r e p r e s e n t s t h e mean e f f e c t s o f non-1 i n e a r i n t e r a c t i o n s o f f l u c t u a t i o n s c h a r a c t e r i z e d by t i m e s c a l e s smaller than t h e p e r i o d o f averaging.
Although these f l u c t u a t i o n s a r e a f f e c -
t e d by t h e s t r a t i f i c a t i o n , t h e y may s t i l l be r e g a r d e d as s u f f i c i e n t l y d i v e r s i f i e d and randomly d i s t r i b u t e d t o c r e a t e a f o r m o f t h r e e - d i m e n s i o n a l t u r b u lence w i t h r a t h e r s i m i l a r e f f i c i e n c y i n v e r t i c a l and h o r i z o n t a l d i f f u s i o n ( N i h o u l , 1980).
I n o t h e r words, i f t h e t u r b u l e n t d i f f u s i v i t i e s a s s o c i a t e d w i t h
-
mesialscale m i c r o s c a l e f l u c t u a t i o n s a r e n o t t h e same as p o s t u l a t e d i n eq. (8), they may be assumed o f comparable o r d e r s o f magnitude. I n t h a t case, t h e c h a r a c t e r i s t i c l e n g t h s c a l e s o f h o r i z o n t a l v a r i a t i o n s being c o n s i d e r a b l y l a r g e r t h a n t h e v e r t i c a l l e n g t h s c a l e s , one may n e g l e c t t h e h o r i z o n t a l d i f f u s i o n as compared t o t h e v e r t i c a l d i f f u s i o n . i m p l y t h a t t h e r e i s no h o r i z o n t a l d i f f u s i o n i n Nature.
T h i s does n o t
It s i m p l y means t h a t ,
a t t h i s stage, t h e main p a r t i s s t i l l concealed i n t h e a d v e c t i o n t e r m which c o n t a i n s i r r e g u l a r and v a r i a b l e h o r i z o n t a l c u r r e n t s r e s p o n s i b l e f o r a f o r m o f h o r i z o n t a l "pseudo t u r b u l e n c e " (e.g.
N i h o u l 1975, Monin and Ozmidov 1985).
16 The discrepancy between h o r i z o n t a l and v e r t i c a l l e n g t h scales however i m poses, i n most cases, numerical g r i d s w i t h much l a r g e r h o r i z o n t a l meshes ( t y p i c a l l y one order o f magnitude l a r g e r than the lengths scale which one would associate w i t h the time average's c u t - o f f by s i m i l a r i t y estimates). The d i s c r e t i z a t i o n o f t h e equations i s then equivalent t o performing a second ( h o r i zontal space) average and non-linear i n t e r a c t i o n s o f sub-grid scale f l u c t u a t i o n s are responsible f o r an a d d i t i o n a l h o r i z o n t a l d i f f u s i o n which i t i s convenient t o introduce e x p l i c i t l y i n the mathematical e v o l u t i o n equations, a n t i c i p a t i n g t h e subsequent d i s c r e t i z a t i o n .
The second term o f t h e r i g h t -
hand s i d e o f eq. (8) i s then w r i t t e n
where
The h o r i z o n t a l d i f f u s i v i t i e s
zy
can be r e l a t e d t o t h e mesh s i z e and t o
the t u r b u l e n t energy d i s s i p a t i o n r a t e i n t h e associated range o f scales using an extension o f Kolmogorov's theory developed by Ozmidov (e.g.
Nihoul 1975,
Monin and Ozmidov 1985). I n many cases, they can be taken as constants. With eqs. (5) and (6), the e v o l u t i o n equations f o r 2, b, e and E obtained from eq. (8) (where the l a s t term i s replaced by 26 and the production r a t e s a r e given by 10, 11 and 32) and eq. (20) r e l a t i n g Y , e and E, t h e system o f marine weather equations i s closed except f o r e m p i r i c a l c o e f f i c i e n t s o r f u n c t i o n s a,
6,
y
... t o be
determined by c a l i b r a t i o n o f the model.
...
Additional parameters (drag c o e f f i c i e n t s , albedo,
) appear i n t h e expres-
s i o n o f the boundary conditions, e s p e c i a l l y a t t h e a i r - s e a i n t e r f a c e , and t h e i r v a l u a t i o n i s a l s o p a r t o f the c a l i b r a t i o n exercices (e.g.
Nihou1,1984).
THE M I X I N G LENGTH APPROXIMATION
The equation f o r t h e t u r b u l e n t d i s s i p a t i o n r a t e the weak p o i n t o f three-dimensional modelling. production r a t e
QE
, it
E
i s , by common consent,
The f a c t t h a t , through the
introduces many empirical parameters i s a demonstra-
t i o n o f i t s l a r g e l y e m p i r i c a l character and, i n t h e same time, an i n d i c a t i o n o f the amount o f parameterizing which has been subjacent t o t h e s e t t i n g up o f t h i s equation.
17
Several authors have t r i e d t o r e p l a c e t h e equation f o r t i o n s f o r d i f f e r e n t combinations o f
E,
e and
, without
E
by s i m i l a r equa-
succeeding i n decrea-
sing t h e volume o f p a r a m e t e r i z a t i o n and empiricism (e.g. Blumberg and M e l l o r , 1985). Faced w i t h t h i s d i f f i c u l t y , one n a t u r a l l y t r i e s t o s i m p l i f y t h e model and the concept o f "mixing l e n g t h " extended from t h e e a r l y work o f P r a n d t l seems, i n t h i s respect, r a t h e r promising. Combining eqs (18), (19) and (20), one o b t a i n s
lmi s thus e q u i v a l e n t t o p r e d i c t i n g
Predicting
E l a b o r a t i n g from
E .
P r a n d t l ' s e a r l y t h e o r i e s o f turbulence, several authors have come t o t h e conclusion t h a t
lm , t h e modern v e r s i o n o f P r a n d t l ' s "mixing l e n g t h " , could,
i n many cases, be determined
-
as a f u n c t i o n o f space and s t r a t i f i c a t i o n
-
by simple i n s p e c t i o n , thus s p a r i n g t h e a n a l y s t t h e s o l u t i o n o f an a d d i t i o n a l
E
( o r e q u i v a l e n t ) equation. I t i s g e n e r a l l y assumed t h a t t h e m i x i n g l e n g t h
1,
can be w r i t t e n i n t h e
form 1, = 1 0 JI where of
lo
i s i t s value i n n e u t r a l c o n d i t i o n s ( n = 0) and
the s t r a t i f i c a t i o n .
1,
i s an a l g e b r a i c f u n c t i o n o f
JI
i s a function
xj
which must be
such t h a t i t respects the c l a s s i c a l l o g a r i t h m i c s i n g u l a r i t i e s i n t h e bottom boundary l a y e r and a d j u s t t o wind-mixed l a y e r c o n d i t i o n s near t h e surface. The f u n c t i o n
Ri.
JI
has been mostly expressed i n terms o f t h e Richardson number
One o f t h e o r i g i n a l i t y o f t h e GHER-model developed a t t h e GeoHydrodynamics
and Environment Research Laboratory o f t h e U n i v e r s i t y o f Liege i s t h e determib n a t i o n o f parametric r e l a t i o n s h i p s o f JI as w e l l as 6 i n terms o f t h e
-
-
f l u x Richardson number Rf i n s t e a d o f R i (Nihoul and D j e n i d i , 1986). The r e s u l t s o f t h e Medalpex Experiment i n t h e Mediterranean ( D j e n i d i e t a l . , 1987) suggest r a t h e r simple formulas f o r
JI
and
gb
o f t h e type
18
gb
‘L
(1 - R f ) 1 ’ 2
(31)
These expressions - however simple they appear - are not t r u l y surprising. For instance, in the case of a stably s t r a t i f i e d environment, eddies a t scale l m have only to transfer, t o the viscous s i n k , via the energy cascade, an energy E ~ ~- R(f ) 1per u n i t time where E, denotes the energy extracted per u n i t time from the mean flow. In the absence of s t r a t i f i c a t i o n , E, would Q
be passed on t o the cascade by eddies a t scale 1 ,
.
The relation
amounts t o requiring t h a t , i n any case, for a given energy level the character i s t i c time of dissipation computed with the turbulent viscosity be the charact e r i s t i c time of evolution of the b i g eddies, i.e.
THE SHALLOW WELL-MIXED SEA APPROXIMATION I f the sea i s s u f f i c i e n t l y shallow and well-mixed ( f o r instance by intense tidal currents) as the North Sea, several simplifying hypotheses can be made, vi z ( i ) negligible buoyancy e f f e c t s , i . e .
( i i ) local balance of turbulent production and destruction r a t e s , i .e.
i.e.
u s i n g eq. (20) and (36),
19
u*
where
denotes the s o - c a l l e d f r i c t i o n v e l o c i t y .
Eq. (28) gives then
w
1 * m
V % U
The f r i c t i o n v e l o c i t y
u*
and the m i x i n g l e n g t h
t i v e l y , the square r o o t o f t h e bottom s t r e s s
lm can be scaled by, respec-
xb ( p e r u n i t mass o f sea water)
and the t o t a l depth. Thus
where
i s the t o t a l depth,
h i s t h e depth, ,c t h e surface e l e v a t i o n , z = x 3 + h i s t h e
a l t i t u d e above the bottom I x 2 = 0 corresponds t o t h e undisturbed f r e e surface). The f o n c t i o n
i s determined e m p i r i c a l l y from experimental data.
u
vations and models have shown t h a t t h e most important requirement on
Obseru
was
i t s a b i l i t y t o t a k e i n t o account t h e l o g a r i t h m i c s i n g u l a r i t i e s i n t h e bottom boundary l a y e r ; t h e exact shape o f t h e p r o f i l e o f u being much l e s s cogent f o r subsequent c a l c u l a t i o n s (e.g. Nihoul 1977, R o i s i n 1977, Nihoul e t a l . 1979). Neglecting buoyancy and u s i n g eq. (39), one can solve t h e c o n t i n u i t y and momentum equations f o r t h e v e l o c i t y f i e l d w i t h o u t determining t h e a d d i t i o n a l variables e and
E.
A f u r t h e r s i m p l i f i c a t i o n i s obtained by i n t e g r a t i n g these equations over depth and s o l v i n g f o r t h e depth-mean v e l o c i t y
i.
The v e r t i c a l l y i n t e g r a t e d equations read
aH
-t at
0.(Hi)
= 0
& ( H i ) + l.(Hii)
t
fHg3Ai =
- HF
(-Pa + g i + t ) +
zs - zb + Q
P0
where pa i s the atmospheric pressure, water) and
Q
xs t h e
wind s t r e s s ( p e r u n i t mass o f sea
a d i f f u s i o n term r e s u l t i n g from n o n - l i n e a r i n t e r a c t i o n s o f sub-grid
scale processes and v e l o c i t y f l u c t u a t i o n s around i t s v e r t i c a l average ("shear
20
e f f e c t " ; e. g. N i houl , 1975). I n most cases, Q can be expressed i n simple Laplacian d i f f u s i o n form i n t r o ducing a new (constant) v i s c o s i t y c o e f f i c i e n t . The bottom stress xb i s r e l a t e d t o t h e mean v e l o c i t y
and t o t h e wind stress.
The most commonly
used formula i s t h e "quadratic f r i c t i o n law"
where (6
2,
D
i s t h e "drag c o e f f i c i e n t " (D
%
2
and 6 an e m p i r i c a l constant
10-1).
Although eq. (43) has been very successful i n marine forecasting, t h e r e are i n d i c a t i o n s t h a t i t i s n o t v a l i d i n periods o f weak mean c u r r e n t s ( a t t i d e reversal, f o r instance). I n such periods, i n f a c t , the mean v e l o c i t y is a poor i n d i c a t i o n o f t h e f l o w f i e l d : t h e r e may be a s u b s t a n t i a l veering o f the v e l o c i t y vector along t h e v e r t i c a l , w i t h q u i t e d i f f e r e n t bottom and surface currents, and t h i s may be important i n some a p p l i c a t i o n s o r a t s p e c i f i c 1ocations. The 2D model must then be complemented by t h e equation f o r the v e l o c i t y deviation o = g
.
-
The l a t t e r i s e a s i l y obtained by s u b s t r a c t i n g the equa-
t i o n f o r the mean (42) from the o r i g i n a l equation f o r g and reads (Nihoul e t al. 1979)
air at
+
fg3AQ
where
+
a
ail
ax3
ax3
= - (7 - )
-
Ls - L b (44)
H
stands i n b r i e f f o r a l l the c o n t r i b u t i o n s o f t h e non-linear terms
(The d e t a i l e d expression i s given i n Nihoul e t al.,
1979).
Eq. (44) must be solved subject t o the f o l l o w i n g boundary conditions
a= - -u
(g = 0)
a t t h e surface
(45)
a t the bottom
(46)
I n a d d i t i o n , one must have a t t h e bottom
(47)
21
Hence L~ i s now determined by t h e model as a f u n c t i o n o f
i,xs ... .
The non-linear terms j l - are important when the c u r r e n t i s strong b u t e g l i g i b l e when i t i s weak i.e., p r e c i s e l y , when 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 i s questionable. current f i e l d may be important and when eq. (43) f o r
1
xb
Most o f the i n f o r m a t i o n r e q u i r e d i s thus contained i n t h e l i n e a r form o f eq. (44) which one can solve, even a n a l y t i c a l l y , i n p a r a l l e l w i t h the 2D model a t a l l points o f interest. Using an a n a l y t i c a l s o l u t i o n based on s e r i e s expansion i n eigenfunctions o f the v e r t i c a l t u r b u l e n t d i f f u s i o n operator, Nihoul (1977) and R o i s i n (1977) have shown f o r instance t h a t t h e bottom s t r e s s could be w r i t t e n , w i t h a good approximation i n the form
where
q
i s a numerical f a c t o r .
The l a s t term i n t h e r i g h t hand s i d e o f eq. (48) turns o u t t o be n e g l i g i b l e as compared w i t h the f i r s t one as l o n g as the mean v e l o c i t y i does n o t approach zero. It becomes important when i i s s u f f i c i e n t l y small and one can see t h a t i t s e f f e c t , associated w i t h the f l o w ' s i n e r t i a , may be regarded as a "memory" e f f e c t i n the determination o f xb. S t a r t i n g from the 2D depth-averaged model and the l o c a l l y 1D l i n e a r model, one can, by successive i t e r a t i o n s , i n c l u d e t h e non-linear terms and construct a f u l l y 3D = 2D + 1D model as shown on the sketch-plan o f f i g u r e 1. One o f the advantages o f t h i s model i s t h a t t h e 2D submodel can be operated s o l e l y whenever one i s s a t i s f i e d w i t h depth-averaged i n f o r m a t i o n and t h a t the whole machinery needs o n l y be r u n when and ( o r ) where the d e t a i l o f the v e r t i cal s t r u c t u r e i s required.
22
30 = 20
+ 1D
Depth i n t e g r a t e d model
+
as functions o f t, xl,
x2
l i n e a r l o c a l l y 1D model
v
t
Y
X l Y x2
nl -
/ Fig. 1. seas.
Sketch o f the 3D = 2D
non l i n e a r 1D model
I
+ 1D model f o r shallow well-mixed c o n t i n e n t a l
EXAMPLES OF APPLICATIONS I n the l a s t years, the GeoHydrodynamics and Environment Research Laboratory (GHER) o f the U n i v e r s i t y o f Li6ge has developed a 3D = 2D
+
1 D model and a
f u l l y 3D ( t u r b u l e n t energy, mixing length-closure) b a r o c l i n i c model.
The f i r s t
one was c a l i b r a t e d f o r the North-West European Continental S h e l f w i t h emphasis on the i l o r t h Sea, the secondone f o r the Mediterranean w i t h emphasis, i n a f i r s t s e t o f simulations, on the A d r i a t i c Sea. shown on the f o l l o w i n g f i g u r e s , i n i l l u s t r a t i o n .
Some exemplary r e s u l t s are
23
Fig. 2. Tidal fronts calculated by the 3D West European Shelf well-mixed water zones of transition stratified water
0
=
2D + 1D GHER model on the North-
24
0. I
0.2
0.3
0. I
0.2
0.3
Fig. 3. E v o l u t i o n w i t h time, a t t i d e reversal, o f t h e two components 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 a t t h e p o i n t 52"30'N 3'50'E i n t h e Southern B i g h t o f the North Sea. (Depth 22m, wind blowing t o t h e North-East, maximum wind s t r e s s o f 2 W 4 m 2 s-~). The curves 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' i n t e r v a l s . The upper curve represents t h e northern component, the lower curve, the eastern component (GHER 3D = 2D + 1D Model).
25
a7
u2 (m
2)
a6 a6 a4
a3 a2
ai
Fig. 4. Evolution w i t h time over t h e f i r s t h a l f t i d a l p e r i o d o f the Ekman diagram showing the v e r t i c a l veering o f t h e horizontal v e l o c i t y vector a t the point 52'30'N 3"50'E i n t h e Southern Bight o f t h e North Sea. The separation between two successive curves i s 18'. (GHER 3D = 2D + 1D Model).
26
Fig. 5 . Residual sumner transport on the North-West European Shelf (GHER 3D = 2D + 1D Model, real winds, stream functions i n 103m3 s- l ) .
27
Fig. 6 . Residual w i n t e r t r a n s p o r t on t h e North-West European S h e l f (GHER 3D = 2D + 1D Model, r e a l winds, stream functions i n 103m3 5 - l ) .
28
82
Fig. 7. Flow p a t t e r n i n the Northern A d r i a t i c Sea a t 3 m (above) and 18m(below) computed w i t h the 3D GHER model f o r J u l y 28, 1979, 4 H 12 m i n n e g l i g i b l e wind conditions. Fig. 8 and Fig. 9 which follow show t h a t the model is a b l e t o reproduce the observed v a r i a b i l i t y w i t h the tendency t o form gyres i n the region of the Pb. (Djenidi e t a l . , 1987).
6Z
29
Fig. 8.
Evolution o f the f l o w p a t t e r n a t 3 m, i n the Northern A d r i a t i c Sea , 1987).
(GHER, 3D Model , D j e n i d i e t a l .
OE
30
F i g . 8'. E v o l u t i o n o f t h e f l o w p a t t e r n a t 3 m, i n t h e Northern A d r i a t i c Sea (GHER, 3D Model, D j e n i d i e t a l . , 1987).
31
Fig. 8". Evolution of the flow pattern a t 3 m y in the Northern Adriatic Sea (GHER, 3D Model, Djenidi e t a l . 1987).
32
Fig. 9. Comparison between t h e p r e d i c t e d seston d i s t r i b u t i o n f o r July 28, 79, 4 H 12 m and remote sensing observations (CZCS data, graduated i n mg chlorop h y l l m 3 ) (GHER 3D Model, D j e n i d i e t a l . , 1987).
33
ACKNOWLEDGMENTS The authors are indebted t o t h e European Atomic Energy Community (Euratom) for
p a r t i a l support o f t h i s research v i a contracts SC-O12B/BIAF/423(SD) and 2831-85/PC ISPB. They wish t o express t h e i r g r a t i t u d e t o t h e i r colleagues o f the Commissariat 1 1'Energie Atomique, Paris and the J o i n t Researchcenter I s p r a f o r many f r u i t f u l discussions and a u t h o r i z a t i o n t o reproduce some o f the figures. REFERENCES Blumberg, A.F. and Mellor, G.L., 1985. A d e s c r i p t i o n o f three-dimensional coastal ocean c i r c u l a t i o n model. I n : N. Heaps ( E d i t o r ) , Three-Dimensional Shelf Models, Coastal and Estuarine Dynamics, 5, American Geophysical Union Publ.. Djenidi, S., Nihoul J.C.J., Clement, F. and Salas de Leon, D., 1987. The MODEM c o n t r i b u t i o n t o Medalpex. Annales Geophysicae, 5; 1-19. Monin, A.S. and Ozmidov, R.V., 1985. Turbulence i n the ocean. D. Reidel Publ. Co. , Dordrecht. Nihoul, J.C.J. , 1975. Modelling o f Marine Systems. E l s e v i e r Publ. Co. , Amsterdam. Nihoul, J.C.J., 1977. Three-dimensional model o f t i d e s and storm surges i n a shallow well-mixed c o n t i n e n t a l seas. Dyn. Atmos. Ocean., 2: 29-47. Nihoul, J.C.J., Runfola, Y. , and Roisin, B. , 1979. Non-linear three-dimensional modelling o f mesoscale c i r c u l a t i o n i n seas and lakes. I n : J.C.J. Nihoul (Editor), Marine Forecasting, E l s e v i e r Publ. Co., Amsterdam, chapter 15, 235-259. Nihoul, J.C.J., 1980. The t u r b u l e n t ocean. I n : J.C.J. Nihoul ( E d i t o r ) , Marine Turbulence, E l s e v i e r Publ. Co., Amsterdam, chapter 1, 1-19. Nihoul, J.C.J. , 1984. A three-dimensional marine c i r c u l a t i o n model i n a remote sensing perspective. Annales Geophysicae, 2: 433-442. Nihoul , J.C.J. and D j e n i d i , S., 1986. Three-dimensional mathematical models f o r "marine weather" p r e d i c t i o n . I n v i t e d paper, E n v i r o s o f t Conference, Los Angeles, USA, November 19-21, 1986. Rodi, W., 1985. Survey o f c a l c u l a t i o n methods f o r f l o w and mixing i n s t r a t i f i e d f l u i d s . I n : Proceedgins IUTAM Symposium on Mixing i n S t r a t i f i e d Fluids, Margaret River Western A u s t r a l i a , August 85, 1-51. Roisin, 8. , 1977. ModBles tri-dimensionnels des courants marins. M i n i s t r y f o r Science P o l i c y Brussels, Rep. ACN3, 124 pp.
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35
ON MODELING THREE-DIMENSIONAL ESTUARINE AND MARINE HYDRODYNAMICS
Y. PETER SHENG University of Florida, 336 Weil Hall, Gainesville, FL
32611 ( U . S . A . )
ABSTRACT Recent advances of a three-dimensional numerical model of estuarine and marine hydrodynamics are described in this paper. In particular, the parameterization of the vertical turbulent transport based on a Reynolds stress model and the adaptation of a generalized curvilinear (or "boundaryfitted") grid to the finite-difference model are highlighted. These two aspects of the three-dimensional numerical model, along with other features, allow accurate simulation of turbulent flows in estuarine and marine waters where complex geometry and bathymetry are generally present.
1. INTRODUCTION Estuarine and marine hydrodynamic processes (e.g.,
tidal currents, front
dynamics, and sediment dispersion) often involve three-dimensional turbulent flow in the presence of complex geometry and bathymetry.
For example, tidal
circulation and salinity transport in such estuaries as Suisun Bay, California (Sheng, et al.,
1985) in Figure 1, and Mississippi Sound, Mississippi (Sheng
and Butler, 1982) are strongly affected by complex geometry and bathymetry. Local bathymetry and geometry significantly affect flow and sediment transport around various estuarine and marine structures (e.g., waters,
and
navigation
channels),
and
dredged
bottom pipelines, breakmaterial disposal mounds.
Hence, to accurately simulate marine and estuarine currents due to tides, winds and density gradients, numerical models must be able to accurately and efficiently resolve (1)
the turbulent transport processes, particularly the
dynamics of various vertical boundary layers shown in Figure 2, and (2) the complex geometry and bathymetry (Sheng, 1986a). Turbulent transport in the various vertical layers as shown in Figure 2 strongly affect many estuarine and marine processes.
For example, oil spill
trajectory is affected by the dynamics of the laminar sublayer and constant flux layer underneath the air-sea interface, while the deposition and erosion of sediments are governed by the laminar sublayer and constant flux layer near the bottom.
Seeking to remove the empiricism contained in simple eddy-
viscosity models, turbulence models such as the Reynolds stress model (e.g., Sheng, 1982 and 1984) and the two-equation (or k-c) model (e.g.,
Rodi, 1980)
36 have been applied to estuarine marine, and riverine environments.
Two simpli-
fied versions of a Reynolds stress model (Sheng, 1 9 8 4 ) have been incorporated into a three-dimensional model of estuarine and marine hydrodynamics by the present author.
A brief description of the Reynolds stress model, and its
simplified versions, along with model applications, will be presented in this paper. Traditional multi-dimensional hydrodynamic models of marine and estuarine currents use the finite difference technique and a uniform Cartesian grid
Figure 1
3-D view of Suisun Bay, California.
@
@
Figure 2
EUNANLAVER
I
\
------CONSTANT FLUX LAVER
Vertical layers within relatively Faep coastal/estuarine waters.
37 Leendertse, 1967) with the shoreline and bathymetry represented by
(e.g.,
numerous stair-steps.
This not only severely limits the model accuracy but
also frequently dictates a prohibitively large number of grid points to resolve a complex environment.
To better resolve the lateral geometry, more
refined lateral grids have been used with various finite-difference models. For example, Sheng (1976) used nested and dynamically coupled Cartesian grids
to study the 3-D
nearshore circulation,
Butler (1978)
utilized stretched
Cartesian grid to study coastal waves, Warnstrath (1977) employed conformal grid to study storm surge, and Waldrop and Tatom (1976) used an orthogonal curvilinear grid to study thermal plume in river.
The 3-D
coastal hydro-
dynamic model of Sheng and Butler (1982) and Sheng (1983) used a laterally exponentially-stretched
Cartesian grid and a vertically
a-stretched
grid,
which is a special form of the "boundary-fitted grid". Recently, Johnson (1982) employed the boundary-fitted grid technique to study
the
2-D
vertically-averaged
riverine
circulation.
Non-orthogonal
boundary-fitted curvilinear grids for the physical domain were first generated.
Equations of motion in the Cartesian coordinates were then transformed
into those in the curvilinear coordinates by performing chain-rule tranaformation.
Although the independent variables (the coordinates) were transformed,
the dependent variables (the velocity components) remained unchanged as in the Cartesian coordinates.
The 'present model, however, solves the transformed
equations of motion in terms of the "contravariant" components of velocity vectors.
This technique allows for simpler equations and boundary conditions
and better representation of complex geometry with relatively few number of grid points, and thus significantly improves the capability of finite difference models.
Highlights and some applications of the curvilinear-grid
hydrodynamic model will be presented. 2. A THREE-DIMENSIONAL CARTESIAN-GRID MODEL 2.1 Mean Equations
The basic equations describing the mean motion of coastal and estuarine waters are consisted of the continuity equation, the equations of motion, the heat equation, the salinity equation and the equation of state. ( 1 ) the
hydrostatic
approximation,
( 3 ) the eddy-viscosity
(2) the
Boussinesq
Assuming
approximation,
and
concept, the various equations can be written with
respect to a right-handed Cartesian coordinate system as:
v.u-0
(1)
38 1 5
aT
+ at
. (UT)
V
a p
-u, represents
(Kv
z)+ V, . (%IT) aT
(3)
where u represents the 3-D velocity vector (u,v,w) in the (x,y,z) directions,
,.
the horizontal velocity vector (u,v) in the (x,y) directions,
z is the unit vertical vector, t is time, f is Coriolis parameter,
V, repre-
sents the horizontal gradient, Pa is atmospheric pressure, g is gravitational acceleration, 5 is surface elevation, P is density, T is temperature, S is salinity, (pt, Dv) represent the vertical turbulent eddy coefficients, and
a,
(%, %, DH) represent the lateral turbulent eddy coefficients. Notice that the dynamic boundary condition at the free surface has already been incorporated into the above equations. 2.2
Boundary Conditions Various layers shown in Figure 2 exist in the water column of relatively
deep estuarine and marine waters.
In relatively shallow waters, the effect of
friction may be so important that the Ekman layers merge.
Dynamics of the
relatively thin sublayer (-1 mm) and the constant flux layer (-1 m) can affect the transport of such materials as heat, sediment, oxygen, nutrient and oil slick, which are often introduced into the water body from the surface or bottom boundaries. The constant flux layer above the bottom is quite similar to that above the free surface. Applying the law of wall at the bottom allows one to relate the bottom stress with the velocity at some distance above the bottom as: T -b
*
w' 'dw
where
; 1 -+I u-+
Zb represents
density, le, is
the bottom stress vector in (x,y) directions, pw is water
the horizontal velocity vector in (x,y) directions at some distance z+ above the bottom, and Cdw is the drag coefficient determined from:
where
K
is the von-Karman constant, zo is the physical roughness height, L is
39 the
Monin-Obhukov
L approaches
OD
similarity
length,
and
is
a
stability
function.
while 4, approaches 0 for neutrally-stratified flows.
Equstions (6) and (7) can be applied to the marine boundary layer above the air-sea
interface to compute the surface wind
velocity at some distance eo),
(2,)
stress from the wind
above the air-sea interface (with roughness
when Pa and cda (subscript a stands for air) are used instead of pW and
cdw. Additional constant flux layer expressions similar to (6) and (7) relate the heat flux and mass flux at bottom or free surface to local mean temperature gradient and mean concentration gradient.
If one is concerned with the
dispersion of non-neutrally buoyant particles such as sediments, planktons, and larvae, the boundary conditions are more complicated.
For instance,
erosion and deposition of sediments must be included in the bottom boundary conditions. 2.3 Turbulence Parameterization In the present (A,,
%,
3-D
model,
the vertical
turbulent eddy coefficients
Dv) are determined from simplified versions of a Reynolds stress
model and require no ad-hoc parameter tuning with data. model will be discussed in the next section.
The Reynolds stress
Since the lateral turbulent
diffusion is generally much less important than the lateral advection and vertical turbulent diffusion in shallow seas, the lateral turbulent eddy coefficients are often taken to be constants to parameterize the sub-grid scale turbulence associated with the large lateral eddies. 2.4
a-Stretching:
A Boundary-Fitted Grid
Sheng et al. (1978) detailed the vertically-stretched (or "a-stretched") version of the above equations invoking the small amplitude approximation, i.e.,
I;
"
fluxes ,
concentration variance ,
concentration-
and turbulence macroscale A.
Because of the
the concentration equation,
salinity equation and
temperature equation, we shall only include temperature as a variable for simplicity
in subsequent
discussion.
Following the procedure originally
outlined by Donaldson (1973), the second-order correlation equations in threedimensional vector form are:
>
a at
+ v,
.
V . IT +
.
(qA~)
-
Lbsq (10) h
where q is the total turbulent intensity defined as
~'~
and A is the
turbulence macroscale representing the average turbulent eddy size. the right-hand-side terms in the above equations (e.g., the buoyancy
terms, and
parameterization.
the rotation terms) are exact, and
require no
The last three terms in Equation ( 8 ) and the last two terms
in Equations ( 9 ) and (10) third-order
Many of
the production terms,
are "modeled" terms representing the effects of
correlations, pressure correlations, and viscous dissipations.
The model coefficients (b, A, vc and
8 )
have been determined from critical
laboratory experiments where only one of the coefficients is important. final coefficients (b = 0.125, A = 0.75, vc = 0 . 3 and s * 2.8)
The
thus determined
have remained fixed for all model applications.
3.2 Estuarine and Uarine Applications of the Reynolds Stress Model The Reynolds stress model described above has been extensively applied to estuarine,
marine
and
atmospheric
environments.
The
1-D
version
(the
variables vary in the vertical direction only) has been used to study wave boundary layer underneath a nearly sinusoidal wave (Sheng, 1982 and 19841, wave boundary layer underneath a cnoidal wave (Sheng, 19841,
current-wave
interaction within bottom boundary layers (Sheng, 1983 and 19841, flow within a vegetation canopy (Sheng, 1982), ocean mixed layer dynamics (Sheng, 1984), and
sediment-laden boundary layer (Sheng and
Villaret,
1986).
The
2-D
Reynolds stress model is currently being used to study flow over a rippled bed. study.
The fully 3-D Reynolds stress model is also being used for a dispersion
As an example, the vertical distribution of Reynolds stress within a
nearly sinusoidal wave boundary layer, a cnoidal wave boundary layer, a current-wave boundary layer, and a vegetation canopy are shown in Figure 6. While eddy viscosity models may match the measured mean velocity profiles reasonably well by adjusting the eddy viscosity coefficients, the Reynolds stress model can predict the mean flow and Reynolds stresses without ad-hoc parameter tuning.
In addition, the Reynolds stress model explicitly computes
the thickness of time varying logarithmic layer in wave boundary layer and clearly reproduces the effects of wave on current such as the increased Reynolds stress, turbulent intensity and apparent roughness.
45
Figure 6(a) Vertical distribution of Reynolds stress within the turbulent wave boundary layer measured by Jonsson and Carlsen (1976).Comparison between model results (solid lines) and data (symbols). at 4 phase angles. Figure 6(b) Vertical distribution of Reynolds stress within the turbulent wave boundary layer under a cnoidal wave at 8 phase angles. Model results only. Figure 6(c) Simulated current-wave bottom boundary layer at a site in the Mississippi Sound. Vertical profiles of Reynolds stress averaged over the wave cycle. zo = 0.1 cm, Uloo = 10 cmfsec, and Tw 2.5 sec.
-
Figure 6(d) Comparison of model predictione with in and above a corn canopy.
46 3.3
Simplified Versions of Reynolds Stress Model Two simplified versions of the Reynolds stress model eliminate some of
the terms in Equations (8) through (11) and are particularly useful.
In the
super-equilibrium version, all the second-order correlations are assumed to be in local equilibrium such that there is no time evolution or spatial diffusion of the correlations.
Equations (8) to (11) are thus simplified to a closed
set of algebraic equations between the second-order gradients of mean velocity and temperature.
correlations and the
In the quasi-equilibrium version,
most second-order correlations are assumed to be in high Reynolds number local equilibrium,
and algebraic relationships hold for these correlations and mean
flow gradients. equations,
The dynamics of the turbulence is carried by two dynamic
one for q2 =
<x'.v,'>
and one for A.
This approximation gives
reasonable results so long as the time scale of turbulence, A/q, is much less than the time scale of mean flow. The two simplified versions of
the Reynolds stress model have been
applied to study various boundary layers in laboratory, estuarine and marine environments.
The quasi-equilibrium version was able to faithfully reproduce
the wave boundary layer experiment described in Sheng (1982 and 1984). addition,
it has been used to study sediment-laden
boundary layers.
In The
super-equilibrium version has been successfully used to simulate storm generated
currents on continental shelf near Grand
Bank
(Sheng,
1986b) and
tropical cyclone generated currents. Both simplified versions have been incorporated into the 3-D hydrodynamic model developed by the present author. 4. A THREE-DIMENSIONAL CURVILINEAR-GRID MODEL 4.1 Boundary-Fitted Grid To model flow within a rectangular domain, Cartesian grid.
cylindrical or spherical grid. and bathymetry (see, e.g., grid,
it is natural to use a
For cylindrical or spherical domains, it is natural to use a Hence, in the presence of complex shoreline
Figure 11, it is natural to use a "boundary-fitted"
or generalized curvilinear grid
to accurately
represent
the model
boundaries. Conformal grid, orthogonal grid and non-orthogonal grid are the various types of "boundary-fitted"
grid with increasing complexity and generality.
For relatively simple geometries, it is possible to generate conformal or orthogonal grids by rather straightforward techniques. coastal applications, however,
For most estuarine and
the shoreline geometries are usually quite
complex and conformal or orthogonal grid cannot be generated unless the shorelines are approximated by simple curves.
Hence, in general, it is essential
47
to generate non-orthogonal grid for estuarine and coastal applications.
The
present 3-D hydrodynamic model employs the elliptic grid generation technique (Thompson, 1982) to generate non-orthogonal boundary-fitted grid in the horizontal directions and o-stretched grid in the vertical direction.
As shown in Figure 7a, the basic problem is to solve: (12)
x
2
n = Q
where
2
(13)
represents the Laplacian operator in x and y directions and P and Q
are forcing functions for achieving desired grid resolution and alignment, with the following boundary conditions:
6 E
-
n
= E(x,y),
constant,
n
constant
on boundaries 1 and 3
= n(x,y)
on boundaries 2 and 4
In practice, however, one actually solves for (x,y) in terms of (6.n) by interchanging the dependent and independent variables in Eqs.
(12) through
(15).
As indicated earlier, the present model solves the transformed equations of motion in the (6,n) plane in terms of the "contravariant" velocity components (Figure 7b) instead of the Cartesian velocity components.
Sheng
(1986~) discussed the numerous advantages of the "contravariant model" over other models that work with covariant, physical or Cartesian velocity c o w ponents.
The model first develops the tensor-invariant form of the equations
of motion and then expands into component equations in 5 and
n directions.
For simplicity, we will list the tensor-invariant form of the verticallyintegrated equations of motion:
where
5 0 % ) . The melding simply weights t h e f o r e c a s t stream f u n c t i o n (J, ) and t h e a n a l y s i s stream f u n c t i o n (qa) i n v e r s e l y with r e s p e c t t o y , i - e . , f I ) =~ y$ ~ + ~( l - ~ ) $Barotropic ~ . simulations with t h i s scheme were explored by T u (1981).
Although i t i s n o t t h e o r e t i c a l l y o p t i m a l , i t i s a s i g n i f i c a n t q u a l i -
t a t i v e and q u a n t i t a t i v e (Rienecker e t a z . , 1987, Table 5 ) improvement of t h e e s t i m a t e which u t i l i z e s r e a d i l y a v a i l a b l e f i e l d s and c o r r e l a t i o n s .
I t seems
reasonable t o implement such s t r a t e g i e s while both developing ocean f o r e c a s t i n g technology and t a i l o r i n g optimal a s s i m i l a t i o n schemes t o oceanic flow f i e l d s and open r e g i o n a l conditions.
I n comparing Figs. 6d and 6e i t i s important t o
recognize t h a t t h e a n a l y s i s o f 6e i s n o t t h e b e s t e s t i m a t e of t h e OPTOMA I1 f i e l d s because it does n o t contain t h e e f f e c t of dynamical i n t e r p o l a t i o n , which Rienecker e t al. (1987) b e l i e v e t o improve t h e e s t i m a t e .
This c o n s i d e r a t i o n
f u r t h e r enhances t h e value of d a t a a s s i m i l a t i o n i n t h e i n t e r i o r of t h e f o r e c a s t domain.
DAY 5522
DAY 5524
DAY 5526
DAY 5528
Fig. 7. Typical s y n o p t i c o b j e c t i v e a n a l y s i s f i e l d s used f o r d a t a a s s i m i l a t i o n . The a r e a shaded i n each box has an expected e r r o r e s t i m a t e of t h e o b j e c t i v e a n a l y s i s g r e a t e r than f i f t y p e r c e n t . Contour i n t e r v a l i s 0.5. (From Rienecker e t aZ., 1987)
4 A GULF-STREAM FORECAST SYSTEM
- GULFCASTING
W e have e s t a b l i s h e d and a r e now o p e r a t i n g a r e a l time Gulf-Stream Descript i v e P r e d i c t i v e System (GS-ODPS) from 50°-700 W longitude i n open ocean domains
one t o two thousand k i l o m e t e r s on a s i d e .
In t h i s region t h e Gulf-
Stream c u r r e n t a f t e r breaking away from Cape H a t t e r a s , a m p l i f i e s g e n e r a l l y eastward propagating meanders which a p e r i o d i c a l l y snap o f f t o t h e n o r t h and south a s warm core o r c o l d core r i n g s (Watts, 1983; Fofonoff, 1981).
Several
rings of both t y p e s t y p i c a l l y e x i s t t h e r e and ( m u l t i p l e ) ring-stream,
ring-ring
i n t e r a c t i o n s occur i n c l u d i n g recoalescence of r i n g s i n t o t h e stream and r i n g mergers (Richardson, 1983).
These v a r i a b l e c u r r e n t s and eddies r e p r e s e n t some
of t h e most e n e r g e t i c mesoscale phenomena known t o e x i s t i n t h e world ocean and they a r e accompanied by a s s o c i a t e d mesoscale f e a t u r e s such a s s h i n g l e s , outbreaks, e x t e r n a l e d d i e s r e l a t e d t o c u r r e n t looping, e t c .
This i s a complex
region both k i n e m a t i c a l l y and dynamically, c h a r a c t e r i z e d by inhomogeneous, nons t a t i o n a r y and a n i s o t r o p i c s t a t i s t i c s .
Several f a c t o r s , however, make t h i s an
a t t r a c t i v e region f o r r e s e a r c h and f o r e c a s t t r i a l s i n c l u d i n g t h e p r a c t i c a l and s c i e n t i f i c importance of t h e r e g i o n , t h e s t r e n g t h of t h e mesoscale s i g n a l s , and h i s t o r i c a l phenomenological knowledge. The l a t t e r two f a c t o r s provide t h e b a s i s f o r a s p e c i a l dynamical model i n i t i a l i z a t i o n scheme c a l l e d f e a t u r e initialization.
The major mesoscale
features, i - e . , t h e Gulf-Stream a x i s and t h e r i n g s a r e i d e n t i f i e d and g e n e r a l l y located by s a t e l l i t e InfraRed (IR) o b s e r v a t i o n s (obtained from NOAA-7).
Several
passes over s e v e r a l days o r a few weeks a r e u s u a l l y necessary t o d e a l with t h e cloud o b s c u r a t i o n problem.
The model i s then i n i t i a l i z e d with s t a n d a r d forms of
f e a t u r e s f o r t h e c u r r e n t j e t and r i n g s t r u c t u r e s .
Each o f t h e s e f e a t u r e models
has a simple a n a l y t i c a l form with a few f r e e i n d i c e s which must be s e t , such a s the maximum s w i r l speed and r a d i u s of a r i n g , e t c .
This approach i s a p p l i c a b l e
because experience i n d i c a t e s t h a t t h e Gulf-Stream p r o f i l e s a r e always remarkably s i m i l a r when viewed along t h e l o c a l and i n s t a n t a n e o u s a x i s of t h e stream. Furthermore, r i n g s have c h a r a c t e r i s t i c
and common s t r u c t u r e s even a s they age.
Figure 8 shows schematically t h e f e a t u r e s "hanging i n p l a c e " i n t h e dynamical model j u s t p r i o r t o a run.
The model w i l l dynamically a d j u s t t h e f e a t u r e s and
i n t e r a c t them, dynamically i n t e r p o l a t e between t h e f e a t u r e s , and then evolve t h e f i e l d s forward i n t i m e .
To e s t i m a t e f u t u r e boundary c o n d i t i o n s w e c o n s t r u c t a
simple propagation model f o r r i n g p o s i t i o n s and f o r meander c r e s t s and troughs based on t h e l a s t few weeks o b s e r v a t i o n s and p r o j e c t forward.
The model was
tuned and v a l i d a t e d f o r t h i s region and scheme by a d e t a i l e d dynamical study f o r t h e p e r i o d November-December 1984 (Robinson, P i n a r d i and S p a l l , 1987) and by f i v e real-time
forecast research exercises c a r r i e d out i n t h e period
November 1985 through June 1986 (Robinson, S p a l l , Walstad and L e s l i e , 1987). During t h e s e e x e r c i s e s AXBT f l i g h t s were used t o o b t a i n d a t a f o r improving t h e l o c a t i o n of c r i t i c a l f e a t u r e s and f o r v e r i f i c a t i o n s .
F o r e c a s t s a r e c a r r i e d out
f o r a week o r two, which can be c h a r a c t e r i z e d e i t h e r by simple propagation o r by major events.
A s t r i k i n g and s u c c e s s f u l l y f o r e c a s t week's development
i l l u s t r a t e d i n Figure 9.
is
The l a r g e amplitude wave grew within t h e region i n t h e
dynamical f o r e c a s t ; it w a s l a t e r observed i n t h e IR, and AXBT's dropped between 65'
and 67O W agreed with t h e f o r e c a s t w i t h i n a i r c r a f t n a v i g a t i o n a l accuracy
(within
* 2 km) .
106
515km
f Resdution
. Fig. 8.
Schematic o f t h e Harvard Open Ocean Model as used i n t h e Gulf Stream
ODPS with t h e v a r i o u s f e a t u r e models (stream, warm eddy, c o l d r i n g ) i n p l a c e .
Actual model l e v e l s and h o r i z o n t a l r e s o l u t i o n are a l s o i n d i c a t e d . Robinson, P i n a r d i and S p a l l , 1987)
(From
Since t h e f a l l of 1986 we have been maintaining t h e GS-ODPS, nowcasting and f o r e c a s t i n g on a r e g u l a r b a s i s . s u r f a c e temperature d a t a l a r g e domain,
The components a r e :
i) S a t e l l i t e derived sea
i i ) dynamical model runs on a supercomputer i n a
iii) subdomain model runs performed t o t e s t s e n s i t i v i t i e s t o
f e a t u r e l o c a t i o n s and d e t a i l s of major i n t e r a c t i v e e v e n t s , and
i v ) AXBT f l i g h t s
t o remove ambiguities i n I R d a t a , t o p i n p o i n t c r i t i c a l f e a t u r e s and t o provide d a t a f o r updating and a s s i m i l a t i o n .
Figure 10a shows a l a r g e domain supercom-
p u t e r f o r e c a s t f o r 2 3 May, 1986 on which i s i n d i c a t e d a subdomain s e l e c t e d f o r s e n s i t i v i t y study and an AXBT f l i g h t t r a c k designed f o r d e f i n i n g f o u r c r i t i c a l a x i s c r o s s i n g s and two r i n g l o c a t i o n s .
The major s e n s i t i v i t y i n question i n -
volved p r i m a r i l y a warm core ring-stream i n t e r a c t i o n which would r e s u l t i n a rapid axis distortion.
Figure 10b shows a v e r t i c a l s e c t i o n o f model output flow
and temperature f i e l d s , which can of course be taken a t any time and o r i e n t a t i o n . In summary. w e have implemented a rudimentary system which has coverage v i a a
107
B,
JANUARY 6
A)
60
65
70
JANUARY13
65
70
40
40
35
35
60
-
MINm-3.94E MAX-4.09E
MIN -4.63E MAX-6.26E
F i g 9. Results of a Gulf Stream f o r e c a s t f o r January 1986. a) Streamfunction for January 6 a t 100m. b ) A s i n a ) b u t f o r January 13. Contour i n t e r v a l f o r both i s 1.0. (From Robinson, S p a l l , Walstad and L e s l i e , 1987.) remotely sensed d a t a component, an i n t e r p r e t a t i v e model component, and a dedicated component f o r t h e a c q u i s i t i o n of c r i t i c a l subsurface roles of t h e dynamical model a r e of course
A)
x)
65
55
The
ii) t o i n t e r p o l a t e sparse
iii) t o e x t r a p o l a t e forward i n t i m e , i . e . ,
60
data.
i ) t o provide subsurface f i e l d s
derived from remotely sensed s u r f a c e o b s e r v a t i o n s , data dynamically, and
in s i t u
t o predict.
50
MIN=-5.82 MAX = 5.10
F i g . 10. a) Streamfunction a t 100m f o r a l a r g e domain supercomputer f o r e c a s t f o r 23 May 1986. The s m a l l e r box d e f i n e s a domain s e l e c t e d f o r s e n s i t i v i t y s t u d i e s . An AXBT f l i g h t t r a c k i s shown by t h e bold l i n e . A and B i n d i c a t e t h e b ) V e r t i c a l s e c t i o n of model end p o i n t s f o r t h e f i r s t l e g of t h e f l i g h t t r a c k . output flow and temperature f i e l d s along t h e l e g of t h e f l i g h t t r a c k i n a ) i n dicated by t h e A and B. S o l i d l i n e s i n d i c a t e p o s i t i v e streamfunction; dashed l i n e s i n d i c a t e negative streamfunction; bold l i n e s a r e contours of temperature. (From Robinson, Contour i n t e r v a l f o r streamfunction i n a ) and b ) i s 1.0. S p a l l , Walstad and Leslie, 1987)
5 CONCLUS IONS The oceanic mesoscale i s t h e analogue of t h e atmospheric s y n o p t i c s c a l e . A v a r i e t y of mesoscale phenomena occur which a r e t h e e n e r g e t i c a l l y dominant flows over much of t h e ocean and a r e t h e " i n t e r n a l weather" phenomena of t h e sea. Forecasting research p r e s e n t s s c i e n t i f i c and t e c h n i c a l problems rooted i n modern n o n l i n e a r mechanics and computational f l u i d dynamics.
Systematic f i e l d
e s t i m a t i o n , e s p e c i a l l y four-dimensional d a t a a s s i m i l a t i o n (involving remotely sensed and
i n s i t u d a t a and r e a l i s t i c numerical dynamical models),
and e s s e n t i a l .
is relevant
Phenomenological s c a l e s on t h e o r d e r of t e n s t o . h u n d r e d s of
kilometers and of s e v e r a l days t o months make r e g i o n a l f o r e c a s t i n g p r e s e n t l y necessary but f a c i l i t a t e d a t a a s s i m i l a t i o n i n oceanic r e a l t i m e .
Mesoscale
f o r e c a s t i n g i s now f e a s i b l e and s u c c e s s f u l r e a l t i m e f o r e c a s t s have been c a r r i e d o u t . Ocean p r e d i c t i o n r e s e a r c h i s being vigorously pursued encompassing: model development and v e r i f i c a t i 0 n ; d a t a a s s i m i l a t i o n and p r e d i c t a b i l i t y s t u d i e s ; and p h y s i c a l process and p r e d i c t i o n experiments. The prospectus f o r ocean mesoscale f o r e c a s t i n g i s e x c e l l e n t , provided t h e systematic approach i s implemented with t h e p o t e n t i a l s both of t h e i n d i v i d u a l components and of t h e i r combination e x p l o i t e d . of r e g i o n a l kinematics and dynamics i s required.
Physically, t h e c l a s s i f i c a t i o n Standard regions f o r opera-
t i o n a l f o r e c a s t s should be i d e n t i f i e d and r e l e v a n t tuned and v e r i f i e d r e g i o n a l models e s t a b l i s h e d f o r use when a c c u r a t e r e g i o n a l f o r e c a s t s a r e d e s i r e d .
On
t h e l a r g e s c a l e (very l a r g e r e g i o n a l , b a s i n o r g l o b a l ) , o b s e r v a t i o n s of s e a s u r f a c e h e i g h t and s e a s u r f a c e temperatures with mesoscale r e s o l u t i o n should be c o n t i n u a l l y a v a i l a b l e v i a s a t e l l i t e coverage.
Also a r e l a t i v e l y coarse
r e s o l u t i o n eddy r e s o l v i n g model should be k e p t running on a supercomputer a s s i m i l a t i n g remotely sensed and
i n s i t u data.
(Scatterometer winds and
remotely sensed a i r - s e a f l u x e s w i l l u l t i m a t e l y be n e c e s s a r y ) .
Data from an
in
s i t u subsurface o b s e r v a t i o n a l network must be telemetered i n r e a l t i m e and should c o n s i s t of an e f f i c i e n t mix o f :
t i m e s e r i e s from f r e e f l o a t i n g and
moored s e n s o r s , repeated s e c t i o n s , and remotely sensed d a t a (e.g., tomographic).
acoustic
Regions of s p e c i a l i n t e r e s t w i l l r e q u i r e denser d a t a sampling
and f i n e mesoscale r e s o l u t i o n models f o r p r e d i c t i o n and d a t a a s s i m i l a t i o n . Accurate runs f o r t h e subregions can draw upon t h e c o a r s e r l a r g e r e g i o n a l model output f o r boundary and i n i t i a l condition d a t a , should be used f o r s e n s i t i v i t y runs, and can be c a r r i e d o u t a t f o r e c a s t i n g c e n t e r s , on s h i p s a t s e a , o r wherever a powerful b u t p o r t a b l e microcomputer can be made a v a i l a b l e .
The
i n t e r a c t i o n among t h e components of t h e f o r e c a s t system i s symbiotic and t h e whole i s much more powerful than t h e sum of t h e p a r t s .
109 ACKNOWLEDGMENTS The general concepts of ocean p r e d i c t i o n s c i e n c e and t h e approach t o t h e regional mesoscale f o r e c a s t i n g problem were f i r s t p r e s e n t e d i n Cambridge, Massachusetts a t t h e OCEAN PREDICTION WORKSHOP i n A p r i l 1986 and then f u r t h e r developed f o r t h e 1 8 t h I n t e r n a t i o n a l Liege Colloquium on Hydrodynamics "Three-dimensional models of marine and e s t u a r i n e dynamics. Professor J . C . J .
"
In Liege
Nihoul presented t h e i n t e r e s t i n g opportunity f o r deep s e a
dynamicists, long three-dimensional, c i s t s , long p r e d i c t i v e .
t o i n t e r a c t with shallow water dynami-
The i n t e r f a c i n g of c o a s t a l and open ocean models
o f f e r s challenging and important o p p o r t u n i t i e s f o r s e v e r a l y e a r s t o come. I t i s a p l e a s u r e t o acknowledge M r .
Michael A.
S p a l l ' s s c i e n t i f i c contri-
butions t o t h e establishment of t h e Gulf-Stream ODPS which were e s s e n t i a l t o
i t s success.
I am g r a t e f u l f o r t h e valued a s s i s t a n c e of M r s .
D'Arcangelo, MS.
Marsha G.
of t h e manuscript.
Cormier and M r .
Wayne G.
Renate
L e s l i e i n t h e production
This r e s e a r c h was supported by t h e O f f i c e of Naval
Research (N00014-84-0461)
and t h e I n s t i t u t e of Naval Oceanography under
contracts t o Harvard University.
REFERENCES ACOS-NSF, 1984. Emergence of a Unified Ocean Science: A Report by t h e Advisory Committee f o r Ocean S t u d i e s -The National Science Foundation. Carton, J . A . , 1987, How P r e d i c t a b l e a r e t h e Geostrophic Currents i n t h e Recirculation Zone of t h e North A t l a n t i c ? J. Phys. Oceanogr, i n p r e s s . 1985. Comprehensive Objective Maps of Carton, J . A . and McGillicuddy, D . J . , POLYMODE Streamfunctions. Reports i n Meteorology and Oceanography, 2 1 , Harvard U n i v e r s i t y , Cambridge, MA. Charney, J . G . , F j o r t o f t , R., and von Neumann, J . , 1950. Numerical I n t e g r a t i o n of t h e Barotropic V o r t i c i t y Equation. T e l l u s , 2 : 237-254. Cornillon, P., Evans, D., and Large, W., 1986. Warm Outbreaks of t h e Gulf Stream i n t o t h e Sargasso Sea. J. Geophys. R e s . , C5, 91: 6583-6596. Davis, R.E., 1985. D r i f t e r Observations of Coastal Surface Currents During CODE. J. Geophys. R e s . , 90: 4741-4755. 1981. The Gulf Stream System. I n : B.A. Warren and C. Wunsch Fofonoff, N . P . , ( E d i t o r s ) , The Evolution of P h y s i c a l Oceanography, MIT P r e s s , Cambridge, MA. G h i l , M . , Cohn, S . , T a v a n t z i s , J . , B u b e , K . , and Isaacson, E . , 1981. Application of Estimation Theory t o Numerical Weather P r e d i c t i o n . In : L. Bengtsson, M. Ghil and E. Kallen ( E d i t o r s ) , Dynamic Meteorology, Data Assimilation Methods, Springer-Verlag, New York/Heidelberg/Berlin, pp. 139-224. ( E d i t o r ) , 1986. SYNOP Synoptic Ocean P r e d i c t i o n Program Cover Hogg, N.G. Document. Woods Hole Oceanographic I n s t i t u t i o n . Holloway, G . , and West, B . J . , 1984. P r e d i c t a b i l i t y of F l u i d Motions, A I P Conference Proceedings No. 1 0 6 , American I n s t i t u t e of Physics, N e w York. 1984. The p o t e n t i a l f o r ocean p r e d i c t i o n and t h e r o l e of Hurlburt, H.E., a l t i m e t e r d a t a . Mar. Geodesy, 8: 17-66. 1983. Shallow Currents i n t h e Caribbean Sea and Gulf of Mexico Kinder, T . H . , a s Observed with S a t e l l i t e - T r a c k e d D r i f t e r s . B u l l . Mar. S c i . , 33 ( 2 ) : 239-246. Malone, T.F. and Roederer, J . G . ( E d i t o r s ) , 1985. Global Change. ICSU Press by Cambridge U n i v e r s i t y P r e s s , Cambridge/New York.
-
-
110 McWilliams, J . C . , Brown, E.D., Bryden, H.L., Ebbesmeyer, C.C., E l l i o t , B.A., Heinmiller, R . H . , Lien Hua, B . , Leaman, K.D., Lindstrom, E . J . , Luyten, J . R . , McDowell, S.E., Owens, W.B., P e r k i n s , H . , P r i c e , J . F . , Regier, L., Riser, S.C., Rossby, H.T., Sanford, T . B . , Shen, C.Y., T a f t , B.A., and 1983. The Local Dynamics of Eddies i n t h e Western North Van Leer, J . C . , Atlantic. I n : A.R. Robinson ( E d i t o r ) , Eddies i n Marine Science. Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, pp. 92-113. Toward t h e Application of t h e Kalman F i l t e r t o Regional M i l l e r , R . N . , 1986. Open Ocean Modeling. J. Phys. Oceanogr., 16: 72-86. Robinson, A.R. and Haidvogel, D.B., 1983. A Baroclinic Miller, R.N., Quasigeostrophic Open Ocean Model. J. Comput. Phys., 50 (1): 38-70. Piacsek, S.A. and Robinson, A.R. ( E d i t o r s ) , 1981. Ocean Mooers, C.N.K.M., P r e d i c t i o n : The S c i e n t i f i c Basis and t h e Navy's Needs, A S t a t u s and Prospectus Report. Proceedings of t h e Ocean P r e d i c t i o n Workshop, Monterey, CA, May 1981. Mooers, C.N.K.M., Robinson, A.R. and Thompson, J . D . ( E d i t o r s ) , 1987. Ocean P r e d i c t i o n workshop 1986, A S t a t u s and Prospectus Report on t h e S c i e n t i f i c Basis and t h e Navy's Needs. Proceedings of t h e Ocean P r e d i c t i o n Workshop: Phase I - Cambridge, MA, A p r i l 1986; Phase 11 - Long Beach, MS, November 1986. 1986. Q u a s i g e o s t r o p h i c Energetics of Open P i n a r d i , N. and Robinson, A . R . , Ocean Regions. Dyn. Atmos. Oceans, 10 ( 3 ) : 185-221. P i n a r d i , N. and Robinson, A . R . , 1987. Dynamics of Deep Thermocline J e t s i n t h e POLYMODE Region. J. Phys. Oceanogr., i n p r e s s . Richardson, P.L., 1983. Gulf Stream Rings. I n : A.R. Robinson ( E d i t o r ) , Eddies i n Marine Science. Springer-Verlag, Berlin/Heidelberg/New York/ Tokyo, pp. 19-45. Rienecker, M . M . , Mooers, C.N.K.M. and Robinson, A . R . , 1987. The Evolution of Mesoscale Features o f f Northern C a l i f o r n i a : Dynamical I n t e r p o l a t i o n and Forecast Experiments. J. Phys. Oceanogr., i n p r e s s . Robinson, A. R . , 1982. Dynamics of Ocean Currents and C i r c u l a t i o n : R e s u l t s of POLYMODE and Related I n v e s t i g a t i o n s . I n : A. Osborne and P.M. R i z z o l i ( E d i t o r s ) , Topics i n Ocean Physics, Society I t a l i a n a d i F i s i c a , Bologna, I t a l y , pp. 3-29. Robinson, A . R . , 1983. Overview and Summary of Eddy Science. I n : A.R. Springer-Verlag, B e r l i n / Robinson ( E d i t o r ) , Eddies i n Marine Science. Heidelberg/New York/Tokyo, pp. 3-15. Robinson, A.R., 1986. D a t a A s s i m i l a t i o n , Mesoscale Dynamics and Dynamical Forecasting. I n : J . J. 0' Brien ( E d i t o r ) , Advanced Physical Oceanographic Numerical Modelling, Proceedings of t h e NATO Advanced S t u d i e s I n s t i t u t e , R. Reidel, Dordrecht, The Netherlands, pp. 465-483. Robinson, A.R. and L e s l i e , W.G., 1985. Estimation and P r e d i c t i o n of Oceanic Fields. Progr. Oceanogr., 1 4 : 485-510. 1987. The Harvard Open Ocean Model: Robinson, A.R. and Walstad, L . J . , C a l i b r a t i o n and Application t o Dynamical Process, F o r e c a s t i n g , and Data Assimilation S t u d i e s . J. Appl. N u m e r . Math., i n p r e s s . Robinson, A.R., Carton, J . A . , P i n a r d i , N. and Mooers, C.N.K.M., 1986. Dynamical Forecasting and Dynamical I n t e r p o l a t i o n : An Experiment i n t h e C a l i f o r n i a Current. J. Phys. Oceanogr., 1 6 : 1561-1579. P i n a r d i , N. and S p a l l , M.A., 1987a. Gulf Stream Simulations Robinson, A.R., and t h e Dynamics of Ring and Meander Process, i n p r e p a r a t i o n . S p a l l , M.A., Walstad, L . J . and L e s l i e , W.G., 1987b. Robinson, A.R., Forecasting t h e I n t e r n a l Weather of t h e Sea: A Real-Time System f o r Gulf Stream Meanders and Rings, i n p r e p a r a t i o n . Sarmiento, J . L . and Bryan, K . , 1982. An Ocean Transport Model f o r t h e North J. Geophys. Res., 87: 394-408. Atlantic. Shapiro, R., 1971. The U s e of Linear F i l t e r i n g a s a Parameterization f o r J. Atmos. S c i . , 2 8 : 523-531. Atmospheric Diffusion. University of C a l i f o r n i a P r e s s , Stommel, H . , 1965. The Gulf Stream. Berkeley/Los Angeles/London.
111 Swallow, J . , 1976. Variable Currents i n Mid-Ocean. Oceanus, 19: 18-25. Tu, K . , 1981. A Combined Dynamical and S t a t i s t i c a l Approach t o Regional Forecast Modeling of Open Ocean Currents. Ph.D. t h e s i s , Harvard University, Reports i n Meteorology and Oceanography, 13. Walstad, L . J . , 1987. The Harvard Quasigeostrophic Model: Hindcasting, Forecasting, and Development of t h e Coupled Quasigeostrophic - Surface Boundary Layer Model. Ph.D. t h e s i s , Harvard U n i v e r s i t y , Cambridge, MA. Watts, D. R., 1983. Gulf Stream V a r i a b i l i t y . In : A. R. Robinson ( E d i t o r ) , Eddies i n Marine Science. Springer-Verlag, Berlin/Heidelberg/New York/ Tokyo, pp. 114-144. Wunsch, C., 1978. The North A t l a n t i c C i r c u l a t i o n W e s t of 50 W Determined by Inverse Methods. Rev. Geophys. Space Phys., 16: 583-620.
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113
PREPARATION OF ESTUARY AND MARINE MODEL EQUATIONS BY GENERALIZED FILTERING METHODS K. W. BEDFORD, J. S. DINGMAN and W. K. YE0 Department o f C i v i l Engineering The Ohio State U n i v e r s i t y , 2070 N e i l Avenue Columbus, Ohio 43210 (USA)
ABSTRACT Higher order averaging procedures which circumvent t h e l i m i t a t i o n s o f t r a d i t i o n a l Reynolds averaging are presented and a p p l i e d t o t h e threedimensional equations used f o r marine and estuary models. These averages o r f i l t e r s can e i t h e r be analog o r d i g i t a l , and a review o f the classes o f such f i l t e r s and t h e i r c h a r a c t e r i z a t i o n i s presented f i r s t . Low pass f i l t e r s i n both s i n g l e and cascaded form are then a p p l i e d t o t h e governing equations, and closures v i a t r a d i t i o n a l and h i g h pass f i l t e r expansions are i d e n t i f i e d . F i n a l l y , t h e r e l a t i o n s h i p between analog f i l t e r s , d i g i t a l f i l t e r s and commonly used higher order numerical schemes i s explored, and i t i s shown t h a t c e r t a i n numerical schemes are indeed d i g i t a l f i l t e r forms o f t h e analog f i l t e r s .
1
INTRODUCTION During t h e l a s t t h i r t y years o f surface water f l o w and t r a n s p o r t modeling
(meteorological models as w e l l ) ,
t h e basic Reynolds averaged form o f t h e mean
flow turbulence
equations has been t h e unquestioned basis f o r t h e governing
model equations.
Accompanying t h i s s i n g u l a r set o f averaged equations has come
an
variety
overwhelming
of
numerical
representations
for
these
equations.
A d d i t i o n a l complexity e x i s t s i n t h e s e l e c t i o n o f which c l o s u r e t o use and, many c l o s u r e forms,
what e m p i r i c a l c o e f f i c i e n t s t o use.
for
Considerable research
a c t i v i t y i s devoted t o c l o s u r e and d i s c r e t i z a t i o n forms w h i l e very l i t t l e e f f o r t
i s expended i n reviewing and p o s s i b l y improving t h e averaging and t h e r e f o r e s t r u c t u r e o f the basic governing equations. Recently a small
body o f research suggests t h a t indeed a cause of
the
m u l t i t u d i n o u s forms o f computational and c l o s u r e forms resides i n d i f f i c u l t i e s w i t h t h e method used t o average and prepare t h e basic t u r b u l e n c e equations.
The
basis f o r t h i s suggestion resides i n t h e requirement t h a t t h e marine equations must be averaged t o account f o r t i m e v a r y i n g average flows and t h a t consistency i n t h e averaging must be achieved.
The c u r r e n t l y used Reynolds average i s n o t Analog f i l t e r s based upon a
a l l t o g e t h e r adequate i n meeting t h i s requirement.
114 g e n e r a l i z e d a v e r a g i n g o r f i l t e r d e f i n i t i o n a r e b e i n g developed t o address these averaging equations
for
complete
and w i l l
requirements
t h e marine
and
to
existing
i n this
and e s t u a r y model
analogy t o d i g i t a l
signal
analog f i l t e r s a r e a v a i l a b l e ; forms
be used
problem.
processing,
however,
numerical
article t o
formulate
Additionally,
digital
filter
new
and i n
forms o f t h e
t h e i r r e l a t i o n t o t h e analog equation
methods
is
unexplored.
Therefore,
the
1) p r e s e n t a r e v i e w o f f i l t e r i n g procedures;
objectives o f t h i s a r t i c l e are t o :
2) summarize t h e r u l e s o r procedures by which t h e s e f i l t e r s a r e a p p l i e d t o t h e equations;
3) r e v i e w t h e t y p e s o f analog a v e r a g i n g o r f i l t e r s used i n marine
models t o date;
4) a p p l y t h e s e procedures t o t h e d e r i v a t i o n o f a new s e t o f
marine and e s t u a r y e q u a t i o n s ;
and 5)
c o n t r a s t t h e analog e q u a t i o n forms and
p r e s e n t l y used h i g h e r o r d e r d i s c r e t i z a t i o n s .
The l a s t o b j e c t i v e seeks t o begin
d e t e r m i n i n g i f many o f t h e proposed numerical methods a r e s i m p l y d i g i t a l forms o f t h e newly r e c o g n i z e d analog f i l t e r terms. 2
BASIC AVERAGING DEFINITIONS
2.1 D e f i n i t i o n o f f i l t e r i n g o p e r a t i o n A continuous f i e l d variable,
say
f(x,t),
can be decomposed i n t o i t s mean
and f l u c t u a t i n g components as (Dakhoul and Bedford, 1986a):
where
x denotes t h e
I n these equations: xi
+ y j + zk);
Cartesian vector s p a t i a l coordinate ( x =
t denotes time;
and G(x,t)
i s a weight
or f i l t e r function
c o n s t r a i n e d such t h a t +m
I
IJ
G(x,t)
d x d t = 1.0
(3)
-m
2.2 A n a l y s i s and t y p e s o f f i l t e r s The d i s t i n c t i o n s between t h e v a r i o u s analog and d i g i t a l f i l t e r s a r e based upon
the
behavior
of
the
filter
in
resolving desirable
portions
of
the
wavenumber o r frequency spectrum o f t h e problem and s u p p r e s s i n g o r e l i m i n a t i n g undesirable portions. transform
-* *
R = f
/f
of
the
*
The f i l t e r response f u n c t i o n ,
filter
function,
is
a measure
R,
of
d e f i n e d as t h e F o u r i e r this
activity,
e.g.,
= G ; where t h e a s t e r i s k denotes t h e F o u r i e r t r a n s f o r m o f t h e
v a r i a b l e o r function.
The response f u n c t i o n may be e i t h e r p a r a m e t e r i z e d as a
115 function o f wavenumber, response
k,
o r frequency,
f u n c t i o n s and t h e r e f o r e
Based upon
W.
filters
are
identified
Rk,Wy
f o u r types o f
according t o which
p o r t i o n o f t h e wave number o r frequency spectrum i s resolved; these are lowpass (L), highpass (H), bandpass (BP) and bandreject (BR) response f u n c t i o n s . Without question,
t h e dominant type o f averaging used i n model p r e p a r a t i o n
i s the low pass f i l t e r wherein h i g h frequency and/or
wavenumber
information
above a c u t o f f value
wC o r k c i s e l i m i n a t e d w h i l e t h e lower frequency motion i s
allowed o r retained.
An a d d i t i o n a l a t t r i b u t e o f t h e low pass f i l t e r i s t h a t t h e
resolved p o r t i o n o f t h e v a r i a b l e o r s i g n a l not possess e i t h e r amplitude o r phase
To date,
distortion.
no use has been made i n surface water wave o r turbulence
modeling ( o r any o t h e r f l u i d s modeling f o r t h a t m a t t e r ) o f t h e o t h e r averaging definitions. high
pass
I n passing, filter
H(k,w) = l-L(k,w).
is
i t should be noted t h a t t h e response f u n c t i o n f o r t h e
simply
Therefore,
related
to
i n principle,
the
low
pass
function,
i.e.,
i f a low pass f i l t e r i s known,
then so i s an e q u i v a l e n t h i g h pass f i l t e r .
2.3 Types o f low pass f i l t e r s A review o f low pass f i l t e r f u n c t i o n s used i n equation p r e p a r a t i o n has been presented i n Dakhoul
and Bedford (1986a) and can be roughly broken i n t o two
major categories; t h e u n i f o r m and t h e Gaussian f i l t e r .
Two f u r t h e r s u b d i v i s i o n s
occur w i t h i n each category i n t h a t e i t h e r s p a t i a l o r temporal versions o f these
A fifth filter,
f i l t e r s are possible.
a generalized spatial-temporal
has a l s o been designed and t e s t e d by Dakhoul and Bedford (1986a,
filter,
1986b),
and
thereby t h e previous f o u r f i l t e r s become s p e c i a l cases o f t h i s general f i l t e r .
i) Uniform f i l t e r .
The d e f i n i t i o n f o r t h e general
u n i f o r m space-time
f i l t e r f o r n = l t o 3 s p a t i a l dimensions i s :
where
6 t and
6 i are averaging scales t o be selected by t h e analyst.
The response f u n c t i o n o f t h i s f i l t e r i s
Spatial o r temporal recognized
that
the
f i l t e r s a r e e a s i l y d e r i v e d from these functions. fixed
interval
s p e c i a l i z e d temporal f i l t e r , i.e.,
Reynolds average
(Reynolds,
1895)
It i s
is a
116
?(El=-
1
6t
t + 6t/2
I
f(x,t')
dt'.
t-dt/2
A s i m i l a r f i x e d i n t e r v a l volume average has been used i n atmospheric models s i n c e t h e work o f Smagorinsky (1963) and Oeardorff (1973). ii)
Gaussian f i l t e r
The generalized Gaussian space-time f i l t e r i s defined
f o r n = l t o 3 s p a t i a l dimensions as:
w i t h a response f u n c t i o n d e f i n e d as:
I n t h i s equation,
y i s a coefficient
which commonly v a r i e s between 1 and 6.
The use o f moving-average h i g h e r order f i l t e r s ,
d e f i n e d by eqns.
(4) and
(7). was f i r s t performed w i t h s p a t i a l f i l t e r s by Leonard (1974) and i n surface water f l o w and t r a n s p o r t models by Bedford (1981) Babajimopolous and Bedford (1980) and Bedford and Babajimopolous (1980).
I n s u r f a c e water f l o w modeling,
Bedford (1981) found i t necessary t o use a n i s o t r o p i c s p a t i a l f i l t e r s and f u r t h e r n o t i c e d t h a t three-dimensional
models based upon h y d r o s t a t i c pressure created,
propagated, and d i s s i p a t e d t u r b u l e n c e as a h o r i z o n t a l two-dimensional even though a three-dimensional 3
flow f i e l d
v e l o c i t y f i e l d i s calculated.
APPLICATION OF ANALOG FILTERING
3.1 Rules o f averaging Using t h e f i x e d i n t e r v a l Reynolds averaging d e f i n i t i o n now i n use today, i t i s p o s s i b l e (Hinze,
1975) t o define a s e r i e s o f averaging r u l e s f o r averaging
v a r i o u s combinations o f f u n c t i o n s and operations; summarizes those operations.
t h e second column i n Table 1
I n t h i s t a b l e t h e overbar stands f o r t h e averaging
o f e i t h e r t h e f u n c t i o n f or g;
furthermore,
t stands f o r t i m e and s i n d i c a t e s a
general spat ia1 dimension. The t h i r d column c o n t a i n s t h e
rules permitted
average i n t e r p r e t a t i o n f o r t h e f i l t e r i n g operation. t h a t r u l e s No. 4, valid.
5 and 6,
i f one assumes a moving It i s n o t i c e d imnediately
v a l i d f o r f i x e d i n t e r v a l averaging,
a r e no longer
As a r e s u l t , c o n s i d e r a b l e d i f f e r e n c e s i n t h e f i n a l average equation form
w i l l result.
117 TABLE 1 Summary o f a v e r a g i n g r u l e s f o r f i x e d and moving average f i l t e r d e f i n i t i o n s Averaging r u l e No.
Fixed i n t e r v a l average
Moving average
a f =s at
at
E
=pf
as
as
4
f
5
-f = f
=
O
f
#
O
? # f
6
3.2 The a p p l i c a t i o n o f Reynolds a v e r a g i n g t o t h e Navier-Stokes e q u a t i o n s The Navier-Stokes e q u a t i o n s a r e w r i t t e n i n t e n s o r n o t a t i o n as: 9
I n t h i s equation
p is
viscosity
is
and
ui
t h e density, the
u 1= u, u2 = v and u3 = w. the indices.
velocity
p i s t h e pressure, vector,
v is
t h e kinematic
ui = uli + u 2j t u ~ f o~ r , which
Repeated i n d i c e s i n a t e r m i m p l y summations over a l l
To a p p l y Reynolds a v e r a g i n g eqn.
(9)
i s averaged (Hinze,
1975).
u s i n g Rules No. 1, 2, and 3, t h e f o l l o w i n g e q u a t i o n r e s u l t s :
The decomposition o f t h e n o n l i n e a r t e r m occurs by a f u r t h e r averaging, i.e.,
After
118
Employing r u l e No. 6 and eqn.
( 1 1 ) g i v e s t h e f i n a l Reynolds averaged form o f the
Navier-Stokes e q u a t i o n s
The
last
terms
in
eqn.
(12)
are
the
Reynolds
stress
terms
about
which
c o n s i d e r a b l e c l o s u r e d i s c u s s i o n occurs. 3.3 A p p l i c a t i o n o f g e n e r a l i z e d a v e r a g i n g t o t h e Navier-Stokes e q u a t i o n s For t h e case o f moving o r g e n e r a l i z e d a v e r a g i n g which d o e s n ' t
restrict
f l o w s t o s p a t i a l l y o r t e m p o r a l l y c o n s t a n t means, t h e f o l l o w i n g d e f e c t s i n t h e Reynolds procedure a r e i d e n t i f i e d and remedied. i )
Inertia
term
decomposition.
provocatively pointed out, Gaussian s p a t i a l
Rule No.
As
Leonard
(1974)
so
clearly
f i l t e r and t h e T a y l o r
series,
and
Using a
6 i n Table 1 i s n o t c o r r e c t .
he was a b l e t o improve t h e
i n e r t i a t e r m decomposition as i n t h e f o l l o w i n g e q u a t i o n :
-uj
2
+
= ui
6 i -4yi ui uj
+ o (6i4)
I n t h i s expansion t h e f i l t e r c o e f f i c i e n t s the
6 i a r e a l l assumed equal
(=6) as are
y. (= y). 1
ii)
Cross c o r r e l a t i o n t e r m decomposition.
Rules No.
6 and 4,
T h e r e f o r e C l a r k e t a l . (1977) found t h a t
iii)
Averaging
D u e t o h e i n a p p l i c a b i l i t y of
t h e c r o s s c o r r e l a t i o n terms
u! a r e no l o n g e r zero. J
i n c o n s i s t e n c i e s and cascade f i l t e r i n g .
R e c e n t l y these
a u t h o r s have noted an i n c o n s i s t e n c y i n t h e a p p l i c a t i o n of t h e moving average d e r i v a t i o n o f t h e b a s i c e q u a t i o n s i n a d d i t i o n t o t h o s e i n 3.3(i)
and 3 . 3 ( i i ) .
It i s n o t e d t h a t t h e decomposition
13)
suggested by Leonard (eqn.
i n e r t i a terms i n v o l v e s a two s t e p o r t w o - f o l d a v e r a g i n g technique.
f o r the
In digital
s i g n a l p r o c e s s i n g l i t e r a t u r e t h i s i s c a l l e d a cascaded f i l t e r ( R a b i n e r and Gold, 1975).
We n o t e t h a t
because R u l e 5 i s
not
v a l i d f o r t h e moving average
approach, t h e n n o t o n l y must t h e n o n l i n e a r / i n e r t i a t e r m be cascade f i l t e r e d , but a l s o t h e l i n e a r d i f f e r e n t i a l terms i n t h e g o v e r n i n g equations. t o be c o n s i s t e n t t h e e n t i r e e q u a t i o n must be cascade f i l t e r e d . temporal a c c e l e r a t i o n t e r m i s cascade f i l t e r e d as f o l l o w s :
I n o t h e r words, Therefore,
the
119
(15) and u s i n g t h e T a y l o r s e r i e s expansion and s p a t i a l Gaussian f i l t e r , eqn.
(15) i s
r e w r i t t e n as:
-
S i m i l a r expansions o c c u r f o r t h e p r e s s u r e g r a d i e n t terms as w e l l as any C o r i o l i s o r s o u r c e / s i nk terms. iv) eqns.
Summary Navier-Stokes
(13-16),
equation.
When combining t h e developments i n
new g e n e r a l i z e d t u r b u l e n t Navier-Stokes e q u a t i o n s emerge which
now p e r m i t t u r b u l e n c e t o be d e f i n e d r e l a t i v e t o a non-constant mean.
I n tensor
n o t a t i o n , t h e e q u a t i o n becomes ( d r o p p i n g t h e viscous t e r m )
where F i r e p r e s e n t s t h e a c c e l e r a t i o n f i l t e r terms
S i represents t h e pressure f i l t e r term
and R i r e p r e s e n t s t h e s u b g r i d s c a l e t e r m
Note t h a t t h i s d e r i v a t i o n has been done w i t h a s p a t i a l d e r i v a t i o n f o r a spatial/temporal
filter,
Gaussian f i l t e r .
as discussed p r e v i o u s l y ,
however, i n l i g h t o f t h e new cascaded f i l t e r approach used t o d e r i v e eqns.
-
( 1 9 ) and t h e d i r e c t
filter,
the
necessary.
use
of
a
r e a l i z a t i o n o f a temporal direct
Such i n v e s t i g a t i o n s
temporal
filter
A
i s possible; (17)
e f f e c t due t o t h e s p a t i a l
component
may no l o n g e r be
a r e b e i n g pursued by t h e second and t h i r d
a u t h o r s of t h i s a r t i c l e as p a r t o f t h e i r D o c t o r a l D i s s e r t a t i o n research.
4
SHALLOW-WATER MODEL EQUATIONS It i s a v e r y s i m p l e t a s k t o extend t h e cascaded f i l t e r o r averaging method
t o t h e d e r i v a t i o n o f s h a l l o w - w a t e r model e q u a t i o n f o r use i n t h r e e - d i m e n s i o n a l
120 marine and e s t u a r y s i m u l a t i o n s .
I f it i s assumed t h a t t h e c o o r d i n a t e s u r f a c e i s
p l a c e d a t t h e s t i l l water l e v e l w i t h z b e i n g p o s i t i v e upwards, weak v e r t i c a l
t h e n f o r very
a c c e l e r a t i o n t h e f o l l o w i n g model e q u a t i o n s f o r c o n t i n u i t y ,
x, y,
and z momentum and a p a s s i v e contaminant c can be w r i t t e n f o r e i t h e r s p a t i a l o r temporal f i 1t e r s as :
aiii
- = o axi
@ =-pg az
-
a({.:)
at
ax.
E+-J
t M = G
J
I n t h e x and y momentum, e q u a t i o n s (eqns.
22 and 2 3 ) , f r e p r e s e n t s t h e e a r t h ' s
a n g u l a r r o t a t i o n frequency f o r a p a r t i c u l a r l a t i t u d e and
q i s t h e f r e e surface
p o s i t i o n measured fran t h e s t i l l w a t e r l e v e l .
4.1 D e f i n i t i o n o f f i l t e r s
-
s p a t i a l forms
The f i l t e r forms f o r a Gaussian s p a t i a l f i l t e r a r e : Fu = Fa = F ( i = l )
Fv =
F8
= F ( i = 2)
1
Here F ( i = l ) and F ( i = 2 ) a r e r e s p e c t i v e l y F i (eqn. 2; and
R8, R$,
18) e v a l u a t e d a t i = 1 and i =
and Gs a r e t h e a p p r o p r i a t e r e s i d u a l o r s u b g r i d s c a l e terms which
must be expressed as a f u n c t i o n o f t h e mean f l o w v a r i a b l e s .
121
-
4.2 D e f i n i t i o n o f f i l t e r s
temporal forms
I f o n l y a temporal f i l t e r i s d e s i r e d t h e n t h e f i l t e r terms become:
6t2 a a2i Fu=F$=-(-) 4y a t at2
-a
NV = N$ = -g
ay
6t
{
2
6
2 - -
+
a 6 t 2 (u ' j ) K ~ { at2
2ar, at2
1-
6 t 2 a 2-v f4yat2
1 t R8, and Gt a r e a l s o e a s i l y d e f i n e d and must be Ru,
The a p p r o p r i a t e forms f o r
expressed i n terms o f t h e averaged v a r i a b l e s . 5
RELATIONSHIP OF ANALOG TO DIGITALLY AVERAGED OR FILTERED MODEL EQUATIONS
It i s n a t u r a l t o ask whether a l l t h e a d d i t i o n a l terms represented by t h e filter
terms
i n eqns.
(21)
to
(35)
have any d i g i t a l
p o s s i b l e t h a t many o f t h e h i g h e r o r d e r numerical fashion,
be
representing
the
analog
filter
equivalent;
schemes might,
terms
derived
o r i s it
i n an ad-hoc
above.
Such
a
comparison r e q u i r e s expanding a l l h i g h e r o r d e r procedures on t h e same g r i d and a comparison w i t h t h e analog f i l t e r e d terms d i g i t i z e d i n a c o n s i s t e n t fashion. Such a comparison
i s beyond t h i s paper,
however,
i l l u s t r a t e t h a t t h i s may i n d e e d be t h e case.
several
examples serve t o
Some t h r e e - d i m e n s i o n a l expansions
f o r a l l t h e g o v e r n i n g e q u a t i o n s a r e q u i t e space consuming, t h e r e f o r e o n l y s i m p l e t e r m expansions a r e p r e s e n t e d here w i t h t h e e x t e n s i o n s t o f u l l three-dimensional d i s c r e t i z a t i o n s a m a t t e r o f a l g e b r a and n o t conceptual d i f f i c u l t y . 5.1 The f i n i t e element a p p r o x i m a t i o n
I f f o r example t h e l i n e a r i z e d averaged x-momentum e q u a t i o n i s t a k e n as:
ai at
then
+ g s -
for
a
fi= 0 two-dimensional
rectangular
finite
element
representation
approximated w i t h b i l i n e a r b a s i s f u n c t i o n s t h e above e q u a t i o n s becomes:
122
Where
the
numerical
subscript
notation
represents
the
evaluation
of
the
dependent v a r i a b l e s a t g r i d p o i n t s i+l, j+l, i, j, i-1, j-1; e t c . I f eqn.
22 i s expanded f o r t h i s two dimensional case w i t h e i t h e r a Gaussian
s p a t i a l f i l t e r ( y =6) o r a u n i f o r m f i l t e r , t h e form becomes: 2
-a
2-
2
2-
at
ax2
2
4-
the
simplest
differential
ax 2 ay2
ay
2 2+ g & { i + & W + % ax2
If
2
{ ,-+iLaU.+++qLLL
2
2
2
in
4-
L + + & $ L L2L 2) aY
second
terms
2
1
ax by
order
centered
eqn.
(36),
approximations
and
are
6x and 6y a r e
used set
for
the
equal
to
AX and 2 ~ y , r e s p e c t i v e l y , t h e n t h e r e s u l t i n g e x p r e s s i o n i s e x a c t l y i d e n t i c a l t o t h e f i n i t e element d i s c r e t i z a t i o n . 5.2
Shuman f i l t e r f a c t o r form: Shuman (1962)
l i n e a r equation
p r e s e n t e d an advanced d i s c r e t i z a t i o n
C o r i o l i s terms i n t h e equations.
Again,
for
t h e p r e s s u r e and
by way o f example, eqn.
(36) i s used.
Shuman's scheme f o r t h i s e q u a t i o n r e s u l t s i n t h e f o l l o w i n g e x p r e s s i o n :
ai +
+g
I
(ill
+
211,
+
11-1)
f
- ig {(ill+ 2iO1+ i-11)
-
611+ 2i-10+ i - 1 - 1 ) l
2ioo+ i-10)
+
+
(il-l+
20-1
+
v-1-1
1l=O
(39)
By u s i n g a Gaussian f i l t e r a p p l i e d o n l y on t h e s p a t i a l d e r i v a t i v e s and w i t h discretizations
again
only
centered,
second
order
finite
d f ference
123 approximations,
an exact equivalence i s obtained.
It i s i n t e r e s t i n g t o note
t h a t t h e averaging has n o t been c o n s i s t e n t l y a p p l i e d t o t h e time d i f f e r e n t i a l term; however i n t h i s technique as used by J e l e n s i a n s k i (1972) i n t h e SPLASH and SLOSH storm surge models, o t h e r ad hoc t i m e f i l t e r i n g was required. 5.3 Pressure averaging technique; d i s s i p a t i v e i n t e r f a c e scheme As suggested by Abbott (1979) and used by Jensen (1983), interface
scheme,
described
on
the
simple
x-momentum
the dissipative
equation
without
the
C o r i o l i s term, t h e expansion i s :
Using a general Gaussian f i l t e r , t h e expansion
and by r e p l a c i n g t h e terms i n eqn.
(41) w i t h second order centered time and
space f i n i t e d i f f e r e n c e approximations, t h e f o l 1owing approximati on occurs :
Equivalences
between
8 = l / y and y
for
y = 4
analog
and
digital
forms
occur
as
follows:
for
> 2 t h e f i l t e r scheme i s e q u i v a l e n t t o A b b o t t ' s (1979) method;
and
8 = 0.25,
(1971); w h i l e f o r
y =
t h e method i s i d e n t i c a l
2.0, and
8 = 0.50,
t o t h e method by Shuman
t h e f i l t e r expansion i s e q u i v a l e n t t o
the method o f McCowan (1978) and Hansen (1983). Many more comparisons o f t h i s n a t u r e are c u r r e n t l y underway by t h e second author f o r h i s d i s s e r t a t i o n research. digital
The correspondence between t h e analog and
forms lends credence t o t h e analog d e r i v a t i o n s performed i n t h e f i r s t
p a r t o f t h i s paper.
It i s a l s o t h e case t h a t s i n c e t h e analog d e r i v a t i o n s a r e
complete and robust, t h a t t h e non-presence o f vdrious d i g i t a l f i l t e r terms i n a numerical model might expose a f l a w i n t h e numerical treatment o f a term i n t h e b a s i c equations.
6
CONCLUSIONS
A g e n e r a l i z e d averaging o r f i l t e r i n g procedure f o r d e r i v i n g t u r b u l e n t f l o w equations accounting f o r c o n t i n u o u s l y v a r y i n g average dependent v a r i a b l e s has been presented and a p p l i e d t o t h e three-dimensional estuary equations.
shallow water marine and
I n t h e course o f these d e r i v a t i o n s ,
questions as t o t h e
124 adequacy o f t h e Reynolds average approach are r a i s e d and a method f o r remedying t h e flaws by t h e f i l t e r expansion approach i s recommended.
Since so many models
a r e based upon these Reynolds average equations, i t i s suggested t h a t a number o f h i g h e r order d i s c r e t i z a t i o n s i n these models are used t o remedy t h e averaging defects.
As a means o f i n i t i a l t e s t i n g o f t h i s hypothesis,
v a r i o u s numerical
schemes are shown t o be e q u i v a l e n t t o t h e analog f i l t e r forms d e r i v e d here. F u r t h e r work i s proceeding on t h i s comparison. 7
ACKNOWLEDGMENTS
This work was supported by t h e National Science Foundation research grant No. CEE 8410522 and t h e i r support i s g r e a t l y appreciated.
8
REFERENCES
Abbott, M. A.,
1979.
Computational Hydraulics.
Babajimopoulos, C., and Bedford, preserve s p e c t r a l s t a t i s t i c s .
K. W., 1980. J. Hyd. Div.,
Pittman, London. Formulating l a k e models which ASCE, 106 (HY1): 1-19.
Bedford, K. W., 1981. Spectra p r e s e r v a t i o n c a p a b i l i t i e s o f Great Lakes t r a n s p o r t models. In: Hugo B. F i s c h e r ( E d i t o r ) , Transport Models f o r I n l a n d and Coastal Waters. Academic Press, New York, 172-221. Bedford, K. W., and Babajimopoulos, C., 1980. V e r i f y i n g l a k e t r a n s p o r t models w i t h s p e c t r a l s t a t i s t i c s . J. Hyd. Div., ASCE, 106 (HY1): 21-38. Clark, R. A., Ferziger, J. H., and Reynolds, W. C., 1977. Evaluation of Subgrid-scale Turbulence Models Using a F u l l y Simulated Turbulent Flow. Rept. TF-9, Stanford U n i v e r s i t y Thermosciences D i v i s i o n . and Bedford, K. W., 1986a. Improved averaging method f o r Dakhoul, Y. M., t u r b u l e n t f l o w simulation. P a r t I; t h e o r e t i c a l development and a p p l i c a t i o n t o Burger's t r a n s p o r t equation. I n t . J. Numer. Meth. F l u i d s , 6: 49-64. and Bedford, K. W., 1986b. Improved averaging method f o r Dakhoul, Y. M., turbulent flow simulation. P a r t 11; c a l c u l a t i o n s and v e r i f i c a t i o n . I n t . J. Numer. Meth. F l u i d s , 6: 65-82. 1973. The use o f s u b g r i d t r a n s p o r t o p e r a t i o n s i n a threeDeardorff, J. W., dimensional model o f atmospheric turbulence. J. F l u i d Eng., 429-438. Hinze, J. O.,
1975.
Turbulence, 2nd ed.,
McGraw H i l l , New York.
Jelesnianski, C. P., 1972. SPLASH 1. (Special Program t o L i s t Amplitudes o f NDAA Tech. Mem. NUS TDL-46, Surges from Hurricanes) L a n d f a l l Storms. S i l v e r Springs, Md. Jensen, R. E., 1983. A Consistent Analysis o f Boussinesq-type Water Wave Equations i n Continuous and D i s c r e t e Form. Ph.D. t h e s i s , Texas A and M U n i v e r s i t y , Col 1ege S t a t i o n .
125 Leonard, A., 1974. Energy cascade flows. Adv. Geophys, 18A: 237-248.
in
large-eddy
simulation
of
turbulent
1978. Numerical s i m u l a t i o n o f shallow water waves. Proc. McCowan, A. D., Fourth A u s t r a l i a n Conference on Coastal and Ocean Engineering, Adelaide, A u s t r a l i a , 132-136. Rabiner, C. R., Processing.
and Gold, B., 1975. P r e n t i c e H a l l , NJ.
Theory and A p p l i c a t i o n o f D i g i t a l Signal
Reynolds, O., 1895. On t h e dynamical t h e o r y o f incompressible viscous f l u i d s and t h e d e t e r m i n a t i o n o f t h e c r i t e r i o n . Philos. Trans. R. SOC. London, Ser. A, 1986: 123-164. Numerical experiments w i t h t h e p r i m i t i v e equations. Proc. Shuman, F. G., 1962. I n t . Symp. Num. Weather P r e d i c t i o n i n Tokyo, November 1960. Meteorological Society of Japan, Tokyo, 85-107. Shuman,
F. G.,
1971.
R e s u s c i t a t i o n o f an i n t e g r a t i o n procedure.
NWC Office
Note 54. General c i r c u l a t i o n experiments w i t h t h e p r i m i t i v e Smagorinsky, J. , 1963. equations: I. The b a s i c experiment, Mon. Weather Rev., 91: 99-161.
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127
A LIMITED AREA MODEL FOR THE GULF STREAM REGION
WILLIAM R. HOLLAND National Center f o r Atmospheric Research P.O. Box 3000, Boulder, Colorado 80307
ABSTRACT Studies of eddy/mean flow i n t e r a c t i o n s i n b a s i n - s c a l e , eddy-resolving numerical models have been c a r r i e d out f o r a decade o r so. Recently, a s a r e s u l t of a need f o r b e t t e r v e r t i c a l and h o r i z o n t a l r e s o l u t i o n , a new generation of models designed t o accomplish c a l c u l a t i o n s i n l i m i t e d a r e a s of a basin, models with open boundary c o n d i t i o n s , have begun t o be developed. These models have been a p p l i e d t o t h e Gulf Stream region, t o t h e Agulhas Current, t o t h e Brazil/Falklands Current confluence region, and t o t h e Tasman Sea region. These r e s u l t s w i l l be discussed, both i n terms o f llphysics" o f t h e s e e n e r g e t i c western boundary c u r r e n t regions and i n terms of t h e "numerics" a s s o c i a t e d with very f i n e r e s o l u t i o n and with open boundary c o n d i t i o n s . The most ambitious undertaking o f t h i s kind by t h e author and h i s colleagues has been a very f i n e r e s o l u t i o n model study of t h e Gulf Stream region from Cape H a t t e r a s t o t h e Grand Banks and bordered on t h e north and south by boundaries a t 50 and 25 Degrees North r e s p e c t i v e l y . The Gulf Stream e n t e r s t h e region o f i n t e r e s t a s a western boundary c u r r e n t with a c e r t a i n s p e c i f i e d inflow. The flow e x i t s t h e region by way o f open boundaries on t h e e a s t , n o r t h , and south. Various p o s s i b l e boundary conditions have been t r i e d and t h e i m p l i c a t i o n s of t h e s e f o r i n t e r i o r physical behavior examined. In p a r t i c u l a r , t h e n a t u r e o f Gulf Stream meander processes and t h e r o l e o f important bottom r e l i e f i n t h e region a r e discussed. 1.
INTRODUCTION S t u d i e s of eddy/mean flow i n t e r a c t i o n s i n b a s i n - s c a l e , eddy-resolved,
numerical models have been c a r r i e d out f o r a decade o r so.
The e a r l y work
focused upon t h e o r i g i n of mesoscale eddies a s a r e s u l t o f b a r o c l i n i c and b a r o t r o p i c i n s t a b i l i t i e s of t h e western boundary c u r r e n t and i t s seaward extension, and o f t h e R e c i r c u l a t i o n nearby (Holland and Lin, 1975a,b; Holland, 1978).
Recent work has begun t o r e f i n e t h e p i c t u r e and has examined various
theoretical issues.
These include s t u d i e s o f t h e homogenization of p o t e n t i a l
v o r t i c i t y (Holland, Keffer, and Rhines,
1984), i n s t a b i l i t y mechanisms
(Haidvogel and Holland, 1978; Holland and Haidvogel, 1980), eddy mixing and gyre e q u i l i b r a t i o n (Rhines and Holland, 1979; Holland and Rhines, 1980), and the p e n e t r a t i o n s c a l e o f t h e Gulf Stream (Holland and Schmitz, 1985). In a d d i t i o n t o t h e s e t h e o r e t i c a l s t u d i e s , comparisons o f model r e s u l t s with observations have played an important p a r t i n model refinement and i n i d e n t i f y i n g important i s s u e s regarding t h e physics o f t h e Gulf Stream system
128 (Schmitz and Holland, 1982; Schmitz et al., 1982; Holland, 1985; Schmitz and Holland, 1986). This work is currently being extended with models of much higher vertical resolution than heretofore to examine the vertical structure of mean and eddy fields in the Gulf Stream and Kuroshio. As illustrations of this kind of comparison, Schmitz and Holland (1986) show several observational/model intercomparisons that are currently being examined. In particular meridional sections of mean zonal flow and eddy kinetic energy in eight layer numerical experiments have much in common with North Atlantic and North Pacific current meter mooring data. The correspondence is by no means exact but it is clear that both vertical and horizontal structure in the numerical experiment has many features in common with the data in both mean and eddy quantities, including approximately correct ratios of surface to deep mean and eddy currents in the intense flow, and similar meridional structure in terms of eastward and westward (recirculating) mean flows. An additional comparison between the data at 55W (in the Gulf Stream) and a similar point in the intense eastward flow of one of these eight layer model calculations indicates that the vertical structure of eddy kinetic energy is remarkably similar to observations, suggesting strongly that we are on the right track regarding the eddy/mean flow interactions that give rise to the intense eddy field in these vigorously eddying western boundary regions. To date, most numerical studies have been highly idealized with respect to the geometry of the ocean basin in question. There are many good reasons For one thing, simpler situations are a vital and necessary part of understanding the more complex ones. When confronting questions requiring detailed comparisons with observations, however, one must ask whether o r f o r this.
where dynamically similarregionsof an idealized basin can be found in the western North Atlantic. Or, turning the question around, where in an idealized basin should one seek to compare observations along 55W (or any other place)? Thus, as models more faithfully reproduce observed features, we are driven toward more faithful inclusion in our models of basin shape, bottom topography, and boundary conditions (e.g., wind stress, buoyancy flux, inflow and outflow across the boundaries of local domains, etc.) Such models, with various physical and geometrical factors successively put in or taken out, allow us to ascertain which features are key to understanding the dynamics and which are not. In the last several years, we have begun to develop models of the North Atlantic (and other basins) that have somewhat 'realistic' geometry. For example, Holland (1983, 1986) examined the wind-driven circulation in the North Atlantic basin from 15'N to 65'N. using a three layer quasigeostrophic model with L degree horizontal resolution. Studies of the role of eddies in the general circulation and studies of the oceanic response to transient
129
Figure 1. The time averaged circulation in the North Atlantic basin using the annual mean Hellerman wind stresses, based upon a three layer QG model. The streamfunctions at 150 m, 650 m, and 3000 m are shown.
130
Figure 2. The instantaneous streamfunctions of the flow in the North Atlantic basin at the same levels as in figure 1. Note the strong Gulf Stream meandering in the upper layer and the important eddying circulations in the deep ocean.
131 wind forcing continue. Figure 1 shows, for example, the time-averaged streamfunction for a particular case with steady wind forcing. The upper layer (a) shows the mean gyre forced by the mean annual Hellerman wind stress; the middle (b) and lower (c) layers show the time-averaged eddy driven components of the flow, with a broader recirculation in the main thermocline, and a very narrow recirculation in the deep water under the mean Gulf Stream. These figures are the result of time averaging over a five year period. Figure 2 shows instantaneous views of the streamfunctions, illustrating the rich eddy field in the western North Atlantic. Figure 3 shows the upper layer instantaneous streamfunction for a case with Hellerman's monthly forcing with the annual component removed. Thus the forcing is purely transient with periods between 2 and 12 months represented. The main response (after a 20 year spin-up time) shows westward propagating, annual period, baroclinic Rossby waves as the primary response. The amplitude of these waves is small compared to the eddy signal shown in the experiment above (figures 1 and 2 ) but away from the intense Gulf Stream region of instability, particularly in the eastern basin, the transient signal would dominate.
Figure 3 . An instantaneous, upper layer streamfunction in the North Atlantic QG model, driven by Hellerman's seasonal winds only (no mean forcing). Westward propagating, annual period, first baroclinic mode Rossby waves dominate the solution.
132 Both these experiments have been run with constant depth oceans.
If
topography had been p r e s e n t and higher frequency wind f o r c i n g included, t h e Ocean response might have shown important deep t r a n s i e n t flows driven d i r e c t l y by t r a n s i e n t f o r c i n g .
Such experiments have y e t t o be c a r r i e d o u t .
Even though t h e s e s t u d i e s can be made with a h o r i z o n t a l r e s o l u t i o n o f
L
degree i n l a t i t u d e and longitude, f o r some purposes even h i g h e r r e s o l u t i o n w i l l be needed.
This i s p a r t i c u l a r l y t r u e when h i g h e r v e r t i c a l r e s o l u t i o n
(more than t h r e e l a y e r s ) i s used. T h i s i s due t o t h e f a c t t h a t with more v e r t i c a l r e s o l u t i o n , higher b a r o c l i n i c modes, with s m a l l e r Rossby r a d i i of deformation, a r e included.
I t should be kept i n mind t h a t such experiments
w i l l e v e n t u a l l y be c a r r i e d out using t h e p r i m i t i v e equations, with t h e i r much
g r e a t e r computational c o s t .
Therefore a new generation o f eddy-resolving
l i m i t e d a r e a models (ELAM's) t h a t can s u c c e s s f u l l y handle open boundaries i s needed, using both quasigeostrophic and p r i m i t i v e equation physics, t h a t w i l l allow us t o t e l e s c o p e i n on l o c a l regions o f a l a r g e r domain. This paper d i s c u s s e s some i n i t i a l attempts t o study a v a r i e t y o f eddyr i c h western boundary c u r r e n t regions, i n c l u d i n g t h e Gulf Stream region, t h e Agulhas Current, t h e Brazil/Falklands Current confluence r e g i o n , and t h e Tasman Sea.
Preliminary r e s u l t s w i l l be shown t o i l l u s t r a t e t h e power and
the limitations of
t h e Limited Area Model approach a s well a s t o i n d i c a t e
some o f t h e i n t e r e s t i n g v a r i e t y o f behaviors t o be found. 2.
WESTERN BOUNDARY CURRENT MODELS Regional ocean models can i n c l u d e very high h o r i z o n t a l r e s o l u t i o n and
r e a l i s t i c geometry a t t h e expense o f having t o deal with open boundary conditions.
There a r e many p o s s i b l e ways t o handle t h e open boundaries but
a l l must deal with two b a s i c problems: ( i ) how does t h e e x t e r n a l ocean i n f l u e n c e t h e domain o f i n t e r e s t , and ( i i ) how do f e a t u r e s o f t h e flow generated i n t e r n a l l y r e a l i s t i c a l l y c r o s s t h e open boundaries, thus l e a v i n g t h e domain o f i n t e r e s t ?
These d i f f i c u l t i e s a r i s e o f course because t h e open
boundary c o n d i t i o n s themselves depend upon t h e c i r c u l a t i o n i n both t h e i n t e r n a l and e x t e r n a l ocean regions. Radiation boundary conditions allow, f o r some problems, t h e second o f these problems t o be addressed.
Extrapolation techniques a r e used t o extend
t o t h e boundary changes d i c t a t e d by i n t e r i o r i n f l u e n c e s propagating toward t h a t boundary.
Such techniques work well f o r some problems and not a t a l l
f o r o t h e r s (Orlanski, 1976; Carmelengo and O'Brien, 1980; Roed and Smedstad, 1984; Chapman, 1985). The i n f l u e n c e o f t h e o u t e r ocean on t h e i n n e r one could be handled i n various ways: by embedding t h e domain o f i n t e r e s t i n a l a r g e r , perhaps coarse r e s o l u t i o n , domain f o r which numerical c a l c u l a t i o n a r e a l s o done; by
133 carrying out s e p a r a t e l y a c o a r s e r domain c a l c u l a t i o n and saving t h e ' a p p r o p r i a t e ' boundary c o n d i t i o n s therefrom t o be imposed on a l a t e r l o c a l c a l c u l a t i o n ; by parameterizing t h e f a r f i e l d i n f l u e n c e i n some fashion, f o r example by imposing c r o s s boundary flow and d e n s i t y information from t h e o r e t i c a l i d e a s such as "Sverdrup balance'' and "geostrophy".
A l l of these
techniques have important d e f i c i e n c e s . The f i r s t technique allows feedback between t h e i n n e r and o u t e r domain but t h e s c a l e s o f motion allowed i n t h e coarse r e s o l u t i o n w i l l not adequately r e p r e s e n t t h e s c a l e s i n t h e f i n e resolution.
The second and t h i r d techniques s i m p l i f y t h e problem by
disallowing feedback; t h e o u t e r domain i s not a f f e c t e d by t h e evolving s o l u t i o n s i n t h e domain o f i n t e r e s t .
This may o r may n o t be a c r u c i a l choice,
depending upon t h e problem and upon t h e r e a l i s m o f t h e e x t e r n a l l y imposed boundary c o n d i t i o n s . F i n a l l y , both problems ( i ) and ( i i ) go away i f good enough observations exist at the
open boundary.
A t t h e p r e s e n t t i m e , t h i s i s u n l i k e l y f o r many
problems o f i n t e r e s t because o f t h e s c a l e o f e f f o r t needed o b s e r v a t i o n a l l y t o f u l l y d e s c r i b e t h e space/time behavior o f t h e flow along a s e c t i o n o r boundary o f any l e n g t h . Western boundary regions a r e e s p e c i a l l y prone t o a l i m i t e d a r e a approach, because t h e western boundary i s a p h y s i c a l one, not r e q u i r i n g t h e approximations described above.
In a d d i t i o n , t h e e a s t e r n s i d e o f t h e domain
is o f t e n l e s s t r a n s i e n t and l e s s i n e r t i a l , and t h e b e t a e f f e c t causes i n t e r n a l l y c r e a t e d t r a n s i e n c e (due t o i n s t a b i l i t i e s ) t o propagate westward, away from t h e open boundary.
This makes t h e Gulf Stream region, with open
boundaries on t h e e a s t a t 40°W and on t h e south a t 25ON, an e s p e c i a l l y a t t r a c t i v e region f o r s u c h a study ( i n a d d i t i o n t o t h e more obvious reasons; t h a t t h e Gulf Stream and
i t s meandering and r i n g formation behavior i s t h e
best known region o f t h e World Ocean). Before looking a t some r e s u l t s from t h i s region, however, l e t us f i r s t examine some models under development f o r o t h e r western boundary regions, a l l from t h e Southern Hemisphere.
These a r e t h e Agulhas Current region, t h e
Brazil-Falklands Currents confluence region, i n c l u d i n g Circumpolar flow through Drake Passage, and t h e region o f t h e Tasman Sea.
In t h e f i r s t two
of t h e s e r e g i o n s , t h e domain i s not t o t a l l y blocked on t h e western s i d e o f t h e domain o f i n t e r e s t , and i n t h e t h i r d , t h e problem i s complicated by t h e presence o f t h e New Zealand land mass.
A l l o f t h e s e l e a d t o s p e c i a l problems
and s i t u a t i o n s t h a t , i n t h e l i m i t e d a r e a model c o n t e x t , r e q u i r e s p e c i a l solutions. The Agulhas Current region i s one o f considerable i n t e r e s t , f i r s t l y because i t i s t h e only western boundary c u r r e n t t h a t runs out o f boundary and secondly because t h e region south o f South A f r i c a i s a crossroads of
134
interocean exchange (Gordon, 1986).
The dynamical nature of that current
thus becomes of larger significance than just the local one, and the manner i'n which the current carries water from the Indian Ocean into the South Altantic and the extent to which some of that water "retroflects" back into the South Indian Ocean is of considerable interest to large scale oceanographers. Figure 4 shows results from two model calculations (Holland, 1987a) using a three layer quasigeostrophic limited area model of the Agulhas region. The upper layer streamfunctions are shown at a particular instant in each calculation to illustrate briefly three points: the technique by which the boundary regions on both east and west 'parameterize' the interaction of this local domain with the rest of the South Indian and Atlantic Oceans; the
Figure 4. Instantaneous upper layer streamfunctions for two numerical experiments covering the region south of Southern Africa. The large scale, counterclockwise circulation is driven near the eastern side of the limited area to produce a given zonal flow near the eastern boundary. The land masses are Madagascar and Southern Africa.
135 transient nature of the flow as the Agulhas Current, flowing down the east coast of Southern Africa, turns westward and partially retroflects back to the east; and the difference between the two calculations when bottom topography is introduced. Both calculations are the result o f a long spin-up until a statistical equilibrium is reached. The local domain is actually closed but the flow near the eastern side is driven by a mass flux in and out of a narrow eastern boundary region in which special forcing functions are imposed. These conditions give a certain zonal flow whose meridional and vertical structure is known. A simple analogy to wind-stress curl forcing, acting upon each layer, is used to create the zonal flow wanted. The same zonal flow is driven in both calculations. The westward flow toward the African continent and Madagascar in the north is unstable and creates a highly transient Agulhas Current formation region. As the boundary current moves southward and turns more and more westward, anti-cyclonic eddies form into ring-like features. The net transport by the Agulhas Current is about 60 Sverdrups, and the rings have similar transports. In the flat bottom case (a), the eddies are quite small scale and have relatively fast westward propagation speeds. In the experiment with bottom topography (b), the eddies and rings are much larger and relatively slow moving. Note that the amount of water that retroflects (returns back to the east, south of South Africa) in the topographic case is somewhat greater than that in the case without topography, suggesting the important role of eddy-topographic interactions in the retroflection process (Holland, 1987a). The rings in both cases move westward and ultimately are absorbed in a region of enhanced friction near the western boundary. The western boundary layer acts as a passive southward recirculation zone in these cases as well as an "eddy absorber." This kind of local calculation, in which the dynamical behavior is determined almost entirely by the instability processesinthe Agulhas formation and retroflection regions, can be carried out without complicated boundary conditions. "Pumps and baffles" can be inserted to produce the large scale circulation desired. The transient eddies are ultimately absorbed near the western edge of the domain without much reflection, so that a simple sponge layer is workable, and the forcing region near the eastern edge of the domain allows for a simple specification of the zonal flow to parameterize the gyre structure further eastward in the South Indian Ocean. A second model study (Holland, 1987b), this time f o r the region of the Brazil/Falklands Current confluence, is illustrated in figure 5.
The model
has open boundaries on the eastern and northern sides of the domain and on the upstream side west of South America where the Circumpolar Current enters
136
Figure 5. The quasigeostrophic streamfunctions at three levels in a limited area model of the Brazil Current/Falklands Current confluence. The Brazil Current and Circumpolar flow enter the region on the north and west boundaries respectively and the combined flow exits on the east boundary.
137
Figure 6 . Three model calculations for the region of the Southwest Pacific using a barotropic model. The highly transient flow is driven near the eastern boundary by forcing terms that create multiple gyres (a parameterization of wind forcing further to the east). Top (a): Subtropical gyre dominates whole region; middle (b): the line separating the subtropical and subpolar gyres is at mid-basin; bottom (c): the line separating the gyres is at the mid-latitude of the New Zealand land mass. Note: the northernmost gyre circulates counterclockwise.
138
Figure 7 . A regional model of t h e Gulf Stream region. The i n s t a n t a n e o u s streamfunctions a t t h r e e l e v e l s a r e shown (150 m, 650 m, and 3000 m r e s p e c t i v e l y ) . The c i r c u l a t i o n i s driven by inflows and outflows only; no wind f o r c i n g p r e s e n t . I n t h i s experiment t h e ocean i s o f constant depth.
the region. The Brazil Current enters from the north and the combined flows exit on the east. In this three layer, constant depth calculation, boundary conditions on the streamfunction in each layer are specified so that the horizontal location and vertical structure of the inflows and outflows is fixed once and for all. The influence of the rest of the Southern Ocean circulation is "parameterized" in these boundary conditions. The location of the eastern boundary outflow is especially important to the interior solution and is chosen here to roughly coincide with the Circumpolar flow at the Prime Meridian (0"E longitude), as suggested by temperature and salinity observations and geostrophic calculations. The vorticity at inflow points is set by the flow structure there but the vorticity at outflow points is gotten by a simple extrapolation procedure from the interior; essentially the longitudinal gradient of vorticity is set to zero. This allows the circulation some freedom to export vorticity from the region and provides an opportunity for transients to be absorbed and/or exported across this boundary. In addition, the western, northern and eastern open boundaries have adjacent narrow zones of enhanced friction that help absorb transients reaching these boundaries. The Circumpolar Current turns northward as it rounds the southern tip of South America and brushes the southern edge of Falklands plateau. (Note that our continental boundary is chosen to be the 200 meter depth contour so that shallow shelves connect the Falklands plateau to South America). The Brazil Current coming southward turns seaward before encountering the topography of the plateau, at least at this instant of the calculation. The entire region is highly transient; the Brazil Current develops strong eddies as it turns eastward and the combined flow downstream exhibits instability-driven meandering almost to the eastern boundary. There is considerable recirculation in the deep ocean south of the Circumpolar Current. A third set of calculations,using the limited area approach for the region of the Southwestern Pacific, shows a variety of possible behaviors for the East Australia Current (figure 6 ) . In particular, the nature of the circulation varies enormously as the parameterized wind gyre forcing to the east of this domain of interest is changed. An examination of the wind stresses over the South Pacific shows very large variability on the seasonal time scale. The boundary between the subtropical and subpolar gyres moves over a wide range of latitudes, suggesting the three calculations shown. Here the important role of the New Zealand land mass and the relationship of the wind-stress curl distribution is explored using a barotropic, constant depth model. The top picture (figure 6a) shows an instantaneous streamfunction when the
140 e n t i r e domain i s "forced" from t h e e a s t , a s i n t h e Agulhas case above, with a s i n g l e anticlockwise ( s u b t r o p i c a l type) wind f o r c i n g .
The zonal flow i n
t h e n o r t h forms a western boundary c u r r e n t (our East A u s t r a l i a Current) t h a t flows southward r i g h t through t h e Tasman Sea and r i g h t around New Zealand. The southern s u b t r o p i c a l gyre boundary, a s demarked by our "wind forcing" region on t h e e a s t , i s n e a r t h e southern boundary of t h e domain.
The middle
p i c t u r e ( f i g u r e 6b) shows t h e o p p o s i t e extreme when t h e boundary between t h e northern s u b t r o p i c a l gyre and t h e southern subpolar gyre i s much f u r t h e r north (near t h e northern t i p o f New Zealand).
F i n a l l y t h e bottom p i c t u r e
( f i g u r e 6c) i l l u s t r a t e s t h e middle ground i n which t h e gyre boundary i s about a t t h e m i d l a t i t u d e o f t h e N e w Zealand land mass. The c i r c u l a t i o n s are very d i f f e r e n t .
In t h e last two c a s e s , much of t h e
East A u s t r a l i a Current s e p a r a t e s from t h e boundary t o flow n o r t h o f N e w Zealand.
The eastward flow i s h i g h l y t r a n s i e n t and, p a r t i c u l a r l y i n t h e
f i n a l c a s e , l a r g e counterclockwise eddies form t o move southward i n t o t h e r e l a t i v e l y q u i e t region west o f N e w Zealand ( a Rossby Wave shadow zone t h a t
i s s h i e l d e d from t h e d i s t a n t f o r c i n g t o t h e e a s t ) .
This f i n a l s o l u t i o n has
t h e f l a v o r o f t h e a c t u a l s i t u a t i o n and i s being i n v e s t i g a t e d i n a b a r o c l i n i c model ocean (Holland, 1 9 8 7 ~ ) . 3.
THE GULF STREAM MODEL
For t h e purpose o f studying t h e meandering and r i n g formation processes i n t h e Gulf Stream region, a q u a s i g e o s t r o p h i c l i m i t e d a r e a model o f t h e region from 30°N t o 55ON and 80°W t o 4OoW has been c o n s t r u c t e d . In t h e numerical experiments shown h e r e , t h e r e i s no wind f o r c i n g a c t i n g upon t h e region o f i n t e r e s t ; t h e c i r c u l a t i o n i s e n t i r e l y driven by t h e s p e c i f i e d inflow o f t h e Gulf Stream as a western boundary c u r r e n t south of Cape H a t t e r a s and by t h e s p e c i f i e d outflow of t h e Stream a c r o s s t h e e a s t e r n boundary j u s t south o f 40'". The model has t h r e e l a y e r s i n t h e v e r t i c a l ; t h e two numerical experiments here have inflow and outflow t r a n s p o r t s o f 30 Sverdrups i n t h e upper l a y e r (300 m t h i c k ) , 32 Sverdrups i n t h e second l a y e r (700 m t h i c k ) , and no inflow
o r outflow i n t h e lowest l a y e r (4000 m t h i c k ) .
The boundary c o n d i t i o n s a r e
s i m i l a r t o t h o s e i n t h e Brazil/Falklands r e g i o n a l problem; streamfunction and v o r t i c i t y s p e c i f i e d on inflow, streamfunction s p e c i f i e d on outflow, and v o r t i c i t y g r a d i e n t s e t t o zero a t outflow p o i n t s .
The l o c a t i o n of both inflow
and outflow i s f i x e d i n time, r e s t r i c t i n g t h e meandering process near t h e e a s t e r n boundary. Figure 7 shows t h e streamfunction a t a p a r t i c u l a r i n s t a n t a f t e r a long spinup process f o r a case o f constant depth.
The meanders seen h e r e a r e
s t r o n g l y time dependent and o c c a s i o n a l l y a warm-core or cold-core r i n g i s shed. The deep l a y e r , not d r i v e n by boundary inflows, i s dominated by an eddy f i e l d
141 t h a t extends from j u s t o f f Cape H a t t e r a s t o 5OoW beneath t h e Gulf Stream. Figure 8 shows t h e upper l a y e r streamfunction 20 days l a t e r , j u s t a s a cold ring i s about t o break o f f a f t e r t h e deep meander development. Figure 9 shows t h e streamfunction a t a p a r t i c u l a r i n s t a n t f o r a case with bottom topography (shown i n f i g u r e 10).
The Stream has a very convoluted
character i n t h e v i c i n i t y o f t h e New England Seamounts, but it i s not c l e a r from any instantaneous p i c t u r e how important t h e topographic i n f l u e n c e might be. The deep eddy f i e l d i s c l e a r l y s t r o n g l y influenced by bathymetry, but analyses of various s t a t i s t i c s a r e r e q u i r e d t o a s c e r t a i n whether t h e eddy f i e l d can communicate upward t h e l o c a t i o n o f t h e Seamounts. One such s t a t i s t i c i s shown i n f i g u r e 11. The c e n t r a l s t r e a m l i n e f o r each o f t h e s e cases i s shown every 20 days f o r a 680 day p e r i o d , t o i n d i c a t e something o f t h e n a t u r e o f t h e envelope o f t h e v a r i o u s p a t h s o f t h e Stream. Figure l l a shows t h e f l a t bottom c a s e , f i g u r e l l b t h e case with bottom topography.
The l a t t e r case shows t h e p r o p e n s i t y f o r c o l d c o r e r i n g
formation j u s t west o f t h e New England Seamounts. southwestward
The r i n g s formed t h e r e move
u n t i l t h e y "feel" t h e c o n t i n e n t a l s l o p e s o u t h e a s t o f Cape
Hatteras, where t h e y decay but a l s o a f f e c t t h e Stream n o r t h o f t h e i r l o c a t i o n .
Figure 8. The upper l e v e l streamfunction 20 days following t h e maps shown i n f i g u r e 7. Note t h a t t h e meander i n f i g u r e 7 has deepened and has j u s t formed a r i n g - l i k e f e a t u r e s o u t h of t h e Stream.
142
Figure 9. Instantaneous streamfunctions at three levels are shown (150 m, 650 m, and 3000 m respectively) for calculation like that in figure 7 except that bottom topography is included (see figure 10). The regional model is driven precisely the same as the flat bottom case in figure 7.
143 Figures 12 and 13 show another s t a t i s t i c f o r t h e s e two c a s e s k i n e t i c energy p a t t e r n s based upon a f i v e year average. the
-
t h e eddy
The maxima follow
Gulf Stream a x i s and a r e w i t h i n a f a c t o r o f two o f observed values a t a l l
levels.
The case with bottom topography ( f i g u r e 13) shows c l e a r l y t h e e f f e c t s
of t h e vigorous c o l d c o r e r i n g development and shedding, as a southward extension o f t h e EKE p a t t e r n .
Note a l s o t h e l a r g e d i f f e r e n c e s i n t h e two
abyssal p a t t e r n s , presumably ' d i r e c t l y r e l a t e d t o t h e presence o r absence of variable depth. These r e s u l t s a r e intended mainly t o i l l u s t r a t e t h e n a t u r e o f t h e l i m i t e d area model approach.
Much deeper analyses i n t o t h e s e numerical experiments
and many more experiments themselves a r e needed t o even begin to understand the dynaniical behaviors found h e r e and t o even begin t o s o r t out t h e dependence upon boundary c o n d i t i o n s , topographic e f f e c t s , l o c a l wind f o r c i n g , and t h e o t h e r various parameters t h a t govern t h e flow ( f r i c t i o n c o e f f i c i e n t , v e r t i c a l and h o r i z o n t a l r e s o l u t i o n , s t r e n g t h o f inflow/out, e t c . , e t c . ) . Studies o f t h i s kind a r e
being vigorously pursued.
Moreover i t i s l i k e l y t h a t p r i m i t i v e equation models with much h i g h e r v e r t i c a l r e s o l u t i o n w i l l have t o p l a y a r o l e i n model s t u d i e s o f t h e Gulf Stream region.
Such s t u d i e s are a l s o underway and new ways of coping with
Figure 10. The bottom topography f o r t h e c a l c u l a t i o n i l l u s t r a t e d i n f i g u r e 9. The topographic v a r i a t i o n s a r e considered t o e x i s t only i n t h e lowest layer. Note t h e l i n e of t h e N e w England seamounts t h a t c r o s s t h e p a t h of t h e Gulf Stream.
144 the open boundary conditions are being examined. In the end, it may be necessary to acquire an observational description of the Gulf Stream, particularly at outflow from the domain where the Stream meanders broadly over several degrees of latitude (say at 5 O o W o r 40'W) t o adequately handle the "prediction" of behavior in this region. It is likely that satellite data (AVHRR,
altimetry) can very usefully help us to initialize models o f this
kind for dynamic calculations and t o develop assimilation schemes f o r predictive calculations.
Figure 11. A time sequence of individual upper layer streamlines marking the middle of the inflowing Gulf Stream. The streamline position every 20 days (for 680 days total) is shown as an indication of the envelope of Gulf Stream paths. Upper:the constant depth case; Lower: the case with variable depth.
145
Figure 12. The p a t t e r n s o f eddy k i n e t i c energy f o r t h e c o n s t a n t depth case, based upon f i v e years o f time averaging. The t h r e e l e v e l s a r e t h e same a s i n f i g u r e 7.
146
Figure 13. The patterns o f eddy k i n e t i c energy f o r the case with variable depth, based upon f i v e years o f time averaging. The three l e v e l s are the same as i n figure 9 .
147 4. CONCLUSION Regional models with open (ocean) boundaries on some sides of a domain of interest look quite promising as members of a hierarchal approach to ocean modelling. Clearly, every model choice involves compromises and tradoffs. Global, basin scale and regional models, used in conjunction with each other, allow us to gain a much broader perspective upon the important influences on large scale ocean circulation. Coarse resolution interbasin models without eddies, medium resolution eddy-resolving basin studies, and high resolution limited area models all have something to contribute to our understanding of the dynamics of ocean circulation. 5. REFERENCES Camerlengo, A.L., and J.J. O'Brien, 1980. Open boundary conditions in rotating fluids. J. Comput. Phys., 35: 12-35. Chapman, David C., 1985. Numerical treatment of cross-shelf open boundaries in a barotropic coastal ocean model. J. Phys. Oceanogr., 15: 1050-1075. Gordon, A.L., 1986. Interocean exchange of thermocline water. J. Geophys. Res., 91: 5037-5046. Haidvogel, D.B., and W.R. Holland, 1978. The stability of ocean currents in eddy-resolving general circulation models. J. Phys. Oceanogr., 8: 393-413. Holland, W.R., 1978. The role of mesoscale eddies in the general circulation of the ocean: Numerical experiments using a wind-driven quasigeostrophic model. J. Phys. Oceanogr., 8: 363-392. Holland, W.R., 1983. Simulation of midlatitude variability. In: The Role of Eddies in the General Ocean Circulation, Proceedings Hawaiian Winter Workship, University of Hawaii, January 5-7. Holland, W.R., 1985. Simulation of mesoscale ocean variability in midlatitude gyres. In: Atmospheric and Oceanic Modeling - Volume 28A of Advances in Geophysics, Academic Press, Orlando. Holland, W.R., 1986. Quasigeostrophic modelling of eddy-resolved ocean circulation. In: Proceedings of the Nato Advanced Study Institute, Banyuls Sur Mer, France (in press). Holland, W.R., 1987a. Numerical studies of the Agulhas Current region using a regional ocean model (in preparation). Holland, W.R., 1987b. The Brazil Current/Falklands Current confluence region: Numerical model studies o f a local region (in preparation). Holland, W.R., 1987c. Geometrical influences on the western boundary current (the East Australia Current) of the South Pacific (in preparation). Holland, W.R., and L.B. Lin, 1975a. On the origin of mesoscale eddies and their contribution to the general circulation of the ocean. I. A preliminary numerical experiment. J. Phys. Oceanogr., 5: 642-657. Holland, W.R., and L.B. Lin, 1975b. On the origin of mesoscale eddies and their contribution to the general circulation of the ocean. 11. A parameter study. J. Phys. Oceanogr., 5: 658-669. Holland, W.R., and D.B. Haidvogel, 1980. A parameter study of the mixed instability of idealized ocean currents. Dyn. Atmos. & Oceans, 4: 185-215. Holland, W.R., and P.B. Rhines, 1980. An example of eddy induced ocean circulation. J. Phys. Oaeanogr., 10: 1010-1031. Holland, W.R., and W.J. Schmitz, Jr., 1985. On the zonal penetration scale Of model midlatitude jets. J. Phys. Oceanogr., 15: 1859-1875. Holland, W.R., T. Keffer, and P.B. Rhines, 1984. Dynamics of the ocean general circulation: The potential vorticity field. Nature, 308: 698-705. Orlanski, I., 1976. A simple boundary condition for unbounded hy-perbolic flows. J. Comput. Phys., 21: 251-269.
148 Rhines, P.B., and W.R. Holland, 1979. A theoretical discussion of eddy-driven mean flows. Dyn. Atmos. 6 Oceans, 3: 289-325. Roed, L.P., and O.M. Smedstad, 1984. Open boundary conditions for forced waves in a rotating fluid. SIAM, J. Sci. Stat. Comput., 5: 414-426. Schmitz, W.J., Jr., and W.R. Holland, 1982. A preliminary comparison of selected numericdl eddy-resolving general circulation experiments with observations. J. Mar. Res., 40: 75-117. Schmitz, W.J., Jr., P.P. Niiler, R.L. Bernstein, and W.R. Holland, 1982. Recent long-term moored instrument observations in the Western North Pacific. Jour. Geophys. Res., 8 7 : 9425-9440. Schmitz, W.J., Jr., and W.R. Holland, 1986. Observed and modeled mesoscale variability near the Gulf Stream and Kuroshio extension. J. Geophys. Res., 91: 9624-9638.
149
STUDY OF TRANSPORT FLUCTUATIONS AND MEANDERING OF THE FLORIDA CURRENT USING AN ISOPYCNIC COORDINATE NUMERICAL MODEL
DOUGLAS B. BOUDRA, RAINER BLECK and FRIEDRICH SCHOTT Rosenstiel School o f M a r i n e and Atmospheric Science, Rickenbacker Causeway, Miami FL 33149, (USA)
University o f Miami,
4600
ABSTRACT An isopycnic c o o r d i n a t e numerical model , u s i n g t h e F1 ux-Corrected T r a n s p o r t a l g o r i t h m t o c o n t r o l i s o p y c n a l o u t c r o p p i n g and i n t e r s e c t i o n w i t h t h e ocean bottom, i s c o s f i g u r e d i n a channel w i t h t h e b o t t o m topography o f t h e F l o r i d a B u l k parameters d e t e r m i n e d from a n a l y s i s o f o b s e r v a t i o n s i n S t r a i t s a t 27 N. t h e S u b t r o p i c a l A t l a n t i c C l i m a t e S t u d i e s (STACS) program a r e combined w i t h a dynamical i n i t i a l i z a t i o n procedure, g e n e r a t i n g a F l o r i d a C u r r e n t - l i k e mass/flow pattern. Ten i s o p y c n a l l a y e r s and 2 km h o r i z o n t a l g r i d s p a c i n g r e s o l v e t h i s cross-sectional flow. The channel i s c r e a t e d by d u p l i c a t i n g t h e c r o s s - s e c t i o n i n t h e downstream d i r e c t i o n . I n v e s t i g a t i o n focuses on whether t h e t r a n s p o r t f l u c t u a t i o n s on t i m e s c a l e s o f a few days t o s e v e r a l weeks and t h e meandering observed i n t h e F l o r i d a C u r r e n t may b e e i t h e r l o c a l l y f o r c e d b y t h e w i n d o r d u e t o i n h e r e n t d y n a m i c a l instabilities. When a s i n g l e c r o s s - s e c t i o n i s f o r c e d w i t h s p a t i a l l y c o n s t a n t b u t t e m p o r a l l y f l u c t u a t i n g d o p s t r e a m wind s t r e s s , t h e b a r o t r o p i c t r a n s p o r t response e x h i b i t s an a l m o s t 90 phase l a g t o t h e wind. The response a m p l i t u d e i s s l i g h t l y l e s s t h a n d i r e c t l y p r o p o r t i o n a l t o t h e a m p l i t u d e and p e r i o d o f t h e The b a r o c l i n i c r e s p o n s e , d e f i n e d a s t h e d i f f e r e n c e b e t w e e n t h g forcing. t r a n s p o r t f l u c u t a t i o n s above and below 200 m depth, e x h i b i t s t h e almost 90 phase l a g t o t h e wind, b u t has an o r d e r o f magnitude l e s s a m p l i t u d e t h a n t h e barotropic. I n t h e t h r e e - d i m e n s i o n a l channel, when t h e c r o s s - s t r e a m d e n s i t y g r a d i e n t i s s t r o n g enough, p e r t u r b a t i o n s w i t h w a v e l e n g t h s g r e a t e r t h a n 60 km e x h i b i t substantial amplification. The c u r r e n t i s most u n s t a b l e t o wavelengths of a l i t t l e more t h a n 200 km when u s i n g t h e d y n a m i c a l l y i n i t i a l i z e d f i e l d s and 150 km when u s i n g analyzed STACS Pegasus d a t a t o i n i t i a l i z e . Events of wave a m p l i f i c a t i o n a r e o f l i m i t e d d u r a t i o n , however, and l e a v e t h e b a s i c s t r u c t u r e o f t h e c u r r e n t unchanged. I n c o r r e s p o n d i n g f l a t b o t t o m channel experiments, t h e p e r t u r b a t i o n c o n t i n u e s t o a m p l i f y u n t i l t h e b a r o c l i n i c s t r u c t u r e has been s u b s t a n t i a l l y modified. From an a n a l y s i s o f energy c o n v e r s i o n s i t i s concluded t h a t t h e p r i m a r y mechanism o f wave a m p l i f i c a t i o n i n a l l cases i s t h e r e l e a s e o f baroclinic instability.
1 INTRODUCTION The F l o r i d a C u r r e n t i s t h a t p a r t o f t h e N o r t h A t l a n t i c western boundary c u r r e n t system w h i c h c o n t i n u e s on f r o m t h e G u l f o f Mexico Loop C u r r e n t t h r o u g h the Florida Straits
--
f i r s t b e t w e e n t h e Keys a n d Cuba and t h e n , t u r n i n g
150 northward,
between t h e mainland and t h e Bahamas
n o r t h F l o r i d a coast.
--
and a l o n g t h e c e n t r a l and
The s t r a i t s g r a d u a l l y narrow downstream f r o m t h e G u l f and
a r e a t t h e i r n a r r o w e s t a t 27'
N, a f t e r w h i c h t h e passageway opens up r a p i d l y on
t h e e a s t s i d e and t h e r e i s no more c o n s t r i c t i o n t h e r e a f t e r . F l u c t u a t i o n s o f t h e t o t a l t r a n s p o r t t h r o u g h t h e s t r a i t s on a wide range o f t i m e s c a l e s have been measured ( N i i l e r and Richardson, e t al,
1985).
1973; Brooks, 1979; Lee,
The temporal mean t r a n s p o r t i s about equal t o t h e annual mean 6 3 30 t o 32 X 10 m s-'.
Sverdrup t r a n s p o r t a t t h e l a t i t u d e o f South F l o r i d a
--
The d i f f e r e n c e between t h e annual maximum, u s u a l l y i n June, and t h e minimum i n October i s 10-15s o f t h e t o t a l .
But l a r g e r f l u c t u a t i o n s have been observed w i t h
p e r i o d s o f s e v e r a l days t o a few weeks. I n a d d i t i o n t o fluctuations i n transport, meander t h r o u g h t h e s t r a i t s .
t h e F l o r i d a C u r r e n t i s known t o
The a m p l i t u d e of t h e meanders seems t o depend
p a r t l y on t h e w i d t h o f t h e s t r a i t s and, t h u s , alongstream p o s i t i o n (Schmitz and Richardson, 1968), owing t o t h e above-mentioned c o n s t r i c t i o n downstream f r o m t h e Gulf.
The p e r i o d range o f t h e meanders i s ,
l i k e t h a t o f some o f t h e l a r g e
t r a n s p o r t f l u c t u a t i o n s , from a few days t o a few weeks. Besides t h e i r p o s s i b l e i n f l u e n c e on t h e h e a t budget o f t h e N o r t h A t l a n t i c and whatever c l i m a t i c impact which t h a t may e n t a i l , t h e s e phenomena a r e i n t e r e s t i n g i n themselves,
and v a r i o u s m e c h a n i s m s h a v e been i n v o k e d t o e x p l a i n t h e i r
e x i s t e n c e and c h a r a c t e r .
The t r a n s p o r t f l u c t u a t i o n s w i t h p e r i o d s o f a few days
t o two weeks a r e most commonly a t t r i b u t e d t o s i m i l a r p e r i o d f l u c t u a t i o n s i n t h e l o c a l o r s y n o p t i c s c a l e wind f o r c i n g (Wunsch and Wimbush, 1977;
Lee,
e t al,
1985).
1977; Duing e t a l ,
The meandering b e h a v i o r has been r e l a t e d t o t h e
t r a n s p o r t v a r i a t i o n s by Duing (1975),
t o c u r r e n t i n s t a b i l i t i e s by N i i l e r and
Mysak (1971), De Soeke (1975), and O r l a n s k i (1969), and t o s h e l f wave modes by S c h o t t and Duing (1976) and Brooks and Mooers (1977). I n 1982-1984,
an i n t e n s e 27 month o b s e r v a t i o n a l experiment was c a r r i e d out
across t h e s t r a i t s a t 27' (STACS) program.
N as p a r t o f t h e S u b t r o p i c a l A t l a n t i c C l i m a t e Studies
The o v e r a l l goal o f t h e experiment was t o d e t e r m i n e t h e most
a p p r o p r i a t e system f o r m o n i t o r i n g t h e t r a n s p o r t f l u c t u a t i o n s and meandering o f t h e current.
But, i n a d d i t i o n , a w e a l t h o f new i n f o r m a t i o n was gathered, u s i n g
t h e PEGASUS c u r r e n t p r o f i l e r s and moored c u r r e n t m e t e r a r r a y s , additional
documentation
and
analysis
of
the
above-mentioned
which a l l o w s phenomena,
Moreover, t h e s e new d a t a s e t s , t h e most complete generated t o date, may be used t o i n i t i a l i z e n u m e r i c a l m o d e l s w h i c h may f u r t h e r a s s i s t i n d e v e l o p i n g an understanding o f F l o r i d a Current behavior.
I t i s t h i s l a s t endeavor w i t h which
t h e c u r r e n t paper i s concerned. I n what f o l l o w s ,
we d e s c r i b e a n u m e r i c a l model o f an i d e a l i z e d F l o r i d a
C u r r e n t and e x p e r i m e n t s i n which we have s t u d i e d 1 ) t h e response o f t h e model
151 c u r r e n t t o f l u c t u a t i o n s i n wind f o r c i n g and 2 ) t h e s t a b i l i t y o f t h e c u r r e n t t o p e r t u r b a t i o n s o f v a r i o u s wavelengths.
Since t h e F l o r i d a C u r r e n t i s one small
component o f an enormous and complex c i r c u l a t i o n system, we cannot a t t h i s stage hope t o model a l l o f i t s observed b e h a v i o r .
But as a s t a r t i n g p o i n t i n s t u d y i n g
F l o r i d a C u r r e n t b e h a v i o r w i t h a n u m e r i c a l model, and p a r t i c u l a r l y because t h e data s e t r e c e n t l y c o m p i l e d f o r t h e 27'
N c r o s s - s e c t i o n i s t h e b e s t a v a i l a b l e , we
choose t o focus o u r a t t e n t i o n on t h e s t r u c t u r e and v a r i a b i l i t y o f a c u r r e n t w i t h the c h a r a c t e r i s t i c s e x h i b i t e d i n t h a t data set.
Here we i l l u s t r a t e t h e model
c u r r e n t b e h a v i o r and g i v e some p r e l i m i n a r y comparison w i t h s t a t i s t i c s from t h e
A more t h o r o u g h comparison w i l l be g i v e n i n a f o r t h c o m i n g
STACS d a t a s e t . paper.
I n S e c t i o n 2, we d e s c r i b e t h e model and a t h e o r e t i c a l l y - b a s e d i n i t i a l i z a t i o n procedure.
We b r i e f l y i l l u s t r a t e t h e t r a n s p o r t response o f t h e model c u r r e n t t o
wind s t r e s s f l u c t u a t i o n s
i n S e c t i o n 3.
I n S e c t i o n 4,
we show how t h e
i n i t i a l i z e d c u r r e n t responds t o presence o f alongstream p e r t u r b a t i o n s o f v a r i o u s wavelengths,
a t t h e same t i m e c o n t r a s t i n g t h i s response w i t h t h a t o f a f l a t
bottom v e r s i o n o f t h e model.
I n S e c t i o n 5, we f i r s t d e s c r i b e an i n i t i a l i z a t i o n
based more on t h e analyzed PEGASUS d a t a ,
which possess a p o t e n t i a l v o r t i c i t y
s t r u c t u r e r a t h e r d i f f e r e n t f r o m t h a t g i v e n by t h e f i r s t procedure.
Experiments
t e s t i n g t h e s t a b i l i t y o f t h e more r e a l i s t i c c u r r e n t a r e t h e n i l l u s t r a t e d .
In
S e c t i o n 6, we summarize. 2 THE MODEL AND INITIALIZATION PROCEDURE 2.1 The numerical model S i n c e western boundary c u r r e n t s a r e c h a r a c t e r i z e d by s u b s t a n t i a l h o r i z o n t a l d e n s i t y g r a d i e n t s which a r e c r u c i a l t o t h e i r s t r u c t u r e , c o o r d i n a t e p r i m i t i v e e q u a t i o n model
recently
we use t h e i s o p y c n i c
d e s c r i b e d by B l e c k and Boudra
Use o f t h i s model g u a r a n t e e s t h a t 1 ) f r o n t a l s t r u c t u r e s w i l l b e
(1986).
o p t i m a l l y resolved f o r a given v e r t i c a l gradients w i l l
r e s o l u t i o n and 2 ) h o r i z o n t a l d e n s i t y
n o t b e smeared o u t w i t h i n t e g r a t i o n t i m e d u e t o l a t e r a l
d i f f u s i o n , which i s g e n e r a l l y t h e case i n a z - c o o r d i n a t e system.
The advantages
of
the
the
isopycnal
coordinate
system
can
be
overwhelmed
by
numerical
d i f f i c u l t i e s a s s o c i a t e d w i t h t h e c o o r d i n a t e s u r f a c e s coming t o g e t h e r w i t h t h e upper s u r f a c e , w i t h each o t h e r and w i t h b o t t o m topography.
The r e c e n t Bleck and
Boudra model, i n c o n t r a s t t o t h e q u a s i - i s o p y c n i c c o o r d i n a t e model o f B l e c k and Boudra
(1981),
algorithm,
collapsing
of
Flux-Corrected
Transport,
developed o r i g i n a l l y by B o r i s and Book
(1973) and extended t o m u l t i - d i m e n s i o n a l
coordinate
layers
using
a
controls
a p p l i c a t i o n b y Zalesak (1979).
special While
space i s n o t a v a i l a b l e h e r e t o g i v e t h e d e t a i l s o f t h e FCT model, we may b r i e f l y d e s c r i b e t h e e s s e n t i a l concept and two a s p e c t s o f t h e model i n t r o d u c e d here.
152
F i r s t o f a l l , t h e a l g o r i t h m i s i n c o r p o r a t e d i n t h e mass c o n t i n u i t y equation, which p r e d i c t s l a y e r t h i c k n e s s .
A h i g h o r d e r scheme, i n t h i s case f o u r t h o r d e r ,
i s u s e d t o e s t i m a t e mass f l u x e s i n t o a n d o u t o f a g r i d b o x p r o v i d e d t h e t h i c k n e s s v a l u e which would r e s u l t f r o m t h e s e e s t i m a t e s i s bounded away from zero.
When t h i s i s n o t t h e case,
t h e e s t i m a t e s a r e combined w i t h t h o s e made
w i t h a f i r s t order schene, through which no values l e s s than zero are generated. I n t h e model c a l c u l a t i o n , t h e e q u a t i o n s a r e i n t e g r a t e d i n f u l l a t massless g r i d p o i n t s as w e l l as t h o s e w i t h g r e a t e r t h a n z e r o l a y e r t h i c k n e s s .
This i s
r e l a t i v e l y t r o u b l e - f r e e i n f l a t b o t t o m c a l c u l a t i o n s when t h e c o o r d i n a t e s u r f a c e s do n o t i n t e r s e c t t h e l o w e r boundary.
For a p p l i c a t i o n s such as t h e F l o r i d a
S t a i t s , w i t h s t e e p l y s l o p i n g topography, a p o t e n t i a l problem i s encountered,
as
more t h a n one l a y e r comes i n c o n t a c t w i t h t h e l o w e r boundary, t h a t i s , unless v e r t i c a l r e s o l u t i o n i s v e r y low. c o i n c i d e a t t h e ocean bottom, potential,
gz + pa,
I f g r i d p o i n t s i n more t h a n one massless l a y e r
the
pressure
force
variable,
t h e Montgomery
which changes v e r t i c a l l y due t o t h e change i n s p e c i f i c
w i l l do so even i n t h e absence o f r e a l f l u i d . Particularly i f this c o n f i g u r a t i o n i s j u s t n e x t t o a r e g i o n where one o f t h o s e l a y e r s has g r e a t e r volume,
than zero thickness,
t h e computation o f horizontal pressure force i n the
momentum e q u a t i o n s w i l l
g i v e an a r t i f i c i a l
result.
Therefore,
f o r c e i s averaged o v e r t h e b o t t o m 30 m e t e r s o f r e a l f l u i d ,
t h e pressure
and t h i s v a l u e i s
assigned t o a l l g r i d p o i n t s i n t h e column c o n f i n e d w i t h i n t h i s l a y e r .
This
problem i s n o t encountered a t t h e upper boundary s i n c e p r e s s u r e i s z e r o there, and, thus,
t h e Montgomery p o t e n t i a l does n o t change f r o m one massless l a y e r t o
t h e next. V e l o c i t y v a l u e s i n massless r e g i o n s o f a l a y e r i n t h e above model can become noisy.
To p r e v e n t t h i s f r o m c a u s i n g n u m e r i c a l p r o b l e m s i n t h e a d j a c e n t
non-zero-thickness
r e g i o n s , we have adopted a w e i g h t i n g o f v e l o c i t i e s which, a t
t h e end o f each t i m e s t e p , r e p l a c e s v e l o c i t i e s a t any g r i d p o i n t w i t h l e s s than
5 m l a y e r t h i c k n e s s b y a 5 m v e r t i c a l average.
I n p a r t i c u l a r , massless g r i d
p o i n t s a t t h e upper and l o w e r boundary a r e r e a s s i g n e d v e l o c i t i e s computed as a mean from t h e 5 m j u s t below and above them,
respectively.
For additional
d e s c r i p t i o n o f t h e model, t h e r e a d e r i s r e f e r r e d t o B l e c k and Boudra (1986).
2.2 Domain shape, boundary c o n d i t i o n s , l a t e r a l f r i c t i o n I n i t s s i m p l e s t form,
t h e model i s c o n f i g u r e d as a two-dimensional
s e c t i o n w i t h t h e approximate b a t h y m e t r y o f t h e F l o r i d a S t r a i t s a t 27'N. F i g . 1.
crossshown i n
A l l g r a d i e n t s p e r p e n d i c u l a r t o t h e c r o s s - s e c t i o n a r e assumed zero.
The
l a t e r a l boundary c o n d i t i o n s a r e n o - s l i p ,
and b o t t o m f r i c t i o n i s i n c o r p o r a t e d
according t o a standard bulk formula
i n a 25 m b o t t o m b o u n d a r y l a y e r .
Formulated as a t h r e e - d i m e n s i o n a l channel, c y c l i c boundary c o n d i t i o n s connect
153 t h e 'ends'
o f t h e channel.
oriented north-south,
A l s o i n t h i s 3-0
form,
t h e channel
i s assumed
b u t t h e v a r i a t i o n o f f i s considered a higher order
e f f e c t , and t h e c o n s t a n t v a l u e t r u e f o r 27'N
i s used.
Internal lateral f r i c t i o n
i s incorporated i n a Laplacian-type term w i t h constant v i s c o s i t y :
*
-
100 k m
The v e r t i c a l i s approximated i n ten
isopycnal
layers
with
an
increment o f .4 uT u n i t s between
-E
I
Florida Straits Cross Section at 27ON
175
consecutive layers. cross-section
I
I 350
I n t h e x-z
the horizontal i s
r e s o l v e d by 51 g r i d p o i n t s w i t h
2 km s p a c i n g and, t h u s , has 100
tw
n 525
km t o t a l w i d t h .
I n the north-
south d i r e c t i o n ,
a t o t a l o f 16
g r i d p o i n t s i s used t o r e s o l v e a p e r t u r b a t i o n wavelength. the
750
north-south
grid
Thus, spacing
v a r i e s as t h e wavelength v a r i e s among t h e experiments.
Fig.1. I l l u s t r a t i o n o f t h e model c r o s s s e c t i o n used t o r e p r e s e n t t h e F l o r i d a S t r a i t s a t 270N.
2.3 I n i t i a l i z a t i o n As
indicated
in
the
Introduction,
two
approaches
have
been
i n i t i a l i z e t h e model w i t h a F l o r i d a C u r r e n t - l i k e mass/flow f i e l d . i s based c l o s e l y on t h e mean s t r u c t u r e o f t h e 27'
taken
to
One o f t h e s e
N c r o s s - s e c t i o n compiled from
t h e PEGASUS c u r r e n t p r o f i l e r measurements d u r i n g t h e STACS o b s e r v a t i o n s and analyzed by Leaman, e t a1 (1987).
Experiments w i t h t h e model i n i t i a l i z e d i n
such a f a s h i o n a r e d e s c r i b e d h e r e i n S e c t i o n 5.
The o t h e r c o n s i s t s o f i n i t i a l
s p e c i f i c a t i o n o f t h e mass f i e l d w i t h an a n a l y t i c f u n c t i o n r e l a t i n g p r e s s u r e and s p e c i f i c volume, f o l l o w e d by a h o r i z o n t a l d e f o r m a t i o n process meant t o s i m u l a t e oceanic f r o n t o g e n e s i s ( H o s k i n s and B r e t h e r t o n , parameters r e p r e s e n t i n g t h e t o p - t o - b o t t o m steepness o f t h e f r o n t , vertical
The f u n c t i o n u s e s
and c r o s s - s t r e a m d e n s i t y change, t h e
i t s p o s i t i o n i n t h e cross-channel d i r e c t i o n ,
scale o f t h e density variation.
flexibility,
1972).
The procedure, t h u s ,
and t h e
allows great
s i n c e one may d e t e r m i n e t h e s t a b i l i t y o f t h e c u r r e n t as t h e s e
parameters a r e v a r i e d .
Space i s n o t a v a i l a b l e h e r e t o g i v e f u l l d e t a i l s .
This
154 w i l l be done i n t h e above forthcoming
mentioned paper.
The
interested
r e a d e r may a l s o f i n d t h e method
detailed
study
of
in
a
mesoscale
frontogenesi s
by
e t a1 (1987).
The i n i t i -
lialized
B1 eck,
cross-section,
well-balanced
with
the
l a t e r a l and bottom bounda r y c o n d i t i o n s and w i t h a downstream
transport
lo6 m3
X
31.7
shown i n F i g u r e 2. after,
Here-
we r e f e r t o t h i s
mass/ f l ow as
of
s-l i s
configuration
our a n a l y t i c
initial
fields. Fig.2. I l l u s t r a t i o n o f t h e a n a l y t i c i n i t i a l f i e l d s . The n o r t h w a r d v e l o c i t y i s c o n t o u r e d u s i n g t h i n s o l i d l i n e s w i t h an i n t e r v a l o f 10 cm s-1. S e l e c t e d i s o t a c h s a r e l a b e l l e d . The p o s i t i o n o f t h e i s o p y c n a l c o o r d i n a t e s u r f a c e s i s i n d i c a t e d by t h e rows o f ' + I s i q n s . The p o t e n t i a l v o r t i c i t y , av + Lf!a x ' i s c o n t o u r e d i n heavy s o l i d ( a ~ ?t ap ap l i n e s a t i n t e r v a l s o f 1 x 10-15 cm-2 s . +
f,
av
3 MODEL RESPONSE TO FLUCTUATING WIND R e c e n t l y , Lee, e t a1 (1985) and S c h o t t and Lee (1986) have found s i g n i f i c a n t coherence between F l o r i d a S t r a i t s t r a n s p o r t and l o c a l wind s t r e s s i n c e r t a i n e n e r g e t i c wind bands, n o t a b l y p e r i o d s o f 3.5 days and 20 days. that,
i n o u r model,
i f f r i c t i o n i s negligible,
application
It can be shown o f a fluctuating
n o r t h - s o u t h wind s t r e s s should y i e l d a b a r o t r o p i c t r a n s p o r t response which i s p r o p o r t i o n a l t o t h e a m p l i t u d e and p e r i o d o f t h e f o r c i n g and has a 90' t o t h e wind. from 90'
phase l a g
As f r i c t i o n i s i n c r e a s e d , t h e phase l a g s h o u l d g r a d u a l l y decrease
and t h e r e l a t i o n s h i p between f o r c i n g a m p l i t u d e and p e r i o d and t h e
response a m p l i t u d e should f a l l f u r t h e r below s t r i c t p r o p o r t i o n a l i t y . To d e t e r m i n e t h e a c t u a l model b e h a v i o r i n t h i s regard, we p e r f o r m experiments
155 w i t h t h e a n a l y t i c i n i t i a l f i e l d s i n which downstream wind s t r e s s i s a p p l i e d i n a l i n e a r l y d e c r e a s i n g f a s h i o n o v e r t h e upper 100 m o f t h e model, as i n Bleck and Boudra (1981, 1986).
I n t h i s case, t h e s t r e s s has no h o r i z o n t a l v a r i a t i o n , b u t
i s a s i n u s o i d a l f u n c t i o n o f time.
A c o n s t a n t e a s t e r n boundary c o n d i t i o n on t h e
mass t r a n s p o r t s t r e a m f u n c t i o n ,
used i n i n i t i a l i z a t i o n ,
i s relaxed.
Mean
6 3 t r a n s p o r t i s m a i n t a i n e d a t a p p r o x i m a t e l y 30 X 10 m s - l by adding a c o n s t a n t s t r e s s of a p p r o x i m a t e l y .2 X In m2 s - ~t o t h e above f l u c t u a t i n g component. these experiments,
t h e e a s t e r n boundary c o n d i t i o n f o r t h e s t r e a m f u n c t i o n i s
computed i n a manner a n a l o g o u s t o t h a t u s e d f o r i s l a n d s b y B r y a n ( 1 9 6 9 ) , according t o t h e methodology developed by Kamenkovitch (1962). The t r a n s p o r t response o f t h e model c u r r e n t t o f l u c t u a t i n g wind i s t e s t e d by successively doubling t h e f o r c i n g p e r i o d w h i l e holding t h e f o r c i n g amplitude constant a t boundaries,
.5
.
m 2 s-'
X
an i n t e r n a l
Our e x p e r i m e n t s w i t h f r e e - s l i p
l a t e r a l f r i c t i o n c o e f f i c i e n t o f 10 m2 s-',
lateral and zero
bottom d r a g e x h i b i t an approximate d o u b l i n g o f t h e b a r o t r o p i c t r a n s p o r t response and a 90'
phase l a g between t h e wind and t h e response, as t h e above argument
implies.
B a r o c l i n i c response, d e f i n e d as t h e t r a n s p o r t above 200 m minus t h a t
below 200 m,
shows a s i m i l a r r e l a t i o n s h i p w i t h t h e wind b u t i s an o r d e r o f
magnitude s m a l l e r . When
a
drag
of
coefficient
.003 f o r
the
bottom
boundary
layer
is
i n c o r p o r a t e d , a l o n g w i t h an i n t e r n a l l a t e r a l v i s c o s i t y o f 65 m2 s-l and n o - s l i p i l l u s t r a t e d i n Table 1, i s l e s s t h a n s t r i c t l y
l a t e r a l boundaries, t h e response,
p o r p o r t i o n a l t o t h e f o r c i n g p e r i o d , as expected.
However, t h e phase d i f f e r e n c e
between t h e f o r c i n g and response decreases o n l y s l i g h t l y f r o m 90'. TABLE 1 Model
Florida
stress.
Current
response
to
Forcing Period (Days)
sinusoidally
f l u c t u a t i n g downstream wind
m2 s-'.
F o r c i n g A m p l i t u d e = .5 X
Response Amp1 it u d e
( l o 6 m3 s-l)-
~
8 16 32 64 128
.55
1.00 1.82
3.4 6.37
The f o r c i n g / r e s p o n s e phase l a g i n t h e STACS d a t a i s 90' days, b u t o n l y 20'
a t t h e 20 day p e r i o d (Lee,
about 1 day l a g i n b o t h cases.
e t al,
a t a p e r i o d o f 3.5
1985).
This represents
I t seems l i k e l y t h a t t h i s p h a s e l a g i s
determined by f a c t o r s o t h e r t h a n f r i c t i o n : f o r i n s t a n c e , r e s i d e n c e t i m e o f f l u i d i n t h e S t r a i t s v e r s u s t h e p e r i o d and s p a t i a l s c a l e o f t h e f o r c i n g .
With our
156 s i m p l e model geometry we a r e l i m i t e d t o d e s c r i b i n g t h e response o f a continuous channel b e i n g f o r c e d u n i f o r m l y .
4 GROWTH OF MEANDERS Coherent, e n e r g e t i c meandering s i g n a l s i n t h e F l o r i d a C u r r e n t a t p e r i o d s o f a p p r o x i m a t e l y 5 and 12 days have been d e t e c t e d by Johns and S c h o t t (1987) through
frequency-domain
observations. o r near 27'
empirical
mode
analysis
of
STACS
current
meter
The c u r r e n t meters were moored i n an a r r a y a c r o s s t h e s t r a i t s a t N from December 1983 t o June 1984.
T h e i r a n a l y s i s suggests t h a t 1
t h e s e meanders have downstream p r o p a g a t i o n speeds and wavelengths o f 36 km d170 km and 28 km d-',
340 km, r e s p e c t i v e l y .
They f i n d no s t r o n g c o r r e l a t i o n
between meandering and t o t a l t r a n s p o r t f l u c t u a t i o n s , are r e l a t i v e l y unrelated. w i t h t h e s e modes,
,
s u g g e s t i n g t h a t t h e two
On examining t h e energy c o n v e r s i o n t e r m s a s s o c i a t e d
t h e y conclude t h a t t h e meanders a r e g i v i n g up energy t o t h e
mean f l o w t h r o u g h b a r o t r o p i c c o n v e r s i o n . i n d i c a t e s small p o s i t i v e c o n v e r s i o n
-
The b a r o c l i n i c c o n v e r s i o n t e r m
f r o m t h e mean t o t h e e d d i e s
magnitude i s a p p a r e n t l y i n s i g n i f i c a n t i n v i e w o f t h e e r r o r bars.
-
but i t s
Also, t h e f l u x
terms c a l c u l a t e d f r o m moored c u r r e n t meter d a t a a r e d e f i c i e n t because t h e t o p 100 m,
where much o f t h e energy c o n v e r s i o n t a k e s p l a c e ,
i s n o t covered by
instrumentation. I n a c o m p u t a t i o n o f mean t o eddy energy c o n v e r s i o n s u s i n g t h e f u l l s e t o f PEGASUS data, t h e r e i s a l s o l i t t l e e v i d e n c e t h a t t h e F l o r i d a c u r r e n t a t 27' unstable.
I n t h i s analysis,
N is
however, Leaman, e t a1 (1987) o b t a i n such a small
n e t c o n v e r s i o n o f energy between t h e mean f l o w and t h e e d d i e s i n b o t h b a r o t r o p i c and b a r o c l i n i c terms t h a t t h e y conclude t h a t t h i s c o n v e r s i o n i s o f i n d e t e r m i n a t e sign.
The p o s s i b i l i t y t h u s e x i s t s t h a t t h e c u r r e n t i n t h e s t r a i t s i s a t t h e
t h r e s h o l d o f i n s t a b i l i t y and t h a t t h i s i s o c c a s i o n a l l y m a n i f e s t e d i n conversion o f energy i n t o meanders w i t h wavelengths such as t h o s e r e p o r t e d by Johns and S c h o t t (1987). I n t h i s s e c t i o n , we w i s h t o d e t e r m i n e whether t h e c u r r e n t i n o u r a n a l y t i c initial
fields exhibits
t e n d e n c i e s t o t r a n s f e r energy i n t o meanders through
b a r o t r o p i c o r b a r o c l i n i c conversion.
Because o f t h e c o m p l e x i t i e s i n v o l v e d i n
t h e i n c o r p o r a t i o n o f downstream v a r i a t i o n s i n channel w i d t h and b o t t o m topography,
we have chosen t o e x c l u d e t h o s e v a r i a t i o n s
mentioned e a r l i e r , t h e i n i t i a l i z e d c r o s s - s e c t i o n t h e downstream d i r e c t i o n t o c r e a t e t h e channel, g r i d points.
for
t h e t i m e being.
As
o f Figure 2 i s duplicated i n
so t h a t t h e r e a r e 16 downstream
The s t a b i l i t y o f t h i s c u r r e n t t o p e r t u r b a t i o n s w i t h wavelengths o f
96 t o 256 km has been t e s t e d by v a r y i n g t h e downstream g r i d s p a c i n g from 6 km t o 16 km a t 2 km i n t e r v a l s .
A t some o f t h e l o n g e r wavelengths,
t h e number o f down-
stream g r i d p o i n t s has been d o u b l e d t o v e r i f y t h a t 16 downstream g r i d p o i n t s i s sufficient.
The solutions show l i t t l e s e n s i t i v i t y t o t h i s change i n resolution.
157 The i n i t i a l p e r t u r b a t i o n i s s p e c i f i e d o n l y i n t h e c r o s s - s t r e a m v e l o c i t y f i e l d a n d has a s i n u s o i d a l v a r i a t i o n i n t h e downstream d i r e c t i o n . the
perturbation
The a m p l i t u d e o f
is
a
velocity
field
o b t a i n e d a t some a r b i t r a r y t i m e d u r i n g t h e i n i t i a l i z a t i o n procedure. s e n t s a cross-channel lack
of
perfect
It r e p r e -
'sloshing'
geostrophic
due t o balance
between t h e i n i t i a l mass and f l o w f i e l d s ,
16.1
as
mys
well
as
l a t e r a l and
their
adaptation
to
the
b o t t o m boundary c o n d i t i o n s .
The p e r t u r b a t i o n i s t h u s a f u n c t i o n o f t h e v e r t i c a l and c r o s s - s t r e a m d i r e c t i o n s and
exhibits
velocities
m s-l a t t h e
a p p r o x i m a t e l y f r o m -.17 upper
surface
to
bottom.
For
eastern
boundary
m
t.23
these
varying
s-l a t t h e
experiments,
condition
for
the mass
t r a n s p o r t streamfunction i s h e l d constant Top 30 m mean f l o w and denExperiment w i t h f l a t bottom. wide channel, and p e r t u r b a = 208 km. D e n s i t y C . I . = .1 u F u l l l e n g t h a r r o w and each addiTiona1 b a r b = 25 cm s-1 speed. Arrows p l o t t e d e v e r y t h i r d p o i n t i n x-direction, every y-point. Fig.3. sity. 100 km tion b
.
a t 31.7
X
physically
lo6 m3 s-'. related
T h i s can be
to
the
downstream
p r e s s u r e head y i e l d e d b y an upper s u r f a c e downstream s l o p e o f a p p r o x i m a t e l y 1 cm km-'. Before
illustrating
the results
for
the Florida Straits, i t i s instructive t o p o i n t out t h e behavior o f a current i n i t i a l i z e d i n t h e same manner as d e s c r i b e d above i n a channel o f 100 km w i d t h b u t It i s found t h a t such a c u r r e n t i s u n s t a b l e t o wavelengths
w i t h a f l a t bottom.
of g r e a t e r t h a n a p p r o x i m a t e l y 60 km and t h a t t h e maximum p e r t u r b a t i o n growth r a t e i s a t a wavelength o f 208 km. n o t i c e a b l y a f t e r s e v e r a l days,
The a m p l i t u d e o f t h i s meander has i n c r e a s e d
and t h e upper 30 m f l o w p a t t e r n a t 16.1
days
( F i g u r e 3) e x h i b i t s a s t r o n g c y c l o n i c eddy t o t h e l e f t o f t h e c u r r e n t core, which i s d r a i n i n g energy from t h e a v a i l a b l e p o t e n t i a l energy f i e l d ( F i g u r e 4a). The e n e r g y c o n v e r s i o n t e r m s f o r t h i s m o d e l , subsequently
a p p l i e d t o an ocean model
d e r i v e d i n B l e c k (1985) and
i n t e r c o m p a r i s o n by B l e c k and Boudra
(1986) can be w r i t t e n as f o l l o w s f o r c o n v e r s i o n between p o t e n t i a l and alongstream (i.e.,
z o n a l ) mean and p e r t u r b a t i o n k i n e t i c e n e r g i e s :
+' *ap
P->KE = - V P->KM =
*
ap
.
MP p,
KE->KM = $' -?I? ($ ap
MP, Montgomery P o t e n t i a l
V,MP
-: 3 v
. vp)?
( - ) a1 ong-stream mass-wei ghted average ( ) ' departure from (-).
158 The c o n v e r s i o n from potent i a l t o eddy k i n e t i c i s t h e one a s s o c i a t e d w i t h baroc l i n i c c o n v e r s i o n , and t h a t from mean k i n e t i c t o eddy kinetic
with
conversion. energy
barotropic The graph o f
conversion
time (Fig.
4b)
versus
shows t h a t
even a t t h e s t a r t o f t h i s experiment t h e r e i s a small P
to
which
KE,
slightly
less
than P t o
KM.
conversion
of
is
magnitude This l a t t e r
is
normally
p o s i t i v e and r e s t o r e s mean k i n e t i c energy l o s t through
As t h e eddy
dissipation.
b e g i n s t o grow r a p i d l y from 8 t o 12 d a y s ,
conversion
baroclinic
climbs
sharply
w h i l e t h e P t o KM changes s i g n due t o t h e l o s s o f potential
energy.
Baro-
t r o p i c conversion likewise becomes s t r o n g l y negative. This scenario s i g n i f i e s the release o f baroclinic ins t a b i l i t y o f the current. By t h e end o f t h i s exp e r i m e n t , t h e energy conversion 4b), TIME (DAYS)
has
peaked
(Fig.
but during the last
several
days,
the
mean
p o t e n t i a l energy has drop4 ped f r o m 7.2 X 10 t o 2.5 X Fig.4. (a)Mean and eddy p o t e n t i a l and k i n e t i c e n e r g i e s , averaged o v e r t h e channel as a funct i o n o f t i m e f o r t h e X = 208 km f l a t bottom (b)Energy c o n v e r s i o n r a t e s as a experiment. f u n c t i o n o f t i m e f o r t h e same experiment.
lo4 J
m-2.
Thus, t h e basic
b a r o c l i n i c s t r u c t u r e o f the channel
has
been
159 s u b s t a n t i a l l y modified.
T h i s suggests t h a t a c u r r e n t w i t h many o f t h e same b u l k
parameters as t h e F l o r i d a C u r r e n t i s b a r o c l i n i c a l l y u n s t a b l e i n a f l a t bottom channel w i t h t h e w i d t h o f t h e F l o r i d a S t r a i t s .
The p e r t u r b a t i o n generated i n
s t a b i l i z i n g t h e f l o w ( F i g . 3) i s much l a r g e r t h a n any meander e v e r observed i n the F l o r i d a S t r a i t s .
27'
It seems u n l i k e l y , t h e n , t h a t t h e model c u r r e n t w i t h t h e
N bottom topography w i l l e x h i b i t such s t r o n g i n s t a b i l i t y . In fact,
t h e model w i t h b a t h y m e t r y does e x h i b i t
through t h e b a r o c l i n i c c o n v e r s i o n term. o f t h e wave g r o w t h a r e more l i m i t e d .
meander growth,
primarily
But t h e h o r i z o n t a l and temporal s c a l e s
I n a d d i t i o n , t h e g r o w t h i s slower.
It i s
a l s o found t h a t t h e g r o w t h r a t e i s s t r o n g l y dependent on t h e t o t a l cross-channel d e n s i t y change a t t h e s u r f a c e ,
a parameter
s p e c i f i c a t i o n which determines,
t o a l a r g e degree,
current.
the baroclinicity o f the
Since t h e surface l a y e r o f t h e F l o r i d a Current i s h o r i z o n t a l l y
well-mixed,
t h e t o t a l e a s t t o west s u r f a c e d e n s i t y i n c r e a s e i n t h e STACS d a t a
(Leaman, e t a l ,
1987) i s b u t .2 uT u n i t s .
t o west i s 1.2 u n i t s . i n t e r m e d i a t e one, depth.
used i n t h e i n i t i a l mass f i e l d
A t 50 m d e p t h t h e i n c r e a s e from e a s t
The v a l u e chosen f o r t h e experiments d e s c r i b e d h e r e i s an
.7 u n i t s ,
c o r r e s p o n d i n g t o t h e a c t u a l v a l u e a t 30 t o 40 m
Values o f .3 o r l e s s l e a d t o v e r y l i t t l e meander growth.
Parameters d e s c r i b i n g t h e i n s t a b i l i t y as a f u n c t i o n o f wavelength a r e g i v e n i n Table 2.
It i s found t h a t a g a i n t h e most u n s t a b l e wavelength i s 208 km.
meander has a p e r i o d o f a p p r o x i m a t e l y 8 days.
The
I n g e n e r a l , more t i m e i s r e q u i r e d
f o r t h e meander t o r e a c h maximum a m p l i t u d e as wavelength i s increased.
The
maximum i n P o c c u r s b e f o r e o r a t t h e same t i m e as t h e KE maximum. The maximum E KE t o KM c o n v e r s i o n g e n e r a l l y o c c u r s a t t h e same t i m e o r s h o r t l y a f t e r t h e maximum P t o KE c o n v e r s i o n . TABLE 2 Energy and energy c o n v e r s i o n Energy i s i n u n i t s o f J m-'
as a f u n c t i o n o f meander wavelength and energy c o n v e r s i o n i n J m-'
s-'.
days. KE A ( km)
96 128 160 192 208 224 256
4.3 4.9 6.6 7.7 8.4 6.3 5.8
KE Time
pE
7.3 9.2 11.2 12.2 15.1 17.0 17.1
2.6 3.0 3.25 3.9 4.4 3.4 3.3
PE
P t o KE
T i me
6.3 6.8 8.8 12.2 15.1 17.0 17.1
P t o KE
KE t o ,K ,,
KE t o ,,K, Time
,007 .013 .016 .019 .028 .017 .007
8.8 7.8 9.8 13.8 14.6 15.6 17.1
T i me
.039 .04 .052 .07 .074 ,059 .045
6.8 8.3 9.8 12.2 14.2 15.6 17.1
(A).
Time i s i n
160 The upper l a y e r f l o w p a t t e r n f o r t h e 208 km wavelength s h o r t l y a f t e r t h e t i m e o f i t s maximum a m p l i t u d e ( F i g . 5) shows a g a i n a c y c l o n i c eddy t o t h e l e f t o f t h e c u r r e n t c o r e b u t w i t h considerably l e s s s p a t i a l e x t e n t than i n the f l a t b o t t o m case.
F u r t h e r , t h e graphs o f
energy and energy c o n v e r s i o n vs.
t i m e (Fig.
6), while they implicate release o f baroclinic
l7,l
instability
DAYS
meander growth,
as
the
physical
of
mechanism
show t h a t t h e mean p o t e n t i a l
and k i n e t i c energy a r e l i t t l e a f f e c t e d by t h e e v e n t o f eddy growth. is
released t h e
After the instability
baroclinic
conversion
term
d e c r e a s e s t o a p p r o x i m a t e l y t h e same v a l u e as at
the
initial
conversion term The system i s ,
time rises
and
therefore,
m a r g i n a l l y unstable.
the
slightly
barotropic above
zero.
one which i s o n l y
Events o f meander growth
l e a v e t h e mean mass/flow s t r u c t u r e r e l a t i v e l y Fig.5. 4s i n Fig.3, b u t f o r t h e experiment w i t h t h e bottom topography i l l u s t r a t e d i n Fig.1.
unchanged.
5 GROWTH OF MEANDERS USING THE STACS ANALYZED DATA 5.1 Development o f i n i t i a l c o n d i t i o n s W i t h i n t h e c o n t e x t o f t h e a n a l y t i c a l l y d e r i v e d F l o r i d a C u r r e n t , as described above,
meander
growth
i n f l u e n t i a l physical
has
been
investigated
as
a
function
of
the
most
Space 1 i m i t a t i o n s p r e v e n t us f r o m d e t a i l i n g
parameters.
a l l o f t h i s e x p e r i m e n t a t i o n here.
The experiments d e s c r i b e d i n t h e previous
s e c t i o n were i n i t i a l i z e d u s i n g t h e parameter
v a l u e s w h i c h d e v e l o p t h e most
r e a l i s t i c l o o k i n g c r o s s - s e c t i o n w i t h r e s p e c t t o t h e STACS d a t a (Leaman, e t a l , 1987).
A f e a t u r e which c o u l d n o t be e a s i l y i n c l u d e d i n t h a t i n i t i a l f i e l d i s
t h e h o r i z o n t a l l y well-mixed surface layer,
which i s perhaps 30 m t h i c k a t t h e
western boundary and 60 t o 70 m a t t h e e a s t e r n boundary.
I n addition,
the
e a s t e r n boundary ( t h e s t e e p s l o p e o f L i t t l e Bahama Bank) has a SSE t o NNW t i l t i n the Straits, maximum
in
f e a t u r e cannot any
rate,
which l e a d s t o
northward be
these
easily special
c o n v e r g i n g f l o w a t mid-depth and a subsurface
velocity
near
included i n
our
characteristics
t h a t boundary. analytic give
Likewise,
initial
the real
this
conditions.
Florida
Current
At
a
161 rather
different
vorticity that
potential
structure
illustrated
from
by
the
t h i c k s o l i d l i n e s o f Figure 2 f o r our a n a l y t i c a l i n i t i a l fields.
Since t h e change i n
sign
of
horizontal
and
vertical
gradients
of
potential
vorticity
are
g e n e r a l l y considered c r u c i a l factors i n the s t a b i l i t y o f c u r r e n t s , i t seems warranted t o explore the s t a b i l i t y o f a model c u r r e n t w i t h a more r e a l is t ic TIME (DAYS)
p o t e n t ia 1
v o r t i c i t y structure. We
begin development o f
t h e i n i t i a l f i e l d f o r these experiments analyzed
with
the
density
vs.
p r e s s u r e data o f Leaman, e t a1 (1987).
The t h i c k n e s s o f
density layers representing increments
of
determined
through
.4
aT
is
1i n e a r
i n t e r p o l a t i o n , given d e n s i t y analyzed
at
i n t e r v a l s. channel
dbar
10
The
spacing
cross-
of
their
data a n a l y s i s i s 1.878
km,
which becomes t h e new model g r i d p o i n t spacing.
In this
case, t h e g r i d p o i n t s on t h e TIME (DAYS)
western s i d e o f t h e s t r a i t s
with bottom Fig.6. As i n Fig.4, b u t f o r t h e experiment w i t h t h e bottom topography o f Fig.1.
less
than
pressure
90
dbar
have
been
discarded, so t h a t t h e t o t a l straits
width
is
82.6 km.
162 The
analyzed
surface
bottom l a y e r assumed
observed
velocity
minus
velocity
is
geostrophical l y
balanced w i t h t h e s u r f a c e Montgomery
p o t e n t i a1
gradient.
This
velocity
d i f f e r e n c e i s a l s o specified
for
velocity
the
top
layer
initially
and
t h a t i n t h e remainder o f the
column
is
computed
i n t e g r a t i n g downward u s i n g t h e thermal wind r e l a t i o n . The model c r o s s - s e c t i o n i s then
integrated
days
of
for
simulated
ten time
w i t h t h e e a s t e r n boundary condition
on
the
mass
transport
streamfunction 6 3 O f 31.7 lo s-l and the same lateral and
Fig.7. As i n Fig.2, b u t f o r t h e i n i t i a l f i e l d s developed f r o m t h e STACS analyzed d a t a . P o t e n t i a l v o r t i c i t y i s c o n t o u r e d a t i n t e r v a l s o f 5 X 10-15 S.
b o t t o m boundary c o n d i t i o n s as used i n S e c t i o n 4.
An
average o f t h e downstream v e l o c i t y and p r e s s u r e f i e l d s o v e r t h e f i n a l 200 t i m e s t e p s y i e l d s a w e l l - b a l a n c e d mass/flow c o n f i g u r a t i o n ( F i g .
7),
which e x h i b i t s
t h e e s s e n t i a l s t r u c t u r e o f t h e STACS mass and v e l o c i t y a n a l y s i s , as w e l l as t h e p r i m a r y p o t e n t i a l v o r t i c i t y tongue, which r e s u l t s from t h e s t r o n g s t r a t i f i c a t i o n a t t h e base o f t h e s u r f a c e mixed l a y e r . 5.2 S t a b i 1it y t e s t As w i t h t h e a n a l y t i c a l l y d e r i v e d c u r r e n t , a c r o s s - s t r e a m v e l o c i t y f i e l d from t h e i n i t i a l two-dimensional
i n t e g r a t i o n i s saved t o p r o v i d e t h e a m p l i t u d e f o r
t h e s i n u s o i d a l p e r t u r b a t i o n used f o r t h e s t a b i l i t y t e s t .
The maximum values i n
t h i s f i e l d a r e somewhat l e s s t h a n i n t h a t used above, b u t a r e c o n s i d e r e d l a r g e enough f o r t h e c u r r e n t purpose.
The s t a b i l i t y c h a r a c t e r i s t i c s o f t h i s c u r r e n t
as a f u n c t i o n o f wavelength a r e summarized i n TABLE 3.
It i s found t h a t t h i s
more r e a l i s t i c c u r r e n t i s n o t i c e a b l y b a r o c l i n i c a l l y u n s t a b l e t o p e r t u r b a t i o n s w i t h wavelength g r e a t e r t h a n 60 km, as i n t h e p r e v i o u s case.
More t h a n t w i c e as
much t i m e i s r e q u i r e d f o r t h e meanders t o reach peak a m p l i t u d e t h a n i n t h e
163 previous
case.
T h i s c o u l d be an i n h e r e n t c h a r a c t e r i s t i c o f t h e dynamical
d i f f e r e n c e between t h e c u r r e n t s t r u c t u r e s o r i t c o u l d be r e l a t e d t o t h e amplitude and s t r u c t u r e o f t h e i n i t i a l p e r t u r b a t i o n . b e f o r e a c o n c l u s i o n can be reached.
More s t u d y i s r e q u i r e d
I n Table 3, t h e t r e n d s w i t h r e s p e c t t o
wavelength a r e n o t as c o n s i s t e n t as w i t h t h e p r e v i o u s d a t a s e t .
Notably, w h i l e
t h e meander r e a c h i n g g r e a t e s t a m p l i t u d e i n eddy p o t e n t i a l and k i n e t i c energy i s
150 km, t h e maximum growth r a t e i s a t a wavelength o f 120 km.
Significantly,
t h e 150 km wavelength meander has a p e r i o d o f a p p r o x i m a t e l y 5 days, g i v i n g i t about t h e same s p a t i a l and temporal s c a l e as t h e s h o r t e r o f t h e two meanders found i n t h e moored c u r r e n t meter d a t a by Johns and S c h o t t (1987). TABLE 3 Energy and c o n v e r s i o n peaks f o r t h e c u r r e n t i n i t i a l i z e d f r o m t h e STACS a n a l y s i s . U n i t s a r e as i n Table 2. A(km)
90 120 150 180 210
KE
KE T i me
8.2 10.8 11.3 7.8 7.93
PE
pE
P t o KE
P t o KE
Time
16.6 22.6 28.7 29.6 27.9
2.7 3.9 3.98 3.13 2.89
KE t o KM
Time
15.8 22.6 28.7 29.6 27.9
.07 .081 .046 .035 .032
KE t o KM Time
15.8 19.2 28.0 24.4 29.4
.035 .042 .024 .012 .015
16.7 20.0 28.7 28.0 28.2
The g r a p h s o f mean and p e r t u r b a t i o n energy and energy c o n v e r s i o n f o r t h e 150 km wavelength ( F i g .
8 ) show t h a t t h e p e r t u r b a t i o n e n e r g i e s i n i t i a l l y decrease
w h i l e t h e P t o KE and KM t o KE c o n v e r s i o n s a r e n e a r zero.
A t approximately 8
days, b o t h c o n v e r s i o n s b e g i n t o r i s e and t h e p e r t u r b a t i o n energy begins t o grow. The b a r o t r o p i c
c o n v e r s i o n t o t h e e d d i e s b r i e f l y r i s e s above t h e b a r o c l i n i c
c o n v e r s i o n a n d t h e n b e g i n s t o f a l l a t 13 d a y s .
There a r e two peaks i n
b a r o c l i n i c c o n v e r s i o n c o r r e s p o n d i n g t o peaks i n n e g a t i v e b a r o t r o p i c c o n v e r s i o n as w e l l , t h e second o f which i s l a r g e s t . encountered i n t h e p r e v i o u s case. t o t h e end o f t h e experiment, peaks w i l l
f o l l o w these.
T h i s double maximum i s s i m i l a r t o one
Because o f t h e p r o x i m i t y o f t h e f i n a l peaks
i t would be p r e m a t u r e t o s t a t e t h a t no f u r t h e r
However,
b o t h p e r t u r b a t i o n e n e r g i e s have begun t o
decrease a t t h e end, f o r t h e f i r s t t i m e s i n c e t h e i r i n i t i a l f a l l o f f .
For t h e
c u r r e n t purposes,
i t i s f e l t t o be s u f f i c i e n t t o e x p l o r e t h e i n i t i a l s t a b i l i t y
o f t h i s current.
Study o f i t s l o n g t e r m s t a b i l i t y i s saved f o r f u t u r e work.
i n t h e p r e v i o u s case, growth,
As
t h e PM and KM remain e s s e n t i a l l y unchanged d u r i n g wave
and t h e i n i t i a l
c u r r e n t can be c l a s s i f i e d a g a i n as o n l y m a r g i n a l l y
unstable. The upper 40 m f l o w p a t t e r n f o r t h e experiment w i t h meander wavelength 150 km a t the time
o f maximum
amplitude e x h i b i t s
a somewhat
e l o n g a t e d c y c l o n i c eddy
164 j u s t west o t t h e c u r r e n t core i n i t s c y c l o n i c t u r n ( F i g . 9). -1
i
/ -
- ===% mean-
8ll82-
80181.
791m 70179-
77178-
76177-
75h674175-
73174-
----'
----
===7 -u 74p3
73p4
--
72/73
72\73. 11h2-
70l7t
---> ill72
east
north
fig. 6: Spectra of the east and north wind stress components over the
model area west of Ireland. Considered winter seasons: 1970/71
-
1981/82
236
be
n o t i c e d t h a t i n t h i s case t h e s t a n d a r d d e v i a t i o n i s determined d e t e r m i n i s t i -
c a l l y due t o t h e p r e s c r i b e d f o r c i n g and subsequent f i l t e r i n g and t h a t i t
should
n o t be understood i n t h e s o l e l y s t a t i s t i c a l sense. 3.4.1 INTRA-ANNUAL VARIABILITY Statistical
analysis
o f a i r - p r e s s u r e and wind s t r e s s f i e l d s i n m i d and
l a t i t u d e s has shown t h a t e n e r g e t i c f l u c t u a t i o n s a r e e v i d e n t i n t h e p e r i o d F i g u r e 5 and 6,
between
20 and 80 days (Madden and J u l i a n 1972).
spectra
o f t h e wind s t r e s s components a t a p o i n t i n t h e c e n t r a l N o r t h
west o f I r e l a n d ,
low range
representing Sea
and
g i v e an example o f these e n e r g e t i c f l u c t u a t i o n s f o r t h e w i n t e r
months ( O c t o b e r - A p r i l ) .
The o v e r a l l mean spectrum,
however,
does n o t
exhibit
these s i g n a l s . I t i s almost w h i t e ( W i l l e b r a n d t 1978). The assumption t h a t s i m i l a r f l u c t u a t i o n s may e x i s t f o r t h e f l o w f i e l d i s near The s p e c t r a o f t h e t r a n s p o r t t h r o u g h a s e c t i o n a t Dover and a t Faeroe-
a t hand.
Shetland c o r r o b o r a t e t h i s c o n j e c t u r e ( f i g u r e 7 and 8 ) . in
t h e same p e r i o d range.
not
They show d i s t i n c t peaks
The mean s p e c t r a averaged o v e r a l l w i n t e r seasons do
e x h i b i t any v a r i a b i l i t y i n t h e low f r e q u e n c y domain as i n t h e case
of
the
wind s t r e s s s p e c t r a . 3.4.2
INTER-ANNUAL VARIABILITY analysis
An
seasonal
o f t h e i n t e r - a n n u a l v a r i a b i l i t y was c a r r i e d o u t by
determining
means o f t h e f l o w f i e l d ( d i r e c t i o n and magnitude) and seasonal
l i e s o f t h e k i n e t i c energy d i s t r i b u t i o n ( a c t u a l mean each o f t h e 14 y e a r s . tribution
of
-
anoma-
c l i m a t o l o g i c a l mean)
The k i n e t i c energy i s here g i v e n as 0.5(u2+ v').
mass was n e g l e c t e d i n o r d e r t o a v o i d t h e d e s c r i p t i o n
for
The d i s -
of
spatial
v a r i a n c e s which a r e o n l y caused b y v a r y i n g water depths. F i g u r e 9 and 10 show examples o f a c t u a l c i r c u l a t i o n p a t t e r n s , demonstrated by the
d i r e c t i o n o f t h e f l o w f i e l d and t h e k i n e t i c energy anomaly.
F i g u r e 11 de-
monstrates t h e c o r r e s p o n d i n g c l i m a t o l o g i c a l means w i t h r e g a r d t o a comparison. During "normal" kinetic
spring
1974
( f i g u r e 9 ) t h e c i r c u l a t i o n was e x a c t l y r e v e r s e
d i r e c t i o n i n t h e s o u t h e r n and c e n t r a l N o r t h Sea.
The anomaly
energy e x h i b i t s l o w v a l u e s w i t h i n t h e whole N o r t h Sea,
to
the
of
the
mainly negative
i n t h e e a s t e r n p a r t s and t h e Orkney- Shetland r e g i o n and p o s i t i v e i n t h e western p a r t s . The alt'ogether low v a l u e s o f t h e anomaly g i v e r i s e t o t h e c o n c l u s i o n t h a t t h e magnitude o f t h e "abnormal" r e v e r s e c i r c u l a t i o n i n s p r i n g 74 was tely
of
t h e same o r d e r as t h e c l i m a t o l o g i c a l mean.
This r e s u l t i s a
approximad i s t nct
example t h a t t h e e f f e c t o f wind and a i r - p r e s s u r e can by a l l means superpose
the
( d e n s i t y and t i d a l induced) r e s i d u a l f l o w and become t h e dominant f o r c i n g o f t h e circulation. The s p r i n g 1979 ( f i g u r e 1 0 ) shows an enhancement o f t h e mean c i r c u l a t i o n . Po-
237
I
I
I
mean-
81182-
aolal7SpO-
-
18\79
nm78177-
n(n7475q74-
74n-
np2971-
ss(70-
dsa-.
hours
f i g . 7: Spectra o f t h e t r a n s p o r t r a t e s through section Dover. Considered w i n t e r seasons: 1968/69-1981/82
hours
f i g . 8: Spectra o f t h e t r a n s p o r t r a t e s through section FaeroeShetland. Considered w i n t e r seasons: 1968/69-1981/82
238
f i g . 9: a ) D i r e c t i o n o f the f l o w f i e l d i n s p r i n g 1974 ( d e p t h mean f l o w ) . b ) Anomaly o f the kinetic e n e r g y i n s p r i n g 1974 ( d e p t h mean f l o w ) ;
units: cm2s-2
239
f i g . 10: a) Direction of the flow f i e l d i n spring 1979 (depth mean flow). b) Anomaly of the kinetic energy in spring 1979 (depth mean flow); units : cm2
fig. 11: Climatological mean: spring (depth mean flow). a) Direction of the flow field b) Kinetic energy
241
sitive values of the kinetic energy anomaly can be almost found in the whole North Sea. In particular in the eastern parts the anomaly reaches very high values . The spring 74 as well as the spring 79 are only two examples of abnormal circulation patterns. However during all considered years one can find similar anomalies which are more or less pronounced. A detailed description of these patterns is given in Hainbucher et. al. (1986).
4. FINAL REMARKS The good stability properties of the scheme allow to apply the horizontal momentum diffusion terms in their physical meaning instead o f a numerical emergency brake. We anticipate that this is an important advantage, especially with regard to the ERMs, since a realistic description of horizontal momentum shear can be crucial for the simulation of eddies. The stability properties together with the possibility of the choice of a large, but still physically realistic time-step were necessary pre-requisites for the long term simulation presented above. To the first time an insight into the low frequency variability of the North European shelf sea was provided by the model estimates. Whereas past simulations almost solely considered (cl imatological) mean conditions o f the circulation the here presented results suggest that deviations from the mean on various time scales are at least of similiar importance. In the international discussion about the 'water-quality' of the marine environment of the North Sea again primarily mean values are considered. However, Hainbucher et.al. (1987) have shown by using the above results in a Lagrangian trajectory-model that the water-qua1 ity of the North Sea may undergo simil iar fluctuations as the circulation. Up to now the inadequate sampling of present environmental monitoring activities prevented the detection o f these fluctuations from observations (typical sampling rates are 1 to max. 3 times per year). Hence the extensive model experiment not only served to increase our knowledge about the circulation but it initiated a discussion about important questions concerning the (better) protection/monitoring o f the marine environment.
242
REFERENCES
J.O.
BACKHAUS,
and E.
Maier-Reimer, 1983: On seasonal c i r c u l a t i o n p a t t e r n s i n
In: North Sea dynamics (Ed.: J. Sundermann e t . a l . ) Heidelberg:
t h e North Sea.
Springer, pp. 63-84
J.O.,
BACKHAUS,
1983a:
A s e m i - i m p l i c i t scheme f o r t h e shallow water equations
f o r a p p l i c a t i o n t o s h e l f sea m o d e l l i n g . Continent. Shelf Res. 2, pp. 243-254
J.O.,
BACKHAUS,
1983b:
On
Coastal Oceanography (Ed.:
the c i r c u l a t i o n o f the s t r a t i f i e d North Gade e t . a l . ) ,
Sea.
In:
Plenum P u b l i s h i n g Corporation,
pp.
93-1 11
J.O., 1985: A three-dimensional model f o r t h e s i m u l a t i o n o f s h e l f sea
BACKHAUS,
dynamics. D t . hydrogr. Z . 38, pp. 165-187
J . Bartsch, D. Quadfasel and J . Guddal, 1985: A t l a s o f monthly
J.O.,
BACKHAUS, surface
fields
o f a i r pressure,
North Eastern A t l a n t i c Ocean:
wind s t r e s s and wind s t r e s s c u r l
1955-1982.
Technical Report 3-85,
over
the
I n s t i t u t fur
Meereskunde der U n i v e r s i t a t Hamburg (unpublished manuscript)
J.O.,
BACKHAUS,
circulation
Hainbucher,
D.
D.
Quadfasel and 3. Bartsch, 1985: North Sea
anomalies i n response t o v a r y i n g atmospheric f o r c i n g .
ICES C.M.
1985/C :29
J.O.,
BACKHAUS,
T.
Pohlmann and D.
Hainbucher, 1986: Regional aspects o f the
c i r c u l a t i o n on t h e North European s h e l f . I C E S C.M. BLUMBERG,
A.F.
and G.L.
Mathematicale M o d e l l i n g
Mellor,
1980:
1986/C:38
A coastal ocean numerical model.
o f E s t u a r i n e Physics (Ed.:
J.
Sundermann
et.
In: al.)
Heidelberg: Springer
M.,
BOEHLICH,
1987: A t h r e e dimensional b a r o c l i n i c model o f t h e western B a l t i c .
( t h i s volume) BOWDEN,
K.F.
circulation
and P. Hamilton, and
mixing
1975: Some experiments w i t h a numerical model o f
i n t i d a l estuaries.
Estuarine
and
Coastal
Marine
Science 3, pp. 281-301 CORTEN, A.,
1986: B i o l o g i c a l i n d i c a t i o n s o f a change i n North Sea c i r c u l a t i o n i n
I C E S C.M.
t h e 70’s.
DAVIES,
A.M.,
1982:
1986/L:10 Three
dimensional models o f North Sea
circulation.
In:
Coastal Oceanography (Ed. Gade e t . a l .) Plenum Press, pp. 65-92
J.D.,
DIPPNER,
1987: P r e l i m i n a r y r e s u l t s o f a t h r e e dimensional eddy r e s o l v i n g
model f o r r i v e r plume f r o n t s . ( t h i s volume) DOOLEY, Fair
H.D.,
1983:
I s l e current.
Seasonal v a r i a b i l i t y i n t h e p o s i t i o n and s t r e n g t h o f
In:
Proceedings o f t h e Symposium on North Sea
Hamburg September 1981, (Ed.:
J. Sundermann e t . a l . ) ,
the
Dynamics,
Heidelberg: Springer, pp.
108-1 19 FISCHER,
G.,
1959:
Ein
numerisches Verfahren z u r Errechnung von Windstau und
243 Gezeiten i n Randmeeren. Tellus.11,
E.,
GOEDECKE, Nordsee,
J.
pp. 60-76
Smed and G. Tomczak, 1967: Monatskarten des Salzgehaltes der
d a r g e s t e l l t f u r verschiedene T i e f e n h o r i z o n t e . D t . hydrogr. Z.,
Erg.-
H. 6.9
GOULD,
W.J.,
J.
Loynes and J.O.
Backhaus, 1985: Seasonality i n s l o p e c u r r e n t
t r a n s p o r t n.w. o f Shetland. I C E S C.M. HAINBUCHER,
1985/C:7
1985: P a r a m e t r i s i e r u n g des v e r t i k a l e n I m p u l s t r a n s f e r s i n einem
D.,
dreidimensionalen Z i r k u l a t i o n s m o d e l l der Nordsee.
Diploma Thesis, U n i v e r s i t a t
Hamburg, I n s t i t u t f u r Meereskunde HAINBUCHER O., and
regions:
-
J.O.
Backhaus and T.
Pohlmann,
1986:
Atlas of climatological
actual seasonal c i r c u l a t i o n p a t t e r n s i n t h e North Sea and adjacent 1969-1981.
Technical Report 1-86,
shelf
I n s t i t u t f u r Meereskunde der Uni-
versi t a t Hamburg (unpubl ished manuscript)
HAINBUCHER,
D.,
T. Pohlmann and J.O. Backhaus, 1987:
Transport o f c o n s e r v a t i v e
passive t r a c e r s i n t h e North Sea: f i r s t r e s u l t s o f a c i r c u l a t i o n and t r a n s p o r t
model. Continental S h e l f Research ( i n press) HALTINER,
1971: Numerical Weather P r e d i c t i o n . John Wiley & Sons, I N C . New
G.J.,
York, London, Sydney, Toronto JAMES,
I.D.,
1986:
A f r o n t - r e s o l v i n g sigma c o o r d i n a t e sea model w i t h a simple
h y b r i d advection scheme. A p p l i e d Mathematical Modelling, Vol JAMES,
1987:
I.D.,
The
.lo,
pp. 87-92
design o f a general t h r e e dimensional eddy
resolving
model f o r s t r a t i f i e d seas. ( t h i s volume) JOHNS,
1983:
B.,
Physical
Oceanography o f Coastal and Shelf
Seas.
Elsevier
Oceanography Series 35 LEVITUS,
S.,
paper No.13, LUTHARDT,
H.
1982:
C l i m a t o l o g i c a l a t l a s o f t h e w o r l d ocean. NOAA p r o f e s s i o n a l
US Government P r i n t i n g O f f i c e Washington DC and
L.
Hasse,
1981:
On
t h e r e l a t i o n s h i p between s u r f a c e
and
geostrophic wind i n t h e German B i g h t area. B e i t r . Phys. Atm. 54, pp. 222-232 LUTHARDT, surface
H.
and L .
wind
Dynamics (Ed.:
MADDEN,
R.A.
Hasse,
1983: The r e l a t i o n s h i p between pressure f i e l d and
i n t h e German B i g h t area a t h i g h wind J. Sundermann e t . a l . ) , and P.R.
Jul ian,
speeds.
E.,
1977:
strome. D t . hydrogr. Z., PRANDLE,
D.,
1978:
North
Sea
Heidelberg: Springer, pp. 341-348
1972:
Description of global-scale c i r c u l a t i o n
c e l l s i n t h e t r o p i c s w i t h a 40-50 day p e r i o d . J. A t m . Sc.,
MAIER-REIMER,
In:
29, pp. 1109-1123
Zum E i n f l u B eines m i t t l e r e n Windschubes auf d i e Rest6.28,
Residual
H.6 f l o w s and e l e v a t i o n s i n t h e southern North
Sea.
Proc. R . SOC. London (A) 359: 189-288
RODI, W.,
1980: Mathematical m o d e l l i n g o f turbulence i n e s t u a r i e s . In Mathemati-
cal M o d e l l i n g o f E s t u a r i n e Physics (Ed.: Springer, pp 14-26
J.
Sundermann e t . a l . ) ,
Heidelberg:
244
TOMCZAK, G. and E. Goedecke, 1962: Monatskarten der Temperatur der Nordsee, dargestellt fur verschiedene Tiefenhorizonte. Dt. hydrogr. Z., Erg. H. B,7 WAIS, R., 1985: On the relation of linear stability and the representation of Coriolis terms i n the numerical solution of the shallow water equations. Universitat Hamburg (unpublished manuscript) WILLEBRAND, J., 1978: Temporal and spatial scales of the wind field over North Pacific and North Atlantic. J. Phys. Oceanogr., 8, pp. 1080-1094
245
A THREE DIMENSIONAL CIRCULATION MODEL OF THE SOUTH CHINA SEA
T. POHLMANN I n s t i t u t f u r Meereskunde d e r U n i v e r s i t a t Hamburg Heimhuder StraRe 71, 2000 Hamburg 13, FRG
ABSTRACT Up t o now t h e r e i s a g r e a t l a c k o f o b s e r v a t i o n a l d a t a i n t h e South China Sea. The
b e s t a v a i l a b l e i n f o r m a t i o n about t h e general hydrography o f t h e r e g i o n
was
t h e Naga Report c o m p i l e d b y W y r t k i a l r e a d y i n 1961. The South China Sea i s an e q u a t o r i a l r e g i o n
w i t h a complex topography. I t i s
a regime dominated b y t h e monsoon and s t r a t i f i c a t i o n i s o f enormous importance. A
prognostic
b a r o c l i n i c c i r c u l a t i o n model was a p p l i e d i n o r d e r t o
increase
our p r e s e n t knowledge and o u r u n d e r s t a n d i n g o f t h i s r e g i o n . The g r i d s i z e o f t h i s
12- l a y e r model i s about 50 km i n t h e h o r i z o n t a l . The l a y e r s have a t h i c k n e s s o f 10 m t o 3000 m, Simulations
increasing w i t h depth. were
respectively.
The
c a r r i e d o u t f o r t h e w i n t e r - and f o r
the
summer
monsoon,
c a l c u l a t i o n o f t e m p e r a t u r e and s a l i n i t y d i s t r i b u t i o n s
which
are c o n s i s t e n t w i t h t h e c i r c u l a t i o n p r o v i d e i n s i g h t i n t o new f e a t u r e s l i k e deepr e a c h i n g up- and d o w n w e l l i n g phenomena. was
carried
out
A f i r s t v a l i d a t i o n o f t h e model r e s u l t s
i n comparison w i t h t h e o b s e r v a t i o n a l d a t a compiled
by
Klaus
Wyrtki. 1 INTRODUCTION The South China Sea w i t h an e x t e n t i o n o f 36 Mio kmxkm i s t h e l a r g e s t m a r g i n a l sea i n t h e Southeast A s i a n Waters ( F i g u r e 1 ) .
I t i s s i t u a t e d between t h e
Asian
c o n t i n e n t , Borneo, t h e P h i l i p p i n e s and Formosa (Taiwan). I t s topography which i s d i v i d e d i n t o two p a r t s i s t y p i c a l f o r m a r g i n a l seas o f t h e Western P a c i f i c . northern
part
seperated
from
i s a deep sea b a s i n where depths exceed 5000 m. the
This
main w a t e r body o f t h e P a c i f i c b y a s t r i n g o f
volcanic o r i g i n ( i n c l u d i n g t h e P h i l i p p i n e s ) .
The
basin
is
islands
The southern p a r t i s a s h e l f
of sea,
where depths range between 50 m and 100 m. One
of
the
main s p e c i f i c a t i o n s o f t h e South China Sea i s i t s
location
t r o p i c a l low l a t i t u d e s , which causes two i m p o r t a n t e f f e c t s on t h e c i r c u l a t i o n .
in
246
a)
By t h e r e d u c t i o n o f t h e C o r i o l i s parameter near t h e e q u a t o r
nonlinear
and
f r i c t i o n a l terms g e t an i n c r e a s i n g importance. b)
The
South China Sea i s s i t u a t e d w i t h i n t h e monsoon regime
s t r o n g l y i n f l u e n c e d b y the, t h e atmosphere.
and
p e r i o d i c a l l y semi- anual r e v e r s i n g ,
is
thereby
circulation of
F i g u r e 2a and 2b ( t h e w i n d s t r e s s d i s t r i b u t i o n i n January and i n
J u l y ) r e f l e c t t h e f u l l y developed w i n t e r , r e s p e c t i v e l y summer monsoon s i t u a t i o n .
In winter
n o r t h e a s t e r l y winds p r e v a i l o v e r t h e whole r e g i o n w i t h
magnitude
o f 9 m/s.
I n summer t h e wind d i s t r i b u t i o n t o t a l l y
an
average
reverses.
Weaker
s o u t h w e s t e r l y winds dominate o v e r most p a r t s o f t h e South China Sea ( 6 m/s)
and
o n l y i n t h e n o r t h e r n p a r t s t h e d i r e c t i o n changes t o more s o u t h e r l y winds. The
large
s c a l e d i s t r i b u t i o n of mass which has a c o n s i d e r a b l e i n f l u e n c e
on
t h e c i r c u l a t i o n o f t h e South China Sea c o u l d b e summarized as f o l l o w s : L i g h t t r o p i c a l s u r f a c e w a t e r w i t h low s a l i n i t y and h i g h t e m p e r a t u r e forms c o n t r a s t t o t h e c o l d and s a l t y deep water.
strong
a
The t r a n s i t i o n between these
two water masses t a k e s p l a c e i n an e x t r e m e l y s t r o n g d i s c o n t i n u i t y l a y e r s i t u a t e d i n a depth o f about 120 m. The
renewal o f t h e deep w a t e r b y water masses f r o m t h e P a c i f i c
takes
place
t h r o u g h narrow passages i n t h e s t r i n g o f i s l a n d s which f o r m t h e w e s t e r n boundary o f t h e South China Sea. Figure 23'
3a and 3b show t h e s u r f a c e temperature d i s t r i b u t i o n i n w i n t e r and The l a t e r a l g r a d i e n t s a r e much s t r o n g e r i n w i n t e r .
sununer.
i n t h e n o r t h t o 28'
C
C i n t h e southern parts,
i n c r e a s e i s much weaker ( f r o m 28 The to
0
whereas d u r i n g
C i n t h e n o r t h t o 29
0
32.4
summer
the
C i n t h e south). f r o m 34.0 i n t h e n o r t h e r n
s u r f a c e s a l i n i t y ( f i g u r e 4a and 4b) decreases i n t h e s o u t h e r n p a r t s of t h e South China Sea.
gradient
in
They i n c r e a s e from
In w i n t e r
the
i s l o c a t e d i n t h e c e n t r a l p a r t w h i l e i n summer i t i s s i t u a t e d
maximum in
the
s o u t h e r n p a r t o f t h e South China Sea. Since 1961, of
the
achieved.
The
particularly reaches the
when W y r t k i had p r e p a r e d h i s r e p o r t ,
no s i g n i f i c a n t improvement
general u n d e r s t a n d i n g of t h e dynamics o f t h e South China Sea sea
as
important
maximum v a l u e s .
South
China Sea.
an
important
p r o t e i n - source
f o r r e g i o n s where t h e growth- r a t e
for of
mankind the
T h i s i s t h e case f o r a number o f n a t i o n s , A l s o m a r i n e p o l l u t i o n as a consequence
has
of
been
becomes
population which share the
growing
p o p u l a t i o n and/ o r i n d u s t r y has become a s e r i o u s problem. The complex topography o f t h e r e g i o n m i g h t have been t h e reason why up t o now
no
attempts
for
r e q u i r e s both, challenge
there
model s i m u l a t i o n s have been made.
a s h e l f sea and a deep ocean model,
In f a c t ,
the
topography
Apart from t h i s (numerical)
seems t o be r e a s o n enough t o a p p l y a numerical
model
South China Sea i n o r d e r t o improve o u r u n d e r s t a n d i n g o f i t s dynamics.
on
the
247
1ooo
1 10°E
120°
20°
20°
1OOP
1OoN
O0
F i g . 1 . C h a r t showing t h e l o c a t i o n o f t h e South China Sea
O0
248
Fig. 2a. Windstress distribution in January ( a f t e r He1 lerman, 1968)
249
F i g . 2b. Windstress distribution in July (after He11erman, 1968)
250
F i g . 3a. S u r f a c e temperature d i s t r i b u t i o n i n w i n t e r ( a f t e r L e v i t u s , 1982)
251
Fig. 3b. Surface temperature distribution in s u m e r (after Levitus, 1982)
252
F i g . 4a. Surface salinity distribution i n winter
(after Levitus, 1982)
253
F i g . 4 b . Surface s a l i n i t y d i s t r i b u t i o n i n summer ( a f t e r L e v i t u s , 1982)
254
F i g . 5. Topography o f the model r e g i o n
255
2 THE MODEL The
i n v e s t i g a t i o n s were
barocl i n i c
model
carried out with the aid
(Backhaus,
several s h e l f seas.
1983).
of a
three
T h i s model has a l r e a d y
dimensional
been
applied
to
( N o r t h Sea, N o r t h European S h e l f , B a l t i c ) (Backhaus, 1985,
Boehlich, 1987, t h i s i s s u e ) . So, o n l y a general d e s c r i p t i o n o f t h e main f e a t u r e s of t h e model i s g i v e n h e r e . The g o v e r n i n g e q u a t i o n s are:
1. The e q u a t i o n o f c o n t i n u i t y o f mass 2. The s h a l l o w w a t e r e q u a t i o n 3. The e q u a t i o n c o n t i n u i t y o f t e m p e r a t u r e and s a l i n i t y
4. The e q u a t i o n o f s t a t e f o r seawater In
equation
1.
hydrostatic
approximation was i n c o r p o r a t e d . and
diffusion
terms,
e q u i l i b r i u m was assumed
and
the
Boussinesq-
E q u a t i o n 3. was s i m p l i f i e d b y n e g l e c t i n g source
as a consequence o f t h e i n s u f f i c i e n t d a t a
amount
which
would p e r m i t an a c c u r a t e d e t e r m i n a t i o n o f t h e s e c o e f f i c i e n t s i n t h i s r e g i o n .
A
semi- i m p l i c i t n u m e r i c a l scheme was used t o 5 0 1 t~h e g o v e r n i n g e q u a t i o n s .
More d e t a i l e d i n f o r m a t i o n about t h i s scheme i s g i v e n i n Backhaus and (1987,
Hainbucher
t h i s i s s u e ) . The g r i d s i z e o f t h e model i s about 50 km i n t h e h o r i z o n t a l .
The 12 l a y e r s have a t h i c k n e s s o f 10 m t o 3000 m i n c r e a s i n g w i t h depth. The
l o c a t i o n o f t h e model r e g i o n i s shown i n f i g u r e 5.
arises
A numerical
f r o m t h e f a c t t h a t t h e South China Sea crosses t h e e q u a t o r .
numerical scheme r e q u i r e s d i v i s i o n by f,
t h e C o r i o l i s parameter.
senseless r e s u l t s d i r e c t l y a t t h e e q u a t o r where f solved
by
equator.
placing
Thereby
problem
The a p p l i e d This leads t o
reaches zero. The problem was
t h e g r i d i n a way t h a t no g r i d p o i n t l i e s
directly
on
the
t h e s m a l l e s t v a l u e o f t h e C o r i o l i s parameter reaches s t i l l
a
reasonable v a l u e o f 3.2 x 10-71/s. The boundary c o n d i t i o n s a r e t h e u s u a l ones f o r p r i m i t i v e e q u a t i o n models.
At
c l o s e d l a t e r a l boundaries a no- f l u x and a semi- s l i p c o n d i t i o n was a p p l i e d .
At
open
of
l a t e r a l boundaries t h e w a t e r e l a v a t i o n i s p r e s c r i b e d and t h e g r a d i e n t s
t h e t r a n s p o r t normal t o t h e boundary a r e s e t equal t o zero. salinity used.
For t e m p e r a t u r e and
a m o d i f i e d Sommerfeld r a d i a t i o n - c o n d i t i o n ( O r l a n s k i ,
The
1976) has
been
wind s t r e s s a t t h e sea s u r f a c e and t h e bottom s t r e s s a r e r e p r e s e n t e d
b y q u a d r a t i c s t r e s s laws. 3 THE SIMULATION Several descriptions simulation
simulations of
the
were
carried
w i n t e r and summer
out
in
order
monsoon
to
get
circulation
representative respectively.
A
p e r i o d o f 15 days f o r each o f t h e phases t u r n e d o u t t o be s u f f i c i e n t
t o approximately reach a quasi s t a t i o n a r y state. The
circulation
in
t h e b a r o c l i n i c South China
Sea
model
is
essentially
256
determined by: 1.
Monthly
averaged
wind s t r e s s d a t a f o r January and J u l y e x t r a c t e d f r o m
the
Hellerman (1980) d a t a s e t . ( f i g u r e 2a, 26) 2. Seasonal averaged t e m p e r a t u r e and s a l i n i t y d a t a f o r t h e w i n t e r and t h e s u m e r season. These d a t a are t a k e n f r o m L e v i t u s (1968) and i n t e r p o l a t e d i n t o t h e model g r i d , ( f i g u r e 3a, 3b, 4a, 4b) as i n i t i a l
fields.
3. The topography o f t h e South China Sea. ( f i g u r e 5 ) The high
t i d e s were n e g l e c t e d i n t h e s e s i m u l a t i o n s because t h e y m a i n l y cause frequency
variability
of
t h e c i r c u l a t i o n and
importance
f o r t h e mean monsoon c i r c u l a t i o n .
(Pohlmann,
1985)
induced shelf
are
Furthermore,
therefore another
of
the minor
simulation
r e v e a l e d t h a t n o n l i n e a r i n t e r a c t i o n s between wind- and
c u r r e n t s o n l y r e a c h s i g n i f i c a n t v a l u e s i n some o f t h e s h a l l o w r e g i o n s where h i g h v e l o c i t i e s appear.
tidal
southern
In most o f t h e o t h e r p a r t s o f
the
model area t h i s i s n o t t h e case. 3.1 Mean w i n t e r and summer c i r c u l a t i o n p a t t e r n s The
summer
f i g u r e s ( 6 - 7 ) show t h e r e s u l t s f o r t h e mean w i n t e r r e s p e c t i v e l y
The f i r s t and t h e t h i r d model l a y e r w i l l be presented, whereby t h e
circulation.
l a y e r has an e x t e n t i o n f r o m t h e s u r f a c e t o 10 m depth and t h e l a t t e r
first
one
f r o m 20 m t o 30 m depth. During
w i n t e r ( f i g u r e 6a and 6 b ) an i n f l o w f r o m t h e P a c i f i c i n t o
the
South
China Sea t h r o u g h t h e Luzon S t r a i t i s w e l l d i s t i n g u i s h e d i n t h e upper l a y e r s . I n the
northern
predominate, the
and
central
parts
t u r n i n g southward
westerly
respectively
northerly
currents
when t h e y r e a c h t h e Vietnam c o a s t . I n t h i s area
c u r r e n t i n t e n s i f i e s t o a narrow band o f a p p r o x i m a t e l y 100 km.
When i t
has
l e f t t h e Vietnam c o a s t i t i s w i d e n i n g again and l a t e r l e a v e s t h e South China Sea t h r o u g h t h e Java Sea. As
i t would be expected f r o m an i n s p e c t i o n o f t h e wind f i e l d f o r
current inflow
pattern from
(figure
the
7a and 7b) t o t a l l y r e v e r s e s i n sumner.
Java Sea i n t o t h e South China Sea.
I n the
July
There
southern
recirculation
c e l l has developed as w e l l as a deep- r e a c h i n g c y c l o n i c
the
part.
northern
currents weaker
I n the central parts easterly
predominate. than
in winter.
The
respectively
the is
an
part gyre
a in
northeasterly
s o u t h - g o i n g flow a l o n g t h e Vietnam c o a s t
is
Here t h e f l o w does n o t r e v e r s e w i t h t h e change o f
much the
mnsoon.
-
- p e r a t u r e and s a l i n i t y d i s t r i b u t i o n s 3.2 Mean w i n t e r and summer t e m
For
winter
temperature
and summer,
respectively the differences
between
the
initial
d i s t r i b u t i o n and t h e f i n a l d y n a m i c a l l y balanced d i s t r i b u t i o n ( a f t e r
about 15 days) a r e shown i n f i g u r e 8a,
8b,
9a and 9b f o r t h e f i r s t and f o r t h e
257
Fig. 6a. Mean winter circulation (0-10 m)
258
F i g . 6b. Mean winter circulation (20-30 m)
259
F i g . 7a. Mean summer circulation (0-10 m)
F i g . 7b. Mean summer c i r c u l a t i o n (20-30 m)
261
F i g . 8a. D i f f e r e n c e between simulated and i n i t i a l temperature d i s t r i b u t i o n i n w i n t e r (0-10 m)
262
Fig. 8b. Difference between simulated and initial temperature distribution in winter (60-100m)
263
Fig. 9a. Difference between simulated and initial temperature distribution in sumner (0-10 m)
264
F i g . 9b. D i f f e r e n c e between simulated and i n i t i a l temperature d i s t r i b u t i o n i n sumner (60-100 m)
265
Fig. 10a. Surface currents in February (from ship drifts, Wyrtki, 1961 )
Fig. lob. Surface currents in August (from ship drifts, Wyrtki, 1961 )
266
fifth model layer. The fifth layer extends from 60 m to 100 m depth. During winter downwelling off the Vietnam and pronounced upwelling off the Philippine coast becomes evident (figure 8a and 8b). In sumner the situation is reversed (figure 9a and 9b). In both seasons these phenomena are obviously much stronger in the fifth than in the first layer. So far deep- reaching up- and downwelling has not been observed in this region. Wyrtki’s results are based on surface measurements only, and therefore he was not able to detect this phenomenon. However, this result could be regarded as a stimulation for oceanographers measuring in the South China Sea. 4 VERIFICATION
A verification of the model results, as far as presently possible, was carried out by comparing observed and simulated transport rates through two sections. Wyrtki has calculated transport rates from observational data through the Java Sea and through a section which runs from the Vietnam coast in southwesterly direction into the South China Sea. Table 1. gives a comparision of these values calculated from observations with those transport rates calculated by the model. For the Java Sea, a shelf region, the simulations agree obviously well with the observations in both seasons. The simulated transports off the Vietnam coast are about 30 percent smaller than the observed values. This disagreement might result from the incomplete information about the location and the extent of this section, not given by Wyrtki. TABLE 1 Comparison between observed and simulated transport rates ( x106 m3/s) two sections. Off Vietnam (northeastwards pos it i ve) Winter: Observation Simulation Summer: Observation Simulation
Java Sea (eastwards pos it i ve)
-6.9 -5.0
4.3
4.5
-3.2
3.2
-3.0
4.2
through
267
both
seasons.
T h i s i s e s p e c i a l l y v a l i d f o r t h e south- g o i n g c u r r e n t a l o n g
the
I t i s v e r y s t r o n g i n w i n t e r and forms a weak c o u n t e r c u r r e n t i n
Vietnam
coast.
summer.
The o n l y c o n s p i c i o u s d e v i t a t i o n between o b s e r v a t i o n and c a l c u l a t i o n
is
t h e s i m u l a t e d b u t n o t observed n o r t h e r n c y c l o n i c g y r e d u r i n g t h e summer monsoon. By
comparing
the
results
of
a baroclinic
barotropic simulation it i s substantiated
simulation
(Pohlmann,
against
those
of
a
1985) t h a t t h i s g y r e i s a
barocl i n i c f e a t u r e . 5 CONCLUSION The
s i m u l a t i o n s have shown t h a t t h e model developed f o r t h e South China
satisfies
the
comparision
requirements
is
possible
which a r e f o r m u l a t e d i n c h a p t e r 2.
the
results of the
model
agree
As
far
Sea as
qualitatively
a and
q u a n t i t a t i v e l y w e l l w i t h W y r t k i ’ s c o m p i l a t i o n . The s i m u l a t i o n s have b y a l l means qualitatively
improved
monsoon
circulation.
Vietnam
and
result
t h e knowledge about 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
The
possible
e x i s t e n c e o f up- and
P h i l i p p i n e c o a s t has been p o i n t e d o u t f o r
can
downwelling
the
first
mean
at
the
time.
be o f h i g h l y economical i n t e r e s t f o r t h e f i s h e r y i n d u s t r y
This
in
this
region. The
reaction,
the
s i n - up,
of the
c u r r e n t system on t h e
p r e s e n t l y i n v e s t i g a t e d b y s e v e r a l oceanographers ( L i g h t h i l l , Quadfasel, dependent
1982)
could
a l s o be s i m u l a t e d
m e t e o r o l o g i c a l data,
with
this
momentary n o t a v a i l a b l e ,
monsoon
1969,
model.
Cox,
onset, 1969,
However,
time
must be s u p p l i e d i n a
reasonable r e s o l u t i o n i n o r d e r t o r u n such a model. 6 ACKNOWLEDGEMENTS
I am i n d e b t e d t o P r o f . throughout t h i s work.
Backhaus
Dr.
Also,
f o r h i s v a l u a b l e a d v i c e and a s s i s t a n c e
I thank my c o l l e a g e 0.
Hainbucher f o r making v e r y
h e l p f u l comments on t h e m a n u s c r i p t .
7 REFERENCES Backhaus,
J.O.,
1983.
A sem,i- i m p l i c i t scheme f o r t h e s h a l l o w w a t e r e q u a t i o n s
f o r a p p l i c a t i o n t o s h e l f sea m o d e l l i n n g . C o n t i n e n t . S h e l f Res. 2: 243-254. Backhaus,
J.O.,
1985.
A
Three- Dimensional Model f o r t h e S i m u l a t i o n of Shelf
Sea Dynamics. D t . h y d r o g r . Z. 38: 165-187. Backhaus,
J.O.,
circulation (unpubl.).
Pohlmann, on
T.,
Hainbucher,
t h e N o r t h European S h e l f .
D., ICES
1986. Regional aspects o f t h e Report
C.M.
1986/
C:
38
268
Back haus circu Boehl ich This Cox, M.
3.0. and Hainbucher, D., 1987. A finite difference general ation model for shelf seas. This issue. M., 1987. A three dimensional baroclinic model of the western Baltic. ssue. D., 1970. A mathematical model of the Indian Ocean. Deep Sea Research
17.
Hainbucher, D. Backhaus, J.O., Pohlmann, T., 1986. Atlas of climatological and actual seasonal circulation patterns in the North Sea and adjacent shelf regions: 1969-1981. Technical Report No. 1, Institut fur Meereskunde, Un i ver s i tat H a m bur 9. Hellerman, S., 1968. An update estimate of the wind stress on the world ocean. Monthly weather review, 96. Levitus, 1982. Climatological atlas of the world ocean. NOAA. Professional Paper No. 13. U.S. Goverment Printing office, Washington D.C. Lighthill, M.J., 1969. Dynamical response of the Indian Ocean to onset of the Southwest Monsoon. Philos. Trans. R. SOC. 265. Orlanski, I . , 1976. A Simple Boundary Condition for unbounded Hyperbolic Flows. Journal of Computational Physics 21: 251-269. Pohlmann, T., 1985. Simulation von Bewegungsvorgangen im Sudchinesischen Meer. Diploma Thesis, Institut fur Meereskunde, Universitat Hamburg. Quadfasel, D.R., Wilson, D., Leetmaa, A., 1982. Development of the flow field during onset of the Somali Currrent. Journal of Physical Oceanography. Vol 12, No. 12. Wyrtki, K., 1961. Scientific results of marine investigations of the South China Sea and the Gulf of Thailand 1959-1961. Naga Report. Volume 2.
269
THE INFLUENCE OF BOUNDARY CONDITIONS ON THE CIRCULATION IN THE GREENLANDNORWEGIAN SEA. A NUMERICAL INVESTIGATION. S. LEGUTKE
Institut Fir Meereskunde, Universitat Hamburg, Troplowitzstr. 7, 2000 Hamburg, F.R.G. ABSTRACT The dynamics of the Greenland-Norwegian Sea are investigated, using a numerical model extending from the Greenland-Scotland Ridge to the Fram Strait and including part of the Barents Shelf. The model is based on a finite difference discretisation of the primitive equations with 12 levels and a horizontal grid size of about 20 km. It is driven by windstress and buoyancy fluxes at the surface; at open boundaries volume, salt, and heat fluxes are specified. Quasi diagnostic computations have been performed using climatological seasonal mean data at the boundaries and as initial stratification. The response of the system to various situations at the inflow boundaries is investigated. The current fields produced are in good agreement with existing observations. It is found, that the bottom pressure torque is the dominant term in the vertically integrated vorticity equation almost everywhere. It causes the deep and vertically integrated flow to separate into several gyres. 1 INTRODUCTION
The Greenland-Norwegian Sea (GNS), connecting the Arctic Ocean with the North Atlantic, plays a key role in climate processes on the Northern Hemisphere. By far the major part of the heat transfer between the Arctic Ocean and its neighbouring seas occurs through the Fram Strait (Aagaard and Greisman 1975). On the other hand, the deep and bottom water formed in the Greenland Sea is the major source of bottom water in the North Atlantic (Swift 1984). The rate of these processes is influenced by various factors. Among them are the circulation and characteristics of the water masses involved. The prevailing cyclonic wind stress distribution drives a northward flow of warm saline Atlantic water entering mainly through the Faeroe-Shetland Channel. Part of the Atlantic water leaves the GNS through the Fram Strait, thus providing a heat source for the Arctic Basin. Part of it returns to the south closing, together with the Polar water of the Eastgreenland Current, a cyclonic circulation in this region. The associated doming of isopycnals in the Greenland Basin, together with winter cooling, results in low stability water masses and bottom water formation. Two models of the GNS have been published so far. Creegan (1976) used a model with two layers of constant densities of the region deeper than 500 m between
270
the Fram Strait and the Greenland-Scotland Ridge to investigate the influence of wind stress distribution and inflow through the Faeroe-Shetland Channel. Obviously such a model has some shortcomings such as the exclusion of shelf areas and of the Eastgreenland Current. The 2-layer structure prohibits a proper consideration of the topographic and thermohaline influences on the circulation, A three dimensional model of the Arctic Ocean and the GNS has been presented by Semtner (1976). Using mean forcing functions he obtained a cyclonic stream function covering the whole GNS. But with the relatively coarse horizontal gridsize of 110 km in a region of strong topographic variations the smaller scale features (see for example figures 2,3,and 4 of Metcalf (1960)) of the current system were not resolved. It has been pointed out by various authors, that the joint effect of topography and baroclinicity has a large influence on the flow field (Sarkisyan and Ivanov 1971, Holland 1973, Holland and Hirschman 1972).
L
Fig.1. Model bathymetry. Heavy contour interval i s 1000 m. Open passages are indicated by thin lines. Sections A,B,I,II are referred to in the text.
In the present paper a model is described, that was developed in an attempt to take into account the influence of topographic variations down to scales of 100 km. The model presented is designed to run long term prognostic calculations with variable boundary conditions both at the lateral open boundaries and
271
at the surface. In the experiments described it is initialized with climatological seasonal mean hydrographic and wind stress data. Prognostic calculations are run for one month in order to establish a circulation field consistent with the input data. The results of these quasi diagnostic calculations will be compared with observed currents. In addition the response to varying inflow situations and anomalous wind stress is discussed. 2 DESCRIPTION OF THE MODEL 2.1 The model equations A model similar to that described below has already been used in a study of equatorial dynamics (Latif et al. 1985). It consists of a finite difference discretisation of the primitive equations using the Boussinesq and hydrostatic approximations :
9'g=Pz
vhytw,=O
(3)
The notation is as usual: y denotes the horizontal velocity vector, p is pressure, y density,?, a constant reference density, f the Coriolis parameter, k an unit vector upwards, g the gravitational constant,and vh the horizontal gradient operator. F(y) represents a parameterization of the eddy viscosity effects. An E-grid (Arakawa and Lamb 1977) in spherical coordinates I,cg,z is used with a zonal grid size of .25'1at. The choice of coordinates is motivated by future plans to embed the GNS-model in a model of the Atlantic and Arctic Oceans (H.Friedrich, in preparation): The poles are placed on South America and South East Asia to avoid singularities inside the world ocean. In particular, this coordinate system reduces the grid deformation due to the convergence of meridians within the area of interest. In what follows the terms zonal and meridional will be used to denote the direction of increasing 1 a n d q in model coordinates, while north and south is used for geographical coordinate directions. The eddy viscosity parameterization is a simple Laplacian diffusion denotes the velocity vector integrated F(y)=Ahvh 2(U)/DZ with Ah=103m 2/sec. over the layer depth DZ. This formulation ensures conservation of momentum away from lateral closed boundaries. At the sea surface source terms Q (T,S) resulting from a Newtonian coupling of observed and computed surface fields with a relaxa-
272
t i o n time scale o f 16 days are specified. prognostic
equation
No e x p l i c i t d i f f u s i o n i s needed i n the
f o r temperature and s a l t since an upwind scheme
is
sea surface e l e v a t i o n Z i s computed from t h e l i n e a r i z e d kinematic
The
condition. inertial
used.
boundary
The numerical scheme e f f e c t i v e l y damps t h e e x t e r n a l g r a v i t y mode and oscillations.
The
time step i s then r e s t r i c t e d by
internal
gravity
waves. A value o f 3 h i s used. l e v e l depths o f t h e model are 7,21,37,57,85,126,209,341,551,851,1501,and
The
2701 m. Since t h e lowest box o f each column has a v a r i a b l e depth, t h e r e s o l u t i o n o f the bathymetric f i e l d does n o t depend on t h e number o f boxes. I n t h i s way the levels
can be concentrated i n t h e upper 800 m where t h e hydrographic f i e l d s are
variable.
Below t h e A t l a n t i c water,
t h e water i s q u i t e homogenous and a l a r g e r
spacing can be chosen.
2.2 Boundary c o n d i t i o n s At
l a t e r a l closed boundaries n o - f l u x and n o - s l i p c o n d i t i o n s
system
are
used.
The
i s f o r c e d a t t h e s u r f a c e by a wind s t r e s s f i e l d L = ( T x , T y ) = v A v - y z and by
t h e buoyancy f l u x described above. At
a q u a d r a t i c s t r e s s law i s a p p l i e d ( p A v y z = e . / y / - y.-Da).
z=-H(d,lp),
turning
matrix
Weatherly
Da and t h e drag c o e f f i e n t e have been
(1972).
With
specified
The
according
to
these values 2-5% o f t h e energy d i s s i p a t i o n i s due
to
bottom f r i c t i o n .
At
open
lateral
barotropic city
boundaries
pressure term.
profiles,
vertical
shear
relation.
The
advection
into
whose
gradients
of Z
are
needed
to
the
These are d e r i v e d by geostrophy from s p e c i f i e d velo-
b a r o t r o p i c p a r t s a r e taken from o b s e r v a t i o n
i s computed from hydrographic sections using t h e same
compute
sections
t h e basin.
are used t o compute t h e heat and
while
thermal salt
Zero advective f l u x through t h e bottom
the wind
flux
and
by
closed
boundaries i s ensured by t h e upwind f o r m u l a t i o n o f t h e advection terms i n ( 4 ) .
2.3 The i n p u t data The model
domain
extends
from
the
Greenland-Scotland Ridge
S t r a i t i n c l u d i n g t h e Barents Shelf west o f 30'E The
bathymetric
Labratory
field
has
Charts, Washington D.C.
anticipating
been
digitized
(1980).
t o t h e Fram
(Fig.1). from
the
U.S.Nava1
Research I t shows l a r g e g r a d i e n t s up t o 10- 2
s t r o n g topographic i n f l u e n c e on t h e c i r c u l a t i o n p a t t e r n .
Open boundaries are assumed i n t h e Denmark S t r a i t , Faeroe-Shetland Channel.
t h e Fram S t r a i t and the
No exchange w i t h neighbouring seas i s allowed f o r over
t h e Iceland-Faeroe Ridge, where t h e t r a n s p o r t s are extremely v a r i a b l e b u t low i n t h e average (Meincke 1983). On
the
They can be neglected i n longer term
calculations.
Barents Shelf t r a n s p o r t s are low and estimates u n r e l i a b l e (Aagaard and
Greisman 1975).
For t h e passage between Norway and Scotland model
computations
273 indicate
a t r a n s p o r t o f about 1 SV being o n l y 1/7 o f the
simulated
transport
through the Faeroe-Shetland Channel (Backhaus e t a l . 1985). These boundaries are treated as closed too. As an i n i t i a l s t r a t i f i c a t i o n and f o r t h e boundary conditions t h e c l i m a t o l o g i cal seasonal and annual mean hydrographic data o f the a t l a s published by Levitus (1982) have been used. A t the open boundaries a b a r o t r o p i c v e l o c i t y taken from observation
(Aagaard e t a l . 1973,
Hanzlick 1984)
o r other model
simulations
(Backhaus e t a l . 1985) i s added. The t o t a l t r a n s p o r t through the open boundaries i s 7 SV i n the Westspitsbergen- and Eastgreenland Current , 5.3 SV inflow through the Faeroe Shetland Channel , an i n f l o w o f 0.6 SV o f A t l a n t i c water West o f Iceland and 5.9 SV o u t f l o w i n the Denmark S t r a i t .
1 \ \ \ - I
,.
Fig.2.
(a)Climatological annual mean wind stress. Maximum : 0.14 N/mL (b)Monthly mean wind s t r e s s f o r J u l y 1980. Maximum : 0.07 N/m
2
I n the present computations seasonal c l i m a t o l o g i c a l mean wind s t r e s s f i e l d s computed from d a i l y mean wind s t r e s s data f o r the period 1955-1982 on a 1" g r i d (Backhaus e t a l . 1985) are used.
The data show a pronounced seasonal cycle w i t h
strong p o s i t i v e wind s t r e s s c u r l f o r the sumner t o w i n t e r months. months the wind s t r e s s i s s i m i l a r t o the annual mean (Fig.2a).
During these I n spring i t i s
reduced by one order o f magnitude and i n some years even reverses. An example i s the mean wind s t r e s s o f J u l y 1980 w i t h the l a r g e s t negative stress c u r l f o r the whole p e r i o d (Fig.2b).
Additionally,
anomaly
of
wind
a number o f runs have
been performed w i t h other data sets t o t e s t the response t o i n f l o w
situations
varying i n the range o f observation. These runs are described below. 3 THE SIMULATION RESULTS Anderson and G i l l (1975) have shown, t h a t the response o f a s t r a t i f i e d f l a t ocean t o a change i n wind s t r e s s can be described i n terms o f p l a n e t a r y Rossby waves which are generated a t t h e coasts i n order t o s a t i s f y boundary conditions.
214
The
time t o e s t a b l i s h a steady s t a t e a t an i n t e r i o r p o i n t i s t h e time
by t h e long wave t o t r a v e l from the western boundary t o t h a t p o i n t . account
the
results
i n a time scale o f some months f o r t h e b a r o t r o p i c mode and
for
the
required
Taking i n t o
h i g h l a t i t u d e and h o r i z o n t a l e x t e n t o f t h e basin i n q u e s t i o n
b a r o c l i n i c mode.
Topographic features,
many
this years
which d i v i d e t h e r e g i o n
into
several sub-basins, add basin mode time scales (Anderson and K i l l w o r t h 1977). Previous work has shown t h a t much o f t h e i n f o r m a t i o n on t h e l o n g forcing
i s s t o r e d i n t h e mean d e n s i t y f i e l d .
(1972) and Backhaus and Maier-Reimer (1983) have shown, wind tic
term mean
For example Holland and Hirschman t h a t switching o f f
the
f i e l d r e s u l t s i n o n l y minor changes o f t h e c i r c u l a t i o n p a t t e r n i n diagnoscalculations.
Thus an i n v e s t i g a t i o n o f seasonal v a r i a t i o n s
over several cycles, u s i n g f u l l y v a r y i n g f o r c i n g f u n c t i o n s ,
should
extend
thereby r e q u i r i n g a
great amount o f computer time. T r y i n g t o s p i n up a r e p r e s e n t a t i v e monthly c i r c u lation
with
monthly mean wind s t r e s s data from a s t a t e o f r e s t
will
lead
to
u n r e a l i s t i c r e s u l t s (Creegan 1976). I n o r d e r t o i n v e s t i g a t e the p o s s i b l e i n f l u e n c e o f v a r y i n g boundary c o n d i t i o n s on t h e general c i r c u l a t i o n t h e f o l l o w i n g experiments have been made. Except f o r one s i m u l a t i o n w i t h a homogeneous model t h e i n i t i a l s t r a t i f i c a t i o n i s always taken from t h e c l i m a t o l o g i c a l a t l a s ( L e v i t u s 1982), i . e . annual and seasonal c l i m a t o l o g i c a l mean. The model i s then f o r c e d w i t h t h e corresponding wind stress and buoyancy f l u x a t t h e surface and v e l o c i t y p r o f i l e s a t t h e open boundaries computed as described above. I t i s then allowed t o a d j u s t t o these f o r c i n g funct i o n s i n a p r o g n o s t i c c a l c u l a t i o n f o r one month. This i s about t h e time needed t o e s t a b l i s h t h e c o n t i n e n t a l s h e l f c u r r e n t s t h a t a r e i n i t i a t e d a t t h e open boundaries, as has been v e r i f i e d by d i r e c t comparison o f t h e r e s u l t s w i t h d i f f e r e n t i n f l o w c o n d i t i o n s . The problem remains whether t h e d e n s i t y data used are s u i t a b l e f o r these d i a g n o s t i c c a l c u l a t i o n s . C1 i m a t o l o g i c a l data tend t o be smooth by averaging moving f r o n t s . Anyhow,
once
a s t r a t i f i c a t i o n has been accepted,
t h e response t o
f o r c i n g f u n c t i o n s l a s t i n g f o r one month can be t e s t e d . TABLE 1 I n i t i a l and boundary c o n d i t i o n s o f t h e experiments.
SOW5CL S5W5CL S5W5OP S3JUOP S5W3SY SlWlOP S2W2OP S3W3OP S4W4OP
Stratification
Wind
homogenous a.m. , c l imat.
a.m. ,cl imat.
II
summer, 'I I1 a.m., winter,climat. spring, 'I summer, 'I fall, I1
Open Boundaries closed
II
I1
II
open,climat.
J u l y 1980 sumner,climat. winter,climat. s p r i n g, sumner , " fall, I1
I1
open,synoptic open,climat. I1 I,
I1
variable
275
Other t e s t c o n d i t i o n s e x c e p t t h o s e d e r i v e d f r o m t h e c l i m a t o l o g i c a l mean a r e : closed l a t e r a l boundaries; open boundary c o n d i t i o n s d e r i v e d f r o m s y n o p t i c hydrographic
sumner
sections
with
l a r g e r v e l o c i t y shear and
(keeping
the
total
t r a n s p o r t c o n s t a n t ) l a r g e r s u r f a c e e l e v a t i o n g r a d i e n t s ; wind s t r e s s o f J u l y 1980 w i t h a l a r g e n e g a t i v e anomaly. These e x p e r i m e n t s a r e l i s t e d i n T a b l e 1. Except f o r t h e r e s p e c t i v e changes mentioned t h e y a l l use t h e annual mean c o n d i t i o n s and can be compared w i t h t h i s case. 3.1 The annual mean case The since
general
p i c t u r e o f t h e s u r f a c e c i r c u l a t i o n i n t h e GNS
has
been
known
l o n g ( M e t c a l f 1960) and most o f i t s f e a t u r e s a r e reproduced b y t h e model.
Fig.3. C u r r e n t f i e l d a t 21 F i g . 3 and 4 show
m f o r t h e annual mean case. Maximum : 32 cm/sec.
t h e h o r i z o n t a l c i r c u l a t i o n p a t t e r n a t 21 m ( 1 / 2 o f
the
grid
p o i n t s a r e shown) below t h e w i n d d r i v e n s u r f a c e c i r c u l a t i o n and a t 1500 m i n t h e deep
water.
A
broad n o r t h w a r d d r i f t o f A t l a n t i c w a t e r appears i n t h e
eastern
276
half of the basin and turns to the east when it encounters the Jan-Mayen Mohn Ridge. Velocities drop from 3 cm/sec at the surface to 0.5 cm/sec at 500 m. Helland-Hansen and Nansen (1909) have reported the advective time scale of temperature anomalies from the Sognefjord to the Barents Shelf to be 2 years. This corresponds to a velocity of 2 cm/sec. The same drift velocity of 2-3 cm/sec has been observed by Dickson and Blindheim (1984) from measurements of the large salinity minimum in the Faeroe-Shetland region in 1976 and near Bear Island in 1978/79.
200
Fig.4. Current field at 1501 m for the annual mean case. At the Barents Shelf break the greater part again turns to the north forming the Westspitsbergen Current (WSC) while one branch flows onto the Barents Shelf. A meridional section of zonal velocities (Fig.5) shows the vertical structure of the WSC. Current speeds drop from 20-30 cm/sec at the surface to 12 cm/sec in 550 m depth. This compares well with the vertical velocity shear reported by Hanzl ick (1984), derived from year long current measurements in 1976-78. The
277
total simulated transport is 6 SV, wich is comparable with the mean value of 5.6 SV of Hanzlick (1984) and 7 SV of Aagaard et al. (1973). Coastal currents have developed at the Norwegian and the Greenland coast. Both have speeds up to 10 cm/sec. It should be kept in mind that most of the Greenland Shelf is covered by ice all year long and no allowance is made for its influence on the surface boundary conditions in the model. On the Greenland side polar water flows southward along the shelf break in the East Greenland Current (EGC). One branch of it turns to the east at the JanMayen Ridge, the Jan-Mayen Current (JMC) with velocities of 2-3 cm/sec down to the bottom (Fig.5) but most of it leaves the basin through the Denmark Strait. Thus the circulation is divided into two large gyres with their center in the Greenland and Icelandic basins. The surface velocities of the EGC over the shelf break increases from 10 cm/sec at 7 P N to about 20 cm/sec at 73'N. This might be compared with direct current measurements from ice islands and drifting buoys. Reported velocities are 4 to 12 cm/sec at 800N and increase to 14 to 24 cm/sec at 700N (Einarsson 1972, Aagaard and Coachman 1968). The simulated transport is 7.6 SV. In both currents the transport values are mainly influenced by the downstream inflow conditions.
200
H
1100
H
600
H
800
H
1000 M
2000 M
3000 H
Fig.5. Vertical section of zonal velocity. Section A. The positions o f the sections are given in Fig.1. Contour interval is 1 cm/sec. The circulation of the deep water masses at 1500 m beneath the Atlantic layer is divided into several larger gyres. In the Greenland Basin the rotation is cyclonic as in the upper layers, but in the north-eastern part of the Norwegian basin it has reversed. This has already been reported by Eggvin (1961). Velocities are less than 2 cm/sec except at steep topographic features. A vertical section of meridional velocities extending from the Greenland coast at 78O N to the Norwegian coast at 65'N shows the position of the EGC, the Norwegian Current (NSC), and the coastal current (NCC) (Fig.6).
218
The formation of gyres in the deep flow can be discussed by means of the vertically integrated vorticity equation. It gives a balance between the time derivative o f relative vorticity Z, advection o f relative vorticity A, advection of planetary vorticity B, dissipation of relative vorticity V, bottom pressure torque P, dissipation of vorticity by bottom friction R, and wind stress curl W (Holland 1973). Two sections of the 5 largest terms A,B,V,P, and W are shown in
200
M
1100
M
600
Pl
800
M
1000 M
2000 M
3000 M
Fig.6. Same as Fig.5 but for section B. Fig.7. Section I, perpendicular to the Greenland shelf break, shows a balance between the bottom pressure torque and viscosity effect modified by advection of planetary and relative vorticity. This picture is typical for regions with large topographic gradients, i.e. almost everywhere. Even with the much smaller slope of the inner part of the Lofoten basin (section 11) the bottom pressure torque 0.9
0.5 0.1 0.9
0.5 0.1 -0.1
0.1
-0.1
-0.5 -0.9
-0. I
-0.5
-0.5 -0.9
NW SE Fig.7. Main terms of the vertically integrated vorticity equation. (a) Section I (b) Section 11. The positions are given in Fig.1. Symbols are explained in the text.
279
--(a)
7 . 0 sv SlWl(1P u . 7 sv S2W20P 6 . 1 SV S3W3bP 6 . 7 SV SuWll(1P
(b)
249 KM
166 K M
Fig.8. Transport i n t h e EGC ( a ) and WSC ( b ) f o r seasonal experiments of Table 1. The
coordinate
represented b y each column)
I
I
6 3
u n i t s are 10 m /sec ( v e l o c i t y i n t e g r a t e d over
the
area
. 1
131
I
D
0 N
D
0
280 is
not negligible.
slope
is
Only i n t h e southern p a r t o f t h e section,
practically
zero,
where the bottom
an approximate Sverdrup balance
holds
(Fig.7b).
- H /H. Y the small value o f R and t h e l a r g e topographic g r a d i e n t s a balance o f the
The r e l a t i v e importance o f the p l a n e t a r y and topographic terms i s R / f With above
mentioned k i n d
forcing i s barotropic.
might be expected as long as the
response
to
variable
Willebrand e t a1 .(1980) have suggested t h a t t h e response
wind varying a t a scale l a r g e r than 100 km i s indeed t o a l a r g e e x t e n t baro-
to
tropic.
A longer term r u n o f two years shows, t h a t t h i s k i n d o f balance remains
v a l i d , though t o a somewhat l e s s e r e x t e n t . 3.2 The seasonal runs The
seasonal
pattern.
hydrographic and wind s t r e s s data g i v e a
similar
circulation
The main d i f f e r e n c e i s i n the s t r e n g t h o f t h e c u r r e n t s . The t r a n s p o r t s
on a section through t h e WSC and EGC are shown i n Fig.8.
No s i g n i f i c a n t d i f f e r -
ence i n the v e r t i c a l v e l o c i t y shear can be detected. 3.3 I n f l u e n c e o f p r e s c r i b e d i n f l o w t h e homogenous case, t h e s t r a t i f i c a t i o n of t h e f o l l o w i n g
Except f o r
experi-
ments i s the annual mean. The homogenous r u n shows low t r a n s p o r t s i n t h e WSC and EGC (Fig.lO), Holland
topography pattern
compared w i t h experiment S5W5CL, i n agreement w i t h t h e
and Hirschman on
the
(1972) concerning t h e i n f l u e n c e o f
t r a n s p o r t values.
f o r experiment S5W3SY.
continental
shelf
Fig.9 shows t h e
2nd
level
of and
circulation
The main d i f f e r e n c e l i e s i n t h e s t r e n g t h o f the
c u r r e n t emanating from t h e open
along t h e
s h e l f break around t h e basin.
case
which r e s u l t s i n reduced c u r r e n t s .
too,
result
baroclinicity
boundaries
and
travelling
This i s t r u e f o r t h e closed
boundary
I t can be seen t h a t t h e variance
induced by t h e c o n d i t i o n s a t t h e open boundaries,
which l i e w i t h i n t h e range of
observation, i s a t l e a s t as l a r g e as t h a t induced by t h e seasonal
wind
stress
p a t t e r n and hydrography (Fig.8 and 10).
---
-Fig.10.
2.1
sv sv
SOWSCL SSWSCL 6.0 SV SSWSOP 8.0 S V S5W3SY ‘4.0 SV S3JUOP
‘4.2
2U9 K M (b) Same as Fig.8 b u t f o r t h e f i r s t 5 experiments o f Table 1.
166
KM
281 The
i n f l u e n c e o f a one month anomalous wind s t r e s s i s shown i n
largest
difference
Norwegian
shelf
as
compared w i t h t h e annual mean case
is
where t h e d i r e c t i o n o f t h e coastal c u r r e n t has
Fig.11.
found
The
on
the
reversed.
The
southward undercurrent o f f t h e Lofoten I s l a n d s extends now t o t h e surface.
Fig.11. Same as Fig.10 f o r experiment S3JUOP. The maximum value i s 24 cm/sec.
4 CONCLUSIONS The
results
o f t h e experiments described show a s t r o n g i n f l u e n c e o f
topo-
graphy on t h e general c i r c u l a t i o n . This can be seen by t h e c i r c u l a t i o n f i e l d s of a s i m u l a t i o n o f one month d u r a t i o n s t a r t i n g w i t h observed d e n s i t y data. A calculation
of
t h e terms o f t h e v e r t i c a l l y i n t e g r a t e d v o r t i c i t y equation shows
dominance o f t h e bottom pressure torque i n regions o f v a r y i n g almost tions.
everywhere.
topography,
The c i r c u l a t i o n f i e l d s are i n good agreement w i t h
the i.e.
observa-
282
The influence of various surface forcing and inflow situations at the passages connecting the GNS with the Artic and North Atlantic Oceans has been investigated. The changes in the transport values of the EGC'and the WSC induced by inflow situations varying in the range of observations are comparable to those caused by seasonal mean and anomalous windstress conditions lasting for one month. 5 REFERENCES Aagaard, K, and Coachman, L.K., 1968. The East Greenland Current North of Denmark Strait:I&II. Arctic, 21 : 181-200,267-290. Aagaard, K. and Greisman, P., 1975. Towards new mass and heat budgets for the Arctic Ocean. J. of Geophys. Res., 80(27) : 3821-3827. Aagaard, K., Darnell, C. and Greisman, P., 1973. Year-long current measurements in the Greenland-Spitsbergen Passage. Deep-sea Res., 20 : 743-746. Anderson, O.L.T. and Gill, A.E., 1975. Spin-up of a stratified ocean, with applications to upwelling. Deep-sea Res., 22 : 583-596. Anderson, D.L.T. and Killworth, P.O., 1977. Spin-up o f a stratified ocean, with topography. Deep-sea Res., 24 : 709-732. Arakawa, A. and Lamb, V.R., 1977. Computational design of the basic dynamical processes of the UCLA general circulation model. Methods in Computational Physics, 17 : 173-265. Backhaus, J.O. and Maier-Reimer, E . , 1983. On seasonal circulation patterns in the North Sea. In: North Sea Dynamics, Sundermann,Lenz (Editors), Springer. Backhaus J., Hainbucher O., Quadfasel, D. and Bartsch, J., 1985. North Sea Circulation anomalies in response to varying atmospheric forcing. I.C.E.S., C.M./C: 29, Hydrography Committee. Backhaus, J., Bartsch, J., Quadfasel, D. and Gudall, J., 1985. Atlas of monthly surface fields of air pressure, wind stress and wind stress curl over the North Eastern Atlantic Ocean: 1955-1982. Technical Report 3-85, Inst. of Oceanography, University of Hamburg, FRG. Carmack, E. and Aaqaard, K.,- 1973. On the deep water of the Greenland Sea. DeepSea Res., 20 : 687-715. Creegan, A., 1976. A numerical investigation of the circulation in the Norwegian Sea. Tellus. 28(51 : 451-459. Dickson, H.D. and Blindheim, J., 1984. On the abnormal hydrographic conditions in the European Arctic during the 1970s. Rapp. P.-v. Reun. Int. Explor. Mer, 185 : 201-213. Eggvin, J., 1960. Some results of the Norwegian hydrographical investigation in the Norwegian Sea during the IGY. Rapp. et Proc.-Verb.149, Cons. Internat. Explor. de la Mer. Einarsson, T., 1972. Sea currents, ice drift, and ice composition in the East Greenland Current. In: Sea Ice, Karlsson (Editor), Nat. Res. Counc. of Iceland, Reykjavik, 23-32. Hanzlick, D.J., 1983. The West Spitsbergen Currents: Transport, Forcing, and Variability. Ph.D.Thesis, University of Washington. Helland-Hansen, B . and Nansen, F . , 1909. The Norwegian Sea. Its physical oceanography based upon the Norwegian researches 1900-1904. Rept. Norw. Fish. Mar. Invest. 2(1,2), 390pp. Holland, W.R., 1973. Baroclinic and topographic influences on the transport in western boundary currents. Geophys. Fluid Dynamics, 4 : 187-210. Holland, W.R. and Hirschman, A.D., 1972. A numerical calculation of the circulation in the North Atlantic Ocean. J. of Phys. Oceanogr., 2 : 336-354. Latif, M., Maier-Reimer, E. and Olbers, D.J., 1985. Climate variability studies with a primitive equation model of the Equatorial Pacific. In: J.C.J.Nihou1 (Editor),Coupled ocean-atmosphere mode1s.p~ 63-81.
283
LeBlond, P.H. and Mysak, L.A., 1978. Waves in the Ocean. Elsevier Scientific Pub1 .Co. Levitus, S., 1982. Climatological atlas of the world ocean. NOAA, Prof.Paper 13. Meincke, J., 1983. The modern current regime across the Greenland-Scotland Ridge. In : Structure and development of the Greenland-Scotland Ridge, Bott, Saxov, Talwani, and Thiede (Editors),pp. 637-649. Metcalf, W.G., 1960. A note on water movement in the Greenland-Norwegian Sea. Deep-sea Res., 7 : 190-200. Sarkisyan, A.S. and Ivanov, V.F., 1971. Joint effect of baroclinicity and bottom relief as an important factor in the dynamics of sea currents. Izv., Atmospheric and Oceanic Physics, 7 ( 2 ) : 173-188. Serntner, A.J., 1976. A numerical simulation of the Artic Ocean circulation. J. of Phys. Oceanogr., 6 : 409-425. Swift, J.H., 1984. The circulation of the Denmark Strait and Iceland-Scotland overflow waters in the North Atlantic. Deep-sea Res., 31(11) : 1339-1355. Weatherly, G.L., 1972. A study o f the bottom boundary layer of the Florida Current. J. Phys. Oceanogr., 2 : 54-72. Willebrand, J., Philander, S.G.H. and Pacanowski, R.C., 1980. The oceanic response to large-scale atmospheric disturbances. J.Phys.Oceanogr., 10 : 411-429.
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A THREE DIMENSIONAL BAROCLINIC MODEL OF THE WESTERN BALTIC
M.J. BOEHLICH I n s t i t u t f u r Meereskunde d e r U n i v e r s i t a t Hamburg, Heimhuder S t r a s s e 71, 2000 Hamburg 13, FRG
ABSTRACT The c i r c u l a t i o n i n t h e w e s t e r n B a l t i c i s s i m u l a t e d b y use o f a t h r e e dimensional b a r o c l i n i c n u m e r i c a l model. Eddies and o t h e r s m a l l s c a l e f l o w f e a t u r e s a r e r e s o l v e d and can be r e l a t e d t o t h e b o t t o m topography o r t o b a r o c l i n i c e f f e c t s . Boundary c o n d i t i o n s a r e p r o v i d e d b y two c o a r s e r g r i d models which a l s o t a k e remote f o r c i n g f r o m t h e N o r t h Sea and t h e B a l t i c p r o p e r i n t o account.
1 THE PROBLEM The
d e p l e t i o n o f oxygen i n t h e deep l a y e r s o f t h e K i e l B i g h t i s a w e l l known
problem ( E r h a r d t and Wenck, it
is
not
clear
atmospheric discharge oxygen
1982,
Gerlach, 1984, M i l j d s t y r e l s e n , 1984). So f a r
whether i t was caused n a t u r a l (e.g.,
o f n u t r i e n t s and oxygen consuming
d e p l e t i o n has a n a t u r a l cause,
conditions
of
mixing,
t h e area ( i . e .
up- and
physically strong
anomalies
There
oxygen d e p l e t i o n .
from land).
i t may be connected w i t h
water exchange w i t h
downwell i n g ) .
induced
substances
are
essentially
(by the
If
the
North two
the
physical
Sea, t u r b u l e n t hypotheses
for
resulting
s t r a t i f i c a t i o n d u r i n g calm weather c o n d i t i o n s may l e a d t o t h e
formation
o f oxygen d e p l e t i o n when m i c r o b i o l o g i c a l all
t h e d i s s o l v e d oxygen
hypothesis (Grass1 and Stengel, establish depletion.
a
degradation o f s i n k i n g organic
i n t h e bottom water.
1985,
Two s t u d i e s
c l e a r c a u s a l l i n k between t h e p h y s i c a l c o n d i t i o n s and i s based on t h e assumption
second
but
moderate and v a r y i n g winds ( e p i s o d e s ) may l e a d t o
upwelling
and
hypothesis
matter of
this
F r e y and Becker, 1986). were n o t a b l e t o
The
conditions sea
First,
the
of
man-made
weak c u r r e n t s and t h e
consumes
open
by
f o r c i n g o r anomalies o f temperature and s a l i n i t y ) o r
s m a l l s c a l e eddies.
the
that
Both processes may
oxygen
not
calm
coastal lead
to
and an
entrainment o f n u t r i e n t r i c h w a t e r f r o m t h e b o t t o m l a y e r i n t o t h e s u r f a c e l a y e r . Hence bottom w a t e r w i t h h i g h n u t r i e n t c o n t e n t b u t low oxygen pumped
into
the
e u p h o t i c zone.
T h i s r e s u l t s i n an
concentration
increase o f
the
is
primary
286
production.
The
degradation
of
t h i s o r g a n i c m a t t e r lowers even
further
the
oxygen c o n c e n t r a t i o n w i t h i n t h e water column and e s p e c i a l l y a t t h e sea bed. The
purpose
of
this
paper i s t o i n v e s t i g a t e t h e
second
hypothesis
o u t l i n e d b y means o f a numerical c i r c u l a t i o n model o f t h e K i e l B i g h t . of
interest
is
shown
in fig.
1.
The d e s i g n o f t h e rode1 i s
f o l l o w i n g known f e a t u r e s o f t h e system.
N
550
F i g . 1. Topography o f t h e w e s t e r n B a l t i c model, (model C ) . Depth i n m e t e r s .
The
based
on
just area the
287 The
K i e l B i g h t i s p a r t o f t h e t r a n s i t i o n area between t h e N o r t h Sea and
Baltic.
The
N o r t h Sea i s a s h e l f sea w i t h h i g h s a l i n i t y
c u r r e n t s d i r e c t l y i n f l u e n c e d by t h e ocean. enclosed
sea
and
The B a l t i c i s a
strong
tidal
continental,
semi-
which has r e l a t i v e l y low s a l i n i t y and n e a r l y no t i d a l
The
system
N o r t h Sea
by
a strait.
The
-
currents.
B a l t i c may be understood as two l a r g e b a s i n s
mean
the
connected
c u r r e n t s i n t h e t r a n s i t i o n area a r e a r e s u l t
of
the
water budget o f t h e B a l t i c . and
E v a p o r a t i o n and p r e c i p a t i o n o v e r t h e B a l t i c b a l a n c e each o t h e r ( D i e t r i c h Schott,
1974),
so
driving
force
for
that
the
f r e s h water s u r p l u s due t o r i v e r r u n o f f
the estuarine c i r c u l a t i o n
in
the
transition
is
area.
the This
c i r c u l a t i o n i s two l a y e r e d w i t h an o u t g o i n g c u r r e n t o f low s a l i n i t y water a t t h e surface and an i n g o i n g c u r r e n t o f h i g h s a l i n i t y w a t e r a t t h e bottom. Coriolis
force
Swedish coast
west
and
t h e o u t g o i n g c u r r e n t i s d e f l e c t e d t h r o u g h t h e Sound coast,
through
transition
the
pressure
over
favorable
whereas t h e i n g o i n g c u r r e n t f l o w s a l o n g t h e
the
area
from
Due t o t h e
Great B e l t . are
Deviations from the
caused b y d e v i a t i o n s
mean
Scandinavia
atmospheric
large
induces an o u t f l o w i n t o t h e
situation
for
flow f i e l d
f r o m t h e mean sea
mean d e n s i t y g r a d i e n t between t h e two
basins.
The
most over
Then,
the
n e g a t i v e anomaly o f t h e sea l e v e l i n
resulting Baltic,
wind stress lead t o a followed
air
pressure
J u t l a n d and low p r e s s u r e over Sweden. western
the
gradient High
Sea.
high
the east
in
level
North
i n f l o w occurs w i t h
along Danish
b o t h t h e atmospheric p r e s s u r e
by a t r a n s p o r t o f water from t h e Kattegat
into
and the the
Baltic. Another
reason
f o r c i n g and t h e
f o r d e v i a t i o n s f r o m t h e mean f l o w f i e l d i s t h e
occurence
local
o f s e i c h e s i n t h e e n t i r e B a l t i c generated b y
wind sudden
changes o f t h e s t r e n g t h o r t h e d i r e c t i o n o f t h e w i n d s t r e s s . main cause f o r t h e l i m i t a t i o n o f water exchange between t h e two
The and
thus
Baltic
f o r t h e oxygen problem o f t h e B a l t i c i s i t s s p e c i a l is
Inflowing
d i v i d e d i n t o a s e r i e s of b a s i n s s e p a r a t e d b y s h a l l o w s water
with
h i g h d e n s i t y creeps a l o n g t h e
deepest c o n n e c t i o n s between t h e b a s i n s .
Moreover,
up t o t h e h e i g h t of t h e s i l l b e f o r e t h e water o f t h e
If
the
inflowing
basins,
Kattegat
barocl i n i c water.
have lowered i t s d e n s i t y . or
and
exchanged
each b a s i n
t h e c e n t r a l B a l t i c i s s h a l l o w and
The
sills. the
must be f i l l e d
n e x t b a s i n can be renewed. situated
u n t i l molecular
Unfortunately,
and
following
i t i s i n j e c t e d a t a l e v e l above t h e b o t t o m determined b y
diffusions the
bottom,
water i s not heavier than t h e water
I n t h i s case, t h e b o t t o m water i s n o t
basins
topography.
in
the
its
density.
and t u r b u l e n t
t h e connection narrow
so
deep
between
that
strong
b a r o t r o p i c p r e s s u r e g r a d i e n t s a r e needed t o exchange t h e
bottom
The topography o f t h e western B a l t i c c o n s i s t s o f narrow trenches, p a r t l y
connected
wi t h
each
o t h e r and o f submarine d e p r e s s i o n s .
This
results
in
a
288
complicated s t r u t u r e o f t h e f l o w f i e l d . To
summarize,
depend
we
see t h a t t h e c u r r e n t s w h i t h i n t h e K i e l B i g h t
not
only
on t h e l o c a l w i n d f o r c i n g b u t a l s o on t h e s t a t e o f t h e system N o r t h Sea
Baltic.
The
local
phenomena and t h e remote f o r c i n g have d i f f e r e n t
scales
in
space and t i m e . The
flow
field
windforcing) small some
of
within
t h e western B a l t i c has
s c a l e topography t h e system N o r t h Sea 10 km and because t h e sytem i s i n e r t ,
These
local
numerical needs
characteristics
model
resolution
in
time
scales
of
the
-
Since
satisfactory
observed boundary values
time scales o f t h e order
Obviously t h e
when
model
the
scales, of
days.
designing
requires
qua1 it i y and r e s o l u t i o n ( i n are
to
B a l t i c has l a r g e s p a t i a l
must be t a k e n i n t o account
K i e l Bight.
(local
Due
space and t i m e t o cover a l l e s s e n t i a l l o c a l phenomena.
boundary values o f
time).
small
some hours and small s p a t i a l s c a l e s (some 100m).
a
a
high
It
also
space
and
l a c k i n g , t h e y w i l l be p r o v i d e d by a
l a r g e r s c a l e c i r c u l a t i o n model.
2 MODEL DESIGN The model system c o n s i s t s o f t h r e e components. needed t o compute boundary values f o r t h e l a s t ,
The f i r s t two components
are
l o c a l component ( r e f e r r e d t o as
"model C"). They cover t h e l a r g e s c a l e remote f o r c i n g (model A), and t h e f o r c i n g modified
b y t h e topography o f t h e t r a n s i t i o n area between N o r t h Sea and
Baltic
(model B). A l l components compute p r o g n o s t i c v a l u e s o f t h e c u r r e n t s , sea surface elevation,
s a l i n i t y and temperature. The
f i r s t model (model A) i s a 12
b a r o c l i n i c model w i t h a h o r i z o n t a l r e s o l u t i o n
o f t h e model concept i s g i v e n by Backhaus (1985) and (1987).
Fig.
2
shows
-
layer
A detailed description
o f 12 nm.
b y Backhaus and Hainbucher
t h e a r e a l e x t e n t o f model A and t h e c u r r e n t s
(depth
mean f l o w ) c a l c u l a t e d a f t e r 50 days r e a l t i m e . I t i s f o r c e d b y a ) t h e t i d a l e l e v a t i o n o f t h e sea s u r f a c e a t t h e open boundaries; b ) t h e r i v e r r u n o f f (mean summer
c o n d i t i o n s , 610 km3/year
(Jacobsen, 1 9 8 0 ) ) i n t o
the Baltic; c ) t h e atmospheric wind and t h e p r e s s u r e f i e l d (mean summer c o n d i t i o n s (Backhaus e t al., d) t h e
1985)) and
mean
salinity
September),
3
Fig.
shows
quasi-steady Dietrich The
temperature f i e l d s f o r t h e
summer
season
(May
-
t h e computed
state.
mean sea s u r f a c e e l e v a t i o n i n t h e B a l t i c i n t h e
They compare w e l l w i t h o b s e r v a t i o n s shown as d o t t e d l i n e s ,
and S c h o t t (1974).
grid
within the
and
t a k e n f r o m Lenz (1971) and f r o m Bock (1971 ) .
resolution transition
o f model A i s t o o coarse t o
resolve
the
circulation
area between t h e N o r t h Sea and t h e B a l t i c i n a r e a l i s t i c
way. However, model A i s a good t o o l t o compute t h e boundary and i n i t i a l values,
b m
V
5
c
U
VI
+ 0
E"
a, U
7
VI
R m
W
5 W 5 L
W
t U
3
-6 .-5 4 W
c
E
U 0 0
4-
3 0
7
5
E
4-
slg a,
Q
L U : -0
W
U U 5 Q
s
V
N
.-m lL
289
290
needed f o r t h e
90
100
t h e medium s c a l e model B.
110
12O 13O
14O
15" 1 6 O 1 7 O
Leo
19O 20° 21" 22' 2 3 O 2 4 O 2 9 26" 27" 28" 29" 30°
F i g . 3. Computed mean sea s u r f a c e e l e v a t i o n o f model A i n cm. The d o t t e d l i n e s show t h e sea s u r f a c e e l e v a t i o n a f t e r D i e t r i c h & S c h o t t , 1974. Model B covers t h e t r a n s i t i o n area between t h e N o r t h Sea and t h e B a l t i c , region
between
Skagerrak and Bornholm w i t h
a
grid resolution of 3
c o n c e p t i o n o f model B i s t h e same as t h a t o f model A. since
the
resolution
deepest p a r t o f t h e r e g i o n i s o n l y 80 m, (5-20m).
Model
nm.
the The
I t a l s o has 12 l a y e r s , b u t
i t has a
better
vertical
B i s used t o d e s c r i b e t h e s p e c i a l f e a t u r e s
of
the
A (narrow trenches, s i l l s ) b u t which s t r o n g l y i n f l u e n c e t h e f l o w and t h u s t h e d i s t r i b u t i o n transition
of
area
which
were
s a l i n i t y and temperature.
not s u f f i c i e n t l y
resolved
by
model
The t a s k o f model B i s t o i n t e r p o l a t e t h e r e s u l t s
o f model A i n a p h y s i c a l way.
To model B
initialize by
model
B,
interpolating the
we use t h e r e s u l t s o f model A f o r results o f the
sea
surface
the
region
elevation,
of the
291 salinity
and t h e temperature.
unrealistic barotropic
and
The advantage o f
this
technique
is,
that
b a r o c l i n i c d i s t u r b a n c e s caused b y t h e i n t e r p o l a t i o n
are removed a f t e r a s h o r t t i m e o f computation. Thus model B p r o v i d e s a dynamical i n t e r p o la t ion. Model B i s f o r c e d b y a) The
sea
the
surface elevation,
open
boundaries
and b y p r o f i l e s o f s a l i n i t y and temperature
(Skagerrak
and Bornholm sea)
which
at
are obtained by
i n t e r p o l a t i n g t h e r e s u l t s o f model A; b) t h e same atmospheric wind and p r e s s u r e f i e l d as i n model A . F i g . 4 shows t h e h o r i z o n t a l i n i t i a l f i e l d o f t h e s a l i n i t y o f model 6.
40
After mentioned
days o f r e a l t i m e computation under t h e i n f l u e n c e o f t h e
above t h e
forcing
5. The
s a l i n i t y has advected t o t h e p a t t e r n shown i n f i g .
c u r r e n t s and t h e sea s u r f a c e e l e v a t i o n c o r r e s p o n d i n g t o t h i s s i t u a t i o n a r e shown figs. 6
in
and
7.
A t t h i s s t a g e o f development t h e model system
suitable
t o d e s c r i b e t h e water exchange between t h e N o r t h Sea and
But
representation
the
downwelling) i s n o t model
C
small s c a l e processes
satisfactory.
(e.g.,
To overcome t h i s
already Baltic.
eddies,
drawback,
up- and
we complete t h e
by appending t h e f i n e s c a l e model C.
system
Model
of
is the
c o v e r s t h e r e g i o n between Fyn and Fehmarn (see f i g . 1 ) w i t h a
grid
r e s o l u t i o n o f 0.5 nm i n t h e h o r i z o n t a l . The v e r t i c a l r e s o l u t i o n i s between 2 and
10 m w i t h 10 l a y e r s . salinity same
The boundary v a l u e s f o r model C (sea s u r f a c e
and t e m p e r a t u r e ) a r e o b t a i n e d f r o m t h e r e s u l t s
windstress
as i n t h e preceeding s t e p s a c t s a t
o f model
the
sea
elevation, B
and
the Fig. 8
surface.
shows t h e c u r r e n t s w i t h i n t h e area a t steady s t a t e (mean summer c o n d i t i o n s ) . The advantage
of
structures
model
C
i s t h a t it i s now
possible
to
reproduce
fine
t o 3 nm) such as t h o s e observed b y remote s e n s i n g
(down
scale
techniques
f r o m s a t e l l i t e s (Horstmann, 1983). L o o k i n g a t t h e s e v e r a l stages of t h e model system,
i t becomes c l e a r t h a t t h e
r e d u c t i o n o f t h e g r i d s i z e n o t o n l y works l i k e a m a g n i f y i n g - g l a s s , allows
also
but i t
t h e i n t r o d u c t i o n o f new p h y s i c s t h r o u g h t h e b e t t e r a p p r o x i m a t i o n o f
topography. influence change
The
results
of
o f t h e topography.
the The
c o n s i d e r a b l y compared t o t h e
below t h a t " l a r g e s c a l e " f l o w ,
f i n e mesh model c l e a r l y "large pattern
scale" c u r r e n t of
show
the
pattern
the
dominant does
not
t h e c o a r s e r mesh model B
but
s t r u c t u r e s appear, t h a t were n o t r e s o l v e d b y t h e
preceeding steps. The
benefit
possible
to
forcing.
of
study
t h i s t e c h n i q u e o f a connected model system i s l o c a l small s c a l e processes w i t h o u t n e g l e c t i n g
that the
it
is
remote
292
F i g . 4a. I n i t i a l s a l i n i t y f i e l d o f model B i n t h e 1 s t l a y e r ( 0
-
5 m) i n I .
293
570
56'
55.
54
100
110
120
13O
Fig. 4b. Initial salinity field o f model B in the 3rd layer (10
14O
- 15 m) in
%o.
f-c F i g . 5a. Computed mean s a l i n i t y f i e l d o f model B i n the 1 s t layer ( 0 - 5 m) i n %o.
,
F i g . 5b. Computed mean salinity field o f model B in the 3rd layer (10 - 15 m ) i n 96,.
.
,
.
,
296
a /4
0.5
;2.0 1.0 1.5
3.0
5,. 5.0 6.0 U.0
O at the northern boundary). If this is not true U is not updated. The expression (11) is also used in FOE with U replaced by its global part, i.e. that part of U which represents the free wave part. The global part is defined so that u=ul+ug
(13)
where U1 and Ug are the local (forced) and global (free) part, respectively. The procedure closely follows that described in Roed and Smedstad (1984) with the bottom friction as an additional term and will therefore not be repeated here. The S P O combines the use of radiation conditions and sponge layers. The use of sponge layers in order to dampen out waves and other disturbances generated in the interior are fairly common in atmospheric models, but Israeli and Orszag (1981) was probably the first to suggest to combine a sponge layer and a radiation condition. This had been reported to be fairly successful in many cases (cf. Chapman, 1985). However, the addition of extra grid elements makes the use of sponges very costly. In the present implementation of SPO, a sponge layer 200 km wide is added to the grid in Fig. 1 at both the southern and northern boundaries. Within these layers the bottom friction coefficient is increased exponentially from its interior value to four times that value at the outer edge of the sponge by use of the formula exp (-bx)
: xl) is characterized by
an
interface rising above the obstacle; while in the case of subcritical flow (?$FF is the mass- transport velocity um, which was first introduced by Longuet-Higgins (1969). Equation (20) is called Stokes' formula. The Lagrangian drift velocity, urn, was only recently revealed and named (Feng et af., 1986a; Feng, 1986a). The second order dynamics shows the dependence of the at which the marked water parcel is released Lagrangian residual velocity on the tidal phase €lo from the fixed point x,. In the previous depth-averaged model, the "two-dimensional" Lagrangian drift velocity traces out an ellipse on a hodograph plane as the initial tidal phase eo varies from 0 to 2 ~ in; other words, when the "marked water columns" are released from a fixed position continuously over a tidal period, the ensemble of the tenninal positions of the "marked water columns'' after a tidal cycle form an ellipse in the "two-dimensional space", the Lagrangian residual ellipse (Cheng er al.. 1986; Feng er uf., 1986a). The Lagrangian residual velocity derived in a threedimensional space, equation (19). or the Lagrangian drift velocity, (21). has a similar but three-dimensional structure (Feng, 1986a). Here it should be pointed out that this unique property of the Lagrangian residual velocity reflects its Lagrangian nature since the Lagrangian residual velocity is born of the Lagrangian mean velocity of a marked water parcel in the tidal field but the latter depends on the trajectory that such a parcel follows and the parcels follow
478
different trajectories depending upon the time, e0, of their release at xo. Noting that (%,O0) is to be selected arbitrarily, and then using (x.0) instead of (xo,B0), the Lagrangian residual velocity can be viewed as a Eulerian field variable and the aggregate of such local velocities may be specified as a Eulerian field of flow provided that the Lagrangian residual velocity expressed by (19)-(21) satisfies the continuity equation for an incompressible fluid. As a matter of fact, by taking the divergence of (19)-(21) and going through some algebraic manipulations, we have
Hence, the definition used here of the Lagrangian residual velocity as the Lagrangian mean velocity of a marked water parcel is valid. The case in which uLR cannot be defined by (18) will be examined and discussed in section 6. Thus, (19)-(21) show that the Lagrangian residual velocity field is similar to the tidal current velocity field in the sense that it is a sum of a tidally periodic fluctuation plus the tidal cycle average since uLhn= . However, given that 0 ( I K ULD I / I uM I ) = K, they are different because the tidally periodic part of the tidal current velocity field is typically greater than the residual part by one or more orders of magnitude. In contrast to the Eulerian residual velocity field which is steady, the Lagrangian residual velocity field is obviously a timedependent field of flow. 4. A SET OF FIELD EQUATIONS FOR THE MEAN LAGRANCIAN RESIDUAL
CIRCULATION INDUCED BY AN MZ-TIDAL SYSTEM As stated in section 1, while much effort has been applied to the study of residual circulation driven by the wind on the sea surface and the variation of water density, only recently has attention been drawn to the tide-induced residual circulation. However, in coastal seas, where the dominant observable motions are tides, the residual circulation is induced not only by the wind on the sea surface and the horizontal gradient of water density but also by the nonlinear coupling of tides, as pointed out by Nihoul and Ronday (1975). A scale analysis on the general circulation in the Bohai Sea and the East China Sea has also shown that the tide-induced residual circulation is, in general, a component of the general residual circulation (Feng et d., 1986). The residual circulation is conventionally derived from current-meter records using filter techniques or time averages of time series records to remove tidal variations, it., the residual circulation is conventionally defined as the Eulerian residual circulation. However, it is becoming increasingly clear that the residual circulation should be related to the Lagrangian residual velocity since the problem of residual circulation is to describe and understand the inter-tidal transport processes (Csanady, 1982; Feng, 1986b). Thus, it might be appropriate to define the (tide-induced) residual circulation as the (tide-induced) mean Lagrangian residual circulation. To describe the tide-induced mean Lagrangian residual circulation and to study some of its characteristics, a set of field equations governing the tidal cycle average of the Lagrangian
479
residual velocity, or the mass-transport velocity, uM, is derived as follows : i) Field equations
v.um=o,
ii) Boundary conditions : atz=O,
wM=O,
and
A=--
auLM avM a2 aZ
atz=-h,
-0;
um=0;
where
a 5 a50 + -)h~ v ~ U O 850 1 h~ &O > + Z< % ax ay aZ + (- ay + -2 -)ax v aZ a 3% 850 au0 +. v (v > aZ < v v .t& aZ > - < aZ $=-
2 < -1 G o . VZQ > a Y 2
a 5 h~ 850 &O + (-ho + -1 -)ab aZ ax 2 ay aZ 2 ay + -)ax v avo a avo +aZ < v v . s o aZ > - < aZ . v (v -)aZ >
+-
In these equations, (n,,n2) represent the nonlinear coupling of astronomical tides and can be naturally named “tide-induced body force”. The tidal force contains two parts. The first term characterizes the nonlinear interaction between the tidal displacement and the tidal elevation, and it is horizontally irrotational. The second part represent the effect of eddy viscosity; this term is rotational. (uM,vLM) and wLM are the horizontal and vemcal components of urn, respectively, and is the residual elevation. The conceptual difference between the set of field equations for the Lagrangian residual circulation derived here, (23)-(28), and that for the Eulerian residual circulation (Nihoul and
480
Ronday, 1975; Feng et al., 1984) is revealed by the differences in the kinematic and kinetic boundary conditions at z = 0. Equation (26) shows that there is no "tidal surface source", or
a , at z = 0. Thus, the handling of the continuity equation is simplified when
the mean Lagrangian residual circulation is used to describe inter-tidal processes. There, are other attractive features to this formulation. If a material surface in the water is specified geometrically by the equation F(x,O) = const., F is a quantity which is invariant for a water parcel on the surface, so that :
-DF =De
aF + K U . V F = O
ae
at all points on the surface. In particular, the equation of any surface bounding the sea water must satisfy (29). F (x,e) can also be written as F, + KF, + 9 F 2 + O ( d ) ,like the other variables. Substituting this expansion into equation (29), and taking a tidal cycle average of the latter, we have
urn. V = E 1 aZ
aZ
to the zero-th order approximation, in which So has been approximated by <S>. The equation derived here, (32), is different from the traditional long-term transport equation (Fischer et af., 1979). On the one hand, in the latter, the convection has been unreasonably represented by the Eulerian residual velocity, but in equation (32) the convective transport is reasonably expressed by the Eulerian mean of the Lagrangian residual velocity, i.e., by the mass-transport velocity. On the other hand, an assumption on the so-called "tidal dispersion" has to be introduced into the conventional long-term transport equation (Fischer er af., 1979). whereas equation (32) can describe correctly the Lagrangian nature of the convection affecting inter-tidal transport processes without introducing any hypothesis for tidal dispersion. Let
denote the depth-averaged quantity of A, or
equation satisfied by
-a&
<s> is derived as follows :
-a&
ax + vm aY
= Pe-'
=
-l oA dz.
An inter-tidal transport
-h
1 V . ( h D . V <s>) h
where P, denotes the Peclet number for the residual motion, Pe = m,L / D,; D denotes the dispersion coefficient tensor due to the vertical shear of the horizontal component of the masstransport velocity (Bowden, 1965), and D, represents its scale. Since Pe is usually a large parameter (Bowden, 1965; Feng et al., 1986b), the convective transport effect is greater than the dispersive one. Thus the equation just derived is reduced to
-a&
-a=
ax +vm--
aY
-0
(33)
Equation (33) suggests that, at the zero-th order approximation, the transport process is determined purely by the convection, so that the concentration isolines for <s> coincide with the streamlines of the depth-averaged mass-transport velocity. As a matter of fact, an equation similar to (33) can be derived under similar conditions for the horizontally two-dimensional tidal problem (Feng et al., 1986b). Of course, equation (33), derived in a three-dimensional space, behaves as the depth-average of a three-dimensional mass-transport velocity. It should
482
be pointed out that this equation is valid in the "interior" of a basin because the diffusion or dispersion becomes more pronounced in the "boundary region" (Feng ef al., 1986b).
6. LONG-TERM TRANSPORT PROCESSES
The conclusions derived above concerning the Lagrangian residual velocity and the intertidal transport processes are based upon a weakly nonlinear dynamical model of tidal flow and intra-tidal transport processes. Several hypotheses have been made, including that O(ICN)= O(K). or 0 0 = 1, in which O(K) < 1. Given that N = 1 + (n-l)K, the condition O(N) = 1 is valid in the cases of O(n) < K-' . If we assume O(K)= lo-', then O(n) = ~ - ~ - l O o ,which implies that the theory and model given above could not be directly applied to the cases of long averaging period of time such as a season or a year. It is worth giving the following approach to these problems of long-term transport processes. Instead of the Lagrangian residual velocity, the original Lagrangian mean velocity of a marked water parcel has to be used here, or go+ 2m1
The definition (34) can be reduced to (18) only when O(n) < C2,of course, including n = 1. Nevertheless. the Lagrangian mean velocity expressed by (34) can be formulated by means of the Lagrangian residual velocity. As a matter of fact, noting that eo + 2m1
uL I uL (xo,80;n) = 2nn
where xj = q + 1?2x
j
u (X(q,e).e) de
90
5uL (xi, e0 + 2ni ;l) ' 1
c=1,2,..A), using (18), and substituting (19)-(21)
i=O
into the expression just derived. we have
uL (x,+0;n)= where
(q;n) + K DuL (q.QO;n)
(35)
483
(38) b
xM,O=XO
and x = O
if b < a
a
Obviously, the formulae (35)-(38) will be exactly reduced to the formulae (19)-(21), using the Lagrangian residual velocity uLR instead of the Lagrangian mean velocity uL (xg,e0;l), if n = 1. It is certainly expected that the formulae (35)-(38) can be approximately reduced to the formulae (19)-(21), using the Lagrangian residual velocity uLR instead of the Lagrangian mean velocity UL (xg,B0;n), if O(n) < K-' even though n > 1. In fact, we have X M , ~= xo as O(n) < K-' in view of the terms ( 6 2 A (38); and then,
) = O ( 6 n) < 1 contained in the formulae (35)j
.
and ( "u; (xo;n), "u; (xg;n)) = ( approximately.
ULD (xg), U;D
(xgN ; and finally, UL (xg,80;n) = ULR (xg,00),
When n is of an order of magnitude equal to or greater than K-', the Lagrangian mean velocity UL (xo,e0;n) cannot be proven to satisfy the continuity equation for incompressible fluids. Thus, the Lagrangian residual velocity cannot be defined as the Lagrangian mean velocity, or UL (xo,00;n). when O(n) 1 K-*. It is evident that the Lagrangian mean velocity UL (xo.eO;n) is, in general, not only a function of (xg,B0) but also dependent on n. Explicitly containing n reflects rationally the experience of the Lagrangian motion of the marked water parcel moving from the initial position at time e0 to the terminal position x, at time 00 + 2x11. Because the Lagrangian mean velocity is directly proportional to the net Lagrangian displacement of the marked water parcel, with a proportionality coefficient (2xn)-' , in view of (35)-(38), the latter behaves well like the former, and the horizontal projection of the net Lagrangian displacement will trace out an ellipse over a tidal cycle. The equations also show that the net Lagrangian displacements, or 2 A n UL (xg,eo;n), of marked water parcels released from the position xg at tidal phase e0 over such a long time as a season can be of an order of magnitude greater than those over one or a few of tidal cycles. Thus, the Lagrangian residual drifts, or 2 n: n K "uL (xo,eO;n). of which the horizontal components trace out an ellipse over a tidal cycle, play a more pronounced role in the dispersion, for example, of pollutants on "long" time scales than on "short" scales, although the Lagrangian drift velocities, or K D ~ (xo,e0;n), L are of the same order of magnitude in both cases. For example, the ratio of the Lagrangian residual drift for O(n) = K-' to that for O(n) = 1 is O(K-'), which suggests the importance, at least the potential importance, of the Lagrangian residual drifts for long-term dispersion phenomena though O( K I D ~ (xg,e0;n) L I / I (x,-,;n) I ) = O(K) < 1. Detailed descriptions of the phenomena and the dynamics mentioned in this section can be found in another paper
484
(Feng, 1986~). 7. AN APPLICATION TO THE BOHAI SEA, CHINA
The dynamic model proposed in the present paper should be verified through a test of the theory against field data in a realistic situation. This three-dimensional model can be conveniently applied to the summer residual circulation and the inter-tidal transport processes in the Bohai Sea, China (Fig. 1). A scale analysis has shown that the tide-induced nonlinear effect and the Huanghai Sea Warm Current which enters the Bohai Sea through the Bohai Strait may be the principal factors contributing to the formation of the summer circulation in the Bohai Sea. The wind stress on the sea surface and the baroclinic effects are negligible and less important, respectively (Feng et al., 1986). Thus, the total transport through the Bohai Strait has to be prescribed, that transport can be obtained from the calculation of the current speed based upon field data. For the tides, we can use an existing three-dimensional nonlinear numerical model of the M2-tide in the Bohai Sea. The calculated depth-averaged masstransport velocity field as the mean residual circulation of the summer in the Bohai Sea has recently been obtained (Sun et al., 1986), and is exhibited in Fig. 2. According to equation (33), the concentration isolines for the depth-averaged tidal cycle mean of the concentration of any conservative and passive tracer must approximately coincide with the streamlines of the depth-averaged mass-transport velocity. The salinity can be conveniently used as such a tracer in the Bohai Sea for the summer. The salinity distribution at the depth of 10 m in the Bohai Sea for June, 1958, as derived from observations, is shown in Fig. 3. This picture can be used qualitatively to represent a typical summer distribution of the depth-averaged salinity in the Bohai Sea. Even without including other factors likely to affect the distribution of salinity, such as stratification, the Huanghe River runoff and the coastal dispersion boundary layer, the patterns of isohahe contours of Fig. 3 and the mass-transport velocity field of Fig. 2 seem to be in surprisingly good qualitative agreement. Furthermore, the pattern of the mean Lagrangian residual circulation in the Bohai Sea for the summer can be used to explain dynamically the existence of an area, situated in the northeastern part of Laizhou Bay, where the water is of relatively high salinity and transparency and of relatively low temperature and concentration of suspended sediments. These conditions explain the presence, in that area, of such bottom fauna as spatangia (Su et al.. 1986). It is of interest to note that the pattern of the summer circulation in the Bohai Sea is only one big counterclockwise gyre if the coupled mean tide-induced Lagrangian residual circulation is not considered (Feng et al., 1986). Such a circulation cannot be used to interpret the existence of the area stated above. The present theory for predicting long-term transport processes and studying their dynamics in coastal seas seems to be satisfactory, at least qualitatively.
485
40
Fig. 1. Depth distribution of the Bohai Sea. China, (m).
Fig. 2. Computed depth-averaged mass-transport velocity field (cdsec).
486
Fig. 3. Salinity distribution ( "/oo) measured at the depth of 10 m in the Bohai Sea in June 1958. Arrows denote observed Eulerian residual currents. 8. CONCLUSION
Based upon a three-dimensional weakly nonlinear theory, the Lagrangian residual velocity is approximately expressed as a sum of a tidally periodic fluctuation, which we call the Lagrangian drift velocity, and of the tidal cycle average, i.e., the mass-transport velocity. The Lagrangian residual velocity is different from the Eulerian residual velocity, which is steady, and it is similar to the tidal current velocity. A set of field equations governing the mass-transport velocity is derived. These equations show that the mean Lagrangian residual circulation is more relevant than the Eulerian one to the description of inter-tidal flow processes. In particular, a new inter-tidal transport equation for the tidal cycle average of the concentration of any conservative and passive tracer is proposed, in which the convective transport is characterized by the mass-transport velocity and there is no need for an ad hoc hypothesis on "tidal dispersion". It is also shown that the Lagrangian residual drift, which is related to the Lagrangian drift velocity, is potentially the most important factor controlling the dispersion or spreading of tracers in the long-term transport processes. And finally, when the theory is preliminarily applied to the Bohai Sea, China, a good qualitative agreement between the present theory and field data is revealed.
487
9. REFERENCES Alfrink, B.J. and Vreugdenhil, C.B., 1981. Residual currents. Delft Hydraul. Lab., Delft, The Netherlands, Rep. R 1469-11,42 pp. Bowden, K.F., 1965. Horizontal mixing in the sea due to a shearing current. J. Fluid Mech.,
21 : 83-95. Bowden, K.F., 1967. Circulation and diffusion. In: G.H. Lauff (Editor). Estuaries. Publ., No. 83,AAAS, Washington, D.C., pp.15-36 Cheng, R.T., 1983. Euler-Lagrangian computations in estuarine hydrodynamics. In : C. Taylor, J.A. Johnson and R. Smith (Editors), Proc. of the Third Intern. Conf. on Num. Meth. in Laminar and Turbulent Flow. Pineridge Press, pp. 341-352. Cheng, R.T. and Casulli, V., 1982. On Lagrangian residual currents with applications in South San Francisco Bay, California. Water Resour. Res., vol. 18, 6 : 1652-1662. Cheng, R.T., Feng, S. and Xi, P., 1986. On Lagrangian residual ellipse. In : J. van de Kreeke (Editor), Intern. Conf. on Physics of Shallow Estuaries and Bays, Lecture Notes on Coastal and Estuarine Studies. Springer-Verlag, pp. 102-113 Csanady, G.T., ,1982. Circulation in the Coastal Ocean. D. Reidel Publ. Comp., Dordrecht/Boston/London, 279 pp. Feng, S., 1977. A three-dimensional nonlinear model of tides. Scientia Sinica, vol. 20, 4 :
436-446. Feng, S., 1986a. A three-dimensional weakly nonlinear dynamics on tide-induced Lagrangian residual current and mass-transport. Chinese J. of Oceanology and Limnology, vol. 4,2 :
139-158. Feng, S., 1986b. On the fundamental dynamics of barotropic circulation in shallow seas. Acta Oceanologica Sinica (submitted for publication). Feng, S., 1986c. On tracer spreading in a long-term transport process (in preparation). Feng, S.,and Cheng, R.T., 1986. Formulation of the governing equations for Lagrangian residual current and residual transport. Intern. Symposium on Physics of Shallow Bays, Estuaries and Continental Shelves. Shandong College of Oceanography, Qingdao, China, November 1986 (in preparation). Feng, S., Cheng, R.T. and Xi, P., 1986a. On tide-induced Lagrangian residual current and residual transport, Part I : Residual current. Water Resour. Res., vol. 22, 12 : 1623-1634. Feng, S., Cheng. R.T. and Xi, P., 1986b. On tide-induced Lagrangian residual current and residual transport, Part II : Residual transport with application in South San Francisco Bay, California. Water Resour. Res., vol. 22. 12 : 1635-1646. Feng, S., Xi, P. and Zhang. S., 1984. The baroclinic residual circulation in shallow seas. Chinese J. of Oceanology and Limnology, vol. 2, 1 : 49-60. Feng, S.,Xi, P. and Zhang, S., 1986. Numerical modeling of the general circulation. In : C.K. Tseng and M. Tomczak (Editors), Oceanography of East China Sea, Huanghai Sea and Bohai Sea, Chapter 6. Lecture Notes on Coastal and Estuarine Studies. Springer-Verlag (submitted for publication). Fischer, H.B., List, E.J.. Koh, R.C.Y., Imberger, J. and Brooks, N.H., 1979. Mixing in Inland and Coastal Waters. Academic Press, New York, 483 pp. Heaps, N.S., 1978. Linearized vertically-integrated equations for residual circulation in coastal seas. Deut. Hydrog. Z., 31 : 147-169. Longuet-Higgins, M.S.,1969. On the transport of mass by time-varying ocean currents. Deep Sea Res., 16 : 431-447. Nihoul, J.C.J. and Ronday, F.C., 1975. The influence of the "tidal stress" on the residual circulation. Tellus, 27 : 484-489. F'ritchard, D.W., 1954. A study on salt balance in a coastal plain estuary. J. Mar. Res., 13 :
133-144. Song, L., 1986. A hydrodynamic velocity-splitting model with a depth-varying eddy viscosity in shallow seas. Acta Oceanologica Sinica (accepted for publication).
488
Stem, M.E., 1975. Ocean Circulation Physics. Academic Press, New York, 246 pp. Su, Z., Wiseman, W.T., Fan, Y., Gao, S., Qian, Q. and Yang, Z., 1986. Analyses of hydrologic characteristics of the area adjacent to the Huanghe Estuary (personal communication). Sun, W., Xi, P. and Song, L., 1986. Numerical calculation of the three-dimensional tideinduced Lagrangian residual circulation in the Bohai Sea. Acta Oceanologica Sinica (submitted for publication). Tee, T.K., 1976. Tide-induced residual current : A 2-D nonlinear numerical model. J. Mar. Res., 31 : 603-628. Zimmennan, J.T.F., 1979. On the Euler-Lagrangian transformation and the Stokes’ drift in the presence of oscillatory and residual currents. Deep Sea Res., 26A : 505-520.
489
THREE DIMENSIONAL NUMERICAL MODEL FOR THERMAL IMPACT STUDIES
M. DARRAS, P. DONNARS and P. PECHON Research Engineers Electricit6 de France, Laboratoire National d'lydraulique Chatou. France
-
ABSTRACT ' The three dimensional numerical model ODYSSEE was developed to study the near-field thermal impact of coastal nuclear projects. A finite difference scheme is used to solve the governing equations. To have a good reproduction of vertical mixing, an accurate turbulence modelling is used involving the effect of vertical thermal gradient. The application of the model to the test case of a flume with a stratified flow shows that the model does fairly well in spite of small discrepancies which occur when the advection term and the diffusion term are not negligible in the equations of turbulent fluxes of momentum. Then the model is applied to the case of the nuclear power station of Gravelines. The measured and computed temperature profiles are very similar ; the vertical thermal front and the horizontal stratification are well represented. Nevertheless the horizontal mixing modelling will be improved in the future because it is not accurate enough in some particular cases when velocity shears are strong.
1
-
INTRODUCTION
The thermal studies of coastal nuclear power stations using a once-through cooling system require a good knowledge of the heated water dilution. The marine currents ensure the mixing of temperature so that the dispersion in the ambient flow far from the outlet can be estimated with a two-dimensional numerical model assuming that current and temperature are homogeneous over the depth. In the near-field of a buoyant discharge outlet, the vertical mixing is not strong enough to force the vertical homogenization of water. Then a three-dimensional numerical model or a physical model is needed in order to represent the buoyancy effects in the fluid. The purpose of the paper is to show the recent developments and applications of the three-dimensional model ODYSSEE, which computes simultaneously curent repartition and heat dilution. 2
-
THE NUMERICAL MODEL ODYSSEE
2.1 The basic equations A three-dimensional modelisation needs a simultaneous solution of the
490
heat dilution and of the velocity field, because the heat repartition creates buoyancy effects which influence the current, which in turn influences the heat repartition. The flow is governed by the unsteady Navier-Stokes equations and the mass conservation equation. Simultaneously the temperature repartition is obtained by solving the advection
-
dispersion equation, the exchange of heat between
sea and atmosphere being negligible here : -dT =
div (K grad T) dt where T : temperature K : coefficient of turbulent diffusion. The temperature distribution act on the velocities through the local water density by the following equation :
_- 1
- =a
p
B
a~
P
where P : local water density
B : coefficient of thermal expansion 2 . 2 The assumptions
The equations are simplified using the following assumptions :
- In the case of
tidal or wind induced currents and when the bottom topography
is gentle, the flow pattern is almost horizontal and the vertical acceleration is small compared with the gravity. The pressure is thus assumed to be
hydrostatic in the Navier-Stokes equations.
-
The variation of the water density in the Navier Stokes equations is
linearized using the Boussinesq approximation. 2 . 3 The turbulence modelling
The turbulent fluxes of momentum and of temperature are modelled with the Prandtl's
mixing
length
hypothesis
by
horizontal
and
vertical
eddy
viscosities. Horizontally, the velocity gradients are generally small, so the advection transfers prevail over the
turbulent exchanges. Therefore an arbitrary
horizontal viscosity coefficient is taken (from a reasonable range of values). On the other hand, the vertical turbulent exchanges are not negligible compared with the vertical advection because the latter is very weak. Consequently, the vertical turbulent transfers require an accurate model, as a result of the substantial interference with thermal phenomena. Moreover, it is not desirable to increase the complexity of the system to be solved by introducing additional multidimensional differential equations.
491 Consequently, a compromise is chosen, as follows. The equations used to describe the correlations between the temperature and
-
-
velocity fluctuations (u' u' and utiT', i and j = 1 to 3 for x, y, z), i j as well as the turbulent energy, k = (u" + v q 2 + w")/2, and TIZ, are
-
described through the Launder (1975) model for unknown terms. Dissipation, E
,
is then modelled as follows : E =
C E C .
1 0.47 and 1 is a length scale. The length scale is assumed to be constant in the fluid except near the bottom and the sea-surface where it varies linearly with the distance from the boundary. The horizontal gradients are assumed to be negligible, compared with the
where CE
=
vertical gradients. Moreover, the classical local equilibrium assumption is adopted :. the energy produced is assumed to be instantaneously absorbed or dissipated at the same place, and therefore the diffusion, transport and unsteady terms are not taken into account. These approximations will be discussed in the first application.
It can be shown (Dewagenaere 1979) that the eddy viscosity can be determined from the resulting system, which can be solved algebraically. The resulting turbulent kinetic energy depends on the Richardson number (fig 1) : g aT
In a stable medium (Ri
hT
515 The form f o r pT, hpw
= pw f o r pw>pT This
equation determines p .
J’
particular
position
under
p i n the interval-[j6t,(j+l)6tI, at the
i.e.
consideration,
in
terms
of
%j’
a, b, a t t h a t
position.
5
HORIZONTAL GRADIENTS of 5 and p
Horizontal’ gradients
a
are r e q u i r e d f o r t h e d e t e r m i n a t i o n o f P
and Q from (3). These may b e e v a l u a t e d as f o l l o w s . Suppose t h e p o i n t a t which c u r r e n t s are t o b e e s t i m a t e d c o i n c i d e s w i t h g r i d point
0
from
the
a
of
and ar;/ay a t
Then
Then v a l u e s o f 5
Thus, a t any p a r t i c u l a r time, l e t 5-3, DISTPM
\ 1600 Moo 2400 1600 Moo 2400 DISTPM
FIG.4 : HORIZONTAL VELOCITIES IN THE 3-D BLOCK A T VARIOUS FRACTIONS OF THE DEPTH (SIGMA COORDINATES) 4 HOURS AFTER HIGH WATER (APPLICATION EXAMPLE OF FIG.2)
553
I '
1600
Moo
2400
I'
1600
Moo
2400
MLocmEs AT 0.024
J
J.
>
FIG.5
2ooo
2400
1600
Moo
2400
1600
Moo
*
DISTPM (E
>
D I S T M (I
DEPTH (K.3)
Moo 2400 MLaXTES AT 0.06) DEPTH (K:6) 1600
4
1600
D
7
2400 MLCUTES NPR THI W K E (K-18)
oIsIps*x ( C
: HORIZONTAL VELOCITIES IN THE 3-D BLOCK A T VARIOUS
FRACTIONS OF THE DEPTH (SIGMA COORDINATES) AT LOW WATER (APPLICATION EXAMPLE OF FIG.2)
This Page Intentionally Left Blank
555
A HIGH RESOLUTION THREE-DIMENSIONAL MODEL SYSTEM FOR BAROCLINIC ESTUARINE DYNAMICS AND P A S S I V E POLLUTANT D I S P E R S I O N
J . KROHN and K. DUWE GKSS Forschungszentrum Geesthacht, P.O. Box 1160, 2054 Geesthacht K.D. PFEIFFER I n s t i t u t f u r Meereskunde, U n i v e r s i t a t Hamburg, T r o p l o w i t z s t r . 7 , 2000 Hamburg 54 ( F e d e r a l Republ i c o f Germany)
ABSTRACT I n o r d e r t o examine t h e dynamics o f t h e E l b e e s t u a r y d i f f e r e n t three-dimens i o n a l models have been developed. I n i t s l o w e r p a r t t h e E l b e e s t u a r y i s c h a r a c t e r i z e d by a s i g n i f i c a n t i n f l u e n c e o f b a r o c l i n i c i t y connected w i t h f r e s h w a t e r d i s c h a r g e and s t r o n g small s c a l e v a r i a t i o n s o f b o t t o m topography, i n t h e upper p a r t p u r e f r e s h w a t e r c o n d i t i o n s p r e v a i l . The system o f models t h e r e f o r e c o n s i s t s o f h i g h r e s o l u t i o n b a r o c l i n i c and b a r o t r o p i c v e r s i o n s , a b a s i c v e r s i o n 100 m o f 250 m r e s o l u t i o n c o v e r i n g t h e t o t a l e s t u a r y and f i n e r segments ( 5 0 r e s o l u t i o n ) b e i n g a p p l i e d t o areas o f s p e c i f i c i n t e r e s t . The s h a l l o w w a t e r equations a r e formulated s e m i - i m p l i c i t l y t o y i e l d a s u i t a b l e d i s c r e t i z a t i o n o f t h e space and t i m e domain; t h e d r y i n g o f t i d a l f l a t s i s handled w i t h o u t g i v i n g r i s e t o shockwave-like d i s t u r b a n c e s i n t h e v e l o c i t y f i e l d . The model has been extended t o t r e a t t h e t r a n s p o r t o f p a s s i v e t r a c e r s by a p p l y i n g a 'Monte-Carlo'-technique. The t r a n s f o r m a t i o n i n t o t h e L a g r a n g i a n f o r m u l a t i o n uses l i n e a r i n t e r p o l a t i o n and some m o d i f i c a t i o n s t a k i n g i n t o acc o u n t t h e h i g h v a r i a b i l i t y o f t h e f l o w f i e l d and t h e complex morphology. t h e b r a c k i s h w a t e r zone and t h e f r e s h w a t e r Two areas have been chosen t o s i m u l a t e t h e d i s p e r s i o n o f a number o f r e zone j u s t seaward o f Hamburg l e a s e d p a s s i v e t r a c e r s under t i d a l i n f l u e n c e . R e s u l t s a r e o b t a i n e d f o r r e l e a s e n e a r t h e s u r f a c e and r e l e a s e e v e n l y d i s t r i b u t e d o v e r t h e w a t e r depth. The c a l c u l a t e d p a r t i c l e p a t h s and p a r t i c l e d i s t r i b u t i o n s g e n e r a l l y show a l a r g e v a r i a b i l i t y , depending on t h e l o c a t i o n and t h e t i d a l phase o f i n s e r t i o n .
-
-
-
1 INTRODUCTION The r i v e r Elbe, f l o w i n g i n t o t h e N o r t h Sea a t t h e s o u t h e a s t e r n c o r n e r o f t h e German B i g h t i s one o f t h e l a r g e s t r i v e r s i n Europe ( a b o u t 1100 km t o t a l length).
I t s d r a i n a g e area i n c l u d e s l a r g e towns (e.g.
Prague, Dresden, B e r l i n ,
Hamburg) and i n d u s t r i a l areas. Consequently t h e w a t e r i s c o n s i d e r a b l y p o l l u t e d by u r b a n sewage and i n d u s t r i a l e f f l u e n t s .
The hazardous e f f e c t o f p o l l u t a n t s
becomes even more dangerous i n t h e t i d a l l y i n f l u e n c e d l o w e r p o r t i o n o f t h e r i v e r where t h e f l o w r e v e r s e s w i t h i n t h e s e m i d i u r n a l t i d a l c y c l e . t h a t t h e same volume o f w a t e r passes a f i x e d p o i n t (e.g. s e v e r a l times.
T h i s means
a sewage i n t r o d u c t i o n )
The r e l a t i v e e f f e c t o f f r e s h w a t e r d i s c h a r g e on t h e v e l o c i t y
f i e l d decreases w i t h i n c r e a s i n g r i v e r c r o s s - s e c t i o n a l
areas.
p a r t i c l e needs a l o n g e r t i m e t o proceed downstream,
a t y p i c a l v a l u e f o r mean
T h e r e f o r e a water
556 discharge conditions
i s 20 days t r a v e l
t i m e f r o m Hamburg t o t h e N o r t h Sea
(125 km) compared t o two days f o r t h e same d i s t a n c e w i t h i n t h e n o n - t i d a l p a r t of
the river. O r i g i n a t i n g f r o m storm-surge p r e d i c t i o n models t h e s e e n v i r o n m e n t a l prnblems
m o t i v a t e d t h e development o f more s o p h i s t i c a t e d models t o examine,
i.e.
to
understand and p r e d i c t , t h e dynamical f e a t u r e s o f t h e whole E l b e e s t u a r y . T h i s paper d e a l s w i t h a d e s c r i p t i o n o f t h e p r e s e n t s t a t e o f e s s e n t i a l l y t h r e e - d i m e n s i o n a l models f o r t h e c i r c u l a t i o n and t r a n s p o r t o f p a s s i v e t r a c e r s and g i v e s some r e s u l t s o b t a i n e d so f a r .
2
HYDROGRAPHY The t i d a l wave progresses i n t o t h e E l b e e s t u a r y up t o t h e w e i r a t Geest-
h a c h t , i.e.
about 140 km.
I n i t s l o w e r p a r t t h e e s t u a r y i s c h a r a c t e r i z e d by a
s i g n i f i c a n t i n f l u e n c e o f barocl i n i c i t y .
N o r t h Sea w a t e r and f r e s h w a t e r a r e
mixed f o r m i n g a pronounced b r a c k i s h w a t e r zone t h a t may be d e f i n e d by i t s s a l i nity, water).
v a r y i n g between 1
( n e a r l y f r e s h w a t e r ) and 34
-
( N o r t h Sea
The h o r i z o n t a l e x t e n t o f t h i s zone ( u p t o 50 km) and i t s p o s i t i o n
depend upon m e t e o r o l o g i c a l c o n d i t i o n s and on t h e f r e s h w a t e r d i s c h a r g e which v a r i e s between 400 m3/s i n September and 1200 m3/s
i n April.
A 15 m deep
n a v i g a t i o n a l channel g i v i n g access t o t h e p o r t of Hamburg i s a d j o i n i n g extensive tidal
f l a t areas l e a d i n g t o s t r o n g t o p o g r a p h i c g r a d i e n t s .
Forced by a
t i d a l range o f up t o 5 m t h i s r e s u l t s i n complex dynamics i n c l u d i n g f l o o d i n g and f a l l i n g d r y w i t h s t r o n g c u r r e n t s exceeding 2 m/s i n t h e deeper p a r t s o f t h e river.
The dynamical f l o w p a t t e r n w i t h s t r o n g h o r i z o n t a l and v e r t i c a l c u r r e n t
shear i s accompanied by v e r t i c a l s t r a t i f i c a t i o n due t o t h e s a l t wedge i n t r u s i o n i n t o t h e e s t u a r y . An example i s i l l u s t r a t e d i n F i g . 1 f o r a s t a t i o n i n a narrow channel w i t h i n t h e t i d a l f l a t s , showing v e r t i c a l s a l i n i t y g r a d i e n t s . o f 10
10-3
o v e r 5 m. These mesoscale phenomena i n c o n t r a s t t o t h e l a r g e s c a l e f l o w i n t h e deep waterways show t h e h i g h v a r i a b i l i t y of t h e e n t i r e f l o w regime t h a t i t s e l f causes
high
rates
and
a
high
variability
of
erosion,
sedimentation
and
r e d i s t r i b u t i o n o f suspended m a t t e r and w a t e r p r o p e r t i e s . The w a t e r l e v e l s and c u r r e n t s depend n o n l i n e a r l y on t h e r i v e r d i s c h a r g e (seasonal v a r i a t i o n ) ,
the
t i d a l range i n t h e German B i g h t ( s p r i n g / n e a p c y c l e , stormsurges) and t h e meteor o l o g i c a l f o r c i n g . So i t seems i m p o s s i b l e t o d e f i n e a mean t i d e as b e i n g r e p r e s e n t a t i v e f o r t h e estuary.
557
1
1
1
~
1
~
1
1
1
1
1
"
~
"
"
'
-
-1
E
-
-2
r
-
I-
n. Y
-3
0
-
-4 .6
7 8 9 10
20
15 1
1
1
1
1
1
1
1
1
1
1
Fig. bserved v e r t i c a l p r o f i l e s o f s a l i n i t y and d e n s i t y (calci1.,ted c o n d u c t i v i t y ) i n a narrow channel, 10 km west o f B r u n s b u t t e l 3
'rom
THE MODEL SYSTEM There a r e a number o f m o t i v a t i o n s t o develop an e s t u a r i n e model system o f
d i f f e r e n t components.
As a consequence o f t h e d e s c r i p t i o n s g i v e n i n chapter 2
i t i s obvious t h a t o n l y a three-dimensional
model can t a k e i n t o account t h e
observed h o r i z o n t a l and v e r t i c a l v a r i a b i l i t i e s . Due t o present t e c h n i c a l l i m i t a t i o n s i t i s impossible t o handle an a l l - p u r p o s e model f o r t h e e n t i r e e s t u a r y w i t h s u f f i c i e n t resolution. been d i f f e r e n t . as, e.g.
I n o r d e r t o overcome t h i s problem t h e s t r a t e g y has
I f one i s i n t e r e s t e d i n a general study o f t h e e n t i r e e s t u a r y
long-term o r storm surge s t u d i e s , a r e l a t i v e l y coarse r e s o l u t i o n would
be s u f f i c i e n t ,
whereas f o r processes l i n k e d t o t h e v e l o c i t y f i e l d l i k e t h e
f o r m a t i o n o f a t u r b i d i t y zone, o f sand,
been t o combine l a r g e s c a l e (250
-
s a l t wedge f o r m a t i o n and erosion/sedimentation
a r e g i o n a l l y l i m i t e d area approach i s appropriate.
-
So t h e idea has
500 m g r i d r e s o l u t i o n ) and s m a l l e r s c a l e (50
100 m r e s o l u t i o n ) models t o form a three-dimensional model system.
3.1 The e s t u a r i n e c i r c u l a t i o n model system The model system i s based on a s e m i - i m p l i c i t ,
two-time-level
formulation o f
t h e shallow water equations f o r b a r o t r o p i c s h e l f seas (Backhaus,
1983).
An
e x t e n s i o n f o r v e r t i c a l l y i n t e g r a t e d f l o w upon t i d a l f l a t s ( v e r y t h i n l a y e r of water,
flooding,
Duwe e t a1
f a l l i n g d r y ) has been g i v e n by Duwe and Hewer (19R?), whereas
. (1983)
i n t r o d u c e d t h e corresponding three-dimensional
v e r s i o n i n c l u d i n g some v e r i f i c a t i o n s .
barocl i n i c
An a p p l i c a t i o n o f t h i s model t o t h e f r e s h
water zone t o g e t h e r w i t h a comparison between t h e t h r e e - and two-dimensional ( v e r t i c a l l y i n t e g r a t e d ) v e r s i o n s w i 11 be r e p o r t e d by Krohn and Lobmeyr (1986).
558
P f e i f f e r and Siindermann (1986) extended t h e model
i n o r d e r t o overcome t h e
t e c h n i c a l r e s t r i c t i o n t h a t o n l y t h e uppermost c o m p u t a t i o n a l l a y e r was a l l o w e d t o f a l l dry. P r e s e n t l y t h i s s p e c i f i c model i s f o r m u l a t e d e x p l i c i t l y i n a c l a s s i c a l FTCS scheme (FTCS
-
f i n i t e d i f f e r e n c e approximation forward-in-time
and
centered-in-space). According
t o the different
problems t r e a t e d ,
four
model
components a r e
a v a i l a b l e ( T a b l e 1). TABLE 1 Overview o v e r t h e f o u r model components. Model numbers r e f e r t o F i g . 2 Model area
Function
Model number E n t i r e estuary 1
O v e r v i e w / P r o d u c t i o n o f boundary v a l u e s Study o f t h e o s c i l l a t i o n system
B r a c k i s h w a t e r zone 2
Study o f v e l o c i t y f i e l d i n b r a c k i s h w a t e r zone F o r m a t i o n o f t u r b i d i t y z o n e / s a l t wedge
N e u f e l d wadden a r e a 3
V e r i f i c a t i o n o f 3D v e l o c i t y f i e l d w i t h v e r t i c a l l y h i g h r e s o l v i n g model
Fresh w a t e r zone 4
V e r i f i c a t i o n o f 3D v e l o c i t y f i e l d ( b a r o t r o p i c ) w i t h e x t e n s i v e measurements
F i g . 2. Schematic map o f t h e E l b e e s t u a r y . Areas o f t h e d i f f e r e n t models a r e i n d i c a t e d by r e c t a n g l e s , t h e e n c i r c l e d numbers r e f e r t o Tab1 e 1.
559
- -&4 . l
0.7 Time i n hours
S U R F A C E SPEED
I
MEASUREMENT
- SlMULATlON
07
I
1:13[77 BOTTOM SPEED
';:r;'--15
Time i n hours
S U R F A C E SALlNlTY
~
~
15
BOTTOM SALlNlTY
Time in hours
Fig. 3. Model no. 2: Observed versus computed s u r f a c e v e l o c i t y , b o t t o m v e l o c i ty, s u r f a c e s a l i n i t y and b o t t o m s a l i n i t y ( f r o m t o p t o b o t t o m ) a t a s t a t i o n i n t h e n a v i g a t i o n a l channel between Cuxhaven and B r u n s b u t t e l
.
I n o r d e r t o i l l u s t r a t e t h e h o r i z o n t a l v a r i a b i l i t y o f t h e f l o w and s a l i n i t y f i e l d s , t h e r e s u l t s o f a s i m u l a t i o n f o r a t y p i c a l autumn s i t u a t i o n (September, low f r e s h w a t e r d i s c h a r g e ) and s p r i n g t i d e a r e g i v e n i n F i g s . 4 t o 6. The dens i t y d i s t r i b u t i o n n e a r l o w w a t e r ( F i g . 4) shows a pronounced v a r i a t i o n between t h e deeper and s h a l l o w e r regions. h i g h water, too.
T h i s t o p o g r a p h i c e f f e c t can be observed near
Here s t r o n g g r a d i e n t s a r e generated e s p e c i a l l y i n areas where
r e l a t i v e l y dense N o r t h Sea w a t e r i s e n a b l e d t o f l o w o v e r f l o o d e d t i d a l f l a t s t o
560
The areas covered by t h e d i f f e r e n t models a r e sketched i n F i g . 2. I n Table 2 some d a t a a r e g i v e n on h o r i z o n t a l
(As) and
mean v e r t i c a l (Az) g r i d r e s o l u t i o n ,
t h e t i m e s t e p ( A t ) and t h e Courant number, i.e.
t h e f a c t o r by w h i c h t h e maximum
t i m e s t e p a l l o w e d f o r an e x p l i c i t l y f o r m u l a t e d n u m e r i c a l scheme, i s exceeded due t o t h e use o f an i m p l i c i t f o r m u l a t i o n . The l a s t t w o columns i n d i c a t e t h e number o f g r i d p o i n t s handled i n t h e c a l c u l a t i o n s and t h e c o m p u t a t i o n a l t i m e needed t o s i m u l a t e one s e m i d i u r n a l c y c l e on a Siemens 7.882 computer. TABLE 2 D e t a i l s o f t h e f o u r segment models Model E n t i r e estuary B r a c k i s h w a t e r zone N e u f e l d wadden a r e a Fresh w a t e r zone
The b a s i c model
(no.
-
As (m)
A z (m)
250 250 2 50 100
4 2 1 1.5
At (s)
CFL factor
150 150 10 60
10 10 10 5
No. o f wet p o i n t s
CPU (h)
000 000 000 000
2 1.5 2.5 1
18 16 9 9
1 ) e x t e n d s o v e r t h e e n t i r e e s t u a r y f r o m t h e seaward
boundary t o t h e w e i r a t Geesthacht where t h e t i d a l wave i s a r t i f i c i a l l y stopped.
T h i s model g i v e s an o v e r v i e w over t h e o s c i l l a t i o n system,
l e v e l movement e s p e c i a l l y f o r s t o r m s u r g e c a l c u l a t i o n s .
i.e.
t h e water
Secondly i t p r o v i d e s
boundary v a l u e s f o r t h e l i m i t e d a r e a "segment models" o f h i g h e r r e s o l u t i o n . The segment model no. 2 has been designed t o s t u d y t h e 3D v e l o c i t y f i e l d i n t h e most v a r i a b l e p a r t o f t h e e s t u a r y , t h e b r a c k i s h w a t e r zone.
It i s essen-
t i a l l y b a r o c l i n i c so t h a t t h e f o r m a t i o n and movement o f a s t r o n g v e r t i c a l dens i t y g r a d i e n t ( ' s a l t wedge') can be simulated. zontal
r e s o l u t i o n as a model no.
A t t h e boundaries,
T h e r e f o r e i t has t h e same h o r i -
1 b u t a h i g h e r v e r t i c a l r e s o l u t i o n ( 2 m).
t h e time s e r i e s o f water l e v e l s i s prescribed,
measured
v a l u e s a t t h e seaward boundary and model d a t a o f t h e b a s i c model a t t h e e a s t e r n boundary.
F o r t h e d e n s i t y c y c l i c boundary c o n d i t i o n s a t t h e seaward boundary
and m o d e l l e d d a t a a t t h e o t h e r one a r e used. There have been some f i e l d e x p e r i m e n t s i n o r d e r t o v e r i f y t h e v e l o c i t y field.
F o r example F i g . 3 shows t h e computed and observed t i m e s e r i e s o f near
s u r f a c e and b o t t o m v e l o c i t i e s a t a s t a t i o n i n t h e deep n a v i g a t i o n a l channel. The model a l s o reproduces t h e observed v e r t i c a l t i m e l a g o f f l o w r e v e r s a l between near b o t t o m and s u r f a c e l a y e r s . Near l o w w a t e r s l a c k time, a t l o w s a l i n i -
t y ( s e e l o w e r p a r t o f F i g . 3 ) , t h e f l o w r e v e r s a l s t a r t s n e a r t h e bottom, due t o friction. surface
A t h i g h water slack time,
b a r o c l i n i c i t y predominates and t h e near
f l o w changes i t s d i r e c t i o n b e f o r e t h e b o t t o q water.
561
LOWER ELBE ESTUARY
Fig. 4. Model no. 2: Density d i s t r i b u t i o n i n sigma-t a t low water i n Cuxhaven f o r mean September discharge and s p r i n g t i d e ( s u r f a c e l a y e r ) .
LOWER ELBE ESTUARY
Fig. 5. Same as Fig. 4 b u t f o r h i g h water. meet r e l a t i v e l y l i g h t water t r a n s p o r t e d by t h e ebb c u r r e n t . For t h e same season and astronomical s i t u a t i o n t h e v e r t i c a l l y i n t e g r a t e d E u l e r i a n r e s i d u a l t r a n s p o r t s a r e d e p i c t e d i n Fig.
6 t o i l l u s t r a t e t h e v a r i a b l e f l o w pattern.
The
highest values a r e l i n k e d t o t h e deeper p a r t s and d i r e c t e d towards t h e sea, whereas on t i d a l f l a t s and i n t h e narrow channels t h e t i d a l mean t r a n s p o r t may be d i r e c t e d up-estuary. A l o n g i t u d i n a l s e c t i o n o f d e n s i t y shows t h a t d u r i n g one
562
LOWER ELBE ESTUARY
Fig. 6. Model no. 2: E u l e r i a n r e s i d u a l t r a n s p o r t s ( v e r t i c a l l y i n t e g r a t e d ) f o r mean September d i s c h a r g e and s p r i n g t i d e .
Fig. 7. Model no. 2: L o n g i t u d i n a l s e c t i o n o f d e n s i t y i n sigma-t a t l o w w a t e r i n Cuxhaven.
F i g . 8. Same as Fig. 7 b u t f o r h i g h water. t i d a l c y c l e t h e e s t u a r y changes f r o m w e l l - m i x e d c o n d i t i o n s ( F i g . w a t e r t o p a r t i a l l y mixed and even s t r a t i f i e d c o n d i t i o n s ( F i g .
7 ) near low 8) n e a r h i g h
w a t e r i n i t s upper and l o w e r p a r t s , r e s p e c t i v e l y . Segment model no. 3 i n c o r p o r a t e s t h e t w o most s i g n i f i c a n t t o p o g r a p h i c feat u r e s , t h e deep n a v i g a t i o n a l channel and an a d j a c e n t t i d a l f l a t , a narrow channel up t o 5 m deep.
surrounded by
T h i s model i s an e x t e n s i o n o f t h e p r e v i o u s
ones as i t i s c a p a b l e o f t r e a t i n g more t h a n one c o m p u t a t i o n a l l a y e r t o t a k e p a r t i n t h e process of f a l l i n g d r y and f l o o d i n g . . T h i s
f e a t u r e has been m o t i v a t -
563 ed by t h e s t r o n g c u r r e n t shear and d e n s i t y g r a d i e n t s i n t h e uppermost meters o f t h e b r a c k i s h w a t e r zone w a t e r column.
The model i t s e l f has been d e s c r i b e d i n
1986). Here some a d d i t i o n a l r e s u l t s
d e t a i 1 r e c e n t l y ( P f e i f f e r and Sindermann,
w i l l be demonstrated. The a b i l i t y o f s i m u l a t i n g t h e f l o w f i e l d i n v e r y s h a l l o w areas i s demonstrated i n F i g . 9, where a v e r t i c a l c r o s s - s e c t i o n f r o m n o r t h t o s o u t h a c r o s s t h e model area i s shown d u r i n g h i g h and l o w water. with
measurements,
data
have
been
selected
from
stations
d u r a t i o n matches t h o s e o f n e i g h b o u r i n g g r i d c e l l s . Fig.
F o r comparison whose
flooding
The r e s u l t s a r e g i v e n i n
10. The q u a l i t a t i v e agreement i s good, e s p e c i a l l y t h e r e p r o d u c t i o n o f t h e
non-harmonic b e h a v i o u r and t h e t y p i c a l peak a f t e r f l o o d begins.
0E Z
+
4'
n 8-
......
W
0
1216-
High water 20-
SOUTH
NORTH
0
E
4.
c
8-
c
.
,2120
16.
1.0
0.5
- 1 . 0 m/s
20.
F i g . 9. Model no. 3: V e r t i c a l c r o s s s e c t i o n f r o m n o r t h t o s o u t h showing c a l c u l a t e d v e l o c i t i e s a t h i g h w a t e r ( t o p ) and l o w w a t e r ( b o t t o m ) . P o s i t i v e (negat i v e ) v a l u e s i n d i c a t e f l o o d ( e b b ) c u r r e n t s . The d i a m e t e r o f t h e symbols i s l i n e a r l y v a r y i n g w i t h t h e magnitude o f t h e v e l o c i t y .
564
-
1
* >
* 0.5 0
I
Normalized
Tidal
1
Period
F i g . 10. Model no. 3: Observed versus c a l c u l a t e d v e l o c i t i e s a t p o i n t s on t i d a l f l a t s . F u l l d o t s i n d i c a t e o b s e r v a t i o n , s o l i d l i n e denotes c a l c u l a t i o n . F i r s t t w o measurements ( f r o m t o p ) were t a k e n on Nov. 6, 1979, second p a i r on J u l y 6, 1981. (Courtesy o f Wasser- und S c h i f f a h r t s a m t Cuxhaven.) Segment model no. t h e Elbe estuary.
4 c o n s t i t u t e s an a p p l i c a t i o n t o t h e f r e s h w a t e r zone o f
I n t h e area c o n s i d e r e d e x t e n s i v e measurements have been p e r -
formed ( M i c h a e l i s and Knauth, 1985) used f o r model t e s t i n g (Krohn, and Lobmeyr, 1986). The topography ( F i g . 1 1 ) c l e a r l y shows t h e deep n a v i g a t i o n a l channel (up t o 15 m) connected w i t h s t r o n g l a t e r a l g r a d i e n t s . The model i s . f o r c e d by p r e s c r i b e d w a t e r l e v e l s a t i t s t w o open boundaries, t a k e n f r o m o f f i c i a l t i d e gauge records. These, however, had t o be c o r r e c t e d i n o r d e r t o reproduce a n o t h e r t i d e gauge s t a t i o n d a t a w i t h i n t h e model area.
I n t h e f u t u r e t h i s problem w i l l be
overcome by u s i n g w a t e r l e v e l boundary v a l u e s produced by t h e b a s i c model o f t h e t o t a l estuary.
A more t h o r o u g h d i s c u s s i o n o f t h i s t o p i c i s g i v e n i n t h e
above mentioned paper.
565
I
1
F i g . 11. Model no. 4: Topography, depths i n d e c i m e t e r s r e f e r r e d t o l o c a l c h a r t datum (mean sea l e v e l minus 1.1 m). S e c t i o n A-B i n d i c a t e s p r o f i l e where model r e s u l t s a r e compared w i t h d a t a ( s e e F i g . 1 3 ) . A f t e r t h i s c o r r e c t i o n , t h e observed and computed c r o s s - s e c t i o n a l averages of t h e v e l o c i t y match q u i t e s a t i s f a c t o r i l y ( F i g . 12). The measured and c a l c u l a t e d n e a r s u r f a c e v e l o c i t y d i s t r i b u t i o n s a c r o s s t h e r i v e r a r e shown i n Fig.
13 a t
f o u r d i f f e r e n t i n s t a n t s d u r i n g one t i d a l c y c l e . The decrease towards t h e r i v e r banks a r e reproduced. E x i s t i n g d i f f e r e n c e s m i g h t be due t o t h e s t i l l u n s u f f i c i e n t l y r e s o l v e d b o t t o m topography.
A t y p i c a l v e l o c i t y f i e l d near l o w w a t e r i s
shown i n F i g . 14. A p a r t f r o m t h e deep n a v i g a t i o n a l channel i n t h e e a s t e r n p a r t , t h e f l o w has a l r e a d y r e v e r s e d i t s d i r e c t i o n , e s p e c i a l l y i n t h e s h a l l o w s o u t h e r n areas.
1
-Po
20
24
TM (GUT + 2 HOURS)
Fig. 12. Model no. 4: Comparison between observed ( d o t s ) and c a l c u l a t e d v e l o c i t i e s of t h e uppermost c o m p u t a t i o n a l l a y e r . Values a r e i n t e g r a t e d o v e r t h e r i v e r width.
566
"1 nu
? 0
1
2
3
4
5
6
7
8
9
F i g . 13. hfodel no. 4: Across r i v e r p r o f i l e o f observed (symbols) and c a l c u l a t e d Profile v e l o c i t i e s i n t h e lippermost l a y e r a t d i f f e r e n t t i m e s (UTC, 24.8.1982). r u n s f r o m A t o R, see Fig. 11.
I
24.8.82
15.40h
F i g . 14. Model no. 4: Computed v e l o c i t y v e c t o r s i n s u r f a c e l.ayer, l o w water s l a c k time. V e c t o r s p l o t t e d e v e r y second g r i d p o i n t . Ebb c u r r e n t t o t h e l e f t . 3.2 The p a s s i v e t r a c e r t r a n s p o r t model
As a f i r s t s t e p towards t h e s i m u l a t i o n of suspended m a t t e r t r a n s p o r t and t h e f o r m a t i o n o f a t u r b i d i t y zone t h e t r a n s p o r t o f a p a s s i v e t r a c e r i s computed, i.e. a c o m p l e t e l y d i s s o l v e d c o n s e r v a t i v e substance. Due t o t h e computation a l e f f o r t t r a c e r methods a r e m a i n l y s u i t a b l e f o r t r a c k i n g a l i m i t e d number o f p a r t i c l e s r e l e a s e d i n t o t h e w a t e r as may be t h e case f o r s h i p a c c i d e n t s o r waste w a t e r discharge.
1
567
Consider a g i v e n c o n c e n t r a t i o n C o f a p a s s i v e substance i n a t h r e e - d i m e n s i o n a l f l o w f i e l d u = (u,v,w).
I t s temporal and s p a t i a l change i s d e s c r i b e d by t h e
well-known a d v e c t i o n - d i f f u s i o n e q u a t i o n
(t
-
time;
-
x,y,z
C a r t e s i a n c o o r d i n a t e s ; kh, k v
-
horizontal, v e r t i c a l d i f f u -
I n o r d e r t o overcome problems a r i s i n g from numerical d i f f u -
sion coefficients.)
s i o n caused by f o r m u l a t i o n s o f t h e n o n l i n e a r a d v e c t i o n terms, a Lagrangian app r o a c h has been used. T h i s method (see, e.g.
Bork and Maier-Reimer,
1978) es-
s e n t i a l l y a v o i d s numerical d i f f u s i o n as t h e n o n l i n e a r terms vanish. The v e l o c i ty
field
i s 'split
into
an
advective p a r t
(calculated velocities)
and
a
s t o c h a s t i c p a r t ( f l u c t u a t i o n s d e f i n e d below). I t s h o u l d be a d m i t t e d t h a t a v e r a g i n g o f v e l o c i t i e s may p r i n c i p a l l y l e a d t o small d i f f u s i o n - l i k e e f f e c t s . T u r b u l e n t d i f f u s i o n i s s i m u l a t e d by a s t a t i s t i c a l random process ('Monte-Carlo m e t h o d ' ) by a d d i n g f o r each s p a t i a l c o o r d i n a t e a random number t o t h e v e l o c i t y component.
Due t o s p e c i a l p r o p e r t i e s o f t h e e s t u a r i n e f l o w f i e l d some m o d i f i c a -
t i o n s t o t h e c l a s s i c a l method have been necessary. Numerical experiments c a r r i e d o u t i n o r d e r t o t e s t d i f f e r e n t f o r m u l a t i o n s o f the turbulent diffusion led t o
t h e f l u c t u a t i o n s b e i n g r e p r e s e n t e d by u'
=
where
a ph
a
E (-
1,l)
a , 0 and y a r e random numbers o u t o f a " t o p h a t " d i s t r i b u t i o n o v e r t h e
g i v e n i n t e r v a l , a n d ph a n d p v d e f i n e t h e b a n d w i d t h o f t h e h o r i z o n t a l and v e r t i c a l f l u c t u a t i o n s . Their values are determined a f t e r
31
ki =
(4t
-
(piI2bt,
i = (h, v )
(6)
t i m e s t e p ) , see Maier-Reimcr and Sijndermann (1982) s o t h a t k h i s o f t h e
o r d e r o f 1 m2/s a n d k v o f t h e o r d e r o f 0.05 observed means.
d / s , both valiies representing
I n t h e v i c i n i t y o f s o l i d boundaries a ' n o - s l i p '
condition f o r
568 t h e v e l o c i t y component p a r a l l e l t o t h e boundary i s a p p l i e d , g i v i n g a more r e a l i s t i c approach w i t h r e s p e c t t o c o n s e r v a t i o n o f t r a n s p o r t s computed i n t h e Eulerian grid.
I f a p a r t i c l e t r i e s t o c r o s s a s o l i d boundary o r t h e f r e e s u r -
face, which can o n l y be caused by t h e random component,
the p a r t i c l e i s trans-
p o r t e d w i t h a n o t h e r random component s a t i s f y i n g t h e n o - f l u x boundary c o n d i t i o n . The s t r o n g v a r i a b i l i t y o f t h e f l o w r e q u i r e s t h a t t h e o r i g i n a l t i m e s t e p (depending on t h e r e l a t i o n s h i p between g r i d s i z e and t i m e s t e p o f t h e c u r r e n t model) be decreased by a f a c t o r o f 4 i n t h e t r a n s p o r t c a l c u l a t i o n .
T h i s may
l e a d t o a temporal i n t e r p o l a t i o n o f t h e v e l o c i t i e s . Fig.
15 shows t h e e v o l u t i o n o f p a r t i c l e d i s t r i b u t i o n s 1, 6 and 10 hours
a f t e r r e l e a s e near t h e s u r f a c e a t a p o i n t near t h e n a v i g a t i o n a l channel i n t h e b r a c k i s h water zone.
Due t o l o n g i t u d i n a l c u r r e n t shear t h e c l o u d i s s t r e t c h e d
c o n s i d e r a b l y whereas l a t e r a l d i f f u s i o n seems t o be n e g l i g i b l e . v e r s a l t h e c o n c e n t r a t i o n moves back and
-
A f t e r f l o w re-
i n t h e case o f hazardous substances
-
c o u l d a f f e c t t h e t i d a l f l a t s and c o a s t a l zone. An a p p l i c a t i o n t o t h e f r e s h w a t e r zone segment model i s d e p i c t e d i n F i g . 16.
A l l c a l c u l a t i o n s have been performed w i t h 10 000 p a r t i c l e s . A d e t a i l e d d e s c r i p t i o n can be found i n Duwe e t a l .
(1986).
569
F i g . 15 a. Model no. 2: p o i n t (XI
P a r t i c l e d i s t r i b u t i o n 1 hour a f t e r r e l e a s e a t marked
F i g . 15 b. Model no. 2: P a r t i c l e d i s t r i b u t i o n 6 hours a f t e r r e l e a s e a t marked p o i n t (XI
570
Fig. 15 c . Model no. 2 : Particle distribution 10 hours after release a t marked point ( X I
Fig.
16. Model no. 4: Particle distribution 2 hours after release a t marked
point ( X I
571 4
CONCLUSIONS On t h e b a s i s o f a s e m i - i m p l i c i t f o r m u l a t i o n o f t h e s h a l l o w water equations a
s e t o f numerical models f o r e s t u a r i n e a p p l i c a t i o n s has been developed.
I t com-
b i n e s h i g h s p a t i a l r e s o l u t i o n and computational e f f i c i e n c y as t h e courant numb e r o f one i s c o n s i d e r a b l y exceeded.
With respect t o t h e g r i d s i z e and a v a i -
l a b l e observations t h e v e l o c i t y f i e l d i s w e l l reproduced.
I t i s obvious,
how-
ever, t h a t more d e t a i l e d observations a r e necessary f o r f u r t h e r v e r i f i c a t i o n , e s p e c i a l l y o f 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 f l o w i n t h e v i c i n i t y o f t h e bottom. Together w i t h improved f o r m u l a t i o n s o f t r a n s p o r t phenomena, e.g. front-like
structures,
this w i l l
r e s o l u t i o n of
l e a d t o a model system t h a t i s capable o f
c a l c u l a t i n g t h e t r a n s p o r t o f d i s s o l v e d as w e l l as suspended matter.
5
REFERENCES
983. .A s e m i - i m p l i c i t scheme f o r t h e sha low water equations f o r Backhaus. J.. a p p l i c a t i o n t o s h e l f sea modelling. Cont. S h e l f Res., 2 ( 4 ) : 243-254. 1978. On t h e spreading o f power p l a n t c o o l i n g Bork, I. and Maier-Reimer E., water i n a t i d a l r i v e r a p p l i e d t o t h e r i v e r Elbe. Adv. Water Res.,..l. 1982. E i n s e m i - i m p l i z i t e s Gezeitenmodell f u r WattgeDuwe, K. and Hewer, R., b i e t e . D. Hydrogr. Zeitschr., 35: 223-238. Hewer, R.R. and Backhaus, J.O., 1983. Results o f a s e m i - i m p l i c i t Duwe, K.C., two-step method f o r t h e s i m u l a t i o n o f markedly n o n l i n e a r f l o w i n coastal seas. Cont. S h e l f Res., 2 ( 4 ) : 255-274. Duwe, K.. Krohn, J., P f e i f f e r , K., Riedel-Lorj6, J.C. and Soetje, K.C., 1986. Ausbreitung .yon wassergefahrdenden S t o f f e n i n der s u d l i c h e n Deutschen Bucht und i m Elbe-Astuar nach F r e i s e t z u n g durch S c h i f f e . Subm. t o D. Hydrogr. Z e i t s c h r . Krohn, J. and Lobmeyr, M., 1986. Comparison o f two- and three-dimensional h i g h r e s o l v i n g e s t u a r i n e models w i t h observations i n t h e E l b e r i v e r . I n preparation. Maier-Reimer, E. and Sundermann, J., 1982. On t r a c e r methods i n computational hydrodynamics. I n : M.B. Abbott and J.A. Cunge ( E d i t o r s ) . Engineering a p p l i c a t i o n s o f computational h y d r a u l i c s , Volume 1. Pitman, Boston/London/Mel bourne, pp. 198-217. 1985. Das Bilanzierungsexperiment 1982 M i c h a e l i s , W. and Knauth, H.-D., (BILEX '82) a u f d e r U n t e r e l be. GKSS Forschungszentrum Geesthacht, Report GKSS 85/E/3 (unpubl.), 212 pp. P f e i f f e r , K.D., Sundermann, J., 1986. Ein dreidimensionales Flachwassermodell m i t v e r t i k a l e r Auflosung i m Tidehubbereich: Entwicklung und e r s t e Anwendungen. D i e Kuste, 43: 149-165.
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573
A THREE DIMENSIONAL CONTINENTAL SHELF
NUMERICAL
MODEL
OF SEMI-DIURNAL TIDES ON THE EUROPEAN
A. M. DAVIES
of
Oceanographic L43 7RA ( E n g l a n d )
Institute Merseyside
Sciences,
Bidston
Observatory,
Birkenhead,
ABSTRACT A t h r e e d i m e n s i o n a l hydrodynamic t i d a l model of t h e N o r t h west European Continental s h e l f is developed u s i n g a staggered f i n i t e d i f f e r e n c e g r i d i n t h e h o r i z o n t a l and a s p e c t r a l method t h r o u g h t h e v e r t i c a l . Some p r e l i m i n a r y c a l c u l a t i o n s of t h e M and S2 t i d e s performed w i t h t h e model are u s e d t o d e m o n s t r a t e t h e s e n s i t i v i z y of computed t i d e s i n t h e North Sea t o t i d a l i n p u t a t t h e c o n t i n e n t a l s h e l f edge. Histograms of e r r o r s between o b s e r v e d and computed M2 and S2 t i d e s , a t o n e h u n d r e d a n d t w e n t y n i n e o f f s h o r e a n d coastal t i d e g a u g e s , show t h a t t h e model c a n a d e q u a t e l y r e p r o d u c e t h e t i d e s i n t h e N o r t h Sea, p r o v i d e d a n a c c u r a t e s p e c i f i c a t i o n of t h e t i d e s a l o n g t h e s h e l f e d g e i s a v a i l a b l e . Some i n d i c a t i o n of t h e a c c u r a c y of t h e computed s p r i n g t i d a l c u r r e n t s is o b t a i n e d by comparing computed s u r f a c e c u r r e n t s w i t h o b s e r v a t i o n s . 1
INTRODUCTION
earlier
In
dimensional the tide,
papers
(Davies
numerical
models
and
D a v i e s ( 1 9 8 6 ) c o m p u t i n g b o t h t h e M2 a n d M4 t i d e s .
with
t h e North-West European s h e l f , t h e S In this
F u r n e s 1 9 8 0 , Heaps and J o n e s 1 9 8 1 ) t h r e e
were u s e d t o s i m u l a t e o n l y t h e M2 component of
paper
the
component of t h e t i d e is also i m p o r t a n t .
and S2 t i d a l c u r r e n t s are computed by i n t e g r a t i n g t h e
M2
t i d a l model w i t h b o t h M
2
However o n
and S2 i n p u t o n t h e o p e n b o u n d a r y , and s e p a r a t i n g t h e
2
two components b y h a r m o n i c a n a l y s i s of t h e r e s u l t i n g computed time series. this
means
combination the
stress
bottom of
these
By
and t u r b u l e n c e i n t h e model are d e t e r m i n e d by t h e To t h e a u t h o r ' s knowledge t h i s is
t i d a l constituents.
f i r s t time t h a t a t h r e e d i m e n s i o n a l t i d a l model h a s b e e n u s e d t o s i m u l a t e
t h e s e t i d a l c o n s t i t u e n t s i n combination. The
model
uses
Galerkin-spectral vertical variation efficient spectral
are of
a
finite-difference
method
in
eigenfunctions eddy
(Davies models
the
and
the
horizontal,
and
a
By t h i s means a n a c c u r a t e a n d c o m p u t a t i o n a l l y
Stephens
(Davies
in
The f u n c t i o n s u s e d t h r o u g h t h e
of a n e i g e n v a l u e problem i n v o l v i n g t h e v e r t i c a l
viscosity. and
grid
vertical.
1 9 8 3 ) model c a n b e d e v e l o p e d .
Furnes
Unlike e a r l y
1 9 8 0 , Heaps and J o n e s 1 9 8 1 ) which were
574
to
restricted model
eddy
certain
viscosity
v e r t i c a l v a r i a t i o n s of e d d y v i s c o s i t y , i n t h e p r e s e n t can
vary
through
the
vertical
and w i t h h o r i z o n t a l
p o s i t i o n and time, i n a n a r b i t r a r y manner. The
three-dimensional
latitude
by
showing
the
the
model
the
deep
1/2O
numerical
longitude
and
model covers
has
a
grid
resolution
of 1/3O
t h e c o n t i n e n t a l s h e l f (see F i g . 1 ,
g r i d a n d t h e l o c a t i o n of some t i d e g a u g e s ) . The o p e n b o u n d a r y of c o i n c i d e s w i t h t h e s h e l f edge.
T h i s is a n a t u r a l boundary between
a n d s h a l l o w e r s h e l f , a n d h a s b e e n c h o s e n a s a n o p e n boundary
ocean
b e c a u s e i t c o i n c i d e s w i t h o f f - s h o r e t i d e g a u g e measurements made by C a r t w r i g h t which form t h e b a s i s of t h e b o u n d a r y i n p u t t o t h e model together w i t h
(19761, data
a two d i m e n s i o n a l N.E.
from
(personal
are
communication).
determined,
A t l a n t i c model of F l a t h e r , P r o c t o r and Wolf
Along t h e s e o p e n b o u n d a r i e s M2 and S2 t i d a l i n p u t
a r a d i a t i o n c o n d i t i o n i s employed t o allow d i s t u r b a n c e s
and
from t h e i n t e r i o r of t h e model t o p r o p a g a t e o u t w a r d s .
Co-tidal
from
the
of M2 and S2 o v e r t h e r e g i o n of t h e s h e l f are c o n s t r u c t e d
charts model
output,
and
the
accuracy
of
the
model i s d e t e r m i n e d by
comparing a m p l i t u d e and p h a s e of t i d a l e l e v a t i o n s w i t h o b s e r v a t i o n s t a k e n a t a number
of
g a u g e s . I n most cases t h e model c a n s a t i s f a c t o r i l y r e p r o d u c e
tide
t h e observed t i d e s . Mean s p r i n g t i d a l c u r r e n t s a t sea s u r f a c e a n d sea bed o v e r t h e whole r e g i o n of
the
are d e t e r m i n e d
shelf
currents
are
in
good
from
agreement
t h e model.
with
similar
S e a s u r f a c e v a l u e s of t h e s e distributions
d e r i v e d from
o b s e r v a t i o n s (Howarth 1982, Howarth and Pugh 1 9 8 3 ) . 2. THREE-DIMENSIONAL SPECTRAL MODEL
In the
s e c t i o n we b r i e f l y d e s c r i b e t h e major s t e p s i n t h e f o r m u l a t i o n of
this three
referred
dimensional
to
Davies
Galerkin-spectral
model.
The i n t e r e s t e d r e a d e r is
and Owen ( 1 9 7 9 1 , D a v i e s ( 1 9 8 0 , 1 9 8 3 a , b ) , Owen ( 1 9 8 0 ) f o r
more d e t a i l s . 2.1 Hydrodynamic e q u a t i o n s
The e q u a t i o n s f o r c o n t i n u i t y and m o t i o n f o r a homogeneous f l u i d , n e g l e c t i n g the
advective
g i v e n by
terms
and
shear
i n t h e h o r i z o n t a l , i n p o l a r c o o r d i n a t e s are
575
x ,
where below
are e a s t - l o n g i t u d e and n o r t h - l a t i t u d e r e s p e c t i v e l y , w i t h Z d e p t h surface. I n t h e s e e q u a t i o n s , t d e n o t e s time,
cp
the
undisturbed
5 e l e v a t i o n of t h e sea s u r f a c e above t h e u n d i s t u r b e d l e v e l , h u n d i s t u r b e d d e p t h of water, R r a d i u s o f e a r t h , y C o r i o l i s p a r a m e t e r , u , v e a s t - g o i n g and north-going
components
of
at
current
depth
a n d g a c c e l e r a t i o n due t o
z,
The c o e f f i c i e n t of eddy v i s c o s i t y IJ. v a r i e s w i t h x , y, z and t. For t i d e s a zero stress s u r f a c e boundary c o n d i t i o n is r e q u i r e d , g i v i n g
gravity.
w i t h s u b s c r i p t zero d e n o t i n g e v a l u a t i o n a t z:O.
a
Applying
slip
condition
at
sea bed and u s i n g a q u a d r a t i c law of
the
bottom f r i c t i o n , g i v e s
where
is
k
the
coefficient
of
bottom
friction,
and s u b s c r i p t h denotes
e v a l u a t i o n a t z=h. 2.2 A p p l i c a t i o n of t h e G a l e r k i n method
W e
now
boundary
briefly
conditions
basis
set
choice
of
basis
( 4 a , b ) a n d ( 5 a , b ) i n t e r m s of t h e G a l e r k i n method w i t h a
In
as modes viscosity
coordinates
functions
Legendre
successful. taken
t h e s o l u t i o n of e q u a t i o n s ( 1 ) t o ( 3 ) s u b j e c t t o
o f f u n c t i o n s f r (r=1,2,
or
Chebyshev
eddy
consider
0,
the
...m )
through t h e v e r t i c a l .
I n general the
f r is a r b i t r a r y and f u n c t i o n s s u c h as B - s p l i n e s ,
polynomials formulation
( D a v i e s and Owen 1979) have proved v e r y d e v e l o p e d h e r e , t h e s e b a s i s f u n c t i o n s are
of a n e i g e n v a l u e problem i n v o l v i n g t h e v e r t i c a l v a r i a t i o n o f
)I.
For
convenience
o v e r t h e i n t e r v a l &
