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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1
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;
(depth).
Later, Sheng and Butler (1982) and Sheng (1983) detailed
the a-stretched equations without the small amplitude approximation.
Both
models treated separately the external (barotropic) modes, which are governed by the vertically integrated equations, and the internal (baroclinic) modes, which are governed by the 3-D simulation.
equations, to allow for efficient numerical
The recent model also uses a semi-implicit scheme to speed up the
external mode simulation.
As shown in Figure 3, the u-stretched grid is a
special form of "boundary-fitted" grid, and hence allows accurate resolution of complex geometry and bathymetry. applied to numerous water bodies, e.g., Mississippi Sound (Sheng and Butler,
The u-stretched
3-D
model has been
Lake Erie (Sheng et al., 1982).
estuaries are briefly described in the following.
Recent
1978) and
applications to two
40
u=o
a. Y
u =I Figure 3
A vertically o-stretched system.
2.5 Suisun Bay, California Suisun Bay, California is part of the San Francisco Bay system.
Results
of tidal harmonic analysis (Cheng and Gartner, 1984) indicate M2 and K1 as the major constituents. Strong bi-directional tendency was observed at most tidal data stations and the basin geometry was found to significantly affect the principal tidal current direction. The tidal current speed can vary up to a factor of two between spring and neap tides. Salinity within the Suisun Bay typically varies from 10-15 ppt at the western end to 0-10 ppt at the eastern end, but exhibit significant temporal variations. Before studying episodic events, model simulations were performed to examine some of the general features associated with the 3-D flow field within Suisun Bay (Sheng et al., 1985). A synthetic tide consisting of M2 and K1 constituents was used to drive the western boundary (with amplitudes of 55 cm and 31 cm for M2 and K1) and the eastern end (with amplitudes of 43 cm and 25 cm for M2 and K1), with a 90 degree phase shift. The salinity was assumed to be 18-20 ppt along the western end and 12-14 ppt along the eastern end. Boundary condition for salinity depends on the flow direction along the open boundary. During outflow, boundary salinity value is computed by considering advection from interior only. During inflow, exterior data are applied. Using an
of 100 m2/sec and a bottom roughness
(2,)
of 0.4 cm, a five day
numerical simulation was performed. The near-surface and near-bottom currents at the end of five days are shown in Figure 4. More detailed comparison with data indicated reasonable agreement with general circulation features including stronger ebb currents, bi-directionality in tidal currents, and flow reversal at some stations. More realistic simulations are being planned.
41 2.6 Charlotte Harbor, Florida
The Charlotte Harbor estuarine system receives discharges from drainage of 16 percent of Florida through the Peace, Myakka and Caloosahatchee Rivers. The estuarine system is connected with the Gulf of Mexico through various The northern area of inlets between the barrier islands on the west. Charlotte Harbor, with a maximum depth of about 7 m, has been modeled recently (Sheng, et al., 1985). SUISUN BAY :
SRLINITY RUh : 120 HBURS VEL0ClTY F0R SIGMR = -0.9
2
* 1
0’ R
B
c
0
L
X 2.5CE+O1
4
mxrnum VECTOR
VELBCITY F0R SIGW = -0.1
A
B
c
E
0
X
?.SX*OI
d
Figure 4
-
taxinur!
VECTBR
Tide and salinity-driven currents in Suisun Bay near the bottom ( a = -0.9) and near the surface ( a -0.1) at 120 hours.
42
As shown in Figure 5, the Peace River inflow (upper right corner) was assumed to be 15,000 cfs and the tidal elevations along the southern boundary of the northern area of Charlotte Harbor was prescribed as a superposition of a strong diurnal and a weaker semi-diurnal
components with the elevation
varying from -17 cm (t = 0) to 28 cm (t = 15 hrs) to -17 cm (t = 24 hrs) over 24 hours (Goodwin, 1986).
Again, salinity along the open boundary follows the
inflow-outflow condition described before.
The tidal currents and salinity
-
-
c
F
L
o\ (D
m
m
I n 0
::
rD
\D
z D
I D -
? rn
rn
................ ................ ..................
U VI
0 0
Y 0 VI
8 0
0
C
R
X
-
D
X
9.13€+00
(cl
C
R
6.33€+01
(dl
MAXIMUM VECTOR
c
A
D
MAXIMW VECTOR
R
Figure 5
7 CONTOURS CONTOUR LEVELS FROM 3.00 TO 21 .o CONTOUR INTERVAL OF 3.00
C
8
0
X
X
(a1
-
0
(b]
1 1 CONTOLRS CONTOUR LEVELS FROM 1 .OO TO 11 .O CONTOUR INTERVAL OF 1 .OO
Computed salinity contour in Charlotte Harbor after 48 hours of model simulation with a 1/2-km grid. (a) near bottom, (b) near surface. Salinities in ppt. Computed 3-D velocity field in Charlotte Harbor after 48 hours of model simulation with a 1/2-km grid. (c) near bottom, (d) near surface. Velocities in cm/sec.
43
distribution at the end of a 48-hour simulation are shown in Figure 5. salinity wedge
A
corresponding to the classical estuarine circulation in a
partially stratified estuary is clearly exhibited. Significant salinity variation exists in the vertical direction and compares reasonably well with data.
3. VERTICAL TURBULENCE PARAMETERIZATION Aimed
at
better
"predicting"
turbulent
flows
and
removing
the
deficiencies of ad-hoc eddy viscosity models, turbulent transport modeling has been an area of active research in the last decade. (e.g.,
k-z) model, one-equation (e.g.,
While two-equation
k) model, and algebraic model have been
adapted to estuarine and marine hydrodynamic models, this paper starts the discussion
on vertical
turbulence
parameterization
with
"Reynolds
stress
model" which is the most complete "second-order closure model'' and hence the origin of the various simplified turbulent transport models. 3.1 Reynolds Stress Model The Reynolds stress model seeks closure of equations at the level of correlations, i.e., through dynamic equations derived for the
second-order
Reynolds stresses represents
, (E' is
ensemble
temperature variance
average),
<TIT'>,
temperature covariance <w'T'>, similarity
in modeling
the fluctuating velocity vector and heat
fluxes , mass
"< >"
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
=
-
. 5 - [ . 21T
+ ( - + . 2
3
I) -*I
a
I
44
a
+ v,
at
.
P_ =
- 2<x1T'> . IT +
.
(qA~)
-
Lbsq (10) h
where q is the total turbulent intensity defined as
.v,'>~'~
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
<
is surface elevation, U is vertically-Integrated velocity, T is
vertically-averaged
temperature,
the
superscript
denotes
a
contravariant
vector component, the subscript denotes a covariant vector component, H is total depth, go is determinant of the metric tensor gi, between the prototype grid (x,y)
and the transformed grid
(C,n),i
denotes a covariant spatial
derivative, I m denotes a contravariant spatial derivative, ckj is the permutation tensor, cd is bottom friction coefficient, Pa is atmospheric pressure, is density,
‘I~ is
and bottom, and
p
surface wind stress, qs and qb are heat fluxes at surface and
% are
lateral turbulent eddy coefficients.
Details of the 3-D transformed equations of motion and their solutions in terms of contravariant velocity components can be found in Sheng (1986~). The
3‘
PROTOTYPE
TRANSFORMED
Figure 7(a)
A boundary-fitted grid in the prototype and transformed system.
Figure 7(b)
Contravariant, covariant and physical components of a vector in the prototype and transformed systems.
49
3-D boundary-fitted hydrodynamic model,
like the finite-element
model, is
capable of resolving complex geometries.
It is, however, more efficient than
finite-element models because it only requires tridiagonal matrix inversions while finite-element models must deal with inversions of large band matrices. It should be noted that the expansion ot the tensorinvariant equations are rather laborious and is best accomplished by means of a "symbolic manipulator" to avoid human errors. details.
The reader is referred to Sheng (1986~) for
It should also be pointed out that, despite the many extra terms,
the transformed equations thus derived do become significantly simplified if conformal or orthogonal grid is used.
The model thus is more general than
those models that only work for conformal or orthogonal grids, particularly because it is extremely difficult to generate conformal or orthogonal grids for compex domains.
As an example, the expanded equation of motion in the E-
direction is:
+
Lateral Turbulent Diffusion Terms
where (U,V) are the contravariant velocity components in (E,n) directions, gij and gij are metric tensor coefficients, and Di is Christopher symbol of the jk second kind. 4.2 Model Applications The 3-D curvilinear grid model has been and is being applied to real estuaries (e.g.,
Chesapeake Bay and James River).
For simplicity here, we
will present some results obtained with the vertically-integrated version of the model.
-
The first example is concerned with tidal circulations in a quarter circular annular wedge driven by a simple sinusoidal tidal forcing of 6 along the outer radius. tion g are set to unity. satisfy 1
<
r
<
2.5 and 0
Eo
For simplicity, depth h and gravitational acceleraThe domain is defined by all (r.0)
<
Q,
<
n/2.
values that
n/3. First of all, a simple polar coordinate was used for the linearized tidal simulation with all the metric
tensor
coefficients
o is equal to
analytically
computed.
Because of
the
orthogonal grid, metric tensors are diagonal and Christopher symbols are
50
identically zero.
The model results of surface elevation and velocity agree
very well with analytical results and both showed little azimuthal variation. Next, a rather skewed grid (Figure 8) by an elliptic grid generator was used.
The Computational 6rid Tidal Forcing Problem
1
I
Figure 8
Figure 9
-
The computational grid for tidal forcing problem. Surface Elevation at t 80 Tidal Forcing Problem
forcing
51 Solving rhe linearized tidal circulation in this 12 x 11 grid yields rather symmetric results for both water elevation and velocity.
Figure 9 shows the
water elevation over the entire domain at an instant of time.
Figure 10 shows the distribution of surface elevation along two cross sections of constant 0
(0-0 and + - 9 0 ° ) .
All the results compare well with the analytical results. Cross Sections of Surface Elevation- T i d a l Forcing
€
L
0l.o
.
i.2
8
i.4
i.e
*
1
.
1.8
0
I
.
2.2
2.0
I
.
2.4
2.8
r
Cross sectional plot of the surface elevation for the tidal forcing problem. The solid lines are the numerical solution along the 0 and 90 degree edges, the dashed line is the analytical solution.
Figure lO(a)
4.0
F
Cross Sections of Radial V e l o c i t y
-
T i d a l Forcing
r
Figure 10(b)
Cross sectional plot of the radial velocity for the tidal forcing problem using the numerical transformation ( 90. along 0 0. ( ) numerical solution along 0 ( ) analytic solution.
--------
-- -- --
-
52
The second example is the refraction and diffraction of a tsunami (T = 4 minutes) by a circular island (r = 10 km) and a parabolic bottom (depth = 0.444 km at shore to 4 km at r = 30 km). Considering an incident monochromatic plane wave along the positive x direction and applying radiation condition at infinity, we presented our results in terms of the wave amplitude and
-
phase lag along the upper half of the shoreline ( + '0 to 9 = 1 8 0 ' ) . As shown in Figure 11, the results agree well with the analytical results of Homma (1950).
b, * 4 km
' Figure ll(a)
1 r,
SO hm
Configuration of a circular island. 340
- Parabolfc Bottom - 4 min
f
4.5
Figure ll(b)
Wave amplitude along the upper shoreline of the circular island computed by the present model and Homma (1950) Ruy
340
- Parabolfc Bottom - 4 .In
blmth
Figure ll(c)
Same as ll(b),
except for phase lag instead of wave amplitude.
53 ACKNOWLEDGEMENT The work reported here has been partially supported by the U.
S. Army
Engineer Waterways Experiment Station under contract DACW 39-80-C-0087,
with
H. L. Butler and B. Johnson as contract monitors, and U. S. Geological Survey under contract 14-08-0001-4730, with R. T. Cheng as contract monitor. 5. REFERENCES Butler, H. L., 1978. Coastal flood simulation in stretched coordinates. Proceedings 16th Int'l. Conf. Coastal Engineering. ASCE/Hamburg, Germany. Cheng, R. T. and J. W. Gartner, 1984. Tides, tidal and residual currents in San Francisco Bay, results of measurements, 1979-1980. U.S.G.S. WRE Report 84-4339. Donaldson, C. duP., 1973. Atmospheric turbulence and the dispersal of atmospheric pollutants. In: D. A. Haugen (Editor) AMS Workshop on Micrometeorology. Science Press, Boston, pp 313-390. Goodwin, C. 1986. Personal communication. Johnson, B. J., 1982. Numerical modeling of estuarine hydrodynamics on a boundary-fitted coordinate system. In: J. E. Thompson (Editor), Numerical Grid Generation. Elsevier Science Publishing Company, Inc., pp 409-436. Leendertse, J. J., 1967. Aspects of a computational model for long water wave propagation. Rand Corp., Rept. RH-5299-RR Rodi, W. 1980. Turbulence Models and Their Application in Hydraulics. IAHR. Sheng, Y. P., 1976. Currents and contaminant dispersion in the near-shore of large lakes. J. Great Lakes Res., vol. 2, no. 2, pp 402-414. Sheng, Y. P., 1982. Hydraulic applications of a turbulent transport model. Proceedings 1982 ASCE Hydraulic Division Specialty Conference on Applying Research to Hydraulic Practice, pp 106-119. Sheng, Y. P., 1983. Mathematical modeling of three-dimensional coastal currents and sediment dispersion. A.R.A.P. Report 486; also Technical Report CERC-83-2, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi. Sheng, Y. P., 1984. A turbulent transport model of coastal processes. Proceedings 19th Int'l. Conf. on Coastal Engineering, ASCE, pp 2380-2396. Sheng, Y. P., 1986a. Finite-difference models for hydrodynamics of lakes and shallow seas. In: Grey, W. G. (Editor), Physics-Based Modeling of Lakes, Reservoirs and Impoundments. ASCE. Sheng, Y. P., 198613. Validation of ocean current model OCMlD for storm current simulations. A.R.A.P. Report No. 580. Sheng, Y.P., 1986c. Modeling coastal and estuarine processes using boundaryfitted grids. In: Wang, Shen, and Ding (Editors), River Sedimentation. Proceedings 3rd Int'l Symposium on River Sedimentation, pp 1416-1442. Sheng, Y. P. and H. L. Butler, 1982. Modeling coastal currents and sediment transport. Proceedings 18th Int'l. Conf. on Coastal Engineering. ASCE, pp 1127-1148. Sheng, Y. P. and C. Villaret, 1986. On the determination of sediment erosion relationship. In preparation. A three-dimensional estuarine Sheng, Y. P., S. F. Parker, D. H e m , 1985. hydrodynamic model (EHSM3D), to be published as a Water Resources Investigation Report by the Water Resources Division of U.S. Geological Survey. Sheng, Y. P., W. Lick, R. T. Gedney, and F. B. Molls, 1978. Numerical computation of the three-dimensional circulation in Lake Erie: A comparison of a free-surface and a rigid-lid model. J. Phys. Oceano., pp 72-73. Thompson, J. E., 1982. General curvilinear coordinate systems. In: J. E. Thompson (Editor), Numerical Grid Generation. Elsevier Science Publishing Company, Inc., pp 1-30.
54
Waldrop, W. R. and F. B. Tatom, 1976. Analysis of the thermal effluent from the Gallatin steam plant during low river flows. Report no. 33-30, TVA. Nearshore numerical storm surge and tidal simulaWarnstrath, J. J., 1977. T. R. €I-77-17. Army Engineer Waterways Experiment Station, tion. Vicksburg, Mississippi.
55
CIRCULATION MODELLING USING ORTHOGONAL CURVILINEAR COORDINATES ALAN F. BLUMBERG'
lEydroQual, I n c .
and H . JAMES HERRINGL
,1
L e t h b r i d g e P l a z a , Mahwah, N e w J e r s e y 07430
ZDynalysis of P r i n c e t o n , 219 Wall S t r e e t , P r i n c e t o n , N e w J e r s e y 08540-1512
ABSTRACT
A n u m e r i c a l e s t u a r i n e and c o a s t a l o c e a n c i r c u l a t i o n model i s developed i n orthogonal c u r v i l i n e a r c o o r d i n a t e s . The g o v e r n i n g e q u a t i o n s c o n s i s t o f t h e e q u a t i o n o f c o n t i n u i t y , t h e t h r e e c o m p o n e n t s o f momentum and c o n s e r v a t i o n equations f o r t h e r m a l energy and s a l t . Other p r o g n o s t i c e q u a t i o n s a r e s o l v e d f o r t h e t u r b u l e n c e k i n e t i c e n e r g y and t u r b u l e n c e m a c r o s c a l e , b o t h of which are p a r t of a t u r b u l e n c e c l o s u r e submodel r e p r e s e n t i n g t h e v e r t i c a l m i x i n g p r o c e s s . The c i r c u l a t i o n model employs s t a n d a r d f i n i t e d i f f e r e n c e t e c h n i q u e s and a c c o m o d a t e s i r r e g u l a r l y shaped c o m p u t a t i o n a l g r i d s w i t h no apparent d i f f i c u l t i e s . Successf u l two and t h r e e - d i m e n s i o n a l s i m u l a t i o n s o f a v a r i e t y o f problems with a n a l y t i c o r well e s t a b l i s h e d numerical s o l u t i o n s demonstrate t h a t t h e model p h y s i c s and n u m e r i c s c a n p r o d u c e m e a n i n g f u l r e s u l t s . The t e s t c a s e s p r e s e n t e d h e r e a l s o could be used i n t h e p r o v i d e a framework f o r o b j e c t i v e c o m p a r i s o n which development of o t h e r numerical models. 1 INTRODUCTION
The t r a d i t i o n i n ocean, e s t u a r i n e
and l a k e c i r c u l a t i o n modelling h a s been t o
use f i n i t e d i f f e r e n c e t e c h n i q u e s on a r e c t a n g u l a r g r i d . There are many i n s t a n c e s however, when t h e c o m p u t a t i o n a l r e s o u r c e s r e q u i r e d become e x c e s s i v e . T h i s s i t u a t i o n a r i s e s i n r e g i o n s where t h e c o a s t l i n e h a s prominent f e a t u r e s o r w h e r e boundary l a y e r d y n a m i c s a r e i m p o r t a n t .
I f t h e number of p o i n t s is not l a r g e
enough t o r e s o l v e t h e c o a s t a l f e a t u r e s o r b o u n d a r y l a y e r , t h e n t h e n u m e r i c a l s o l u t i o n i s l i k e l y t o h a v e g r o s s e r r o r s e v e n i n t h e i n t e r i o r regions. As an example, c o n s i d e r t h e c i r c u l a t i o n i n t h e c o a s t a l waters o f f C a l i f o r n i a , a s e t t i n g which p r o v i d e d t h e m o t i v a t i o n f o r t h e development o f t h e model t o be described h e r e i n . The d o m i n a n t f e a t u r e s o f t h e c i r c u l a t i o n o f f t h e c o a s t o f C a l i f o r n i a are c o a s t a l u p w e l l i n g i n t h e n e a r s h o r e and t h e C a l i f o r n i a C u r r e n t o f f s h o r e . Upwelling o c c u r s i n a n a r r o w r e g i o n a d j a c e n t t o t h e c o a s t and h a s a t y p i c a l offshore l e n g t h s c a l e o f 10-20 km ( t h e b a r o c l i n i c r a d i u s o f d e f o r m a t i o n ) . The C a l i f o r n i a C u r r e n t , on t h e o t h e r hand, i s r e l a t i v e l y broad w i t h a t y p i c a l width of 500 km and flows southward roughly p a r a l l e l t o t h e c o a s t . I n o r d e r t o r e s o l v e upwelling p r o c e s s e s i n a numerical c i r c u l a t i o n model, t h e s p a t i a l g r i d should b e
56 about 5 km, whereas a r e l a t i v e l y c o a r s e r g r i d , of p e r h a p s 2 5 t o 50 km,
suffices
f u r t h e r o f f s h o r e . The t r a d i t i o n a l uniform r e c t a n g u l a r g r i d i s i n a d e q u a t e h e r e , b e c a u s e i t would r e q u i r e a n e x t r a o r d i n a r i l y l a r g e number of p o i n t s t o c o v e r a domain encompassing t h e c o a s t a l and o c e a n i c r e g i o n s o f i n t e r e s t . I t becomes i m p o r t a n t t h e n t o h a v e a v a i l a b l e a c i r c u l a t i o n model K i t h t h e a b i l i t y t o r e s o l v e a r b i t r a r y topography and a l s o t h e f l e x i b i l i t y t o r e f i n e t h e g r i d i n regions of p a r t i c u l a r i n t e r e s t o r i n regions of l a r g e g r a d i e n t s . Numerous models have been developed o v e r t h e y e a r s which a l l o w f o r some f l e x i b i l i t y i n t h e g r i d s p e c i f i c a t i o n . E a r l i e r models used r e c t a n g u l a r g r i d s w i t h transformed [Reid et a l . ,
19771 or s t r e t c h e d [ P e f f l e y a n d O ' B r i e n ,
19761
c o o r d i n a t e s . O t h e r m o d e l s s u c h a s t h a t o f T h a c k e r [19771 employed i r r e g u l a r g r i d s w i t h t r i a n g u l a r c o n f i g u r a t i o n s . The e x p l o s i v e g r o w t h
of f i n i t e element
t e c h n i q u e s [ P i n d e r and Gray, 19771 h a s been m o t i v a t e d by t h e s e a r c h f o r f l e x i b i l i t y i n t h e d e s i g n o f t h e c o m p u t a t i o n a l g r i d . In more r e c e n t t i m e s , t h e u s e o f non-orthogonal boundary-fitted
c o o r d i n a t e s ( J o h n s o n , 119821 and S p a u l d i n g ,
[19841) h a s come i n t o vogue. The purpose o f t h e paper is t o p r o v i d e a n a l t e r n a t i v e a p p r o a c h t o t h o s e e n u m e r a t e d a b o v e a n d t h u s make p o s s i b l e long t e r m i n t e g r a t i o n s u s i n g more
modest c o m p u t a t i o n a l r e s o u r c e s .
What f o l l o w s i s a d e t a i l e d d e s c r i p t i o n o f a n u m e r i c a l e s t u a r i n e and c o a s t a l ocean c i r c u l a t i o n model t h a t a l l o w s c o n s i d e r a b l e l a t i t u d e i n t h e d e s i g n o f t h e c o m p u t a t i o n a l g r i d . T h i s a d d i t i o n a l freedom
i s accomplished through t h e u s e of
a n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s y s t e m on t h e h o r i z o n t a l c o o r d i n a t e s u r f a c e s . When c a s t i n t h e s e c o o r d i n a t e s t h e g o v e r n i n g e q u a t i o n s r e t a i n much of t h e a n a l y t i c a l s i m p l i c i t y of t h e f a m i l i a r C a r t e s i a n e q u a t i o n s . One b e n e f i c i a l r e s u l t of t h e s i m p l i c i t y is t h a t t h e computational c o s t f o r t h e equations i n o r t h o g o n a l c u r v i l i n e a r form i s o n l y m a r g i n a l l y i n c r e a s e d o v e r t h a t f o r t h e i r C a r t e s i a n c o u n t e r p a r t and i s ponding b o u n d a r y - f i t t e d
c o n s i d e r a b l y less t h a n t h e c o s t f o r t h e c o r r e s -
equations.
Through a series of model t e s t c a l c u l a t i o n s , t h e v i a b i l i t y and v e r s a t i l i t y of t h i s new f o r m u l a t i o n w i l l b e d e m o n s t r a t e d . C o n s t r u c t i o n of t h e h o r i z o n t a l g r i d mesh used by t h e model i s
i n c l u d e d i n t h e d i s c u s s i o n . The t e s t c a s e s which f o l -
low are examples i n which a n a l y t i c and o t h e r n u m e r i c a l s o l u t i o n s a r e a v a i l a b l e f o r c o m p a r i s o n a n d a s s e s s m e n t of model performance. Both two and three-dimens i o n a l cases w i l l be considered.
2 MODEL FORMULATION The
e q u a t i o n s which d e s c r i b e t h e c i r c u l a t i o n i n e s t u a r i e s and i n c o a s t a l and
open o c e a n s are t h e e q u a t i o n s o f c o n t i n u i t y , t h e t h r e e c o m p o n e n t s o f momentum a n d e q u a t i o n s f o r t h e c o n s e r v a t i o n o f t h e r m a l e n e r g y and s a l t . These e q u a t i o n s , t o g e t h e r w i t h a n e q u a t i o n of s t a t e and a s u i t a b l e f o r m o f t u r b u l e n c e c l o s u r e ,
57 are sufficient
t o determine primitive variables consisting of
the three
components of v e l o c i t y , t h e e l e v a t i o n o f t h e f r e e s u r f a c e , t h e t e m p e r a t u r e , t h e s a l i n i t y and t h e d e n s i t y . S e v e r a l s i m p l i f y i n g assumptions have been made i n f o r m u l a t i n g t h e e q u a t i o n s .
The h y d r o s t a t i c a p p r o x i m a t i o n i m p l i e s t h a t t h e l o c a l p r e s s u r e a t a p o i n t i s a f u n c t i o n only of t h e weight o f t h e w a t e r column above i t and t h u s t h e t r a n s p o r t of v e r t i c a l momentum i s n e g l i g i b l e .
The B o u s s i n e s q a p p r o x i m a t i o n i s t h a t t h e
v a r i a t i o n s i n t h e d e n s i t y about a mean v a l u e , Po s a y , are d y n a m i c a l l y n e g l i g i b l e except i n t h e d e t e r m i n a t i o n o f t h e l o c a l h y d r o s t a t i c p r e s s u r e .
2.1 Equations o f Motion The e q u a t i o n s w h i c h form t h e b a s i s o f t h e c i r c u l a t i o n model a r e w e l l e s t a b l i s h e d i n g e n e r a l o r t h o g o n a l c o o r d i n a t e s ( s e e E r i n g e n [ 19621 f o r e x a m p l e ) . Consider a system of orthogonal c u r v i l i n e a r coordinates with h o r i z o n t a l coordin a t e s ( & I , &2) and v e r t i c a l c o o r d i n a t e
(2)
a s shown i n F i g u r e 1. The m e t r i c
Z
Fig. 1. The o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s y s t e m u s e d i n t h e c i r c u l a t i o n model.
c o e f f i c i e n t s , h i and h 2 , are d e f i n e d so t h a t a d i s t a n c e increment s a t i s f i e s t h e relation
58 The d i f f e r e n t i a l a r c l e n g t h s a l o n g €1 and = hzdt2.
€ 2 at p o i n t P a r e d s l = h l d t l and ds2
The h o r i z o n t a l v e l o c i t y v e c t o r h a s components
i n t h e € 1 and € 2 d i r e c t i o n s , Arakawa and Lamb [I9771
respectively.
F o l l o w i n g M e r i l e e s [ 1 9 7 6 ] and
t h e c o n t i n u i t y equation is
where w i s t h e v e r t i c a l v e l o c i t y .
The h o r i z o n t a l momentum e q u a t i o n s c a n b e
w r i t t e n as
and t h e v e r t i c a l momentum e q u a t i o n w i t h t h e h y d r o s t a t i c assumption i s ap P g = - z
(6)
t
w i t h P o , t h e r e f e r e n c e d e n s i t y ; PI t h e i n s k u d e n s i t y ; g , t h e g r a v i t a t i o n a l a c c e l e r a t i o n ; f , t h e C o r i o l i s p a r a m e t e r , and P, t h e p r e s s u r e . approximation h a s been u s e d i n d e r i v i n g E q u a t i o n s
The B o u s s i n e s q
( 4 ) and ( 5 ) . The p r e s s u r e a t
d e p t h z can b e o b t a i n e d by i n t e g r a t i n g E q u a t i o n ( 6 ) from z t o t h e f r e e s u r f a c e ,
z =
(€1,tZ,t),
and i s 0
59
In t h e d i s c u s s i o n s w h i c h f o l l o w t h e a t m o s p h e r i c p r e s s u r e ,
Patm,
i s assumed t o
contribute l i t t l e t o t h e pressure gradient. The d i f f u s i o n t e r m s , r e p r e s e n t e d by
rl
and r 2 i n E q u a t i o n s ( 4 ) and ( 5 ) ,
can
be w r i t t e n as
ah
and
The v e r t i c a l eddy d i f f u s i v i t y f o r t u r b u l e n t momentum mixing i s denoted a s KM. F i n a l l y , the s h e a r stress components i n E q u a t i o n s (8) and ( 9 ) a r e
Tll
t
+
221 2u ah
hlh2
%r< *El
'21
+
(10)
,
(11)
'12 qq[$]] h2 a
and
The h o r i z o n t a l d i f f u s i v i t y i s d e n o t e d a s AM.
The momentum e q u a t i o n s p r e s e n t e d
h e r e a r e q u i t e s i m i l a r t o t h e more u s u a l C a r t e s i a n c o o r d i n a t e system, e x c e p t f o r t h e a d d i t i o n a l t e r m s w h i c h a c c o u n t f o r t h e c u r v a t u r e of t h e c o o r d i n a t e s y s t e m itself.
The C a r t e s i a n system e q u a t i o n s ( s e e Blumberg a n d M e l l o r , [ 1 9 8 6 1 ) a r e
recovered by t h e t r a n s f o r m a t i o n h l d t l + dx and h p d t p 4dy. The c o n s e r v a t i o n e q u a t i o n s f o r t e m p e r a t u r e , w r i t t e n i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s as
8,
and s a l i n i t y , S, may b e
60
ae
(h2Ule) t
a (hlU28) at2
la t
(We)
and
T h e v e r t i c a l eddy d i f f u s i v i t y f o r t u r b u l e n t mixing o f h e a t and s a l t i s KH w h i l e t h e h o r i z o n t a l d i f f u s i v i t y i s denoted as AH.
Using t h e t e m p e r a t u r e and
s a l i n i t y , t h e d e n s i t y i s computed a c c o r d i n g t o t h e e q u a t i o n o f s t a t e P = p(e.s),
(15)
due t o Fofonoff [ 1 9 6 2 ] . The h o r i z o n t a l m i x i n g c o e f f i c i e n t s A M a n d AH are used t o p a r a m e t e r i z e a l l motion which is n o t r e s o l v e d on t h e n u m e r i c a l g r i d .
Since t h e g r i d i s non-uni-
f o r m , t h e h o r i z o n t a l m i x i n g c o e f f i c i e n t s must v a r y p r o p o r t i o n a l l y i n o r d e r t o m a i n t a i n an uniform g r i d Reynolds number. The r e l a t i o n employed i s
w h e r e A,
i s t h e e q u i v a l e n t d i f f u s i v i t y f o r an uniform g r i d with a g r i d spacing
o f ho. The boundary c o n d i t i o n s a t t h e f r e e s u r f a c e , z = ~ ( € 1 ,( 2 ) ,
au,
au
are:
61
and
where ( ~ ~ 1~ , ~ i s2 t h) e s u r f a c e wind s t r e s s v e c t o r . n,
and t h e s a l i n i t y f l u x a t t h e s u r f a c e i s
H
A t the bottom of t h e basin,
The l o c a l n e t h e a t f l u x i s
5.
t h e normal g r a d i e n t s of
8
and S a r e z e r o .
A d d i t i o n a l l y , at t h e bottom boundary, b ,
and
where H({l,
stress.
+’b
’o
€2)
i s t h e bottom topography and ( T b l , r b 2 ) i s t h e bottom f r i c t i o n a l
A s i n Blumberg and M e l l o r [19861 t h e bottom s t r e s s i s determined from
cDI 3bl 3b .
(19a)
The value of t h e d r a g c o e f f i c i e n t CD i s g i v e n by
+
where zb a n d v b
a r e t h e d e p t h of and c o r r e s p o n d i n g v e l o c i t y a t t h e g r i d
point n e a r e s t t h e bottom and
K
is t h e von Karman c o n s t a n t .
is t h a t t h e c a l c u l a t i o n s w i l l y i e l d
+ vb
The p u r p o s e o f ( 1 9 )
= (?b/KUrb)ln(Z/Zo)
boundary r e g i o n i f enough r e s o l u t i o n i s p r o v i d e d .
i n t h e lower
In, f o r example, 100 m w a t e r
t h e log l a y e r is w e l l r e s o l v e d , whereas i n much d e e p e r water, it i s n o t . latter instance,
In t h e
i t i s a d v a n t a g e o u s t o abandon (19b) and s p e c i f y CD = 0.0025.
S p e c i f i c a l l y , t h e f i n a l a l g o r i t h m i s t o s p e c i f y t h e l a r g e r v a l u e o f t h e two v a l u e s g i v e n b y ( 1 9 b ) and 0.0025.
The p a r a m e t e r zo is t h e l o c a l bottom rough-
ness. A v a l u e of 1 cm is used as s u g g e s t e d by G r a n t and Madsen [ 1 9 7 9 1 .
This
v a l u e r a t h e r e f f e c t i v e l y p a r a m e t e r i z e s t h e d i s s i p a t i o n produced by s h o r t p e r i o d
swells and t i d e s when t h e y are n o t e x p l i c i t l y i n c l u d e d i n t h e model f o r c i n g . somewhat s m n l l e r v a l u e o f zo ( - 0 . 2
A
cm) s h o u l d be used when t h e swells a n d / o r
t i d e s a r e in,: ruded. L a t e r a l b o u n d a r y c a n d i t i o n s f o r t h e c u r v i l i n e a r c o o r d i n a t e system model a r e formulated i n a s i m i l a r manner t o t h o s e o f a r e c t a n g u l a r c o o r d i n a t e s y s t e m model. On c l o s e d , l a n d b o u n d a r i e s t h e normal component o f v e l o c i t y i s set t o
62 z e r o . The n o r m a l g r a d i e n t s o f t e m p e r a t u r e and s a l i n i t y a r e a l s o z e r o s o t h a t t h e r e a r e no a d v e c t i v e and d i f f u s i v e h e a t
and
salt
fluxes across these
b o u n d a r i e s . Open l a t e r a l b o u n d a r y c o n d i t i o n s must b e p r e s c r i b e d w i t h s p e c i a l care since these conditions represent a parameterization of t h e environment e x t e r n a l t o t h e domain under c o n s i d e r a t i o n . C o n s i d e r a b l e a t t e n t i o n i s c u r r e n t l y being devoted t o t h e development o f r o b u s t t e c h n i q u e s f o r s p e c i f y i n g t h i s parameterization.
In most of t h e model s i m u l a t i o n s t o f o l l o w i n S e c t i o n 3 , t h e
s i m p l e s t p o s s i b l e f o r m u l a t i o n f o r t h e open boundary c o n d i t i o n i s u s e d . T h a t i s , t h e s e a s u r f a c e e l e v a t i o n i s s p e c i f i e d a s a f u n c t i o n o f t i m e . There i s one s i m u l a t i o n , however, which u s e s a r a d i a t i o n c o n d i t i o n t o p a s s t r a n s i e n t s through t h e boundary and t h i s c o n d i t i o n i s d e s c r i b e d i n S e c t i o n 3 . 4 .
The "u" c o o r d i n a t e system proposed by P h i l l i p s [ 1 9 5 7 1 i s u s e d i n t h e model f o r m u l a t i o n t o o v e r c o m e t h e c o m p u t a t i o n a l problems which a r i s e i n t h e v i c i n i t y of l a r g e b a t h y m e t r i c i r r e g u l a r i t i e s .
Following Blumberg a n d M e l l o r [ 1 9 8 3 ] , a
new s e t o f i n d e p e n d e n t v a r i a b l e s t h a t t r a n s f o r m s b o t h t h e s u r f a c e and bottom i n t o c o o r d i n a t e s u r f a c e s is introduced.
The number o f v e r t i c a l g r i d p o i n t s i n
t h e t r a n s f o r m e d s y s t e m i s t h u s t h e same f o r t h e s h a l l o w c o a s t a l r e g i o n s as f o r t h e d e e p e r o c e a n i c r e g i o n s . The g o v e r n i n g e q u a t i o n s a r e t r a n s f o r m e d from (€1, &2, z, t ) t o
(&?, &$,
u , t*) c o o r d i n a t e s where
In t h i s system u r a n g e s from u = 0 a t z = 7) t o u = - 1 a t z = -H.
A new v e r t i c a l
v e l o c i t y can now be d e f i n e d a s
v
- .;.;[h2u1 1
(%
+
2).
h1U2
(
2 9)] +
which t r a n s f o r m s t h e boundary c o n d i t i o n s , E q u a t i o n s ( 1 7 ~ )and ( 1 8 b ) , i n t o o
(E:,
E;,
0 , t*)
-
0 and w
* E,,*
(El,
-1, t*)
0
.
T h e g o v e r n i n g e q u a t i o n s may now b e w r i t t e n ( a l l n o t a t i o n a l convenience) as
(22a,b)
*
w i l l be dropped f o r
63
- -U-2I D- -ahl- '1
h:
--
U 2I D
hlh2
U2D 2 ah2
hlh2
fDU2
a',
ahl
U2D 2 ah2
"2
hi
(24)
- _ -t fDUl
=
-
-a'
aE2
h2PO
a[,
t
Dr;
,
(25)
f Z[K"E] . The pressure gradient terms i n
0
VP = o 0 g h
where
v
t gDV
,/o
pdo
-
(27)
( 2 4 ) and ( 2 5 ) are g i v e n , i n u coordinates, by
lo 2 0
gVD
o
do
,
i s the gradient operator i n orthogonal c o o r d i n a t e s .
(28)
The h o r i z o n t a l and
v e r t i c a l d i f f u s i o n terms f o r momentum are d e f i n e d according t o :
64
.+
&
ahl
[=21
Tq
$1 ah
-
T22
and
(30)
The t r a n s f o r m e d z-coordinates,
s h e a r s t r e s s c o m p o n e n t s r e t a i n t h e i r same f o r m a s i n
E q u a t i o n s ( l o ) , ( 1 1 ) and (121, i f t h e i d e a s d e v e l o p e d b y M e l l o r
and Blumberg [ 1 9 8 5 ] a r e used.
These i d e a s permit t h e r e l a t i v e l y s i m p l e mathe-
m a t i c a l forms f o r t h e s h e a r stress components and t h e h o r i z o n t a l d i f f u s i o n terms i n E q u a t i o n s ( 2 6 ) , ( 2 7 ) , ( 2 9 ) and ( 3 0 ) , y e t produce a n a c c u r a t e computation of bottom boundary l a y e r dynamics even on s l o p i n g bottoms.
2.2 Turbulence C l o s u r e Model The
governing e q u a t i o n s c o n t a i n p a r a m e t e r i z e d Reynolds s t r e s s and f l u x t e r m s
which account f o r t h e t u r b u l e n t d i f f u s i o n o f momentum, h e a t a n d s a l t b y s m a l l s c a l e p r o c e s s e s n o t d i r e c t l y i n c l u d e d i n t h e model.
The p a r a m e t e r i z a t i o n of
t u r b u l e n c e i n t h e c i r c u l a t i o n model i s based on t h e work o f M e l l o r a n d Yamada (19741. The v e r t i c a l mixing c o e f f i c i e n t s , KM and KH, i n E q u a t i o n s
(a),
( 9 1 , ( 1 3 ) and
( 1 4 ) a r e o b t a i n e d by a p p e a l i n g t o a s e c o n d o r d e r t u r b u l e n c e c l o s u r e scheme which c h a r a c t e r i z e s t h e t u r b u l e n c e by e q u a t i o n s f o r t h e t u r b u l e n c e k i n e t i c e n e r g y , q2/2, and a t u r b u l e n c e m a c r o s c a l e , l
,
according t o , ( i n
Q
coordinates),
65
and
where s w a l l p r o x i m i t y f u n c t i o n d e f i n e d as
= 1
+
,
E2[$I2
(33)
with
(L)-'
= (q
-
z)-'
t
(H
t
,
z)-'
(34)
has been i n t r o d u c e d . With t h e u s e o f prescription of
t h e c l o s u r e assumptions it is p o s s i b l e t o reduce t h e
t h e m i x i n g c o e f f i c i e n t s KM, K H a n d K q t o t h e f o l l o w i n g
expressions,
% Fss,,
K,.
E
FqSH
and
K
q
E
Fqsq.
(35a,b,c)
The s t a b i l i t y f u n c t i o n s , SM, SH a n d S q a r e a n a l y t i c a l l y d e r i v e d , a l g e b r a i c r e l a t i o n s f u n c t i o n a l l y dependent upon These r e l a t i o n s f o l l o w
aUl/ao, aU2/au,
gp;'ap/ao
,
q,
and L .
f r o m a c l o s u r e h y p o t h e s i s f i r s t d e s c r i b e d by M e l l o r
[1973] and r e c e n t l y summarized by M e l l o r and Yamada [19821. It i s convenient t o d e f i n e
and
Then t h e s t a b i l i t y f u n c t i o n s become
66
[6AlA2GM]
t SH
[ l t 6A1G, 2
-
S,
[1 - 2A2B2GH - 12A1A2 H] G
(37)
A2
and S,
9A A G
-
SH [12A1GH 2 t 9AlA2GH] = A1(l
-
35)
,
(38)
which are r e a d i l y s o l v e d f o r SM and SH as f u n c t i o n s of GM and GH, and
s
9
= 0.20
.
39)
A n e c e s s a r y c l o s u r e assumption i s t h a t a l l l e n g t h s are p r o p o r t i o n a l t o each other, thus, (40) By a p p e a l i n g t o l a b o r a t o r y d a t a [ M e l l o r a n d Yamada,
19821
the empirical
c o n s t a n t s were a s s i g n e d t h e v a l u e s : = ( 0 . 9 2 , 0.74,
(A1,A2,B1,B2,C1)
16.6,
1 0 . 1 , 0.08)
(41a)
and
(El, E2) = (1.8,
1.33)
.
The s u r f a c e and bottom boundary c o n d i t i o n s on q2 and 9. a r e
213 u2 rs
2
= B1
qLE=
0
and
2 = B1213 u2r b w h e r e urs and
(42~)
’
and Urb a r e t h e f r i c t i o n v e l o c i t i e s a s s o c i a t e d w i t h t h e s u r f a c e wind
bottom f r i c t i o n a l stresses, r e s p e c t i v e l y .
2 . 3 Orthogonal C u r v i l i n e a r Grid G e n e r a t i o n The s p e c i f i c a t i o n o f a n o r t h o g o n a l c u r v i l i n e a r g r i d i s c o n s i d e r a b l y more involved t h a n t h a t o f a C a r t e s i a n g r i d .
Both t h e s h a p e o f t h e d o m a i n and t h e
r e l a t i v e s p a c i n g o f t h e g r i d l i n e s throughout t h e domain must b e s p e c i f i e d i n t h e case of a c u r v i l i n e a r g r i d .
In a C a r t e s i a n g r i d , on t h e o t h e r h a n d , a l l
g r i d l i n e s a r e s t r a i g h t and t h e i r s p a c i n g remains uniform from end t o end.
The
67 a d d i t i o n a l freedom i n h e r e n t i n a c u r v i l i n e a r g r i d makes i t p o s s i b l e t o a d a p t t h e g r i d t o t h e f e a t u r e s o f t h e f l o w so t h a t many g r i d l i n e s f a l l i n r e g i o n s o f rapid change i n f l o w p r o p e r t i e s . There a r e a l s o c o n s t r a i n t s which p r e v e n t an o r t h o g o n a l g r i d from b e i n g completely a r b i t r a r y , as a r e f i n i t e e l e m e n t a n d b o u n d a r y f i t t e d g r i d s , f o r instance.
An o r t h o g o n a l g r i d c o n s i s t s o f f a m i l i e s o f l i n e s i n t e r s e c t i n g
orthogonally.
Each g r i d l i n e of one f a m i l y must c r o s s e a c h g r i d l i n e o f t h e
o t h e r f a m i l y o n c e , a n d o n l y o n c e ; e n t e r i n g on one boundary and e x i t i n g on t h e opposite boundary.
A l s o , l a r g e l o c a l c u r v a t u r e i n one f a m i l y o f g r i d l i n e s must
be avoided, s i n c e
it causes a focusing of t h e o t h e r , orthogonal, family of g r i d
l i n e s t o form what i s r e f e r r e d t o a s a s i n g u l a r i t y i n t h e g r i d . Many c o n t i n e n t a l s h e l f r e g i o n s , a n e x a m p l e o f w h i c h w i l l b e d e s c r i b e d i n Section 3 , e x h i b i t much more r a p i d changes i n f l o w p r o p e r t i e s and b a t h y m e t r y o n t r a n s e c t s n o r m a l t o t h e c o a s t t h a n on a l o n g s h o r e t r a n s e c t s . In o r d e r t o model t h i s b e h a v i o r e f f i c i e n t l y , i t i s advantageous t o u s e a c a l c u l a t i o n g r i d w i t h a band o f c l o s e l y s p a c e d g r i d l i n e s p a r a l l e l t o t h e c o a s t .
T h i s f i n e mesh w i l l
r e q u i r e t h e smallest number of l i n e s i f t h i s g r i d domain i s b e n t s o t h a t t h e lines e s s e n t i a l l y follow t h e coast. F o r c i n g t h e g r i d t o match t h e e x a c t s h a p e of t h e c o a s t i s i m p r a c t i c a l , however.
The a c t u a l c o a s t l i n e may b e rough and may i n c l u d e b a y s a n d p r o m o n t o r i e s
which a r e n o t r e p r o d u c i b l e on t h e g r i d s c a l e chosen. In any c a s e , t h e c o a s t a l p r o f i l e o f t e n does n o t r e f l e c t t h e s h a p e o f t h e t o p o g r a p h y b e l o w t h e s u r f a c e t h a t i s more i m p o r t a n t i n d e t e r m i n i n g t h e c i r c u l a t i o n .
Therefore, the inner
boundary of t h e g r i d i s b e s t chosen a s some mean c o a s t l i n e p r o f i l e . The o u t e r b o u n d a r y may b e c h o s e n a p p r o x i m a t e l y t h e r e q u i r e d d i s t a n c e o f f shore, b u t may b e smoother t h a n t h e s h o r e b o u n d a r y .
The g r i d s p a c i n g i n t h e
o f f s h o r e d i r e c t i o n i s d e t e r m i n e d by e r e c t i n g a set o f c u r v i l i n e a r l i n e s between t h e i n n e r and o u t e r b o u n d a r i e s which are spaced i n t h e r e q u i r e d p r o p o r t i o n .
If
t h e s e l i n e s are d e s c r i b e d as l i n e s o f c o n s t a n t € 1 , t h e y may b e e x p r e s s e d i n t h e form f(X, 9 ; € $
where
= 0
,
(43)
and ' p a r e t h e l o n g i t u d e and l a t i t u d e , r e s p e c t i v e l y .
Another f a m i l y of g r i d l i n e s f o r constant l i n e s of c o n s t a n t €1.
&z,
may be e r e c t e d normal t o t h e
The c o n d i t i o n o f o r t h o g o n a l i t y i s p r o v i d e d b y t h e
Cauchy-Riemann e q u a t i o n s , i n v e r t e d t o r e l a t e X a n d q~ a s f u n c t i o n s o f t h e g r i d c o o r d i n a t e s , [ 1 and
&z,
68 and
E l i m i n a t i n g t h e m e t r i c s b e t w e e n E q u a t i o n s ( 4 4 ) and (45) y i e l d s a r e l a t i o n between t h e a n g l e s o f t h e (1 and (2 c o o r d i n a t e l i n e s ,
E q u a t i o n (46) i s b o t h n e c e s s a r y a n d s u f f i c i e n t t o i n s u r e o r t h o g o n a l i t y . E q u a t i o n s (43) and (46) may be s o l v e d f o r t h e l o c a t i o n o f l i n e s o f c o n s t a n t € 2 i n physical space.
Beginning w i t h t h e d e s i r e d g r i d s p a c i n g a t one b o u n d a r y t h e
s o l u t i o n p r o c e e d s a c r o s s t h e d o m a i n t o t h e o p p o s i t e b o u n d a r y . The r e s u l t s , w r i t t e n i n t h e form
and
r e p r e s e n t a smooth, one t o o n e , mapping o f t h e c o m p u t a t i o n a l g r i d o n t o t h e p h y s i c a l s u r f a c e of t h e e a r t h e x p r e s s e d i n l o n g i t u d e and l a t i t u d e . Although t h e p h y s i c a l l o c a t i o n of t h e g r i d i e required f o r i n t e r p r e t a t i o n a n d p l o t t i n g of r e s u l t s , o n l y t h e metrics of t h e c o m p u t a t i o n a l g r i d , h l and h 2 , a r e a c t u a l l y required f o r t h e s o l u t i o n of t h e equations of motion. E q u a t i o n s (44) a n d
From
(45) i t c a n b e shown t h a t t h e m e t r i c s a r e g i v e n by t h e
relations
= R
h:
2
(49)
and
2
h2
-
R2
(50)
where R i s t h e r a d i u s of t h e e a r t h . The c o n s t r u c t i o n , t e s t i n g and s e l e c t i o n o f a s a t i s f a c t o r y g r i d i s a n i t e r a t i v e process.
It begins with t h e s p e c i f i c a t i o n of s e v e r a l b a s i c parameters; t h e
r e g i o n t o b e c o v e r e d b y t h e p h y s i c a l g r i d and t h e m a t r i x s i z e of t h e computat i o n a l g r i d . A l l o f t h e c o m p u t a t i o n a l g r i d s used f o r t h e m o d e l s i m u l a t i o n s t o f o l l o w have been g e n e r a t e d w i t h t h e p r o c e d u r e s d e s c r i b e d above.
69 When e v a l u a t i n g t h e g r i d d i s t r i b u t i o n i t i s i m p o r t a n t t o c o n s i d e r t h e r o l e o f the grid i n the solution of t h e equations o f motion.
In a f i n i t e d i f f e r e n c e
s o l u t i o n t h e v a r i a b l e s a t a g r i d p o i n t r e p r e s e n t some a p p r o p r i a t e l y d e f i n e d average o v e r t h a t g r i d box.
If the properties of t h e physical flow a r e rela-
t i v e l y u n i f o r m , o r a t l e a s t m o n o t o n i c a l l y v a r y i n g o v e r a g r i d box, t h e n t h e numerical s o l u t i o n can c l o s e l y r e p r e s e n t t h e a c t u a l c i r c u l a t i o n . when t h e p h y s i c a l v a l u e s v a r y w i d e l y w i t h i n a g r i d box,
Alternatively,
t h e e f f e c t s of t h i s
s p a t i a l v a r i a b i l i t y a r e not e x p l i c i t l y included i n t h e n u m e r i c a l s o l u t i o n and must b e
using a
introduced
subgrid
scale
parametrization of missing
physics. There i s y e t
a n o t h e r c o n s i d e r a t i o n i n c h o o s i n g a g r i d b e s i d e s i t s adequacy
in representing t h e solution.
The impact of g r i d s p a c i n g o n t h e t i m e s t e p o f
t h e c a l c u l a t i o n must a l s o be assessed. step
( s e e S e c t i o n 2.4)
The approximate maximum a l l o w a b l e t i m e
applies for the e n t i r e g r i d .
Therefore t h e smallest
value anywhere i n t h e f i e l d i s c o n t r o l l i n g .
2.4 Numerical Techniques The e q u a t i o n s w h i c h form t h e c i r c u l a t i o n model t o g e t h e r w i t h t h e i r boundary c o n d i t i o n s are s o l v e d by f i n i t e d i f f e r e n c e t e c h n i q u e s . A s t a g g e r e d g r i d ( " C " g r i d , see Arakawa a n d Lamb, [ 1 9 7 7 ] ) i s used f o r h o r i z o n t a l d i f f e r e n c i n g .
The
a r r a n g e m e n t o f p o i n t s h a s U1 a t p O i n t s ? A & 1 / 2 away from t h e p o i n t where H , 7) a r e d e f i n e d and Up a t p o i n t s k A < p / 2 away from t h e H and 7) p o i n t s .
i s d e f i n e d a t t h e Up p o i n t and hp i s d e f i n e d a t t h e U1 p o i n t .
The m e t r i c h i The v e r t i c a l
d i f f e r e n c i n g remains i d e n t i c a l t o t h a t used by Blumberg and Mellor [1983]. More d e t a i l s o f t h e d i f f e r e n c i n g t e c h n i q u e s c a n b e found i n t h a t a r t i c l e .
It s h o u l d
be p o i n t e d o u t t h a t t h e f i n i t e d i f f e r e n c e e q u a t i o n s e m p l o y e d c o n s e r v e e n e r g y ,
mass and momentum, and t h e y i n t r o d u c e d no a r t i f i c i a l h o r i z o n t a l d i f f u s i o n . A mode s p l i t t i n g t e c h n i q u e similar t o t h a t d e s c r i b e d by Simons 119741 i s used for computational
efficiency.
An
e x t e r n a l mode d e r i v e d b y v e r t i c a l l y
i n t e g r a t i n g E q u a t i o n s ( 3 ) , ( 4 ) , and ( 5 ) c a n b e w r i t t e n as,
at and
t
a hl
a5
-
fUZD
-
Al
(52)
70 w h e r e t h e p r e s s u r e h a s b e e n e l i m i n a t e d u s i n g E q u a t i o n s (7) and t h e v e r t i c a l l y i n t e g r a t e d v e l o c i t i e s are d e f i n e d as n
The terms A 1 a n d A 2 c o n t a i n v e r t i c a l i n t e g r a l s of t h e d e n s i t y g r a d i e n t , advect i v e and d i f f u s i v e terms.
The c o m p u t e d b o t t o m f r i c t i o n a n d p r e s c r i b e d wind
s t r e s s e s a r e a l s o i n c o r p o r a t e d i n t o t h e A's. The c o m p u t a t i o n a l s t r a t e g y is t o s o l v e e q u a t i o n s f o r t h e e x t e r n a l mode, t h a t
i s , t h e s h a l l o w w a t e r wave e q u a t i o n s ( 5 1 ) , (52) and ( 5 3 ) , w i t h a s h o r t t i m e s t e p t o resolve high frequency motions. Equations (52)
The terms on t h e r i g h t h a n d s i d e o f
a n d ( 5 3 ) a r e s u p p l i e d f r o m t h e i n t e r n a l mode a n d a r e h e l d
c o n s t a n t i n time o v e r t h e e x t e r n a l mode i n t e g r a t i o n p e r i o d . p r o v i d e s d?)/d.$l and dq/d&
The e x t e r n a l mode
f o r i n s e r t i o n i n t o t h e i n t e r n a l mode e q u a t i o n s which
are t h e n s o l v e d w i t h a much l o n g e r t i m e s t e p , t h e r a t i o
between t h e e x t e r n a l
a n d i n t e r n a l wave s p e e d s . Once t h e v e r t i c a l s t r u c t u r e h a s been d e t e r m i n e d , t h e terms on t h e r i g h t hand s i d e o f E q u a t i o n s ( 5 2 ) and ( 5 3 ) a r e updated a n d a n o t h e r e x t e r n a l mode s o l u t i o n b e g i n s . The t i m e d i f f e r e n c i n g scheme c o n s i s t s o f c e n t e r e d d i f f e r e n c e s ( l e a p f r o g ) . The numerical scheme i s q u a s i - i m p l i c i t at t h e forward t i m e level.
i n that vertical diffusion is evaluated
This makes p o s s i b l e s m a l l v e r t i c a l s p a c i n g n e a r t h e
s u r f a c e and bottom w i t h o u t t h e need t o r e d u c e t h e t i m e increment o r r e s t r i c t t h e magnitude of t h e mixing c o e f f i c i e n t s . From a v i e w p o i n t o f c o m p u t a t i o n a l s t a b i l i t y , t h e Courant-Friedrichs-Levy c o n d i t i o n on t h e v e r t i c a l l y i n t e g r a t e d , e x t e r n a l mode, t r a n s p o r t e q u a t i o n s governs t h e s i z e of the t i m e step.
An e x t e n s i v e F o u r i e r s t a b i l i t y a n a l y s i s i s
n o t p o s s i b l e w i t h t h e p r e s e n t model e q u a t i o n s because o f t h e s p a t i a l l y v a r y i n g
metrics.
T h e c r i t i c a l t i m e s t e p t h u s c a n no l o n g e r b e g i v e n i n e x p l i c i t
a l g e b r a i c form e v e n f o r a l i n e a r i z e d s u b s e t .
However, by a n a l o g y t o t h e
a n a l y s i s t e c h n i q u e s used w i t h l i n e a r e q u a t i o n s i n a C a r t e s i a n c o o r d i n a t e , a r u l e of thumb f o r t h e c r i t i c a l t i m e s t e p i s found t o b e
Atc
<
5[
(hi2
+ h;2]r1'2
(55)
e
which h o l d s l o c a l l y . averaged velocity. used.
Here c e = 2 ( g H ) f +
i
where
i
is t h e f a s t e s t v e r t i c a l l y
The e n t i r e domain must now b e examined and t h e smallest Atc
The i n t e r n a l mode i s r e s t r i c t e d by a s i m i l a r c o n d i t i o n e x c e p t t h a t ce,
r e p l a c e d by
Ci,
a v a l u e twice
current velocity.
is
t h e f a s t e s t i n t e r n a l wave speed p l u s t h e l a r g e s t
For most c o a s t a l and open o c e a n s i t u a t i o n s , c e / c i i s from 80
t o 100, w h i l e i n e s t u a r i e s t h e r a t i o i s somewhaG s m a l l e r , p e r h a p s 10.
71 2.5 V e r t i c a l l y I n t e g r a t e d Model The e x t e r n a l mode e q u a t i o n s c a n b e c a s t
i n t o a s t a n d a l o n e model by
e x p l i c i t l y i n t e g r a t i n g t h e terms w h i c h a p p e a r on t h e r i g h t h a n d s i d e o f I t w i l l b e a p p a r e n t from e x a m i n a t i o n o f t h e s t a n d -
Equations (52) and ( 5 3 ) .
alone, v e r t i c a l l y i n t e g r a t e d model and t h e e x t e r n a l mode e q u a t i o n s t h a t t h e y a r e q u i t e s i m i l a r . By d e m o n s t r a t i n g t h e v a l i d i t y o f t h e c u r v i l i n e a r c o o r d i n a t e approach with t h e v e r t i c a l l y i n t e g r a t e d model, t h e p e r f o r m a n c e o f a n i m p o r t a n t component of t h e t h r e e - d i m e n s i o n a l model i s a l s o b e i n g proven a t computing c o s t s f a r below t h o s e o f t h e f u l l t h r e e - d i m e n s i o n a l
model. For completeness t h e s e
new, i n t e g r a t e d , e q u a t i o n s a r e p r e s e n t e d h e r e ,
ah 1 a h l -2 - -UID - -- E i D h21
hlh2
a%
t @
hl
a%
- fU2D - D-r l
= 91
(63)
and
a Iiap t I t hlhi
[+[81!2Dhi] "
- - -1 hlh2
t @
ahl -2
a t 2 'lD
?!L + fc,D
h2
t $-[$Dhlh2]]
1 ah2 -2
- -a t 2 U2D - D-r 2
-
a2
.
(64)
The t e r m s a l a n d m 2 c o n t a i n t h e bottom f r i c t i o n ( p a r a m e t e r i z e d w i t h a q u a d r a t i c drag law) and t h e s p e c i f i e d wind s t r e s s ,
and
where CD is a d r a g c o e f f i c i e n t u s u a l l y t a k e n t o be 0.0025 and
7';
and T!
are the
wind a t r e s s c o m p o n e n t s i n t h e €1 a n d € 2 d i r e c t i o n s a t t h e s u r f a c e , The quan-
t i t i e s 7' and T 2 a r e v e r t i c a l i n t e g r a l s o f t h e h o r i z o n t a l momentum d i f f u s i o n and a r e w r i t t e n as
12
t--[
41
ah1
hlh2 at2
and
3
MODEL APPLICATIONS
To establish confidence in t h e curvilinear coordinate model physics and computer coding, a hierarchy of numerical experiments has been conducted.
The
numerical experiments have been designed to convincingly test the model at the same time minimizing computational costs.
T h e initial numerical experiments
involve the external mode portion a s a stand alone model. Other simulations involve the full three-dimensional model used t o investigate specific oceanic processes.
T h e test cases which follow are not meant t o be exhaustive but
rather to illustrate the versatility and viability of the curvilinear coordinate system approach. T h e examples are cases in which analytic and other numerical solutions are available for comparison and assessment of model performance.
73 3.1 Wedge Shaped Domain Lynch and G r a y ( 1 9 7 8 1 h a v e d e r i v e d a n a l y t i c s o l u t i o n s f o r a v a r i e t y o f l i n e a r i z e d , s h a l l o w w a t e r e q u a t i o n s problems.
C o n s i d e r t h e p h y s i c a l s e t t i n g and
g r i d c o n f i g u r a t i o n shown i n F i g u r e 2 w h i c h i n v o l v e s t h e p r o p a g a t i o n o f waves i n t o a wedge shaped domain.
A t t h e w a l l s , r = r,,,
8= 0
and 8 = r r / 2 , no flow i s
Fig. 2 . The g r i d c o n f i g u r a t i o n used i n t h e wedge shaped domain e x p e r i m e n t s .
allowed and a t r = r l , t h e open boundary,
the surface elevation i s prescribed
t o vary s i n u s o i d a l l y i n t i m e w i t h uniform a m p l i t u d e and phase.
Tne C o r i o l i s a s
well as n o n - l i n e a r and h o r i z o n t a l d i f f u s i o n terms a r e n e g l e c t e d i n t h i s a p p l i c a tion.
A l i n e a r bottom f r i c t i o n f o r m u l a t i o n i s u s e d i n t h e s e c a s e s i n s t e a d o f
E q u a t i o n s (65) and (66) w i t h a f r i c t i o n a l c o e f f i c i e n t o f
s-l.
The m e t r i c s
f o r t h i s geometry, h l = 1 and h2 = r , h a v e b e e n n u m e r i c a l l y computed and solutions obtained.
The computed e l e v a t i o n s and v e l o c i t i e s t o g e t h e r w i t h t h e i r
a n a l y t i c a l c o u n t e r p a r t s a r e shown i n F i g u r e s 3 f o r b o t h a f l a t bathymetry and one t h a t s l o p e s i n a q u a d r a t i c f a s h i o n .
The a g r e e m e n t i s e x c e l l e n t . I n a d -
d i t i o n , t h e c o m p u t e d v e l o c i t i e s and e l e v a t i o n s
a r e p e r f e c t l y r a d i a l a t every
g r i d p o i n t a s a r e t h e a n a l y t i c s o l u t i o n s . Other c a s e s i n v o l v i n g t h e s e g e o m e t r i e s with wind
produce s i m i l a r , r a t h e r good, comparisons.
3.2 I s l a n d C i r c u l a t i o n Now, c o n s i d e r t h e c i r c u l a t i o n around a c i r c u l a r i s l a n d w i t h a narrow s h e l f i n v e s t i g a t e d n u m e r i c a l l y by Wang [ 1 9 8 2 ] . F i g u r e 4.
The g r i d c o n f i g u r a t i o n i s shown i n
Two c a s e s are c o n s i d e r e d ; t h e s t e a d y r e s p o n s e t o a uniform wind and
14
-
0
0
\
F
3
\
Y
F
>.
Y
t
W
V
s
0 3
t
W
>
J
a I
J
4 0 a
a
E
1.00
R A D I A L DISTANCE ( r / r o )
1.75
2.50
R A D I A L DISTANCE ( r / r o )
Fig. 3. A comparison of the model's computed (0) surface elevation (left) and for flat and quadratic circulation (right) with the analytical solution (-) sloping bottoms in a wedged shaped domain.
Fig. 4 . The grid configuration used in the circular island domain experiments.
75 the g e n e r a t i o n and propagat i o n of quasi-geostrophic,
s h e l f waves.
The i s l a n d
r a d i u s i s 5 0 h , t h e s h e l f w i d t h i s 16km a n d t h e b o t t o m d e p t h i s l i n e a r ( 4 0 m d e p t h a t t h e i s l a n d t o 2 0 0 m a t t h e s h e l f ) f o r t h e s t e a d y s t a t e c a s e and f l a t
(loom) f o r t h e wave o n e .
The C o r i o l i s p a r a m e t e r i s
s-',
t h e non-linear
terms a r e i n c l u d e d and t h e f r i c t i o n p r o v i d e d b y E q u a t i o n s ( 6 5 ) and ( 6 6 ) a r e u s e d . The b o u n d a r y c o n d i t i o n s on t h e s h e l f b r e a k assume t h e s h e l f motion t o b e completely t r a p p e d i n t h e s h e l f r e g i o n , t h a t i s ,
7]= 0.
In a d d i t i o n , t h e
non-linear terms i n t h e momentum e q u a t i o n a r e a l s o n e g l e c t e d a t t h e s h e l f break. Figure 5 i l l u s t r a t e s t h e steady
stress.
The s u r f a c e
response t o an eastward
e l e v a t i o n with i t s antisymmetric
2 dyne cm-'
patterns,
wind
maximum
Fig. 5 . The s t e a d y - s t a t e r e s p o n s e of a n i s l a n d w i t h a s l o p i n g c o n t i n e n t a l s h e l f t o a uniform e a s t w a r d wind: t h e s u r f a c e e l e v a t i o n ( l e f t ) and t h e v e r t i c a l l y averaged c u r r e n t ( r i g h t ) . The i s l a n d r a d i u s i s 50 km and t h e s h e l f width i s 16 km. a m p l i t u d e s a t t h e n o r t h e r n a n d s o u t h e r n t i p s o f t h e i s l a n d and t h e v e l o c i t y f i e l d a g r e e w e l l w i t h Wang's numerical r e s u l t s . The i s l a n d c o n f i g u r a t i o n w i t h a f l a t bottom p e r m i t s q u a s i - g e o s t r o p h i c waves which p r o p a g a t e c l o c k w i s e around t h e i s l a n d w i t h a phase speed of 9Okm day-'
.
Figure 6 i l l u s t r a t e s t h e s u r f a c e e l e v a t i o n a n d v e l o c i t y f i e l d s d r i v e n b y a westward wind stress o f 1 dyne
a t 20 h r i n t e r v a l s . The computed phase and
d i r e c t i o n are c o r r e c t and i n agreement w i t h Wang's r e s u l t s , i f n o t e i s t a k e n t h a t t h e r e i s a d i s c r e p a n c y b e t w e e n what p a r a m e t e r s Wang used and what were reported i n h i s a r t i c l e .
9L
76
"I F i g . 6. The s u r f a c e e l e v a t i o n ( l e f t ) and v e r t i c a l l y averaged c i r c u l a t i o n ( r i g h t ) f o r a n i s l a n d a r e a responding t o a westward wind. The t o p d i s t r i b u t i o n s a r e a t 1 0 h o u r s a f t e r t h e o n s e t o f t h e wind a n d t h e bottom o n e s are 20 h o u r s l a t e r . The i s l a n d r a d i u s is 50 km and t h e s h e l f width i s 16 km.
3.3
Enclosed B a s i n The n e x t e x a m p l e i s concerned w i t h t e s t i n g t h e c u r v i l i n e a r c o o r d i n a t e model
i n a s i t u a t i o n where t h e h o r i z o n t a l g r i d s p a c i n g s change r a d i c a l l y . C o n s i d e r t h e "Bow T i e " shaped c l o s e d b a s i n i l l u s t r a t e d i n F i g u r e 7.
The o r t h o g o n a l i t y o f t h e
g r i d was i n s u r e d by s e l e c t i n g t h e m e t r i c s b a s e d upon a n e l l i p t i c c y l i n d r i c a l c o o r d i n a t e system.
F o r t h i s s e t t i n g t h e North-South
width a t t h e most narrow p o r t i o n is-17km. a l a t i t u d e o f 45"
(f =
from 1.2 km t o 17 km.
s-').
a x i s i s -1OOkm
and t h e
The b a s i n i s 10m deep and s i t u a t e d a t
The s p a t i a l r e s o l u t i o n v a r i e s s u b s t a n t i a l l y ,
The s t e a d y - s t a t e r e s p o n s e t o a uniform wind i s e x a m i n e d .
Here t h e s i m p l e w i n d s e t u p o n a c o m p l i c a t e d g r i d i s examined.
The s o l u t i o n s
should n o t depend o n t h e g r i d c o n f i g u r a t i o n b u t r a t h e r o n t h e p h y s i c a l f o r c e s d r i v i n g t h e system.
To a c h i e v e a f a s t s p i n - u p o f t h e b a s i n , t h e n o n - l i n e a r
bottom f r i c t i o n c o e f f i c i e n t i s i n c r e a s e d by a f a c t o r o f 1 0 0 .
The m o d e l s t a r t s
f r o m r e s t and i s d r i v e n i n t h e f i r s t experiment by a n e a s t w a r d 1 d y n e cm-2
wind s t r e s s of
a n d i n t h e s e c o n d e x p e r i m e n t by a w i n d o f s i m i l a r magnitude b u t
directed t o the north. A t s t e a d y s t a t e , which o c c u r s a f t e r - 6 0
h o u r s , t h e v e l o c i t i e s should be z e r o
and t h e e l e v a t i o n d i s t r i b u t i o n p e r f e c t l y s y m m e t r i c .
The r e s u l t s o f t h e two
F i g . 7 . The g e o m e t r y and g r i d c o n f i g u r a t i o n i n t h e "Bow T i e " experiments. The b a s i n h a s a n o r t h s o u t h e x t e n t of -100 km a n d a t t h e n a r r o w e s t p o r t i o n , t h e width i s -17 km. experiments are shown i n F i g u r e 8.
The amount o f s e t u p and t h e d i r e c t i o n a g r e e
almost p e r f e c t l y w i t h what one o b t a i n s a n a l y t i c a l l y . Also, t h e v e l o c i t i e s h a v e vanished as expected.
3.4 C h a r l e s t o n
bum^
As a l a s t e x a m p l e o f t h e t w o - d i m e n s i o n a l c a s e s , t h e v e r t i c a l l y i n t e g r a t e d model h a s been run i n a reduced g r a v i t y mode t o s i m u l a t e t h e i n t e r n a l d y n a m i c s of t h e flow as i t p a s s e s around a t o p o g r a p h i c f e a t u r e . r e d u c e d g r a v i t y model c o n s i s t s o f two l a y e r s o f
The t w o - l a y e r ,
s t r a t i f i e d f l u i d with a fixed density contrast. deep so t h a t t h e evolution
stably
The bottom l a y e r i s i n f i n i t e l y
b a r o t r o p i c mode i s f i l t e r e d o u t . The model d e s c r i b e s t h e
o f t h e f i r s t i n t e r n a l ( b a r o c l i n i c ) mode o f t h e water column ( s e e
[ G i l l 19821 f o r a d e r i v a t i o n ) .
The d o m a i n h a s a l a r g e bump i n t h e w e s t e r n
boundary c o a s t l i n e s i m i l a r t o t h e " C h a r l e s t o n bump", t h e r i d g e and bottom t r o u g h f e a t u r e o f f C h a r l e s t o n , South C a r o l i n a a t about 32".
The C h a r l e s t o n bump was
c h o s e n h e r e b e c a u s e i t i s a r e g i o n w h i c h e x h i b i t s r e c u r r i n g meander and eddy activity
.
The model d o m a i n a n d g r i d are
shown i n F i g u r e 9 .
The l a t i t u d i n a l width o f
t h e domain i s 800km and t h e l o n g i t u d i n a l width i s 300km w i t h t h e r e g i o n b e i n g c e n t e r e d a b o u t 32.N.
The domain h a s been r o t a t e d by 45',
counterclockwise.
The
g r i d r e s o l u t i o n o f t h e domain i s f i n e s t n e a r t h e w e s t e r n c o a s t w h e r e t h e o f f -
F i g . 8 . The s t e a d y s t a t e e l e v a t i o n i n response t o a) an eastward 1 dyne c m - 2 wind s t r e s s and b ) a northward 1 dyne cm-2 wind s t r e s s . The u n i t s o f t h e e l e v a t i o n are i n cm.
DISTANCE ( k m )
F i g . 9 . The c o m p u t a t i o n a l simulations.
g r i d used f o r the
i d e a l i z e d C h a r l e s t o n bump
79 shore spacing is-6km
and i n c r e a s e s to-10km
spacing v a r i e s from-3km
away from t h e bump.
n e a r t h e bump to-15km
The a l o n g s h o r e
f u r t h e r away. With t h e s e g r i d
i n t e r v a l s , t h e p h y s i c a l p r o c e s s e s c h a r a c t e r i z e d by t h e b a r o c l i n i c r a d i u s should be well r e s o l v e d and mesoscale eddy a c t i v i t y s h o u l d b e c a p t u r e d . The c o e f f i c i e n t o f t h e h o r i z o n t a l d i f f u s i o n i s 100mZs-’ i n t h i s s i m u l a t i o n . The i n i t i a l d e p t h o f t h e u p p e r l a y e r i s 500m and t h e r e d u c e d g r a v i t y i s so t h a t t h e r a d i u s o f d e f o r m a t i o n i s about
O.OZms-‘,
chosen h e r e
since
40km. A depth o f
i t b e s t c h a r a c t e r i z e s t h e Charleston
is
500m
bump r e g i o n . The model
bump shown i n F i g u r e 9 h a s roughly t h e same amplitude as t h e p e r t u r b a t i o n found i n t h e a c t u a l 500m i s o b a t h b u t i s somewhat l a r g e r i n l a t i t u d i n a l e x t e n t .
This
d i f f e r e n c e i s o f minor c o n s e q u e n c e t o t h e r e s u l t s o f t h e s e i d e a l i z e d experiments; i f a n y t h i n g t h e r e s u l t s w i l l b e smoother t h a n t h e o b s e r v a t i o n s s i n c e t h e model bump i s l e s s a b r u p t . Coriolis parameter
The a s s o c i a t e d g r a v i t y wave speed is-3ms-1
i s f o + B y , where f o = 7.8 x
6-l
,p
, the
= 1.936 x
and y i s t h e l a t i t u d e i n k i l o m e t e r s .
10-llm-ls-l,
One of t h e major d i f f i c u l t i e s i n m o d e l l i n g g e o g r a p h i c a l l y l i m i t e d r e g i o n s , as mentioned p r e v i o u s l y , is t h e problem a s s o c i a t e d w i t h t h e o p e n l a t e r a l b o u n d a r y conditions.
E x t e r n a l l y p r e s c r i b e d o r g r a d i e n t boundary c o n d i t i o n s are u s u a l l y
employed a l t h o u g h i t c a n be shown t h a t t h e s e c o n d i t i o n s c a n p r o d u c e m i s l e a d i n g model r e s u l t s . [19851
A r e c e n t l y developed boundary c o n d i t i o n by Blumberg and Kantha
seems t o circumvent t h e problems i n c e r t a i n p h y s i c a l s e t t i n g s . I t i s a
m o d i f i e d form o f t h e Somnerfeld r a d i a t i o n c o n d i t i o n which p e r m i t s t r a n s i e n t s t o pass o u t through t h e boundary and a t t h e same time a l l o w s t h e i n f l o w and o u t f l o w a t t h e b o u n d a r i e s t o v a r y about some mean.
av av K+cK
Mathematically, t h e c o n d i t i o n is:
4xv-vk3
(69)
where C i s t h e phase speed o f t h e g r a v i t y wave and n i s t h e d i r e c t i o n n o r m a l t o t h e p l a n a r boundary.
The term o n t h e r i g h t r e p r e s e n t s damping t h a t f o r c e s t h e
v a l u e o f V a t t h e b o u n d a r y t o some known vk w i t h a t i m e s c a l e of t h e o r d e r t o
Tf.
T f = 0 c o r r e s p o n d s t o a n e x t e r n a l l y f i x e d boundary where no d i s t u r b a n c e s
a r e a l l o w e d t o p a s s o u t through t h e boundary and Tf’*represents
a pure r a d i a -
t i o n c o n d i t i o n which r e n d e r s a boundary t r a n s p a r e n t t o waves t r a v e l l i n g i n t h e p o s i t i v e n d i r e c t i o n w i t h phase speed C .
A p a r a b o l i c v e l o c i t y p r o f i l e i s chosen t o r e p r e s e n t t h e Gulf Stream,
80 and Vo i s 60cms-1.
w h e r e L i s t h e h a l f w i d t h o f t h e Gulf Stream,-45km, t r a n s p o r t a s s o c i a t e d w i t h t h i s p r o f i l e i s 20 x 106m's-', t h e t o p 50010 o f t h e water column.
The v e l o c i t y p r o f i l e g i v e n by (70) i s used
a l o n g both t h e n o r t h e r n and s o u t h e r n b o u n d a r i e s . for Tf;
The
a r e a l i s t i c amount f o r
A v a l u e o f 25 days i s s e l e c t e d
i t i s a p p r o x i m a t e l y t h e average a d v e c t i v e t r a n s i t t i m e f o r a p a r c e l t o
t r a v e r s e t h e domain. e a s t e r n boundary.
The p u r e r a d i a t i o n c o n d i t i o n (TP-0)
i s used a l o n g t h e
A t t h e c o a s t , a no normal f l o w c o n d i t i o n i s u s e d .
k'or t h e
i n i t i a l s t a t e , m o t i o n i s s p e c i f i e d i n t h e i n t e r i o r o f t h e domain a c c o r d i n g t o (70).
The i n t e r f a c e i s t a k e n t o h a v e no i n i t i a l d i s p l a c e m e n t .
During t h e
s p i n - u p p r o c e s s e s , t h e system a t t a i n s a q u a s i - g e o s t r o p h i c b a l a n c e w i t h i n a few i n e r t i a l p e r i o d s (-2
days).
A sequence of s n a p s h o t s from t h e bump experiment o f t h e upper l a y e r t r a n s p o r t and i n t e r f a c e d i s p l a c e m e n t i s shown i n F i g u r e 1 0 .
Positive values f o r the
i n t e r f a c e displacement i n d i c a t e r e g i o n s where t h e l a y e r t h i c k n e s s h a s i n c r e a s e d . The s n a p s h o t s t a k e n a t 20 day i n t e r v a l s c l e a r l y i l l u s t r a t e m e s o s c a l e e d d i e s o f -120km
d i a m e t e r a p p e a r i n g s p o n t a n e o u s l y in t h e v i c i n i t y o f t h e b m p .
In s p i t e
of t h e t i m e i n d e p e n d e n t s p e c i f i c a t i o n f o r t h e b o u n d a r y c o n d i t i o n s , a t i m e d e p e n d e n t f i e l d o f e d d i e s d e v e l o p s . By present.
d a y 10
two a n t i c y c l o n i c e d d i e s a r e
A s t i m e p r o g r e s s e s t h e s e e d d i e s move t o t h e n o r t h w e s t , p r o b a b l y t h e
r e s u l t o f a d v e c t i o n by t h e mean flow, and e x i t t h e domain.
F i g u r e 10b shows t h e
f o r m a t i o n of a c y c l o n i c eddy j u s t d o w n s t r e a m o f t h e bump a n d a n a n t i c y c l o n i c e d d y u p s t r e a m o f t h e bump.
Both e d d i e s a l s o grow t o a d i a m e t e r ofN120km and
move northwest.
In c o n t r a s t , a p a r a l l e l n u m e r i c a l experiment u s i n g a l l t h e same parameters b u t i n a domain w i t h o u t a bump p r o d u c e d no e d d i e s . r e a c h e d s t e a d y s t a t e and w a s i n g e o s t r o p h i c e q u i l i b r i u m . flow h a d b e e n e s t a b l i s h e d .
The f l o w e s s e n t i a l l y A simple inflow/out-
It should be noted t h a t by i n c r e a s i n g t h e flow
R e y n o l d s number (ReN2LV/AM) b y a f a c t o r o f 2 ,
to-450,
d i d however produce
eddies i n t h i s case.
3.5 Three-Dimensional I d e a l i z e d C a l i f o r n i a S h e l f Mddel The f i n i t e d i f f e r e n c e c o m p u t a t i o n a l code o f t h e c u r v i l i n e a r model i s q u i t e intricate.
A d d i t i o n a l c o m p l e x i t i e s a r e i n t r o d u c e d by t h e mode s p l i t t i n g
technique.
It i s t h e r e f o r e n e c e s s a r y t o e s t a b l i s h t h e s e l f - c o n s i s t e n c y o f t h e
computer program and t o d e m o n s t r a t e t h a t t h e g o v e r n i n g e q u a t i o n s a r e p r o p e r l y coded. The f i r s t o f t h e s e t e s t s i s t h e n u l l c a s e where t h e model (30 x 40 x 16) i s n o t e x t e r n a l l y f o r c e d and i s c l o s e d on i t s l a t e r a l b o u n d a r i e s .
Any i n i t i a l l y
p r e s c r i b e d h o r i z o n t a l l y homogeneous ( i n z space) d e n s i t y d i s t r i b u t i o n s h o u l d n o t r e s u l t i n any motion f o r a r b i t r a r y topography.
This i s a r a t h e r good t e s t
t o i d e n t i f y c o d i n g e r r o r s i n a "z" c o o r d i n a t e model b u t i s a n even more impor-
81
Fig. 10. A t i m e sequence of upper l a y e r t r a n s p o r t s and p y c n o c l i n e d i s p l a c e m e n t s a t 10 day i n t e r v a l s b e g i n n i n g a t day 10. t a n t t e s t when u s i n g t h e u f o r m u l a t i o n .
A s e v e r a l hundred t i m e s t e p s i m u l a t i o n
w i t h r e a l i s t i c C a l i f o r n i a C o a s t t o p o g r a p h y p r o d u c e d no v e l o c i t y
fields.
Symmetry o f t h e code was e s t a b l i s h e d by f o r c i n g a s q u a r e b a s i n with a n o r t h e r l y wind and t h e n w i t h t h e C o r i o l i s p a r a m e t e r
rotated,
a n e a s t e r l y wind.
The
n u n e r i c a l s i m u l a t i o n s showed t h e same s o l u t i o n s a f t e r t h e second set was r o t a t e d through 9 0 ' .
A h o s t o f o t h e r symmetry t e s t s were conducted.
The u s e o f g l o b a l d i a g n o s t i c s a l s o provided a means o f checking t h e computations.
The programmed f i n i t e d i f f e r e n c e e q u a t i o n s l i k e t h e i r d i f f e r e n t i a l
82
c o u n t e r p a r t s c o n s e r v e mass, volume, h e a t , s a l t and t o t a l energy. The global (over t h e computational domain) budgets of t h e s e q u a n t i t i e s were m o n i t o r e d i n each of t h e above mentioned experiments and i n every case t h e r e was conservation t o within computer round-off The t h r e e - d i m e n s i o n a l
error. c i r c u l a t i o n model was t h e n used t o i n v e s t i g a t e t h e
response of a region c h a r a c t e r i s t i c of t h e C a l i f o r n i a Coast t o wind f o r c i n g . The c o m p u t a t i o n a l g r i d employed h e r e i s i l l u s t r a t e d i n Figure 11. An enlargement of the near c o a s t a l r eg i o n al o n g c e n t r a l C a l i f o r n i a i s a l s o provided i n t h a t figure.
The g r i d s p a c i n g v a r i e s from-4km
n e a r t h e c o a s t to-25km
near t h e
42 4i 40
39
38 37 36
35 34
33 32 31 30 128
126
124
122
120
lie
Fig. 11. The C a l i f o r n i a coast g r i d ( l e f t ) used i n t h e t h r e e - d i m e n s i o n a l model experiment. The f o r c i n g zone i s i n d i c a t e d . A magnified view of t h e subregion B i s on t h e r i g h t . w e s t e r n p o r t i o n o f t h e domain.
The alongshore r e s o l u t i o n v a r i e s f r o m - 1 6 h
50km with h i g h e s t r e s o l u t i o n o c c u r r i n g n e a r t h e c o a s t a l p r o m o n t o r i e s .
to The
r e g i o n h a s a c o n t i n e n t a l s h e l f - s l o p e w i t h c h a r a c t e r i s t i c s t y p i c a l of c e n t r a l C a l i f o r n i a and i s
5.61 x
r t 30m
x 10m2(r
-
r l ) t 20Om
r
5rl
r1 5 r S r2
r
5r2
,
,
( 7 1)
83 where r i s t h e d i s t a n c e from t h e c o a s t and r1 = 3 0 h , r2 = lOOkm ( s e e F i g u r e 14 f o r a p l o t of t h e t o p o g r a p h y ) . The d o m a i n i s r e p r e s e n t e d b y a 30 x 40 x 2 1 l a t t i c e o f p o i n t s w i t h t h e v e r t i c a l d i s t r i b u t i o n of p o i n t s h a v i n g a n i r r e g u l a r spacing a s shown i n T a b l e I . bottom i s a p p a r e n t .
The i n c r e a s e d r e s o l u t i o n n e a r t h e s u r f a c e and
The h o r i z o n t a l m i x i n g c o e f f i c i e n t s a r e c h o s e n t o b e
Table I: V e r t i c a l R e s o l u t i o n o f t h e C a l i f o r n i a S h e l f C i r c u l a t i o n M d e l . i n meters.
k -
U
1 2
0.000 0.0006 0.0012 0.0030
3
4 5 6 7 8 9 10
0.0060 0.0120 0.0240 0.0360
0.0600 0.1000
.o
.o .6 1.2
.12
.6
.30 .60 1.20 2.40 3.60 6.00
1.5 3.0
17 18 19 20
26.00 34.00 42.00 50.00 60.00 70.00 80.00 88.00 94.00
21
1.0000
100.00
16
,a
l m ~
.3
0.2600 0.3400 0.4200 0.5000 0.6000 0.7000 0.8000 0.8800 0.9400
13
z
.oo
0.1800
14 15
‘500 m
.06
10.0 18.0
11 12
loom2 s-’
‘100 m
6.0 12.0 18.0 30.0 50.0
90.0 130.0 170.0
210.0 250.0 300.0 350.0
400.0 440.0 470.0 500.0
Depths
~
~
3.0
6.0 12.0 24.0 36.0 60.0
100.0 180.0 260.0 340.0 420.0 500.0
600.0 700.0 800.0
880.0 940.0 1000.0
v a l u e s m a l l enough n o t t o smooth o u t i m p o r t a n t f e a t u r e s .
At the
edges of t h e domain, t h e n o r m a l c o m p o n e n t s o f t h e v e l o c i t y a r e s e t t o z e r o . While t h i s p r e s c r i p t i o n of t h e v e l o c i t y f i e l d i s not g e n e r a l , i t s u f f i c e s h e r e as long as only a s h o r t d u r a t i o n experiment is c o n s i d e r e d .
The i n i t i a l s t r a t i f -
i c a t i o n employed i s h o r i z o n t a l l y homogenous w i t h a v e r t i c a l s t r u c t u r e as shown i n Figure 12. Alongshore winds i n a 300km band ( s e e F i g u r e 1 1 f o r t h e l i m i t s ) a r e imposed w i t h a m a g n i t u d e o f 1 dyne cm-2 equal t o 34 ppt properties.
.
The s a l i n i t y i s set everywhere
and i s u s e d as a c h e c k o f t h e c o m p u t e r c o d e ’ s c o n s e r v a t i o n
The ocean i s i n i t i a l l y a t r e s t .
The response o f t h i s ocean b a s i n o v e r a n 6 day p e r i o d t o t h e sudden o n s e t o f alongshore winds i n t h e 300km band i s now examined. The e v o l u t i o n o f t h e s u r f a c e c i r c u l a t i o n i s i l l u s t r a t e d i n F i g u r e 13. Here t h e a l o n g s h o r e v e l o c i t y o n a l i n e about 6 km f r o m t h e c o a s t i s u s e d .
It i s e v i d e n t t h a t a s t r o n g a l o n g s h o r e
c o a s t a l j e t d e v e l o p s q u i c k l y w i t h i n t h e f o r c i n g r e g i o n and s p r e a d s p o l e w a r d a s time p r o g r e s s e s .
By t h e e n d o f t h e s i m u l a t i o n t h e c u r r e n t h a s reached t h e
84 TEMPERATURE ("C 1 0
0
500
2
4
6
I
8
12
10
14
16
8
10 12
I
4
-
6
14
16
E
I I-
n
I000 200
W
0
300 I500
F i g . 1 2 . The i n i t i a l t e m p e r a t u r e d i s t r i b u t i o n u s e d i n t h e p r o g n o s t i c model e x p e r i m e n t s . The d i s t r i b u t i o n i s t y p i c a l o f t h a t o b s e r v e d o f f t h e c o a s t o f C a l i f o r n i a . The i n s e r t is a d e t a i l o f t h e upper 300 m. northernmost boundary w h i l e n o t p e n e t r a t i n g southward of t h e f o r c i n g zone.
Thus
i n a g r e e m e n t w i t h t h e r e c e n t l y d e v e l o p e d i d e a s of c o a s t a l t r a p p e d waves [ s e e P h i l a n d e r and Yoon, 1982; Suginohara, 1982 and Yoon and P h i l a n d e r , 1 9 8 2 1 , t h i s m o d e l p r o d u c e s d i s t u r b a n c e s t h a t t a k e p l a c e i n i t i a l l y a t t h e s o u t h e r n edge of t h e f o r c i n g zone and e x t e n d poleward.
As d e m o n s t r a t e d by o t h e r s and r e p r o d u c e d
h e r e , t h e equatorward v e l o c i t y i n i t i a l l y i n c r e a s e s w i t h t i m e and s t o p s i n c r e a s i n g from t h e s o u t h e r n edge. North o f t h e f o r c i n g zone t h e t i m e e v o l u t i o n shows t h e f r e e p r o p a g a t i o n o f c o a s t a l t r a p p e d ( t h e f i r s t mode) waves t r a v e l l i n g w i t h a speed of
-225km
day-'
. The
t i m e e v o l u t i o n o f t h e c i r c u l a t i o n a t 200111 depth
a l s o i s shown i n F i g u r e s 1 3 . The d e v e l o p m e n t o f a p o l e w a r d u n d e r c u r r e n t i s e v i d e n t f r o m t h e a l o n g s h o r e v e l o c i t y a t t h i s d e p t h . The poleward u n d e r c u r r e n t d e v e l o p s a t a slower r a t e with
t h a n does t h e c o a s t a l j e t t h r o u g h waves t r a v e l l i n g
phase s p e e d s o f - 7 5 b day-'
.
The phase s p e e d s o f t h e waves a p p e a r i n g i n
F i g u r e 13 a g r e e w i t h t o t h o s e computed u s i n g t h e t e c h n i q u e s d e s c r i b e d b y B r i n k 119821 f o r c a l c u l a t i n g f r e e c o a s t a l - t r a p p e d wave modal s t r u c t u r e s and d i s p e r s i o n c u r v e s . Within t h e f o r c i n g z o n e t h e u p w e l l i n g i s q u i t e i n t e n s e w i t h s u r f a c e t e m p e r a t u r e s b e c o m i n g -1
1/2'C
c o o l e r by t h e end o f day 6 , as shown i n F i g u r e
14. It s h o u l d b e n o t e d t h a t F i g u r e 1 4 a l s o shows t h e f o r m a t i o n o f a w e l l d e f i n e d b o t t o m b o u n d a r y l a y e r on t h e s h e l f . I n a d d i t i o n , downwelling i s p r e s e n t a l o n g
85 the s h e l f b r e a k a t d e p t h s below 300m.
325 SOUTM
-
t
i
6 50
975
1300 NORTH
325
0
050
911
SOUTH
DISTANCE ALONGSHORE (kml
I300 NORTM
DISTANCE ALONGSHORE (km)
F i g . 1 3 . The t i m e e v o l u t i o n o f t h e s u r f a c e a l o n g s h o r e v e l o c i t y (cm 8 - l ) a l o n g a l i n e - 6 km from t h e c o a s t ( l e f t ) and t h e 200 m d e p t h a l o n g s h o r e v e l o c i t y (cm s - l ) along a line-20 km from t h e c o a s t ( r i g h t ) .
m
e
o
s
o
4
o
3
o
z
o
10
0
Distance Offshore (Km) Fig. 14. The t e m p e r a t u r e d i s t r i b u t i o n a l o n g a v e r t i c a l s e c t i o n n e a r t h e c e n t e r I n i t i a l l y t h e temperature d i s t r i b u t i o n is o f t h e wind f o r c i n g zone ( C I = 1.C). only a f u n c t i o n o f depth.
4 CONCLUSIONS A n u m e r i c a l c i r c u l a t i o n model h a s been developed u s i n g a n o r t h o g o n a l c u r v i -
l i n e a r c o o r d i n a t e system.
A s e r i e s of n u m e r i c a l e x p e r i m e n t s have been p r e s e n t e d
f o r c a s e s w h e r e a n a l y t i c and n u m e r i c a l s o l u t i o n s a r e a v a i l a b l e f o r comparison. These t e s t c a s e s i n d i c a t e t h a t t h e c u r v i l i n e a r m o d e l i s a v i a b l e o p t i o n when m o d e l l i n g a n a r e a where d e t a i l s of t h e c o a s t l i n e s a r e i m p o r t a n t as i n c o n t i n e n t a l s h e l f and e s t u a r i n e r e g i o n s .
No numerical problems have been e n c o u n t e r e d a s
f a r a s t h e g r i d a s p e c t r a t i o ( v a l u e s of 1:15 have been u s e d ) and d e s i g n o f t h e computational g r i d are concerned. I d e a l i z e d experiments with a v e r t i c a l l y C
i n t e g r a t e d model, t h a t i s , t h e e x t e r n a l mode component o f t h e t h r e e - d i m e n s i o n a l model,
r u n i n a reduced g r a v i t y mode have i l l u s t r a t e d t h e d r a m a t i c i m p a c t o f
t h e C h a r l e s t o n bump o n t h e i n t e r n a l d y n a m i c s o f t h e G u l f Stream. The r e s u l t s d e m o n s t r a t e t h a t when a Gulf Stream f l o w i s f o r c e d around a t o p o g r a p h i c f e a t u r e , i n t e n s e e d d i e s d e v e l o p s p o n t a n e o u s l y ; however, when t h e same flow t r a v e r s e s a r e g i o n w i t h o u t such a f e a t u r e no e d d i e s a r e produced. To f u r t h e r e s t a b l i s h c o n f i d e n c e i n t h e t h r e e - d i m e n s i o n a l model, t h e r e s p o n s e of a s t r a t i f i e d o c e a n w i t h a c o a s t l i n e c h a r a c t e r i s t i c o f t h e C a l i f o r n i a C o n t i n e n t a l S h e l f r e g i o n t o a n a l o n g s h o r e u p w e l l i n g f a v o r a b l e wind i s examined.
The
r e s u l t i n g c i r c u l a t i o n w i t h u p w e l l i n g and t h e p r o p a g a t i o n o f v a r i o u s modes o f c o a s t a l t r a p p e d waves i s i n s u b s t a n t i a l agreement w i t h p r e v i o u s s t u d i e s . I n a c a s e n o t p r e s e n t e d h e r e , a year-long m o d e l s i m u l a t i o n f o r t h e y e a r 1981
has
b e e n c o n d u c t e d w i t h t h i s s h e l f model. The model w a s d r i v e n w i t h s y n o p t i c wind f i e l d s d e r i v e d from t h e N a t i o n a l Weather S e r v i c e p r e d i c t i o n s and c l i m a t o l o g i c a l h y d r o g r a p h y a n d c i r c u l a t i o n f o r model i n i t i a l i z a t i o n and boundary c o n d i t i o n s . The i n t e r e s t e d r e a d e r i s r e f e r r e d t o Blumberg e t a l . [1985].
In c o n s i d e r i n g whether t o u s e a c u r v i l i n e a r model o r a C a r t e s i a n model f o r a p a r t i c u l a r problem, i t s h o u l d b e n o t e d t h a t t h e c u r v i l i n e a r c o o r d i n a t e model h a s a d d i t i o n a l terms i n t h e g o v e r n i n g e q u a t i o n s which a r e c t a n g u l a r c o o r d i n a t e model does n o t have.
The r e s u l t i s about a 60% i n c r e a s e i n computer time f o r t h e same
number o f g r i d p o i n t s , w h i c h t e n d s t o some e x t e n t t o o f f s e t t h e s a v i n g s from improved g r i d r e s o l u t i o n . T h e r e f o r e , a C a r t e s i a n c o o r d i n a t e m o d e l may c o n t i n u e t o b e more a t t r a c t i v e f o r c a l c u l a t i o n s w h e r e t h e c o m p u t a t i o n a l domain h a s a r e g u l a r shape and t h e d e s i r e d r e s o l u t i o n i s r e l a t i v e l y u n i f o r m t h r o u g h o u t . The p r i n c i p a l a p p l i c a t i o n o f t h e o r t h o g o n a l c o o r d i n a t e s y s t e m model i s t o o b t a i n high
r e s o l u t i o n where i t i s r e q u i r e d w i t h o u t p a y i n g t h e p e n a l t y o f u n n e c e s -
s a r i l y h i g h r e s o l u t i o n i n o t h e r p a r t s o f t h e modeled r e g i o n . There a p p e p r t o be
no a d d i t i o n a l l i m i t a t i o n s imposed by t h e u s e o f c u r v i l i n e a r c o o r d i n a t e s . F i n a l l y , i t i s hoped t h a t t h e
c a s e s s e l e c t e d f o r examples of t h e a p p l i c a t i o n
of t h e model i n S e c t i o n 3 w i l l p r o v i d e a framework f o r t e s t i n g and comparing t h e r e s u l t s of o t h e r models as w e l l .
5
ACKNOWLEDGEMENT T h i s work was c o m p l e t e d w h i l e AFB was w i t h D y n a l y s i s o f P r i n c e t o n . The
a s s i s t a n c e o f D r Lakshmi H . K a n t h a i n c o m p u t i n g t h e a n a l y t i c a l wave s p e e d s , d i s c u s s e d i n t h e C a l i f o r n i a s h e l f model s i m u l a t i o n , i s g r a t e f u l l y acknowledged. Funding f o r t h e s t u d y w a s p r o v i d e d by t h e M i n e r a l s Management S e r v i c e o f t h e U.
S . Department of
t h e I n t e r i o r u n d e r c o n t r a c t Numbers
AA851-CT1-67
and
14-12-0001-29113.
6
REFERENCES
Arakawa, A . a n d Lamb, V.R., 1977. Computational d e s i g n o f t h e b a s i c dynamical p r o c e s s of t h e UCLA G e n e r a l C i r c u l a t i o n Model. Methods i n C o m p u t a t i o n a l P h y s i c s , 17. Academic P r e s s , pp. 173-265. Blumberg, A.F. a n d M e l l o r , G . L . , 1 9 8 3 . D i a g n o s t i c a n d p r o g n o s t i c n m e r i c a l c i r c u l a t i o n s t u d i e s o f t h e S o u t h A t l a n t i c B i g h t . J . Geophys. R e s . , 8 8 : 4579-4592. Blumberg, A.F. a n d M e l l o r , G . L . , 1 9 8 6 . A d e s c r i p t i o n of a three-dimensional c o a s t a l ocean c i r c u l a t i o n model. I n : N . Heaps ( E d i t o r ) , T h r e e - D i m e n s i o n a l S h e l f M o d e l s , C o a s t a l and E s t u a r i n e S c i e n c e s , 5 , American Geophysical Union (AGU). Blumberg, A.F. , K a n t h a , L.H. , H e r r i n g , H . J . and M e l l o r , G.L., 1985. C a l i f o r n i a s h e l f p h y s i c a l oceanography c i r c u l a t i o n m o d e l : F i n a l r e p o r t . D y n a l y s i s o f P r i n c e t o n Report No. 88, 368 pp. Blumberg, A.F. and Kantha, L.H., 1985. Open boundary c o n d i t i o n s f o r c i r c u l a t i o n models. J. o f H y d r a u l i c Engin., 111: 237-255. Brink, K . H . , 1 9 8 2 . A c o m p a r i s o n o f l o n g c o a s t a l t r a p p e d wave t h e o r y w i t h o b s e r v a t i o n s o f f P e r u . J . Phys. Oceanogr., 12: 897-913. E r i n g e n , A . C . , 1 9 6 2 . N o n l i n e a r t h e o r y o f Continuous Media. McGraw-Hill Book Company , N e w York. F o f o n o f f , N.P. , 1 9 6 2 . P h y s i c a l p r o p e r t i e s o f sea-water. I n : N.M. H i l l , Ed., The Sea. Vol. 1 , I n t e r s c i e n c e P u b l i s h e r s o f J o h n W i l e y a n d S o n s , N e w Y o r k , pp. 3-30. G r a n t , W.D. and Madsen, O . S . , 1979. Combined wave and c u r r e n t i n t e r a c t i o n w i t h a rough bottom. J . Geophys. Res., 8 4 : 1797-1808. G i l l , A.E., 1982. Atmosphere-Ocean Dynamics. Academic P r e s s , New York. Johnson, B.H., 1982. Numerical m o d e l l i n g o f e s t u a r i n e hydrodynamics on a boundary f i t t e d c o o r d i n a t e system. I n : J . Thompson ( E d i t o r ) , N u m e r i c a l G r i d G e n e r a t i o n , E l s e v i e r , pp. 409-436. Lynch, D . R . a n d G r a y , W.G. , 1 9 7 8 . A n a l y t i c s o l u t i o n s f o r computer f l o w model t e s t i n g . J. o f H y d r a u l i c s Div., ASCE, 104: 1409-1428. M e l l o r , G.L. , 1 9 7 3 . Analytic p r e d i c t i o n of t h e p r o p e r t i e s of s t r a t i f i e d p l a n e t a r y s u r f a c e l a y e r s . J . Atmos. S c i . , 30: 1061-1069. Mellor, G.L. and Blumberg, A.F., 1985. M o d e l i n g 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 v i t i e s w i t h t h e s i g m a c o o r d i n a t e s y s t e m . Mon. Wea. Rev. , 1 1 3 : 1379-1383. Mellor, G.L. and Yamada, T., 1974. A h i e r a r c h y of t u r b u l e n c e c l o s u r e models f o r p l a n e t a r y boundary l a y e r s . J. Atmos. S c i . , 31: 1791-1806. M e l l o r , G.L. a n d Yamada, T . , 1 9 8 2 . Development o f a t u r b u l e n c e c l o s u r e model f o r g e o p h y s i c a l f l u i d problems. Rev. Geophys. a n d S p a c e P h y s . , 2 0 : No. 4 , 851-875. Merilees, P . E . , 1 9 7 6 . F u n d a m e n t a l s o f n u m e r i c a l w e a t h e r p r e d i c t i o n . I n : A. Murphy and D. Williamson ( E d i t o r s ) , W e a t h e r F o r e c a s t i n g a n d W e a t h e r F o r e casts: M o d e l s , S y s t e m s , a n d Users, NCAR Tech. Note NCARICA-5 + 1976-ASP, Boulder, Colorado, 1, 2-138. P e f f l e y , M.B. a n d O ' B r i e n , J . J . , 1 9 7 6 . A t h r e e - d i m e n s i o n a l s i m u l a t i o n o f c o a s t a l u p w e l l i n g o f f Oregon. J. Phys. Oceanogr., 6 : 164-180.
88 P h i l a n d e r , S.G.H. and Yoon, J . - H . , 1982. E a s t e r n boundary c u r r e n t s and c o a s t a l upwelling. J. Phys. Oceanogr., 12: No. 8, 862-879. P h i l l i p s , N . A . , 1957. A c o o r d i n a t e system having some s p e c i a l advantages f o r numerical f o r e c a s t i n g . J. o f Meteorology, 14: 184-185. P i n d e r , G.F. and G r a y , W . G . , 1977. F i n i t e Element S i m u l a t i o n i n S u r f a c e and Subsurface Hydrology. Academic P r e s s , N e w York. Reid, R.O., Vastano, A.C., Whitaker, R.E. and Wanstreth, J . J . , 1977. Experiments i n storm s u r g e s i m u l a t i o n . I n : E.D. Goldberg, I . N . McCave, J . J . O ' B r i e n and J . H . S t e e l e ( E d i t o r s ) , The Sea, Vol. 6, John Wiley. N e w York. S p a u l d i n g , M.L., 1984. A v e r t i c a l l y averaged c i r c u l a t i o n model u s i n g boundaryf i t t e d c o o r d i n a t e s . J. Phys. Oceanogr., 14: 973-982. Suginohara, N . , 1982. C o a s t a l upwelling: Onshore-offshore c i r c u l a t i o n , equatorward c o a s t a l jet and poleward u n d e r c u r r e n t o v e r a c o n t i n e n t a l s h e l f - s l o p e . J . Phys. Oceanogr., 12: 272-284. Thacker, W.C., 1977. I r r e g u l a r g r i d f i n i t e - d i f f e r e n c e t e c h n i q u e s : s i m u l a t i o n s of o s c i l l a t i o n s i n s h a l l o w c i r c u l a r b a s i n s . J. Phys. Oceanogr., 7: 284-292. Wang, D-P. , 1982. Development o f a t h r e e - d i m e n s i o n a l , l i m i t e d - a r e a ( i s l a n d ) s h e l f c i r c u l a t i o n model. J. Phys. Oceanogr., 12: 605-617. Yoon, J - H . and P h i l a n d e r , S.G.H., 1982. The g e n e r a t i o n o f c o a s t a l u n d e r c u r r e n t s . J. Oceanogr. SOC. of J a p a n , 38: 215-224.
89
P R E D I C T I N G OPEN OCEAN CURRENTS, FRONTS AND E D D I E S
ALLAN R. ROBINSON Division of Applied S c i e n c e s , Harvard U n i v e r s i t y , Cambridge, Massachusetts
ABSTRACT
The concept of ocean p r e d i c t i o n s c i e n c e , r e l a t e d t o t h e f o r e c a s t i n g of t h e " i n t e r n a l weather of t h e s e a " i s introduced a s important f o r ocean s c i e n t i f i c research g e n e r a l l y and f o r p r a c t i c a l a p p l i c a t i o n i n t h e a r e a s of marine operat i o n s and p l a n e t a r y environmental management. Systematic f i e l d e s t i m a t i o n , melding observations obtained by remotely l o c a t e d and in situ s e n s o r s with dynamical model f o r e c a s t s (4-dimensional d a t a a s s i m i l a t i o n ) i s advocated. The c h a r a c t e r i s t i c s of t h e oceanic mesoscale (analogous t o t h e atmospheric synoptic s c a l e ) a r e reviewed. Forecasting t h e mesoscale (0(10-102 km) and 0(10-102 days) ) i s f e a s i b l e and has been i n i t i a t e d , b u t a r e g i o n a l approach i s now necessary. Examples of r e c e n t p r o g r e s s i n mesoscale p r e d i c t i o n r e s e a r c h a r e presented, including an ongoing f o r e c a s t system which i s nowcasting and f o r e c a s t i n g i n t h e Gulf-Stream region i n real time (GULE'CASTING).
1 INTRODUCTION
- OCEAN
PREDICTION SCIENCE
Knowledge of t h e kinematical s t r u c t u r e and dynamical c h a r a c t e r i s t i c s of flows i n t h e deep ocean h a s accrued r a p i d l y i n r e c e n t years.
The e n e r g e t i c a l l y
dominant flow i s v a r i a b l e i n space and t i m e , g e n e r a l l y extending smoothly throughout t h e water column from s u r f a c e t o bottom with t h e most i n t e n s i v e flow i n t h e main thermocline region.
This s o - c a l l e d oceanic mesoscale v a r i a b i l i t y
i s t h e oceanic analogue of t h e atmospheric s y n o p t i c s c a l e weather phenomena, and i s t h e " i n t e r n a l weather of t h e s e a " (Robinson, 1983).
Nowcasting and
f o r e c a s t i n g oceanic mesoscale phenomena i s i n t e r e s t i n g and compelling simply because it i s t h e dominant phenomena i n most of t h e ocean.
From a p r a c t i c a l
viewpoint, p r e d i c t i o n i s of c r i t i c a l importance t o a p p l i e d marine o p e r a t i o n s and t o s c i e n t i f i c r e s e a r c h a t sea.
Although t h e n e c e s s i t y of p r e d i c t i o n has
been r e a l i z e d f o r some time (Mooers, Piacsek, and Robinson, 1981) it i s only now t h a t f o r e c a s t i n g has become f e a s i b l e and i n i t i a l e f f o r t s have begun.
The
b a s i c p h y s i c a l f i e l d s of v e l o c i t y , p r e s s u r e , d e n s i t y , temperature and s a l i n i t y i n t h e ocean a r e l i n k e d ; f o r e c a s t i n g any of them implies e s s e n t i a l l y t h e forec a s t i n g of a l l of them.
A d d i t i o n a l l y , nowcasting and f o r e c a s t i n g of t h e b a s i c
physical f i e l d s allows by r e l a t i v e l y simple e x t e n s i o n , t h e p r e d i c t i o n both of associated f i e l d s (e.g.,
h e a t f l u x , sound speed) and a c t i v e and passive
d i s p e r s i n g m a t e r i a l s (e.g.,
n u t r i e n t s , t r a c e r s , and p o l l u t a n t s ) .
Directed and concerted u t i l i z a t i o n of t h e s e a and i t s resources i s i n c r e a s i n g , and t h e i n a d v e r t e n t impact of an ongoing t e r r e s t r i a l t e c h n i c a l / i n d u s t r i a l s o c i e t y on t h e oceans i s a s e r i o u s concern (Malone and Roederer, 1985). Deep open ocean f o r e c a s t i n g i s a valuable a i d t o marine technology and t o l a r g e s c a l e environmental management.
Underwater o p e r a t i o n s a s s o c i a t e d with resource
exploration and e x p l o i t a t i o n (e.g.,
o i l , m i n e r a l s , hydrothermal energy,
f i s h e r i e s ) , t r a n s p o r t a t i o n and defense, do o r would b e n e f i t s u b s t a n t i a l l y from u s e f u l p r e d i c t i o n s from an economic viewpoint and f o r human s a f e t y . i n c r e a s i n g l y o p e r a t e underwater.
Navies
For marine environmental management nowcasts
and f o r e c a s t s i n c r e a s e t h e e f f e c t i v e n e s s i n coping with and a s s e s s i n g t h e a c t u a l o r p o t e n t i a l e f f e c t s of a c c i d e n t a l o r planned r e l e a s e s of p o l l u t a n t s and foreign material.
Chemical and sludge dumps are r o u t i n e , a c c i d e n t a l o i l s p i l l s occur
and must be d e a l t w i t h , low-level n u c l e a r waste dumps e x i s t , and s e v e r a l n a t i o n s a r e considering d i s p o s i n g of high l e v e l n u c l e a r wastes i n t h e seabed. P r e d i c t i o n s a r e important t o ocean science research,
i ) f o r p h y s i c a l oceanographic
ii) f o r research s t u d i e s i n which p h y s i c a l f i e l d s i n f l u e n c e b i o l o g i -
c a l , chemical o r g e o l o g i c a l p r o c e s s e s , and
iii) as a means f o r g e n e r a t i n g
resources f o r otherwise unobtainable d a t a s e t s . p h y s i c a l research.
Consider f i r s t t h e impact on
Mesoscale phenomena, such as b u r s t s of midocean b a r o c l i n i c
i n s t a b i l i t y (McWilliams e t a z . , 1983; P i n a r d i and Robinson, 1987) a r e notoriously i n t e r m i t t e n t and e v e n t f u l .
Pinpointing t h e a n t i c i p a t e d occurrence of such
events f o r focused research reduces resource requirements and a c c e l e r a t e s r e search progress.
The a l t e r n a t i v e i s w a i t i n g f o r such e v e n t s t o occur a t a f i x e d
mooring s i t e o r w a i t i n g f o r an adequate a r r a y of f l o a t s improbably but f o r t u i t o u s l y t o e n t e r t h e r e l e v a n t space-time domain.
Moreover, r e s e a r c h on r e g i o n a l
mesoscale dynamical processes i n t h e ocean o v e r l a p s s i g n i f i c a n t l y r e g i o n a l forec a s t i n g research.
Hypothesis t e s t i n g a s s o c i a t e d with t h e theory of dynamical
processes involves t h e i d e n t i c a l r e g i o n a l i n i t i a l / b o u n d a r y condition problem c e n t r a l t o f o r e c a s t i n g and h i n d c a s t i n g (Robinson and Walstad, 1987).
Energy
and v o r t i c i t y balances r e q u i r e t h e i n f e r e n c e from measurements and o b s e r v a t i o n s of accurate f i e l d e s t i m a t e s which can be achieved by a dynamical f i l t e r i n g process t e c h n i c a l l y akin t o f o r e c a s t i n g procedures ( P i n a r d i and Robinson, 1 9 8 7 ) . B i o l o g i c a l , chemical and g e o l o g i c a l oceanographers ( t h e l a t t e r involved f o r example with sediment and/or bottom boundary l a y e r p r o c e s s e s ) have become i n c r e a s i n g l y aware of t h e mesoscale p h y s i c a l environment and of t h e n e c e s s i t y f o r knowing t h e s y n o p t i c t r a n s p o r t f i e l d s .
These are of c r u c i a l importance i n r e a l
time f o r t h e e f f i c i e n t design and execution of experiments a t sea and a r e u l t i mately necessary f o r t h e i n t e r p r e t a t i o n of biogeochemical process measurements which g e n e r a l l y w i l l a l s o e x h i b i t v a r i a b i l i t y on t h e mesoscale.
Finally w e note
t h a t s i n c e synoptic/mesoscale p r e d i c t i o n i s i n i t s i n f a n c y , dynamical r e s e a r c h i s s u e s and p r a c t i c a l f o r e c a s t i n g i s s u e s a r e impossible t o s e p a r a t e now from t h e
p r a c t i c i n g viewpoint. Thus cooperation between fundamental r e s e a r c h e r s and marine o p e r a t o r s can be of s u b s t a n t i a l mutual b e n e f i t .
Not only can t h e
c a p a b i l i t y of a c c u r a t e and e f f i c i e n t f o r e c a s t s be achieved r e l a t i v e l y r a p i d l y , but well-designed i n i t i a l i z a t i o n , updating and v e r i f i c a t i o n d a t a can a l s o provide an extensive ongoing dynamical experimental database of a scope generally unavailable f o r fundamental research s t u d i e s . The e l u c i d a t i o n of oceanic mesoscale phenomena and t h e i n i t i a t i o n of deep s e a predictions are o c c u r r i n g i n a broadly based supportive i n t e l l e c t u a l and techn i c a l context.
P h y s i c a l oceanography has evolved r a p i d l y during r e c e n t years.
New sensors o f t e n borne on new platforms and sampling f o r t h e f i r s t t i m e a t d e f i n i t i v e r a t e s have revealed new phenomena and a r e providing a q u a n t i t a t i v e , d e f i n i t i v e and permanent kinematic database.
New i d e a s and t h e o r i e s a r e being
brought t o b e a r and l a r g e computer-based numerical models a r e a new component of t h e science.
The comprehensive and a b s t r a c t b a s i s of geophysical f l u i d
dynamics h e l p s t o focus dynamical oceanographic problems and r e l a t e them t o analogues which a r i s e i n s i s t e r s c i e n c e s such a s meteorology, a s t r o p h y s i c s , and engineering f l u i d dynamics.
Contemporary techniques of a p p l i e d mathematics,
including numerical a n a l y s i s and i n c o r p o r a t i n g computational p h y s i c s , a r e now available t o grapple with e s s e n t i a l n o n l i n e a r i t i e s .
A g e n e r a l methodology f o r
dealing with q u a s i d e t e r m i n i s t i c and quasirandom n o n l i n e a r dynamical s y s t e m s i s slowly evolving.
New technology, new i d e a s and a c c e l e r a t e d e f f o r t have evolved
ocean science so r a p i d l y t h a t n i n e t y p e r c e n t of our "corpus" of c r e d i b l e knowledge today has accrued during t h e l a s t decade. A most s i g n i f i c a n t element of t h e new methodology f o r nowcasting, f o r e c a s t i n g
and d e s c r i p t i v e - s y n o p t i c oceanography g e n e r a l l y i s systematic field estimation. Here t h e t e r m "systems" r e f e r s t o a multicomponent approach, b u i l t out of ongoing and h i s t o r i c a l o b s e r v a t i o n s , a v a l i d a t e d g e n e r a l dynamical model, and various r e g i o n a l s t a t i s t i c s .
A l l e s t i m a t e s have e r r o r s , b u t a combination of
two independent e s t i m a t e s can be made which has an expected e r r o r lower than e i t h e r of t h e two component e r r o r s .
This technique i s e f f i c i e n t and i s u t i l i z e d
by engineers and astronomers under t h e terminology of optimal e s t i m a t i o n .
One
independent e s t i m a t e can be derived from o b s e r v a t i o n s and another from t h e dynamical model.
Oceanographers have t r a d i t i o n a l l y combined d a t a with model
c o n s t r a i n t s v i a geostrophic computations and more r e c e n t l y i n terms of diagn o s t i c numerical computations (e.g.,
Sarmiento and Bryan, 1 9 8 2 ) .
The i n v e r s e
method i s a modem technique of c o n s t r a i n i n g d a t a by models which oceanographers (e.g., Wunsch, 1978) s h a r e with g e o p h y s i c i s t s .
The most powerful and r e l e v a n t
systematic f i e l d e s t i m a t i o n methods f o r a p p l i c a t i o n t o ocean p r e d i c t i o n occur i n meteorology where they a r e known a s four-dimensional d a t a a s s i m i l a t i o n and a r e used f o r melding numerical f o r e c a s t s and asynoptic observations. data a s s i m i l a t i o n r e q u i r e s real-time
Real-time
a c q u i s i t i o n , t e l e m e t e r i n g , and transmission
of d a t a , t h e r a p i d movement of l a r g e d a t a s e t s , t h e accessing of computers remotely and/or t h e use of powerful m i n i a t u r i z e d cpu devices a t sea.
Micro-
e l e c t r o n i c s and networking advances make real-time deep ocean f o r e c a s t i n g a r e a l i t y and r a p i d t e c h n o l o g i c a l progress i n t h e s e a r e a s i s t o be expected. Overall, ocean s c i e n t i s t s a r e aware t h a t t h e g r e a t e s t impact of contemporary technology on t h e i r d i s c i p l i n e w i l l come from two s o u r c e s :
s a t e l l i t e s and
supercomputers.
(ACOS-NSF,
A s t e p f u n c t i o n of progress i s a n t i c i p a t e d
t h e i n f l u e n c e s a r e , of course, already f e l t .
1984) and
Both of t h e s e devices s p e c i f i c a l l y
bear upon o p p o r t u n i t i e s f o r progress i n ocean p r e d i c t i o n .
Remote s e n s i n g from
s a t e l l i t e s provides an extensive coverage of measurements i n t h e space-time domain not a t t a i n a b l e previously n o r now by any o t h e r means.
Supercomputers,
e s s e n t i a l f o r t h e development and a p p l i c a t i o n of ocean c u r r e n t and c i r c u l a t i o n models are t h e f a s t e s t and most powerful machines a v a i l a b l e ( e . g . , t h e Crays and Cybers) and a r e s t i l l s e v e r e l y o v e r s t r e s s e d by t h e demands o f ocean models. Here t h e f r o n t i e r s of hardware development and i n t e l l e c t u a l p r o g r e s s go hand i n hand.
A s i n o t h e r branches of modem physics and e n g i n e e r i n g , r e a l i s t i c simula-
t i o n s of ( r e g i o n s o f ) t h e ocean a r e now p o s s i b l e .
The use of such s i m u l a t i o n i n
s c i e n t i f i c research (numerical experimentation, s e n s i t i v i t y and process s t u d i e s , e t c . ) i s thought by many t o r e p r e s e n t t h e f i r s t major s t e p forward i n t h e b a s i c
s c i e n t i f i c method s i n c e t h e seventeenth century.
Science i s now a t r i p a r t i t e
endeavor with Simulation added t o t h e two c l a s s i c a l components. Experiment and Theory.
2 THE MESOSCALE FORECAST PROBLEM The synoptic/mesoscale phenomena i n t h e ocean encompasses a wide range of features including:
t h e meandering and f i l a m e n t i n g o f c u r r e n t s and j e t s , f r o n t s ,
p l a n e t a r y waves, midocean eddy f i e l d s , s o l i t o n s , e t c .
The pervasive h o r i z o n t a l
s p a t i a l s c a l e i s on t h e o r d e r of t h e i n t e r n a l Rossby r a d i u s of deformation ( t h e depth s c a l e t i m e s t h e r a t i o of t h e buoyancy frequency t o t h e C o r i o l i s frequency) with t h e a s s o c i a t e d t i m e s c a l e s r e l a t e d t o advections, p l a n e t a r y wave propagat i o n r a t e s , o r p o s s i b l y forcing.
The v e r t i c a l s c a l e s a r e u s u a l l y a s s o c i a t e d
with t h e f u l l water column and t h e main thermocline depth, b u t may be as s h o r t a s a few hundred meters.
Aspects of t h e s t r o n g e s t mesoscale v a r i a b i l i t y (e.g.,
Gulf Stream meanders and r i n g s ) have been known f o r decades (Stommel, 1965; Richardson, 1976) b u t it i s only s i n c e t h e 1970's t h a t a r e a l i s t i c s y n o p t i c p i c t u r e of t h e ocean i s becoming a v a i l a b l e (Robinson, 1983).
Dedicated experiments
defined t h e Northwestern A t l a n t i c midocean eddies and provided t h e b a s i s f o r e x t e n s i v e geographical e x p l o r a t i o n .
Recent d i s c o v e r i e s i n c l u d e Caribbean
(Kinder, 1983) and Mediterranean (Robinson e t
al., 1987a) e d d i e s .
I t i s impor-
t a n t t o n o t e t h a t new sampling and i n t e r p r e t a t i v e techniques a r e s t i l l r e v e a l i n g q u a l i t a t i v e l y new f e a t u r e s .
Such new f e a t u r e s may e x i s t i n r e l a t i v e i s o l a t i o n
such a s sub-mesoscale
" l e n s e s " (McWilliams e t U l . ,
1985) o r may be a s s o c i a t e d with major u
1983) and s q u i r t s (Davis,
priori known v a r i a b i l i t y such a s Gulf
Stream s h i n g l e s and outbreaks ( C o r n i l l o n , Evans and Large, 1986).
The c h a r a c t e r i s t i c s , d i s t r i b u t i o n and s t a t i s t i c s of t h e eddy f i e l d of t h e global ocean a r e emerging. heterogeneous a s p e c t s .
The g l o b a l eddy f i e l d has both homogeneous and
First,
consider t h e dynamically homogeneous ocean.
Mesoscale eddies a r e u b i q u i t o u s , o c c u r r i n g almost everywhere they a r e sought and u s u a l l y with dominant energy.
Quiescent r e g i o n s e x i s t ( e . g . , t h e mid North
East P a c i f i c and t h e b e t a t r i a n g l e region of t h e North A t l a n t i c ) , b u t they a r e t h e exception r a t h e r than t h e r u l e .
Moreover, t y p e s o f eddy f e a t u r e s r e c u r i n
t h e world ocean i n similar circumstances, consider e.g.
t h e c h a r a c t e r i s t i c s of
t h e v a r i a b i l i t i e s of t h e Gulf Stream, Kuroshio, and o t h e r major Western boundary current systems (meanders, r i n g s extension r e g i o n s , etc.
.
Dynamically analogous
mesoscale phenomena may appear s u p e r f i c i a l l y d i s s i m i l a r only because of e f f e c t s due t o l o c a l circumstances i n c l u d i n g s t r a t i f i c a t i o n , topography, and i n t e r a c t i o n with o t h e r s c a l e s .
There a r e , however, important h e t e r o g e n e i t i e s .
Certain
f e a t u r e s occur only i n r e s t r i c t e d regions and may be t i e d t o l o c a l d e t a i l s of topography and c o a s t a l geometry.
Coastal and i s l a n d mesoscale f e a t u r e s and
e f f e c t s a r e d i s t i n c t i v e , have deep s e a i n f l u e n c e s and i n t e r a c t i o n s , and r e q u i r e more dedicated study.
The kinematics and physics of e q u a t o r i a l mesoscale
phenomena d i f f e r from t h a t a t mid and h i g h e r l a t i t u d e s , and t h e p o l a r ocean v a r i a b i l i t y i s t i e d t o i c e dynamical processes.
The remainder of t h i s discussion
w i l l be concerned with mid-latitude mesoscale v a r i a b i l i t y and i t s f o r e c a s t i n g .
This problem i s c l o s e s t t o t h e atmospheric weather p r e d i c t i o n problem.
Many
phenomenological f e a t u r e s a r e s t i l l i m p l i c a t e d , and from a s y n o p t i c viewpoint, a degree of s t a t i s t i c a l h e t e r o g e n e i t y e x i s t s r e l a t e d t o t h e space-time i n t e r mittency of e n e r g e t i c e v e n t s , Several time and space s c a l e s r e l e v a n t f o r t h e p r e d i c t i o n problem a r i s e : from t h e phenomena i t s e l f , from t h e space-time domain of i n t e r e s t , and from t h e techniques adopted f o r t h e p r e d i c t i o n procedure,
The s p a t i a l s c a l e s , somewhat
l a r g e r than t h e i n t e r n a l deformation r a d i u s , range from t e n s t o hundreds of kilometers.
The temporal s c a l e s range from s e v e r a l days t o weeks and months.
Compared t o t h e atmosphere (thousands of k i l o m e t e r s , hours-days), t h e problem of i n t e r n a l weather p r e d i c t i o n i n t h e ocean involves very many s m a l l f e a t u r e s extending over v a s t regions b u t evolving slowly. vantages.
Q1
There a r e disadvantages and ad-
t h e one hand, d a t a c o l l e c t i o n and computer r e s o l u t i o n requirements
make i t impossible now f o r t h e oceanographer t o adopt t h e g l o b a l o r hemispheric approach of t h e meteorologists. t i o n i s necessary.
The r e g i o n a l approach t o mesoscale ocean predic-
Cm t h e o t h e r hand, updating and t h e a s s i m i l a t i o n of d a t a i n
"oceanic r e a l time" i s f a c i l i t a t e d .
The l o c a t i o n , e x t e n t , and d u r a t i o n of
regional f o r e c a s t s a r e d i c t a t e d of course by p r a c t i c a l o r s c i e n t i f i c i n t e r e s t .
Homogeneity f a c t o r s should be e x p l o i t e d as much as p o s s i b l e by t h e development of f l e x i b l e and p o r t a b l e techniques and of r e g i o n a l models which can be e a s i l y relocated.
The p r e d i c t i o n domain may need t o be l a r g e r than t h e p r a c t i c a l
i n t e r e s t domain, so as t o include v o r t i c i t y i n t e r a c t i o n s which i n f l u e n c e t h e evolution of t h e mesoscale w i t h i n t h e domain of i n t e r e s t , o r so a s t o remove boundary e f f e c t s t o a d i s t a n c e of n e g l i g i b l e i n f l u e n c e , e t c . There a r e two b a s i c p h y s i c a l types of r e g i o n a l mesoscale f o r e c a s t problems i n
( i )e v o l u t i o n occurs i n t h e domain v i a i n t e r n a l oceanic dynamical pro-
which:
(ii)e v o l u t i o n i n t h e domain involves t h e response t o l o c a l and
cesses only and
atmospheric forcing.
In t h e l a t t e r case ( i i ) ,a knowledge of t h e s u r f a c e f l u x e s
of momentum, h e a t , energy, and d e n s i t y must be known, and a f o r e c a s t of t h e atmospheric winds i s a c r i t i c a l i n p u t t o t h e oceanic f o r e c a s t .
Thus r e g i o n a l
atmospheric s k i l l s and atmospheric p r e d i c t a b i l i t y time s c a l e s c o n s t r a i n t h e ocean f o r e c a s t e r .
The t y p e ( i )problem however appears t o be t h e s i g n i f i c a n t
one f o r most of t h e oceanic mesoscale.
The e n e r g i z a t i o n of t h e mesoscale within
t h e region o f i n t e r e s t may have occurred remotely i n both space and t i m e ( e . g . , r i n g s snap o f f t h e Gulf Stream which i t s e l f i n t e g r a t e s wind energy slowly W e w i l l f u r t h e r r e s t r i c t our
accumulated over t h e open region of t h e g y r e ) .
discussion here t o t h e type ( i )problem, again t h e c l o s e s t analogue t o meteorol o g i c a l forecasting. (a
Two l i m i t i n g subcases a r e noteworthy.
In t h e f i r s t case
propagation region problem) s i g n a l s e s s e n t i a l l y propagate through t h e domain;
f e a t u r e s e n t e r through one boundary and e x i t through another. case (an
interaction region
In t h e second
problem) f e a t u r e s evolve under t h e i n f l u e n c e of l o c a l
dynamical processes i n t h e domain, n o n l i n e a r conversion occurs r e s u l t i n g i n b i r t h , growth, t r a n s f o r m a t i o n , decay, of e d d i e s , e t c .
3
MESOSCALE P R E D l C T I ON RESEARCH Operational and research nowcasting and f o r e c a s t i n g of t h e mesoscale r e q u i r e s
running r e a l i s t i c deep oceanic dynamical models w i t h r e a l ocean d a t a and t h e implementation of r e g i o n a l f o r e c a s t systems.
The r e g i o n a l dynamical models
should be a s f l e x i b l e and p o r t a b l e a s p o s s i b l e .
components of an h i e r a r c h y of ocean models. both
i ) dynumical and
i i ) regional bases.
c o n s i s t of quasigeostrophic,
We regard such models a s
modular
Conceptually, t h e h i e r a r c h y has
F i r s t l y , t h e i n t e r n a l dynamics may
i n t e r m e d i a t e o r p r i m i t i v e e q u a t i o n s ; secondly, t h e
mesoscale model may have i n t e r c o n n e c t e d s u r f a c e and bottom boundary l a y e r models and must be e x p l i c i t l y o r p a r a m e t r i c a l l y embedded i n a l a r g e r s c a l e b a s i n o r general c i r c u l a t i o n model. and u l t i m a t e l y v e r i f i e d . are essential.
The dynamical model must be c a l i b r a t e d , v a l i d a t e d Model-model,
a s w e l l a s model-data intercomparisons
Observational network research and development, i n c l u d i n g
o b s e r v a t i o n a l system s i m u l a t i o n s , i s a major t a s k .
Adequate and f e a s i b l e d a t a
bases must be e s t a b l i s h e d w i t h an e f f i c i e n t mix o f remotely sensed and
in situ
95 measurements.
Data a s s i m i l a t i o n methods need a d a p t a t i o n from meteorology and
fundamental research f o r s p e c i a l oceanic circumstances.
E r r o r sources and
s t r u c t u r e s which a r i s e from computation, p h y s i c s , and observations must be determined and s e n s i t i v i t i e s a s c e r t a i n e d , component-wise and s y s t e m a t i c a l l y . These i s s u e s r e p r e s e n t a challenging and enduring range of research problems but progress i s occurring.
A u s e f u l and comprehensive review and overview of t h e
mid-1986 s t a t u s of research from t h e viewpoint of t h e growing community of American ocean p r e d i c t i o n s c i e n t i s t s i s provided by Mooers, Robinson and Thompson (1987), and Hurlburt h a s a s s e s s e d a s p e c t s . W e have been pursuing a l i n e of r e s e a r c h involving s y s t e m a t i c f i e l d estimat i o n conceptualized i n terms of an Oceanic Descriptive P r e d i c t i v e System (ODPS) schematized i n Figure 1.
In t h i s context r e a l ocean s t u d i e s and f o r e c a s t t r i a l s
THE DESCRIPTIVE- PREDICTIVE SYSTEM
-
INITIALIZATION
-
L
DATA ASSIMILATION
r-l I
I
DYNAMICALLY FORECAST FIELDS
DATA ASSIM.
t
I
VIA OPTIMAL ESTIMATION THEORY (MINIMIZE SELECTED ERROR NORM) OPTIMAL FIELD ESTIMATE: THE OCEANIC FORECAST, PHYSICAL PROCESS SrUDlES A schematic of t h e components of an oceanic d e s c r i p t i v e - p r e d i c t i v e Fig. 1. system (ODPS) (From Robinson and L e s l i e , 1985)
.
have been c a r r i e d o u t by a number of i n v e s t i g a t o r s u t i l i z i n g t h e quasigeostrophic version of t h e Harvard open Ocean Model i l l u s t r a t e d i n Figure 2 . s t r o p h i c model i s a powerful and r o b u s t t o o l .
The quasigeo-
The fundamental i n i t i a l / b o u n d a r y
condition problem f o r t h e model r e q u i r e s some means of s p e c i f i c a t i o n a t each
96
HARVARD OPEN OCEAN MODEL PORTABLE
-
-
-a(+ a J ( J I ,
O +PJ.x =Jwr
at
5
=v;~+r2(0.~z)z (5-15 km Grid
-
& ; 1 II
Y
Arbitrary Boundary
I II
/ma
V
OG Mountains 1
F i g . 2 . A schematic of t h e Harvard open Ocean Model w i t h t h e governing equations included. The h o r i z o n t a l and v e r t i c a l g r i d s are i n d i c a t e d . Arrows i n d i c a t e t h e c u r r e n t flow. A t t h e base of t h e model domain i s an i d e a l i z e d bottom topography. (From Robinson and Walstad, 1987)
t i m e s t e p of t h e inflow/outflow f i e l d w i t h v o r t i c i t y on t h e inflow (Charney, F j o r t o f t , and von Neumann, 1950). The model s u b g r i d s c a l e d i s s i p a t i o n (Fpqr) i s a Shapiro (1971) t y p e f i l t e r i n g w i t h a r b i t r a r y s t r e n g t h and frequency, of t h e vorticity field.
The c a l i b r a t i o n of t h e model and i t s a p p l i c a t i o n t o dynamical
p r o c e s s , f o r e c a s t i n g and d a t a a s s i m i l a t i o n s t u d i e s is p r e s e n t e d by Robinson and Walstad (1987) and Robinson (1986) reviews t h e d a t a a s s i m i l a t i o n problem and r e l a t e d research progress.
Here w e w i l l b r i e f l y summarize t h e background and
then c i t e f o u r r e c e n t examples of r e s e a r c h f i n d i n g s r e l a t e d t o t h e p r e d i c t i o n and p r e d i c t a b i l i t y of mesoscale e d d i e s and t o t h e theory and p r a c t i c e of d a t a
91 assimilation. The POLYMODE Synoptic Dynamics Experiment (SDE) c a r r i e d o u t i n t h e western North A t l a n t i c i n t h e l a t t e r h a l f of t h e 1970's cooperatively between s c i e n t i s t s from t h e USA and t h e USSR (Robinson, 1982, 1983) provided t h e f i r s t continuous data s e t of
(almost) s y n o p t i c o b s e r v a t i o n s i n t h e ocean over s e v e r a l independent
r e a l i z a t i o n s i n t h e space-time domain.
The SDE l a s t e d f o r more than a year and
subsets of d a t a were c o l l e c t e d over a r e a l regions extending from about two t o
s i x hundred k i l o m e t e r s on a s i d e .
This d a t a s e t , t h e f i r s t of i t s kind e v e r ,
provided a unique opportunity f o r t h e development and c a l i b r a t i o n of t h e regional open model, i n c l u d i n g s e n s i t i v i t y s t u d i e s with r e s p e c t t o i n i t i a l and boundary condition e r r o r s i n both flow and v o r t i c i t y .
Regional simulation metho-
dologies, r e a l d a t a i n i t i a l i z a t i o n procedures, and EVA, a model c o n s i s t e n t Energy and V o r t i c i t y Analysis scheme f o r t h e i n f e r e n c e of r e g i o n a l dynamical processes (Pinardi and Robinson, 1986) were i n i t i a l l y developed i n t h e SDE d a t a context. During t h e f i r s t h a l f of t h e 1980's t h e Harvard group s c i e n t i s t s c o l l a b o r a t e d with Prof.
CNK Mocer's group from t h e Naval Postgraduate School a t Monterey i n a
program c a l l e d OPTOMA (Ocean P r e d i c t i o n Through Observations Models and Analysis) dedicated t o mesoscale r e g i o n a l p r e d i c t i v e and dynamical methodological research. The s e t t i n g w a s t h e region of i n t e n s e j e t s
f i l a m e n t s eddies and f r o n t s i n t h e
C a l i f o r n i a Current system, b a s i c a l l y a deep water open ocean environment, but l o g i s t i c a l l y convenient and e x t e n s i b l e f o r c o a s t a l i n t e r a c t i o n s t u d i e s .
Repeated
sampling of a c e n t r a l domain has provided an adequate set of p a r t i a l l y connected synoptic r e a l i z a t i o n s and r e l a t e d s t a t i s t i c s .
The OPTOMA d a t a context continues
t o provide f o r t h e s y s t e m a t i c development of t h e ODPS concepts, t h e development of d a t a a s s i m i l a t i o n methods, and t h e t e s t i n g and a p p l i c a t i o n o f EVA.
Success-
f u l real-time p r e d i c t i o n experiments have been c a r r i e d out and t h e r o l e of dynamical model i n t e r p o l a t i o n i n modern s y n o p t i c / d e s c r i p t i v e oceanography exemp l i f i e d (Robinson, Carton, P i n a r d i and Mooers, 1986).
P r e s e n t l y a major
research e f f o r t of t h e Harvard group i s d i r e c t e d towards t h e Gulf Stream meander and r i n g region extending eastward of Cape Hatteras p a s t t h e Grand B a n k s , p a r t l y as a c o n t r i b u t i o n t o t h e ONR m u l t i - i n s t i t u t i o n a l research i n i t i a t i v e SYNOP, f o r which t h e main f i e l d program w i l l be f o r two y e a r s s t a r t i n g i n t h e f a l l of 1987.
(Hogg, 1986).
Nowcasting and f o r e c a s t i n g a r e now performed r o u t i n e l y
(see Section 4 ) and research on t h e v o r t i c i t y dynamics and e n e r g e t i c s on t h e b i r t h and death of r i n g s ( v i a ring-stream i n t e r a c t i o n s ) (Robinson, P i n a r d i and S p a l l , 1987) i s continuing. The f i r s t example i s concerned w i t h t h e p r a c t i c a l problem of producing a regional f o r e c a s t i n an open domain r e q u i r i n g , t h e r e f o r e , t h e s p e c i f i c a t i o n of The d a t a employed (XBTs and c u r r e n t 2 meters) i s a POLYMODE-SDE s u b s e t c a l l e d Mark-2 mapped over t h e i n n e r 300 km f u t u r e boundary c o n d i t i o n s (Walstad, 1987).
region v i a a m u l t i v a r i a t e a n a l y s i s ( i n Fig.
1, t h e upper left-hand and c e n t r a l
boxes i n p u t t i n g t o t h e lower left-hand box) and used f o r t h e c a l i b r a t i o n and i n i t i a l i z a t i o n s t u d i e s mentioned above.
A r e c e n t l y completed study of t h e major
e n e r g e t i c events i n t h e Mark-I1 d a t a ( P i n a r d i and Robinson, 1987) provides t h e d e t a i l e d dynamical context f o r t h e i n t e r p r e t a t i o n of t h e p r e d i c t i o n study. technique adopted f o r a Regional Dynamical Forecast
(RDF
- the
The
right-hand
column of Fig. 1) involves a p r i o r s t a t i s t i c a l f o r e c a s t of t h e boundary condit i o n s around t h e edge of t h e domain by forward e x t r a p o l a t i n g i n time w i t h a mixed space-time c o r r e l a t i o n function i n an o b j e c t i v e a n a l y s i s scheme.
This we
c a l l an RDF with S t a t i s t i c a l l y Forecast Boundary Condition (RDF-SFBC).
Compari-
son f i e l d s of i n t e r e s t a r e t h e Regional S t a t i s t i c a l Forecast (RSF
- the
central
column of Fig. l), and t h e RDF r e s u l t i n g from updating t h e boundary c o n d i t i o n s c o n t i n u a l l y with t h e b e s t a v a i l a b l e d a t a (Benchmark Boundary Condition, RDFBBC).
The RDF-SFBC procedure has been applied t o t h e d a t a 3 times w i t h a
s l i d i n g s t a r t i n g t i m e , with and without t h e e f f e c t s of bottom topography.
The
r e s u l t s f o r f o u r examples i n terms o f t h e domain averaged d i f f e r e n c e f i e l d between t h e analyzed o b s e r v a t i o n and t h e f o r e c a s t ( c a l l e d t h e normalized r o o t mean square e r r o r ) a r e shown i n Figure 3a. I n t h i s region where t h e s t a t i s t i c a l model i s good, t h e RDF-SFBC g i v e s r e s u l t s a f t e r 15 days with an e r r o r l e v e l of about 45% which compares with 25% f o r t h e benchmark RDF-BBC and 62% f o r t h e purely s t a t i s t i c a l procedure. Thus model dynamics g e n e r a l l y improves t h e f o r e c a s t s ' accuracy and t h e accuracy of f o r e c a s t s made from only one independent synoptic d a t a r e a l i z a t i o n a r e u s e f u l g e n e r a l l y f o r a t l e a s t 15 days. Examining t h e r e s u l t s i n more d e t a i l i n d i c a t e s : 1) a q u a n t i t a t i v e dependence on t h e types of f e a t u r e s propagating through t h e domain, ii) a dominant e r r o r a r i s i n g from boundary c o n d i t i o n e r r o r s , and t h e r e f o r e lii) a c o r r e l a t e d degradation of t h e RDF-SFBC and t h e RSF, i v ) t h a t topographic e f f e c t s g e n e r a l l y h e l p a f t e r about t h r e e weeks b u t may be d e t r i m e n t a l t o s h o r t e r f o r e c a s t s of t h e thermocline f i e l d s i f t h e deep f i e l d s a r e i n e r r o r . I t i s noteworthy t h a t v ) good pred i c t i o n s v i a t h e RDF-SFBC procedure can be made even i n an interaction region problem. This i s t h e c a s e f o r days 3640-3670 shown i n Fig. 3b, d u r i n g which cyclone i n t e n s i f i c a t i o n accompanied by a b u r s t of b a r o c i i n i c energy conversion
i s occurring.
( I n Fig. 3b and a l l subsequent f i g u r e s streamfunctions a r e non-
dimensional. See primary r e f e r e n c e s f o r d e t a i l s . ) Comprehensive maps of t h e POLYMODE region have been constructed over t h e 500 km sq. domain of t h e Synoptic Dynamics Experiment (SDE) by Carton and McGillicudy (1985) from SDE c u r r e n t meter and hydrographic d a t a and a l s o incorp o r a t i n g Local Dynamics Experiment (LDE) and SOFAR f l o a t d a t a .
Analysis e r r o r s
are e s t i m a t e d which a r i s e from gappy sampling, from extending measurements i n t o t h e deep water, and from t h e lack of a b s o l u t e v e l o c i t y measurements around much of t h e 200 km o u t e r r i m .
This a n a l y s i s provides a unique s e t of mesoscale
ocean d a t a over a l a r g e domain w i t h continuous s y n o p t i c r e a l i z a t i o n s over several
99
W
1 .o
t
RSF
0
z W
3640
3648
3650
3655
3663
3660
MIN=- 2.77 MAX=3.51
3670
3670
MIN=-3.11 MAXz3.79
MIN=-2.98 MAXz3.22
MIN=-3.65 MAX=3.49
MIN=-3.27 MAXs4.1I
3. a ) Comparison of t h e domain averaged d i f f e r e n c e between analyzed observation f i e l d s and. t h e numerically f o r e c a s t f i e l d s f o r 30-day f o r e c a s t s . RDF i n d i c a t e s Regional Dynamical F o r e c a s t ; RSF i n d i c a t e s Regional S t a t i s t i c a l Forecast; BBC i n d i c a t e s Benchmark Boundary Conditions and SFBC i n d i c a t e s S t a t i s t i c a l l y Forecast Boundary Conditions. b ) Comparison of t h e analyzed observation f i e l d s ( t o p row) with t h e numerically f o r e c a s t f i e l d s (bottom row) Contour i n t e r v a l i s 0.7. a t ten-day i n t e r v a l s from t h e POLYMODE d a t a s e t . (From Walstad, 1987)
-Fig.
eddies from d e d i c a t e d
in situ
data.
In t h e second example c i t e d Carton (1987)
u t i l i z e s t h i s a n a l y s i s t o provide i n i t i a l c o n d i t i o n s , boundary conditions and v e r i f i c a t i o n d a t a f o r a s e t of now-,
fore-,
and h i n d c a s t i n g experiments i n a
100 f i r s t month of a dynamical f o r e c a s t ;
i v ) t h a t benchmark f o r e c a s t s can maintain
a 60% e r r o r l e v e l f o r a t l e a s t s e v e r a l months; and
v) t h a t i n benchmark f o r e -
c a s t s , major e r r o r s propagate i n from t h e boundaries. The p r e d i c t a b i l i t y question i s approached by attempting t o minimize t h e r o l e of e r r o r s i n t h e o b j e c t i v e l y mapped d a t a i n a study of t h e comparison of forec a s t s which have t h e same boundary c o n d i t i o n s b u t d i f f e r e n t i n i t i a l c o n d i t i o n s . D i f f e r e n t model-derived i n i t i a l c o n d i t i o n s with s u c c e s s i v e l y g r e a t e r d i f f e r e n c e a r e achieved by s t a r t i n g a new f o r e c a s t every t e n days i n t o a sixty-day f o r e c a s t o r i g i n a l l y i n i t i a l i z e d w i t h mapped data.
The r e s u l t s a r e summarized i n Fig. 4 L ,
which d e p i c t s tlie ensemble averaged model-model d i f f e r e n c e f i e l d s .
Forecasts
i n i t i a l l y c l o s e diverge t o an expected e r r o r l e v e l of 35% b u t f o r e c a s t s i n i t i a l l y f a r a p a r t decrease t h e i r d i f f e r e n c e s .
These r e s u l t s a r e i n t e r p r e t e d a s i n d i c a -
t i n g t h a t t h e combination of t h e model, t h e domain and t h e p r e d i c t i o n scheme contains an i r r e d u c i b l e e r r o r l e v e l , t i e d t o i n t e r n a l i n s t a b i l i t i e s , o f about
30- 40%. The Kalman f i l t e r i s an optimal s e q u e n t i a l e s t i m a t i o n technique used by engineers and researched by meteorologists which i s t h e o r e t i c a l l y optimal b u t demanding of computer resources
(Ghil
e t a l . , 1981).
Our n e x t example i s an
i d e a l i z e d study by M i l l e r (1986) e x p l o r i n g t h e p o t e n t i a l of t h i s method f o r r e g i o n a l d a t a a s s i m i l a t i o n i n quasigeostrophic models.
A s e r i e s of t e s t s was
run with t h e K a l m a n f i l t e r and a one-dimensional l i n e a r i z e d b a r o t r o p i c quasigeos t r o p h i c model.
This one-dimensional scheme w a s t i e d t o t h e Harvard Open Ocean
Model by choice of dynamical model parameters and by t y i n g t h e e r r o r models t o t h e c a l i b r a t e d computational e r r o r of t h e three-dimensional model ( M i l l e r 1983).
e t al.,
An u n s t a b l e numerical scheme w a s d e l i b e r a t e l y chosen f o r t h e one-
dimensional computational model.
These t e s t s had a twofold purpose:
i ) t o de-
termine whether t h e Kalman f i l t e r using s p a r s e d a t a would allow a model t o t r a c k an unstable process without diverging i n time from a r e f e r e n c e s o l u t i o n , and
i i ) t o t e s t t h e s u i t a b i l i t y of t h e Kalman f i l t e r f o r open boundary problems by means of examples designed t o determine whether updating with d a t a confined t o t h e i n t e r i o r of t h e model region could c o n t r o l t h e e r r o r i n t h e i n t e r i o r , l e a v i n g t h e g r e a t e s t e r r o r n e a r t h e boundary. Figure 5 shows a summary of r e s u l t s from t h i s p i l o t study.
Figures 5a and 5b
show r e s u l t s from an experiment with p e r i o d i c boundary c o n d i t i o n s which focuses on q u e s t i o n s ( i )and f i g u r e 5c, an experiment with open boundary c o n d i t i o n s adding t h e complications of
( i i ) . The r e f e r e n c e s o l u t i o n was a sum of Rossby
waves contaminated by n o i s e i n t h e form of F o u r i e r components w i t h Gaussian random amplitudes.
The t o t a l RMS amplitude of t h e s o l u t i o n was normalized t o 1;
t h e p e r i o d of t h e f a s t e s t wave was was
4T/32.
where.
471, and t h e temporal r e s o l u t i o n i n t h e model
Without updating, t h e e r r o r v a r i a n c e s r a p i d l y grow t o u n i t y every-
In t h i s experiment, t h e f o r e c a s t s o l u t i o n was updated by sampling t h e
101 study s l a n t e d towards q u e s t i o n s of r e g i o n a l ocean p r e d i c t a b i l i t y .
What a r e t h e
l i m i t s t o t h e p o t e n t i a l accuracy of a p r e d i c t i o n scheme s e t by i n t e r n a l i n s t a b i l i t i e s , and how do t h e e r r o r s they generate compare t o t h o s e a r i s i n g from d a t a inadequacies and model e r r o r s ?
P r e d i c t a b i l i t y q u e s t i o n s a r e fundamental
(Holloway and W e s t , 1984) and provide a u s e f u l viewpoint even though ocean prediction i s a t an e a r l y s t a g e .
S i x experiments a r e c a r r i e d o u t each invoking 34
model i n t e g r a t i o n s r e p r e s e n t i n g an attempt t o produce ensemble s t a t i s t i c s f o r a variety of f o r e c a s t i n g examples based on independent s y n o p t i c r e a l i z a t i o n s . Carton c a l l s t h e FDF-BBC a "hindcast with a c c u r a t e boundary conditions.
The
I'
e r r o r s a f t e r both 10 days and 60 days f o r 34 d i f f e r e n t such h i n d c a s t s are shown i n Fig. 4a, whi.ch t h u s i l l u s t r a t e s t h e degree of v a r i a t i o n of i n d i v i d u a l r e s u l t s t h a t go i n t o
t h e e r r o r statistics.
The error here i s again defined a s t h e
domain averaged normalized r o o t mean square d i f f e r e n c e between t h e mapped observa t i o n s ( c o n t a i n i n g d a t a sampling and a n a l y s i s e r r o r s ) and t h e model runs (cont a i n i n g model and computational e r r o r s )
.
R e s u l t s of t h i s study i n d i c a t e :
i ) t h a t a f o r e c a s t by simply p e r s i s t i n g t h e i n i t i a l f i e l d s everywhere l e a d s t o a 60% expected e r r o r i n 10 days and continues t o grow r a p i d l y beyond t h a t t i m e ;
ii) t h a t a dynamical model f o r e c a s t made w i t h p e r s i s t e n t bo'undary c o n d i t i o n s maintains an expected e r r o r of 60% i n t h e i n n e r t h i r d of t h e domain f o r t h r e e weeks;
iii) t h a t t h e major i n f l u e n c e of i n i t i a l c o n d i t i o n s i s f e l t during t h e
O0.1S 2 Y
3380
3460
3550
STARTING DAYS
3650
u
0
10
20
30 40
60
60
DAYS
F i g . 4 . a) Domain averaged d i f f e r e n c e between analyzed observation f i e l d s and the numerically f o r e c a s t f i e l d s a f t e r t e n days ( s o l i d l i n e ) and s i x t y days (dahsed l i n e ) f o r 34 s e p a r a t e f o r e c a s t s ; each with a d i f f e r e n t s t a r t i n g d a t e . b) Domain average d i f f e r e n c e between f o r e c a s t s w i t h i d e n t i c a l boundary condit i o n s but d i f f e r e i n g i n i t i a l conditions. K i n d i c a t e s t h e number of days i n t o a s i x t y day f o r e c a s t t h e subsequent f o r e c a s t begins. (From Carton, 1987)
102
'
a
W
' 0 2 4 6 X
-
0.5
-0.5
10 20 TIME
30
...
l.Or
JI
'0
-_
-I
0
I
I
I 2
1
I
I 3 n
X
-
F i g . 5. a) The expected e r r o r variance as a f u n c t i o n of space f o r a onedimensional l i n e a r i z e d b a r o t r o p i c q u a s i g e o s t r o p h i c model using a Kalman f i l t e r . b) The e v o l u t i o n of t h e e r r o r variance i n a region upstream of t h e d a t a dense region. c ) A comparison of streamfunction values f o r a sample r e a l i z a t i o n ( s o l i d l i n e ) and t h e r e f e r e n c e s o l u t i o n (dashed l i n e ) over t h e mode 1 domain. The computed expected e r r o r b a r s a r e one s t a n d a r d d e v i a t i o n wide. (From M i l l e r , 1986).
reference s o l u t i o n a t two p o i n t s i n t h e i n t e r i o r every f o u r t h t i m e s t e p . a r e 19 g r i d p o i n t s i n t h e domain and p o i n t s 8 and 1 2 a r e updated.
There
The expected
e r r o r variance a s a function of space converged r a p i d l y t o t h e p a t t e r n shown i n Fig. 5a; Fig. 5b shows t h e e v o l u t i o n of t h e e r r o r variance i n a region upstream of t h e d a t a dense region. frequency.
The o s c i l l a t i o n s i n t h a t graph r e f l e c t t h e updating
Note t h a t t h e e r r o r i s c o n t r o l l e d around t h e 0 . 3 l e v e l ; t h e s e a r e
worst-case r e s u l t s s i n c e Fig. 5a is j u s t before t h e n e x t updating time s t e p and Fig. 5b i s upstream of updating. The second experiment involves t h e s p e c i f i c a t i o n of open boundary conditions; t h e boundary condition p r e d i c t i o n scheme i n t r o d u c e s an a d d i t i o n a l e r r o r source. Figure 5c shows a sample r e a l i z a t i o n compared t o t h e r e f e r e n c e s o l u t i o n t o g e t h e r with computed expected e r r o r b a r s .
Updating was performed each s t e p a t f o u r
103 The s o l i d curve i s t h e f o r e c a s t and t h e dashed curve
points i n t h e i n t e r i o r .
is t h e r e f e r e n c e s o l u t i o n .
The e r r o r b a r s a r e one s t a n d a r d d e v i a t i o n wide.
Note t h a t t h e e r r o r i n t h e i n t e r i o r i s w e l l c o n t r o l l e d , d e s p i t e t h e r e l a t i v e l y large expected e r r o r near t h e boundaries which a r e f a r from t h e d a t a l o c a t i o n s . Ten experiments w e r e s t u d i e d and i t w a s g e n e r a l l y found t h a t s p a r s e d a t a could r e s u l t i n reasonable f o r e c a s t s , and t h a t i n t e r i o r updating could c o n t r o l t h e interior.
In a d d i t i o n , although Kalman f i l t e r i n g i s n o t o r i o u s l y demanding of
computer r e s o u r c e s , t h e s e r e s u l t s i n d i c a t e t h a t t h e resource demand i s expected
t o be manageable on modern computers f o r t y p i c a l m i d l a t i t u d e eddy p r e d i c t i o n problems. A v a r i e t y of s t r a t e g i e s of f i e l d e s t i m a t i o n and p r e d i c t i o n have been a p p l i e d -
t o t h e OPTOMA d a t a and intercompared by Rienecker, Mooers, and Robinson (1987). Of p a r t i c u l a r i n t e r e s t i s a simple and computationally e f f i c i e n t d a t a assimilat i o n scheme which melds o b j e c t i v e l y analyzed s y n o p t i c o b s e r v a t i o n s ( * 1 day) with t h e model dynamical f o r e c a s t , t h e dynamical f o r e c a s t u t i l i z e d f o r t h e a s s i m i l a t i o n i s w i t h s t a t i s t i c a l l y f o r e c a s t boundary c o n d i t i o n s (RDF-SFBC)
.
The flow i n t h e 150 km square study domain during t h e p e r i o d i l l u s t r a t e d con- s i s t e d of a j e t e n t e r i n g from t h e e a s t , flowing westward and then t u r n i n g southward.
The s t a n d a r d o b j e c t i v e mapping i s shown i n Fig. 6e.
a
b
d
*Fig. 6 .
Figure 6a i s
C
e
A comparison of t h e s t r a t e g i e s of f i e l d e s t i m a t i o n and p r e d i c t i o n f o r J u l i a n day 5522 (6 J u l y , 1983). a) S t a t i s t i c a l f o r e c a s t , b) dynamical forec a s t with p e r s i s t e n t boundary c o n d i t i o n s , c ) dynamical f o r e c a s t with benchmark boundary c o n d i t i o n s , d ) dynamical f o r e c a s t w i t h d a t a a s s i m i l a t i o n , and (From Rienecker e t d., Contour i n t e r v a l i s 0.5. e ) objective analysis. 1987).
104 a purely s t a t i s t i c a l f o r e c a s t based on a time-forward e x t r a p o l a t i o n v i a object i v e a n a l y s i s , Fig. 6b i s a f o r e c a s t with p e r s i s t e n t boundary c o n d i t i o n s , and Fig. 6c i s t h e dynamical f o r e c a s t .
Our focus i s on Fig. 6d, which i s t h e
dynamical f o r e c a s t with d a t a a s s i m i l a t i o n .
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 f o r a s s i m i l a t i o n a r e shown i n Figure 7 ; t h e a r e a shaded h a s , y , t h e 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 f (y > 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.
-
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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
/ -
-
the
current
edge
of
the
varies
in
position
by
core
cross-stream r o u g h l y 8 km.
T h i s value
i s w e l l w i t h i n t h e range o f t h o s e t y p i c a l l y observed i n association Current
1 loll oo
I n t h i s pattern,
western
with
meanders
Florida at
this
latitude
(Schmitz
Richardson,
1968; Johns and
and
S c h o t t , 1987).
An i n s t r u c t i v e way t o I do 8 20 24 28 32 12 16 TIME (DAYS)
describe
the
cross-
s e c t i o n a l s t r u c t u r e o f the
(a 1
p e r t u r b a t i o n i s through the alongstream .06 .05
1
I
-1
mean
perturbation
of
the
kinetic
e n e r g y (PKE).
I n Figure
10, we compare t h i s f i e l d , computed
at
maximum
wave
with
that
the
time
of
amplitude,
computed
by
Leaman, e t a1 ( 1 9 8 7 ) f o r the
full
spectrum
p e r t u r b a ti o n
of
energy
a n a l y z e d f r o m t h e PEGASUS -.01 -
i' 5;
-
data.
By f a r , t h e g r e a t e s t
concentration
of
PKE
in
b o t h model and o b s e r v a t i o n i s i n t h e c y c l o n i c shear zone, of
n e a r and t o t h e l e f t
t h e c u r r e n t c o r e , and
above 350 m. Fig.8. As i n Fig.4, f o r t h e experiment i n i t i a l i z e d w i t h analyzed STACS o b s e r v a t i o n s .
The maximum
i n p e r t u r b a t i o n energy i n t h e o b s e r v a t i o n s i s about t w i c e t h a t i n o u r f i e l d and is
slightly
below
the
165
surface,
whereas o u r s i s a t t h e u p p e r
surface.
The f a c t o r o f 2 i s p r o b a b l y
accounted f o r i n t h a t t h e a n a l y s i s o f observations i n c l u d e s f l u c t u a t i o n s on a l l time and s p a t i a l
scales,
and our model
f i e l d i s f o r a s i n g l e component.
Both
i l l u s t r a t i o n s suggest f l u c t u a t i o n energy near t h e eastern model.
slope,
sections ( n o t shown) the
especially
the
current
shifting
in
back
shows t h e a x i s of
the and
lower forth
two
layers
several
grid
A
p o i n t s along t h e meander wavelength. bottom
28.7 DAYS
I n s p e c t i o n o f instantaneous c r o s s
trapping
suggested,
of
and
wave
energy
Rhines
(1970)
is has
suggested t h a t c o n d i t i o n s a r e f a v o r a b l e i n t h e F l o r i d a S t r a i t s f o r generation of bottom trapped waves.
The amplitude of
the f e a t u r e ,
however,
along t h e e a s t e r n
slope
475-350
m
from
depth
may
Fig.9. Instantaneous upper 40 m mean f l o w p a t t e r n and d e n s i t y f o r experiment in i t i a1 i z e d w i t h anal y z e d STACS observations. Arrows D e n s i t y i s conare as i n Fig.3. t o u r e d i n i n t e r v a l s o f .05 uT.
be
exaggerated from t h e numerical s t r a i n of dealing
with
such
a
steeply
sloping
bottom. N o t a b l y a b s e n t f r o m t h e model reaching across t h e s t r a i t s .
PKE d i a g r a m i s t h e s u r f a c e l a y e r t o n g u e
This i s most l i k e l y due t o t h e l a c k o f seasonal
and l o c a l wind stress-induced f l u c u t a t i o n s , which are, o f course, present i n t h e analyzed o b s e r v a t i o n s and l i k e l y i n t r o d u c e t h i s degree o f v a r i a b i l i t y , although t h e l a t t e r i s much more apparent i n t h e p e r t u r b a t i o n p o t e n t i a l than k i n e t i c energy (Leaman, e t a1 , 1987). 6 SUMMARY
An i s o p y c n i c c o o r d i n a t e numerical model has been c o n f i g u r e d i n a channel w i t h t h e F l o r i d a S t r a i t s bottom topography a t 27'N transport tluctuations
f o r study o f F l o r i d a Current 1 )
w i t h p e r i o d s o f a few days t o several weeks,
meanders which may r e s u l t from weak i n s t a b i l i t i e s c o n d i t i o n s a r e developed, and d e n s i t y ,
first,
i n the current.
and 2) Initial
u s i n g 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 pressure
which i n c o r p o r a t e s b u l k parameters from STACS observations,
and
which i s p u t t h r o u g h a dynamical i n i t i a l i z a t i o n , and, second, u s i n g a c t u a l g r i d p o i n t analyzed d a t a from STACS. Using t h e s i n g l e c r o s s - s e c t i o n from t h e f i r s t s e t o f i n i t i a l c o n d i t i o n s , t h e current
is
subjected
to
horizontally
uniform,
but
temporally fluctuating,
166 a l o n g s t r e a m wind s t r e s s . T h i s evokes a b a r o t r o p i c t r a n s p o r t response which, with
weak
varies
friction,
linearly
with
f o r c i n g p e r i o d and l a g s 90'
i n phase b e h i n d t h e
As f r i c t i o n i s
forcing. increased,
the
response
exhibits
a
noticeable
decrease
from
linear
proportionality forcing
with
period,
but
it
v a r i e s o n l y s l i g h t l y from t h e 90'
phase l a g .
This
l a s t result d i f f e r s with the
roughly
one
day
observed phase l a g (Lee, et
al,
1985), and
the
d i s c r e p a n c y i s l i k e l y due to
the
two-dimensional
structure
of
compared
with
time
of
the
fluid
in
Straits
versus
temporal
and
scale forcing
.
of
When
model
residence the the spatial
the
wind
the
cross-
sectional
current
is
duplicated
16
in
times
t h e downstream d i r e c t i o n t o form a channel, DISTANCE (km)
(b) Fig.10. (a)The a l o n g s t r e a m mean o f p e r t u r b a t i o n KE (computed as t h e d i f f e r e n c e between t h e t o t a l KE and KM) a t t h e t i m e when t h i s parameter has i t s b a s i n averaged maximum. (b)Mean p e r t u r b a t i o n KE f r o m t h e f u l l spectrum o f p e r t u r b a t i o n s measured w i t h t h e PEGASUS p r o f i l e r s d u r i n g t h e STACS e x p e r i m e n t i n t h e F l o r i d a S t r a i t s ( a f t e r Leaman, e t a l . , 1987).
flow
is
found
b a r o c l in i c a l l y to
to
the be
unstable
perturbations
of
g r e a t e r t h a n about 60 km wavelength.
I n t h e range
6 0 t o 256 km, t h e m o s t unstable
wavelength
is
167 208 km.
The i n s t a b i l i t y i s manifested n o t a b l y i n development o f a c y c l o n i c eddy
t o t h e west o f t h e c u r r e n t core.
The wave a m p l i f i c a t i o n ,
however, i s o f l i m i t e d
h o r i z o n t a l and temporal s c a l e and, when complete, t h e mean k i n e t i c and p o t e n t i a l energies a r e l a r g e l y unchanged. Likewise, (Leaman,
t h e c u r r e n t developed from t h e a n a l y s i s o f STACS PEGASUS d a t a
et
al,
1987)
exhibits
release
of
baroclinic
p e r t u r b a t i o n s w i t h wavelengths g r e a t e r than 60 kin.
instability
for
I n t h i s case, t h e maximum
p e r t u r b a t i o n energy i s reached f o r a wavelength o f 150 km and p e r i o d about 5 days,
s i m i l a r t o one o f two s i g n i f i c a n t meandering modes d e t e c t e d by Johns and
Schott
(1987) i n STACS moored c u r r e n t meter data.
p e r t u r b a t i o n k i n e t i c energy f o r t h i s meander i n t h e x
The alongstream mean o f
-
z plane i s found t o bear
t h e same p a t t e r n s as t h a t obtained f o r t h e PEGASUS data, observed f l u c t u a t i o n s
--
that is,
i n c l u d i n g a l l of t h e
except f o r those f e a t u r e s l i k e l y associated
w i t h seasonal and l o c a l wind event-induced f l u c t u a t i o n s .
F i n a l l y , t h e meander
growth i s again o f l i m i t e d h o r i z o n t a l and temporal s c a l e and, when complete, t h e mean k i n e t i c and p o t e n t i a l energies have changed l i t t l e . These r e s u l t s , a l b e i t w i t h a much s i m p l i f i e d model, suggest t h a t t h e c u r r e n t flowing
through
instability,
the
Florida
Straits
is
at
the
threshold
of
baroclinic
and t h a t occasional conversion o f energy from p o t e n t i a l t o eddy
k i n e t i c may account f o r t h e meandering behavior o f t h e F l o r i d a Current.
The
i n s t a b i l i t y i s a p p a r e n t l y n o t s t r o n g enough t o cause f o r m a t i o n o f a l a r g e eddy i n the Straits.
But i t may, f o r example, l e a d t o b r i e f s u r f a c e f l o w r e v e r s a l s
along t h e western edge i n t h e c y c l o n i c a l l y curved p o r t i o n o f t h e meander. ACKNOWLEDGEMENTS We wish t o acknowledge h e l p f u l d i s c u s s i o n s w i t h Drs. Tom Lee, B i l l Johns, and Kevin Leaman, and t o thank Leaman and Peter Vertes f o r making t h e i r a n a l y s i s o f PEGASUS d a t a a v a i l a b l e . This work has been supported by NSF g r a n t s OCE 83-11510 C a l c u l a t i o n s were performed on t h e Cray-1 computers a t t h e and OCE 86-00593. National Center f o r Atmospheric Research, which i s sponsored by t h e National Science Foundation. 6 REFERENCES and Boudra, D.B. 1981. I n i t i a l t e s t i n g o f a n u m e r i c a l ocean B l e c k , R., J. c i r c u l a t i o n model u s i n g a h y b r i d ( q u a s i - i s o p y c n i c ) v e r t i c a l coordinate. Phys. Oceanogr., 11: 755-770. B l e c k , R., 1985. On t h e c o n v e r s i o n between mean and eddy components o f p o t e n t i a l and k i n e t i c energy i n i s e n t r o p i c and i s o p y c n i c coordinates. Dyn. Atmos. and Oceans, 9: 17-37. Bleck, R. and Boudra, D., 1986. Wind-Driven Spin-up i n eddy-resolving ocean J. Geophys. Res. , models formulated i n i s o p y c n i c and i s o b a r i c coordinates. 91(C6): 7611-7621. B l e c k , R., Onken, R., and Woods, J.D., 1987. A t w o - d i m e n s i o n a l model of mesoscale f r o n t o g e n e s i s i n t h e ocean. Submitted t o Quart. J. Roy. Met. SOC.
168 Boris, J.P., and Book, D.L., 1973. F l u x - c o r r e c t e d t r a n s p o r t . I. SHASTA. A f l u i d t r a n s p o r t a l g o r i t h m t h a t works. J. Comput. Phys., 11: 38-69. 1977. Free, s t a b l e c o n t i n e n t a l s h e l f waves i n Brooks, D.A. and Mooers, C.N.K., J. Phys. Oceanogr., 7: 380-388. a sheared b a r o t r o p i c boundary c u r r e n t . Brooks, I.H., 1979. F l u c t u a t i o n s i n t h e t r a n s p o r t o f t h e F l o r i d a Current a t p e r i o d s between t i d a l and two weeks. J. Phys. Oceanogr., 9: 1048-1053. A numerical method f o r t h e study o f t h e c i r c u l a t i o n o f the Bryan, K., 1969. World Ocean. J. Comput. Phys., 4: 347-376. 1975. Some e f f e c t s o f b o t t o m t o p o g r a p h y on b a r o c l i n i c De Soeke, R.A., s t a b i l i t y . J. Mar. Res., 33: 93-122. Duing, W.O., 1975. Synoptic s t u d i e s o f t r a n s i e n t s i n t h e F l o r i d a Current. J. Mar. Res., 33: 53-73. Mooers, C.N.K., and Lee, T.N., 1977. Low-frequency v a r i a b i l i t y i n Duing, W.O., t h e F l o r i d a Current and r e l a t i o n s t o atmospheric f o r c i n g from 1972 t o 1974. J. Mar. Res., 35: 129-161. Hoskins, B.J. and Bretherton, F.P. , 1972. Atmospheric f r o n t o g e n e s i s models: Mathematical f o r m u l a t i o n and s o l u t i o n . J. Atmos. Sci 29: 11-37. Johns, W.E. and Schott, F., 1987. Meandering and t r a n s p o r t v a r i a t i o n s o f t h e F l o r i d a Current. J. Phys. Oceanogr. I n press. Kamenkovitch, V.M., 1962. Trudy I n s t i t u t a Okeanologii, Akad. Nauk SSSR, 56: 241. M o l i n a r i , R.L.d Vertes, P.S., 1987. S t r u c t u r e and v a r i a b i l i t y o f Leaman, K.D., t h e F l o r i d a Current a t 27 N: A p r i l 1982 J u l y 1984. J. Phys. Oceanogr. In press. Lee, T.N., Schott, F., and Zantopp, R., 1985. F l o r i d a Current Low-frequency v a r i a b i l i t y as observed w i t h moored c u r r e n t meters d u r i n g A p r i l 1982 t o June 1983. Science, 227: 298-301. N i i l e r , P.P. and Mysak, L.A., 1971. B a r o t r o p i c waves a l o n g an e a s t e r n c o n t i n e n t a l s h e l f . Geophys. F l u i d Dyn., 2: 273-288. N i i l e r , P.P. and Richardson, W.S., 1973. Seasonal v a r i a b i l i t y o f the F l o r i d a Current. J. Mar. Res.. 31: 144-167. Orlanski, I.,1969. The i n f l u e n c e o f bottom topography on t h e s t a b i i t y o f j e t s i n a b a r o c l i n i c f l u i d . J. Atmos. Sci., 26: 1216-1232. Rhines, P.B., 1970. Edge-, bottom, and Rossby waves i n a r o t a t i n g s t r a t i f i e d f l u i d . Geophys. F l u i d Dyn., 1: 273-302. Schmitz, W.J., Jr., and Richardson, W.S., 1968. On t h e t r a n s p o r t o f t h e F l o r i d a Current. Deep-sea Research, 15: 679-693. Schott, F. and Diiing, W.O., 1976. C o n t i n e n t a l s h e l f waves i n t h e F l o r i d a S t r a i t s . J. Phys. Oceanogr., 6: 451-460. Schott, F. and Lee, T.N., 1986. V a r i a b i l i t y o f s t r u c t u r e and t r a n s p o r t o f the F l o r i d a Current i n t h e p e r i o d range o f days t o seasonal. I n Preparation. and Wimbush, M., 1977. Simulataneous pressure, v e l o c i t y , and Wunsch, C., temperature measurements i n t h e F l o r i d a S t r a i t s . J. Mar. Res., 35: 75-104. Zalesak, S.T., 1979. F u l l y dimensional f l u x - c o r r e c t e d t r a n s p o r t a l g o r i t h m f o r f l u i d s . J. Comput. Phys., 31: 335-362.
.,
-
169
DYNAMICS OF AGULHAS RETROFLECTION AND RING FORMATION I N A QUASI-ISOPYCNIC COORDINATE NUMERICAL MODEL
E.P. CHASSIGNET AND D.B. BOUDRA Rosenstiel School o f Marine and Atmospheric Science, U n i v e r s i t y of Miami , 4600 Rickenbacker Causeway, Miami FL 33149-1098, (USA)
ABSTRACT The Agulhas r e t r o f l e c t i o n r e g i o n o f t h e i d e a l i z e d South A t l a n t i c - I n d i a n ocean model described by De R u i j t e r and Boudra (1985) and Boudra and de R u i j t e r (1986) I S analyzed i n d e t a i l . F i r s t , t h e p h y s i c a l mechanism o f t h e model r e t r o f l e c t i o n i s presented through i l l u s t r a t i o n o f t h e Agulhas' v o r t i c i t y balance among various experiments, and second, t h e r i n g f o r m a t i o n process i s described i n terms o f i t s v e r t i c a l s t r u c t u r e and t h e associated energy conversions. A n a l y s i s o f t h e v o r t i c i t y b a l a n c e i n a o n e - l a y e r model shows t h a t b o t h i n e r t i a and i n t e r n a l f r i c t i o n may account f o r a p a r t i a l r e t r o f l e c t i o n where a l i n e a r , w e a k l y v i s c o u s s y s t e m has none. I n a s e r i e s o f one-, two-, and it i s shown t h a t three-layer nonlinear, weakly viscous experiments, r e t r o f l e c t i o n i s accomplished t h r o u g h a f r e e i n e r t i a l boundary l a y e r , as suggested o r i g i n a l l y by De R u i j t e r (1982). and, furthermore, t h a t i t i s t h e p l a n e t a r y v o r t i c i t y advection, r a t h e r than t h e i n e r t i a l overshooting distance, H a l v i n g h o r i z o n t a l g r i d spacing ( f r o m 40 t o 20 which i s of g r e a t e s t importance. km) i s shown t o have a m i n o r i m p a c t on t h e v o r t i c i t y b a l a n c e and t h e r e t r o f l e c t i o n strength. The importance o f a s u b s t a n t i a l viscous s t r e s s c u r l along t h e coast o f A f r i c a , as p r o v i d e d by t h e n o - s l i p c o n d i t i o n , i s i l l u s t r a t e d through comparison w i t h a s l i p p e r y A f r i c a experiment. Because o f t h e apparent importance o f t h e p l a n e t a r y v o r t i c i t y advection i n t h e r e t r o f l e c t i o n regime, an experiment w i t h a more r e a l i s t i c South A f r i c a n c o a s t a l geometry i s described. It i s shown t h a t t h e r e t r o f l e c t i o n i s s t i l l s t r o n g b u t t h a t t h e associated r e c i r c u l a t i o n i s l e s s intense. The primary terms o f t h e v o r t i c i t y equation have smaller magnitude b u t t h e y e x h i b i t t h e same b a s i c balance as i n t h e n o n l i n e a r , weakly viscous, r e c t a n g u l a r A f r i c a experiments: between p l a n e t a r y v o r t i c i t y advection and viscous s t r e s s c u r l a l o n g t h e coast, and between p l a n e t a r y and r e l a t i v e v o r t i c i t y advection on t h e seaward s i d e o f t h e coastal Agulhas and elsewhere throughout t h e r e t r o f l e c t i o n . The mean e n e r g e t i c s o f t h e e x p e r i m e n t s a r e examined i n o r d e r t o g a i n a d d i t i o n a l understanding o f t h e model r e t r o f l e c t i o n . Also, t h e signatures o f b a r o t r o p i c and b a r o c l i n i c i n s t a b i l i t y i n t h e r i n g f o r m a t i o n process f o r t h r e e experiments a r e s t u d i e d i n d e t a i l u s i n g eddy-mean energetics. Ring formation i s accompanied by development o f a pronounced c y c l o n i c c i r c u l a t i o n i n t h e lower layer. However, 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 conversions reach a maximum a t t h e moment o f r i n g c u t o f f . Therefore, a mixed i n s t a b i l i t y i s suggested.
170 1 INTRODUCTION One o f t h e most i n t r i g u i n g f e a t u r e s i n w o r l d ocean c i r c u l a t i o n i s t h e r e t r o f l e c t i o n o f t h e Agulhas Current south o f A f r i c a , and an important phenomena associated w i t h i t i s r i n g f o r m a t i o n a t i t s western edge.
The Agulhas con-
s t i t u t e s t h e p r i m a r y boundary c u r r e n t of t h e southwestern I n d i a n Ocean, and as i t separates from t h e southern t i p o f A f r i c a
i n a s o u t h e r l y o r southwesterly
t h e major p a r t o f i t r e t r o f l e c t s , r e t u r n i n g eastward t o r e j o i n t h e
direction, subtropical
gyre.
Several times a y e a r ,
i n retroflecting,
the current cuts
i t s e l f o f f and a r i n g o f warm, s a l t y I n d i a n Ocean water i s formed.
These r i n g s
d r i f t westward and, along w i t h o t h e r leakage of Agulhas water around t h e t i p o f A f r i c a , l i k e l y have an important impact on t h e heat and s a l t budget o f t h e South At1 a n t i c . I n t h i s paper, we present some b a s i c dynamical ideas and numerical r e s u l t s concerning t h e Agulhas r e t r o f l e c t i o n and t h e r i n g formation. t o p i c s i n c l u d e , 1 ) t h e r e t r o f l e c t i o n v o r t i c i t y balance,
The dynamical
2) t h e mean e n e r g e t i c s
3) t h e n a t u r e o f t h e i n s t a b i l i t y associated w i t h r i n g formation.
and
2 DESCRIPTION OF THE MODEL AND THE EXPERIMENTS The n u m e r i c a l models used a r e d e s c r i b e d i n De R u i j t e r and Boudra (1985) and Boudra and de R u i j t e r (1986)(henceforth r e f e r r e d t o as DB and BD, r e s p e c t i v e l y ) . The authors i n c l u d e d temporal v a r i a b i l i t y , f r i c t i o n , i n e r t i a and t h e b e t a - e f f e c t i n a study o t t h e South A t l a n t i c - I n d i a n one-layer
simplification
of,
and
then
numerical model o f Bleck and Boudra (1981).
-2
nes cm-* r t
1000km
Ocean c i r c u l a t i o n , u s i n g f i r s t a
the
full
-
quasi-isopycnic
coordinate
Perhaps o v e r l y i d e a l i z e d aspects o f
1440 k
+I-AFRICA 0
INDIAN OCEAN
Fig.1. Geometry o f t h e f l a t b o t t o m ocean b a s i n used i n a l l t h e n u m e r i c a l experiments, except E l l . The wind s t r e s s p r o f i l e i s shown a t t h e l e f t .
171 t h e i r model were t h e s i m p l e r e c t a n g u l a r shape f o r A f r i c a and t h e s m a l l t o t a l I n DB and BD, t h e p r i m a r y f o c u s was on parameters a f f e c t i n g t h e
basin size.
It was surmised t h a t t h e n o - s l i p
exchange o f f l u i d between t h e two b a s i n s .
A f r i c a n c o a s t and t h e magnitude o f t h e p l a n e t a r y v o r t i c i t y a d v e c t i o n a t separation
figured
prominently
in
the
retroflection
vorticity
I n f l u e n c e o f s e v e r a l o t h e r model parameters among t h e i r experiments, i n t e r n a l 1a t e r a l f r i c t i o n ,
balance. including
horizontal resolution , s t r a t i f i c a t i o n , incorporation
o f bottom drag, and i n t e r a c t i o n w i t h t h e o t h e r w i n d - d r i v e n c u r r e n t s , were a l s o examined. The p a r a m e t e r s f o r t h e experiments d i s c u s s e d i n t h i s s t u d y a r e presented i n Table 1.
Many o f t h e s e a r e e x p e r i m e n t s d i s c u s s e d i n DB and BD, b u t s e v e r a l new
TABLE 1 Parameters f o r t h e experiments d i s c u s s e d i n t h i s study. Blanks i n d i c a t e no change from t h e p r e v i o u s experiment. El
E2
1
2
3
1000
1000 4000
600 700 3700
.02
.02 .005
Experiment
L I N l NLINl
LINE
NLIN2
Nonlinear terms
NO
NO
YES
Number o f 1a y e r s Thickness o f the layers (m)
YES
.005 Viscosity
330
3300
E3
E8
E9
El0
FS
NS
Ell
300 900 3800
330
(m2s-l) Bottom d r a g c o e f f . (s- ) Bounda ry condition? on A f r i c a
NS
Horizontal resolution (km)
40
Africa's geometry
A
'NS=No-slip;
FS=Free-slip
20
B 'A=Rectangular
( F i g . 1); B=More r e a l i s t i c (Fig.5)
172 ones have been i n t r o d u c e d t o c l a r i f y a s p e c t s o f t h e v o r t i c i t y b a l a n c e and one t o give the Beta-effect
more r e a l i s t i c i m p o r t a n c e .
The b a s i n u s e d i n t h e
experiments i s d e s c r i b e d i n DB and BD, b u t i s i l l u s t r a t e d h e r e f o r convenience (Fig.1).
The steady wind s t r e s s p r o f i l e i s shown a t t h e l e f t o f t h e f i g u r e ,
The d i r e c t e f f e c t o f t h e wind s t r e s s decreases l i n e a r l y t r o m i t s maximum near t h e s u r f a c e t o z e r o a t a p r e s s u r e o f 100 db.
I n t h e m u l t i l a y e r cases, t h e wind
f o r c i n g i s t h e r e f o r e p a r t i t i o n e d among t h e t o p two l a y e r s i f t h e upper l a y e r t h i c k n e s s becomes l e s s t h a n 100 db.
The c o n d i t i o n s on m e r i d i o n a l boundaries a r e
g e n e r a l l y n o - s l i p (u=v=O) and t h o s e f o r zonal b o u n d a r i e s f r e e - s l i p
(v=du/dy=O),
except i n one experiment where t h e c o a s t o f A f r i c a i s t r e a t e d homogeneously as a f r e e - s l i p boundary.
The c h a r a c t e r o f r e t r o f l e c t i o n and r i n g f o r m a t i o n i n t h e
experiments i s summarized i n Table 2. TABLE 2 D e s c r i p t i o n o f t h e upper l a y e r f l o w p a t t e r n o f t h e model r e t r o f l e c t i o n r e g i o n LINl
Small r e c i r c u l a t i o n c e l l east o f A f r i c a . into the Atlantic.
Most o f t h e f l o w ( 3 5 Sv) goes
NLINl
S u b s t a n t i a l r e t r o f l e c t i o n e x t e n d i n g southwest o f t h e t i p o f A f r i c a . Approx. 15 Sv leakage i n t o t h e A t l a n t i c . High t e m p o r a l v a r i a b i l i t y .
LIN2
E s t a b l i s h m e n t o f a r e t r o f l e c t i o n b u t w i t h 20-25 Sv leakage.
NLIN2
S i m i l a r t o LINE. The r e t r o f l e c t i o n i s moved s o u t h and west o f t h e t i p o f A f r i c a . Steady s o l u t i o n .
El
S i m i l a r t o NLIN1, b u t w i t h 25 Sv leakage and l e s s v a r i a b i l i t y . formed c o n t i n u o u s l y a t a r a t e o f about 5 p e r y e a r .
E2
S i m i l a r t o E l , b u t w i t h 15 Sv leakage and t h e p o s i t i o n o f t h e r e t r o f l e c t i o n closer t o the t i p o f Africa. The r a t e o f r i n g f o r m a t i o n i s about 4 p e r y e a r .
E3
The c u r r e n t r e t r o f l e c t s v e r y soon a f t e r i t s e p a r a t e s f r o m t h e coast. S m a l l l e a k a g e ( l e s s t h a n 5 Sv). High eddy energy i n t h e r e t r o f l e c t i o n . No r i n g s .
E8
S i m i l a r t o E3, b u t t h e c i r c u l a t i o n i s more i n t e n s e .
E9
Most o f t h e f l o w t u r n s westward and t h e n n o r t h w a r d a l o n g t h e c o a s t i n t o the Atlantic.
El0
S i m i l a r t o E3, b u t w i t h l e s s i n t e n s e r e t r o f l e c t i o n and a s t r o n g downstream meander. Only 4 r i n g s a r e formed i n 10 y e a r s , each o f which requires a strong i n t e r a c t i o n w i t h t h e subpolar f r o n t .
Ell
The b o u n d a r y c u r r e n t i s b r o a d e r w i t h s m a l l e r v e l o c i t i e s t h a n E10. The r e t r o f l e c t i o n i s s t i l l s t r o n g , b u t i s l e s s i n t e n s e and t h e r e i s a 5-10 Sv leakage. About 3 r i n g s a r e formed p e r y e a r , b u t n o t i n such a r e g u l a r f a s h i o n as i n E l .
Rings a r e
No r i n g s .
173 3 THE V O R T I C I T Y BALANCE 3.1 Method o f a n a l y s i s T a k i n g an averaye o v e r t i m e ,
assuming s t a t i s t i c a l
equilibrium,
the f u l l
v o r t i c i t y e q u a t i o n can be w r i t t e n
3 * W S RVA
i:
(vs
*
x
L) a3 as
+ k’
-
(vs=
VERT
x
VSP)
SOL
=o WIND
VISC
where s i s t h e g e n e r a l i z e d v e r t i c a l c o o r d i n a t e , d e n s i t y here;
t, i s t h e r e l a t i v e
v o r t i c i t y e v a l u a t e d a t s=constant; c u r l z i s t h e v e r t i c a l component o f t h e c u r l ; A i s t h e c o n s t a n t h o r i z o n t a l eddy v i s c o s i t y and t h e o t h e r symbols a r e conventional.
(RVA)
i s t h e r e l a t i v e v o r t i c i t y advection,
v o r t i c i t y advection,
(STRCH) t h e s t r e t c h i n g term,
c u r l , (WIND) t h e wind s t r e s s c u r l , o f the vertical
advection.
(BETA) t h e p l a n e t a r y
(VISC) t h e v i s c o u s s t r e s s
(SOL) t h e s o l e n o i d a l t e r m and (VERT) t h e c u r l
The a c t u a l f i n i t e d i f f e r e n c e f o r m u l a t i o n of t h e
v o r t i c i t y e q u a t i o n w h i c h we u s e i n c o m p u t a t i o n o f t h e s e t e r m s c o n s e r v e s p o t e n t i a l v o r t i c i t y and p o t e n t i a l e n s t r o p h y ( s e e Bleck,
1981).
1979; B l e c k and Boudra,
F i n d i n g t h e mean v a l u e s o f t h e terms f o r t h e f i n a l f i v e y e a r s o f each
experiment,
a b a l a n c e was o b t a i n e d .
VERT a r e z e r o o r n e g l i g i b l e . coordinates.
Our computations have shown t h a t SOL and
T h i s can be expected f r o m t h e use o f i s o p y c n a l
I n a d d i t i o n , STRCH i s equal t o z e r o i n t h e one l a y e r case and has
v e r y small temporal mean v a l u e s i n t h e m u l t i l a y e r case. addressed i n o u r a n a l y s i s .
It, t h e r e f o r e ,
i s not
174 3.2 The v o r t i c i t y balance i n a one-layer model It i s i n s t r u c t i v e t o begin our d i s c u s s i o n o f t h e model v o r t i c i t y balance w i t h
a weakly
viscous,
l i n e a r experiment.
equations o f motion,
Removing t h e
inertial
terms
from t h e
t h e v o r t i c i t y equation i s s i m p l i f i e d and t h e balance i s
among BETA, V I S C and WIND.
Our steady s o l u t i o n L I N l ( n o t i l l u s t r a t e d here) has
q u a l i t a t i v e l y t h e same c h a r a c t e r i s t i c s as t h e a n a l y t i c a l s o l u t i o n o f De R u i j t e r (1982).
Along t h e n o - s l i p coast o f A f r i c a i n t h e Munk l a y e r , l a t e r a l f r i c t i o n
balances t h e
@-induced i n p u t o f
relative vorticity.
l a t t e r i s d e t e r m i n e d by t h e s o u t h w a r d v e l o c i t y ,
The magnitude o f t h e
which i s governed b y t h e
U
BETA Fig.2.
Experiment
LINP.
(a)
The mean mass t r a n s p o r t streamfunction.
6 3 1 contour i n t e r v a l i s 5 X 10 m svorticity
VISC
.
( b ) The t i m e mean d i s t r i b u t i o n o f p l a n e t a r y
a d v e c t i o n (BETA) around t h e southern t i p o f A f r i c a .
i n t e r v a l i s 2 x 10-12s'2.
The
The contour
( c ) As i n ( b ) , except viscous s t r e s s c u r l ( V I S C ) .
175 combination o f i n t e g r a t e d wind c u r l and l a y e r depth, t h e low v i s c o s i t y case ( L I N l ) ,
f r i c t i o n becomes v e r y s m a l l a f t e r s e p a r a t i o n , planetary v o r t i c i t y
as w e l l as f r i c t i o n .
In
t h e b o u n d a r y l a y e r i s r e l a t i v e l y t h i n and
advection.
The o n l y way
and t h e r e f o r e , so d o e s t h e for the flow t o satisfy t h i s
c o n d i t i o n a t t h e s o u t h e r n t i p o f A f r i c a i s t o innnediately t u r n westward o r e a s t ward.
The wind c u r l maximum i s o n l y 120 km n o r t h o f t h e l a t i t u d e o f A f r i c a ' s
tip.
Therefore, t h e r e i s v e r y l i t t l e l i n e a r r e t u r n f l o w t o w a r d t h e I n d i a n Ocean
interior north o f that latitude.
Thus, most o f t h e f l o w branches westward i n t o
the Atlantic. I f t h e v i s c o s i t y i s i n c r e a s e d b y an o r d e r o f m a g n i t u d e ( L I N 2 ) ( F i g . 2 ) ,
f r i c t i o n becomes i m p o r t a n t i n t h e a r e a s o u t h o f A f r i c a . (Fig.2~).
The c u r r e n t c a n
can be balanced by V I S C
c o n t i n u e southward f o r some d i s t a n c e s i n c e BETA (Fig.2b)
I n o t h e r terms, t h e f r i c t i o n a l boundary l a y e r i s broadened enough SO
t h a t more s t r e a m l i n e s o f t h e s e p a r a t i n g A g u l h a s a r e a b l e t o c o n n e c t w i t h returning
Sverdrup
retroflection.
lines
to
the
south
and,
thus,
establish
a
partial
Important i n e r t i a l e f f e c t s are, therefore, n o t required i n order
t o o b t a i n r e t r o f l e c t i o n ; so t h a t t h e l a t t e r may be o b t a i n e d i n h i g h l y d i f f u s i v e , coarse-resolution
models
primarily
because o f
the
large
lateral
viscosity
r e q u i r e d b y t h o s e model s. I n t r o d u c i n g t h e i n e r t i a l terms adds a new t e r m t o t h e v o r t i c i t y balance, t h e r e l a t i v e v o r t i c i t y a d v e c t i o n RVA.
Experiments NLINl (Fig.3)
and NLIN2 ( n o t
i l l u s t r a t e d h e r e ) correspond r e s p e c t i v e l y t o t h e n o n l i n e a r r u n s o f L I N l and LIN2.
I n t h e h i g h v i s c o s i t y case (NLINE), most o f t h e r e t r o f l e c t i o n i s moved
south and west o f t h e s e p a r a t i o n p o i n t , b u t t h e g e n e r a l c h a r a c t e r i s t i c s o f t h e f l o w p a t t e r n a r e unchanged.
I n c l u s i o n o f t h e i n e r t i a l terms r e s u l t s i n an
increase i n t h e d i f f u s i o n o f t h e negative r e l a t i v e v o r t i c i t y . the t o t a l frictional,
velocity
i s a b i t s t r o n g e r and t h e boundary l a y e r ,
i s thinner.
T h i s i s because now i n e r t i a l -
BETA does n o t i n c r e a s e , however, s i n c e t h e o r i e n t a t i o n
of t h e r e t r o f l e c t i o n g a i n s a SW-NE t i l t . The i n c r e a s e i n V I S C i s balanced by t h e new t e r m RVA.
The m a j o r conceptual d i f f e r e n c e from LIN2 i s t h a t t h e f l u i d may
now f l o w southward f r o m t h e t i p o f A f r i c a due t o i t s own i n e r t i a .
Frictional
e f f e c t s are, however, s t i l l dominant. I n t h e c o r r e s p o n d i n g l o w v i s c o s i t y case ( N L I N l ) ( F i g . 3 ) layer i s relatively thin,
where t h e boundary
f r i c t i o n i s u n i m p o r t a n t a f t e r s e p a r a t i o n and t h e
e f f e c t o f i n e r t i a i s predominant.
A s u b s t a n t i a l r e t r o f l e c t i o n i s s t i l l obtained
as RVA now becomes t h e p r i m a r y t e r m t o b a l a n c e BETA.
Inclusion o f the i n e r t i a l
terms i n t h e weakly v i s c o u s case s u b s t a n t i a l l y m o d i f i e s t h e f l o w p a t t e r n and t h e
176
Fig.3. E x p e r i m e n t NLIN1. AS i n F i g . 2 f o r ( a ) and ( b ) . ( c ) Time mean d i s t r i b u t i o n o f r e l a t i v e v o r t i c i t y advection (RVA). The contour i n t e r v a l i s 2 x
( d ) As i n F i g . 2 ~ .
10-12s-2.
d i s t r i b u t i o n o f terms o f t h e r e t r o f l e c t i o n v o r t i c i t y balance.
The c i r c u l a t i o n
east o f A f r i c a i s southward i n t e n s i f i e d due t o i n e r t i a , and weaker along t h e coast i t s e l f . I n each o f LIN2, NLINl and NLIN2, t h e r e i s a s u b s t a n t i a l r e t r o f l e c t i o n . similarity
among a l l
t h r e e i s t h e establishment of
separation i n which t h e beta e f f e c t major p o r t i o n o f t h e c u r r e n t .
a boundary l a y e r a f t e r
i s balanced by t h e t u r n i n g eastward o f a
I n LINE t h e boundary l a y e r i s p u r e l y f r i c t i o n a l ,
i n NLINE i t i s i n e r t i a l - f r i c t i o n a l , layer.
The
and i n NLINl i t i s a f r e e i n e r t i a l boundary
While t h e p h y s i c a l mechanism o f r e t r o f l e c t i o n i s n o t t h e one proposed by
De R u i j t e r (1982)
--
t h e i n e r t i a l overshooting d i s t a n c e
r e t r o f l e c t i o n being accomplished
within
a free
--
inertial
h i s concept o f t h e boundary
layer i s
177 apparently a p p l i c a b l e .
I n t h e experiments remaining t o be discussed,
r e t r o f l e c t i o n i s accomplished i n t h i s f r e e boundary l a y e r .
the
NLINl possesses a
s i m i l a r d i s t r i b u t i o n o f t h e terms as t h e weakly n o n l i n e a r m u l t i l a y e r cases and
w i l l be discussed f u r t h e r i n p a r a l l e l w i t h them. 3.3 The v o r t i c i t y balance i n t h e m u l t i l a y e r model As described i n BD, i n experiments E l , E2 and E3,
( i ) Variation o f inertia.
t h e s t r e n g t h o f t h e r e t r o f l e c t i o n depends on t h e i n e r t i a o f t h e currents, which can be q u a n t i f i e d b y t h e Kossby number (Ro) o f t h e o v e r s h o o t i n g boundary For E l and E2, t h e t o p l a y e r v o r t i c i t y balance i s s i m i l a r t o t h a t f o r
current.
There i s a change i n t h e s p a t i a l d i s t r i b u t i o n o f t h e
NLINl above (Fig.3). terms,
however,
for
E3
(Fig.4).
These
experiments
v a r i a b i l i t y , which i s n o t t h e case i n LIN1, LINE and NLIN2.
have
high
temporal
RVA, t h e r e f o r e , can
be separated i n t o two components, one r e l a t e d t o t h e mean f l o w and t h e o t h e r one t o the transients.
The RVA p a t t e r n i s l a r g e l y dominated by t h e component
associated w i t h t h e mean flow.
However, t h e component due t o t h e t r a n s i e n t s can
have a canceling e f f e c t on t h a t due t o t h e mean flow,
e s p e c i a l l y i n NLINl and
El. Along t h e coast o f A f r i c a and c l o s e t o t h e boundary, important.
viscous e f f e c t s are
Negative r e l a t i v e v o r t i c i t y i s generated by t h e n o - s l i p boundary
c o n d i t i o n and then t r a n s p o r t e d eastward by d i f f u s i o n . Ro does from E l t o E3 and they balance each other.
BETA and VISC increase as Away from t h e boundary, t h e
e f f e c t o f d i f f u s i o n decreases and t h e balance i s m a i n l y between RVA and BETA. J u s t a f t e r t h e f l o w leaves t h e coast, flow i s s t r o n g l y sheared.
V I S C i s s t i l l important because t h e
Before t h e separation,
maximum a t t h e f i r s t g r i d p o i n t e a s t o f t h e coast. such a p r o f i l e has a minimum a t t h i s g r i d p o i n t .
t h e southward v e l o c i t y i s The viscous s t r e s s c u r l o f I f t h e boundary c u r r e n t were
very w e l l resolved, a p o s i t i v e maximum i n V I S C would appear near t h e coast. t h e c u r r e n t overshoots t h e t i p o f A f r i c a , resolved by t h e g r i d l a t t i c e . side o f the free j e t ,
As
t h e c u r r e n t broadens and i s b e t t e r
A p o s i t i v e maximum does appear on t h e western
and t h i s f e a t u r e i s present i n most o f t h e n o n l i n e a r
experiments discussed (NLIN1, NLIN2, E l , E2, E3 and E10). A f t e r separation,
t h e experimental v o r t i c i t y balances d i f f e r .
Changes i n
V I S C leave BETA dominant, l e a d i n g t o generation o f p o s i t i v e r e l a t i v e v o r t i c i t y . From here on RVA balances BETA. For each case, a t l e a s t a p o r t i o n o f t h e c u r r e n t t u r n s eastward i n t o t h e I n d i a n Ocean.
On t h e westernmost s i d e o f t h e c u r r e n t ,
i n e r t i a i s n o t s t r o n g enough t o dominate t h e l i n e a r tendency, and t h i s p a r t o f t h e f l o w d r i f t s westward i n t o t h e A t l a n t i c .
This leakage i s very small i n t h e
h i g h Rossby number cases, since o n l y a very small p a r t o f t h e c u r r e n t ' s western edge has weak motion.
178
d)
........ ....... ‘ I ........ I .:.:.:.:.:.:.:. ..............I I ........ ............... ............... I I ............... ;$;;;;$I I. ....... 2::::::::;:T .............. :::::A: I ..A%+..
;:;:;$
.......
Fig.4.
Experiment E3. As i n Fig.3,
VlSC
’
\I
except t h e contour i n t e r v a l i n 10 X 10- 12 ,-2
i n ( b ) , ( c ) and ( d ) . The
p l a n e t a r y v o r t i c i t y advection
(BETA),
proportional t o the
southward
component o f t h e v e l o c i t y , i s o f primary importance i n c o n t r o l l i n g t h e shape and the strength o f the r e t r o f l e c t i o n .
As t h e Rossby number o f t h e separating
c u r r e n t increases, so does BETA and t h e tendency t o t u r n eastward.
Therefore,
t h i s t u r n moves c l o s e r t o t h e t i p o f A f r i c a as southward i n e r t i a increases, i n c o n t r a d i c t i o n t o t h e i n e r t i a l overshooting concept, and has no d i r e c t connection w i t h t h e d i s t a n c e t o t h e r e t u r n i n g Sverdrup l i n e s . I n c l u s i o n o f bottom drag i n E8 ( n o t i l l u s t r a t e d here) decreases t h e eddyinduced mean c i r c u l a t i o n i n t h e bottom l a y e r , b u t increases t h e i n t e n s i t y o f the upper l a y e r c i r c u l a t i o n , since t h e bottom l a y e r f l o w regime i n E3 i s a primary mechanism f o r d r a i n i n g energy from t h e r e t r o f l e c t i o n area.
I n t h i s experiment,
t h e terms o f t h e model v o r t i c i t y balance a r e s l i g h t l y l a r g e r i n magnitude, b u t t h e i r h o r i z o n t a l d i s t r i b u t i o n remains as i n E3.
179 As p o i n t e d o u t by BD, 40 km g r i d r e s o l u t i o n can-
( i i ) Horizontal resolution.
They performed El0
n o t adequately d e s c r i b e t h e d e t a i l s o f boundary l a y e r f l o w s .
w i t h g r i d spacing h a l v e d i n o r d e r t o t e s t t h e s e n s i t i v i t y o f t h e r e t r o f l e c t i o n to
improved
resolution
of
the
boundary
layer
and
release
of
baroclinic
The r e t r o f l e c t i o n f l o w p a t t e r n o f El0 ( n o t i l l u s t r a t e d h e r e ) i s
instabilty.
s i m i l a r t o t h a t o f E8.
I n E10, t h e b o t t o m l a y e r c i r c u l a t i o n i s r e v i v e d by t h e
improved r e s o l u t i o n o f i n s t a b i l i t i e s and t h e upper l a y e r c i r c u l a t i o n i n t h e retroflection
a r e a i s l e s s i n t e n s e t h a n i n E3 and E8.
There i s a s i m i l a r
h o r i z o n t a l d i s t r i b u t i o n o f t h e terms o f t h e v o r t i c i t y e q u a t i o n i n El0 as i n E3 and E8.
BETA has a maximum near t h e c o a s t as i n E3, b u t V I S C i s even l a r g e r
because t h e maximum v e l o c i t y i s g r e a t e r and c l o s e r t o t h e c o a s t w i t h t h e improved r e s o l u t i o n .
An a d d i t i o n a l c o n t r i b u t i o n from RVA balances t h i s enhanced
VISC.
R e g a r d i n g t h e p r i n c i p a l b a l a n c e t h e same c o n c l u s i o n as i n S e c t i o n 3.3.a be drawn.
can
As t h e c u r r e n t s e p a r a t e s a t t h e t i p , V I S C decreases and t h e r e m a i n i n g
dominant term,
BETA,
generates p o s i t i v e r e l a t i v e v o r t i c i t y .
As i n t h e 40 km
r e s o l u t i o n case, RVA balances BETA as t h e c u r r e n t t u r n s eastward r e j o i n i n g t h e I n d i a n Ocean s u b t r o p i c a l gyre. ( i i i ) Boundary c o n d i t i o n a l o n g t h e A f r i c a n c o a s t .
BD o b t a i n e d a v e r y
d i f f e r e n t two-basin f l o w c o n f i g u r a t i o n when a f r e e - s l i p boundary c o n d i t i o n a l o n g t h e c o a s t o f A f r i c a was s u b s t i t u t e d f o r t h e n o - s l i p c o n d i t i o n .
In the
corresponding n o - s l i p case (E8) o n l y a small amount o f f l u i d escapes i n t o t h e A t l a n t i c , t h e m a j o r p a r t o f t h e c u r r e n t r e t r o f l e c t i n g i n t o t h e I n d i a n Ocean.
In
t h e f r e e - s l i p e x p e r i m e n t , most o f t h e f l u i d goes around t h e southern t i p of Africa i n t o the Atlantic. I n t h e t o p l a y e r v o r t i c i t y b a l a n c e f o r E9 ( n o t i l l u s t r a t e d h e r e ) ,
VISC i s
n e g l i g i b l e i n comparison t o RVA and BETA a l o n g t h e c o a s t and n o r t h o f t h e t i p o f Africa.
The p r i n c i p a l b a l a n c e i s between t h e s e l a s t two terms.
n o - s l i p case, boundary l a y e r
Contrary t o t h e
t h e r e i s no c y c l o n i c v o r t i c i t y p r e s e n t a l o n g t h e coast. i s primarily inertial
boundary l a y e r o f t h e n o - s l i p case.
The
as opposed t o t h e i n e r t i a l - f r i c t i o n a l
I n o r d e r t o o v e r s h o o t t h e t i p (and a f t e r -
ward r e t r o f l e c t ) , t h e c u r r e n t must c o e x i s t w i t h a r e g i o n beyond i t s western edge which i s l a r g e l y m o t i o n l e s s .
I n t h e n o - s l i p case,
t h e w e s t e r n boundary
c o n d i t i o n f o r t h e c u r r e n t a l o n g t h e c o a s t i s a l r e a d y a m o t i o n l e s s one and, t h u s , p r o v i d e d n o n l i n e a r e f f e c t s a r e s t r o n g enough, t h e c u r r e n t s h o o t s p a s t t h e t i p i n a r e l a t i v e l y n a t u r a l fashion.
I n c o n t r a s t , t h e c o n d i t i o n a t t h e western edge of
t h e c o a s t a l c u r r e n t i n E9 r e s u l t s i n t h e c u r r e n t h a v i n g i t s maximum v e l o c i t y there.
I n an o v e r s h o o t i n g c o n f i g u r a t i o n i t would have t o a d j u s t t o a s t r o n g l y
sheared c o n d i t i o n .
180 3.4 South A f r i c a n c o a s t a l geometry The shape and o r i e n t a t i o n o f A f r i c a i n t h e p r e v i o u s e x p e r i m e n t s i s one which maximizes t h e i m p o r t a n c e o f BETA i n t h e model r e t r o f l e c t i o n .
Inasmuch as t h i s
t e r m p l a y s a dominant r o l e , g i v i n g i t a more r e a l i s t i c l e v e l o f importance seems A f t e r l e a v i n g t h e South A f r i c a n c o a s t , t h e r e a l Agulhas f l o w s along
warranted.
t h e Agulhas Bank s h e l f break, w h i c h i s o r i e n t e d a p p r o x i m a t e l y f r o m n o r t h e a s t t o southwest,
before flowing out
i n t o deep water.
I n t h i s configuration,
the
component o f v e l o c i t y a d v e c t i n g p l a n e t a r y v o r t i c i t y i s about 70% o f t h e t o t a l
loo%,
r a t h e r than
as w i t h t h e m e r i d i o n a l l y o r i e n t e d c o a s t o f BD's model.
,
The
b a s i n geometry shown i n Fig.5a has been chosen, t h e r e f o r e , as a n e x t s t e p toward model r e a l ism. The t i m e mean t o p l a y e r f l o w p a t t e r n (Fig.5a)
f o r t h i s 20 km r e s o l u t i o n
experiment ( E l l ) b e a r s many o f t h e same c h a r a c t e r i s t i c s as t h o s e o f E l 0 ( f o r comparison,
see Fig.13
i n BD).
However,
e s p e c i a l l y i n t h e I n d i a n Ocean s e c t o r . l e s s intense.
t h e r e a r e some n o t a b l e d i f f e r e n c e s , The r e t r o f l e c t i o n i s s t i l l s t r o n g , b u t
The mean Agulhas a l o n g t h e c o a s t i s somewhat broader,
and i t s
With t h e t i l t e d c o a s t , t h e energy o f t h e
maximum v e l o c i t y i s l e s s t h a n i n E10.
boundary r e g i o n has more o f a tendency t o l e a k i n t o t h e A t l a n t i c r a t h e r t h a n t o be t r a p p e d i n t h e I n d i a n Ocean. The r e t r o f l e c t i o n v o r t i c i t y b a l a n c e o f E l l (Fig.5)
i s mostly similar t o that
i n t h e h i g h Rossby number r e c t a n g u l a r geometry e x p e r i m e n t s w i t h weak f r i c t i o n . Along t h e c o a s t BETA (Fig.5b)
and V I S C (Fig.5d)
e s s e n t i a l l y b a l a n c e each o t h e r .
Again, however, a t c e r t a i n p o i n t s , V I S C i s p a r t i c u l a r l y s t r o n g , and RVA makes up f o r what BETA does n o t s u p p l y t h e r e .
Over t h e remainder o f t h e r e t r o f l e c t i o n ,
BETA and RVA ( F i g . 5 ~ ) a r e i n balance. geometry magnitude
has no m a j o r of
BETA a t
I n c o r p o r a t i o n o f t h e more r e a l i s t i c
impact on t h e r e t r o f l e c t i o n separation
still
vorticity
balance.
The
r e q u i r e s an e a s t w a r d t u r n and thus,
retroflection. C o n s p i c u o u s l y absent f r o m t h e diagram i s t h e f e a t u r e i n V I S C j u s t beyond and t o t h e west o f t h e t i p , which i s p r e s e n t i n t h e o t h e r m u l t i - l a y e r experiments d i s c u s s e d here.
T h i s i s a p p a r e n t l y because t h e c o a s t a l t u r n a t s e p a r a t i o n i s
n o t n e a r l y so s h a r p as w i t h r e c t a n g u l a r geometry and t h e r e i s much more motion of
f l u i d around
Atlantic.
this
turn,
p r i n c i p a l l y w i t h i n t h e r i n g s d r i f t i n g i n t o the
I n f a c t , t h e r e t r o f l e c t i o n o f t h e Agulhas i s u s u a l l y upstream o f t h e
t i p o f A f r i c a and t h e d e p i c t e d f l o w p a t t e r n o f Fig.5a averages t h r o u g h many r i n g f o r m a t i o n events.
The f i e l d o f RVA e x h i b i t s a l a r g e r c o n t r i b u t i o n f r o m t h e
t r a n s i e n t s t h a n i n E3, E8 and E l 0 because o f t h e much g r e a t e r frequency o f r i n g formation. S e c t i o n 4.
More d e t a i l s c o n c e r n i n g t h e t r a n s i e n t a s p e c t s o f E l l a r e p r o v i d e d i n
181
880 k m
800 k m
840 k m
I
SOUTH AFRICA/
SOUTH AFRICA/
RVA
GULHAS BANK
Fig.5.
Experiment E l l .
the l a s t f i v e years.
( a ) Upper l a y e r mean mass t r a n s p o r t streamfunction f o r The C. I . i s 5 x 106m3s-1;
( b ) , ( c ) and ( d ) as i n Fig.3.
182
4 ENERGETICS AND R I N G FORMATION 4.1 F o r m u l a t i o n o f t h e energy c o n v e r s i o n terms A d e t a i l e d d e r i v a t i o n o f t h e t i m e mean and t r a n s i e n t eddy energy conversion terms i n i s o p y c n i c coordinates
i s g i v e n by B l e c k (1985).
Following his
are as f o l l o w s :
f)
vs .(
p P ) -0 ap “V, as
S
v’dp.(,f.vs+~‘ as
where
I#
i s the geopotential;
p
Lp as
( - ) i s t h e t i m e average;
(-)
i s t h e mass weighted
( ) ’ t h e d e p a r t u r e from t h e mass weighted t i m e average; KM i s t h e
t i m e average;
k i n e t i c energy a s s o c i a t e d w i t h t h e mean f l o w and KE, w i t h t h e t r a n s i e n t eddies; P
is
the
total
available
potential
energy
and
the
other
symbols
are
conventional. B l e c k (1985) showed t h a t i n t h e i s o p y c n i c framework a c o n v e r s i o n t e r m appears i n t h e e n e r g e t i c s s u g g e s t i n g an exchange between t h e eddy p o t e n t i a l energy and .K,
To a v o i d c o n f u s i o n ,
components.
therefore,
P i s n o t decomposed i n t o mean and eddy
It i s assumed t h a t t h e exchange between P and KE i s a s s o c i a t e d w i t h
b a r o c l i n i c c o n v e r s i o n and t h a t t h e t r a n s f e r f r o m KM t o KE
s associated with
b a r o t r o p i c conversion. 4.2 Mean e n e r g e t i c s I n o r d e r t o g a i n some a d d i t i o n a l i n s i g h t i n t o t h e i m a c t o f t h e model parameters
most
energetics
of
influential
on t h e
retroflection,
we now examine t h e mean
some o f t h e e x p e r i m e n t s d i s c u s s e d above,
diagram f o r m i n Fig.6.
illustrated
i n box
The t i m e means o f t h e e n e r g i e s and c o n v e r s i o n s terms a r e
computed over t h e l a s t r i v e y e a r s o f a t e n y e a r r u n f o r E l ,
E2, E3, E8, E Y and
E l l and o v e r s e l e c t e d y e a r s f o r E10. C o n s i d e r i n g t h e i n f l u e n c e o f i n e r t i a on t h e b a s i n averaged e n e r g e t i c s , f i r s t compare E l and E2. thinner
i n E2 than i n E l ,
we
The l a y e r r e c e i v i n g most o f t h e d i r e c t wind f o r c i n g i s leading t o
increased k i n e t i c energy i n t h e boundary
183
P
P
1.7
4.6
7.5
3.4
U
WI
WI t
7 BDD
+
WI
KM
P
1.2
&$
-
.LD
15 LD
LD
1.1 LD
1
el
f)
BDD
WI
KM 1.1
t -
BDD WI
+LD
. LD
P 14.8 . LD
9)
t
BDD
Fig.6. Eddy-mean energetics f o r ( a ) E l ; ( b ) E2; ( c ) E3; (d) E8; (e) E9; ( f ) E10, y e a r 4 ; ( 9 ) E10, y e a r 7 ; ( h ) E l l . W I i s t h e wind i n p u t ; LD, t h e l a t e r a l dissipation and BDD t h e bottom drag dissipation.
184 currents,
even though t h e b a s i n average k i n e t i c energy does n o t change much
(Fig.6aYb). shear.
As t h e t o p l a y e r f l o w i n c r e a s e s i n speed,
From t h e thermal wind r e l a t i o n ,
i s o p y c n a l s u r f a c e s and,
thus,
a v a i l a b l e p o t e n t i a l energy,
t h i s l e a d s t o more s t r o n g l y t i l t e d
t o more energy exchange between KM and P.
however,
opposed t o t h e k i n e t i c energy, c u r r e n t s (Fig.3 o f BD).
so does t h e v e r t i c a l The
i s d i s t r i b u t e d o v e r t h e whole domain, as
which i s c o n c e n t r a t e d m o s t l y i n t h e boundary
I n t h e b a s i n averaged sense, then, t h i s parameter has a
much l a r g e r i m p a c t on t h e p o t e n t i a l
e n e r g y t h a n on t h e k i n e t i c e n e r g y .
H o r i z o n t a l p l o t s show t h a t most o f t h e c o n v e r s i o n f r o m KM t o KE o c c u r s i n t h e r e t r o f l e c t i o n area.
Rings a r e c o n t i n u o u s l y b e i n g formed t h e r e and, t h e r e f o r e ,
t h e i n c r e a s e i n s t r e n g t h o f t h e boundary c u r r e n t s f r o m E l t o E 2 a p p a r e n t l y r e s u l t s i n an i n c r e a s e i n k i n e t i c energy exchange between t h e mean f l o w and t h e eddies i n t h i s h i y h l y t r a n s i e n t r e g i o n . The u p p e r l a y e r i s t h i n n e r y e t i n E3 ( F i g . 6 ~ ) so ~ t h a t i n e r t i a i s even g r e a t e r i n t h e boundary c u r r e n t s , p o t e n t i a l energy f r o m E l .
l e a d i n g t o an i n c r e a s e o f 300% i n a v a i l a b l e
T h i s i n c r e a s e i s accentuated by t h e f a c t t h a t t h e
c i r c u l a t i o n s i n t h e s u b t r o p i c a l I n d i a n and A t l a n t i c oceans a r e almost separated, The boundary c u r r e n t s ' k i n e t i c energy i n c r e a s e s , b u t no r i n g s a r e formed i n t h i s experiment, and so t h e r e i s l e s s c o n v e r s i o n between KM and KE t h a n i n E2, even though t h e r e i s s u b s t a n t i a l l y more KE i n E3. (Fig.6d) 3.3,
I n c l u s i o n o f b o t t o m d r a g i n E8
increases d i s s i p a t i o n i n t h e bottom layer.
As mentioned i n Section
t h i s l e a d s t o a more s t a b l e and i n t e n s e r e t r o f l e c t i o n i n t h e upper l a y e r .
Therefore, t h e r e i s l e s s K E Y b u t t h e c o n v e r s i o n s a r e e s s e n t i a l l y as i n E3. I n t h e f r e e - s l i p case ( E 9 ) (Fig.6e),
t h e t o t a l k i n e t i c energy i s 50% h i g h e r
t h a n i n E8, p r i m a r i l y because o f t h e r e d u c t i o n o f k i n e t i c energy d i s s i p a t i o n along t h e coast o f A f r i c a .
The boundary c u r r e n t s and t h e f r e e j e t i n t h e South
A t l a n t i c become s t r o n g e r , and t h r o u g h g e o s t r o p h i c adjustment, between k i n e t i c
and p o t e n t i a l
particularly turbulent
energies increase also.
P and t h e exchange
The f l o w regime i s
s o u t h o f A f r i c a i n t h i s experiment
leading t o large
values o f KE and C(KM,KE). The i n c r e a s e d h o r i z o n t a l r e s o l u t i o n o f E l 0 b e t t e r r e p r e s e n t s t h e r e l e a s e o f baroclinic instability.
I n c r e a s e s a r e n o t i c e d i n P,
and i n t h e p o t e n t i a l t o
k i n e t i c energy c o n v e r s i o n s because t h e boundary c u r r e n t s a r e even s t r o n g e r than i n E3 and E8.
Since f o u r r i n g s were formed d u r i n g t h e t e n y e a r experiment, i t
i s a l s o i n t e r e s t i n g t o n o t e t h e d i f f e r e n c e i n t h e e n e r g e t i c s f o r y e a r s w i t h and without r i n g formation.
I n t h e l a t t e r case,
p r i m a r y energy p a t h ( F i g . 6 f ) .
t h e KM-P-KE
appears t o be t h e
During a year w i t h a r i n g ,
t h e b a r o c l i n i c and
b a r o t r o p i c t r a n s f e r s a r e o f t h e same o r d e r o f m a g n i t u d e ( F i g . 6 9 ) . comparison
strongly
formation o f a r i n g .
suggests
the
a s s o c i a t i o n between C(KM,KE)
transfer
This and
S u p p o r t i n g t h i s a s s e r t i o n i s t h e f a c t t h a t t h e change i n
185 South A f r i c a '
s geometry ( E l l ) r e s u l t s i n more b a r o t r o p i c c o n v e r s i o n .
t h e dominant t r a n s f e r i s t h e one f r o m KM t o KE (Fig.6h).
I n fact,
The m a j o r d i f f e r e n c e
between t h e two e x p e r i m e n t s i s t h e much g r e a t e r number o f r i n g s formed i n E l l . Also,
P i s approximatively
15% s m a l l e r i n E l l t h a n i n E l 0 because t h e r e t r o -
f l e c t i o n i s l e s s i n t e n s e and t h e Agulhas a l o n g t h e boundary i s somewhat broader. The q u e s t i o n a r i s e s as t o w h e t h e r b a s i n - a v e r a g e d e n e r g e t i c s a r e t r u l y relevant i n discussing
t h e dynamical
processes i n t h e r e t r o f l e c t i o n r e g i o n .
H a r r i s o n and Robinson (1978) showed t h a t energy t r a n s f e r averaged o v e r a model domain may n o t b e c h a r a c t e r i s t i c o f a n y i m p o r t a n t s u b r e g i o n .
BD showed,
however, t h a t e s p e c i a l l y i n t h e h i g h Rossby number experiments, most o f KM and
KE a r e l o c a l i s e d i n t h e r e t r o f l e c t i o n area. almost a l l t h e C(KM,KE) region.
Our c a l c u l a t i o n s have shown t h a t
i s i n t h e Agulhas r e g i o n and C(P,KE)
i s maximum i n t h i s
T h e r e f o r e , t h e e n e r g e t i c s averaged o v e r t h e whole b a s i n g i v e a good
i n d i c a t i o n o f e v e n t s o c c u r i n g i n t h e r e t r o f l e c t i o n a r e a f o r t h e s e cases. 4.3 Ring f o r m a t i o n A g u l h a s r i n g s have been observed and s t u d i e d b y Gordon (1985) and Olson and Evans (1986).
Lutjeharms (1981) was a b l e t o c a p t u r e t h e f o r m a t i o n of one r i n g
i n an a n a l y s i s o f s a t e l l i t e imagery, b u t l i t t l e has been e s t a b l i s h e d about t h e frequency o f t h e i r f o r m a t i o n and about t h e dynamics t h a t d r i v e t h e shedding, m o s t l y because t h e r i n g s a r e s t r o n g l y n o n l i n e a r f e a t u r e s .
F o r example, m i g h t
t h e f o r m a t i o n o f t h e r i n g s and t h e a s s o c i a t e d leakage be s t r o n g l y i n f l u e n c e d b y seasonal f a c t o r s ( w i n d - s t r e s s o r i n f l o w ) o r b y n o n l i n e a r o s c i l l a t i o n s of t h e r e c i r c u l a t i o n r e g i o n o f f South A f r i c a ?
Observational e f f o r t s t o f i n d time-
dependence i n t h e Agulhas C u r r e n t t r a n s p o r t b y Pearce and G r u n d l i n g h (1982) were n o t c o n c l u s i v e and,
i n o u r model, r i n g s f o r m w i t h c o n s t a n t f o r c i n g .
In their
study o f t h e dynamics o f t h e l o o p c u r r e n t and eddy shedding o f t h e G u l f o f Mexico,
H u l b u r t and Thompson (1980) o b t a i n e d a s i m i l a r r e s u l t w i t h a q u a s i -
annual eddy shedding.
Even w i t h r e a l i s t i c t i m e v a r i a t i o n o f t h e u p p e r - l a y e r
i n f l o w , t h e eddy-shedding p e r i o d was dominated b y t h e quasi-annual p e r i o d r a t h e r than by t h e f o r c i n g period,
a l t h o u g h . t h e i n f l u e n c e o f t h e l a t t e r was n o t
negligible. I n o r d e r t o g e t some u n d e r s t a n d i n g o f r i n g f o r m a t i o n i n t h e m o d e l , we d e s c r i b e a t y p i c a l e v e n t o f E l l , i l l u s t r a t e d h e r e i n Fig.7. a t day 2950 (Fig.7a),
I n t h e upper l a y e r ,
t h e f l o w p a t t e r n i s i n a quasi-steady configuration.
An
a l r e a d y formed r i n g i s c e n t e r e d j u s t s o u t h e a s t o f t h e t i p o f A f r i c a and t h e r e t r o f l e c t i o n o f t h e Agulhas p r o p e r i s a s h o r t d i s t a n c e up t h e coast.
Thirty
days l a t e r , t h e meander e a s t o f t h e r e t r o f l e c t i o n has begun t o a m p l i f y and t h e c e n t e r o f t h e r e t r o f l e c t i o n i s moving southwest ( F i g . 7 ~ ) . r i n g has begun moving toward t h e A t l a n t i c .
The a l r e a d y formed
The l o w e r l a y e r a t day 2950 shows a
186
4
DAY 2950
LAYER 1
b)
DAY 2950
LAYER 3
Fig.7.
Ring f o r m a t i o n i n E l l .
streamfunctions.
Upper and l o w e r l a y e r mean mass t r a n s p o r t 6 3 1 The c o n t o u r i n t e r v a l i s 5 x 10 m s-
.
c o u p l e t o f c y c l o n i c g y r e s l o c a t e d under t h e meander (Fig.7b). o f t h e upper l a y e r meander grows,
As t h e a m p l i t u d e This
t h e y merge and i n t e n s i f y (Fig.7f).
i n t e n s i f i c a t i o n reaches a maximum when t h e r i n g c u t s o f f .
I t s i n t e n s i t y weakens
r a p i d l y t h e r e a f t e r (Fig.7j). The d r a m a t i c g r o w t h o f t h i s c y c l o n i c c i r c u l a t i o n i n t h e t h i r d l a y e r i s s u g g e s t i v e t h a t b a r o c l i n i c i n s t a b i l i t y i s b e i n g r e l e a s e d d u r i n g r i n g formation. I n h i s r e c e n t model s t u d y of t h e Gulf of Mexico Loop C u r r e n t ,
H u l b u r t (1986)
a l s o found t h a t an i n t e n s e c y c l o n i c c i r c u l a t i o n develops i n t h e l o w e r l a y e r when an eddy i s shed.
I n o r d e r t o g e t more i n s i g h t i n t o t h e i n s t a b i l i t i e s i n v o l v e d ,
187
F i g .7.
(Continued)
we now l o o k a t t h e t i m e e v o l u t i o n o f t h e energy c o n v e r s i o n t e r m s d u r i n g t h i s and t h e subsequent e v e n t ( F i g . 8 ) . averaged o v e r a y e a r .
The mean components i n t h e e n e r g e t i c s a r e
The two s t r o n g s i g n a l s i n C(KM,KE) i n Fig.8 correspond t o
t h e two e v e n t s o f r i n g f o r m a t i o n .
I n b o t h cases, b e f o r e t h e meander b e g i n s t o
grow, t h e b a r o c l i n i c c o n v e r s i o n t e r m i s n e g a t i v e .
Then, as t h e a m p l i t u d e of t h e
meander i n c r e a s e s , C(P,KE) r i s e s above z e r o and reaches a maximum when t h e r i n g i s formed.
The b a r o t r o p i c c o n v e r s i o n t e r m i s l a r g e r and reaches a maximum a t
t h e same time.
It i s n o t c l e a r , however, t h a t C(KM,KE)
barotropic i n s t a b i l i t y .
i s directly related t o
P o s s i b l y , i t i s s i m p l y s u g g e s t i v e t h a t k i n e t i c energy
which has been r e s i d i n g i n t h e t i m e mean f l o w f i e l d i s b e i n g t r a n s f e r e d i n t o a
188
8 tl3AVl
0
066Z AVQ
Fig.7.
(Continued)
d i f f e r e n t flow configuration.
Therefore,
no c o n c l u s i o n can be reached as t o
whether b a r o t r o p i c o r b a r o c l i n i c i n s t a b i l i t y i s dominant.
We can o n l y say t h a t
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 c o n v e r s i o n s have s i g n i f i c a n t peaks d u r i n g r i n g formation.
It would appear t h a t t h e i n s t a b i l i t y i s o f t h e mixed t y p e .
The c h a r a c t e r and f r e q u e n c y o f r i n g f o r m a t i o n i n t h e e x p e r i m e n t s depend on t h e c h o i c e o f parameters,
as shown i n Table 2.
Analysis o f t h e v o r t i c i t y
balance i n S e c t i o n 3 showed how i m p o r t a n t i s t h e magnitude o f t h e p l a n e t a r y v o r t i c i t y advection i n t h e strength o f t h e r e t r o f l e c t i o n .
As t h e Rossby number
o f t h e boundary c u r r e n t s i s i n c r e a s e d , more f l u i d r e t r o f l e c t s and fewer r i n g s a r e formed.
With 40km g r i d r e s o l u t i o n ,
r i n g s a r e formed i n t h e l o w Rossby
189
DAY 3000
LAYER 1 NNVP WHlfIDV IV31U4V HlllOS
c - - - - -
E tl3AVl
OOOE AVQ
(4 Fig.7.
(Continued)
number cases ( E l and E2) and none i n t h e h i g h Rossby number cases (E3 and E8). But i n t h e l a t t e r case, retroflection
a change i n t h e geometry o f A f r i c a i n w h i c h t h e
i s 'less i n t e n s e (same as E l l w i t h 40km g r i d r e s o l u t i o n , n o t
presented here) leads t o formation o f r i n g s .
A few r i n g s were a l s o formed w i t h
an increase i n h o r i z o n t a l r e s o l u t i o n (E10) i n which t h e release o f b a r o c l i n i c i n s t a b i l i t y i s improved. The a n a l y s i s o f r i n g s formed i n E l and E l 0 supports the r e s u l t found i n E l l . I n both experiments,
as shown i n Fig.9
and Fig.10
respectively,
C(K,,,,KE)
C(P,KE) reach maxima o f t h e same magnitude a t t h e moment o f r i n g formation.
and As
i n E l l , a c y c l o n i c c i r c u l a t i o n develops i n t h e lower l a y e r and reaches maximum
190
061
F i g .7 (Continued) i n t e n s i t y when t h e r i n g separates from t h e gyre.
To conserve space here,
instantaneous streamfunctions f o r those two experiments a r e n o t presented. El,
In
r i n g s a r e tormed i n a continuous manner a t t h e southern t i p o f A f r i c a a t a
r a t e o f about f i v e per year.
As soon as an a l r e a d y formed r i n g begins t o d r i f t
toward t h e A t l a n t i c , t h e c e n t e r o f t h e r e t r o f l e c t i o n moves southward and s h o r t l y t h e r e a f t e r a new r i n g c u t s o f f .
The two peaks i n Fig.9
consecutive events o f E l i n y e a r 9.
The formation process i s q u i t e d i f f e r e n t i n
E10,
where o n l y f o u r r i n g s a r e formed i n t e n years.
correspond t o two
The c o r e o f t h e Agulhas
Current possesses more p l a n e t a r y v o r t i c i t y advection t h a n i n E l .
I n E10, f o r a
r i n g t o c u t o f f , t h e c e n t e r o f t h e r e t r o f l e c t i o n must be r a t h e r f a r south and a
191
TIME [DAYS] Fig.8.
Experiment E l l .
The dashed-dotted
V a r i a t i o n s o f t h e energy c o n v e r s i o n terms w i t h t i m e .
l i n e corresponds t o C(KM,
t h e d o t t e d l i n e t o C(P,KM). dome o f
low p o t e n t i a l
KE), t h e s o l i d l i n e t o C(P,KE) and
Rings a r e formed a t day 3000 and 3100.
vorticity
fluid
from t h e s u b p o l a r g y r e must i n t r u d e
northward a l o n g t h e e a s t s i d e o f t h e r e t r o f l e c t i o n r e g i o n .
I n addition, t h e
f r o n t between t h e g y r e s needs t o be f a r enough s o u t h so t h a t t h e c u t o f f r i n g can escape i n t o t h e A t l a n t i c b e f o r e b e i n g r e c a p t u r e d by t h e Agulhas. presented i n Fig.10
For t h e e v e n t
o c c u r i n g a t t h e b e g i n n i n g o f y e a r 9 ( i l l u s t r a t e d i n Fig.14
of DB), t h e f i r s t peak i n C(KM,KE) corresponds t o t h e i n i t i a l r i n g c u t o f f ,
the
second one t o a temporary r e a b s o r p t i o n and t h e t h i r d one t o t h e f i n a l c u t o f f o f the ring.
No b a r o c l i n i c conversions are associated w i t h t h e l a s t pinching off.
5 SUMMARY The v o r t i c i t y balance, t h e mean e n e r g e t i c s and energy c o n v e r s i o n s have been used as t o o l s w i t h which t o d e s c r i b e t h e dynamics o f Agulhas r e t r o f l e c t i o n and r i n g f o r m a t i o n i n a q u a s i - i s o p y c n i c c o o r d i n a t e n u m e r i c a l model. analysis
of
some
new
experiments
has i n c r e a s e d
I n addition,
understanding o f
t h e model
192
-2.01 ' ' ' ' 3050
'
3100
' '
' ' ' 3150
'
' ' ' ' 3200
I
'
' '
3250
'
I
TIME [DAYS1 Fig.9.
As i n Fig.8 f o r experiment E l .
Rings a r e formed a t day 3080 and 3210.
parameters most i n f l u e n t i a l i n t h e r e t r o f l e c t i o n . The a n a l y s i s
o f t h e v o r t i c i t y balance o f a one-layer s i m p l i f i c a t i o n o f the
Bleck and Boudra (1981) numerical model i n an i d e a l i z e d South A t l a n t i c - I n d i a n Ocean basin showed t h a t e i t h e r s t r o n g i n t e r n a l f r i c t i o n o r i n e r t i a can b r i n g about a p a r t i a l r e t r o f l e c t i o n where a l i n e a r model w i t h weak f r i c t i o n has none. With t h e f u l l quasi-isopycnic coordinate model, i t was f i r s t shown how important i s the magnitude o f t h e p l a n e t a r y v o r t i c i t y advection i n t h e s t r e n g t h o f the retroflection.
As t h e Rossby number o f t h e boundary c u r r e n t s i s increased, more
f l u i d r e t r o f l e c t s and r e t r o f l e c t s e a r l i e r a f t e r separation.
When horizontal
r e s o l u t i o n i s doubled t o b e t t e r r e s o l v e t h e boundary c u r r e n t s and i n s t a b i l i t i e s , t h e magnitude o f t h e terms of t h e v o r t i c i t y balance increases as t h e current becomes narrower and i t s maximum v e l o c i t y increases. unchanged.
The importance o f t h e f r i c t i o n a l boundary c o n d i t i o n was brought out
through a n o - s l i p / f r e e - s l i p adopted,
comparison.
A new South A f r i c a n geometry was also
i n which t h e p l a n e t a r y v o r t i c i t y advection i s l e s s important.
r e t r o f l e c t i o n was shown t o be l e s s i n t e n s e , unchanged.
Otherwise, t h e balance i s
The
b u t t h e primary balance i s
193
16 14
,
1
1
1
,
,
,
1
,
,
,
,
,
,
,
,
,
1
,
,
1
-
-
2800
2850
2900 2950 TIME [DAYS1
3000
Fig.10. As i n Fig.8 f o r experiment E10. The r i n g i s formed a t day 2880, t h e n reabsorbed a t day 2900. The f i n a l c u t o f f o c c u r s a t day 2930. A d d i t i o n a l i n s i g h t on t h e model r e t r o f l e c t i o n was g a i n e d t h r o u g h an a n a l y s i s o f t h e mean e n e r g e t i c s o f t h e e x p e r i m e n t s .
I t was shown t h a t as i n e r t i a
increases, more k i n e t i c energy i s p r e s e n t i n t h e boundary c u r r e n t s , even though t h e b a s i n averaged k i n e t i c energy does n o t change much.
T h i s l e a d s t o more
s t r o n g l y t i l t e d i s o p y c n a l s u r f a c e s and t o more energy exchange between KM and P. Energy exchange between KM and
KE i s e s p e c i a l l y l a r g e i n t h o s e experiments i n
which r i n g s a r e
As t h e
formed
often.
Rossby
number o f t h e boundary c u r r e n t s
increases, more f l u i d r e t r o f l e c t s and fewer r i n g s a r e formed. experiments were examined i n d e t a i l .
C e r t a i n aspects o f
different
each e x h i b i t s
C(P,KE) c u t s off.
among t h e experiments,
but
Events o f t h r e e
r i n g formation are
a maximum i n C(KM,KE),
and a pronounced c y c l o n i c development i n t h e l o w e r l a y e r when a r i n g While 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 c o n v e r s i o n s have peaks, whether
one p r i n c i p a l f o r m o f i n s t a b i l i t y l e a d s t o r i n g f o r m a t i o n i s n o t c l e a r . I n s t a b i l i t y of t h e mixed t y p e seems more l i k e l y .
194 ACKNOWLEDGEMENTS We w i s h t o acknowledge h e l p f u l d i s c u s s i o n s w i t h Prof. Claes Rooth Rainer Bleck’s a s s i s t a n c e i n c l a r i f y i n g t h e f i n i t e d i f f e r e n c e form o f v o r t i c i t y equation. T h i s work has been s u p p o r t e d b y t h e O f f i c e Research Contract No. N00014-85-C0020. Computations were performed Cray-1 computers a t t h e National Center f o r Atmospheric Research, sponsored by t h e National Science Foundation.
and Prof. t h e model o f Naval u s i n g the which i s
REFERENCES Finite-difference equations i n g e n e r a l i z e d v e r t i c a l K., 1979. Bleck, P o t e n t i a l v o r t i c i t y conservation. Contrib. Atmos. coordinates. P a r t 11: 360-372. PhyS., BlecR., 1985. On t h e c o n v e r s i o n between mean and eddy components o f p o t e n t i a l and k i n e t i c energy i n i s e n t r o p i c and i s o p y c n i c coordinates. Atmos. Oceans, 9, 17-37. 1981. I n i t i a l t e s t i n g o f a numerical model ocean Bleck, R., and koudra, U.B., . c i r c u l a t i o n model u s i n g a h y b r i d ( q u a s i - i s o p y c n i c ) v e r t i c a l coordinate. J Phy Oceano r., 11 755-770. 5:R u i j t e r , W., 1986. The wind-driven c i r c u l a t i o n i n the BoduSouth A t l a n t i c - I n d i a n Ocean. 11. Experiments u s i n g a m u l t i - l a y e r numerical
2,
of t h e Agulhas and B r a z i l Current 361-373. system. J. Phys. Oceanogr., de H u i j t e r , W., and Boudra, D.B., 1985. The wind-driven c i r c u l a t i o n i n the I. Numerical experiments i n a one-layer model. South A t l a n t i c - I n d i a n Ocean. Dee Sea Hes., 32, 557-574. Gor&l9m I n d i a n - A t l a n t i c t r a n s f e r of t h e r m o c l i n e water a t the Agulhas R e t r o f l e c t i o n . Science, 227, 1030-1033. and R o b i n s r n . , 7 9 7 8 . Energy a n a l y s i s o f open regions o f Harrison, D.E., t u r b u l e n t f l o w s - mean eddy e n e r g e t i c s of a numerical ocean c i r c u l a t i o n 185-211. experiment. Dyn. Atmos. Oceans, and Thompson, J.D., 1980. A numerical study o f Loop Current Hulburt, H.E., i n t r u s i o n s and eddy shedding. J. Phys. Oceanogr., 10, 1611-1651. Hulburt, H.E., and Thompson, J.D., 1982. The d y n a m i c T o f t h e Loop Current and shed eddies i n a numerical model o f t h e G u l f o f Mexico. Hydrodynamics of Nihoul (ed.), 243-298. semi-enclosed seas. By J.C.J. 1981. S p a t i a l scales and i n t e n s i t i e s o f c i r c u l a t i o n o f the Lutjeharms, J.R.E., ocean areas adjacent t o South A f r i c a . Dee -Sea Res 28a, 1289-1302. and Evans, R.H., 1986. Ring-’haX Deep-sea Res., 2, Olson, D.B., 27-42. Pearce, A.F., and Grunlingh, M.L., 1982. IS t h e r e a seasonal v a r i a t i o n i n the Agulhas Current? J. Mar. Res., 40, 177-184.
12,
2,
195
MODELLING OF MESOSCALE OCEANIC INSTABIL TY PROCESSES Aike Beckmann I n s t i t u t f u r Meereskunde an d e r U n i v e r s t a t K i e l Dusternbrooker Weg 20 D-2300 K i e l 1 (FRG)
ABSTRACT Using a q u a s i g e o s t r o p h c mu t i - l e v e l model on t h e B-p ane f o r a l i m i t e d area of o r d e r 1000 * 1000 km t h e n f l u e n c e o f t h e v e r t i c a l d i s c r e t i z a t i o n on t h e e v o l u t i o n o f mesoscale i s t a t l i t y s t r u c t u r e s i s s t u d ed. A s p e c i a l way of chosinq l e v e l s w i t h oDtimal r e p r e s e n t a t i o n o f shear-mode g r o w t h - r a t e s i s presenfed. Two cases ( i o c a l p e r t u r b a t i o n i n s t a b i l i t y and l a r g e s c a l e meander i n s t a b i l i t y ) a r e t a k e n as a t e s t , comparing t h e s o l u t i o n s w i t h o b s e r v a t i o n s i n t h e Canary b a s i n and w i t h o t h e r n u m e r i c a l model r e s u l t s . ~
1 INTRODUCTION An i m p o r t a n t p a r t o f t h e r e s e a r c h - p r o j e c t "Warmwatersphere o f t h e A t l a n t i c " a t t h e I n s t i t u t f u r Meereskunde K i e l i s t h e o b s e r v a t i o n o f l a r g e s c a l e phenomena and mesoscale v a r i a b i l i t y i n t h e e a s t e r n N o r t h A t l a n t i c . Several c r u i s e s t o v a r i o u s areas t o o k p l a c e i n t h e l a s t few years. As a r e s u l t f r o m t h e s p r i n g 1982 s u r v e y d e t a i l e d measurements o f t h e hydrography o f t h e Canary b a s i n a r e a v a i l a b l e . Since then, r e p e a t e d c r u i s e s r e v e a l e d t h e p i c t u r e o f a permanent A z o r e s - c u r r e n t w i t h 20 cm/s maximum j e t speed and a c r o s s - j e t s c a l e o f o r d e r 100 km. F i g 1. shows t h e q u a s i - s y n o p t i c dynamic h e i g h t f i e l d f r o m Kase e t a l .
(19851,
i l l u s t r a t i n g the
meandering s t r u c t u r e o f t h e f r o n t . Recent s t u d i e s i n c o n n e c t i o n w i t h t h e TOPOGULF experiment a l s o show meandering j e t behaviour west o f t h e M i d A t l a n t i c Ridge and a more Rossby-wave dominated regime eastward o f t h e r i d g e (M. Arhan, p e r s . comm.). There i s evidence t h a t t h e f r o n t a l band can be f o l l o w e d back t o t h e b r a n c h i n g o f t h e G u l f - s t r e a m i n t h e Newfoundland b a s i n (e.g.
i n Gould, 1985).
A f i r s t g l a n c e a t t h e i n v o l v e d s c a l e s y i e l d s U = 20 cm/s, L = 50 km, H = 500 m
and W = 2*10-4 m/s;
t h e s e v a l u e s r e s u l t i n g i n t h e p r i n c i p a l parameters:
Rossby-number
:
Ro = 0.04
B-Ross by-number
:
RB = 4.0
Burger-number
:
Bu = 1.0
r a t i o divergence/rotation
:
= 0.02
Thus t h e f l o w i s c l e a r l y l i n e a r w i t h r e s p e c t t o momentum advection, o n l y s l i g h t l y d i v e r g e n t , b u t s t r o n g l y n o n l i n e a r and b a r o c l i n i c i n t h e v o r t i c i t y balance.
196
Fig. 1. Objective analysis of the geopotential anomaly field 25/1500 dbar (m2/sZ) with the approximate centre o f the frontal band marked by the 13.5 m Z / s 2 isoline (after Kase et al., 1985). For the modelling of such a current regime and the dominant dynamical processes (like Rossby wave motion, nonlinear vorticity advection and barotropic and baroclinic instability) a quasigeostrophic model with high resolution in the horizontal as well as the vertical direction was developed. 2 THE QUASIGEOSTROPHIC MULTI-LEVEL MODEL 2.1 Model design As we have seen we are concerned with spatial scales of 1 0 t o 1000 km and associated time-scales of order days to months. According t o this we treat any larger scale phenomenon as a background field, climatologically generated and influenced. This background field can be described by the thermal wind relation a1 one.
In this part of the subtropical gyre the mean current is directed eastward, resulting from the increase of density t o the north. In the quasigeostrophic approximation the equations t o deal with are the nonlinear vorticity and density equation, which can be combined t o the well known baroclinic vorticity-equation; however we use a diagnostic relation for the vertical velocity resulting from elimination o f the time-derivative in the vorticity and density equation.
197 Together t h e y f o r m o u r s e t o f e q u a t i o n s : V2Yt + J ( Y , V 2 Y )
+ BY
-
fowz = O ( Y )
A s u i t a b l e p a r a m e t r i z a t i o n o f s u b - g r i d - s c a l e processes D(Y) i s i n c l u d e d u s i n g
t h e h o r i z o n t a l Austausch-concept ( b i h a r m o n i c f r i c t i o n ) f o r t h e r e l a t i v e v o r t i c i t y AV'S.
The c h o i c e of t h e f r i c t i o n parameter i s due t o n u m e r i c a l reasons o n l y , i.e.
t o a v o i d u n a c c e p t a b l e growth o f t h e s m a l l e s t s c a l e s a v a l u e o f 2-10 'm'/s
is
necessary. T h i s corresponds t o a b i h a r m o n i c f r i c t i o n f o r t h e s t r e a m f u n c t i o n o f
Ah = 50 m2/s f o r t h e 40 km-wave. F o r t h e p r e s e n t s t u d y v e r t i c a l d i f f u s i o n i s neg l e c t e d . T h i s i s p o s s i b l e even i n a n u m e r i c a l model i f t h e h o r i z o n t a l p r o j e c t i o n o f any v e r t i c a l process can be r e s o l v e d b y t h e h o r i z o n t a l g r i d . I n t h e h o r i z o n t a l d i r e c t i o n s t h e p s e u d o s p e c t r a l approach i s i n t r o d u c e d t o m i n i m i z e t h e phase e r r o r s and t o a s u r e a good r e p r e s e n t a t i o n of t h e n o n l i n e a r a d v e c t i o n terms
(Merilees
& Orszag,
1979 f o r a d e t a i l e d d e s c r i p t i o n ) .
a s s o c i a t e d b o u n d a r y - c o n d i t i o n s a r e p e r i o d i c i t y i n b o t h x and y d i r e c t i o n ; 32
The
*
32
wavenumbers a r e used. The p r o d u c t terms a r e e v a l u a t e d e n e r g y - c o n s e r v i n g on t h e p h y s i c a l g r i d u s i n g FFT f o r t h e t r a n s f o r m a t i o n s . E n s t r o p h y - c o n s e r v a t i o n was dropped because i t appears t o have o n l y a r e l a t i v e l y s m a l l i n f l u e n c e on t h e h o r i z o n t a l f i e l d s w i t h i n 50 days o f simulation.
In
the
frame
of
this
numerical
concept
mass-conservation
is
automatically f u l f i l l e d . A t o t a l l y p e r i o d i c model i s n o t q u i t e what i s d e s i r e d , s i n c e t h e r e has t o be a
n e t t r a n s p o r t t h r o u g h t h e r e g i o n under c o n s i d e r a t i o n . To i n c o r p o r a t e t h i s a s e p a r a t i o n o f t h e mean zonal v e l o c i t y f r o m t h e F o u r i e r - e x p a n s i o n was performed according t o :
where u0 i s t h e m e r i d i o n a l l y averaged zonal v e l o c i t y . The v e r t i c a l d i s c r e t i z a t i o n i s done u s i n g a l e v e l - c o n c e p t w i t h c o m p u t a t i o n o f and w a t s t a g g e r e d l e v e l depths ( f o l l o w i n g f o r example Bengtsson & Temperton, 1979). The b o u n d a r y - c o n d i t i o n s a r e v a n i s h i n g v e r t i c a l v e l o c i t i e s t o g e t h e r w i t h Yz = 0 a t t h e t o p and a t t h e b o t t o m ( t h u s o m i t t i n g any e f f e c t s o f bottom-topography
and wind-induced f o r c i n g ) . Another more d e t a i l e d l o o k a t t h e model a p p l i e s t o t h e energy budget: t h e s e p a r a t i o n o f t h e mean z o n a l t r a n s p o r t f r o m t h e p u r e l y p e r i o d i c p a r t has t o be
198
performed at,each Y-level. This results in a mean zonal transport profile to be treated separately. Writing down the energy-balance-relations for this modified quasigeostrophic system (friction neglected) we find that there is a net gain of energy from the climatological background flow field. dE/dt
=
I T1
( f $ / N a ) uoZ
II Yi Y;
dx dy
dz
(5)
Unfortunately there is no additional prognostic equation for the time dependence of the mean flow velocity in the frame of the quasigeostrophic concept; so this system has still to be called a linearized one. In order to take into account even the influence of the periodic part on the horizontally integrated part we change the vertical shear of the mean flow according to the loss of its available potential energy, keeping the vertically integrated transport through the basin constant. This results in a set of equations for each w-level:
It is easily seen from eqn. (6) that if the initial shear i s zero it remains unchanged. This additional equation leads to an energetically consistent model of local mesoscale dynamics including arbitrary mean zonal flow profiles. 2.2 Vertical Calibration Since the quasigeostrophic approximation refers to the mean stratification of the region under consideration the measured mean density-profile from the Canary-basin and the corresponding Brunt-Vaisala-frequencies are the basis forthis numerical model (fig. 2a). For these given profiles the vertical structure of the first and second baroclinic mode are shown in fig. 2b. The first mode zero-crossing happens at roughly 1000 rn depth. The first and second baroclinic Rossby-radius of deformation are found to be 26.3 and 8.9 km respectively. The initial flow-profile is chosen to be dominated by the first and second baroclinic mode. For the maximum velocity of 20 cm/s at the surface the total transport through the basin reaches 8 Sverdrups. For a mean depth of approximately 4400 m and a typical value of 2*10-3s-’ for N o we find a reasonable vertical resolution of order 500 m for horizontal grid-spacing of a x = a y = 10 km (in order to fix the Burger-number at unity) so there will be 9 level in the vertical. A higher vertical resolution requires vertical diffusion for numerical stability reasons. The first approach in chosing the vertical grid would be to fix the levels equidistant, but for a better representation of the vertical structure found in
199
the ocean (e,g. fig. 2a) it seems more convenient to concentrate the layers in the upper part of the water-column; for instance a grid spacing of 100, 5 * 200, 300, 500 and 2500 m depth with Y-level depths at 50, 200, 400, 600, 800, 1000, 1250, 1650 and 3150 m respectively.
2000-
b) computed vertical structure of first and second baroclinic mode, c) 8th baroclinic eigenfunction used for the vertical discretization, d ) configuration of staggered I - and w-level.
60
LOOO-
@ I
I
0
Z/m
2ool
2000
LO00
om* I w4-
Z/m
20004
I
4000
Instead, it is not a priori clear how such a vertical discretization represents the vertical structure o f dynamical processes in a numerical model. So we make use of the measured density-profile in form of the higher order vertical eigenfunction (fig 2c). For a 9-degrees-of-freedom model the 8th baroclinic vertical mode yields the desired 8 zero-crossings where the layer-boundaries are fixed. The streamfunction-levels are located in the center of the layers (fig. 2d).
200
..........
-
~
.....
.
\
,......-.--.-_.__
l./y--
7
0 krn 200
..
I
0 krn 200
Fig. 3. An example for the different results of models with a) conventional vertical discretization and b) choice of levels according to 8th eigenfunction of the density profile. Shown is the density in the main thermocline (ca. 700 m depth) after 50 days of integration. A first comparison of these two distinct level-choices is shown in fig. 3 for the density in the main thermocline P700 m) after 50 days of integration. Several features can be recognized (the detached eddies and enhanced fronts) in both versions. The dominant processes seem to be unchanged but the details are rather different.
3 LINEAR INSTABILITY PROCESSES 3.1 Unstable shear-modes To test the sensitivity of the solutions (e.g. instabilities) on different ways of vertical discretization in a more quantitatively way we first look at the linearized system to determine the shear-mode growth-rates of the system; solving an eigenvalue problem for the baroclinic vorticity-equation neglecting every meridional variation and using the following approach: I'(x,y,z,t)
= F(z)*exp(i(kx
-
wt))
;
w = w
+ i wi
(7)
The computation of the continuous (that means 20 m vertical resolution) shear modes for the maximum velocities yields a growth-rate maximum at wavenumber 8 (80 km wavelength) with 4.5 days exponential time scale. The conventionally truncated system shows a significant different curve for waves shorter than that maximum, while the eigenfunction level choice compensates the effect of reduction of vertical degrees of freedom to a large amount (fig. 4a).
201
0 Z/m 1000
I
-
I
I I I
\
1I
I
I
2000
I
-
I I
I
I I
I
I
I I
I I
3000 -
I
I I I I I
I
I I
I
LO00
SM
-
I LL
I
L 7112 0
L 0 12 ZONAL WAVENUMBER
V2
F i g . 4. a) Shear-mode g r o w t h - r a t e s f o r c o n t i n u o u s (CONT), c o n v e n t i o n a l (CONY) and e i g e n f u n c t i o n ( E I F U ) v e r t i c a l d i s c r e t i z a t i o n ; b ) a m p l i t u d e s o f t h e most u n s t a b l e t h e r m o c l i n e shear-mode
(TSM)
and b o t t o m - i n t e n s i f i e d shear-mode
(BSM)
of the
c o n t i n u o u s model. The dashed l i n e s show t h e phase.
I n t h e 9 - l e v e l model o f t h e Canary b a s i n t h e r e a r e two d i f f e r e n t u n s t a b l e modes t o d i s t i n g u i s h : t h e f i r s t i s t h e t h e r m o c l i n e shear-mode (TSM), b o t t o m - i n t e n s i f i e d shear-mode (BSM);
t h e second t h e
each b e l o n g i n g t o a n o t h e r d e p t h where t h e
necessary c o n d i t i o n f o r b a r o c l i n i c i n s t a b i l i t y ( Beff
=
8 + ((f$/N a)uoz)z changes
s i g n ) i s f u l f i l l e d . O f course, c r i t i c a l l e v e l s cannot be s u f f i c i e n t l y r e s o l v e d b y any o f t h e s e d i s c r e t e models. The d o m i n a t i n g growth r a t e s correspond t o t h e t h e r m o c l i n e shear-mode ( T S M ) . For wavelenghts s h o r t e r t h a n 50 km t h e r e i s no u n s t a b l e TSM p o s s b i l e , t h e d o t t e d curves r e p r e s e n t t h e b o t t o m - t r a p p e d o r b o t t o m - i n t e n s i f i e d shear-mode (BSM). F l i e r 1 (1978) p o i n t e d o u t t h a t a modal r e p r e s e n t a t i o n o f t h e v e r t i c a l i n a quasigeo s t r o p h i c model y i e l d s b e t t e r r e s u l t s compared w i t h l e v e l / l a y e r approach i n case o f low v e r t i c a l r e s o l u t i o n ; t h i s can a l s o be concluded f o r t h e s e s i m u l a t i o n s : r e f e r i n g t o t h e modal s t r u c t u r e o f t h e v e r t i c a l p r o b l e m t h e d i s c r e t e model seems t o be c l o s e r t o t h e c o n t i n u o u s case.
202
3.2 Growth-rates o f u n s t a b l e waves As an e x t e n s i o n we c o n s i d e r t h e l i n e a r i z e d system w i t h i n c l u d e d y-dependency c o n s i s t i n g of a zonal b a r o c l i n i c j e t and superimposed s m a l l s c a l e p e r t u r b a t i o n s i n i t s 9-level-version.
Y'(x,y,z,t)
=
F(z)*M(y)-exp(i(kx
uo(y,z) = U(z)-exp(-u2y2)
-
wt))
;
;
w = w ).I
+ iw.
(8) (9)
= 1/50 krn
TheanalysisfollowsHolland&Haidvogel (1980)withthreedifferences:ninelevels a r e used i n s t e a d o f two, no f r i c t i o n i s i n c l u d e d and p e r i o d i c i t y i n t h e m e r i d i o n a l d i r e c t i o n i s i n t r o d u c e d t o g e t c l o s e t o t h e n o n l i n e a r model. Examples 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 e i g e n f u n c t i o n s a r e g i v e n i n f i g . 5. The most u n s t a b l e mode i s shown i n f i g . 5a w i t h l a r g e a m p l i t u d e s i n t h o s e depths, where
the
necessary
condition
for
baroclinic
instability
is
fulfilled
( p r e d o m i n a n t l y i n t h e t h e r m o c l i n e ) ; t h e most u n s t a b l e b o t t o m - i n t e n s i f i e d wave i s d e p i c t e d i n f i g . 5b. There a r e m e r i d i o n a l l y h i g h e r o r d e r u n s t a b l e s o l u t i o n s b u t t h e s e have reduced g r o w t h - r a t e s .
0
KM
640
0
KM
640
F i g . 5. a ) M e r i d i o n a l and v e r t i c a l s t r u c t u r e o f t h e most u n s t a b l e e i g e n f u n c t i o n from t h e l i n e a r i z e d 9 - l e v e l v e r s i o n , b ) another u n s t a b l e mode w i t h b o t t o m - i n t e n s i f i e d a m p l i t u d e . S o l i d l i n e s show t h e amplitude, b r o k e n l i n e s d e n o t e t h e phase.
203 R e s u l t s on t h e s e n s i t i v i t y o f i n s t a b i l i t y processes i n q u a s i g e o s t r o p h i c l e v e l models may be sumnarized as f o l l o w s :
-
a moderate change o f t h e assumed deep sea g r a d i e n t (beyond 2000 m d e p t h ) has
s t r o n g e f f e c t s on t h e growth o f t h e b o t t o m - t r a p p e d mode. S m a l l e r g r a d i e n t s enable t h e system t o be u n s t a b l e even a t s m a l l e r wavelengths l e a v i n g t h e g r o w t h - r a t e s o f t h e u n s t a b l e thermocline-wave u n a l t e r e d .
-
t h e e i g e n f u n c t i o n - d i s c r e t i z a t i o n has s t r o n g e f f e c t s i n c e r t a i n s p e c t r a l bands:
t h e g r o w t h - r a t e s a r e l a r g e r and t h e maximum i s s h i f t e d towards s m a l l e r wavelengths. Again i t seems t h a t f i r s t - c h o i c e - l e v e l l i n g c u t s o f f c e r t a i n waves j u s t w i t h i n t h e range o f i n t e r e s t f o r s i m u l a t i o n s o f t h e mesoscale. Even i n t h e n o n l i n e a r model we can p r e s c r i b e a s i n u s o i d a l p e r t u r b a t i o n o f one p a r t i a l wave i n zonal d i r e c t i o n on a p u r e l y zonal c u r r e n t o f an exp(-p2y2)shape and l o o k f o r t h e growth o f t h i s s p e c t r a l component w i t h i n t h e f i r s t 10 days ( w i t h o u t f r i c i t o n ) t o determine t h e "growth-rates". O b v i o u s l y t h e r e w i l l be an e n e r g y - t r a n s f e r t h r o u g h t h e wavenumber spectrum t o s m a l l e r s c a l e s due t o t h e e n e r g y cascade and a l s o t h e r e s e r v o i r of t h e a v a i l a b l e p o t e n t i a l energy a t t h e p a r t i c u l a r wavenumber i s l i m i t e d , so t h e s e g r o w t h - r a t e s a r e expected t o be s m a l l e r t h a n t h e l i n e a r i z e d ones. But as f i g . 6 shows t h e shape o f t h e c u r v e i s i n good agreement and t h e maximum l i e s a t wavenumber 8 (80 km) i n b o t h
Fig.
6.
Amplitude
growth-rates
of
d i f f e r e n t configurations o f linearized
systemscomparedtothenonlinearmodel as a
function
of
zonal
wavenumber.
E F l denotes t h e c o n v e n t i o n a l l e v e l l i n g , EF2 o u r
eigenfunction
l e v e l l i n g and
NL-MODEL t h e n o n l i n e a r model w i t h EF2
discretization.Wavenumberlmeans640km wavelength, wavelength. ZONAL WAVENUMBER
wavenumber 16 means 40 km
204
4 NONLINEAR QUASIGEOSTROPHIC INSTABILITY OF A JET
4.1
Local p e r t u r b a t i o n i n s t a b i l i t y As an example o f t h e i n s t a b i l i t y r e s u l t i n g f r o m a l o c a l p e r t u r b a t i o n o f 100 km
m e r i d i o n a l e x t e n t f i g 8. shows t h e e v o l u t i o n o f t h e e x t e r n a l mode s t r e a m f u n c t i o n a t
10 days i n t e r v a l . Note t h e growth o f t h e m e r i d i o n a l a m p l i t u d e o f t h e p e r t u r b a t i o n , t h e downstream meandering and t h e mushroom-type i n s t a b i l i t y a t day 30 ( f i g . 8d), f o r c i n g t h e e d d y - p a i r s t o d e t a c h b y means o f enhanced a d v e c t i o n . W i t h i n t h e 50 d a y s - s i m u l a t i o n t h e maximum l o c a l t r a n s p o r t has r a i s e d f r o m 8 t o
36 Sverdrups;
t h e maximum v e l o c i t y a t t h e s u r f a c e r u n s up t o about 45 cm/s.
As expected t h e dominant meander Rossby-radius ( i . e .
scale
i s r o u g h l y of
order
2 n times the
150 km). The d e n s i t y - f l u x d i r e c t e d southward due t o b a r o c l i n i c
i n s t a b i l i t y ( w i t h a n o r t h w a r d h e a t f l u x ) has a maximum about day 30; exchanging water-mass p r o p e r t i e s across t h e f r o n t a l j e t . The momentum t r a n s p o r t has a l a r g e southward component. Effects o f periodicity model
, i.e.
r e g i o n f r o m t h e west
significantly.
t h e p e n e t r a t i o n o f downstream meanders i n t o t h e does
not
change t h e s t r u c t u r e s
until
day 50
An experiment u s i n g an extended r e g i o n was performed
and no
fundamental d i f f e r e n c e s were found. A s h o r t l o o k a t t h e change o f t h e mean t r a n s p o r t p r o f i l e t h r o u g h t h e f i r s t 50
days ( f i g . 7 ) shows a n e t decrease o f v e r t i c a l shear as expected, e s p e c i a l l y i n t h e upper l a y e r s , and moreover a mean-current g e n e r a t i o n i n t h e deep l a y e r .
I
I
2000-
:
-u,(z, ---uo
1.0) (2, t-50d)
I
I I
I
4000-
Fig.
7.
I
Change o f t h e mean z o n a l t r a n s p o r t p r o f i l e b y r e d u c t i o n of t h e mean
p o t e n t i a l energy w i t h i n 50 days i n t e g r a t i o n .
205
Fig.
8.
i n a 640
Evolution o f
*
external
640 km box ( u n i t s m 2 / s ) .
mode
streamfunction
at
10 days
interval
206
F i g . 9. a ) surface s t r e a m f u n c t i o n , b ) s u r f a c e d e n s i t y , c ) t h e r m o c l i n e d e n s i t y and d ) deep sea d e n s i t y a f t e r 50 days o f i n t e g r a t i o n . The f l o w f i e l d a t t h e s u r f a c e e x h i b i t s an a x i s y m e t r i c r i n g f o r m a t i o n ( f i g . 9a) b u t t h e d e n s i t y - f i e l d i s d i f f u s e ( f i g . 9b); i n m i d - t h e r m o c l i n e depths t h e c h a r a c t e r o f t h e f r o n t i s much more o b v i o u s and a f i r s t mode v e r t i c a l s t r u c t u r e of t h e eddies i s i n d i c a t e d from a comparison of f i g . 9b and 9c. The double e d d y - s t r u c t u r e e v o l v e d i n t h e b a r o t r o p i c mode i s m a i n l y due t o t h e deep sea r e c t i f i c a t i o n due t o t h e b o t t o m - i n t e n s i f i e d u n s t a b l e mode ( f i g . 9d).
207 R e s u l t s o f I k e d a & Ape1 (1981) u s i n g a t w o - l a y e r q u a s i g e o s t r o p h i c model of t h e G u l f - s t r e a m ( d e a l i n g w i t h t h e same l e n g t h s c a l e s b u t 1 m/s maximum v e l o c i t y ) shows an analogous eddy-detachment-process
w i t h i n 36 days.
Moreover p r i m i t i v e - e q u a t i o n - m o d e l r e s u l t s f r o m Kielmann & Kase (1986) a p p l i e d t o t h e same r e g i o n and observed q u a n t i t i e s do n o t propose a d d i t i o n a l processes t o be important. 4.2 L a r g e - s c a l e meander i n s t a b i l i t y Kase e t a l . Rossby-wave
(1985) decomposed t h e dynamic h e i g h t f i e l d o f f i g .
field
and
linear
trend
and
a
residual
1 into a
perturbation
field
( f i g . 10a,b).
36O
N
32O
I L 00 I
220
240
w
I
200
I
F i g . 10. a) S u p e r p o s i t i o n o f t h e l i n e a r s p a t i a l t r e n d and t h e Rossby-wave f i t f o r t h e g e o p o t e n t i a l t o p o g r a p h y 25/1500 dbar o f f i g . 1. b ) O b j e c t i v e a n a l y s i s of t h e r e s i d u a l mesoscale p e r t u r b a t i o n f i e l d 25/1500 dbar a f t e r removal o f t h e composite mean f i e l d ( a f t e r Kase e t al.,
1985).
I n o r d e r t o s i m u l a t e such a f l o w - p a t t e r n , dominated b y s t r o n g nonzonal f l o w s we i n i t i a l i z e d t h e model w i t h a l a r g e s c a l e meander o f t h e same v e r t i c a l s t r u c t u r e as i n s e c t i o n 4.1 ( f i g . l l a ) . T h i s i d e a l i z e d meander o f 640 km z o n a l wavelength and 200 km m e r i d i o n a l a m p l i t u d e s p l i t s w i t h i n t h e 50 days i n t e g r a t i o n t o waves o f zonal wavenumber 3 (213 km wavelength),
b u t even a f t e r t h e i n i t i a l s t r u c t u r e i s r e c o g n i z a b l e from t h e
arrangement o f t h e e d d i e s and i n d i c a t e s an average p r o p a g a t i o n o f 2 cm/s t o t h e east, i . e .
i t i s n e a r l y s t a t i o n a r y ( f i g . 11). The l o c a l t r a n s p o r t s and s u r f a c e
v e l o c i t i e s have r a i s e d b y a f a c t o r o f t h r e e . outside t h e jet-regime.
Note t h e r a d i a t i n g Rossby-waves
208
Fig.
11.
i n a 640
Evolution
*
of
external
640 km box ( u n i t s m2/s).
mode
streamfunction
at
10 days
interval
209
-! Fig. 12. a) surface streamfunction, b) surface d e n s i t y , c ) thermocline d e n s i t y and d) deep sea d e n s i t y a f t e r 50 days o f i n t e g r a t i o n . The f l o w a t t h e s u r f a c e shows t o t a l l y m e r i d i o n a l l y arms o f a continuous flow band; s t r o n g f r o n t a l zones can be seen i n t h e mid-depth d e n s i t y ( f i g . 12 c ) . The eddy f i e l d i s d r i v i n g t h e deep sea c u r r e n t s ( f i g . 12d). Comparisons between day 30 ( f i g . l l d ) o f t h e s i m u l a t i o n s and t h e observed f i e l d (fig.
l o b ) show s t r u c t u r a l s i m i l a r i t i e s .
From t h i s we may conclude t h a t t h e
hydrographic survey took a snapshot o f an emerging i n s t a b i l i t y of such a l a r g e scale meander. The reason f o r t h e occurrence o f t h e observed l a r g e - s c a l e f e a t u r e ( p o s s i b l y an i n t r u s i o n o f dense water f r o m t h e n o r t h o r a r e s u l t of previous i n s t a b i l i t y processes) cannot be explained w i t h our model.
210 5 CONCLUSIONS Quasigeostrophic model-results
concerning mesoscale i n s t a b i l i t y processes
depend c r i t i c a l l y on t h e used v e r t i c a l d i s c r e t i z a t i o n f o r a g i v e n h o r i z o n t a l grid-spacing. Best r e s u l t s are obtained from an eigenfunction-based choice o f t h e l e v e l - d e p t h s as may be seen from shear-mode growth-rates ( s e c t i o n 3). A s u i t a b l y c a l i b r a t e d quasigeostrophic m u l t i - l e v e l model ( u s i n g v e r t i c a l modes
f o r t h e choice of t h e l e v e l s ) o f h i g h h o r i z o n t a l r e s o l u t i o n i s a b l e t o s i m u l a t e frontal
dynamics
( n e g l e c t i n g a1 1 thermodynamic processes) v e r y accurate and
e f f i c i e n t . Test-computations w i t h t h e 9 - l e v e l Canary b a s i n model show reasonable time-scales f o r t h e growth and propagation o f f r o n t a l s t r u c t u r e s . A corresponding t w o - l e v e l model i s p r i n c i p a l l y o v e r e s t i m a t i n g t h e v e r t i c a l shear and a m p l i f i e s t h e b a r o c l i n i c i n s t a b l i t y process. Another important r e s u l t i s t h a t t h e detachment o f eddies i s n o t r e s t r i c t e d t o Gulf-stream-parametws w i t h l a r g e h o r i z o n t a l v e l o c i t i e s . T h i s eddy-pair-separation process r e q u i r e s about 50 days ( s e c t i o n 4.1).
Yet no r i n g - f o r m a t i o n l i k e i n t h e
boundary c u r r e n t r e g i o n s c o u l d be observed i n t h e model. T h i s i s p o s s i b l y due t o t h e d i f f e r e n t v e r t i c a l s t r u c t u r e o r t h e l a r g e r s u r f a c e k i n e t i c energy o f t h e Gulf-stream. The q u a s i - s y n o p t i c measurements i n t h e Canary-basin c a r r i e d through by Kase e t a l . (1985) may be e x p l a i n e d as an e a r l y s t a t e o f an e v o l v i n g i n s t a b i l i t y ( s e c t i o n 4.2).
6 REFERENCES Bengtsson, L. Temperton, C., 1979. D i f f e r e n c e Approximations t o Quasigeostrophic Models. I n : Numerical Methods Used i n Atmospheric Models. GARP P u b l i c a t i o n Series, No 17, Vol 2, 340-380. F l i e r l , G.R., 1978. Models o f V e r t i c a l S t r u c t u r e and t h e C a l i b r a t i o n o f Two-Layer Models. Dyn. Atmos. Oceans, 2, 341-381. Gould W.J., 1985. Physical Oceanography o f t h e Azores F r o n t . Progr. Oceanogr., 14, 167-190. Haidvogel, D.B., 1980. A Parameter-Study o f t h e Mixed Holland W.R., I n s t a b i l i t y o f I d e a l i z e d Ocean Currents. Dyn. Atmos. Oceans, 4, 185-215. Ikeda M., Apel, J.R., 1981. Mesoscale Eddies Detached f r o m S p a t i a l l y Growing Meanders in an Eastward Flowing Oceanic Jet Using a Two-Layer Quasigeostrophic Model. Journ. Phys. Ocean., 11, 1638-1661. Zenk, W., Sanford, T.B., H i l l e r , W., 1985. Currents, F r o n t s and Kase, R.H., Eddy-Fluxes i n t h e Canary Basin. Progr. Oceanogr., 14, 231-257. Kielmann, J., Kase, R.H., 1986. Numerical M o d e l l i n g o f meander and eddy f o r m a t i o n i n t h e Azores-Current f r o n t a l zone. Submitted t o Journ. Phys. Ocean. Orszag, S.A., 1979. The Pseudospectral Method. I n : Numerical Merilees, P.E., Methods Used i n Atmospheric Models. GARP P u b l i c a t i o n Series, No 17, Vol 2, 278-301.
211
AN EDDY-RESOLVING MODEL FOR R I V E R PLUME FRONTS
J.W. DIPPNER Deutsches Hydrographisches I n s t i t u t , Bernhard-Nocht-StraRe D-2000 Hamburg 4, FRG
78,
ABSTRACT A b a r o c l i n i c e d d y - r e s o l v i n g model o f a r i v e r plume f r o n t i s developed. The n u m e r i c a l framework i s c a r r i e d o u t w i t h a s e m i - i m p l i c i t two s t e p d i f f e r e n c e scheme and t h e t r a n s p o r t e q u a t i o n o f d e n s i t y i s s i m u l a t e d w i t h an a c t i v e L a g r a n g i a n t r a c e r technique. P r e l i m i n a r y r e s u l t s o f f r o n t a l and eddy f o r m a t i o n a r e presented.
1 INTRODUCTION I n t h e German B i g h t e x i s t t h r e e d i f f e r e n t t y p e s o f f r o n t s which have been c l a s s i f i e d b y Krause e t a1.(1986).
The f i r s t i s a seasonal thermal f r o n t
between t h e s t r a t i f i e d and w e l l mixed area i n t h e German B i g h t ; t h e second i s an u p w e l l i n g f r o n t i n t h e r e g i o n o f t h e O l d E l b e V a l l e y o c c u r r i n g d u r i n g e a s t e r l y winds; and t h e t h i r d i s a r i v e r plume f r o n t due t o t h e r i v e r runoff of t h e r i v e r s E l b e and Weser, w h i c h i s t h e s u b j e c t o f t h i s paper. The dynamics and t h e developement o f r i v e r plume f r o n t s depend s t r o n g l y upon t h e c h a r a c t e r i s t i c s o f t h e e s t u a r y .
The main f e a t u r e s a r e t h e amount
o f r i v e r r u n o f f , the v e l o c i t y o f t i d a l currents,
and g e o m e t r i c a l c o n s t r a i n t s
such as w i d t h and d e p t h o f t h e e s t u a r y which a r e r e s p o n s i b l e i n f o r m i n g a h i g h l y s t r a t i f i e d , m o d e r a t e l y s t r a t i f i e d , o r v e r t i c a l l y homogeneous e s t u a r y ( P r i t c h a r d 1955). R e c e n t l y , a l o t o f plume models have been developed (Kao e t a l . 1977, Bork 1978, S t r o n a c h 1981), b u t t h e s e models a r e two-dimensional n u m e r i c a l models. Only few models o f s h a l l o w w a t e r f r o n t s a r e t h r e e - d i m e n s i o n a l (Harashima and Oonishi 1981, James 1984). Recent measurements, sensing techniques, plume f r o n t ,
by t h e use o f modern p r o f i l i n g equipment,
remote
and experiments w i t h dye patches o f rhodamine B a t t h e
have r e v e a l e d a r a t h e r c o m p l i c a t e d s t r u c t u r e of
meanders (Becker e t a l .
eddies and
1983, Franz and K l e i n 1986). I n g e n e r a l , f r o n t s a r e
characterized by two d i f f e r e n t l e n g t h scales.
In t h e c r o s s f r o n t d i r e c t i o n ,
t h e span i s o f t h e o r d e r o f a few b a r o c l i n i c r a d i i o f d e f o r m a t i o n which i s o f t h e o r d e r of 5 t o 20 km i n t h e German B i g h t . I n t h e a l o n g f r o n t d i r e c t i o n , t h e span i s determined b y t h e wavelength of
meander, which i s determined
212
by bathymetric scales, atmospheric forcing scales, wavelength of coastally trapped waves, and baroclinic instability scale (Bowman 1978). To get a better understanding of the complicated structure of the river plume front and the involved processes in the German Bight, a three-dimensional eddyresolving model is developed. This process model and the model philosophy will be outlined in the following, and preliminary results are shown.
2 EQUATIONS AND BOUNDARY CONDITIONS The model is based upon the well known primitive equations using hydrostatic and Boussinesq approximation:
i-
L
V) + fu
P, =
-
9 P
Vt
=
-
l/po
py
i-
(KM
v,), +
+,, V 2v (3)
(41
L(1) = 0
Here u, v, p and P represent the horizontal velocities, pressure, and a reference density and g the and density, f the Coriolis parameter, p 0 the gravity acceleration. The operator L i s defined by: L(a)
=
(ua), + (va)Y
i-
(wa),
(61
The governing equations are solved with a semi-implicit two-step method in a schematic funnel-shaped estuary model with a linear bottom topography, shown in Fig. 1.
213
D E P T H CONTOUR
3
INITIRL DENSITY = 1 SIGMQ T
CI
Fig. 1 Model area and topography (left), contour interval (CI) 1 m, and initial density (right) CI = lut
The boundary conditions are as follows: 1) No flow through the lower boundary and no flow through the lateral
coastal boundaries. 2) Free surface kinematic boundary condition. 3) A tidal signal of an M, tide at the open boundary. 4) A quadratic bottom stress. 5) Half slip condition at the lateral boundaries. 6 ) A river runoff of 1000 m3/s, which is in the order of the River Elbe. 3 DESCRIPTION OF THE NUMERICAL MODEL Equations 1 to 4 are vertically integrated over the layer thickness of the
model and discretized forwards in time and central in space in a staggered C-grid. The vertical derivative is handled with an implicit difference scheme. The barotropic pressure gradient in the equations of motion and the horizontal divergence o f the mass transport in the continuity equation are discretized with the Crank-Nicolson method. Substituting the equations of motion in the continuity equation, integrating the continuity equation over the
214 e n t i r e w a t e r column, tion,
and u s i n g t h e f r e e surface k i n e m a t i c boundary c o n d i -
r e s u l t s i n an e l l i p t i c e q u a t i o n f o r
solved economically w i t h description of al.
this
t h e water elevation,
successive o v e r r e l a x a t i o n
semi-implicit
two-step
technique.
method
which i s
A detailed
i s g i v e n b y Duwe e t
(1983) o r by Backhaus (1985). A c c o r d i n g t o Arakawa and Lamb (1977), t h e
n o n - l i n e a r terms i n t h e e q u a t i o n s o f m o t i o n a r e d i s c r e t i z e d w i t h an energy and e n s t r o p h y c o n s e r v i n g scheme. The h o r i z o n t a l g r i d r e s o l u t i o n i s two k i l o metres and t h e v e r t i c a l r e s o l u t i o n i s f i v e metres. The model works w i t h a t i m e s t e p o f 745 seconds.
The model parameters a r e a c o n s t a n t h o r i z o n t a l
eddy v i s c o s i t y o f 2 m 2 / s ,
a c o n s t a n t v e r t i c a l eddy v i s c o s i t y o f 30 cm2/s,
and a c o e f f i c i e n t o f b o t t o m f r i c t i o n o f 0.0015. The t r a n s p o r t
of
d e n s i t y ( E q u a t i o n 5)
Lagrangian t r a c e r technique,
i s computed w i t h
an a c t i v e
which i s e x p l a i n e d i n t h e f o l l o w i n g .
From t h e
numerical p o i n t o f view, t h e main disadvantage o f t h e E u l e r i a n system i s t h e a d v e c t i v e p a r t o f t h e t r a n s p o r t e q u a t i o n y i e l d i n g t h e w e l l known numerical problems, e s p e c i a l l y i n t h e presence o f s t r o n g g r a d i e n t s . By t r a n s f o r m a t i o n o n t o Lagrangian c o - o r d i n a t e s
i t s i m p l y vanishes. F o r t h e d i f f u s i o n p a r t o f
t h e e q u a t i o n which i s a p r i o r i assumed t o be s m a l l i n r e l e v a n c e and unknown i n detail,
a s i m p l e procedure which reproduces q u a n t i t i e s o f s t a t i s t i c a l
s i g n i f i c a n c e o n l y , may be regarded as s u f f i c i e n t . A w a t e r body i s i n t e r p r e t e d n o t as a continuum b u t as a f i n i t e s e t o f
water p a r t i c l e s w i t h d e f i n i t e physical properties. property of
an E u l e r i a n g r i d c e l l
a l l particles i n the cell. co-ordinate
x=x(a,t),
F o r t h a t reasons,
i s t h e ensemble
Each p a r t i c l e can be i d e n t i f i e d by i t s a c t u a l
which depends upon t h e i n i t i a l c o - o r d i n a t e
t h e t i m e , where x(a,O)=a.
the
averaged p r o p e r t y o f a and
The L a g r a n g i a n v e l o c i t y i s d / d t x ( a , t ) = u L ( a , t )
and t h e r e l a t i o n t o t h e E u l e r i a n v e l o c i t y i s u L ( a , t ) = u E ( x [ a , t ] , t ) .
Hence,
t h e t r a n s p o r t and d i f f u s i o n o f t h e p a r t i c l e s a r e c a l c u l a t e d by: X.
1
t+atD x . t + a t [
UL
1
+
(1-1 - 0.5)
P]
(7)
where i i s t h e i - t h p a r t i c l e , uL t h e a d v e c t i o n v e l o c i t y o b t a i n e d by t r a n s formation onto Lagrangian co-ordinates,
P i s t h e "band-width''
o f turbulent
f l u c t u a t i o n , and 1-1 i s a random number u n i f o r m l y d i s t r i b u t e d between 0 and 1. A r e f l e c t i n g boundary c o n d i t i o n f o r t h e random walk i s used. T h i s t e c h n i q u e
has o f t e n been s u c c e s s f u l l y a p p l i e d t o d i f f e r e n t p a r t i c l e p r o p e r t i e s , such as r a d i o a c t i v i t y (Maier-Reimer
1975),
h e a t (Bork
and Maier-Reimer
1978),
zooplankton ( D i p p n e r 1980), o r c r u d e o i l ( D i p p n e r 1983,1984). I n t h e s e a p p l i c a t i o n s , t h e p a r t i c l e s a r e d y n a m i c a l l y p a s s i v e and r e p r e s e n t a c e r t a i n amount
of
substance.
For dynamically
active properties,
this
215
kind of method will not be sufficient. For example, if a particle represents a certain amount of salt, an unrealistically high number of particles would be needed to ensure a sufficient degree of accuracy. In this case, a redefinition of the particle property is necessary. To overcome this problem, the particle should be defined as a water blob with a given salinity. In the present case, each particle has a density, and the density of a grid cell of the model, which drives the circulation model, is the ensemble average of the densities of the particles in the cell. In equation 7, only two processes are considered: the advection by the velocity field and turbulent transport by a stochastical change of the coordinates. Additionally, two processes should be considered; namely, the buoyancy and irreversible mixing. Since the density of a cell is the ensemble average of the density of the particles in the cell, the individual particles are the subject of buoyant forces. Taking friction into account, in analogy to Stokes' Law, a constant velocity is assumed, balancing buoyancy and friction. Therefore, the vertical particle co-ordinate yields: Z. 1 +At =
z1 . +~A t
y ( pt
- Pit)
/P
where p is the density of the cell, P i is the density of the particle, and is a threshold value of a sinking or rising velocity. Irreversible mixing in a cell is simulated by changing the individuality of the particle. This yields an adjustment of the property of the particle to the property o f the cell:
y
Pi + A t = p i t + A t
K
(2r/h)2 ( p t - p i t )
where h is the grid size and K is the coefficient of eddy diffusivity. A detailed description of this technique is given by Maier-Reimer and Sundermann (1981). How many particles are necessary? There are two criteria: An empty cell should be avoided. In this case the density of the cell is averaged from the density of the surrounding cells. The second criterion is the desired accuracy. The model starts with ten particles in each grid box, which guaranties both criteria,and at the river mouth there is a continuous inflow of particles with a density of 8 u t . The initial condition of the density field is shown in Fig. 1. If a particle is leaving the model through the open boundary, it is extrapolated with the velocity on the open boundary. Thus, the particles can move in and out through the open boundary. The parameters of the tracer model are a horizontal diffusion velocity of 5 cm/s, which is proportional to an eddy viscosity of 0.56 m2/s, no
216 vertical diffusion velocity,
a Stokes'
1 mm/s,
velocity of
and an eddy
d i f f u s i v i t y of 6 m2/s. 4 RESULTS AND DISCUSSION The model a c t u a l l y runs on a Cyber 180-840 and needs 100 seconds CPUtime per t i d a l period. periods,
The model reaches a steady s t a t e a f t e r t h r e e t i d a l
and a f t e r t e n t i d a l p e r i o d s a f r o n t i s formed. The spin-up process
o f t h e k i n e t i c energy shows a s t a t i s t i c a l l y steady s t a t e ( F i g . 2 ) .
10
0 ) 0
I
I
20
I
I
I
40
I
I
I
80
60
I
I
100
t i d a l periods Fig. 2
Spin-up o f t h e k i n e t i c energy over 100 t i d a l c y c l e s
Two d i f f e r e n t experiments a r e presented. bottom w i t h a depth o f t e n metres;
The f i r s t v e r s i o n has a f l a t
i n t h i s case,
t h e d e n s i t y adjustment
(Equation 9) i s neglected. The r e s u l t s show a f o r m a t i o n o f a f r o n t w i t h v e r y strong surface density gradients (Fig.
3),
beside t h e t y p i c a l c i r c u l a t i o n p a t t e r n
-
and t h e s t r e a m f u n c t i o n shows
-
an enhancement o f t h e v e l o c i t i e s
i n t h e f r o n t a l zone and a t y p i c a l meandering ( F i g . 3).
The second r u n i s a
c a l c u l a t i o n w i t h a l i n e a r funnel-shaped topography (Fig. l ) , and t h e d e n s i t y adjustment i s simulated w i t h an eddy d i f f u s i v i t y o f 6 m 2 / s .
That means, i f a
p a r t i c l e has a d e n s i t y d i f f e r e n c e o f 1 u t t o t h e d e n s i t y o f t h e c e l l , t h e p a r t i c l e i s adjusted a f t e r 23.8
hours.
I n t h i s case a f r o n t i s a l s o formed
w i t h much weaker g r a d i e n t s o f t h e s u r f a c e d e n s i t y and p a r a l l e l t o t h e depth contour ( F i g .
4).
In t h i s c a l c u l a t i o n t h e r e i s t no r e l a t i o n between t h e
streamfunction p a t t e r n s ( F i g . 4) and t h e d i s t r i b u t i o n of t h e s u r f a c e d e n s i t y .
217
2/
Fig. 3
DENS1 T Y S l G H A 7
S u r f a c e d e n s i t y ( l e f t ) CI = 2 0 t, and t o t a l s t r e a m f u n c t i o n ( r i g h t ) CI = 500 m 3 / s a f t e r 19 t i d a l p e r i o d s ( f l a t b o t t o m )
CI Fig. 4
-
STREAtlFUNC7ION CI 500 M * * 3 / S
-
1
SIGMA 7
-
STREfltlFUNCTION C1 500 M + + 3 / S
S u r f a c e d e n s i t y ( l e f t ) CI = 10 and t o t a l s t r e a m f u n c t i o n ( r i g h t ) CI = 500 m3/s a f t e r 25 t i d a l t ’ p e r i o d s ( l i n e a r topography)
218
STREAMFUNCTION
C I = 1 0 0 fl++3/S Fig. 5
Eddy s t r e a m f u n c t i o n a f t e r 45 t i d a l p e r i o d s C I = 100 m3/s
The reason f o r t h i s ,
i s t h a t t h e v e l o c i t y f i e l d i s c o n t r o l l e d much more
s t r o n g l y by t h e b o t t o m topography t h a n b y t h e d e n s i t y f i e l d . D e f i n i n g an eddy s t r e a m f u n c t i o n f u n c t i o n and after
5 is
t h e model
$ I = $
-7, where
$ i s t h e t o t a l stream-
t h e mean s t r e a m f u n c t i o n averaged o v e r 97 t i d a l p e r i o d s
A t y p i c a l r e s u l t (Fig.
has reached a s t e a d y s t a t e .
i s t h e development o f an a n t i c y c l o n i c and a c y c l o n i c eddy. a r e more o r l e s s p e r s i s t e n t o v e r t h e whole s i m u l a t i o n ,
5)
These eddies
and t h e y seem t o
be produced by t h e topography.
5 CONCLUSION It
is
clearly
demonstrated t h a t
it
i s p o s s i b l e t o c a l c u l a t e eddies
and f r o n t s w i t h t h e a c t i v e L a g r a n g i a n t r a c e r t e c h n i q u e . F o r t h e f u t u r e , an i n t e n s i v e parameter
test
i s p l a n n e d t o o b t a i n b e t t e r knowledge o f
the
model response. A f t e r t h e t e s t t h e model w i l l be extended t o t h e dimensions o f t h e German B i g h t w i t h r e a l i s t i c topography, d e n s i t y f i e l d s , and wind i n p u t . ACKNOWLEDGEMENT T h i s work i s s u p p o r t e d b y t h e Deutsche Forschungsgemeinschaft. The a u t h o r would l i k e t o thank h i s c o l l e a g u e s I .
R. Wais f o r f r u i t f u l d i s c u s s i o n s .
Bork,
K.
Duwe, E.
Maier-Reimer and
219 REFERENCES Arakawa, A. and Lamb, V.R., 1977. C o m p u t a t i o n a l d e s i g n o f t h e b a s i c d y n a m i c a l p r o c e s s e s o f t h e UCLA g e n e r a l c i r c u l a t i o n model. Meth. Computat. Phys. 16: 173-263. 1985. A t h r e e - d i m e n s i o n a l model f o r t h e s i m u l a t i o n o f Backhaus, J.O., s h e l f sea dynamics. D t . h y d r o g r . Z. 3 8 ( 4 ) : 165-187. Becker, G.A., F i u z a , A.F.G. and James, I.D., 1983. Water mass a n a l y s i s i n t h e German B i g h t d u r i n g MARSEN, Phase I . J . Geoph. Res. 88 ( C 1 4 ) : 9865-9870. Bork, I . , 1978. P r e l i m i n a r y model s t u d i e s o f s i n k i n g plumes, SMHI Rapp No. RHO 14. Bork, I . and M a i e r - R e i m e r , E., 1978. On t h e s p r e a d i n g o f power p l a n t c o o l i n g w a t e r 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 E l b e . Adv. Wat. Res. l ( 3 ) : 1 6 1- 168. Bowman, M.J., 1978. I n t r o d u c t i o n and h i s t o r i c a l p e r s p e c t i v e . in: M.J. Bowman and W.E. E s a i a s ( E d i t o r s ) , Oceanic f r o n t s i n c o a s t a l p r o c e s s e s , S p r i n g e r Verlag, B e r l in/Heidelberg/New YorklTokyo. 1980. N u m e r i s c h e S i m u l a t i o n h o r i z o n t a l e r V e r d r i f t u n g v e r t i D i p p n e r , J.W., k a l w a n d e r n d e r Z o o p l a n k t e r . M i t t . I n s t . Meeresk. U n i v . Hamburg, 23: 63-113. D i p p n e r , J.W., 1983. A h i n d c a s t o f t h e B r a v o E k o f i s k b l o w - o u t . V e r o f f . I n s t . M e e r e s f o r s c h . Bremerh. 19: 245-257. D i p p n e r , J.W., 1984. E i n Stromungs- und o l d r i f t m o d e l l f u r d i e D e u t s c h e B u c h t . V e r o f f . I n s t . M e e r e s f o r s c h . Bremerhaven, S u p p l . B. 8. Duwe, K.C., Hewer, R.R. and Backhaus, J.O., 1983. R e s u l t s o f a s e m i - i m p l i c i t t w o s t e p method f o r t h e s i m u l a t i o n o f m a r k e d l y n o n l i n e a r f l o w i n c o a s t a l seas, C o n t i n e n t a l S h e l f Res. 2 ( 4 ) : 255-274. 1986. Some r e s u l t s o f a d i f f u s i o n e x p e r i m e n t F r a n z , H. and K l e i n , H., a t a r i v e r plume f r o n t . D t . h y d r o g r . Z, 3 9 ( 3 ) : 91-112. Harashima, A. and O o n i s h i , Y . , 1981. The C o r i o l i s e f f e c t a g a i n s t f r o n t o q e n e s i s i n s t e a d y s t a t e b u o v a n c v d r i v e n c i r c u l a t i o n . J. Oceanoqr. SOC. Japan, 37: 49-59." 1984. A t h r e e - d i m e n s i o n a l n u m e r i c a l s h e l f - s e a f r o n t model w i t h James, I.D., v a r i a b l e eddy v i s c o s i t y and d i f f u s i v i t y . C o n t i n e n t a l S h e l f Res. 3 ( 1 ) : 69-98.- . Kao, T.W., P a r k , C. and Pao, H.S., 1977. Buoyant s u r f a c e d i s c h a r g e and s m a l l s c a l e o c e a n i c f r o n t : a n u m e r i c a l s t u d y . J. Geoph. Res. 8 2 ( 1 2 ) : 1747-1752. Budeus, G., Gerdes, D., Schaumann, K., and Hesse, K., 1986. Krause, G., F r o n t a l s y s t e m s i n t h e German B i g h t and t h e i r p h y s i c a l and b i o l o g i c a l e f f e c t s . I n : J.C.J. N i h o u l ( E d i t o r ) , M a r i n e I n t e r f a c e s Ecohydrodynamics. E l s e v i e r Oceanography Ser. 42, E l s e v i e r Amsterdam/Oxford/New Y o r k l T o k y o . 1975. Zum E i n f l u O e i n e s m i t t l e r e n Windschubes a u f d i e M a i e r - R e i m e r , E., R e s t s t r o m e d e r Nordsee. D t . h y d r o g r . Z., 2 8 ( 6 ) : 253-262. M a i e r - R e i m e r , E. and Sundermann, J . , 1981. On t r a c e r method i n c o m p u t a t i o n a l h y d r o d y n a m i c s , i n : M.B. A b o t t and J.A. Cunge ( E d i t o r s ) , E n g i n e e r i n g A p p l i c a t i o n o f c o m p u t a t i o n h y d r a u l i c s , P i t m a n London. P r i t c h a r d , D.W., 1955. E s t u a r i n e c i r c u l a t i o n p a t t e r n s . P r o c . Amer. SOC. C i v . Eng. 8 1 ( 7 1 7 ) : 1-11. S t r o n a c h , J.A., 1981. The F r a z e r r i v e r plume f r o n t , S t r a i t o f G e o r g i a . oc. Manag. 6: 201-221. 1
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221
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 0. HAINBUCHER 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 StraBe 71, 2000 Hamburg 13, F.R.G.
ABSTRACT The
basic
ideas b e h i n d t h e s h e l f sea c i r c u l a t i o n model o f t h e I n s t i t u t
Meereskunde
Hamburg (IFMH) a r e o u t l i n e d .
performance
of
rithms
The s t a b i l i t y and
the
t h e n u m e r i c a l scheme a r e e s s e n t i a l l y based upon i m p l i c i t
algo-
o r second o r d e r a p p r o x i m a t i o n s i n t r o d u c e d f o r those terms i n t h e
t i v e e q u a t i o n s which a r e l i k e l y t o produce i n s t a b i l i t y .
fur
computational primi-
As a r e s u l t t h e scheme
t u r n s o u t t o be r a t h e r s t a b l e and much f a s t e r t h a n c o n v e n t i o n a l , s o l e l y e x p l i c i t numerical schemes. The a p p l i c a t i o n o f t h e IFMH-model on t h e d e t e r m i n a t i o n o f t h e low f r e q u e n c y v a r i a b i l i t y o f t h e N o r t h European s h e l f sea p r o v i d e s i n s i g h t
into
a frequency domain where i n f o r m a t i o n f r o m o b s e r v a t i o n a l d a t a i s r a t h e r scarce.
1. INTRODUCTION The
s t o r m surge- and
t y p i c a l g r i d s i z e o f p r e s e n t s h e l f sea models ( t i d a l - ,
circulation-models)
i s o f the order o f ten kilometers.
These models a l l o w f o r a
decent r e s o l u t i o n o f t h e e x t e r n a l Rossby r a d i u s o f d e f o r m a t i o n . However, t h e i n ternal
d e f o r m a t i o n r a d i u s a t m i d - l a t i t u d e s i s w e l l below t e n
kilometers.
This
s c a l e i s n o t r e s o l v e d b y most o f t h e p r e s e n t l y a v a i l a b l e s h e l f sea models. Similiar
as i n ocean m o d e l l i n g we can i n t r o d u c e a s e p a r a t i o n o r
classifica-
t i o n o f b a r o c l i n i c s h e l f sea models a c c o r d i n g t o t h e i n t e r n a l d e f o r m a t i o n s c a l e : general
circulation
models (GCMs) and e d d y - r e s o l v i n g
models
(ERMs).
In
the
f o l l o w i n g we assume t h a t b o t h a r e t h r e e - d i m e n s i o n a l models. Parallel the scale
t o t h e developments i n deep sea m o d e l l i n g t h e r e i s a t r e n d
development (local)
M.Boehlich,
towards
and a p p l i c a t i o n o f ERMs i n s h e l f seas i n o r d e r t o s t u d y phenomena (see f o r example t h e work o f
I.James,
J.Dippner
small and
1987 a l l t h i s volume) and i n o r d e r t o develop s u i t a b l e p a r a m e t e r i z a -
t i o n s o f these processes i n GCMs.
On t h e o t h e r hand GCMs a r e necessary t o
pro-
v i d e t h e ERMs w i t h m o d e l - c o n s i s t e n t boundary c o n d i t i o n s , which c o n t a i n t h e dyna-
222 mics
of
the f a r f i e l d flow.
I t i s a n t i c i p a t e d t h a t t h e f u t u r e development
s h e l f sea models as a whole w i l l p r o f i t f r o m t h i s b i - f o c a l the
approach.
environmental problems i n s h e l f seas c o n s t i t u t e a s e r i o u s demand f o r
rate
and q u a n t i t a t i v e model e s t i m a t e s on v a r i o u s s c a l e s .
often
called
of
Moreover, accu-
I n t h e f i e l d which i s
" w a t e r - q u a l i t y m o d e l l i n g " GCMs and ERMs a r e a b s o l u t e l y
essential
constituents. The
model and i t s a p p l i c a t i o n d e s c r i b e d i n t h i s c o n t r i b u t i o n belongs t o
c l a s s o f t h e GCMs.
the
However, we b e l i e v e t h a t t h e numerical scheme serves a l s o as
a sound b a s i s f o r t h e development o f an ERM s i n c e t h e numerical problems,
which
are d e l t w i t h here, do a l s o appear f o r an ERM. T h i s statement i s v e r i f i e d b y t h e course o f t h e advanced work o f I.James ( t h i s volume,
own c i t a t i o n s t h e r e i n ) and
b y B o e h l i c h ' s f i r s t r e s u l t s on a n e s t e d system o f models f o c u s s i n g on t h e B a l t i c ( t h i s volume), where t h e IFMH-scheme was used f o r h i s ERM and h i s GCM. Since
a d e t a i l e d d e s c r i p t i o n o f t h e IFMH-model and i t s development i s
i n Backhaus (1983a, ponents part
given
1985) we h e r e o n l y b r i e f l y o u t l i n e t h e b a s i c n u m e r i c a l com-
which d i f f e r f r o m commonly used schemes.
T h i s w i l l be t h e
content
I n p a r t B a time-dependent s i m u l a t i o n o f t h e c i r c u l a t i o n o f t h e
A.
European
of
North
s h e l f sea f o r a p e r i o d o f 14 y e a r s i s d e s c r i b e d and analysed w i t h
re-
gard t o i n t r a - and i n t e r - a n n u a l v a r i a b i l i t y o f t h e f l o w . 2. PART A, OUTLINE OF THE IFMH-MODEL The
f o l l o w i n g d e s c r i p t i o n o f o u r model does n o t r e l y upon a s p e c i f i c d i s c r e -
t i s a t i o n o f t h e space domain, t h e model g r i d , because t h e s t a b i l i s i n g a l g o r i t h m s and
o p e r a t i o n s a r e s o l e l y a p p l i e d t o t h e time-domain.
level nates.
We c o n s i d e r a two
scheme based upon f i n i t e d i f f e r e n c e s a p p l i e d t o a l l t h r e e The scheme,
equations
space
timecoordi-
a g r i d - b o x model, i s a p p l i e d on t h e p r i m i t i v e s h a l l o w water
(1.1 and 1.2),
which a r e denoted below i n momentum-form f o r a
fluid
l a y e r with a t h i c k n e s s h. The e q u a t i o n o f motion:
The e q u a t i o n o f c o n t i n u i t y :
Aw =
The
- (UX+VY)
,
shear s t r e s s
g i v e n by:
a t t h e upper and lower boundaries o f t h e d e p t h r a n g e h i s
P p Y = AV( ( u , v ) / h ) ,
223 In
t h e above e q u a t i o n s (U,V)
a r e t h e components o f t h e t r a n s p o r t
i n t e g r a t e d o v e r t h e d e p t h range h;
vertically
E = g p g i s t h e e x t e r n a l ( b a r o t r o p i c ) and I t h e
i n t e r n a l ( b a r o c l i n i c ) p r e s s u r e , s e p a r a t e d a c c o r d i n g t o t h e Boussinesq-approximation;
N(U,V)
and D(U,V)
momentum d i f f u s i o n terms,
r e p r e s e n t t h e n o n l i n e a r a d v e c t i v e and r e s p e c t i v e l y ; AT
shear s t r e s s T ; Av t h e v e r t i c a l eddy v i s c o s i t y ;
tical
the
horizontal
the v e r t i c a l d i f f e r e n c e o f t h e verp the density;
the
f
C o r i o l i s parameter; g t h e a c c e l e r a t i o n due t o g r a v i t y ; s p a t i a l o r temporal d e r i v a t i v e s a r e i n d i c a t e d b y r e s p e c t i v e i n d i c e s . I n t h e e q u a t i o n o f c o n t i n u i t y (1.2)
Aw i s t h e v e r t i c a l d i f f e r e n c e o f t h e v e r t i c a l c u r r e n t v e l o c i t y ; 5 i s t h e e l e v a t i o n o f t h e f r e e s u r f a c e ; an o v e r b a r i n d i c a t e s a ( t o t a l ) d e p t h mean.
In a two t i m e - l e v e l scheme, tially
three
a p p l i e d t o t h e above e q u a t i o n s t h e r e a r e essen-
sources f o r l i n e a r n u m e r i c a l i n s t a b i l i t y a r i s i n g
from
different
terms and combinations o f terms w i t h i n t h e e q u a t i o n s . These are: A ) The o s c i l l a t i o n e q u a t i o n , f o r example: U t = f V . A f o r w a r d i n t i m e a p p r o x i mation
o f t h i s component o f t h e p r i m i t i v e e q u a t i o n s i s a
priori
instable,
as
pointed o u t already b y Fischer (1959). I t i s l i k e l y t h a t t h i s l a t e n t i n s t a b i l i t y damped o u t i n a g r e a t number o f models o f t h e p a s t by an i n c r e a s e
was
horizontal might
momentum
have
diffusion.
Also
i n s h a l l o w water t h e
been accounted f o r b y t h e e f f e c t o f t h e bottom
once t h e l o c a l water d e p t h exceeded ( s a y ) s h e l f values,
necessary friction.
of
the
damping However,
the i n s t a b i l i t y
became
apparent. We
cure
t h i s weakness by i n t r o d u c i n g a second o r d e r a p p r o x i m a t i o n
for
the
C o r i o l i s r o t a t i o n a c c o r d i n g t o Wais ( 1 9 8 5 ) . T h i s a p p r o x i m a t i o n i n c l u d e s t h e ( e x t e r n a l and i n t e r n a l ) p r e s s u r e g r a d i e n t terms i n o r d e r t o m a i n t a i n g e o s t r o p h y .
6)
The
w e l l known s t a b i l i t y l i m i t a t i o n due t o t h e p r o p a g a t i o n
g r a v i t y waves i n a f r e e - s u r f a c e model; rion.
This
t h e Courant-Friedrichs-Lewy
c r i t e r i o n r e q u i r e s a Courant-number,
of
external
(CFL) c r i t e -
which s h o u l d be s m a l l e r
than
unity,
i n o r d e r t o m a i n t a i n l i n e a r c o m p u t a t i o n a l s t a b i l i t y . The r e s u l t i n g l i m i -
tation
o f t h e t i m e - s t e p becomes p a r t i c u l a r l y s t r i n g e n t when deep ocean
regions
are t o be i n c l u d e d i n a s h e l f sea model. We c u r e t h i s weakness b y i n t r o d u c i n g an i m p l i c i t a p p r o x i m a t i o n f o r t h e e x t e r nal
gravity
waves.
T h i s r e q u i r e s t h e s o l u t i o n o f an e l l i p t i c problem f o r
f r e e s u r f a c e e l e v a t i o n a t each t i m e - s t e p .
required f o r t h i s s o l u t i o n ( o f a scalar q u a n t i t y ( t h e surface e l e v a t i o n )
is
t h e h o r i z o n t a l p l a n e ) i s e a s i l y compensated b y t h e advantage o f t h e almost choice
of
t h e time-step.
the
The e x t r a c o m p u t a t i o n a l e f f o r t , which
T h a t i s t o say t h e t i m e consuming
in free
three-dimensional
equations o f m o t i o n need t o be s o l v e d l e s s o f t e n t h a n i n an e x p l i c i t scheme. F o r d e t a i l s see Backhaus ( l o c . c i t . ) . C) also
The
s o l u t i o n o f t h e v e r t i c a l shear s t r e s s terms i n a g r i d - b o x
limited
by a s t a b i l i t y c r i t e r i o n ,
which s t a t e s
a
quadratic
model
is
dependency
224 between t h e v e r t i c a l increment,
the time-step,
and t h e v e r t i c a l eddy v i s c o s i t y
c o e f f i c i e n t : h 2 t 28 t Av. Once t h e space- and time-increments a r e s e t , e s p e c i a l l y i n view o f t h e p o s s i b i l i t y o f u s i n g a l a r g e t i m e - s t e p due t o B ) ,
t h e c r i t e r i o n l i m i t s t h e choice o f
t h e v e r t i c a l eddy v i s c o s i t y c o e f f i c i e n t Av. There i s good reason, based upon t h e a v a i l a b l e l i t e r a t u r e on t h e v e r t i c a l momentum t r a n s f e r w i t h i n t h e sea, t o assume that
this
forcing
coefficient
i s space- and time-dependent.
If
under
t h e c o e f f i c i e n t i s l i m i t e d b y an upper v a l u e t o m a i n t a i n
f e a t u r e l i k e ' c a v i t a t i o n ' may occur.
extreme
wind
stability,
a
As a r e s u l t o f t h e u n r e a l i s t i c d e s c r i p t i o n
o f t h e momentum t r a n s f e r t o o much o f t h e wind induced momentum m i g h t be r e t a i n e d in
t h e upper model l a y e r s ,
As
a whole,
which t h e n may " s l i p " away f r o m t h e l a y e r s beneath.
any n u m e r i c a l l i m i t a t i o n o f t h e v e r t i c a l eddy v i s c o s i t y
means
an
unwanted r e s t r i c t i o n o f t h e p h y s i c s o f t h e v e r t i c a l momentum t r a n s f e r . We
cure
vertical
t h i s weakness b y i n t r o d u c i n g an a d d i t i o n a l i m p l i c i t scheme
for
turbulence
t h e shear s t r e s s terms.
c l o s u r e scheme i n t h e model,
in
the
This allows f o r t h e o p t i o n t o include o r any o t h e r a l g o r i t h m t h a t
a
describes
t h e s p a t i a l and temporal changes o f t h e v e r t i c a l momentum t r a n s f e r parameter Av. We c i r c u m v e n t t h e t h r e e numerical o b s t a c l e s d e s c r i b e d above under A t o C i n t r o d u c i n g a two t i m e - l e v e l scheme, which has t h e f o l l o w i n g g e n e r a l form:
The future
t i m e - l e v e l s a r e i n d i c a t e d b y t h e upper i n d e x n, level.
Time
by
where n+l i n d i c a t e s t h e
averages between t h e s e l e v e l s a r e i n d i c a t e d b y
the
index
n+1/2. The i n t r o d u c t i o n o f t h i s c e n t e r e d t i m e - l e v e l l e a d s t o t h e i m p l i c i t system components f o r t h e e x t e r n a l g r a v i t y waves ( i n 2.1 and 2.2) and f o r t h e
vertical
shear s t r e s s terms i n ( 2 . 1 ) .
I t should be n o t e d t h a t t h e c e n t e r i n g i n t h e t i m e -
domain
which i s e s s e n t i a l l y n e u t r a l w i t h
leads
to
a scheme,
damping o f amplitudes. rators
(on
regard
to
the
In t h e e q u a t i o n s (2.1) and (2.2) s p a t i a l d i f f e r e n c e ope-
any g r i d ) a r e symbolized b y an under-bar i n d i c a t i n g t h e
respective
coordinate.
In ( 2 . 1 ) two o p e r a t o r s T1,T2 appear. t h e C o r i o l i s and t h e p r e s s u r e
These a r e r o t a t i o n m a t r i c e s , a p p l i e d
on
225
where a = cos f At; f3= sin fA t; y= 1 - a . The application of the operator T2 on the pressure gradient terms in (2.1) leads to additional, 'secondary' pressure gradient terms. For this second order approximation of the Coriolis-rotation stability will be maintained, when f a t 5 n, where IT is the circle number. The elliptic system for the free surface is obtained by replacing the momentum divergence terms in the vertically integrated equation of continuity (1.2) by the equations of motion (1.1). To demonstrate this analytically, we differentiate (1.2) with respect to time and arrive at: Stt+
-
-
Utx+Vty
= 0
In this equation the time-derivatives of the total depth mean momentum are replaced by the vertically integrated equation of motion (1.1). The result is the elliptic system, which has the general form: 5 t t +aSxx+b 5 y y + c 5,
= a
Here the cross-derivative of the free surface elevation was added, in order to demonstrate the effect of the rotation operator T 2 in the numerical approximation. Boundary conditions enter the system via the coefficients a to d. It should be noted, however, that the secondary pressure gradient terms may require additional boundary conditions, depending upon the choice of the model-grid. The numerical solution of the above system is equivalent to the solution of a linear system of equations for the free surface. We obtain this solution by means of an iteration, where the method of 'successive over-relaxation' (SOR) was applied with good success. Matrix inversion seems to be not appropriate for this application in view of the time-dependent coefficients. The above sketched procedure to obtain the scalar equation for the free surface is much more straight forward in the numerical approximation. There the solution at time-level n+l of the vertical integral of the equation of motion (2.1) is directly inserted into the approximation for the momentum divergence in (2.2). This is only possible, because the not yet known shear stress terms at level n+1/2 cancel in the vertical integral. Having obtained a solution for the free surface at level n+l, an interim solution for (2.2) can be computed for the pressure gradient terms and for all other explicitly treated terms. The final solution for (2.1) is then obtained by solving the vertical implicit system for the shear stress terms. The succession of operations, i.e. first the solution for the free surface, and then the interim and the final solution, is the only way to integrate the entire scheme in view of the implicit algorithms which involve all three space
226
coordinates.
The scheme as a whole i s s e m i - i m p l i c i t , because t h e i n t e r n a l baro-
pressure g r a d i e n t s ( I x , I y )
clinic
diffusion
and t h e terms D(U,V) f o r h o r i z o n t a l
and t h e n o n l i n e a r advective terms N(U,V) are solved
momentum
explicitly.
The
l a t t e r are approximated b y a vector-upstream a l g o r i t h m ( H a l t i n e r 1971), i n order t o avoid i n s t a b i l i t y due t o c e n t r a l d i f f e r e n c i n g i n a two t i m e - l e v e l scheme. As
pointed
o u t already above,
we r e q u i r e a space- and time-dependent
v i s c o s i t y c o e f f i c i e n t Av f o r r e a l i s t i c s i m u l a t i o n s o f t h e c i r c u l a t i o n . to
include
a two-equation turbulence-closure scheme (reviewed
by
eddy
Attempts
Rodi
1980)
w i t h i n t h e model have f a i l e d f o r a number o f reasons. The main reason might have been a l a c k i n g v e r t i c a l r e s o l u t i o n , which caused t o o l a r g e d i s c r e t i z a t i o n e r r o r s ( t h e v e r t i c a l increments were o f t h e o r d e r o f 10 m, t y p i c a l f o r s h e l f sea GCMs). Further, these have
(time-dependent)
boundary c o n d i t i o n s f o r t h e p r o g n o s t i c v a r i a b l e s
schemes are n o t u n i q u e l y d e f i n e d and t h e y depend upon not
turbulence published 1oc.cit.)
y e t been t e s t e d t h o r o u g h l y under s h e l f sea c o n d i t i o n s modellers pretend t h a t t h e y are u n i v e r s a l ) . applications
concern simple problems,
Most o f
(however, the
in
which the
presently
l i k e open channel f l o w
o r t i d a l f l o w over topography (Johns 1983).
(Rodi
Very seldom a wind s t r e s s
a t t h e sea s u r f a c e i s considered (Blumberg and M e l l o r 1980).
A simplified
turbulence model was a p p l i e d by James (1986) w i t h i n
equation
(equilibrium)
ERM.
any r a t e t h e i n c l u s i o n o f a complete and r e a l l y and
At
parameters,
reliable
onean
working
turbulence m d e l w i t h i n a s h e l f sea model needs f u t u r e research. But a l r e a d y now we can e s t i m a t e t h a t i t would mean a d o u b l i n g o f t h e computational e f f o r t . P r e s e n t l y we circumvent t h i s disadvantage by i n c o r p o r a t i n g a simple a l g o r i t h m based
upon t h e Richardson number,
which a l l o w s f o r an (almost n o n l i n e a r ) feed-
back between t h e momentum t r a n s f e r c o e f f i c i e n t Av and t h e a c t u a l and l o c a l s t a t e of
t h e f l o w and t h e s t r a t i f i c a t i o n .
parameterizations
that
Hainbucher (1985) has t e s t e d a
number
appeared i n t h e 1i t e r a t u r e and found t h e b e s t
matching
w i t h observations f o r a s t a b l y s t r a t i f i e d North Sea f o r t h e parameter-set, posed by Bowden and Hamilton (1975).
The main f e a t u r e o f t h i s
of pro-
parameterization
i s t h e separation o f t h e eddy v i s c o s i t y c o e f f i c i e n t i n a p o r t i o n due t o
neutral
(Avo) and one due t o s t a b i l s t r a t i f i c a t i o n (Avs), r e s p e c t i v e l y : Av = Avmin + Avo * Avs, whereby: Avo = K*H*/Uj/ Avmin cmfs)
,
and Avs = ( I + aRij) - p
i s a minimum l i m i t o f t h e eddy v i s c o s i t y c o e f f i c i e n t ( i n our K a constant,
case
25
H t h e t o t a l water depth, U j t h e c u r r e n t v e l o c i t y i n the
model l a y e r j , R i J t h e Richardson number, a and p are constants, which a r e chosen a c c o r d i n g l y t o Bowden and Hamilton (0=7, p =0.25). T h i s s e p a r a t i o n a l l o w s on t h e one hand t o v a r y t h e eddy v i s c o s i t y , i . e . t h e v e r t i c a l momentum t r a n s f e r , due t o v a r y i n g atmospheric f o r c i n g by t h e p o r t i o n Avo o r on t h e o t h e r hand t o determine t h e r a t e o f momentum t r a n s f e r due t o v e r t i c a l s t r a t i f i c a t i o n by t h e p o r t i o n Avs.
227
3. PART B, AN APPLICATION The concept of the model experiments carried out in this investigation is based upon the assumption that the variability of the circulation in a shallow shelf sea is to a very high extent induced by the action of the atmosphere. In this study the energy transfer between atmosphere and ocean will be restricted to the transfer of momentum. Suitable data which would also allow to include thermodynamics is as yet not available. In order to obtain estimates of the flow for longer time scales, a number of diagnostic simulations were carried out for the North Sea and adjacent shelf regions by prescribing a monthly or a seasonal or even an annual climatological mean of the atmospheric forcing (Prandle 1978, Maier-Reimer 1977, Davies 1982, Backhaus and Maier-Reimer 1983, Backhaus 1983b). The resulting circulation obtained under the prescribed stationary forcing is then assumed to represent the respective climatological mean. This diagnostic approach, frequently applied also for estimates of the oceanic circulation, relies upon the assumption that the momentum transfer between atmosphere and ocean is linear. However, contrary to the deep ocean where it is justified to assume linearity (Willebrandt 1978), in a shallow shelf sea nonlinear bottom friction and nonlinear tidal interactions are dominant features. Thus, there is reason for doubt on the validity of the assumption of linearity as was demonstrated by Backhaus et. al. 1986. Further, the diagnostic simulations do not provide information on the fluctuations of the flow for time scales smaller than the period of the respective mean. As a result practically no information exists which concerns low frequency variability o f the flow induced by the transient activity of the atmosphere. We define the low frequency variability by fluctuations of the flow with time scales which are well above the average response time of the system to atmospheric forcing. For the North European shelf sea the response time ranges between one and about five days, depending upon the rate of frictional dissipation and thus, mainly upon the local amplitude of the tides and the water depth. Since this frequency band coincides with that of synoptic scale atmospheric disturbances one may conclude that a single storm will not contribute significantly to the low frequency flow. We shall therefore call fluctuations of the flow with time scales between weeks and years "subsynoptic" fluctuations. The obvious lack of information on the low frequency variability of the circulation and the doubt concerning the existing diagnostic means provides the main motivation for this investigation; Therefore we will place our focus on the determination of a mean circulation by considering nonlinear processes and on the determination of the order of magnitude and the time scales of atmospherically induced sub-synoptic fluctuations of the circulation, i .e. deviations from the mean.
228
fig.
1:
Mean w i n t e r l y s u r f a c e s a l i n i t y d i s t r i b u t i o n
229
3.1 THE FORCING In the model simulations the effects of a time dependent atmospheric forcing, of the tide and of stratification were incorporated. The circulation will result from a superposition of these three forcing components. The data which was used to determine the atmospheric forcing was kindly provided by the Norwegian Meteorological Institute. The record of 28 years (19551982) of six-hourly sea surface air-pressure distributions on a 150x150 km grid covers the north-eastern North Atlantic Ocean, the adjacent European shelf and some continental margins. The air-pressure fields were used to estimate the wind stress from the geostrophic wind by means of the nonlinear relationship by Luthardt and Hasse (1981, 1983). These parameters (air-pressure and wind stress) constitute the atmospheric forcing for our model (Backhaus et. al. 1985). The considerable horizontal density gradient which is present throughout the year is demonstrated by the climatological mean of the winterly surface salinity displayed in figure 1 for the first model layer. With regard to the simulations it would be desirable to have at least an individual mean of the actual density distribution for each season within the simulation. However, the data coverage of presently available observations is too insufficient for this purpose which in particular applies for the shelf and ocean regions apart from the North Sea. Thus, presently we prescribe a climatological sumner respectively winter mean of the stratification. For the North Sea region the data for temperature and salinity was digitized and interpolated from the charts of climatological monthly means published by Tomczak and Goedecke (1962) and by Goedecke et. al. (1967). For the rest of the model area the climatological means on a 1x1 degree grid prepared for general ocean circulation models by Levitus (1982) were taken. Substantial spatial interpolation was necessary due to the coarse grid in order to obtain a first order approximation of the three-dimensional density field in the model domain. As a consequence the structure of the baroclinic jet at the continental shelf edge (Gould et. al. 1985) is not adequately resolved by the data. A final dynamical interpolation was carried out by means of the baroclinic circulation model in order to eliminate inconsistencies caused by the extensive interpolation. A prognostic advection equation for the baroclinic fields was incorporated in the model scheme and an approximate geostrophic and hydrostatic balance of the density field was obtained by prescribing the forcing of the tide and of a winter respectively summer mean of the wind forcing. The baroclinic pressure fields derived from the dynamically balanced climatological means of the density distributions were prescribed as a constant forcing in all simulations in order to include at least the mean influence of the stratification on the circulation.
230 The M2- t i d e i s p r e s c r i b e d as sea s u r f a c e e l e v a t i o n a t t h e open boundaries o f the
model domain.
40
The r e s o l u t i o n o f t h e t i d e w i t h a r a t h e r coarse t i m e s t e p o f
minutes t u r n e d o u t t o be s u f f i c i e n t enough i n r e g a r d t o t h e
the
question.
density
and
formulation of
I n p a r t i c u l a r t h e n o n l i n e a r i n t e r a c t i o n s between t h e t h e wind induced f l o w which s t r o n g l y i n f l u e n c e t h e
tide,
low
the
frequency
( r e s i d u a l ) f l o w f i e l d were t a k e n i n t o account. Furthermore, a n o t h e r t i m e dependent boundary c o n d i t i o n i s g i v e n by t h e i n v e r se
b a r o m e t r i c e f f e c t due t o a i r - p r e s s u r e v a r i a t i o n s .
A b a r o c l i n i c component o f
t h e boundary c o n d i t i o n r e f l e c t s t h e e f f e c t o f t h e g e o s t r o p h i c a l l y balanced w i n t e r l y r e s p e c t i v e l y summerly d e n s i t y f i e l d . line-integral boundary, empirically that
the
of
t h e v e r t i c a l l y integrated pressure gradients along
where
the
mean
T h i s component i s o b t a i n e d f r o m a
i n t e g r a t i o n c o n s t a n t which f i x e s t h e mean sea
the
open
level
was
determined b y a d j u s t i n g t h e mean sea l e v e l a t t h e open boundary mean l e v e l a t t i d e gauges a t t h e c o n t i n e n t a l c o a s t agreed
within
so an
o v e r a l l r e l a t i v e e r r o r o f 20%. 3.2 MODEL EXPERIMENTS An ensemble o f 14 y e a r s was s i m u l a t e d (1969-1982) i n o r d e r t o o b t a i n a r e p r e sentative
data
b a s i s f o r t h e c a l c u l a t i o n o f c l i m a t o l o g i c a l seasonal means
and
f o r t h e a n a l y s i s o f low f r e q u e n c y v a r i a t i o n s . The m e r i d i o n a l d i s t a n c e o f t h e s p h e r i c a l model g r i d i s 12 minutes, t h e l o n g i I n t h e v e r t i c a l t h e model i s r e s o l v e d b y 7
layers
s i m u l a t i o n s o f t h e w i n t e r seasons (October t o A p r i l ) and b y 12
layers
t u d i n a l d i s t a n c e 20 m i n u t e s . for
the
for
t h e sumner s i m u l a t i o n s .
(suner)/20m
(winter)
The l a y e r t h i c k n e s s i n c r e a s e s w i t h d e p t h f r o m
i n t h e upper l a y e r s o f t h e model a r e a t o
2500m
10m
in
the
whereas
the
deeper l a y e r s o f t h e model area o u t s i d e t h e s h e l f r e g i o n . The s i m u l a t i o n were c a r r i e d o u t w i t h a t i m e s t e p o f 40 minutes, model
o u t p u t was reduced t o d a i l y v a l u e s by i n t e g r a t i n g o v e r two t i d a l p e r i o d s ,
removing t i d a l and i n e r t i a o s c i l l a t i o n s .
A
s t a t i s t i c a l a n a l y s i s o f t h e d a t a was c a r r i e d o u t due t o
low f r e q u e n c y t i m e s c a l e s .
a
separation
of
The d a t a was low-pass f i l t e r e d w i t h a c u t - o f f p e r i o d
o f 6,5 days and s u b s e q u e n t l y w i t h a c u t - o f f p e r i o d o f 90 days. The d i f f e r e n c e o f both
filtered
data
sets contains the intra-annual v a r i a b i l i t y i n
range f r o m about 10 t o 90 days, N o r t h Sea (Backhaus e t .
the
period
S t a t i s t i c a l analysis o f transport rates in
the
a l . 1985) has shown t h a t e v i d e n t e n e r g e t i c f l u c t u a t i o n s
e x i s t i n t h i s p e r i o d range. F l u c t u a t i o n s m a i n l y due t o t h e annual c y c l e o r great e r p e r i o d s a r e i n c l u d e d i n t h e 90-days-low-pass f i l t e r e d data. T h i s c u t - o f f period
is
somewhat
a r b r i t r a r y and cannot be l i n k e d t o any l o w energy
spectral f l u c t u a t i o n estimates. inter-annual v a r i a b i l i t y .
level
in
However, we use t h i s d a t a b a s i s t o e v a l u a t e t h e
23 1
Spring
Autumn
Winter
Summer
fig.2: Seasonal climatological means of the circulation represented as streamfunctions in Sverdrup ( 106m3/s).
232 3.3 CLIMATOLOGICAL MEANS By a v e r a g i n g t h e 90-days-low-pass February,
March-May,
calculated
f i l t e r e d d a t a o v e r t h r e e months
(Oecember-
June-August, September-November ) c l i m a t o l o g i c a l means were
f o r each season.
F i g u r e 2 shows t h e v e r t i c a l l y i n t e g r a t e d
seasonal
means r e p r e s e n t e d as s t r e a m - f u n c t i o n s i n Sverdrup ( 106m 3 / s ) . For
the
e n t i r e s h e l f r e g i o n we o b t a i n t r a n s p o r t r a t e s which a r e w e l l
one Sverdrup,
below
i n w i n t e r and autumn t h e y r e a c h h i g h e r v a l u e s t h a n i n s p r i n g
and
The 200111c o n t o u r , which i s t h e s h e l f edge, r o u g h l y c o i n c i d e s w i t h t h e 1
summer.
Sv c o n t o u r . About
75% o f t h e A t l a n t i c water masses e n t e r i n g t h e N o r t h Sea b a s i n i n e s p e c i a l l y t h r o u g h t h e Orkney-Shetland passage,
north,
the
a r e f l o w i n g back t o t h e
A t l a n t i c v i a t h e Norwegian c o a s t a l c u r r e n t a t t h e e a s t e r n boundary o f t h e basin. This
open
re-circulation cell,
which o n l y weakly i n f l u e n c e s t h e
exchange
of
water masses i n t h e s o u t h e r n p a r t o f t h e N o r t h Sea, can be v i s u a l i z e d b y t h e 0.6 Sverdrup c o n t o u r . I n sumner and s p r i n g t h e r e - c i r c u l a t i o n c e l l i s removed f a r t o the
n o r t h and t h e eastward g o i n g f l o w o f t h e c e l l i s much more pronounced
i n t h e two o t h e r seasons.
than
T h i s r e s u l t agrees w e l l w i t h measurements o f t h e F a i r
I s l e c u r r e n t (Dooley 1983). The western approaches o f t h e N o r t h European s h e l f sea a r e c h a r a c t e r i z e d by r a t h e r weak t r a n s p o r t v a l u e s i n t h e C e l t i c Sea and i n t h e I r i s h Sea, whereas t h e f l o w o b t a i n e d i n t h e a r e a around t h e H e b r i d i e s i s o f s i m i l a r magnitude as i n t h e southern
N o r t h Sea.
Beyond t h e s h e l f a w e l l o r g a n i z e d f l o w o f about 1-5
Sver-
drup, a p a r t o f t h e N o r t h A t l a n t i c e a s t e r n boundary c u r r e n t system, c l o s e l y f o l lows
the
s h e l f edge f r o m I r e l a n d towards t h e n o r t h e r n boundary
of
the
model
domain. These w a t e r masses f e e d t h e r e - c i r c u l a t i o n c e l l w i t h i n t h e N o r t h Sea. 3.4 THE DETERMINATION OF VARIABILITY The
determination
effort. marine
The
o f low f r e q u e n c y f l u c t u a t i o n s i s n o t o n l y
an
academical
d i s t r i b u t i o n o f p o l l u t a n t s i n t h e sea as w e l l as t h e t r a n s p o r t
organismns
and t h e r e b y t h e i r f u r t h e r f a t e a r e e x t e n s i v e l y dependent
of on
t h e l o n g p e r i o d i c a l v a r i a b i l i t y o f t h e c i r c u l a t i o n . The abnormal d i s t r i b u t i o n o f h e r r i n g l a r v a e and s p r a t s t o c k s i n t h e s e v e n t i e s i s an example o f t h i s dependenc y ( C o r t e n 1986).
A
general view about i n t e r - and i n t r a - a n n u a l v a r i a b i l i t y o f t h e
i s g i v e n by two t i m e s e r i e s o f t r a n s p o r t r a t e s ( f i g u r e 3: Strait, shows
f i g u r e 4:
circulation
s e c t i o n t h r o u g h Dover
s e c t i o n t h r o u g h t h e Faeroe-Shetland passage). The s o l i d l i n e
t h e i n t e r - a n n u a l f l u c t u a t i o n s ( f i l t e r c u t - o f f 90 days) d u r i n g
1969 t o 1981.
the
years
The shaded a r e a demonstrates t h e i n t r a - a n n u a l f l u c t u a t i o n s (band-
pass: 6.5 t o 90 days) d u r i n g t h e same p e r i o d . These f l u c t u a t i o n s can be regarded as t h e s t a n d a r d d e v i a t i o n 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 .
I t should,
however,
233
fig. 3: Time series o f transport rates through section Dover (1969 - 1981). Solid line: Inter-annual variability, shaded area: Intra-annual variability
fig. 4: Time series of transport rates through section Faeroe-Shetland (1969 - 1981). Solid line: Inter-annual variability, shaded area: Intra-annual variability
234
mean. mear 8118: 81182-
8481
~
-
-
- 7 4
8OlSl
79/80-
791aa 78/79-
78\79 77i78-
----' - 7
---A q ' ---y
--17178
78177-
761 77
rr176-
7.51 78.
74757q74-
---\I
74\75. 73174
72h3.
11172-
7
7
7 i L::::: -
7d71-
-
~
103
I:::::::
lb2
days
:
I::!::::
:
Ibl
101
north
east
fig. 5:
Spectra o f the east and north wind stress components over the central Xorth Sea. Considered winter seasons: 1970/71 - 1981/82
235
mean-
8ll82eop31-
----> ===% 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
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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
<
0.5
-
2.0 3.0
-
11.0 12.0
-
1.0 1.5 U.0
5.0 6.0 7.0 y 7.0 8.0 8.0 - 9.0 9.0 - 10.0
5p l O . 0
510
p11.0
412.0 615.0
4
15.0 20.0
cn/?s 2 0 - o
56O
550
540
100
I10
120
13O
F i g . 6a. Computed mean currents o f model 6 in the 1st layer ( 0
140
-
5 m)
297
f
5 5 51°
0.5 1.0 1.5 2.0 3.0 9.0 5.0 6.0 7.0 8.0
5 9.0
510.0
R11.0 g12.0 615.0
4
-
10.0 11.0 12.0 15.0 20.0
-
10 m ) .
1.0 1.5 2.0 3.0
9.0 5.0 6.0
7.0 8.0 9.0
cn/'s20*o
56O
55'
54'
F i g . 6b. Computed mean currents o f model B i n the 2nd layer ( 5
298
<
I"
5 0.5 --1.0 1.5
f, 2.0 -
'
$
5
5
51'
2.0 3.0 4.0
3.0 11.0 - 5 . 0 5.0 - 6.0 6.0 - 7.0 7.0 - 8 . 0 8.0 - 9.0 9.0 - 10.0
$10.0 111.0 g12.0 415.0
4
0.5 1.0 1.5
-
11.0 12.0 15.0 20.0 cn/>s2 0 - o
56'
55
54 100
I10
120
F i g . 6 c . Computed mean c u r r e n t s o f model
13"
140
B i n t h e 3 r d l a y e r (10 - 1 5 m).
299
<
. I
5 0.5 1.0 1 5 ' 5 210 3.0 u.0 E 5.0 6.0 -
0.5 1.0 1.5 2 0
3:O 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 15.0
7.0 8.0 9.0 p10.0 J11.0 g12.0 415.0 - 20.0 4 2omo
;
51°
CH/>S
c
56"
c
54' 'I 1
13'
F i g . 6 d . Computed mean c u r r e n t s o f model 8 i n the 4th layer ( 1 5
-
20 m).
300
5Ia
S6O
c
55'
54'
I 100
fJ
110
120
13O
F i g . 7. Computed mean sea s u r f a c e e l e v a t i o n o f model B i n cm.
140
301
11'
F i g . 8a. Computed mean c u r r e n t s o f model C i n the 1st layer ( 0
- 2 m)
302
550
~~
100
F i g . 8b. Computed mean c u r r e n t s o f model C i n t h e 2nd layer ( 2
110
-
4 m).
303
100
110
F i g . 8c. Computed mean currents o f model C in the 3rd layer (4 - 6 m).
304
3 OUTLOOK The next step that will be done is to investigate the time dependent flow and stratification within the Kiel Bight area with regard to frequently occuring oxygen depletions caused by telluric discharge of nutrients.
4 ACKNOWLEDGEMENTS I am indebted to Prof. Dr. J. Backhaus for his valuable advice and assistance throughout this work. This work was funded by the Umweltbundesamt of the Federal Republic of Germany. 5 REFERENCES Backhaus, J. O., 1985. A three-dimensional model for the simulation of shelf sea dynamics. Deutsche Hydrographische Zeitschrift, 38: 165-187. Backhaus, J. 0. and Hainbucher, D., 1987. A finite difference general circulation model for shelf seas. This issue. Backhaus, J., Bartsch, J., Quadfasel, D., Guddal, 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. Institute of Oceanography, University of Hamburg. Bock, K.-H., 1971. Monatskarten des Salzgehaltes der Ostsee, dargestellt fur verschiedene Tiefenhorizonte. Deutsche Hydrographische Zeitschrift. Erganzungsheft Reihe B y Nr.12. Dietrich, G. and Schott F., 1974. Wasserhaushalt und Stromungen. In: L. Magaard and G. Rheinheimer (Editors), Meereskunde der Ostsee. Springer, Berlin/Heidelberg/New York, pp. 33-41. Erhardt, M. and Wenck, A.,1982. Wind pattern and hydrogen sulfide in shallow waters of the western Baltic, a cause effect relationship? 13th Conference of Baltic Oceanographers, Helsinki, 1982: 221-234 Frey, H. and Becker, G., 1986. Untersuchung der langzeitigen Variation der hydrographischen Schichtung in der Deutschen Bucht. Forschungsbericht 102 04 215/15 im Auftrage des Umweltbundesamtes. Gerlach, S.A.,1984. Oxygen depletion 1980-1983 in coastal waters of the Federal Republic of Germany. First report of the working group "Eutrophication o f the North Sea and the Baltic". Berichte aus dem Institut fur Meereskunde an der Christian Albrechts Universitat Kiel, 130. Grassl, H. and Stengel ,M., 1985. Fur chemisch-biologische Prozesse in deutschen Kustengewassern wichtige Wetterlagen. Forschungsbericht 102 04 215/25 im Auftrag des Umweltbundesamtes. Horstmann, U., 1983. Distribution patterns of temperature and water colour in the Baltic sea as recorded in satellite images: Indicators for phytoplankton growth. Berichte aus dem Institut fur Meereskunde an der Christian Albrechts Universitat Kiel, 106. Jacobsen, T.S.,1980. The Belt project. Sea water exchange of the Baltic, measurements and methods. The National Agency of Environmental Protection, Denmark. Lenz, W., 1971. Monatskarten der Temperatur der Ostsee, dargestellt fur verschiedene Tiefenhorizonte. Deutsche Hydrographische Zeitschrift. Erganzungsheft Reihe B y Nr. 11. Miljdstyrelsen, 1984. Iltsvind og Fiskeddd i 1981. Omfang og drsager. Kopenhagen.
305
A STUDY OF VARIOUS OPEN BOUNDARY CONDITIONS FOR WIND-FORCED BAROTROPIC NUMERICAL OCEAN MODELS
L.P. RBED and C.K. COOPER Det norske Veritas, Section for Oceanography, Hovik (Norway) and Conoco Inc., Production Research Dept., Ponca City, OK 74603 (USA)
ABSTRACT This study focuses on the s e n s i t i v i t y of the i n t e r i o r uind-forced response of a n w r i c a l barotropic Ocean model t o changes i n the open boundary conditions (OKs) uhere the term “open” implies a sea boundary uhere the s o l u t i o n i s unknown and must be a s s w d or extrapolated from the i n t e r i o r solution.
Seven d i f f e r e n t OECs
are applied along the tuo l a t e r a l boundaries of a rectangular basin with a f l a t bottom.
One of the longer
sides of the basin i s a s t r a i g h t coast line, u h i l e the other i s clamped (temporal d e r i v a t i v e o f the sea surface i s set t o zero).
Each OEC
i s studied using three wind forcing schemes which t e s t the the boundary un-
der strong local forcing, under weak l o c a l forcing, and u i t h a r e a l i s t i c uind f i e l d (moving cyclone).
The
d i f f e r e n t OECs are compared u i t h each other as u e l l as u i t h a “correct1mcase based on e i t h e r an analytic solut i o n (when available) or model r e s u l t s from an expanded model domain. Level elevation, v o l m e fluxes and excess mass. s i t i v e t o the implemented O K ,
Comparisons are shoun i n t e r m o f sea
I t i s found t h a t the i n t e r i o r response i s o f t e n h i g h l y sen-
that the performance of some OKs i s a strong function of the local wind f o r -
cing a t the boundary, and that a method based on an i n t e g r a t i o n along the c h a r a c t e r i s t i c s p e r f o r m well, and i s generally superior t o the others. The discussion also i d e n t i f i e s the ueaknesses and strengths of the OECs.
1 INTRODUCTION
Open boundary conditions are imposed along the edge of the model domain where the solution is unknown. Open boundaries contrast to other boundaries where the solution can be specified from data, models, or safely assumed (i.e. land). Since the solution is unknown along open boundaries an assumption must be made or the interior solution extrapolated. The importance of the OBC on the interior solution is clearly shown in previous work, e.g. Raed and Cooper (1986). OBCs are receiving increasing attention within the modelling community as suggested by a wave of recent publications: Beardsley and Haidvogel (1981), Harper and Sobey (1983), Chapman (1985), Hayashi et al. (1986), and Martinsen and Engedahl (1986). The general approach of this work is to study the detailed response of several OBCs in a simple ocean basin subject to wind forcing. A finite difference, barotropic numerical model is used. Seven OBCs are considered: (1) a clamped sea surface (dynamic sea surface is set to zero), (2) a zero slope sea surface, ( 3 ) a
306
combination of sponge and radiation condition (Israeli and Orszag, 1981) , (4) a Sommerfeld radiation condition, ( 5 ) a forced/free wave condition (Raed and Smedstad, 1984), (6) an oblique radiation condition after Raymond and Kuo (1984), and (7) a condition constructed by integrating the compatibility equations along the characteristics (originally proposed by Hedstrom, 1979). This study follows the same general approach of Chapman (1985), R0ed and Cooper (1986), and Hayashi et al. (1986). However, this work significantly extends the earlier work in two ways. First, it considers several additional OBCs including: the forced/free wave condition by Raed and Smedstad (1984), the oblique radiation condition by Raymond and Kuo (1984), and the characteristic method by Hedstrom (1979). Second, the present study considers three wind fields: a uniform wind with strong winds on the open boundaries, a bell-shaped wind with weak forcing on the open boundaries, and a moving cyclone. The first and second test the OBCs with strong and weak local forcing, respectively. The third tests them for a realistic case with time varying wind forcing on the open boundaries. The geometry of the model domain is chosen to simulate a shallow continental shelf. Thus, the integration area consists of straight coast, two lateral open boundaries and an offshore open sea boundary running parallel to the coast. The seven OBCs are imposed on the the two lateral boundaries. Along the offshore boundary the dynamic sea surface is set to zero (clamped). This is done because: (1) it is frequently found to be justified along the edge of the continental shelf (Csanady, 1982), and (2) the condition insures that the problem is mathematically well-posed for all seven OBCs. The model and the OBCs are described in Section 2. Section 3 develops the *Istandard1'results or bench marks for the three alternative wind cases. The bench marks are developed from analytic solutions, where possible, or numerical solutions derived from an extended model domain. In the latter case the cross-shore boundaries are moved so far away that they do not affect the original domain for the integration period ( 4 days). The most important part of this study is provided in Section 4 where the results from the test cases are compared with the bench marks. Comparisons are made in terms of snapshots of sea surface elevation and time series of the sea surface elevation, alongshore depth mean currents and excess mass. A discussion and some final remarks are provided in Section 5 . 2 FORMULATION 2.2 The model,
The model employed is one commonly used for simulation of shelf circulation and storm surge predictions, i.e. the wind-forced shallow water equations. Let U,V be the volume flux components along the (x,y) axes and the sea surface elevation above the equilibrium depth. Then the governing equations are
307
and
where t denotes time, g the gravitational acceleration, the density of sea water, f the Coriolis parameter, H the constant equilibrium depth, T s x ~ ythe components of the wind stress at the sea surface and TbXfY the similar components of the bottom stress. Subscripts x,y and t indicate differentiation with respect to subscript. A linear bottom stress formulation will be used in which the bottom stress is proportional to the depth mean current, viz., Tbx=eRU/H , and
Tby=tRV/H
(4)
where R is the bottom friction coefficient. This simple bottom stress parameterization is preferred because it allows analytic solutions to be derived as bench marks. Also the emphasis is on the sensitivity of the response to the changes in the imposed OBC and not on the ensuing circulation itself. TABLE 1
Parameters and physical constants used i n the nunerical experiments
...................................................................................................... Svmbol
Parameter
Va Iue
Unit
...................................................................................................... f
C o r i o l i s parameter
0
Density of sea uater
9
Gravitational acceleration
R
Bottom f r i c t i o n c o e f f i c i e n t
L
Alongshore length of basin
bS
Grid s i z e (between adjacent elevation points)
At
l i m e incremmt
H
E q u i l i b r i u n depth
1.2-10.~ 1025.0 9.81 2.4'10.3 1000.0
20.0
m-
3
kg/m
m/s2
m/s km km
180.0
S
50.0
m
......................................................................................................
The numerical approach closely follows that of Martinsen at al. It is an explicit, finite difference foreward-backward scheme using a staggered grid (Arakawa C-grid). The various parameters used in the simulation are given in Table 1, and the model grid is shown in Fig. 1. The grid resolution and time step chosen is well within the CFL constraint, and typical shelf waves are well resolved (Wajsowicz, 1986). The boundary condition applied at the coast (no flux) and at the outer offshore boundary (clamped) are common to all cases and may be expressed as follows, (1979).
308
V=O at y=O,
(5)
q = O at y=L/2.
(6)
Here L is the length of the regular grid (Fig. 1 and Table 1).
-N N
N-I
N-2
j+1
j
j -1
3 2
1
2
i-I
3
i
Fig. 1. The regular g r i d used i n the n w r i c a l experiments. km.
The applied g r i d i s an Arakaua C-grid.
U-point, and
I
a cross-shore or V-point.
i+1
M-2 M-I
The dimension of the rectangle i s 1000 by 500
o denotes an elevation or ?point,
-
an alongshore f l u x or
Note that the boundaries located t o the l e f t (southern) and t o the
r i g h t (northern) are along U-points, u h i l e the offshore boundary and the c o a s t l i n e f a l l s along V-points.
2.2 The O B C s Seven O B C s have been considered in this study as summarized in Table 2. The notation used by Chapman (1985) has been followed. The first four (i.e. CLP, GRD, MOE and S P O ) were part of his
study, but different numerical analogues have been used here. The Including these two in this study provides little in the way of new findings, but does provide a common grounds for comparison with the previous studies. The details of the numerical analogues of the various O B C s are given Appendix A, while their analytic expressions are given below. CLP and GRD were also studied by Hayashi et al. (1986).
M
309
First, note that CLP, GRD, MOE, FOE, and ORC are essentially only variations of the Sommerfeld radiation condition expressed in the form Qt+cXQx+cYQy=O
at
x = O , L.
(7)
Here Q is any of the three dependent variables n,U,V and cx,y are the projections of the wave celerity along the x,y axes. TABLE 2 OBC types
....................................................................................................... OBC type
Abbreviation
Source
....................................................................................................... Clamped
CLP
Various authors
Gradient
GRD
Various authors
Free wave r a d i a t i o n
HOE
Camerlengo and O'Brien (1980)
Mixed spangelfree wave
SPO
I s r a e l i and Orsrag (1981)
Forced wave r a d i a t i o n
FOE
R0ed and Smedstad (1984)
Oblique r a d i a t i o n
ORC
Raymond and KUO (1984)
Characteristics
HOC
Hedstrm (1979)
Ana Iy t ic
ANA
Various authors
Extended g r i d
EXT
This paper
......................... _..............................................................................
CLP and GRD have historically been the most popular OBCs in ocean models. For CLP, both cx and cy are zero, so that Q (in this case the surface elevation) does not change in time. The analytic expression becomes ')=O
at
x=O,L.
(8)
For GRD the sea surface slope is set to zero at the boundary, which is tantamount to let cy=O and c X + w in (7) (with a=?), so that vx=O
x=O,L.
at
(9)
MOE (suggested by Orlanski, 1976 and later modified by Camerlengo and O'Brien, 1980) is the closest to the classical Sommerfeld radiation condition. It is assumed that the wave celerity in the along-boundary direction, cy, is zero in which case the expression for cx may be computed from (7), or cX'-Qt/Qx
-
(10)
In the implementation of MOE, Q is replaced by the alongshore volume flux, and so the analytic expression for MOE becomes Ut+cUUx=O
at
X=O ,L,
(11)
310
with the wave celerity computed by solving (11) with respect to cuI viz., cU=-Ut/Ux. It is important to realize that in order to derive a nontrivial OBC, interior values of U has to be used in (12). Following the approach of Camerlengo and O'Brien (1980), (11) is only used when 'c signifies an outgoing disturbance (i.e. when cu>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)
: x
exp(b(x-L))
: xlL,
Rs=R
where Rs is the bottom friction coefficient within the sponges and b is a factor determined so as to give Rs=4R at the outer edge of the sponges. Any disturbances left at the edge of the sponges are subject to MOE. Note that any OBC may be used for this purpose. For ORC the projections of the wave celerity are computed from the expressions (cf. Raymond and KUO, 1984),
ORC was suggested by Raymond and Kuo (1984) in order to handle waves hitting the boundary at an oblique angle. It differs from
311
the other radiation conditions in that both cx and cy are computed. The rational for ORC is based on the fact that a wave hitting the boundary at an angle may give values for cx by means of The (12) far in excess of the numeric constraint that cx<&/At. analytic expression for ORC is therefore given by (7) with Q replaced by U, viz., ut+cXUx+cYUY=O at X=O,L,
(16)
with the wave celerities computed from
HOC differs drastically from the above OBCs in that it is based on an integration of the compatibility equations derived from the governing equations. Integration of the compatibility equations rather than the primitive equations are usually referred to as the method of characteristics. This method was explored in the fifties by Hartree (1953) and Freeman and Baer (1957a,b) for use in atmospheric models and was employed by O'Brien and Reid (1967) to find the oceanic response due to a stationary hurricane. Hedstrom (1979) used this method to derive a nonreflecting OBC for nonlinear, one-dimensional and nonrotational hyperbolic problems. His condition has been modified here to include two dimensions, rotation and wind forcing. This modified version of the Hedstrom condition appears to resemble the weaklyreflective boundary condition discussed by Verboom and Slob (1984). The compatibility equations for the system (1)-(3) are D1(U+co~)/dt=Fl+coF3
along Dlx/dt=+co
D2(U-co~)/dt=F1-coF3
along D2x/dt=-co
D3V/dt=F2
along D3x/dt=0
where , F1=fV+TsX/ p-RU/H
,
FZ=-fU-co21)y+TsY/f-RV/H
,
F3=-Vy Here co2=gH and the operators Dj ()/dt are defined by Dj ()/dt=()t+(Djx/dt)
Ox
;
j=1,2,3.
In order to minimize reflections, the slope of the incoming characteristic (defined by D /dt) are forced to zero at the boundary. For instance, at zte northern boundary (x=L) the
312
incoming characteristic is given by Dzx/dt. Thus integrating (19) along D2x/dt=0 and using (18) and (20) to solve for 7tl Ut and Vt it follows that, (25) (26) (27)
A similar procedure
to be satisfied at the southern (x=O) boundary. Expressions (26) and (29), which give the time rate of change of the alongshore flux, form the analytic basis for HOC. 3 STANDARD CASES
The section is broken into three subsection describing the bench marks for the three wind fields. For the first case it is possible to derive an analytic steady-state solution as well as an approximate analytic transient solution. The other two cases can not be solved analytically so bench marks are derived from grid systems extended so that the boundaries of the extended grid does not affect the solution within the regular grid during the integration period. It should be noted that in the numerical implementation of the wind stress, the staggering of the grid is taken into account by specifying TsX at U-points and Tsy at V-points (Fig. 1). 3.1 Uniform alonsshore wind
The goal of this case is to test the OBCs with strong persistent wind forcing on the boundaries. The wind stress consists of a positive alongshore wind stress which decays offshore. It is instantaneously applied at model start up, or: TsX=T=Toexp (-ay), T,Y=O
,
(31)
(32)
where the maximum wind stress, To, is 0.1 Pa, and the decay factor, a, is chosen to give an e-folding of 200 km. The latter means that the offshore boundary experience only 8% of the coastal wind forcing
.
313
In this simple case it is possible to derive analytic or at least some approximate analytic solutions to the governing eqs. (1)-(3). Initial conditions are assumed quiescent, i.e. U=V=O and q = O for all x,y at t=O. Along the coast, the normal flux is set to zero, i.e. V=O at y=O. Along the offshore boundary, y=L/2, the elevation is clamped (?=O) as will be required in the numerical simulations. Details on the derivations of the analytic solution is given in Appendix B . Because of bottom friction a steady-state solution exists in the limit as t+oo and is given by, U=HT/pR,
(33)
Thus the steady-state solution is characterized by a wind driven alongshore flux as given by (33). This in turn is in geostrophic balance with the cross-shore pressure gradient determined by the sea surface elevation slope. Using the typical values given in Table 1 a maximum elevation of 9.1 cm is reached at the coast with a maximum depth mean current, (U/H), of 4.1 cm/s. No general transient solution exists for this problem, but a limited solution can be found using Laplace transforms (cf. Appendix a ) . For instance, the alongshore flux at the coast (y=O) is given by U(y=O,t)=(HTo/eR) El-exp(-Rt/H)]
.
(35)
Only an approximate solution for the sea surface elevation is found and is given by
Thus the temporal scale has an e-folding scale equal (H/R) at coastal locations. With the values used this implies that essentially U already has reached its steady-state, to within a few percent, after 20 hrs. However, a somewhat longer time scale has to be expected at offshore locations, because it will take some time for the coast to signal its presence. The approximate solution (36) clearly satisfies both the steady-state solution dictated by (34) and the initial conditions. It does ignore inertial oscillations but experience show these to be of minimal importance in the present cases. The temporal response of the alongshore depth mean current is shown in Fig. 6(a) (look at the dashed curves). A synoptic picture of the steady-state solution is shown in Fig. 5 (ANA), which clearly shows the exponential offshore decay. 3.2 Bell-sharJed win4
The goal of this case is to test the OBCs in the case of a strong wind forcing in the interior and a weak wind forcing close
314 to the boundaries. For this case, the wind stress is bell-shaped in the alongshore direction, and decreases exponentially offshore. The wind is switched on at 0 hrs then shut off after 48 hours or for Octc48h TsX=Toexp(-a2x2 exp (-ay),
(37)
TsY=0 , and for t>48h
Values for To and a are the same as before. This means both the alongshore and cross-shelf e-folding scales are 200 km. Thus at the offshore boundary the value is again 8% of the value at the similar coastal location. However, at the lateral boundaries it is only 0.2% of the maximum wind forcing. No analytic solution is possible for this case, so a numerical solution was derived by extending the model domain by 4 0 0 0 km to the north and south. Since the propagation speed of the fastest wave in the basin is about 22 m / s , this means that the solution within the regular grid can not be affected by any lateral boundary reflections within the 96 hr integration period. The response of this extended grid solution is shown in Fig. 2 , which shows the time series of the alongshore (north-south) depth mean current. As expected the solution for the spin-down period is a mirror image of the spin-up period. Kelvin waves pass through the northern boundary at roughly 9 and 57 hr (Fig. 2, site C ) These are generated by the impulsive wind forcing at 0 and 4 8 hr. The forced solution (Fig. 2, site B ) , is seen to have a temporal response exponential in nature toward a steady-state solution. The solution is characterized by a depression in the southern part and a crest in the northern part as depicted by Fig. 7 after 4 8 hrs of integration (look at the frame marked EXT).
.
3.3 Movina storm
The goal of this case is to test the O B C s for a practical realworld wind field. The wind stress is generated by a cyclone translating at 15 m/s diagonally across the grid from southwest (SW) to northeast (NE) as depicted in Fig. 3. The translation speed is slow enough for Kelvin waves to propagate ahead of the storm. The cyclone center passes the SW corner approximately 38 hrs into the integration. At 4 0 hrs it is located in the middle of the regular grid and leaves it approximately 10 hrs later. The wind stress components are given by T,~=-T, (Y/R~)exp TsY=T, (X/Rc) exp (4/2 1 I
,
(40)
(41)
315
where the argument is given by
PI-r/Rc) (
,
r2=x2+y2,
T0=3.0 Pa, and Rc is the distance from the center of the storm to maximum wind stress (R,=200 km). X and Y are given by
X=X-xO-UOt,
(43)
Y=y-yo-vot.
(44)
2 .0 1. 8 16
14
-.
12
v)
I
u I-
z
10
.8
W
(L (L
.6
3
u
. z
.4
v)
.2 0 -. 2
-.4
-.6
Fig. 2. S o l i d curves depict the tenporal response of the alongshore (north-south) depth mean current (U/H) for the bell-shaped uind forcing using the extended g r i d .
time series a t the f o l l o u i n g g r i d elements: A=(2,2),
The Letter attached t o each curve correspoms t o
8=(25,2), and C=(50,2),
where ( i , j ) denotes g r i d element
location as depicted by Fig. 1.
Here xo,yo denote the initial position of the storm center and uo,vo the velocity components of the center. No analytic solution is possible for this case, so a numerical solution was derived by extending the model domain as in the bell-shaped wind case. At 36 hr the presence of the storm is becoming visible in the form of a Kelvin wave propagating into the regular grid (Fig. 3). At 4 8 hr the storm center is in the middle of the regular grid as indicated by the depression following in the wake of the storm. The Kelvin wave has just reached the
316
;\,
,\, ,
,, ,
,
, , ,, ,, , ,
T I M E = 42 H O U R S
T I M E = 48 H O U R S
T I M E = 57 H O U R S
+ Y u Y Y u Y Y u y +
Fig. 3. Snapshots of sea surface elevation using the extended g r i d . regular g r i d i s shorn.
Onty the subdamin corresponding t o the
Contour i n t e r v a l i s 20 cm betueen l i n e s u i t h 100 cm betueen heavy l i n e s .
track of the moving cyclone i s shoun i n the upper r i g h t panel. l o c a t i o n o f storm center a f t e r s t a r t of simulation. stress (200 km).
T I M E = 60 HOURS
M a x i m uind stress i s 3.0 Pa.
track i s indicated by the dashed rectangle.
The storm
Nunbers along t r a c k ( i n hours) i n d i c a t e
The dashed c i r c l e i n d i c a t e the radius t o m a x i m uind
The l o c a t i o n o f the regular g r i d u i t h respect t o the storm
317
northern boundary at this time and starts to propagate out of the regular grid. By 54 hr the center is located close to the NE corner and the Kelvin wave is nearly gone from the domain. By 72 hr the disturbances created by the storm have essentially disappeared. Fig. 4 shows the situation from another vantage point - the time series of surface elevation at sites A, B and C.
25 *OOt
I:
/AA
t
l l V \ \
3j
t 0
10
20
30
40
50
60
70
80
90
100
TIME IN HOURS Fig. 4. The temporal response i n terms of the sea surface elevation f o r the moving storm case a t the s i t e s A, 8, and C.
Solution i s based on the extended g r i d sinrrlation.
4 SIMULATIONS WITH OBCS
Considered in the present section is the performance of the seven OBCs for the different wind-forced cases. The solutions are generated with the wind forcing described in Section 3 and with the seven OBCs imposed at the two lateral boundaries of the regular grid. The performance is studied by comparing the numerical results with the standard cases described in Section 3 thought to represent the solution with a perfect OBC. The criteria whereby the performance of each OBC may be evaluated is based on the definition of an open boundary suggested by Roed and Cooper (1986). Thus an open boundary is one which allows disturbances originating in the regular grid to leave it without disturbing or deteriorating the solution within the regular grid. Comparisons are made using time series of sea surface elevations, depth mean currents and snapshots of the sea surface
318
elevation. In addition the term excess mass is often used in the comparisons. The excess mass is defined as the difference between the initial mass and the total mass at any instant. By dividing this difference with the area covered by the regular grid and the density the excess mass may be computed from the expression,
4
EM= ( 1/A)
Pdxdy ,
(45)
where A is the area covered by the regular grid. Thus the excess mass is a measure of the mean sea level deviation away from the equilibrium depth. 4.1 Uniform alonsshore wind The solutions after 96 hours of integration are displayed in Fig. 5, which also includes the analytic solution derived in Section 3. GRD, MOE, FOE and HOC all provide reasonable responses, while CLP, SPO and ORC perform poorly. CLP performs poorly mainly because it is unable to produce the correct wind-forced alongshore currents. The analytic steady-state solution (33) and (34) shows this current to be in geostrophic balance with the cross-shore pressure gradient. The condition imposed by CLP effectively prohibits this wind-forced balance at the boundaries. The consequences of this failure extend to the interior of the regular grid as may be seen from the time series depicted in Fig. 6 at about 7 hours. At this time the coastal elevation levels off and reach a steady-state solution which is only 30% of the analytic solution giben by eq. (34). The poor performance of SPO in this case may be explained by the incorrect mass fluxes generated in the sponge when strong forcing is present at the boundaries. By inspection of the steady-state solution (33), the cross boundary flux is decreased in the sponge due to the increased bottom drag coefficient there. Hence it fails to balance the wind forced flux within the regular grid. This creates a false divergence in the alongshore flux which in turn is responsible for the alongshore gradient depicted in Fig. 5 (look at the frame marked SPO). ORC performs poorly in this case because it carries information in the cross-shelf direction by means of free waves. Thus the information about the no-flux condition at the coast propagates along the open boundary in the form of a free wave rather than a forced wave. This information is thus passed along with the speed given by cy of (17), and this is not fast enough to give the correct cross-boundary fluxes. Divergences and convergencies are created at the boundaries which in turn create the false alongshore gradient in the sea surface elevation indicated after 96 hrs in Fig. 5 . Curiously enough MOE offers an acceptable response in this case despite its apparent similarity to ORC. This is because MOE by virtue of the Camerlengo and O'Brien (1980) modified form lacks
319
ANALYTIC CLP
GRD
MOE
Fig. 5 . The solution a f t e r 96 hours of i n t e g r a t i o n i n terms of contours of surface elevation f o r the uniform alongshore wind case using the seven OBCs. The domain corresponds t o the regular g r i d .
Dashed curves indicate depressions.
Contour i n t e r v a l i s 0.5 cm.
The OBC notation follous that of Table 2 .
OZI
320
Fig. 6. Tim evolution of the depth mean current (upper panel) and excess mass (louer panel) a t s i t e B for the uniform alongshore wind case. r i g h t of the time series.
The dashed curve i s the a n a l y t i c solution.
OEC used i s indicated t o the
Scales a r e indicated by the bottom l e f t numbers along the v e r t i c a l axis.
321
the radiative nature of ORC. more rapidly.
Thus MOE updates the alongshore flux
Bell-shaDed wind The solution at 48 hrs is shown in Fig. 7 just as the wind has been shut off. Comparing the different OBCs with the extended grid solution indicates that CLP, SPO, ORC, and to some degree also FOE and HOC provide reasonable responses. GRD and MOE are clearly inferior to the others. These conclusions are supported by the temporal responses shown in Fig. 8 . ORC is superior to the other radiation conditions in this case. As discussed in Section 3 , Kelvin waves are created in the interior by the impulsive wind forcing. These waves are essentially free waves for this case. As noted by Raymond and Kuo ( 1 9 8 4 ) ORC is carefully constructed so as to radiate free waves. Hence it is expected to be superior to most other radiation conditions for this case. Inspection of the CLP temporal response as depicted in Fig. 8 reveals relatively small reflections despite the fact that CLP is purely reflective. This is because the bottom friction effectively dampens out these reflections. Hence the energy of the reflected waves are small in the middle of the basin. The impact of CLP is therefore controlled largely by the dissipation in the system. So for shallow shelf regions with large bottom friction, CLP may be acceptable if the site of interest is far from the boundaries. This is supported by the response shown in Fig. 7 (look at the frame marked CLP). Like CLP, GRD is also purely reflective. But in this case this plays a minor role. More importantly, GRD causes major changes in mass as shown in the lower panel of Fig. 8 . This is most apparent as the Kelvin waves hits the lateral boundaries; mass is extracted from the regular grid at a constant rate. This may be explained by GRD's inherent ability to maintain (or create) a cross-shore geostrophic balance due to the imposed zero alongshore slope. This ability was crucial for the success of GRD in the previous case, but seems to be responsible for its poor performance in this case. The poor performance of MOE is also linked to its excess mass response. Again the difference in response between ORC and MOE is obvious and strongly suggests that ORC is superior to MOE when there is little or no forcing at the boundary. In this particular case one would expect FOE to give a response similar to that of MOE because the local part is close to zero. Certainly, several of the features present in the MOE response are reflected in the FOE response (cf. Figs. 7 and 8 ) . There is, however, one important difference. As the Kelvin wave leaves the boundary to the north, FOE recovers and stabilizes in contrast to MOE. This is especially true in terms of the excess mass behavior (cf. Fig. 8 ( b ) ) , but is also apparent in the alongshore depth mean current (Fig. 8 (a)) 4.2
.
322
EXTENDED GRID
GRD
MOE
SPO
FOE
ORC
HOC
Fig. 7. Snapshots of the sea surface elevation a t 48 hrs f o r the bell-shaped wind case. as i n Fig. 5 .
Contour i n t e r v a l i s 0.5 cm between lines, with 2 cm between heavy l i n e s .
Notation otherwise
c
323
Fig. 8. Same as Fig. 6 for the
bell-shaped w i n d case.
324
HOC reproduces the alongshore depth mean current well (Fig. 8(a)), but not the excess mass. This is supported by the snapshot in Fig. 7 which shows the problem to be linked to the response at the southern boundary. Indeed, looking at the time series of the alongshore depth mean current at this boundary (Fig. 9) shows that it consistently deviates from the extended grid solution in such a way as to provide less flux. This is not the case at the northern boundary. 2.0(
I
-
. I v)
1.0
-
I
1
I
I
I
1
-
HOC
Extended Grid
I
------
9
-
TIME IN HOURS Fig. 9. The time evolution of the depth mean current at s i t e s A and C, respectively, for the bell-shaped uind case.
Dashed l i n e indicates the response f o r the extended g r i d simulation u h i l e the s o l i d curve corresponds
t o the response using HOC.
4.3 Movina storm
The response of the different OBCs for this case is shown in Figs. 10-12. CLP is not presented for this simulation because of its proved poor behavior both in the previous cases and in other cases including realistic wind fields (cf. Beardsley and Haidvogel, 1981; Chapman, 1985; Hayashi et al., 1986). All the OBCs are less than perfect in this case, but GRD, S P O and HOC seem to provide reasonable responses. In order to evaluate the OBCs it is important to realize that some of the features in the extended grid simulation may not be included in the regular grid simulation. For example, a Kelvin wave is generated when the winds are initially started. This wave
325
T I M E l L4,?,HOv,RSL
EXTENDED GRID
GRD
MOE
SPO
FOE
ORC
Fig. 10. Snapshots of t h e sea surface e l e v a t i o n a t d i f f e r e n t OBCs.
N o t a t i o n otherwise 8s i n Fig. 5.
42 and 48 h r s f o r the moving storm case u s i n g t h e Contour i n t e r v a l and parameters as i n Fig. 3.
326
GRID
GRD
MOE
SPO
FOE
ORC
HOC
Fig. 11. Same as Fig. 10 only snapshots are a t 54 and 60 hrs.
327
propagates in front of the storm and enters the regular grid at about 32 hrs (Fig. 12(b)). As is evident by inspection of Fig. 12(b) all the regular grid simulations are unable, except FOE, to provide this Kelvin wave. This delay is caused by the inherent inability of the regular grid simulations to form the correct Kelvin wave outside of the grid domain. It is interesting to note that GRD does fairly well in this case. This may be explained by the conditions inherent ability to create and maintain a cross-shore geostrophic balance. It is able to, somewhat belatedly, to pick up the incoming Kelvin wave due to its strong cross-shelf geostrophic balance (cf. Figs. 10 and 12). The same is true for the depression which follows in the wake of the storm which hits the northern boundary at about 56 hr. GRD, however, is clearly reflective as depicted by Fig. 12(b). In this case, with a shallow sea the energy of the reflected wave is quickly eroded by bottom friction and is far less obvious in the middle of the basin. This is supported by the temporal response of the alongshore depth mean current as depicted in Fig. 12(a). S P O also gives reasonable results. As revealed by Figs. 10-12 the interior response is only slightly distorted by the strong forcing at the boundaries. This is explained by the short time span that forcing is present at the boundaries, and by the fact that the forcing varies strongly in time. The presence of the sponge layer does impact the response somewhat and adversely affects the mass balance. Despite this deviation from the extended grid simulation, especially close to the boundaries, the response in terms of the depth mean current is satisfactorily. HOC also performs reasonably well. Consistent with the previous simulation with weak forcing at the boundary, the deviations are most pronounced at the southern boundary when the storm has left the regular grid domain (Figs. 10 and 11). MOE, FOE and ORC performs poorly for this case. For instance, both MOE and ORC do not provide the incoming Kelvin wave at all. This explains the exceptional long delay in the excess mass response (Fig. 12(b)). Due to the forced constituent of FOE this condition is able to create the incoming Kelvin wave faster than any of the other OBCs, but otherwise it behaves like MOE. Their respective responses are consistent with their responses in the previous two simulations. For instance, as the Kelvin wave hits the boundary to the north it is a free wave. Hence ORC initially transmits the wave almost perfectly, but as the forcing increases ORC deteriorates (Figs. 10 and 11). 5 DISCUSSION AND FINAL REMARKS
The response of a barotropic, numerical ocean model to different open boundary conditions (OBCs) is studied. Emphasis is on the sensitivity of the models interior wind-forced response to changes in the OBC. Seven OBCs are compared with each other as well as with benchmark simulations. These are either analytical solutions or numerical solutions derived from simulations with an
328
Fig. 12. Same as Fig. 6 f o r the m v i n g storm case.
329
extended grid. The OBCs are summarized in Table 2. Three windforced cases have been studied with: 1) strong, 2) weak and 3 ) time varying wind forcing at the models open boundaries. The results are summarized in Table 3 . These evaluations are based upon an intercomparison of the response in terms of integrated quantities such as: 1) excess mass, and 2 ) snapshots of the sea surface elevation as well as time series of: 1) elevation and 2) depth mean current. Evaluation of performance of each OBC is based on the definition of an open boundary as a computationally boundary at which disturbances originating in the interior of the regular domain is free to leave it without disturbing or deteriorating the interior solution (Rsed and Cooper, 1986). TABLE 3 Surrnary of results.
Y(es) indicates reasonable results u h i l e N ( o ) indicates a less satisfactory response.
Conparisons a r e made a t the southern
The a n a l y t i c or extended g r i d solution i s used as a conparison.
boundary (S) and northern boundary ( N ) and i n t e r m of time series of the alongshore depth m a n current a t s i t e B (U) and excess mass (EM).
OBC type
F i n a l l y an o v e r a l l indicator (1) i s provided.
Uniform uind, strong forcing S
N
U E H
T
Bell-shaped uind, ueek forcing
S
N
U E H
1
Roving storm, varying forcing
S
N
U E H
T
.......................................................................................................... . . . CLP N N N N N Y Y Y Y Y GRO
Y
Y
Y
Y
Y
N
Y
N
N
N
Y
N
Y
Y
Y
HOE
N
Y
Y
Y
Y
N
N
N
N
N
N
N
N
N
N
SPO
N
Y
N
N
N
Y
Y
Y
Y
Y
Y
N
Y
Y
Y
FOE
Y
Y
Y
Y
Y
N
Y
Y
Y
Y
N
N
N
N
N
ORC
N
N
N
N
N
Y
Y
Y
Y
Y
N
N
N
N
N
HOC
Y
Y
Y
Y
Y
N
Y
Y
Y
Y
Y
Y
Y
Y
Y
..........................................................................................................
In general the response is highly sensitive to changes in the imposed OBC. There is no I1bestl1solution. For instance, OBCs that provide reasonable responses with strong forcing at the boundaries, are found to perform poorly in cases with weak forcing. This indicates that the choice of OBC depends upon the particular application that the modeler has in mind. The results also indicate that incorrect responses are most likely to show up at an upstream boundary first. Upstream is here with respect to the wind direction (here the southern boundary). The following are the main conclusions: (1) HOC is the only OBC which provides a reasonable response in all cases studied. (2) SPO may prove viable, especially in cases when weak or variable forcing is applied. In cases with strong and persistent forcing, problems are likely to arise and is due to flux constraints imposed by the sponge. ( 3 ) GRD, one of the most widely used OBCs, provides reasonably responses away from the boundary if the domain is shallow, frictionally dominated and relatively large.
330 ( 4 ) ORC is superior to the other radiation conditions when it comes to handling free waves at the boundaries. The results indicate that use of ORC rather than MOE may improve the FOE response. Exploratory experiments (not shown here) support this conclusion. ( 5 ) CLP should be avoided in most applications. However, it is easy to implement and may be used in short term simulations in which disturbances originating at the boundary do not have time to travel back and adversely affect the interior solution. (6) MOE and FOE should also be avoided in most cases. However, FOE could be improved by incorporating a better radiation condition for its global part. The above conclusions are limited to a study of OBCs for a barotropic, linear model. Nonlinearities, stratification and topography have been neglected in this study. Most of the above OBCs are easily extended to nonlinear models, and more complicated topography. Further studies with a shelf break within the regular grid (not shown) suggest that our conclusions are unchanged. Problems will arise in extending the radiation conditions to baroclinic models. In this case the disturbance will be a mixture of baroclinic and barotropic waves and this complicates the numerical computation of the wave celerity of the disturbance. A decomposition into normal modes may possibly resolve this problem. None of the above considerations seem to constrain the applicability of HOC as long as a set of compatibility equations may be derived. There are probably several ways whereby the response due to a particular OBC may be improved. The obvious is to experiment with combinations of different OBCs. Specifically, exploratory experiments (not shown here) with FOE in which MOE was replaced by ORC for the global part was promising. Another suitable combination is to substitute HOC for MOE in SPO. Moreover, the flow relaxation scheme suggested by Martinsen and Engedahl (1986) which combines SPO and the philosophy of FOE, is also reported to have some success.
APPENDIX A: NUMERICAL FORM OF THE SEVEN OPEN BOUNDARY CONDITIONS All the OBCs are applied at alongshore flux or U-points (cf. Fig. 1). Only the implementation at the northern (x=L) boundary are given in detail. 1. CLP
This is the most cost efficient OBC. U-point gives (linear interpolation),
for all time levels n.
Application of
(8)
at a
331
2. GRD
This OBC is based on (9) with the numerical analogue,
for all time levels n. 3. MOE
An upstream differencing as outlined by Miller and Thorpe (1981) has been used in order to compute 'c from (12), viz. ,
Once 'c is computed (11) may be used to find U"+l (Orlanski, 1976). In the present application, however, the modified form suggested by Camerlengo and OIBrien (1980) has been used. Hence r 7
4"
: cu > 0
.
: j=2,3,. ,N.
(A41
4. SPO A s explained in Section 3 the main feature of SPO is to add a sponge outside the southern and northern boundaries. Within the sponge the governing eqs. (1)-(3) are solved as within the regular grid with due regard to the increasing bottom friction coefficient as given by eq. (14). At the outer edge of the sponge MOE is used. The sponge in the present implementation is 200 km wide which corresponds to adding 10x26 grid elements to the regular grid at the northern and southern ends. This increases the computational effort by 20%.
5. FOE
The local part of U is computed from eqs. (1)-(3) by neglecting the terms containing differentials with respect to x. It is important to realize that this subset of equations may be solved without any reference to an OBC. It provides the locally generated Ekman fluxes due to the wind and the local fluxes due to a cross shelf geostrophic current. The ltfreel1or global part at points adjacent to the boundary is found by subtracting the local solution from the total solution. To find the global part at the boundary for time level n+l MOE is used. Finally Un+l at the boundary is found by adding the global and local parts.
332 6. ORC
The numerical form chosen is based on an upstream differencing of (16)I viz. I
'MI
jn+l= ( l-rx)uMIjn+rXUM-l ,j n - r Y ~ y ~jn. M,
(A51
Here the coefficients rx and rY are given by,
and
The celerities cx and cy are computed by a numerical analogue of (17) in a fashion similar to (A3), e.g.
where I
The expressions (A5)-(A12) are all evaluated for j=2,3,.,N and one-sided differences are used for j=2 and j=N whenever necessary. 7 . HOC
In order to find UM,jn+' a finite difference form of (26) has been implemented. The terms containing derivatives with respect to x have been replaced by one-sided differences. Thus,
333
and z=ht/2As. Note that all the expression (A13) and (A16) are evaluated for j=2,3,..,N. APPENDIX B: THE ANALYTIC SOLUTION 1. Steadv-state solution Since bottom friction is included, it is reasonable to look for a steady-state solution in the limit as t+m. Furthermore, since the wind stress only depends on y, the dependent variables will have no x-dependence. With these observations the governing equations become,
fU=-gHVy-RV/H ,
(B2)
vY=o.
(B3)
Integration of the last equation in conjunction with the boundary condition at the coast yields V=O everywhere. Using this along with the clamped condition at the offshore boundary then yields the closed-form steady-state solution (33) and (34). 2. Transient solution No general transient solution exists for this problem. However, a limited solution can be found by transforming the governing equations by invoking Laplace transforms. Let a circumflex denote a transformed quantity. Applying the transforms in the (y,s) space to eqs. (1)-(3) then gives f i n
h
sU-fV=(T/p) -RU/H,
F1
s +v =o. &Y
The first of these equations may be solved with respect to give 6=[fG+(T/p)]/
[s+(R/H)]
G
to
.
Since the transformed $ satisfies the no flux condition at the coast (y=O) (35) follows by inversion of (B7). n NOW, solving (B5) and (B6) with respect to and V gives
i
334
A
v = ( ~ ~ p / ( a ~ -(exp(-ay)-exp(-Ay) ~~)) 1, where i\=(s((s+ (R/H)) 2+f2}/gH[~+(R/H)]
)
'I2 ,
Formally the solutions for U, V, and 3 may now be found by inverting the transformed solutions (B7)-(B9) back to (y,t) space. In general this is not possible because of the essential singularity at the point s=-R/H. Moreover, the approach taken to find U at theccoast does not avoid this singularity in the inversion of 9 . However, it is possible to obtain a good approximation for 7 everywhere by assuming the coastal elevation to be in geostrophic balance or, ?y (Y=oI t)=- (f/gH)u (y=OI t) I
(B12)
.
Assuming the offshore decay to be where (35) specifies U(y=O,t) exponential in nature and (B12) to hold for all y, (36) follows by integration of (B12) with respect to y. ACKNOWLEDGMENTS The authors would like to thank Dr. R.B. Gordon for stimulating and encouraging discussions. This research was supported by Conoco Norway Inc. through a contract with Det norske Veritas. The support is gratefully acknowledged.
REFERENCES Beardsley, R.C., and D.B. Haidvogel, 1981. Model studies of the wind-driven transient circulation in the Middle Atlantic Bight. Part 1: Adiabatic bounflary conditions, J. Phys. Oceanogr., 11: 355-375. Chapman, D.C., 1985. Numerical treatment of cross-shelf open boundaries in a barotropic coastal ocean model, J. Phys. Oceanogr., 15: 1060-1075. Csanady, G.T., 1982. Circulation in the coastal ocean. D. Reidel Publishing Co., 279. Freeman, J.C. Jr., and L. Baer, 1957a. Pseudo-characteristics. Trans. Amer. Geophys. Union, 38: 65-67. Freeman, J.C. Jr., and L. Baer, 195733. The method of wave derivatives, Trans. Amer. Geophys. Union, 38: 483-494. Harper, B.A., and Sobey, R.J., 1983. Open boundary conditions for open-coast hurricane storm surge, Coastal Engineering, 7: 41-60.
335
Hartree, D.R., 1953. Some practical methods of using characteristics in the calculation of non-steady compressible flow, Rep.AECU-2713, U.S.Atomic Energy Comm., Washington, D.C. Hayashi, T., Greenberg, D.A., and Garrett, C.J.R., 1986. Open boundary conditions for numerical models of shelf sea circulation. Cont. Shelf Res., 5: 487-497. Hedstrom, G.W., 1979. Nonreflecting Boundary Conditions for Nonlinear Hyperbolic Systems. J. Comp. Phys., 30: 222-237. Martinsen, E.A., and Engedahl, H., 1986. The flow relaxation scheme as an open boundary condition in a barotropic model, Submitted to Coastal Engineering. Martinsen, E.A., Gjevik, B., and RBed, L.P., 1979. A numerical model for long barotropic waves and storm surges along the western coast of Norway. J. Phys. Oceanogr., 9: 1126-1138. O'Brien, J.J., and R.O. Reid, 1967. The non-linear response of a two-layer, baroclinic ocean due to a stationary, axially symmetric hurricane, 1: Upwelling induced by momentum transfer, J. Atmos. Sci., 24: 197-207. Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic flows, J. Comp. Phys., 21: 251-269. Raymond, W.H., and KUO, H.L., 1984. A radiation boundary condition for multi-dimensional flows, Quart. J. Roy. Met. SOC., 110: 535-551. R ~ e d ,L.P., and Cooper, C.K., 1986. Open boundary conditions in numerical ocean models. In: J.J.O'Brien (Editor), Advanced Physical Oceanographic Numerical Modelling, NATO AS1 Series C, Vol. 186, D. Reidel Publ. Co.: 411-436. RBed, L.P., and Smedstad, O.M., 1984. Open boundary conditions for forced waves in a rotating fluid, SIAM J. Sci. Stat. Comput., 5: 414-426. Sommerfeld, A., 1949. Partial differential equations: Lectures in Theoretical Physics, Vol. 6, Academic Press, 1949. Verboom, G.K., and Slob, A . , 1984. Weakly-reflective Boundary Conditions for Two-Dimensional Shallow Water Flow Problems. Paper presented at the 5th International Conference on Finite Elements in Water Resources, Burlington (VT), U.S.A., June 1984. Wajsowicz, R.C., 1986. Free planetary waves in finite-difference numerical models. J. Phys. Oceanogr., 16: 773-789.
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337
COASTAL CURRENTS. INTERNAL WAVE COLLAPSES AND TURBULENCE IN STRAIT OF MESSINA ZONE
THE
E. SALUSTI INFN, Dept. of Physics, Universita' "La Sapienza" P.le A.Moro 2, 00185 Rome, Italy R.SANTOLER1 Istituto di Fisica dell'Atmosfera, P.le Sturzo 31, 00144 Rome, Italy
CNR
ABSTRACT Physical phenomena occurring in the area of the Strait of Messina, related to the current and interface rising on the sill are analyzed on the basis of recent experimental observations. The measurements o f the interface time evolution on the sill have revealed for the first time the presence of turbulent columnar disturbances. The formation of a peculiar front in the Gulf of Gioia, in the southern Tyrrhenian Sea, is described as the result of turbulent mixing due to the breaking of nonlinear internal waves on the shelf of this Gulf, directly oriented toward the Strait of Messina where these waves are generated. 1. INTRODUCTION In this note we outline some physical phenomena occurring in the area of the Strait of Messina, on the basis of new experimental observations,
made by the Department of Physics of the
Uni-
versity of Rome in the years 1980-1985, and of historical data. The Sicily nian
strait of Messina separates the Italian peninsula
from
(Fig.1) and is the natural connection between the TyrrheSea
narrowest
and the Ionian Sea.
The width of the
Strait
at
the
point is approximately 3 km where the average depth is
75 m. The sea bottom slopes down to approximately 1 3 0 0 m towards the Ionian Sea and to 300-600 m on the Tyrrhenian side. Thus the Strait
constitutes
a
submarine
barrier or sill t o
the
marine
water
flowing through it. From
Vercelli's
1922 and 1923 measurements,
the
following
338
Oetail Map
20'
Fig.1. Bathymetric et al., 1986).
15'3dE
40'
map of the Strait of Messina (from Guardiani
picture
of water mass distribution in the Strait of Messina
emerged
(Vercelli,
1925;
Hopkins
et al., 1984). A t the narro-
west point there are two layers of marine water: surface nian 27.93)
water
of
Atlantic
origin
has
(T
16.6oC,
Tyrrhe-
S=38.0 X O ,
and deeper Levantine Ionian water (T=14.2OC,
Sz38.6
r t
=
XO,
339
Gt=28.94). 15
The time-averaged velocity profile (over a period of
(v=lO
days) shows a steady surface current flowing southward
cm/s) while a steady bottom current flows northward (v=13 cm/s). A semidiurnal tidal
adjacent seas,
current,
originating from the tides of
is superimposed on these steady currents. In fact
this channel is an amphidromic point for the semidiurnal tides of the
Ionian
rather
and
small
Tyrrhenian Seas.
amplitudes (
Tides in
these
10 cm south of Messina and
north
of
along
the 10 km length of the Strait (as observed
1925).
Messina) but their phase change is
This
abrupt
change
basins
about
16
five
by
is the cause of large
have cm
hours
Vercelli,
sea
surface
slopes, of 1-2 cm/km, which set up strong currents, up to 2-3 m/s (Vercelli, 1925 and Defant, 1940). The and
time-averaged interface depth is
Tyrrhenian Seas,
sill.
strong
A
-
150 m in the
but rises to a mean depth of ~
oscillation of the interface
with
periodicity is superimposed on the steady pattern. on
the
surface
sill alternatively reaches the bottom
3 m 0at the
semidiurnal The interface
and
during the two opposite tidal phases (Del
Ionian
the s e a Ricco,
1982;
Hopkins et al., 1984). Therefore when high tide is present in the Tyrrhenian sea the current flows southward ("rema scendente") and only
Tyrrhenian surface water is found on the sill.
On the con-
trary, during the Tyrrhenian low tide,the current moves northward from the
the Ionian to the Tyrrhenian Sea ("rema montante") sill
only Ionian Intermediate Levantine Water
is
and
on
present.
This water mass has the thermohaline characteristics of Levantine Ionian w a t e r a t a d e p t h o f a p p r o x i m a t e l y 1000 m . For t h i s r e a s o n the temperature difference between the waters present on the sill during summer,
the
two
while
tidal phases can even be as high as
10
O C
in winter it amounts only to several tenths of
in a
340
degree waves
(Vercelli and Picotti, due
to
1925).
the strong tidal action give
mixing of surface and Levantine waters: teristics
Large-amplitude rise
internal
to
extensive
the hydrological charac-
of the Ionian and Tyrrhenian Seas in the neighbourhood
of the Strait of Messina are affected by this water exchange. During surements University new
data
the period 1980-1985 a number of oceanographic were carried out by the Department of Physics of Rome in the area of the Strait of
revealed
t h e c u r r e n t and o f
of
Messina.
some peculiar characteristics o f
r e l a t e d t o t h e b e h a v i o r of
meathe These
this
zone
t h e i n t e r f a c e over
the sill.
Measurements of the interface time-evolution on the sill due to semidiurnal tide are analyzed in Section 2. These measurements revealed for the first time the presence of a columnar disturbance downstream from the sill, which can b e related to the presence of a subcritical flow on the sill. The above tidal mixing generates a new water mass, water
in the following,
called C
whose hydrological characteristics
are
intermediate between LIW and water of Atlantic origin. This water forms a long narrow cold surface water strip flowing south of the Strait along the east coast of Sicily. This strip was detected by using The deep
both hydrological and satellite data (Bohm et al., southward
1983).
flow o f the C water is analyzed in Section
sea vein of warm salty water along the Calabrian coast
3.
A was
also observed. This water,coming from the Strait during the "rema montante" phase,flows northward at an average depth of 130-400 m. The
water follows the bathymetry (Marullo and
Santoleri,
1986)
and can b e considered the anolog of the C water ( t 3 ) . Another interesting phenomenon observed during these cruises was
the existence of large-amplitude internal waves (Alpers
Salusti,
1983;
Criffa et al.,
1986;
and
Sepia et al., 1986). They
341
consist in packets of internal solitary waves characterized by an amplitude of 10-20 m , a
a phase velocity of approximately 0.9 m/s,
wavelength of 100-400 m and a total horizontal wave
ofw2000 distances,
m.
These waves are very stable and often
thickness
travel
before running into a solid obstacle. In our case the
first obstacle met by the northward-travelling waves is the Vatican0 these
long
promontory.
In Section 4 we show that the
Cape
breaking of
internal waves in the vicinity of this cape is responsible
for the formation of the thermal front ( w l 0 km large) observable in this area.
Fig.2. A photograph of a Koden image during the "rema montante" in which s o m e columnar disturbances are visible downstream from the region where the flow is subcritical (Dd250 m, from Di Sarra et al., 1986).
342 2 . LONG INTERNAL WAVES AND TURBULENT COLUMNAR DISTURBANCES IN TEE STRAIT OF MESSINA
The tidal currents in the Strait of Messina and their effect on
the
displacement of the water-mass interface have
scussed by several
Fig.
3a.
authors using Vercelli's data.
Reconstruction
been
di-
Defant (1961)
of the interface from the Koden images corresponding interface position, as computed by Del Ricco (1982)' and by Hopkins et al. (1984). Columnar disturbances are clearly visible downstream from the sill in both flows (from Di Sarra et al., 1986). ( x x x ) during the "rema montante" compared with the
343
studied the tides inside the strait using a barotropic model; more
recent
times Del Ricco (1982) used a numerical
in
baroclinic
nonlinear model to determine the time evolution of the
interface
between
data.
these
two water layers for Vercelli's
March
He
3b. Reconstruction of the interface from the Koden images during the "rema scendente'' compared with the corresponding interface position as computed by Del Ricco (1982)' and by Hopkins et al. (1984). Columnar disturbances are clearly visible downstream from the sill in both flows (from Di Sarra et al., 1986). Fig.
(xxx)
344
found
an internal wave
o f ~ 4 0 0m amplitude in a zone immediately
south of the sill. Hopkins et al. (1984) used Vercelli’s original data
to reconstruct the interface time-evolution using a
simple
between the interface slope and the acceleration of
balance
baroclinic
component.
These two models show that the
interface
has
oscillations oft.tl00 m with a semidiurnal
and
it alternatively touches the bottom and the air-sea surface.
Moreover, large during
tidal
the
periodicity
these models can be used to predict the existence of a
amplitude the
rema
internal
wave
immediately south of
montante. The amplitude
of this
the
wave was
sill as
high as 400 m for Del Ricco (1982) while Hopkins found a value of 100 m.
20
nl of
depth
18.6 sea bottom
at 2 5 0 m
184
18.2 18.0
17.8 17.6 0
5
10
1s
TIME [mind Fig. 4 . Temperature as a function of tire at 20 m of depth during the passage of the columnar disturbance (from Di Sarra- et al.; 1986).
345
A
JANE 84 (October 1984),
pilot cruise,
was organized
to
study the interface time-evolution due to semidiurnal tide, using casts
and a KODEN fish finder (28 and 200 Khz).
KODEN fish
finder
images (Fig.2) were used to infer the explicit time
evo-
lution
of the interface (Farmer et al.,
ship
moved the
1983) whilst
a t w 8 knots along the midline of the Strait. reconstruction
images
of
the interface position
from
the
In figure 3 the
Koden
is shown in comparison with the results of the Del
Ricco
and Hopkins models for the two different tidal periods. It can be seen
that the behavior of the interface is i n general
agreement
with the model results (Di Sarra et al., 1986). Moreover, some of these Koden images showed phenomenon: columns, These
an
interesting
the presence of columnar disturbances, called Bignami
in general visible downstream from the sill ( F i g s . 2 - 3 ) .
disturbances ran from the surface to the bottom (D=250 m)
Outside
-2 -
A
- - - - - - .I n s i d e A \
20
I 1
rn
of
deya
-3
-4
-5
-6
I -3
-2
-1
0
e I
Log (f)
Fig. 5. Spectrum of the segments of temperature moored data shown in figure 4 (from Di Sarra et al., 1986).
346
and
were characterized by a diameter of 400 m and a
ofu0.7
O C
less than the surroundings,
temperature data (Fig.4).
as
temperature
revealed
by
moored
The Fourier spectrum of the temperature
versus time inside and outside the Bignami column showed the same behavior but the energy of the disturbance was
10 times greater
N
(Fig.5). Flow over a submerged mountain has been studied theoretically for a rather
compared
with
parameters:
long time
in time-independent formulations
tank experiments.
stratification,
and
Its behaviour depends on
many
tidal currents and geometrical fac-
tors such as sill shape and size.
However, it is possible to use
a
parameter:
simple
(Long,
but useful descriptive
1955).
horizontal
the Froude
number
This parameter can b e defined as the ratio of the
velocity,
U,
to the phase velocity of the
internal
wave, c. Some general features of Long’s results are of interest. For instance,
supercritical flow (?tF>l) is characterized by
an
interface rising above the obstacle; while in the case of subcritical flow (?$FF
( ’)t Fn=l,
the
sill.
Recent studies have revealed other
interesting
fea-
tures: Smith (1976) found that if the obstacle height reaches 1/9 of the lower layer depth, nonlinear effects become very important and
turbulent
2JC F < 1
phenomena occur.
Mc Intyre (1972) noted that for
some columnar like disturbance appeared downstream
from
the obstacle and was propagated upstream for some time. The formation of the Bignami column can be explained in light of the Mc Intyre’s results. over
the
For Messina, the Froude number
the sill is greater than 1 and the flow
is
supercritical,
but becomes subcritical immediately south and north of the where the mean tidal velocity decreases.
sill,
This could explain both
347
the
existence of the Bignami column and why this phenomenon
was
never observed on the sill. 3 . DETECTION OF COASTAL CURRENTS NEAR THE STRAIT OF MESSINA
A coastal water,
shelf CTD
called C water,
flowing over the Sicilian
south of Messina was detected in satellite imagery and carried out during
casts
oceanographic
cruise
(Bohm
et al.,
TIROS
N
year.
At the same time a cold water spot on the sill and a
water
strip which extended southward from the sill following the
oriental
and
1986).
the prime
by
16 AVHRR cloud-free images obtained
from
NOAA 6 were examined for different periods of
shelf of Sicily were also observed (Fig.6).
the cold
This strip
was usuallyrrr100 km long but sometimes seemed to extend as far as Syracuse. strip that
Its
i s not
C
water
width was between 4 and 10 km. always visible
may
This
in satellite data,
sometimes disappear or flow
cold
water
which suggests below
the
sea
Fig. 6. Satellite image (Ch. 4 AVHRR o f NOAA 6 . June 2 1981 7 : 3 2 CMT) with the cold strip along the ; Sicilian shelf (from Bohm et al. 1986).
348
surface. ted
Bohm et al. analyzed also the hydrological data collec-
during
zone. From the
the PRIME oceanographic cruise of May 1982
in
the analysis of the vertical T-S diagrams they
presence of two different water masses,
C
this found
water (closer
to
the Sicilian coast) and Ionian water (further offshore). The main difference between these two waters is that the C water is tified
in
both
temperature and salinity while
stratified
only
in temperature.
types
the
stra-
latter
The border between these
of waters was found at 5-10 km offshore.
is two
On the basis
of
the similarity between T-S diagrams of the C water and Tyrrhenian
T ("C)
st. 1 .la
15.50
/
010
.aa
// 6;."
40. 50-
.w 70. w. w.
14.50
law
c water only
I 50.
1 w . m
13.50
13.50
1
I
I
st.:
st.3
c water
15.50
/
Ionian waters
14.50
wswo -7a
?a/
13.501 38.10
Fig.
,/
I
38.30
7.
38.50
IW. .1M
I
38.70
38.90
38.10
38.30
38.50
38.70
38.90
%O
Typical T-S diagrams observed during Prim cruise. Station N shaws a typical T-S diagrams a t Tyrrhenian Sea. Station 1 , 2 and 3 concern the Ionian Sea. the
Station 1 shows
typical C water T-S diagram: Station 3 s h m The Ionian
water T-S diagran and Station 2 sham the mixing of Wter and Ionian water mses.
349
Fig. 8. Map of the isotherms in the vertical cross-sections between Messina and Capo Vaticano, (from Marullo et al., 1986).
350
water
observable in the range o f 3 0 - 100 m
masses
panels of Fig.7) Bohm et al.
upper
(1986) indicate Tyrrhenian water as
possibly being the main component o f C water. this
(see
They consider that
water could b e the result o f the mixing that occurs
during
the rema scendente near the sill between the Tyrrhenian water and the local water. A
similar
analysis for t h e data collected during the
cruise (July 1983) north o f the strait o f between
Messina
in
the
the sill and C a p e Vatican0 has been made by Marullo
Santoleri (1986).
Vera zone and
Their temperature cross-sections s h o w the exi-
stence of a warm water mass between 130-400 a of depth. Above and below this deep (Fig.8). in
layer the temperature decreases fairly regularly
This layer o f deep water is characterized b y an increase
temperature
from
37.8%
from 13.9
to
38.3-38.6
O C
to 14.12-14.33 ko.
OC and
Analysis of
the
in
salinity
T-S
diagrams
revealed that this internal vein has t h e hydrological characteristics
o f t h e Ionian Levantine water entering the Tyrrhenian
during
the rema montante.
this water
Marullo and Santoleri have shown that
m a s s moves northward,
in the vicinity o f
sea
following t h e topography,
the Calabrian coast
and
seems to mix with super-
ficial and Levantine Tyrrhenian water.
4. TURBULENCE DUE T O THE COLLAPSE O F LAROE WAVES: GENERATION OF THE C A P O VATICAN0 FRONT As are
is well known,
generated
AMPLITUDE
INTERNAL
large amplitude nonlinear internal waves
in t h e Strait o f Messina b y
the
interaction
of
tides with the bottom topography of t h e sill (Alpers and Saluati, 1983; and
Griffa et al., therefore
dependent
on
northward
and
the tide
1986; Sapia et al., energy that they can intensity.
southward
1985).
transport is strongly
All these waves
a s a sequence of
Their amplitude
propagate
depressions
of
both the
351
interface
organized in packets o f solitary waves and
conserving
their energy and s h a p e until they run into an obstacle. When this happens,
they break, generating turbulent mixing o f the adjacent
stratified waters (Knickerbocker et el., Recently
1980; Kao et al., 1985).
Marullo and Santoleri (1986) suggested that breaking o f
these internal waves could occur in t h e G u l f o f Gioia,
and could
be
the ther-
responsible for t h e formation and disappearance o f
mal front observed south o f C a p o Vaticano both from satellite and in situ measurements. The lies
first experimental evidence o f t h e existence of
in the z o n e was given by a synthetic aperture
image taken o n September 15, lite
from
(area
anoma-
radar
(SAR)
1978, 0 8 1 7 G M T by SEASAT SAR satel-
a height o f 800 km.
This image revealed a dark
area
o f low backscattering properties)dlO km thick south o f the
Capo Vaticano Promontory (Alpers and Salusti,
1983) which can b e
interpreted as t h e signature of t h e deep water spun u p by mixing. To extend knowledge o f this phenomenon, Marullo and Santoleri a
(1986) examined thermal satellite images from NOAA 7 covering period o f approximately t w o years (1982-1983,
Centre de Meteorologie Spatiale, Lannion, that,
It
resulted
28% the presence o f a warm front while
16% o f the cases n o front was observable (Table
same in
France).
1).
period t w o additional oceanographic cruises were the
During made
of
out o f 181 clear-sky days satellite images, 56% showed the
presence o f a cold front, in
by courtesy
zone:
the 1982 P r i m e cruise and the 1983
the P r i m e cruise, using
measurements
a
In
the
performed
Vera
cruise.
measurements around Capo Vaticano were
towed Guildne CTD.
The
primary
aim
of
was t o detect packets of nonlinear internal
these waves,
although analysis o f the data recorded also revealed the existence
of
a front at a distance o f 10 km from
Capo
Vaticano.
The
352
observed thermal gradient was almost 4 OC over 7 km, and positive in the North direction (Sapia,
1986). On the contrary there were
no signs of this front in the Vera cruise. To obtain more details o f the type of front, two additional cruises were performed
zone: Jane
in the
the 1984 Jane cruise and the 1985 Greta cruise. During the cruise
measurements between salinity
(October were made.
1984) CTD casts
and
the
temperature
The analysis of TS profiles showed that
4 0 and 100 m of depth the behavior at
moored
of
innermost stations of the Gulf
temperature of
and
Gioia i s
different from the surroundings,indicating a patch o f mixed water (Fig.9).
The
observed
front
has an
exteneion
ofdl0 K m with a
TABLE 1 Summary of observed satellite images relative to the Capo Vaticano zone for the period 1982-1983. Column TOT indicates numbers of clear-sky observed images for every month. Columns COLD and WARM indicate the number o f these monthly observations in which a warn or a cold front, respectively, was observed in the zone south of Capo Vaticano, while column ABS contains the number o f cases in which no presence of fronts was observed.
YEAR
MONTH
TOT
COLD
1982
January February March April May June July Augs t September
3 7 11
1 3 3 4 8 10 10 7 7
January February March April May June July August September October
2 10 9
1983
9 8 13 17 14 9
a
12 6 11 9 13 10
2 0 2 5 8 4 5 7 9 7
WARM
AB S
0
7 1 1 3 1 5 1 3 1
1 1 2
353
gradient of 3
OC/10
km.
TEMPERATURE ('C)
DENSITY
(2.)
0
0
0
20
20
20
(7
40
40
rn 40 D -I I
D -I
I A
SALINITY
60
A
"
v
D -i
I
60
z
h
60
80
80
a0
100
100
100
E 2 "INTERNAL"
--__
E3 "EXTERNAL"
Fig. 9. Temperature, salinity and density profiles as observed at an internal station (E2) and an external station (E3) during the Jane cruise in the Gulf of Gioia (from Guardiani et al., 1986).
Moored temperature measurements were made at 30 and 8 0 m of depth both outside and inside the front area. The resulting temperature time
series show
data
taken
minutes
a similar behaviour at 8 0 m while at 30 m
inside
(1
O C
the front show greater variation
in 10')
to the front (Fig.10).
in
than the records of the stations The spectral analysis of the
a
the few
external
temperature
time series showed that the spectral function S(f) has a decrease like %':-f
c.
, typical of a Kolmogorov turbulent spectrum
(Fig.11).
The Jane field measurements were repeated during the Greta cruise
(2-4 August casts the
1985).
During this cruise the analysis of the
revealed that in the neighbourhood of a rising sea
bottom
density profile displays an inversion at a depth of 60-90
(Fig.12).
CTD
The power spectra of the moored measurements show
m the
same behaviour as in the Jane cruise, but no front was observable
354 TrCI
PZ
STATION
('INlTRNAL'
; DEPTH' Jon)
A 1, 0
iaa
F W l t1411ED
1
.
ia6
A
1114
''
182
'
180
'
C
*t
FROM A
lv 176
I I
0
15
10
1:
TKI
PS taTERIIALj
STATION! ~
~
p
-
21 3
irci
T
'Or,
p4
m .
*:I
30
&xnnw*r~') PCI
STATION
hn1
STATION n
I
25
20
TIM
80-
,
35
-
12 (irnm~) OOPTH 80-
1x8
21 2
21 1
(31
lY6
?lo
(IS
5
0
TIRE
10
Crnl
0
5
TlnE
bml
w
o
5
Tint
(0
C,.J
Fig. 10. Temperature time-series at different depths and different stations during Jane 1984 cruise (from Guardiani et al., 1986). (Guardiani et al., 1986). These (1985)
that
Vatican0 is from
new the
data support the idea of Marullo
Santoleri
front observed at the Southern coast
due to a mixing process in which energy is
of
Capo
received
the breaking of nonlinear internal waves generated by semi-
diurnal tides in the Strait of Messina. The most tary
and
mechanism
of breaking occurs when the
two
natural elemenlayers
become
equal:
the
solitary waves change polarity.
shore,
the
height of the internal solitary wave increases until
it
breaks
generating turbulence
On approaching
(Knickerboker
et
al.,
the
1980;
355
Helfrich et el., 1984). On the contrary, tank experiments (Kao et al., 1985) seem to suggest thatrwhen the depression internal wave intrudes
over
the
shelf, its
Helmholtz instabilities Unfortunately
in
are
shape
is
responsible
the light of our data,
deformed for
and
Kelvin
the energy
it
is
loss.
impossible
to
identify the explicit elementary breaking mechanism. L.)(ll
L.,ITI
-1
__ PZ
______
\
-1
(INTI
P3 I E X T I
-2
-2
1
-3
-3
-4
-4
-5
-5
-6
-6 -3
-2
0
1
1
-3
-2
-1
0
1
L.,W
L.,ff)
Lqlr) D t P l H : 311-
P2
'INTERNAL'
--C) - - - - -.. A)-
8)
- 5 .
-1
-2
-1
0
1
L.#ffl
Fig. 11. Discrete Fourier Transform of temperature at various Stations and at various depths. The last graph shows the timeintervals A , B , C in Fig.10, relative to station P2, at 30 m of when a depth. The spectrum is larger for the time-interval " A " , high amplitude wave is present. This fact is in agreement with the considerations of Kao et el., 1985 (from Guardiani et al., 1986).
356
SALINITY
TEMPERATURE l'C1
37.5
12 14 18 18 28 22 24 26
[%.I 38.5
30.8
80
188 128
"i 8F 28.5
28
2Q. 5
30.5
31. 5
G
t
22
u
I
+
10
II
50
I
14
I
12437.5
38.8
SALINITY
Stat,,,
38.5 [XI
AI
Fig. 12. Typical Temperature, Salinity and Density profiles and T-S diagram during Greta cruise at the stations located on the shelf break showing an inversion in the T-S diagram. In the upper panel the bottom depth behaviour between Messina and Capo Vaticano is also shown (from Guardiani et el., 1986).
6. DISCUSSION To summarize, of
w e show that a small marine strait like
that
Messina can be a remarkable geophysical laboratory of general
interest. In this Strait the sill is shallower than the interface
357
of Atlantic and Levantine Waters layers; moreover it is an amphidromic
point for the tides of the two basins.
tidal
phenomena
give rise to a peculiar process that is
easy to observe since, are
no
strong
currents
Consequently
in the neighbourhood o f the
tides.
originating
distance from the
These on
phenomena are
sill,
surface
the
rather there
and
deep
t h e s i l l and f l o w i n g t o a considerable
Strait. The study of the time evolution of the
interface around the sill can explain most of the peculiar phenomena
observed
nonlinear
in the Messina zone in terms of the
internal waves,
presence
columnar disturbances and
that have been examined in this paper.
of
turbulence
These phenomena are known
on theoretical grounds and their observations are of considerable interest. O f course, these phenomena are novel and would call for much their
more
observation and speculation.
However,
we feel
interest is so great that even these pilot data can b e
that of
use as a stimulus.
7. REFERENCES Alpers, W., Salusti E., 1983. Scylla and Charybdis Observed from Space, J. Geophys. Res., 88, 1800-1808. Bohm, B., Salusti E . , 1984. Satellite and Field Observations of Currents on the Eastern Sicilian Shelf, Remote Sensing of Shelf Sea Hydrodynamics, Elsevier Sciences publisher Amsterdam, pp.51-68. Bohm, B., Magazu’, G., Wald, L., Zoccolotti, M.L., 1986. Coastal Currents on the Sicilian Shelf South of Messina. Oceanologica Acta (in press). Defant, A., 1940. Scilla e Cariddi, e le correnti di marea nello Stretto di Messina, Geof. Pura Appl., 2, 93. Defant, A., 1961. Physical Oceanography, Vol.11. Pergamon Press, 5 8 9 pp. Del Ricco, R., 1982. A Numerical Model of the Vertical Circulation of Tidal Strait and its Application to Messina Strait. Nuovo Cimento, Vol. 5C, n.1, 21-45. Di Sarra, A., Pace, A., Salusti, E., 1986. Long internal waves and turbulent columnar disturbances. (Preprint, Phys. Dept., Rome University). Farmer, D.M., Freeland, H.J., 1983. The Physical Oceanography of Fjords; Progress in Oceanogr., Vol. 12, 147-219. Guardiani, G., Pace, A., Salusti, B., 1986. Turbulence due to the collapse of internal solitary waves (Preprint, Phys. Dept., Roma University). Gargett, A., 1980. Turbulence Measurements through a Train of
358 Breaking Internal Waves in Knight Inlet, B.C. In: Fjord Oceanography, H.J. Freeland, D.M. Farmer and C.D. Levings, eds., Plenum Press, New York. Griffa, A., Marullo, S., Santoleri, R., Viola, A., 1986. Note on Internal Tidal Waves Generated at the Strait of Messina. Continental Shelf Res. (in press). Helfrich, K.R., Melville, W.K., Miles, J., 1984. On the Solitary Waves over Slowly Varying Topography, J. Fluid Mech., Vol. 149, 305-317 Hopkins, T., Salusti, E., Settimi, D., 1984. Tidal Forcing of the Water Mass Interface in the Strait of Messina, J. Geophys. Res., 89, 2013-2024. Kao, T.W., Fu-Shing, P., Renouard, D., 1985. Internal Solitons on the Pycnocline Generation, Propagation, Shoaling and breaking over a Slope. J. Fluid Mech., Vol. 159, 19-53 Knickerbocker, C.J., Newell, A.C., 1980. Internal Solitary Waves near a Turning Point. Phys. Lett., 75 A, 326-330 Long, R.R., 1955. Same Aspect of the Flow of the Stratified Fluids, 111. Continuous Density Gradients. Tuellus, 7 , 341367. Marullo, S., Santoleri, R., 1986. Fronts and Internal Currents at the Northern Mouth of the Strait of Messina. Nuovo Cimento (in press). Mc Intyre, M.F., 1972.0n Long’s Ipothesis of no Upstream Influence in uniformly Stratified Fluid. J. Fluid Mech., 52, 209-243. Osborn, A.R., Burch, T.L., 1980. Internal Solitons in the Adamann Sea, Science, 208, n.4443, 451-460. Sapia, A., 1986. Observation of Nonlinear Internal Solitary Waves Train at the Nourthern and Southern Mouth of the Strait of Messina, submitted to Deep Sea Res.. Smith, J.D., Farmer, D.M., 1977. Nonlinear Internal Waves and Internal Hydraulic Jumps in a Fjord. In Geofluidynamical Wave Mathematics, 42, University of Washington, Seattle. Vercelli, F., 1925. Crociere per lo Studio di Fenomeni nello Vol.1: I1 Stretto di Messina (R.N. Marsigli, 1922 - 1923). Regime delle Correnti e delle Maree nello Stretto di Messina. Commissione Internazionale del Mediterraneo, Venezia, 205 pp. 1925. Crociere per lo Studio dei Vercelli, F., Picotti, M . , Fenoneni nello Stretto di Messina (R.N. Marsigli, 1922 1923). Vo1.2: I1 Regime Chimico Fisico delle Acque dello Stretto di Messina. Commissione Internazionale del Mediterraneo, Venezia, 103 pp.
359
A THREE-DIMENSIONAL FINITE ELEMENT MODEL FOR THE STUDY OF STEADY AND NON-STEADY NATURAL FLOWS J.-L. ROBERT and Y. OUELLET Civil Engineering department, Research Center on Computer application in Engineering, Lava1 University, Sainte-Foy, Quebec, CANADA, G1K 7P4 ABSTRACT This paper will present the basic concepts of a tridimensional finite element model for the simulation of natural flows. In a first step, the fundamentals of the formulation will be described with an emphasis on initial assumptions, mobile layer concept, type of element, discrete variables of the system. Then, special attention will be taken to expose the original technique developed to include, on the entire depth of water, the effect of the bottom friction and some justifying examples will be shown. Afterwards, a brief discussion on the strategy of resolution for such non linear system will point the specific importance of friction and inertial convective terms. This will be followed by the presentation of some examples showing the good general behaviour of the model, particularly its capability to well simulate the vertical circulation of a flow. 1 INTRODUCTION
After working since many years with finite difference and more recently with finite element depth average two dimensional models, some specific environmental problems in coastal and estuarine zones, involving the effect of density and wind on the hydrodynamic circulation, raise the need of a three dimensional model. The finite element approach was retained for mainly two reasons. The first one was its ability to discretize with variable density of computational points an irregular domain, as it can be found in the nature. This is an important advantage in a way that it allows us to choose the position of artificial boundary conditions far enough to minimize their influence and without using a prohibitive number of computational points. The second reason was linked with the possible future development of the model, specially the possibility of its coupling with existing 2-D F.E. hydrodynamic model. The major goal we kept during the development of the model was its capability to simulate the largest possible range of free surface natural flows, large or small scale, steady or non-steady, tide or wind driven. The domain of application itself has a direct influence on the formulation, in fact natural flows have a small verticalhorizontal aspect ratio. Then, the discretization technique, as it will be shown later, has been chosen differently depending on the vertical and the horizontal directions.
360 2 FORMULATION OF THE 3-D MODEL 2 . 1 Governing Equations
The physical behaviour of a fluid may be expressed by a set of equations based on the forces equilibrium and the principle of mass conservation. In a three dimension (x, y, z) domain the momentum equations are:
where: u, v, w : the velocity components; p : the pressure; p : the density; X, Y, Z : the external forces (Coriolis, gravity) T~~... :
the Reynolds stresses
The Reynolds stresses express the viscous similarity of the effect of the turbulence. In the model, they are included using the Boussinesq assumption using an eddy viscosity coefficient A:
T
x x = 2A 11
= A
T
(4)
(5)
xixj
Note must be taken that in the vertical direction, those terms will receive special treatments. The mass conservation in an incompressible fluid gives the continuity equation:
361 2.2 Vertical Integration
Since the model is specially designed to simulate natural domains, some characteristics of them can be observed and then stated:
- small vertical dimension in respect of horizontal ones; -
vertical acceleration of the flow negligible; small vertical velocity component.
Those statements eliminate from the range of applications high slope super critical flows, closed conduits and turbomachinery flows. With this restriction the assumption of hydrostatic pressure can be taken and the acceleration and stresses term of the third momentum equation vanish keeping only the pressure and the gravity terms. In this form the set of governing equations (1). ( 2 ) , (3) and (4) is integrated in the vertical z direction from the bottom position -h to the water level n. But, in order to get some information about the vertical structure of the flow, the integration process is split into several terms:
by the introduction of intermediate integration limits (fig. 1) which have their position located by:
Lk =
n-(e) k
k = 1, 2 ,
... b
The presence of 11 in the equation (8) means that the position of those intermediate level must move proportionnally with the water level, consequently the relative thickness of a layer is constant. An uniform distribution of those level has been chosen here; however, it is possible to change those relative position by any other relationship including the water level to get another distribution. It is important to note that the intermediate integration surfaces do not have any physical meaning and allow the mass exchange between them. This is specifically prescribed by the intermediate boundary condition (Robert 1983):
a n = k+l
Y
362
Fig. 1 Definition of the vertical integration boundaries. where Un and Vn are mean velocity components in a layer n. At the extremity of the vertical domain boundary conditions must be defined using a streamline condition: atl
atl
at
w(-h)
=
-u(-h)
ah - v(-h) ax
ah
The introduction of those conditions in the int gration pro ess using th Leibniz rule, does not eliminate all boundary velocity terms at the intermediate level, the assumption that those terms are included in the internal stresses term is made (Robert 1983). Then the model is defined as:
- Continuity equation
363
- Movement equations +
+ U k aauk x+Vkay auk -fV
+
e [>+
( TxLk-1 -
a
T
xLk
U kavk T + Vk avk + fUk
k = 1
))]
+
g
=
ahkTxyk aY
0
-- -
Y
The vertical velocity component is computed from equation ( 9 ) . The formulation of the horizontal stresses will be discussed in section 3. 2 . 3 Finite Element Formulation
This short overview of the finite element formulation will point out the specific choices made to obtain the discrete form of the model. The integral form of equations ( 1 2 ) , (13) and (14) is obtained by a Galerkin weighted residual method.
As one can see from the model equations, two dimensional elements can be used by associating at each node the water level
rl
and the two horizontal
components of the mean velocity in each layer Uk and Vk. Two types of interpolation functions were then considered. Based on the work of Cochet 1979 on 2-D hydrodynamic model, a quadratic function for the velocities and a linear function for the water level were tested. The result shown that in a multilayer configuration some instabilities appeared fig. 2. Then a full quadratic function was used and improved considerably the performance of the model. Two full quadratic elements have been developed, a six nodes triangular and a nine nodes quadrilateral elements. Special attention must be kept in the discretization of movement equations (13) and (14). Since the interpolation functions applie only on discrete parameter associated with one layer and are zero for any other parameters, the assembly process is equivalent to the sommation present in those equations. Then the resulting set of algebraic equation will have as much equations as discrete variables. The resulting non linear system in non steady form can be expressed by:
[Ml{il + CK ($)I {$I
=
0
The solution method will be presented In section 4.
364
.
-
I
I
Quadratic element Quadratic-linear element
5
Fig. 2
Computed velocities in the center of a schematic estuary presented at Fig. 5.
3 HORIZONTAL STRESSES 3.1 Formulation
One of the most important factor in the general behaviour of a natural mass of water is the horizontal shear stress. The predominant one is the effect of the bottom roughness over the entire depth of water and in some cases, the effect of the wind at the free surface may be determinant. In such case, this specific stress component may be expressed by:
365 Tx L = CtPa wx IWI 1 0 i where Ct is a drag coefficient, pa the air density and W the component of wind xi velocity W in the xi direction. For the bottom friction term, ChLzy formulation has been chosen:
3 vx i Ivl
=
TxiLb
Here, in the case of a two dimensional model the Ch6zy coefficient C , takes in account the overall effect of the bottom roughness on the flow over all the thickness of the water layer. However, when this depth is discretized, this term will affect only the bottom layer and the coefficient C must be corrected in way such as the instantaneous discharge must stay the same whatever the number of layer. To ensure the transfer of the bottom roughness to the upper layer, an internal shear stress component must appear at the interface of each layer. This may be expressed by discrete form of the Prandtl internal shear stress formulation; in the x direction we have at the upper interface of a layer k :
Auk 4 AU; + AVk2
=
TXLk-1
hk
where Auk and AV are the differences of the flow velocity between the layer k k and k-1 and II is the mixing length. Many authors (Wang 1977, Kokayashi 1980, Leenderstee 1975) use a similar formulation, but the II coefficient is considered as an experimental adjustable coefficient. 3.2 Mixing Length Parameter
As we do not really know what is the value of the mixing length, expect near the bottom where Prandtl assume that it may be proportional to the distance from the bottom, we must find an usable formulation of the mixing length up to the free surface. Montgomery 1943 suggests, from some experiments in particular flow that:
x
is the Von-Karman coefficient which is defined by x = 0.4. As shown on fig.
3 , this distribution presents a plane symetry on the z/h = 0,5 plane. This
366
involves the assumption that the distribution of the horizontal stress is similar near the free surface and near the bottom, which is not really satisfactory. To remedy such situation, Yalin 1977 suggests a distribution of the mixing length based on the assumption of a logarithmic velocity profile:
A s the Montgomery 1943 distribution, this one agrees with the Prandtl assump-
tion near the bottom:
a
=
x z
(21)
Fig. 3 Mixing length vertical distribution. The major advantages of the Yalin 1977 distribution are: a) the mixing length decreases less quickly near the free surface than near the bottom, assuming that the turbulent fluctuation in the neighbourhood of the surface are less damped than by the presence of a rigid boundary at the bottom. b) the horizontal shear stress becomes zero without the necessity of assuming a zero vertical velocity gradient at the surface. This formulation was retained and adapted to the present model; in a discrete form, for the lower interface of a layer k, equation ( 2 0 ) becomes: k'
=
x
hk (b
-
k)
(22)
367
and the corrected bottom friction factor can be written:
where Cr is the Ch6zy reference coefficient for a two dimensional model. In order to verify the good overall behaviour of this approach, a test was run on the model to simulate a steady state uniform turbulent flow with a five level configuration. The velocity profile obtained is presented at the figure 4 .
t
L
y'm
K
..I
-*-
Fig. 4
.
-
-.u, m/s
Velocity profile for a uniform steady flow.
4 SOLUTION METHOD
In steady state configuration, the model of the flow is dominated by a quadratic friction term. The Newton Raphson procedure must be chosen rather than a substitution scheme to solve the non linear system. In fact, it is obvious that the later will not converge because the quadratic term. The incremental form of the method is:
368
where the tangent matrix is:
The non-steady flow is based on an implicit Euler scheme with an iterative Newton-Raphson stabilization procedure:
These techniques need much CPU time, and some possibilities have been set to reduce it. The tangent matrix can be, with a little loss of accuracy, recomputed at the beginning of each time steps rather than each iteration. In some case, computing time reduction may reach a factor three. 5 APPLICATION Many numerical tests have been performed with this model. The simulation of
the tridimensional structure of the flow in a 180'
curved flume in steady
state has been compared with a scale model in laboratory. The dimensions used in the mathematical model were the same than for the physical model to prevent any scale effect on the results. The test bring us the confirmation of the rightness of the formulation chosen here. The results of this comparison can be found in Robert 1983.
5.1 Schematic estuary In the testing process of the model, one goal was to verify that the vertical behaviour of the simulation agrees with the results of a two dimensional vertical models (width averaged) particularly with the Sunderman 1976 formulation based on cubic interpolation function in the vertical direction. The test case can be described as a schematic estuary closed at one end and submitted at the other one to a sinusoidal tidal cycle of 1 meter of amplitude. The flume has a length of 92 km and an width of 4.6 km. The depth is 50 meters except for a central portion where the bottom raise up to a depth of 20 meters. The finite element grid is composed of 10 quadrilateral elements and three layers were specified. The results of some time steps of a complete cycle (fig. 5) show a good agreement with those presented by Sundermann 1976 ones despite the fact that the vertical discretisation of the present model is far less accurate. The time interval of computation was. in this case, 1200 sec. for a full implicit scheme. This case was an occasion to test the validity of the mixing length distribution. The same test was run with a constant interfacial friction factor
369
Fig. 5 Vertical flow structure in a schematic estuary.
and shown some irregularities of the vertical velocity profile as an erroneous position of the maximum velocity during the ebb flow. That situation was cured by the use of the distribution presented by eq. 20. 5 . 2 Wind induced circulation
The second example focuses on wind driven circulation. Some preliminary numerical tests agreed with the Leenderstee 1975 samples in the same condition. As for preceeding example the simulation was performed as a propagation problem; a constant 40 km/h s-W wind blowing over a 10 km x 5 km x 1Gm rectangular bassin. The result, shown in fig. 6 , gives the circulation over the five layers, the water surface elevation and a transversal cross section in the middle of the bassin after six minutes of real time. At this time, the steady state is not totally achieved and the free surface is still moving. The corresponding computing time for this non-linear non-steady 2 2 2 1 degrees of freedom was about 2h40 on a VAX 850 for a full implicit scheme. Nevertheless, some techniques as the computation of the global matrix only at the beginning of each time step can reduce the computing time to one third and the use of a vectorial computer may be usefull. 6 . CONCLUSION
One of the most important concluding remark of this work, is that the automatic adjustment of the interlayer friction factor by the use of a well chosen mixing length distribution gives excellent vertical behaviour of the model. This vertical pattern of the flow is then relatively accurate and the formulation is well adapted to the natural flows. The use of the model is made easy by the fact that the finite element grid is only two dimensions and consequently a coupling with a two dimensional model is provided. This two and three dimensions models should be used to predict the vertical structure of the flow only where it is necessary.
371
-0,6 mrn
d
-
\
.
,
L
I
r
-
r
c
Y
c
e
c
-
L-
c
+ 0,51
Fig. 6
Wind driven c i r c u l a t i o n i n a rectangular bassin.
312
7. REFERENCES Cochet. J.F., 1979. Modglisation d'gcoulements stationnaires et non stationnaires par la mlthode des 6lLments finis. ThPse de Docteur-Inghieur, Universitg de Technologie de CompiPgne, France, 142 p. Kobayashi, M., Nakota, K. and Kawahara, M., 1980. A Three Dimensional MultiLeveled Finite Element Model for Density Current Analysis. In: Third Int. Conf. on Finite Element in Flows Problems, Vol. 11. Banff, Canada, pp. 80-92. Leenderstee, J.J., Liu, S.K., Alexander, R.C. and Nelson, A.B., 1975. Threedimension Model for estuaries and Coastal Seas, Vol 11, Aspects of Computation, R-1764-OWRT, The Rand Corporation, California, USA, 165 p. Montgomery, R.B., 1943. Generalization for Cylinder of Prandtl's Linear Assumption for Mixing Length. Annuals of the New York Academy of Sciences, XLIV: 89-103. Robert, J.L., 1983. ModElisation tridimensionnelle des Ccoulements a surface libre, permanents et non permanents, par la mgthode des Blgments finis. ThSse de doctorat, Universitg Laval, Qudbec, Canada, 233 p. Sunderman, J., 1976. Computation of Barotropic Tides by the Finite Element Method. In: Finite Element in Water Resources, First Int. Conf. Gray, Pinder and Brebbia ed., Princeton University, Pentech Press, London, U.K.. pp. 4.31-4.67. Wang, H.P., 1977. Multi Leveled Finite Element Hydrodynamic Model of Block Island Sound. In: Finite Element in Water Resources, First Int. Conf., Gray Pinder and Brebbia ed., Princeton University, Pentech Press, London, U.K. Yalin, M.S., 1977. Mechanics of Sediment Transport, Second ed., Pergamon Press, Oxford, U.K., 259 p.
373
REAL AND SPURIOUS BOUNDARY LAYER EFFECTS IN THREE-DIMENSIONAL HYDRODYNAMICAL MODELS Bruno M. JAMART and JosC OZER Management Unit of the Mathematical Models of the North Sea and the Scheldt Estuary (MUMM). Institut de MathCmatique, Avenue des Tilleuls, 15, 4000 Likge, Belgium.
ABSTRACT On the basis of a simple test problem, three examples of the effect of boundary layers on large scale flows in three-dimensional hydrodynamical models are discussed. In the first case, an artificial, numerically induced residual flow is shown to arise from an inappropriate lateral boundary condition and the staggered nature of the computational grid. The second example demonstrates that the lack of sufficient horizontal resolution along the lateral boundaries can have a deleterious effect on the speed of propagation of long gravity waves. The last example concerns the effects on the speed of propagation of long gravity waves of the parameterization of the bottom stress in 3-D and 2-D models. 1. INTRODUCTION The present volume bears witness that three-dimensional hydrodynamical models are rapidly coming of age. Among the many reasons why 3-D models sometimes have to be resorted to is the opportunity they offer to better describe boundary layer phenomena. Particular care is required in modelling and/or interpreting boundary layer effects, both from a numerical and from a theoretical point of view. On the one hand, the numerical procedures need to be accurate and to have "high enough" resolution. On the other hand, the mathematical formulation of the problem needs to be "sufficiently complete". In this paper, three examples of the effect of boundary layers on large scale flows are discussed on the basis of a simple test problem. The first two examples concern spurious effects and erroneous results. The third case deals with a real, frictional rather than fictional, boundary layer, namely the bottom boundary layer (BBL). There is no implication other than the necessity of carefulness in the relative proportion of incorrect results (2/3) versus correct ones. In the next two sections, the two 3-D models whose results are used later and the benchmark calculation are briefly described. In sections 4 and 5, two examples are given of spurious effects introduced in the solution by mishandled lateral boundary conditions. In the first case, a spurious boundary layer develops along the lateral solid boundary of the basin, resulting in an artificial, numerically induced residual flow. This case is discussed at length in a recent paper (Jamart and Ozer, 1986) and only a summary is presented here. The second example demonstrates that the lack of sufficient horizontal resolution in the vicinity of a solid vertical boundary can have a deleterious effect on the speed of propagation of long gravity waves, even
374
when the resolution is more than adequate for the wave itself. In section 6, we investigate the relationship between the speed of propagation of long waves and the parameterization of the bottom stress in mathematical models. In particular, the differences between 2-D and 3-D models, and, in the latter case, between a slip and a no-slip bottom boundary condition are emphasized. 2. MODEL EQUATIONS AND SOLUTION PROCEDURES
The numerical experiments discussed in this paper were performed as part of a model intercomparison exercise described in several earlier publications (Jamart and Ozer, 1985, 1986; Jamart er al. , 1986). Hence, only a brief expos6 of the "materials and methods" suffices here. Two independently developed three-dimensional models are considered. The models shall be referred to as "FD-model" and "FU-model", respectively, where FD stands for Finite Difference and FU for Function. The justification for these denominations will ensue from the description of the numerical procedures.
2.1.. The finite difference ("FD")model The FD-model is a version of the numerical model developed by Paul and co-worken (Paul and Lick, 1981). The governing equations are the 3-D, time-dependent, nonlinear Navier-Stokes equations. It is assumed that the pressure is hydrostatic, the fluid homogeneous, and the Coriolis parameter constant. The turbulent fluxes of momentum are parameterized by means of eddy coefficients, denoted Ah in the horizontal plane and A, in the vertical direction. The latter may be a function of time, depth, and/or flow variables. In conventional notations, and using the vertically integrated hydrostatic equation, the equations read : auv
+-auw ay aZ
au2 au+-+-
-fv = -g
ax auv av2 - + -+ -+ at ax ay aZ + fu = -g av aw aU +-+-=o ax ay aZ at
a +Ah ( -a% +-aZu) + a (A, aU ) ax ax2 ay2 a% a% ) + a ( A, av ) a + Ah ( - aY ax2 ay2 az +
For this model, the boundary conditions are : (i) along solid, vertical boundaries, u =v =0
for all t and z;
(4)
(ii) at the bottom, z = - H, u=v=w=O
forallt;
(5)
(iii) at the surface, defined as the time-mean position of the air-sea interface, and denoted z = 0, at
=w
for all t;
375
1
(a) B-Grid
(b) C - G r i d
Fig. 1. Computational grids : (a) B-grid used in the FD-model; (b) C-grid used in the FU-model.
and
where 2: and T$ are the components of the prescribed wind stress. The numerical procedure used to integrate these equations is described in detail by Paul and Lick (1981) and summarized by Jamart et al. (1986). For the following discussion, we only need to note that :
- all derivatives, both spatial and temporal, are approximated by finite difference analogs (hence the denomination as "FD-model");
- the spatial (horizontal) distribution of the variables is such that u and v are calculated at the same points (the vertices of a rectangular mesh), whereas w and q are defined at the center of each cell (see Fig. l(a)); this type of grid arrangement is often called the Arakawa B-grid (Arakawa and Lamb, 1977). The free surface position is computed by a semi-implicit scheme (Paul, personal communication), thus allowing for a longer time step than that constrained by the C-F-L criterion.
376
2.2. The spectral ("FU")model The second 3-D model used in the numerical experiments is, in its present version, limited to the solution of the linearized equations of continuity and motion. Moreover, lateral friction is ignored so that equations (1) and (2) reduce to
The continuity equation (3), integrated vertically, and the kinematic conditions at z = 0 and - H combine to yield :
-*+ - a at
ax
0
0
J u d z + L Jvdz=O -H aY -H
The dynamic boundary condition at the free surface is the same as for the FD-model, i.e., equation (7). At z = -H , the bottom stress is assumed proportional to the bottom current
a condition from which the no-slip case of equation ( 5 ) can be recovered by taking the limit for K+-.
The specification of an appropriate boundary condition along the coastline is trickier than appears at frst sight. A conventional choice for the 3-D linear model consisting of equations (8) to (10) is to set equal to zero the component of the velocity normal to the wall at all depths. Serious difficulties associated with this boundary condition and alternative formulations will be discussed in a later section. In the FU-model, the depth dependence of the velocity variables is not approximated by the classical, finite difference type of representation but rather by means of a truncated expansion in a series of continuous functions. This approach was first proposed by Heaps (1972). It has since been used and expanded upon by Heaps himself (1981) and by many other researchers ( e.g., Nihoul, 1977, Davies, 1977, 1980, and subsequent papers, Owen, 1980, Cooper and Pearce, 1982). Different sets of basis functions have been used in conjunction with this approach. In the FU-model, following Heaps (1972), we choose the eigenfunctions of the vertical diffusion operator. The discretization of the equations in the horizontal plane is accomplished by a finite difference procedure. The disposition of the variables on the computational grid is different in the FU-model from that used in the FD-model. In the function model, the two components of the velocity vector and the elevation are all calculated at distinct locations. This spatial grid, referred to 'as a C-grid, is displayed in Fig. l(b).
377
Finally, the time-stepping procedure used to integrate the equations of the FU-model is a semi-implicit, alternating-direction method patterned after that proposed by Beckers and Neves (1985) for a vertically integrated model. More details on the numerics of the FU-model are given by Jamart and Ozer (1986).
3. THE TEST PROBLEM The boundary layer effects discussed in this paper are, we think, of rather general relevance. Yet, we find it useful for demonstration purposes to use one of the benchmark calculations of a model intercomparison experiment currently in progress. All the numerical results to be displayed were obtained by running the models just described to solve the following problem. The domain is a rectangular basin, closed on all sides, of dimensions 600 x 1200 km, and of uniform depth H = 100 m. The long axis of the basin runs from south to north and is aligned with the y-axis. The latitude is taken as 55' N. The motion starts from rest (u = v = w = 11 = 0) and it is forced by a constant, impulsively applied, southward wind stress ( 2; = - 0.1 N m-2). To our knowledge, the only complete analytical solution to this problem is that of Ekman (1905) for steady state and linear equations, and that solution is actually valid for flat-bottomed
enclosed basins of any shape. At steady state, the two components of the transport are zero everywhere. The surface slope can be calculated from the vertically integrated momentum balance which involves only
vh
q ,2, , and
2b
= p (A,
aUIz z
=-
.
The bottom stress can be
related to v h 11 and z , by solving the vertical problem. Ekman (1905) assumes that the eddy viscosity A, is independent of the depth and he uses a no-slip condition at the bottom. A similar solution is shown to also hold for a linear slip condition and any positive, arbitrarily zdependent A, by Jamart and Ozer (1987). For the time-dependent part of the solution, the presence of sharp comers in the domain boundary complicates the mathematical analysis of the problem. Indications as to the type of wave motion to be expected can be found in the studies by Taylor (1921) and Rao (1966) and in the monograph of LeBlond and Mysak (1978). The effects of a constant vertical eddy Viscosity on Kelvin waves are discussed by Mofjeld (1980). Precious as are such elements of the system response, we mostly rely on model intercomparison to test the codes of the 3-D models. 4. SPURIOUS LATERAL BOUNDARY LAYER ON A C-GRID The equations of the FU-model have a well-known steady solution (Ekman, (1905)) for constant depth and wind stress. The residual transport vanishes everywhere. The surface slope and the velocity profiles are horizontally uniform. However, the numerical solution obtained by running to steady state the FU-model as described in Section 2 differs from the exact solution.
378
In particular, as is shown by Jamart and Ozer (1986, Figs. 4 and 5). the depth-averaged currents are more or less organized in two gyres rotating in opposite directions, with strong flows along the edges of the basin. This pattern is spurious. The solution should be horizontally uniform. The building up of the erroneous residual currents also affects the timedependent part of the solution. The source of the error in the numerical procedure is twofold. First, the computational grid used in this model (i.e., the C-grid) requires that averages be perfomed everywhere to evaluate the Coriolis term in the momentum equations. Second, the boundary condition implemented along the solid walls (i.e., the vanishing of the normal velocity at all depths) is inap propriate for the model's equations. The inadequacy of the boundary condition u . n = 0 at all z stems from the fact that the equations of the FU-model do not include the necessary terms (horizontal diffusion) to allow a smooth vanishing of the velocity in a coastal boundary layer. In the absence of rotation, this results in a singularity in the w-field (the vertical velocity becomes infinite at the wall in the continuous solution). With rotation, the conventional evaluation of the Coriolis term on a C-grid (averaging the values at the four closest points) underestimates the deflecting "force" along solid boundaries because at least one of the points entering the mean is located on the boundary. The inadequacy of the boundary condition for the equations of the FU-model, combined with the properties of the C-grid, is responsible for the numerically induced residual current observed in the results of this model. Different tacks are possible to circumvent the difficulty. The first would be to modify the governing equations so as to model properly the behavior of the fluid in the side-wall friction layers. The requirements for such an approach, and elements of the solution, are discussed by Pedlosky (1979, sections 4.13, 5.12, and 8.3). The second possible tack, which we make here, is to modify the lateral boundary condition so that it is sufficient for the problem to be wellposed and to implement that condition without introducing unwanted effects. As discussed by Greenspan (1968) for steady motion, by Mofjeld (1980) for Kelvin waves, and by Pedlosky (1979), a natural requirement for the flow near a vertical wall is that the normal transport be zero at some distance from the coast. The thickness of the unresolved boundary layers depends at least on viscosity, depth and latitude, and it may be time-dependent. Nevertheless, those layers are in general quite thin for the scales we are considering in the present discussion. Hence, we set 0
I u . n dz = 0
-n
along the closed boundaries of the computational domain. From a theoretical point of view, equation (12) is a sufficient boundary condition for the equations of the FU-model. Indeed, as is discussed, e.g., by Davies (1987), the "3-D (x,y,z,t)" problem can always be decomposed in a "2-D (x,y,t)" problem plus a "1-D (z,t)" problem. The
379
Fig. 2. Time series of sea surface elevation in the southwest comer of the basin. The solid line shows the results of the 3-D FU-model. The horizontal line indicates the elevation of the exact solution at steady state. The dashed line corresponds to the results, discussed in section 6, of a 2-Dmodel.
2-D problem is similar to the classical, vertically-integrated, shallow water wave model for which (12) is an appropriate condition. From a computational point of view, however, the solution of the 3-D equations does require the evaluation of the Coriolis term at all depths, or, equivalently, for all vertical modes in the function model. In particular, if a C-grid is used, some kind of averaging is necessary. Hence, something must be done, in addition to using (12) in the continuity equation. We have shown (Jamart and Ozer, 1986) that a simple stratagem to eliminate the spurious boundary currents observed in the results of the FU-model is to retain in the calculation of the Coriolis term only those points that are not located on the boundary (the "wet-points-only'' approach). Other devices to the same effect, which can be referred to as "computational boundary conditions", are presently under study. When the wet-points-only approach, together with equation (12), is implemented in the FU-model, the solution of the test problem described in Section 3 converges towards the exact steady state solution. Sample results are shown for a particular location (the southwest comer of the basin) in Figs. 2 (elevation) and 3 (depth-averaged speed). For this calculation, A, is m2 s-l ; a no-slip condition is used at the bottom; the constant and equal to 65 x
380
SPURIOUS LATERAL BOUNDARY LAYER ON A C-GRID Simulation # A1 : flat bottom ; southward wind Depth-mean 0.81
current (cm/sec)
near the southwest corner
0.6
0.4
0.2
0.0
-wet -points-
on ly
I________ convent.
~~
approach
Fig. 3. Time series of the depth-averaged current in the southwest comer of the basin. The solid line shows the result of the 3-D FU-model using the wet-points-only method for the evaluation of the Coriolis term. The dashed line shows the erroneous result obtained when the conventional approach is used. computational grid is uniform, with Ax = Ay = 40 km, 10 modes are retained in the velocity expansions. When the Coriolis term is evaluated in the conventional way along the boundary - or, in other words, if the condition u . n = 0 at all z, which also comprises (12), is used in the horizontal averaging procedure - the depth-averaged velocity does not tend to zero as indicated by the dashed line on Fig. 3. The elevation, however, is hardly affected by the spurious residual current. 5. SPURIOUS LATERAL BOUNDARY LAYER ON A B-GRID
With the aim of verifying the time-dependent part of the results of the FU-model, for which no analytical solution appears available, the test problem was solved with the FD-model using the same parameters as for the FU-model. Previous experiments with a similar model reported by Jamart et af. (1986) have indicated that a rather fine vertical spacing is required to obtain a correct evaluation of the bottom stress. Hence, we chose Az = 2 m to perform the calculations , which may be somewhat of an overkill considering the unrealistically large value assigned to 4.
381
SPURIOUS LATERAL BOUNDARY LAYER ON A B-GRID Simulation # A1 : flat bottom ; southward wind Elevation (cm) 12
near the southwest corner
10 8 6
4 2 0
0
6
12
-FD-model
(case 1)
-..-...FD-model
(case 2)
18
24
30
36 42 Time (hours)
48
________ FU-model
Fig. 4. Time series of sea surface elevation in the southwest comer of the basin. The solid line shows the result of the 3-D FD-model using a uniform grid. The dashed line shows the result of the 3-D FU-model. The dotted line shows the result of the 3-D FD-model using a variable grid. Preliminary results obtained with the FD-model are shown in Figs. 4 and 5 , wherein the corresponding time series calculated with the FU-model are also displayed for comparison. Obviously, there are large discrepancies (mostly phase differences) between the results of the two models. The analysis of the results of the benchmark calculation, which was also performed with a 2-D (vertically integrated) model. indicates that the transient response of the system can be viewed as a superposition of :
- the building up, in an asymptotic fashion, of the steady state solution for the sea surface slope;
- fairly rapidly damped inertial oscillations of the velocities at the various depth-levels; - propagating and more slowly damped long gravity waves and the associated quasibarompic currents.
Both the shape and the phase speed of the gravity waves are affected by rotation and by bottom friction (see next section). The first mode, which dominates the transient response, is akin to a Kelvin wave along the straight portions of coastline. This wave does not progress at the same
382
iPURlOUS LATERAL BOUNDARY LAYER ON A B-GRID iimulation # A1 : flat bottom ; southward wind Depth-mean 0.6, 0.5
current (cm/sec)
near the southwest corner
i
0.4
0.3 0.2 0.1 0.0
0
6
12
-FD-model
(case 1)
.-..-..FD-model
(case 2)
18
24
30
36 42 Time (hours)
48
________ FU-model
Fig. 5. Time series of the depth-averaged current in the southwest comer of the basin. Solid dashed and dotted lines are as defined for Figure 4. speed in the FU- and the FD-model. Experiments performed with the FD-model have shown that the observed differences cannot be attributed to :
- the presence or absence of the advective terms; - either the timestep or the degree of implicitness with which various terms are calculated;
- differences in the numerical treatment of variations in the vertical direction; the bottom stresses calculated by the two models in the absence of rotation are the same. Hence, the only possible explanations of the discrepancy are :
- the effect of the horizontal viscous terms (included in the FD-model only); - the difference in computational grids (B-grid in the FD-model; C-grid in the FU-model); - the difference in boundary conditions.
These effects have been studied on the shallow water wave equations by Davey er al. (1983) and Hsieh et af. (1983),whose results are applicable to the present 3-D calculations. First, with a Rossby radius L = c/f - 263 km,the horizontal Ekman number Eh = Ah I fL.’ is of order lo4 for Ah - o(1d m2 s-’). Therefore, using the length of the basin perimeter normalized by L to calculate the nondimensional wavenumber I of the dominant mode, the
383
parameter E = E, / I introduced by Davey et al. (1983) to estimate the importance of lateral viscosity is found to be O(104). Horizontal friction should not affect the phase speed. This is confirmed by numerical experiments : the results of the FD-model are not sensitive to variations of A,. Second, Hsieh et al. (1983) show that, in the absence of lateral viscosity, the phase speed of a free Kelvin wave calculated using a C-grid is independent of the "resolution parameter", defined as A = Ax/L assuming Ay = Ax. For the B-grid, however, provided there is any horizontal friction at all, the phase speed is i) always smaller than in the inviscid case; ii) dependent upon A, and strongly so if &/Az I1; iii) different whether one uses a free-slip or a no-slip lateral boundary condition. For the two grids, the effect of bottom stress only (assumed to be a linear function of the depth-averaged current in those studies) is basically the same as long as A I0.3. Neglecting this bottom stress effect, we find from Fig. 4 of Hsieh et al. that for a uniform grid spacing of 40 km yielding A - 0.15, a free Kelvin wave propagates on a B-grid with no-slip boundary condition at a speed approximately equal to 92% of that calculated on a Cgrid. This is precisely the ratio of the times elapsed between the fist two maxima in elevation calculated with the FU-and with the FD-model. In physical terms, the no-slip boundary condition and the B-grid used in the FD-model create a spurious boundary layer wherein the velocities (and hence the transports) are brought to zero over the cell adjacent to the boundary notwithstanding the fact that the viscous layer is not properly resolved with a meshsize of 40 km. In particular, the longshore transport of the Kelvin waves, which should increase exponentially up to a rather short distance from the coast, is significantly underestimated, leading to a decrease in the phase speed.
As was the case in the previous section, there are several ways to eliminate the spurious boundary layer. First, a free-slip lateral boundary condition is shown by Hsieh et al. (1983) to affect to a much lesser extent than a no-slip condition the phase speed of a free Kelvin wave calculated on a B-grid. Such an approach is certainly useful for the shallow water wave equations but it could be troublesome for the 3-D calculation of the wind-driven part of the motion. We have not attempted to use a free-slip condition in the FD-model. A "safe bet" approach is to reduce the uniform meshsize to the point where the lateral viscous layer is well resolved and/or where the long gravity waves are unaffected by the boundary condition. This remedy is computationally expensive. A third approach is to refine the grid locally, i.e., to use a small A only near shore and in the direction normal to the coast. We have repeated the test calculation with the FD-model using a variable grid spacing (Ax = 5, 10, 15, 20, 30,40 km going from the boundaries towards the center of the basin). The results of that computation are shown in Figs. 4 and 5. They are
in excellent agreement with those of the FU-model. We conclude that the effects of the spurious lateral boundary layer, which is still present, have been effectively reduced to noise level.
384
6. EFFECTS OF THE BOTTOM BOUNDARY LAYER The modelling of the bottom boundary layer (BBL) is no easy task. The question of how the turbulent processes occuring in the BBL should be parameterized is certainly not yet settled. The choice of appropriate values for whatever parameters are included in a given model is also not trivial and the answer is usually site- and/or model-specific. Nevertheless, it is of interest to study the effects on model results of some parameterizations. This is indeed one of the main goals of the model intercomparison exercise mentionned earlier. This exercise includes i) the FD-model; ii) the FU-model; iii) a conventional 2-D (vertically integrated) storm surge model. In both 3-D models, the turbulent mixing of momentum is represented by means of an eddy coefficient. The bottom stress used in the 2-D model, for the experiments reported here, is a linear function of the depth-averaged current, x b = y p ii. For the test problem described in section 3, and provided that the spurious lateral boundary layers discussed in sections 4 and 5 are taken care of, the results of the FD- and FUmodels are almost undistinguishable. The results of the 2-D model, however, are significantly different from those of the 3-D models. For example, the time series of the elevation in the southwest corner calculated with the 2-D model is compared to that of the FU-model in Fig. 2. The friction coefficient y used for this computation is taken as 2.4 x m s-'. The 3-D solution is seen to be always higher than the 2-D solution, to have a first mode oscillation of larger amplitude and longer period, and to have a "smoother" evolution than the elevation calculated with the 2-D model. These three characteristics are even more obvious when the same comparison is done with f = 0 (see Figs. 2 and 3 of Jamart and Ozer, 1986). Heaps (1972) has already observed and commented on such differences, and so have we in several papers. In particular, we have contrasted in some detail the role of the bottom stress in 2-D versus 3-D models for steady state wind-driven circulation in shallow basins (Jamart and Ozer, 1987). This aspect is not discussed here. We have also investigated the effects of bottom stress parameterization on the free oscillations in an enclosed basin (Jamart er al., 1986) using a "semi-analytical" approach. That theory is hereafter improved and expanded upon. For convenience, and because it is sufficient to demonstrate our main points, we restrict the analysis to the case of a constant A,. We shall also ignore rotation in the present study of bottom stress effects. The effects of the Coriolis term on waves in a rectangular basin can only be studied numerically. For an inviscid, homogeneous fluid, Rao (1966) has shown that the effect of rotation on the free gravitational oscillations in a 2 x 1 rectangular basin is to decrease the frequency of the first positive antisymmetric mode and to increase the frequency of all the other modes. The first mode is dominant in the results of our calculations. For the test prob lem discussed here, the period of the first longitudinal mode (equal to 21.3 h in the absence of friction and of rotation) is lengthened by a factor of about 1.1 (T,- 23.4 h) on the rotating earth. This frequency shift due to rotation only is different from that due to bottom stress only which we discuss hereafter.
385
SOLUTION OF THE DISPERSION EQUATION Vormalized period of the gravest free mode Normalized Deriod
1.141.121
.lo-
1.081.061.04
.
0.981 0
.
I
200
I
400
.
I
600
.
I
800
. 1000 . 1200 . 1400 . 1600 I
I
I
1
Av (cm*cm/sec)
I________ slip
-no-slip
-,....-
I-----
slip (K--0.2)
( ~ ~ 0 . 1 )
slip (K--0.4)
Fig. 6. Period of the first longitudinal free mode, normalized by the period of the frictionless solution, as a function of the vertical eddy coefficient. The solid line corresponds to the case of a no-slip bottom boundary condition. Short dashed, dotted, and long dashed lines show the results in the case of a linear slip condition, with K = 0.1, 0.2 and 0.4 cm/s, respectively. For the 2-D, linear model, assuming q dispersion equation is
w2+ i w
5=I
k
gH
- exp [i (k . x - w t)] , it is easily shown that the
.
(13)
Consider now the 3-D model defined by equations (8) to (lo), with f = 0. The vertical problem (a diffusion equation with boundary condition (11) at the bottom and zero shear at the surface) can be solved separately with the pressure gradient as a forcing term. This also holds true for the case f # 0 (see Mofjeld, 1980, for a no-slip bottom boundary condition). The divergence of the bottom stress can then be. calculated analytically, and the dispersion equation for plane waves is found to be :
w2=lkfgH where
[ tanip I--
A = ( l -i)
[ l + * AK' t a n h A I - I ] ]
d z.
The analysis of equations (13) and (14) is most easily done, in
physical terms, by considering separately the effects of the bottom stress on i) the frequency
-no-slip slip ( K s O . 2 )
________ _____
slip (K-0.1) slip (K=0.4)
Fig. 7. Bottom stress induced damping coefficient as a function of the vertical eddy coefficient, Line types are the same as in Figure 6 shift with respect to the frictionless case and ii) the damping of the free waves. For the 2-D model and for reasonable values of y, (13) indicates that the frequency shift is extremely small for the lowest modes present in our test calculations. For example, with y = 2.4 x m s-l , the period of the first longitudinal mode is increased by less than 0.25 hour. The damping coefficient (i.e., the imaginary part, q ,of the frequency), given by - y / 2H, is independent of the wavenumber. For the 3-D model, by contrast, the dispersion equation shows that .both the frequency shift and the damping coefficient depend on the wavenumber as well as on the depth and the friction parameters (A, and K). The solution of (14) for the gravest mode, calculated numerically through an iterative procedure, is shown in Fig. 6 (normalized period as a function of A, for different values of K, the no-slip solution being the limit for K + 00 ) and Fig. 7 (damping coefficient for the same parameters). These figures illustrate the sensitivity of the bottom stress effects to the choice of boundary condition. The phase lag observed in Fig. 2 between 2-D and 3-D results (the latter obtained under the rather unrealistic assumption of no-slip with A, = 65 x m2 s-l) is fully accounted for by the lengthening effect displayed in Fig. 6. Similarly, the difference in damping coefficients (q= - 1.2 x s-l in the 2-D model, versus - 0.8 x s-' in the 3-D model) explains
381
jOLUTlON O F THE DISPERSION EQUATION lamping coefficient Absolute value of the damping coefficient (1 O**-6/sec)
21
I
1
-
3
no-slip
---+slip
(K-0.4)
1
5 .-------.slip
- ,2-D
I
9 11 Mode number
7
(K-0.1) eq.
,
.slip
(KnO.2)
Fig. 8. Bottom stress induced damping coefficient for the first six longitudinal free modes (only odd modes can exist). Line types are the same as in Figure 6. The horizontal line shows the damping coefficient of the 2-D model. the difference in the amplitude of the dominant oscillation. The relative smoothness of the 3-D time series compared to those of the 2-D results is also consistent with equations (13) and (14). Indeed, as is shown in Fig. 8, the damping coefficient in a model that resolves the vertical structure of the flow is dependent on the wavenumber ( I q I increases with I k I , rapidly so if K is large). In a 2-D model, on the other hand, at least for the case of linear friction, the higher modes are not damped more rapidly than the lowest ones, resulting in more complicated time evolutions. Our experience is that this last remark also applies to a 2-D model with a quadratic bottom stress.
7. SUMMARY AND CONCLUSIONS In this paper, three examples of interactions between boundary layer phenomena and large scale flow are discussed. These effects of the boundary layers are either real or spurious. We have shown how the implementation of an inappropriate lateral boundary condition in a 3-D model using a staggered ("C") grid results in the building up of a spurious boundary layer. We have also shown that the lack of resolution near the solid boundary in a model using a B-grid not only prevents the boundary layer phenomena from being properly modelled but
388
also affects to a significant extent the speed of propagation of long gravity waves. Remedies are proposed to eliminate such spurious boundary layers, or at least, their effects on the large scale flow. The parameterization of the turbulence in the bottom boundary layer can be a determinant factor not only for the steady state wind-driven circulation in shallow basins (Jamart and Ozer, 1987) but also for the characteristics of long gravity waves. Dispersion equations taking into account the effects of bottom stress in 2-D and 3-D formulations yield a satisfactory qualitative explanation for the different responses observed in numerical experiments. The dispersion equation for models resolving the vertical structure of the flow also indicates the sensitivity of the results to the choice of bottom boundary condition (slip versus no-slip). Boundary layer phenomena can play an important role in the outcome of threedimensional hydrodynamical calculations. The complexity of the physics and of the numerical procedures dictates that caution be exercised in interpreting the results of models. Simple numerical experiments - homework type calculations - can be useful in gaining insight and in discriminating physical effects from numerical artefacts.
8. ACKNOWLEDGMENTS This work was supported by Det norske Veritas within the frame of an industry research and development project called the Bottom Stress Experiment (BSEX) funded by Fina Exploration Norway. We thank Ms Y. Spitz for help with the calculations and comments on the manuscript, Ms C. Coolen for assistance with the computer graphics and Ms A.F. Lucicki fm typing the manuscript.
9. REFERENCES Arakawa, A. and Lamb, V.R., 1977. Computational design of the basic dynamical processes of the UCLA general circulation model. In : Methods of Computational Physics, Vol. 17, Academic Press. New York, 173-265. Beckers, P.M., and Neves, R.J., 1985. A semi-implicit tidal model of the north European continental shelf. Applied Mathematical Modelling, 9 : 395-402. Cooper, C.K., and Pearce, B., 1982. Numerical simulations of humcane-generated currents. Journal of Physical Oceanography, 12 : 1071-1091. Davey, M.K., Hsieh, W.W., and Wajsowicz, R.C., 1983. The free Kelvin wave with lateral and vertical viscosity. Journal of Physical Oceanography, 13 : 2182-2191. Davies, A.M., 1977. The numerical solution of the three-dimensional hydrodynamic equations using a B-spline representation of the vertical current profile. In : J.C.J. Nihoul (Editor), Bottom Turbulence, Oceanographic Series, Elsevier, Amsterdam, 19 : 1-25. Davies, A.M., 1980. Application of the Galerkin method to the formulation of a threedimensional nonlinear hydrodynamic sea model. Applied Mathematical Modelling, 4 : 245-256. Davies, A.M., 1987. On extracting current profiles from vertically integrated numerical models. Submitted for publication. Ekman, V.W., 1905. On the influence of the Earth’s rotation on ocean-currents. Arkiv fir matematic, astronomi, och fysik, Band 2, No.ll. Greenspan, H.P., 1968. The Theory of Rotating Fluids. Cambridge University Press, Cambridge, 328 pp.
389
Heaps, N.S., 1972. On the numerical solution of the three-dimensional hydrodynamic equations for tides and storm surges. Mtmoires de la SocittC Royale des Sciences de Litge, 6 (11), 143-180. Heaps, N.S., 1981. Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers - I. Mathematical formulation. Geophysical Journal of the Royal Astronomical Society, 64 : 291-302. Hsieh, W.W., Davey, M.K., and Wajsowicz, R.C., 1983. The free Kelvin wave in finite difference numerical models. Journal of Physical Oceanography, 13 : 1385-1397. Jamart, B.M., and Ozer, J., 1985. Hydrodynamical modelling of time-dependent flows in 2, 2.5, and 3 dimensional space. In : R. Van Grieken and R. Wollast (Editors), Progress in Belgian Oceanographic Research, The University of Antwerpen, 1-14. Jamart, B.M., Ozer, J., and Spitz, Y., 1986. Bottom stress and free oscillations. In: J.J. O'Brien (Editor), Advanced Physical Oceanographic Numerical Modelling, D. Reidel Publishing Company, 581-598. Jamart, B.M., and Ozer, J., 1986. Numerical boundary layers and spurious residual flows. Journal of Geophysical Research, 91 (C9): 10, 621-10, 631. Jamart, B.M., and Ozer, J., 1987. Comparison of 2-D and 3-D models of the steady winddriven circulation in shallow waters. Submitted for publication. LeBlond, P.H., and Mysak, L.A., 1978. Waves in the Ocean. Oceanographic Series, 20, Elsevier, Amsterdam, 602 pp. Mofjeld, H.O., 1980. Effects of vertical viscosity on Kelvin waves. Journal of Physical Oceanography, 10, 1039-1050. Nihoul, J.C.J., 1977. Three-dimensional model of tides and storm surges in a shallow wellmixed continental sea. Dynamics of Atmospheres and Oceans, 2 : 29-47. Owen, A.A., 1980. A three-dimensional model of the Bristol Channel. Journal of Physical 'Oceanography, 10 : 1290-1302. Paul, J.F., and Lick, W.J., 1981. A numerical model for three-dimensional variable density hydrodynamic flows. U.S. Environmental Protection Agency Report. Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer, New York, 624 pp. Rao, D.B., 1966. Free gravitational oscillations in rotating rectangular basins. Journal of Fluid Mechanics, 25 : 523-555. Taylor, G.I., 1921. Tidal oscillations in gulfs and rectangular basins. Proceedings of the London Mathematical Society, 20 : 148-181.
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391
A TROPHIC-DIFFUSION 3D MODEL OF THE VENICE LAGOON C. DEJAK and G. PECENIK(*) Dept. o f Physical Chemistry, U n i v e r s i t y o f Venice, Dorsoduro 2137, Venice. (*)Dept. o f Environmental P r o t e c t i o n and Security-PAS, MONTEDIPE S.p.A. , v i a d e l l ' E l e t t r i c i t a 41, Marghera, Venice, I t a l y . ABSTRACT A mathematical 3D d i f f u s i o n model o f t h e Venice lagoon's c e n t r a l area i s presented; i t s . s u b s t a n t i a l correspondence w i t h a model o f t i d a l a g i t a t i o n enables t h e e s t i m a t i o n o f a p p r o p r i a t e eddy d i f f u s i v i t y c o e f f i c i e n t s . The cond i t i o n s s p e c i f i e d a t the'lopen boundaries" a r e d e r i v e d from a l o g a r i t h m i c f i n i t e differences a n a l y s i s o f a d e l o c a l i z e d Gaussian d i s t r i b u t i o n : t h i s approach perm i t s a more r e a l i s t i c r e p r e s e n t a t i o n o f c o n t i n u i t y and o f outward f l u x e s . O p t i m i z a t i o n o f t h e model's program f o r execution on p a r a l l e l supercomput e r s y i e l d s marked t i m e and costs savings, and a l l o w s f o r c o n s i d e r a t i o n o f t h e r mal , chemical and b i o l o g i c a l s t a t e v a r i a b l e s and o f t h e i r interconnections,thus enabling t h e model t o describe t h e e u t r o p h i c a t i o n phenomenon i n t h i s p e c u l i a r ecosystem c h a r a c t e r i z e d by h i g h t u r b u l e n t d i s p e r s i o n . Heat f l u x e s are analyzed i n t h e b u l k and a t t h e a i r - w a t e r i n t e r f a c e , t a k i n g i n t o account t h e r e l a t e d processes o f i n s o l a t i o n , r e - i r r a d i a t i o n , conduction and evaporation. The thermal s t r a t i f i c a t i o n i n t h e deeper lagoon channels i s analyzed i n o r d e r t o describe t h e d i f f u s i o n o f heat from t h e u n d e r l y i n g s a l t wedge i n thermal e q u i l i b r i u m w i t h t h e sea, and t o evaluate t h e sea-lagoon exchanges, although t h e model does n o t account f o r c u r r e n t s a t t h e seaward boundaries.
1. INTRODUCTION Since t h e desastrous f l o o d i n g o f t h e Venice Lagoon i n 1966, many e f f o r t s have been made t o develop t h e o r e t i c a l and p r a c t i c a l instruments t o prevent the d e t e r i o r a t i o n o f b o t h t h e h i s t o r i c a l town and t h e lagoon environment.
Because
of the pressing need f o r an o v e r a l l understanding o f t h e lagoon's hydrodynamics, and i n order t o p r e d i c t t i d a l e l e v a t i o n s a c c u r a t e l y , several f i n i t e - d i f f e r e n c e mathematical hydrodynamical 1D ( D i S i l v i o and D'Alpaos, 1972) and 2D ( G h e t t i , 1973; Sguazzero e t a l . ,
1978) models have been developed, i n c l u d i n g a s t i l l par-
t i a l l y o p e r a t i v e p h y s i c a l 3D model b u i l t a t Voltabarozzo (Padova). Other problems have r e c e n t l y emerged, r e l a t e d m a i n l y t o t h e chemical and thermal p o l l u t i o n and t h e consequent e u t r o p h i c a t i o n , p a r t i c u l a r l y i n t h e cent r a l lagoon, which i n c l u d e s t h e h i s t o r i c a l town and t h e h i g h e s t i t a l i a n indust r i a l concentration.
To face t h i s s i t u a t i o n , a g l o b a l i n t e r d i s c i p l i n a r y pro-
j e c t was i n i t i a t e d (Dejak e t al.,
1985a), w i t h t h e aim o f developing a model
s u i t a b l e n o t o n l y f o r s t u d i e s o f p o l l u t a n t s dispersion, b u t a l s o a b l e t o reproduce the s p a t i a l and temporal d i s t r i b u t i o n s o f r e l e v a n t water q u a l i t y parameters, up t o steady s t a t e achievement.
392 2. CONSTRUCTION OF THE THREE-DIMENSIONAL MODEL 2.1 M o d e l l i n g approach and p r e l i m i n a r y d i s p e r s i o n models As an a l t e r n a t i v e t o t h e b u i l d i n g o f e i t h e r inadequate advection o r impractical
a d v e c t i o n - d i f f u s i o n models, t h e 3D model was conceived on account t h a t
the hydrodynamical regime o f t h e lagoon i s predominantly governed by t i d a l mot i o n , and hence t h e p o l l u t a n t s t r a n s p o r t caused by t i d a l m i x i n g may be considered as e q u i v a l e n t t o t h a t e s s e n t i a l l y induced by a t u r b u l e n t d i f f u s i o n phenomenon.
This approach i s j u s t i f i e d f o r t h i s p a r t i c u l a r body o f water because "re-
s i d u a l c u r r e n t s " have n o t y e t been e x p e r i m e n t a l l y evidenced, and t h e r e f o r e the t i d a l movements averaged over more than one t i d a l c y c l e have a zero average ve1o c i ty. The model was developed through successive stages, i n v o l v i n g t h e b u i l d i n g o f i n t r o d u c t o r y d i s p e r s i o n models designed t o overcome b o t h s t r i c t l y computat i o n a l and general methodological and mathematical problems.
I n i t i a l l y , a 2D
d i f f u s i o n model was prepared, u s i n g a very f i n e mesh o f 100x100 m., conform t o t h e w i d t h o f t h e main lagoon channels.
i n order t o
T h i s model was subsequently
extended t o i n c l u d e t h e advection term, a f t e r e l a b o r a t i o n o f a bathymetric map a s s o c i a t i n g t o each o f t h e 17920 g r i d ' s squares t h e average depth (surveyed from a lagoon topographic c h a r t , scale 1:5000) o f t h e corresponding a r e a . Given t h a t t h e number o f measurements o f s u r f a c e and e s p e c i a l l y o f underwater velocit i e s was r a t h e r small ( G h e t t i e t al.,
1979) t h e c u r r e n t s were f i r s t t e n t a t i v e l y
computed using a f u n c t i o n a l r e l a t i o n s h i p , and then by i n t e r p o l a t i o n using an e x i s t i n g hydrodynamical model (D'Alpaos and D i S i l v i o , 1979) which had been cal i b r a t e d w i t h t h e same v e l o c i t y data and w i t h measured t i d a l e l e v a t i o n s . The i n t e g r a t i o n o f t h e d i f f u s i o n and o f t h e advection terms, r e s p e c t i v e l y , was performed w i t h an e x p l i c i t method and a m o d i f i e d second o r d e r "upstream" scheme which ensures convergence, consistency and s t a b i l i t y .
This scheme a l s o presents
conservative and t r a n s p o r t i v e p r o p e r t i e s r e l a t e d t o t h e c a p a b i l i t y t o damp out d i s c o n t i n u i t i e s (Roache, 1972).
These models c o n s t i t u t e d t h e f i r s t attempts t o
i n c l u d e c o n t r i b u t i o n s due t o t h e t h i r d dimension : i n f a c t , accarding t o the mod i f i e d numerical scheme, t h e c a l c u l a t i o n o f t h e i n t r a c e l l a d v e c t i v e - d i f f u s i v e f l u x accounts f o r the d i f f e r e n c e i n depth between two contiguous 3D c e l l s . 2.2 "Open boundary" c o n d i t i o n s The d i f f i c u l t y t o describe t h e r a t h e r complex t r a n s p o r t processes occuring a t t h e sea entrances and t h e need t o r e s t r i c t t h e modelled area r e q u i r e dealing w i t h "open boundaries" across which a continuous f l u x o f p o l l u t a n t occurs.These s i t u a t i o n s a r e commonly handled e i t h e r by assuming a very f a r boundary where c o n c e n t r a t i o n ( o r d e n s i t y , temperature) i s zero a t a l l times, o r by s e t t i n g equal t o zero, a t a l l times, t h e g r a d i e n t s normal t o t h e boundary.
393
When applying these almost i d e n t i c a l methods, however, an excessive and u n r e a l i s t i c outward mass f l u x was observed.
To avoid t h i s , d i f f e r e n t procedures were
t r i e d , including: a macroscopic balance i n a d d i t i o n t o the p o i n t by p o i n t microscopic balance given i n the d i f f u s i o n equation; the s o l u t i o n o f a time dependent d i f f e r e n t i a l equation a t t h e boundaries, w i t h e x t r a p o l a t i o n s using from three t o s i x e c c e n t r i c p o i n t s (Abramowitz and Stegun, 1964).
The most s a t i s f a c t o r y
solution was obtained by approximating t h e t a i l o f a Gaussian d i s t r i b u t i o n a c r o s the boundary, through a f i n i t e d i f f e r e n c e a n a l y s i s o f t h e logarithm o f the concentration, t h a t i s using a d e l o c a l i z e d Gaussian d i s t r i b u t i o n o r a " f i n i t e r a t i d ' method. n=O c =c 0 1
n=l c =const 1
n=2
n=3 c0=c3R 3
c =C R 0 1
Figure 1. Schematization o f possible e x t r a p o l a t i o n s w i t h n p o i n t s i n v o l v e d : a) no f l u x , b) impossible e x t r a p o l a t i o n , c ) exponential smoothing, d) Gaussian. The adopted method has been shown t o s a t i s f y c o n t i n u i t y o f outward mass flux, as checked against commonly proposed c o n d i t i o n s (Dejak e t a1
. , 1985b).
By means o f a simple numerical procedure, the method can be used t o extrapolate
boundary concentrations ( c o y o u t s i d e the boundary) a t a l l times and a t a l l points, i n terms o f t h e values a t t h r e e i n s i d e g r i d p o i n t s (c,, the boundary).
c2, c3, w i t h i n
Cases appearing i n p r a c t i c a l a p p l i c a t i o n o f t h i s method are
shown i n Fig.1, where R = cl/c2
( o r c2/c1 i f cl > c2).
2.3 Achievement o f steady s t a t e and r e l a t e d mathematical conditions With both 2D a d v e c t i o n - d i f f u s i o n models, some o f t h e major features of p o l l u t a n t d i s p e r s i o n i n the lagoon were reproduced. steady s t a t e was never reached (Dejak e t al.,
Yet, w i t h t h e f i r s t model
1985c), even a f t e r covering a
high numberof cycles, probably because o f the inaccuracies i n t h e v e l o c i t y field, but mainly as a consequence o f the s t r i c t b i d i m e n s i o n a l i t y o f t h a t model.
In
contrast, by using more r e a l i s t i c t i d a l v e l o c i t i e s , and hence i n c l u d i n g c o n t r i butions coming from the t h i r d dimension, a steady s t a t e was a t t a i n e d w i t h the other model a f t e r a reasonable number o f semidiurnal t i d a l cycles, (Dejak e t al.,
1985d).
394
The difficulty of attaining steady state with either the 2D diffusion model or the earlier advection-diffusion model with an exactly time averaged tide in two dimensions, challenged the possibility to reach steady state through a purely 30 diffusion model or through its equivalent. Mathematically it may be shown (Dejak et al., 1985d) that steady state cannot be attained through a purely diffusive process either in one or in two dimensions. In fact, considering a n-dimensional system (n = 1,2 or 3) and a point-source, assuming that an amount Q of pollutant is released instantaneously, the concentration c, setting IJ~ = r / m , is given by
If the point-source emits continuously with polluting strength Q, the solutions for n equal to 1 and 2, are both diverging and are given respectively by
Standard notations for El, erfc(.), y = l n r , are used (Abramowitz and Stegun, 1964). Finally, for n=3, the following solution is obtained, which tends to the well known solution of the diffusion equation at steady state:
&
n=3 c , = n r erfq =&[+(I
- p+ & -
..)I+
Q
By performing a further integration, it is also easily demonstrated that, within a sphere of radius 6r however small centered at the source, in the first two cases, the amount of pollutant grows indefinitely, while for n=3, it assumes the finite value Q6r2/2k, as time tends to infinity. Hence, contrary to what is sometimes asserted (Diachishin, 1963), the accumulation of a conservative pollutant does not continue indefinitely. This indicates that steady state through a purely diffusive process may be achieved only in three dimensions (Dejak et a1 . , 1985d). 2.4 Choice of the numerical scheme and estimation of 3D eddy diffusivity Steady state pollutant distributions emerging from the advection-diffusion model were observed to be substantially comparable to those produced by the
395
pure d i f f u s i o n model i n t h r e e dimensions (Dejak e t al.,
1985e).
This outcome
f u r t h e r supported the approach, whereby the t i d a l d i s p e r s i v e mechanism might be accounted f o r by means o f a global 3D t u r b u l e n t d i f f u s i o n , i n v o l v i n g an e f f e c t i v e eddy d i f f u s i v i t y l a r g e r (by about one order o f magnitude) than t h a t which would be needed t o reproduce o r d i n a r y v o r t i c i t y a t lagoon scale.
As a f i r s t
approximation, space independent eddy d i f f u s i v i t i e s were included, and t o conform w i t h experimental evidence (Kullenberg, 1971; Kullenberg e t a l . , 1973) the v e r t i c a l d i f f u s i v i t y was assumed t o be one order o f magnitude smaller than the horizontal d i f f u s i v i t y . and v e r t i c a l steps ( 1 m.),
Because o f the d i s p a r i t y between h o r i z o n t a l (100 m.) and o f t h e r e s u l t i n g l a r g e magnitude o f the d i f f u -
sion number, t h e use o f an i m p l i c i t scheme f o r v e r t i c a l d i f f u s i o n was unavoidable t o guarahtee s t a b i l i t y f o r reasonably small time increments.
Among the
many a v a i l a b l e two and t h r e e time l e v e l s schemes (Richtmyer and Morton, 1967) t h a t o f Laasonen (1949) was selected a f t e r comparative 1D a n a l y s i s w i t h respect t o the a n a l y t i c a l s o l u t i o n (Baule, 1970). We chose t h i s scheme because, despit e d i s c o n t i n u i t i e s , i t presents a higher s t a b i l i t y (as proved by F o u r i e r analys i s ) , and a b e t t e r convergence w i t h continuous input, than the o t h e r i m p l i c i t methods.
I n i t i a l l y , t h e i m p l i c i t Laasonen scheme was o n l y adopted f o r t h e c m Later, a f t e r the i n t r o d u c t i o n i n t h e pro-
putation o f the v e r t i c a l d i f f u s i o n .
gram o f a d d i t i o n a l loops r e q u i r e d by the nonlinear "open" boundary conditions, the model was made f u l l y i m p l i c i t , thus achieving a f a s t e r computation even f o r a large d i f f u s i o n number.
The m o d i f i e d model was t e s t e d against the o r i g i n a l
more r e l i a b l e e x p l i c i t model, taken as reference; comparable d i s t r i b u t i o n s were obtained f o r a d i f f u s i o n number up t o a value o f ten. The e f f e c t i v e eddy d i f f u s i o n constant embodying t i d a l a g i t a t i o n was estimated by comparing the steady-state surface d i s t r i b u t i o n s obtained r e s p e c t i v e l y w i t h the 30 model and w i t h 2D a d v e c t i o n - d i f f u s i o n model.
The optimal value f o r
the h o r i z o n t a l d i f f u s i v i t y determined i n t h i s way i s equal t o 22 m2/sec ( o r 7.92 hm2/h).
With t h i s value, both d i s t r i b u t i o n s are i n reasonable agreement.
This r e s u l t i s o f g r e a t s i g n i f i c a n c e as i t j u s t i f i e s the u t i l i z a t i o n o f the 3D model, which includes 128x140~20g r i d points, as a r e l i a b l e s u b s t i t u t e f o r the advection-diffusion model f o r the d e s c r i p t i o n o f p o l l u t a n t dispersion i n the
1agoon. 2.5 Eutrophication model To account b e t t e r f o r the main p e c u l i a r i t i e s o f t h e lagoon ecosystem, the b i o l o g i c a l compartment was conceived p r i m a r i l y on t h e basis o f i n f o m a t i o n provided by " i n s i t u " f u n c t i o n a l i n v e s t i g a t i o n s . A r a t h e r l i m i t e d study o f t h i s type l e d t o the d e f i n i t i o n o f a f i r s t " t r o p h i c r e a c t o r " (Cescon e t al., o r i g i n a l l y adopted t o describe t h e b i o l o g i c a l processes.
1976),
Eight state variables
are presently considered : phytoplankton, zooplankton, degradable carbonaceous
396 d e t r i t u s , d i s s o l v e d oxygen, n i t r o g e n oxydes, ammonia, orthophosphorus, and temperature.
A f u r t h e r r e f o r m u l a t i o n o f t h e b i o l o g i c a l compartment was based upon m u l t i d i s c i p l i n a r y f i e l d experimentationsinitiated i n 1978 under t h e c o o r d i n a t i o n of a Regional Commission and w i t h t h e c o l l a b o r a t i o n o f t h e ENEL-DCO, Environmental Dept. (Piacenza).
The research, whose main o b j e c t i v e was t o study t h e e f f e c t s
of t h e m a l discharges upon the lagoon ecosystem, was soon extended t o i n v e s t i gate o t h e r r e l a t e d biochemical and physicochemical processes, p a r t i c u l a r l y : phytoplankton growth r a t e as a f u n c t i o n o f l i g h t , n u t r i e n t s and temperature, phytoplankton r e s p i r a t i o n , h e t e r o t r o p h i c and n i t r i f i c a t i o n a c t i v i t y , k i n e t i c o f release o f n i t r o g e n and phosphorus from sediments, b e n t h i c oxygen demand, BODdissolved oxygen r e l a t i o n .
So f a r , r e l i a b l e parameter values have been detetmi-
ned f o r phytoplankton maximum growth r a t e and h a l f - s a t u r a t i o n constants f o r n i trogen and phosphorus ( B e r t o n a t i e t a1
., 1985) , as
growth temperature and l i g h t i n t e n s i t y .
we1 l as optimal and maximum
Parameters n o t y e t e x p e r i m e n t a l l y de-
termined were d e r i v e d by e l a b o r a t i o n o f l i t e r a t u r e data (Joergensen, 1979) and from model c a l i b r a t i o n . An example o f two steady s t a t e d i s t r i b u t i o n s provided by t h e model u n i f y i n g d i s p e r s i v e and trophic-dynamic processes, i s shown
i n Fig.2.
The v a r i a b l e s
displayed are phytoplankton and orthophosphorus. 3. COMPUTATIONAL IMPLEMENTATIONS
The computer program was o r i g i n a l l y conceived and organized t o be executed on t r a d i t i o n a l a r c h i t e c t u r e computers.
B r i e f l y , i t i s structured i n three
p r i n c i p a l p o r t i o n s : i n i t i a l i z a t i o n ( i n c l u d i n g parameter d e f i n i t i o n , bathymetry reading, computation o f a u x i l i a r y f u n c t i o n s ) ; t h e main
iteration(continu0us
i n p u t , computation o f 3D d i f f u s i o n o f the e i g h t v a r i a b l e s , computation o f biol o g i c a l e q u i l i b r i a ) ; o u t p u t o f d i s t r i b u t i o n maps a t preassigned time. O f these p a r t s , t h e c e n t r a l one, p e r t a i n i n g t o 3D d i f f u s i o n , c a r r i e d o u t with t h e f r a c t i o n a l step method f o r each d i r e c t i o n , i s t h e most t i m e consuming.
On a CDC
7600, one s i n g l e d i f f u s i v e step f o r a l l e i g h t s t a t e v a r i a b l e s takes two hundred seconds.
With almost f o r t y c y c l e s needed t o achievesteady s t a t e , such a compu-
t a t i o n a l time i s unacceptable f o r r o u t i n e runs.
Furthermore, t h e modelled wa-
t e r body presents f o r a l l l a y e r s an extremely i r r e g u l a r shape, w i t h indented boundaries associated w i t h t h e a b r u p t l y changing bathymetry.
Also, computa-
t i o n a l complications a r i s e from t h e n o n l i n e a r "open boundaries" c o n d i t i o n s as these i m p l y a u x i l i a r y e x t r a p o l a t i o n loops and from t h e huge memory requirements which n e c e s s i t a t e t h e programming o f l a b o r i o u s swapping o f data t o and from secondary memories. The program was a l s o r u n on t h e supercomputers H045480/4 ( H i t a c h i , Tokyo)
397
Figure 2 . Steady s t a t e d i s t r i b u t i o n s o f a) phytoplankton ,b)orthophosphorus
and CRAY-1 a t CIS1 (Clamart sur Seine) where some attempts t o p a r t i a l l y vector i z e the code were made, y i e l d i n g o n l y marginal b e n e f i t s .
I n order t o e x p l o i t
new p a r a l l e l a r c h i t e c t u r e computers, o n l y r e c e n t l y accessible i n I t a l y , an extensive r e o r g a n i z a t i o n o f t h e program was untertaken t o enhance v e c t o r i a l code generation. Consequently, a number o f c o n d i t i o n a l expressions contained w i t h i n
398
"DO-LOOPS" and o r i g i n a l l y programed t o deal w i t h the many possible situations, were removed and replaced by a l g e b r a i c expressions i n v o l v i n g m u t u a l l y excluding terms a c t i v a t e d by l o g i c a l c o e f f i c i e n t s .
By analysing and grouping the many
d i f f e r e n t c o n d i t i o n a l typologies, a l i m i t e d number o f l o g i c a l c o e f f i c i e n t s could be i d e n t i f i e d and packed i n one s i n g l e m a t r i x : the m a t r i x i s c o d i f i e d once, a t s t a r t - u p time, and i t i s decodified, w h i l e o p t i m i z i n g the number o f a u x i l i a r y vectors required, i n s i d e each d i f f u s i o n subroutine. The s t o r i n g capacity was s t i l l i n s u f f i c i e n t t o handle simultaneously the e n t i r e model. This problem was overcome by removing a l l c e l l s n o t representing r e a l water volumes, and by packing o n l y the e f f e c t i v e elements i n e i g h t wide vectors, which replaced the e i g h t 3D matrices o r i g i n a l l y required.
By so doing,
one 3D m a t r i x o f p o i n t e r s i s now needed f o r recovering t h e s p a t i a l p o s i t i o n s of each element: vectors are scattered, depending on each coordinate, w i t h special procedures t o minimize the sizes o f t h e a u x i l i a r y arrays. The reorganized program, as t e s t e d on the CRAY X/MP,
n o t o n l y permits a
reduction o f the computing time t o o n l y t e n seconds per d i f f u s i v e step, but i t a l s o prevents memory s a t u r a t i o n .
The p o s s i b i l i t y o f maintaining time and expew
ses a t a reasonable l e v e l w i l l a l l o w f u r t h e r important m o d i f i c a t i o n s o f the model, p a r t i c u l a r l y i n the b i o l o g i c a l compartment, where t h e i n c l u s i o n o f addit i o n a l v a r i a b l e s i s prospected as are more d e t a i l e d formulations o f t h e i n t e r connections. 4. MODEL APPLICATIONS
E i g h t years ago, w h i l e they were s t i l l being developed, t h e models described above had t o be a p p l i e d t o urgent p r a c t i c a l problems, mainly connected t o the energy sector.
Such problems o f t e n r e q u i r e a m u l t i d i s c i p l i n a r y approach.
For t h e f i r s t study, which concerned the dispersion o f heat released from the i n d u s t r i a l zone (Dejak e t al., 1977), a p r e l i m i n a r y model was used i n which the 3D g r i d j u s t served t o create t h e equivalence between the r e a l bathymetry and a simple sloping bottom t o permit t h e a n a l y t i c a l 3D i n t e g r a t i o n o f the d i f f u s i o n equation (Dejak e t al., 1975). From t h i s e a r l i e r a p p l i c a t i o n i t was possible t o design r a t i o n a l l y t h e f i e l d experiments required t o assess the thermal impact on t h e lagoon o f the c o o l i n g water discharged from a 1 GW power plant. The f i v e year long research t h a t followed, coordinated by a Scientific-Technic a l Commission purposely appointed by the Regional Authority, represents even now the o n l y r e a l l y long term m u l t i d i s c i p l i n a r y and systematic i n v e s t i g a t i o n o f the lagoon ecosystem a c t u a l l y performed ( B a t t a g l i a e t a1
. , 1983).
Moreover,
based upon i n d i c a t i o n s provided by t h i s work, new i n t e g r a t i v e d i r e c t i v e s were t e n t a t i v e l y p u t forward, r e g u l a t i n g thermal p o l l u t i o n i n t h e lagoon.
The 3D
model developed could thus be designed by s t r i c t l y coupling experimental and
399
t h e o r e t i c a l works, as f i e l d data allowed f o r a l e s s a b s t r a c t modelling, and conversely t h e o r e t i c a l work promoted those experimental studies most needed t o proceed i n the modelling e f f o r t s .
Indeed, a t l e a s t one extensive t r a c e r d i f f u -
sion experiment i s r e q u i r e d i n order t o f u l l y implement a combined dispersion eutrophication model.
Such an experiment i s , a t present, f i r m l y impeded by the
local Health A u t h o r i t y . To circumvent t h i s obstacle, t h e model i s applied,with the support o f the National Research Council, t o determine optimal s i t e s f o r a lagoon water q u a l i t y continuous monitoring network.
The r e a l i z a t i o n
of this
network i s indispensable both f o r t h e c o l l e c t i o n o f a l l data r e q u i r e d f o r a decisive comparison between theory and experience, and f o r the p r e d i c t i o n o f the e f f e c t s o f t h e programmed h y d r a u l i c i n t e r v e n t i o n s on the lagoon ecosystem.
5 . MODEL EXTENSION The improvement most r e q u i r e d o f t h e present model i s the i n t r o d u c t i o n o f the thermal i n f l u e n c e on chemical and b i o l o g i c a l processes.
To achieve t h i s ,
the c u r r e n t ' f i e l d studies focus on t h e temperature dependence o f r a t e processes, and the modelling work seeks t o make i t possible t o f o l l o w t h e d a i l y as w e l l as the seasonal e v o l u t i o n o f temperature a t each p o i n t o f t h e modelled area. Even w i t h the seven hundreds o r so temperature p r o f i l e s now a v a i l a b l e , surveyed i n varying depths a t selected lagoon s i t e s and f o r d i f f e r e n t seasons, attempts t o derive temperature values
f o r such a wide model d i r e c t l y form experimental da-
t a appear u n l i k e l y t o succeed, even w i t h t h e support o f sophisticated s t a t i s t i cal treatments.
The d e f i n i t i o n o f a complete s e t o f temperature values, i n
fact, may n o t be derived by c o r r e l a t i o n o f e x i s t i n g h i s t o r i c a l series o f a i r and water temperatures, both because a l a r g e p o r t i o n o f the modelled area i s affected by c o o l i n g water discharges and because o f t h e h i g h l y varying n a t u r a l heat exchanges between the lagoon and the sea.
More p r e c i s e l y , the s o l u t i o n t o t h i s problem i n v o l v e s a d e t a i l e d q u a n t i f i c a t i o n o f the global lagoon heat budget, based on f l u x e s a t i t s boundaries, and p a r t i c u l a r l y a t t h e a i r - w a t e r i n t e r f a c e where t h e processes o f i n s o l a t i o n , backscattering , conduction and evaporation occur ( M a r i o t t i and Dejak, 1982).
Moreover, i n t e r n a l processes
a f f e c t i n g the heat d i s p e r s i o n must be analyzed, p a r t i c u l a r l y i n the v e r t i c a l d i r e c t i o n where thermocline formation and s t r a t i f i c a t i o n add f u r t h e r compl i c a tions. The model 1i n g approach aims a t determining temperature values a t a l l times and a t a l l g r i d ' s nodes, and a t f i n d i n g a d e s c r i p t i o n , consistent w i t h t h e pur e d i f f u s i o n model discussed above, o f the heat exchange w i t h the atmosphere as w e l l as between t h e sea and t h e lagoon.
To handle t h e f i r s t problem, a
monthly v i r t u a l e q u i l i b r i u m temperature, i.e.,
the temperature associated t o
a zero n e t average heat f l u x , has been calculated.
From these, assuming an
400 i n i t i a l a r b i t r a r y punctual temperature value, the e q u i l i b r i u m value i s d e t e m i ned by computing heat fluxes, and considering the thermal i n e r t i a o f water col m n s o f varying depths through an i t e r a t i v e procedure, under t h e c o n s t r a i n t t h a t the y e a r l y sum o f the f l u x e s has t o be equal t o zero.
By successivelyre-
f i n i n g the i n i t i a l a r b i t r a r y temperature, the i t e r a t i o n converges t o a set o f values.
unique
The values o f the meteoclimatic parameters used i n the calcula-
t i o n s were the o r i g i n a l monthly averaged data c o l l e c t e d over a t h i r t y year survey, as w e l l as l i n e a r l y i n t e r p l o l a t e d ; more r e a l i s t i c r e s u l t s were obtained T e n t a t i v e l y , the same temperature was through F o u r i e r i n t e r p o l a t i o n (Fig.3). i n i t i a l l y assumed f o r water columns o f d i f f e r e n t depths (Fig.4), b u t r e a l i s t i c r e s u l t s were obtained o n l y by accounting f o r thermal s t r a t i f i c a t i o n and by adopting depth varying d i f f u s i v i t i e s .
The l a t t e r are estimated, assuming a
steady v e r t i c a l heat f l u x , from t h e r e c i p r o c a l o f experimental temperature grad i e n t s obtained by a best f i t w i t h generalized Gamma functions.
I n t h i s way,
the t y p i c a l e v o l u t i o n o f t h e thermocline f o r the summer and w i n t e r months was obtained (Fig.5),
showing a temperature d i f f e r e n c e o f about 4OC between top and
bottom l a y e r s and an i n f l e c t i o n p o i n t corresponding t o the highest g r a d i e n t and minimal d i f f u s i v i t y ; during intermediate seasons, t h e temperature p r o f i l e s are steeper.
Figure 3. Monthly surface temperature obtained through d i f f e r e n t i n t e r p o l a t i o n procedures o f meteoclimatic parameters.
401
Figure 4. Annual temperature variation with depth.
0 23: I..........:
0
-z
?3 !
19...........
15-
:
:
I
........... ...........t ........... ........... ...........:..:’ septem be r
......
-.. ..... June .................................................. ............................................... ..... december .. .... ..* ..... a.
w......
a a E a 7 ............. march .............................................................
CI
L
1
4
7
10
13
16 depth m.
Figure 5. Temperature profiles f o r typical months. The distribution of temperature profiles, i n f a c t , suggests the presence , in the deeper lagoon channels of an upper layer and of a lower s a l t wedge i n thermal dynamic equilibrium respectively w i t h the atmosphere and the sea. T h u s , when the sea water temperature d i f f e r s greatly from t h a t of the lagoon, consistently w i t h the model’s conception, i t i s assumed t h a t sea-lagoon heat exchanges occur across a t h i n v i r t u a l layer permitting a very slow, yet continuous heat flux and w i t h which the minimal eddy d i f f u s i v i t y i s associated. Actually, t o define a discontinuity between internal and external temperature, a t the open boundaries, would be incompatible w i t h a pure diffusion model because, as
402
the computation proceeds, such a d i s c o n t i n u i t y would induce i n s i d e the modelled area the progressive and u n c o n t r o l l a b l e formation o f u n r e a l i s t i c h o r i z o n t a l temperature gradients.
We b e l i e v e t h a t even a 3D advection model could n o t ade-
quately reproduce the complex v e l o c i t y f i e l d s r e q u i r e d t o compute the heat exchanges a t the lagoon mouths, n o t included i n t h i s model, b u t represented by open boundaries.
Here, according t o the s p e c i f i e d conditions, an automatic ad-
justment occurs between l o c a l l y computed and sea water temperatures, r e s p e c t i v e l y above and below the v i r t u a l separation l a y e r .
Based upon t h e many expe-
rimental temperature p r o f i l e s available, i t i s now possible t o complete the model c a l i b r a t i o n and thus t o estimate more adequate v e r t i c a l eddy d i f f u s i v i t i e s . The i n t r o d u c t i o n o f seasonally varying d i f f u s i v i t i e s w i l l increase the model cap a b i l i t y t o describe both t h e appearance and t h e e v o l u t i o n o f t h e eutrophicaticn phenomenon during the most c r i t i c a l periods. 6. CONCLUSIONS The construction o f the 3D t r o p h i c - d i f f u s i o n model described here has been successfuly achieved owing t o : i)
the absence i n the lagoon o f experimentally s i g n i f i c a n t c u r r e n t s having a non zero long term average v e l o c i t y ;
ii) conditions enabling t h e s i m u l a t i o n o f t i d a l a g i t a t i o n by mere second o r der equations w i t h constant eddy d i f f u s i v i t y ; iii) the c a p a b i l i t y t o c a l i b r a t e the 3D model, up t o t h e steady s t a t e achievement, w i t h a convergent advection-diffusion model i n c l u d i n g features o v e r coming i t s s t r u c t u r a l two-dimensional i t y ; the implementation o f a simple i n t e g r a t i o n scheme (Laasonen, 1949),insens i t i v e t o d i s c o n t i n u i t i e s and p a r t i c u l a r l y f i t t e d f o r use w i t h continuous input ; t h e e l a b o r a t i o n o f p r a c t i c a l and s u f f i c i e n t l y accurate "open boundaries" conditions, such as t h e d e l o c a l i z e d Gaussian d i s t r i b u t i o n . This treatment of t r a n s p o r t processes has the f o l l o w i n g advantages: t o account, w i t h a very f i n e computational g r i d , f o r a bathymetry t h a t i s n o t r e d u c i b l e t o more r e g u l a r geometries due t o the i n t e r m i n g l i n g o f the lagoon's narrow and deep channels; t o work w i t h "moderate" computing time and memory requirements and hence a t reasonable expenses, notwithstanding the small mesh s i z e and the need t o i n t e g r a t e e i g h t interconnected second order d i f f e r e n t i a l equations; t o enable t h e s i m u l a t i o n o f complex e c o l o g i c a l processes l i n k e d t o phys i c a l phenomena having 3 D c h a r a c t e r s i t i c s , and a l s o r e l a t e d t o the concomitant t r a n s p o r t o f t h e v a r i a b l e s involved;
403 4)
t o reproduce, w i t h adequate approximations, t h e seasonal thermal behavior, i n s p i t e o f complications d e r i v i n g from heat f l u x balance, thermal s t r a t i f i c a t i o n and very i r r e g u l a r sea-lagoon exchanges, avoiding t o r e s o r t t o t h e awkward and uncommon treatment i n v o l v i n g t e m p e r a t u r e - s a l i n i t y - d e n s i t y system. The model, owing t o i t s f l e x i b i l i t y ,
constitutes a practical tool f o r the
i n v e s t i g a t i o n o f t h e main f a c t o r s r e s p o n s i b l e f o r t h e d e t e r i o r a t i o n o f the Venice lagoon and a u s e f u l instrument t o h e l p f i n d a more r a t i o n a l p l a n f o r the now unpostponable i n t e r v e n t i o n s r e q u i r e d t o safeguard i t s ecosystem.
7. ACKNOWLEDGEMENTS The authors thank Sig. Sergio Manzi f o r t h e assiduous c o l l a b o r a t i o n o f f e r e d i n program v e c t o r i z a t i o n . The work has been supported by t h e N a t i o n a l Research Council , P r o g e t t o F i n a l i z z a t o "Energetica 2". 8. REFERENCES Abramowitz, M. and Stegun, I.,1964. Handbook o f mathematical f u n c t i o n s . Dover p:887,914,991. Pub. I n c . , N.Y.; B a t t a g l i a , B., Datei, C., Gambaretto, G., Guarise, G., Perin, G., V i a n e l l o , E. and Zingales, F., 1983. Indagine p e r l a v a l u t a z i o n e dei r i f l e s s i ambientali d e l funzionamento a piena potenza d e l l a c e n t r a l e t e r m o e l l e t r i c a d i Fusina. Commissione Tecnico S c i e n t i f i c a per l a sperimentazione ed i c o n t r o l li p e r i o d i c i s u l l a Centrale T e r m o e l e t t r i c a ENEL s i t a i n l o c a l i t a Fusina d i Porto Marghera, Venezia. Provvedimenti Regione Veneto n.810 and n.811, May, 10, 1979. Baule, 6. , 1970. Die Mathematik des Naturfarschers und Ingenieurs., Band I V Y P a r t i e l l e D i f f e r e n t i a l g l e i c h u n g e n , S. H i r z e l Verlag, L e i p z i g . B e r t o n a t i , M., Dejack, C., Mazzei L a l a t t a , I . and Pecenik, G., 1985. Eutrophic a t i o n model o f t h e Venice Lagoon: s t a t i s t i c a l treatments o f phytoplankt o n growth parameters. (Ecol. Model l i n g , i n p r e s s ) . Cescon, B., De Angeli , U. , I o v e n i t t i , L. , I s o l a t i , A., A l f a s s i o Grimaldi , S. , 1976. The c a l i b r a t i o n o f a t r o p h i c model o f t h e Venice Lagoon. Tecneco S.p.A. , Environmental Study Group, S. I p p o l i t o (Pesaro), I t a l y . D'Alpaos, L. and D i S i l v i o , G., 1979. U t i l i z z a z i o n e d e i r i s u l t a t i per l a t a r a t u r a d e i m o d e l l i matematici. 1n:"Le c o r r e n t i d i marea n e l l a laguna d i Venezia". M i n i s t e r 0 d e i L a v o r i P u b b l i c i .Comitato p e r l o s t u d i o dei provvedimenti a d i f e s a d e l l a c i t t a d i Venezia ed a salvaguardia d e i suoi caratt e r i a m b i e n t a l i e monumentali. Eds. I s t i t u t o d i I d r a u l i c a d e l l ' U n i v e r s i t a d i Padova: pp.82-95. Dejak, C., Mazzei L a l a t t a , I . , Lavagnini, I . and Saba, G., 1975. D i f f u s i o n mod e l s f o r chemical and thermal p o l l u t i o n . I n : Proc. o f t h e 3d I n t . Congr. I r a n i a n Chemical Society, - - March 2 8 - A ~ r i l 1 .Pahlavi U n i v e r s i t -y -, Shiraz, I r a n . pp.261-266. Dejak, C., Mazzei L a l l a t a , I. and M e r e g a l l i , L., 1977. D i f f u s i o n o f p o l l u t a n t s i n lagoon and i n t h e sea. I n : Proc. o f t h e I n t . Conar.Louvain-la-Neuve: E4: py5. Dejak, C., 1979. Problemi r i g u a r d a n t i g l i s c a r i c h i d i acque d i raffredamento d i c e n t r a l i t e r m o e l e t t r i c h e e d i a l t r i i m p i a n t i i n d u s t r i a l i n e l l a laguna veneta. Report f o r Ente Zona I n d u s t r i a l e Porto Marghera. n.A2: 1-17. Dejak, C., 1979. V e r i f i c a sperimentale s u g l i s c a r i c h i t e r m i c i d e l l a Centrale d i Fusina, Ibidem, A3:l-4.
404
Dejak, C., Mazzei L a l a t t a , I.and M e r e g a l l i , L., 1981. V e r t i c a l averages i n a three-dimensional d i f f u s i o n o f p o l l u t a n t s . Nuovo Cimento, 4:493-510. Dejak, C., Mazzei L a l a t t a , I.,Pecenik, G. and Rossi, A., 1985a. Development o f a mathematical e u t r o p h i c a t i o n model o f t h e Lagoon o f Venice. (Ecol. Modelling: i n p r e s s ) . Dejak, C., Mazzei L a l a t t a , I.,M e r e g a l l i , L., Messina, E. and Pecenik, G.,1985b. A two-dimensional d i f f u s i o n model o f t h e Venice Lagoon and r e l a t i v e open boundaries c o n d i t i o n s . (Ecol. Modelling: i n p r e s s ) . Dejak, C., Mazzei L a l a t t a , I.,Messina, E., and Pecenik,G.,1985c. An advectiond i f f u s i o n p o l l u t i o n model o f t h e lagoon o f Venice ( E c o l . Model1ing:in press). Dejak, C., Mazzei L a l a t t a , I.,Molin, M., Messina, E. and Pecenik, G., 1985d. Steady s t a t e achievement by i n t r o d u c t i o n o f t r u e t i d a l v e l o c i t i e s i n a p o l l u t i o n model o f t h e Venice Lagoon. (Ecol. Modelling: i n p r e s s ) . Diachishin, A.N., 1963. Waste d i s p o s a l i n t i d a l waters. J. San. Eng.Div., 89 (SA4) :23-43. D i S i l v i o , G. and D'Alpaos, L.,1972. I 1 comportamento d e l l a laguna d i Venezia esaminato con il metodo p r o p a g a t o r i o unidimensionale. Eds. I s t i t u t o Veneto d i Scienze L e t t e r e ed A r t i . Commissione d i S t u d i o d e i Provvedimenti per l a Conservazione e D i f e s a d e l l a Laguna e d e l l a c i t t a d i Venezia. G h e t t i , A., 1973. The h y d r a u l i c problems o f t h e Lagoon o f Venice. Giornale economico. C.C.I.A., Venezia, 1:34-38. G h e t t i , A., B a r t o l e t t i , U., Lippe, E., D'Alpaos, L. and D i S i l v i o , G., 1979. Le c o r r e n t i d i marea n e l l a laguna d i Venezia. I n t r o d u z i o n e e p a r t e l a . M i n i s t e r 0 d e l L a v o r i P u b b l i c i . Comitato p e r l o S t u d i o d e i Provvedimenti a D i f e s a d e l l a c i t t . 3 d i Venezia ed a Salvaguardia d e i suoi c a r a t t e r i ambient a l i e monumentali. Eds. I s t i t u t o d i I d r a u l i c a ; U n i v e r s i t a d i Padova; pp. 3-66. Joergensen, S.E. Handbook o f environmental data and e c o l o g i c a l parameters. Pergamon Press, New York. Kullenberg, G.E., 1971. Results o f d i f f u s i o n experiments i n t h e upper region o f t h e sea.Report 12, Kobenhavns U n i v e r s i t e t , I n s t i t u t f o r F y s i s k Oceanografi. Kullenberg, G.E., Murthy, C.R. , and Westerbuy, H. 1973. An experimental study o f d i f f u s i o n c h a r a c t e r i s t i c s i n t h e t h e r m o c l i n e and hypolimnion regions o f Lake Ontario. I n : Proc. 1 6 t h Conf. Great Lakes Res., pp:774-790. Laasonen, P., 1949.Ueber e i n e Methode z u r Losung der Warmeleitungsgleichung. Acta Math.; pp. 81-309. M a r i o t t i , M. and Dejak, C. , 1982. Valutazione d e i F l u s s i e n e r g e t i c i a l l ' i n t e r f a c c i a acqua-aria n e l l a laguna d i Venezia. I n : Proc. o f t h e 1 0 t h Convegno "Ambiente e Risorse", Bressanone Aug. 30-Sept.4. Richtmyer, D.R. and Morton, K.W., 1967. D i f f e r e n c e methods f o r i n i t i a l value problems , 2nd Ed.,Interscience Publishers, London. Roache, P.J. , 1972. Computational f l u i d dynamics. Hermosa P u b l i s h e r s Albuquerque, New Mexico 87108, USA. Sguazzero, P., G h i g n o l i , C . , R a b a g l i a t i , R. and V o l p i , G., 1978. Hydrodynamic numerical m o d e l l i n g o f t h e lagoon o f Venice. IBM J. Res. Dev., 22 ( 5 ) : 472-480.
405
THREE DIMENSIONAL CONTINENTAL SHELF HYDRODYNAMICS MODEL INCLUDING WAVE CURRENT INTERACTION M.L. SPAULDING AND T. I S M I Applied Science Associates, Inc., 70 Dean Knauss Drive, Narragansett, Rhode Island 02882
ABSTRACT A three dimensional circulation model, suitable for continental shelf and coastal sea regions, has been modified to consider wave current interaction in the bottom boundary layer. The wave current interaction processes are described using a simplified version of the Grant-Madsen approach. The hydrodynamic equations are solved by a weighted residual method in the vertical and by a forward in time, centered in space finite difference technique in the horizontal. The vertical variations in horizontal velocity are represented by an expansion of Legendre polynomials. A split mode computational technique is employed to minimize computational time. The model has been applied to predict the flow and surface elevation fields for tropical storm Delia (Gulf of Mexico) and compared to available field observations. Sensitivity studies to two and three dimensional representations of the vertical eddy viscosity, constant bottom friction coefficient, and wave current interaction have been performed. Results of these simulations in terms of the velocity and surface elevation fields, local force balances and velocity structure are described. 1 INTRODUCTION When the wind blows over the water, the applied shear stress at the surface induces a "steady" current in the water column. This steady current then interacts with the sea floor forming a bottom boundary layer which dissipates energy through frictional losses at the sea bed. The wind, at the same time, generates waves at the sea surface which in turn result in a wave boundary layer at the bottom. This wave boundary layer, interacting with the steady current, over a rough bottom, modifies the effective bottom stress and therefore alters the steady current field. While the interaction between the wave induced and steady current fields has been recognized as important in predicting the current fields near the sea floor (Grant and Glenn, 1983; Glenn and Grant, 1983; Grant and Madsen, 1979; Wiberg and Smith, 1983; Cooper and Pearce, 1980, 1982; Pearce et al, 1979) existing continental shelf and coastal sea circulation models (Jaycor, 1979; Gordon and Spaulding, 1974; ASCE, 1980; Swanson and Spaulding, 1978; Hinwood and Wallis, 1975) do not account for the presence of the wave bottom boundary layer.
406 Grant and Glenn (1983) undertook the extension of earlier work (Grant and Madsen, 1979; Grant and Madsen, 1982) to develop a continental shelf bottom boundary layer model (BBLM) which included the behavior of (1) combined wave and current interaction, (2) moveable bed effects (ripples, near bed transport, bioturbation), (3) stratification due to temperature and salinity, ( 4 ) stratification due to suspended sediment and (5) planetary rotation.
This represented a first step toward developing an accurate methodology to estimate wave current interaction effects. The next step, and the goal of this study, is to integrate the BBLM with an existing continental shelf circulation model in order to develop a model system capable of estimating the interaction of currents and waves for a coastal or shelf region. The specific study objectives within this general context are to: (1) Develop and implement a methodology to predict wave current interaction by integrating the BBLM into a shelf circulation model, and interfacing the resulting model to wind and wave-swell models suitable for storm conditions. ( 2 ) Compare the resulting model system predictions against a storm event for which field observations exist.
Section 2 presents an overview of the model system and linkages and a brief description of the model components. In Section 3, the model is applied to simulate tropical storm Delia.
A total of thirteen cases are run to
explore the sensitivity of model predictions to the vertical representation of the eddy viscosity, to bottom friction coefficient (with and without wave current interaction) and to two and three dimensional descriptions of the flow field. A summary of the study is included in Section 4 .
2 THEORETICAL DEVELOPMENT OF THE MODEL SYSTEM
2.1 The design of a model system capable of accurately predicting the three dimensional current structure during a storm must include not only a three dimensional continental shelf hydrodynamics model, modified to incorporate the BBLM, but models of the wind and wave fields as well. A conceptual flow chart
of the model system is given in Figure 1. Given estimates of the storm parameters (in terms of the pressure field, radius to maximum winds, forward velocity, storm track and deflection angle), the storm wind model predicts the spatial and temporal structure of the wind velocity field. This information is subsequently used as input to the storm wave model which estimates the resulting wave environment in the study region. The wave spectra can then be employed to predict the wave induced bottom currents by well known linear or nonlinear wave theory.
Using the wind
velocity and atmospheric pressure data predicted by the storm wind model, the wave induced bottom currents estimated by the wave model, and the sea bottom sediment characteristics and roughness, a three dimensional continental shelf
407
circulation model is solved iteratively with the BBLM to predict the spatial current structure and surface elevation distributions.
The solution is
advanced in time by simply updating the storm parameters and repeating the calculation sequence. In the text to follow, each model system component is described in detail. This presentation gives a brief overview of the theoretical foundation of the approach and its integration into the model system.
Figure 1.
Overview of model system components and linkages.
2.2 Model Svstem ComDonents
(i) Continen tal Shelf Hv drodvnamic Model.
The three dimensional numerical
hydrodynamic model employed for the present study follows the development originally given in Owen (1980) and reported in our earlier work (Isaji and Spaulding, 1984; Isaji et al, 1982).
Since the approach is well documented in
the refereed literature, we present only a brief overview here. The three dimensional
conservation equations for water mass and momentum
with the Boussinesq and hydrostatic assumptions invoked form the basis for the model.
The three dimensional conservation equations suitable for limited
shelf waters in spherical polar coordinates are written as:
Conservation of momentum
408
-
2Un sin4
-
1 aPa a (Nz -)av - R C O S ~an az az
-
0
following notation has been used: Spherical polar coordinate system with 4 and
n
measured in the
latitudinal and longitudinal direction, respectively.
z is measured
vertically upward from mean sea level. Components of the current in the (6,
directions,
respectively.
n,
and z
Overbar indicates vertically averaged values.
Pressure Atmospheric pressure Density Vertical eddy viscosity Coriolis parameter ( 2
sin4), assumed constant, where n is the
angular speed of the earth’s rotation and 4 the latitude angle. Gravitational acceleration Radius of the earth equations are solved subject to the following boundary conditions. (a) At land boundaries the normal component of velocity is set to zero. (b) At the open boundaries the sea surface elevation is specified as a series of sine waves each with its own amplitude and phase or by appropriate gradients of the local surface elevation or normal velocity field. (c). At the sea surface the applied stress due to the wind is matched to the local stress in the water column and the kinematic boundary condition is satisfied. (d) A t the sea floor, a quadratic stress law, based on the local bottom velocity, is used to represent frictional dissipation and a friction coefficient determined by the BBLM parameterizes the loss rate. Anticipating the use of a weighted residual method, in which vertical variations are represented in terms of a set of basis functions, a new set of independent variables is introduced that transforms both the surface and bottom onto coordinate surfaces. This transformation represents a simple nondimensionalization with the local water column depth. A detailed presentation of the transformed equations can be found in Owen (1980). The numerical solution methodology follows that of Davies (1977a, b) and Owen (1980). The vertical variations in horizontal velocity are described by an expansion of Legendre polynomials.
The resulting equations are then solved
by a Galerkin weighted residual method in the vertical and by an explicit finite difference algorithm in the horizontal.
409 A space staggered grid scheme in the horizontal plane is used to define the study area. Sea surface elevation and vertical velocity afe specified in the center of each cell while the horizontal velocities are given on the cell faces. The u and v velocities are defined on the cell faces normal to the and n directions, respectively.
To reduce computational costs, a "split-mode" or "two-mode" formulation is used (Simons, 1975; Owen, 1980; Gordon, 1982). In the split-mode approach, the free-surface elevation is treated separately from the internal, threedimensional flow variables. The free-surface elevation and vertically integrated velocities are calculated using the vertically integrated equations of motion (external mode) for which the Courant-Friedrichs-Levy (CFL) limit must be met. The vertical structure of the horizontal components of the current may then be calculated such that the effects of surface gravity waves are separated from the three-dimensional equations of motion (internal mode). Long period surface gravity waves, therefore, no longer limit the internal mode calculations and much longer time steps are possible. The external mode equations are approximated by a forward in time, centered in space (FTCS) finite difference scheme. The internal mode is solved using a centered in time technique with vertical diffusive terms centered in time, to ease the vertical time step restriction from the standard FTCS procedure. (ii) Bottom Boundarv Layer Model. Given the extensive computational resources required to perform time dependent three dimensional simulations of storms on a relatively fine mesh and the iterative nature of the full BBLM solution approach (Grant and Madsen, 1979), the hydrodynamic model's computational time would be excessive. A decrease in computational time is clearly required if the proposed approach is to be a viable model. It was therefore decided to simplify the BBLM. The goal of this effort was to maintain the accuracy of the BBLM for conditions present during storm events but drastically reduce the computational time per simulation. The interested reader is referred to Grant and Madsen (1979, 1982), Grant and Glenn (1983), and Spaulding and Isaji (1985) for a thorough presentation of the full and simplified BBLM. The model is meant to be applied primarily for storm conditions. As a result, there are several assumptions inherent in the model development. These include: (1) Wave velocities at the bed (at a height less than the wave boundary layer thickness) are large compared to the current velocity. Grant and Madsen (1979), indicate that this criteria is formalIy met for current to wave velocity ratios of 0.25 or less. Practically, however, much larger ratios do not greatly change the predicted drag coefficients (Spaulding and Isaji, 1985). (2) The bed is always rough, either because of bioturbation or because of sediment transport. These simplifying assumptions allow a substantial reduction in the computational effort to calculate the
410 drag coefficient.
These limitations are not severe for storm or swell
conditions and the model can safely be assumed to be generally applicable. The major limitation for the simplified BBLM case is that stratification is not permitted.
Although stratification is not the typical case, it may be
important in selected areas and at specific times (e.g., self stratification due to suspended sediment transport). The drag coefficient calculated by the model is defined in terms of the mean bottom stress
where CD is the drag coefficient and U l i s the mean current velocity at 1 rn from the bottom. As a note on the computational time, the simplified BBLM is approximately 50 times faster than the full BBLM. (iii) Storm Wind Model. The storm wind model incorporates the symmetric pressure field suggested by Harris (1958).
where P PO P
Pressure at radial location r (mb) Central pressure (mb) Peripheral pressure (mb)
R Radius to maximum winds (km) The wind field was then modeled using the approach proposed by Jelesnianski (1974). v(r,e)
The velocity at any location (r) is given by
- Vmax(r)
[1 -
Vf 2
(1
-
cose)BmaxI
(7)
where Vf is the forward velocity of the storm (knots) and 0 is the angle measured clockwise from the locus of maximum winds. The locus of maximum winds is located at an angle of (90+ 0 ) to the right of the direction of forward motion where the deflection angle is defined as the angle between the true wind direction and a tangent to a circle with the center at the hurricane center. The maximum wind is given by
where fc is the Coriolis parameter (lfir.). The velocity at any distance, r, along the locus of maximum winds is given by
where the units of r and R must be consistent.
411
Although Myers and Malkin (1961) compiled observed data that suggests the deflection angle displays significant temporal and spatial variability for a particular storm, it will be assumed as a constant for the present application. An angle of 5 O is normally selected (Cooper and Pearce, 1982). The wind stress was related to the storm wind using the formulation of Smith and Banke (1975): cd
- (0.63 + 0.66 UlO)
X
(10)
where U1o is the 10 m wind speed in m/s. This formula gives similar results to those of Garratt (1977), Large and Pond (1981) and Wu (1980) over the wind speed range of interest but is substantially lower than the earlier work of Wu (1969) at wind speeds above 70 m/s. While the'wind field model proposed here is relatively simpiistic, it has been selected for the present application because the procedure is well documented (CERC, 1984) and has been extensively used in describing hurricane wind fields for several storms in potential areas of model application (Pearce et al, 1979'; Tetra Tech, 1981). (iv) Storm Wave Model. To hindcast sea states for tropical and extratropical storms, a one dimensional parametric spectral model has been selected based on its simplicity and its successful application to past storm events. This approach is based on an analysis of the JONSWAP energy spectrum, E(f) E(f)
-
a
where u , the width of the spectra, is given as for f 5 fm for f > fm fm is the frequency of the spectral peak, u the Phillips constant, 6 the peak enhancement factor (ratio of JONSWAP Spectrum peak/Pierson Moskowitz Spectrum peak), g gravity and f wave frequency. Hasselman et a1 (1976) proposed that fetch limited wave spectra can be normalized in such a way that there is an approximate invariance of normalized spectral shape with fetch. They proposed that the nonlinear regime wave-wave interactions were responsible for the shape invariance and that it characterized not only idealized fetch limited situations, but also growing wind seas in general. This parametric approach exploits the shape invariance by representing the spectrum in terms of the five JONSWAP parameters ( u , 4 , oa, q,, fm). Noting evidence of the shape invariance of hurricane generated wave spectra measured by a laser altimeter in hurricane Ava, from wave measurements in hurricane Camille (Hasselman et al, 1976) and from meteorological buoy data during hurricane Elise (Withee and Johnson, 1976), Ross (1976) proposed that the advecting wave field for a hurricane can be determined by the local wind and a
412 fetch given by the local radius of curvature of the wind field. Following Kitaigorodskii's (1961) procedure that for fetch limited wind seas all wave variables, when nondimensionalized in terms of gravity, g and wind speed, U, should be functions only of the single nondimensional fetch parameter E gXfl2, where X is the fetch, Hasselman et a1 (1976) represented the nondimensional peak frequency, u Uf,/g where fm is the peak frequency, Eg2/v4 as a function of E . and the nondimensional total energy c Ross (1975) replaced the fetch, X, by r, the radial distance to the eye of the hurricane, to parameterize fetch. The data sets from hurricane Ava, Camille, and Eloise were combined to give the following power law fits to the data
-
v = 0.97
E
-
6-0.21 -5 0.45
= 2.25 x 10
5
if E < 3 y =
x lo4
(12)
-0.13
4.7 6
a = 0.035 v
0.82
-
where E gr/U2. The spectral shape parameters ua and Ub were found to scatter about an average of 0.1 but indicated no consistent dependence on 6. Application of this technique to Ava, Camille, and Eloise showed reasonable comparison to observations. A subsequent comparison of this approach and a full spectral model to Belle showed good agreement to the observed data (Ross and Cardone, 1977). Using the JONSWAP spectrum E(f) we can calculate the significant wave height, H,, as HS
-4
K
and the mean zero up crossing wave period, T,, as Tz
-
m2
where the nth order moment of the spectrum is denoted as
- so m
mn
f" E(f)df
(15)
Important limitations of this simple one dimensional parametric approach are that swell is not included, fetch limited seas due to coastal boundaries are not possible and the shallow water effects of refraction, percolation, frictional losses and change in nonlinear wave-wave interaction are not represented. Its key advantages are its simplicity and successful experience in representing the wave field in the high wind areas of hurricanes. 3 APPLICATION OF THE MODEL SYSTEM TO TROPICAL STORM DELIA
The model system was applied to tropical storm Delia as a test of the methodology under conditions where wave current interaction is expected to be
413 significant. There also existed wind, wave, and current observations from an offshore platform with which to compare model predictions and to better understand model performance. A weak cyclonic circulation in the southwestern Caribbean, noted in 31 August 1973 satellite photographs, strengthed to become tropical storm Delia on 2 September 1973 with its center located at approximately 24O N, 88O W. Delia's central pressure deepened during the night of the 2nd and the morning of the 3rd to 987 mb as the storm traveled in a roughly northwesterly course toward Galveston, Texas (Figure 2). Delia continued on its northwesterly course during the afternoon of the 4th. Later on that day the storm crossed the Texas coast near Galveston and became erratic with an ill defined center. MODEL BATHYMETRY (METERS) I
STORM D E L I A
28
+-
I
2
28
27
I
I
I
I
I
$7
$6
a5
$4
$3
Figure 2. Model grid system, bathymetry and storm track for tropical storm Delia. The wind field for Delia was calculated using the simple wind model. The radius to maximum winds was assumed at 64 km with a maximum wind speed of 25 m/s. The time dependent central pressure was defined using the curve given in Forristall et a1 (1977) based on Navy and Air Force dropsonde data. The lowest central pressure was taken as 987 mb with a peripheral pressure of 1013 mb. The storm was assumed to have a forward velocity of 6.1 m/s and a deflection angle of 50. During the passage of the storm, the currents were recorded at three depths; 4, 10, and 19 m below the surface in 20 m water depth at the Buccaneer The currents were recorded by electromagnetic current station (Figure 2 ) .
414 meters that were moored on taut wires suspended between the bridge of the platform and a steel anchor. While the instruments functioned throughout the storm period it was noted that a large scour hole had developed in the vicinity of the anchor and caused the meters to rotate about 31O. Since it is impossible to know exactly when this rotation took place, the data is presented here in its original form.
It therefore must be recognized that
there is, at the minimum, this uncertainty in the reported currents. Forristall et a1 (1977) give a detailed discussion of the collection and analysis of this data. Wave staff data were also recorded from 3 September 1973 through the storm and provide a measurement of the wave height and period. The model system was applied to the study area using a 22 km square grid system (Figure 2 ) where the orientation is along lines of constant latitude and longitude. Longitude is positive to the east and latitude positive to the north.
The bathymetric data was derived from the NOAA/NOS charts and the
National Geophysical Data Center (NGDC) digitized bathymetric data tapes. The model domain was selected so the offshore boundary was in deep water and the alongshore boundaries were sufficiently removed from the observation site to minimize the problem of accurately specifying the open cross shelf boundary conditions.
The boundary conditions for the simulation were an inverted
barometric pressure condition on the offshore boundary and no gradient of. the vertically averaged velocity normal to the cross shelf boundaries.
The
procedure in locating the cross shelf boundaries was to perform several simulations where the boundaries were moved progressively closer to the area of interest and determine the smallest domain size where the boundary specifications did not markedly influence the model predictions over the time period of interest. These approximations were similar to those of earlier simulations by Forristall et a1 (1977) and Cooper and Pearce (1982). Although these boundary conditions are known to reflect waves (Reid, 1975), sensitivity studies showed that this was not a problem for these relatively short duration simulations. The vertical structure was represented by five Legendre polynomials as a compromise between vertical resolution and computational time.
This
assessment was made by comparing model predictions with varying numbers of polynomials to an analytic solution for wind driven flow in a rectangular channel with specified surface and bottom stresses. A total of thirteen cases (Table 1) were investigated using two and three dimensional simulations, with and without the BBLM included. Sensitivity runs were also performed to investigate the impact of alternate bottom drag coefficient values and fixed (constant value) and variable vertical eddy viscosity on modal predictions. Drag coefficient values were selected to bracket the typical range for coastal hydrodynamic modeling while the vertical
415 eddy viscosity range was based on earlier simulations of Delia by Forristall et a1 (1977) and Cooper and Pearce (1982).
TABLE 1 Simulation cases of trouical storm Delia. GASE 3D/2D VERTICAL EDDY BOTTOM FRICTION NUMBER VISCOSITY FORMULATION cml/s) (NON-DIMENSIONAL) 1 3DW CONSTANT 100 BBL(C) 2 3D CONSTANT 150 BBL 3 3D CONSTANT 200 BBL 4 3D VARIABLE IN TIME & SPACE(D) BBL 5 3D PARABOLIC IN VERTICAL (E) BBL 6 3D CONSTANT 150 0.003 7 3D CONSTANT 150 0.0005 a 3D CONSTANT 300 0.0005 9 3D CONSTANT 300 0.001 10 2D(B) 0.0005 11 2D 0.001 12 2D 0.002 . 2D 0.03 13 (A) Three dimensional (horizontal and vertical) simulation with five Legendre polynomials (B) TWO dimensional vertically integrated simulation (C) Simplified Bottom Boundary Layer Model (BBLM) surface wind friction velocity, (D) Visocisty W*H/RE, where W H local water depth RE 50, the flow Reynolds number (E) Viscosity Vm * a1 * (z+a2) * (z+a3) 150, the maximum viscosity, where Vm z vertical coordinate, al,a2,a3 constants, adjusted to produce the maximum at 10 meters below the surface, and the minimum ( 5 0 cm2/s) at the bottom. For depths shallower than 20 m , the maximum value was used as a constant.
-
-
--
To illustrate the simulations, Case 2 is presented in detail. Other cases will only be summarized. The surface currents are a combined response to the wind stress and the local pressure gradient. As the storm builds in intensity, the surface induced currents increase reaching values greater than 2 m/s during the height Of the storm. The pattern, as expected from the forward motion of the storm, displays an intensification in the right front quadrant. As the storm induced flows reach the coast they are modified such that the
currents are parallel to the coast. As the storm passes southwest of Galveston, the central pressure increases and the storm dissipates. There is a corresponding reduction in the wind induced surface flow. The bottom velocities are, in general, relatively weak throughout the simulation.
Studying the surge response as the storm approaches the shore, water piles up against the coast, east of Galveston, and the southwesterly directed vertically averaged velocities increase in magnitude flowing in a direction
416 parallel to the coast in the nearshore area and parallel to the isobaths in the offshore region. A maximum surge height of 2.25 m is predicted at 37 hours into the simulation. 1973.)
(The simulation starts at 0.0 hrs. on 3 September
As the storm dissipates, the surge height decreases, although
southeast of Galveston Bay it is still present at 49 hours. To docunient the vertical structure, Figure 3 shows the east-west and north-south profiles of the current at the Buccaneer measurement site as a function of depth during the storm passage.
The response is clearly
barotropic with increasing wind stress resulting in increased flow from
ro-:To -:To -;yo
surface to bottom. The selection of the constant vertical eddy viscosity, in this case, minimizes the near surface vertical current gradient while the BBLM enhanced stress gives significant shear in the water column, particularly near
71
the bottom, during the height of the storm. STORM C E N E R l l E O V E R I I C A L V E L O C I T Y
P R O F I L E T I U E 4 A 1 00 00 3 SEP 1973
DELIA CASE
2
METER/SCC
-2 0
I0
I
t
I 15
E N
6111
I
10
I
10
I
15
\
15
w s
10
I
TIUE(HRS)-
21
TIYE(HRS1- 2 5
TIUE(HRS)-
15
I
I
19
TIUELHRSI- 33
Figure 3. Model predicted vertical current profiles at the Buccaneer observation platform at selected times for Hurricane.Delia. Figure 4 shows a comparison between model predictions and observations (when they exist) for wind velocity, atmospheric pressure, wave height and period, surge height, current velocity at the surface, mid depth and bottom and the bottom drag coefficient (noted as bottom friction in the figure) as a function of time through the entire storm simulation.
All are given at the
Buccaneer site. Figure 4 also gives the vertically averaged time dependent momentum balance at the site for the N - S and E-W directions, respectively.
The
momentum balance includes: local acceleration, free surface elevation gradient, Coriolis force, bottom and surface stress and atmospheric pressure gradient. The nonlinear convective acceleration terms are small in comparison
417
Figure 4.
(a)
Comparison of observations and model predictions, Delia, for wind, atmospheric pressure, wave height and period, surge height, water velocity (Buccaneer Site) and bottom friction (b) Vertically averaged momentum at the Buccaneer Platform
to all other terms and have therefore been neglected.
This assessment was
made by performing simulations with and without the convective acceleration terms included and comparing the results. Comparing model predictions to observations shows that the simplified storm wind and wave models reasonably reproduce the observed patterns at the Buccaneer platform.
The agreement between predicted and observed wind fields
is comparable to that of Forristall et a1 (1977) and Pearce and Cooper (1982). Additional tuning of the wind field description was not appropriate since there was only one point for calibration. The perturbations of the predicted wave field are clearly seen to be in direct response to the changing path and hence fetch of the wind field. The model predicted currents are shown in comparison to the 4 m, below the surface, current observation. This vertical location was selected as representative of the water column since the observed vertical structure of the horizontal currents was quite uniform. The model predicted surface current is seen to be in good agreement with data both in terms of magnitude and direction.
The east-west velocity, however, peaks before and dissipates
faster than the observed. The strong westward current observed before the passage of the storm (40-50 cm/s) is typical of the autumn semi-permanent current present in the area (Nowlin, 1971; Estes and Scrudato, 1977; EPA, 1982). EPA (1982) reports currents of 46 cm/s at a heading of 282O for September in the region bounded by 94-95O W longitude and 29-30° N latitude.
418 The model however is not able to reproduce this feature due to the limited size of the domain and the absence of forcing parameters related to this semipermanent mean flow.
The predicted mid-depth and bottom currents are
considerably smaller than observed with the bottom currents 25% of the surface values compared to 95% for the data.
The model predicted shear is mainly
attributed to the enhanced bottom roughness due to wave-current interaction. The peak bottom friction coefficient is approximately 0.030, an order of magnitude larger than is conventionally used for storm surge modeling.
The
fluctuations in the bottom friction coefficient are seen as a direct response to changes in the predicted wave environment.
In terms of the force balance in the north-south (approximately cross shelf) direction, the surface slope is roughly balanced by the Coriolis force and the bottom stress. wind stress.
Local acceleration approximately balances the surface
This balance
is maintained throughout the storm with the
magnitudes of the forces dependent on the storm strength.
The balance in the
east-west (along shelf) direction is between the bottom stress, the Coriolis force, and wind stress terms.
The surface slope only becomes important after
the storm has passed the observation site and is traveling over land.
The
atmospheric pressure gradient terms play only a minor role in the force balances.
If viewed in a coordinate system that is parallel to shore, the
primary force balance is geostrophic with a correction for bottom friction. This is increasingly the case as the storm strength builds. To further illustrate the comparison, Figure 5 shows detailed plots of the currents for each of the three vertical locations for both observations and model predictions.
The lower two graphs present the model predicted bottom
shear velocity and the roughness height.
Although Forristall et a1 (1977)
have derived estimates of the shear velocity and roughness heights for the storm, we have chosen not to use their analysis for comparison to the model because (1) the use of only three points to estimate these values leads to extremely large errors and (2) the assumption of a log velocity profile used to derive their values may not be valid. The model shows a generally increasing shear velocity peaking at approximately 4 cm/s during the height of the storm and decreasing gradually The roughness height is generally on the order of 12 cm but displays substantial peaks of approximately 6 - 8 cm during the 32-26
as the storm dissipates.
and 44-49 hour time periods.
The peaks correspond to increases in the wave
height and period during these times (Figure 4). Although not shown here, a detailed analysis of the other cases is presented in Spaulding and Isaji (1985). summarized here.
The principal results will be
For the three dimensional simulations, with the BBLM
included, increasing the vertical eddy viscosity decreases the surface current and increases the mid-depth current. constant.
The bottom current remains relatively
The surface elevation decreases slightly.
The force balance
419
V E L O C I T Y PROFILE & BOlTOU SHEAR
A1 NCELL-178
2
DELIA CASE
o c ,
2
-100
1: 1-
Figure 5.
UUDEL
Comparison of model predictions and observations for velocity, bottom shear velocity and bottom roughness height at the Buccaneer Site.
remains basically as discussed earlier with the primary effect of increasing eddy viscosity to lower the surface slope and Coriolis terms and to increase the bottom friction. The variable eddy viscosity formulations, Cases 4 and 5, give solutions much like those for the constant cases with similar mean values. Cases 6 through 9, respectively, employ the three dimensional model with two values of constant eddy viscosity and selected constant values of bottom
friction coefficient. Increasing the eddy viscosity has the same effect as noted earlier, that is to decrease the vertical gradient of the velocity field.
Increasing the bottom friction coefficient increases the bottom
stress, reduces the surface slopes and reduces the currents at all levels. The larger the friction coefficient, the stronger the vertical current shear. Comparing a three dimensional simulation, with and without the BBLM, for the same vertical eddy viscosity, 150 cm2/s (Case 2 and 6), the principal difference is the magnitude of the bottom stress.
For the BBLM simulation,
the bottom friction coefficient is approximately an order of magnitude larger than for the conventional three dimensional simulation with constant drag coefficient. There is also a reduction in surface elevation and velocity.
420
The vertical shear rate however remains approximately the same, particularly near the peak of the storm (39 hrs.). The increase in bottom friction coefficient combined with the decreased velocity leads to only a factor of two increase of the BBLM vertically averaged bottom stress compared to the constant drag coefficient case.
The force balances are accordingly modified
with the constant drag three dimensional case having larger Coriolis and surface slope terms than the BBLM simulation. Cases 10 through 13 explore model predictions assuming that a two dimensional vertically averaged approach is suitable to describe the flow. This would appear to be a reasonable assumption given the lack of significant water column stratification due to salinity and temperature in the area and the vertical homogeneity of the current observations.
Cases 10 through 13
correspond to increasing bottom drag coefficient covering the range (0.0005 to 0.003) that typifies coastal areas.
Increasing this friction factor results
in a reduction in both the vertically averaged velocity and the surface elevation. The compensating effect of decrease in velocity to increase in drag coefficient results in about constant bottom stress. The reduction in the free surface height and slope is compensated by a reduction in the Coriolis force. Comparison of model predictions (with the BBLM included) to observations, as shown in Figure 5 , indicate that the model under predicts the resulting vertical velocity distribution, especially near the bottom. The model clearly over predicts the bottom friction. The question arises as to why this happens considering that the BBLM gives such good comparisons to observations in other areas, such as the CODE site (Grant et al, 1984). To investigate this problem, the simplest version of the near bottom stratified (suspended sediment) bottom boundary layer model (Grant and Glenn, 1983) was used to predict the current structure at the Buccaneer site. The reference current was the 3 m depth measurement from Forristall’s data (U3,). Waveheight (H,)
and period (T) were also extracted from Forristall et al. Table
2 summarizes the input data.
The times have been selected to span the most
active period during the storm event. Two cases were selected for simulation. Case 1 assumes a medium sand with
a mean diameter of 0 . 0 5 cm while Case 2 uses a fine sand, with a mean diameter of 0.01 cm. Both have specific gravities of 2 . 6 5 . Based on EPA (1982) and Estes and Scrudato (1977), the fine sand is likely to be more representative of the area. Both cases, however, were studied to illustrate the effect of sand grain size on the predictions. For both cases, the model was run with (stratified) and without (neutral) suspended sediment induced stratification.
The results of the simulation, in terms of the friction velocity u * ~ , roughness height, zoc,and drag coefficient, CD3m, referenced to the 3 m depth are shown in Table 3.
Figure 6 shows a comparison between model predictions
and observations for current at the three vertical stations as a function of
421
time for the (a) neutral and (b) stratified flows. Only Case 2 is shown since it shows slightly better comparison to observations than Case 1. This seems reasonable since the fine sand approximation is likely to better represent the sediments found at the site.
TABLE 2 InDut data for B B M simulation o f troDical storm Delia. Time (hr)
U3m (cm/s)
HS
(m)
T
h/Lo
Ab (m)
(S)
ub (4s)
zr -
Ab 2.6 1.526 0.0735 2.280 30 71.06 3.67 5.51 1.224 2.842 3.0416 0.0647 32 87.14 4.19 5.87 3.4333 1.072 0.0632 3.245 35 112.56 4.72 5.94 4.675 4.457 0.744 0.0514 37 149.38 5.96 6.59 3.7111 0.943 0.0571 3.691 38 141.28 5.03 6.25 2.540 2.776 1.370 5.75 0.0675 40 156.54 3.86 0.0677 2.083 2.28 1.671 42 122.81 3.17 5.74 2.2407 1.721 44 92.42 3.13 0.0694 2.022 5.67 2.273 1.73 0.0661 2,102 47 56.34 . 3.15 5.81 Note O for all times. Wave-current relative angle dc Lo - deep water wave length, h - water depth, Ab - maximum wave excursion, Ub - maximum wave orbital velocity, 2, - height of current reference velocity .
-
10'
0
NEUTRAL
I
STRATIFIED
Figure 6. Comparison of BBLM predictions of current versus depth for Case 2, neutral and stratified simulations. The analysis indicates that during the peak of the storm (hours 35-41), the flow becomes stratified. This stratification results in a reduction in by a factor of 3-5.
The bottom drag coefficient referenced to 3 m lowers
from 0.027-0.037 to about 0.001.
This value of CD is in good agreement with
422
that empirically determined using the two dimensional vertically averaged simulation. The simple stratified model does well at reproducing the currents at the other levels as well.
As the storm dissipates, the flow no longer
shows strong stratification, even though the potential exists.
The observed
velocity profiles at this time are in good agreement with the neutral simulations. This change in response from a stratified to a neutral behavior could b e due to bed armoring effects.
This explanation appears quite
reasonable given the geology of the area where fine sand overlays a silt clay bottom. We cannot however directly confirm this behavior since no suspended sediment concentration data are available during the storm. As the flow returns to a neutral configuration, the friction factor returns to a value on the order of 0.02 and the shear velocity again increases. TABLE 3 BBLM predictions o f friction velocity, roughness and drag coefficient for troDical storm Delia. CASE I* TIME (hr) 35 37 38 40 44 47
U*C 20.04 31.37 25.73 22.82 12.75 8.32
=oc CD3m n e u t r a l 32.23 1.0317 45.37 0.0441 33.90 0.0332 19.60 0.0213 16.77 0.019 21.24 0.022
U*C
zoc CD3m s t r a t i f i e d 9.93 42.23 0.0078 12.61 4.21 0.0071 11.94 46.44 0.0071 12.89 25.68 0.0068 8.34 20.48 0.013 5.58 24.64 0.0098
!asu 18.66 35 27.3 .0275 3.82 45.69 28.67 .0368 4.66 37 37.92 68.41 38 23.83 28.46 .0285 4.52 51.12 40 15.99 .0184 21.24 5.08 30.05 11.95 44 13.82 .017 3.43 23.23 7.86 18.09 .0195 47 * Case 1 Medium Sand, d 0.05 cm, Cb 0.6, Specific gravity - 2.65 Case 2 Fine Sand, d 0.01 cm, Cb 0.6, Specific gravity - 2.65
--
0.0012 0.00097 0.001 0.0011 0.0014
--
Comparing all simulations to observations suggests that the two dimensional vertically averaged case with a drag coefficient of 0.001 gives an extremely reasonable prediction.
If we roughly adjust the reference level to
account for the presence of the semi-permanent mean flow (45 cm/s) which has not been modeled, then a slightly larger value of drag coefficient appears more appropriate.
The use of the BBIM approach, assuming neutral near bottom
stratification significantly underestimates the mean current magnitude in comparison to observations. A n independent analysis of the current structure using the BBIM with the observed 3 m currents as input shows that if suspended sediment induced stratification is considered then the predicted and observed current profiles are in good agreement throughout most of the storm.
423 4 CONCLUSIONS AND RECOMMENDATIONS
Application of the model system was made to tropical storm Delia. The best model predictions, in comparison to the available Observations, were obtained using a constant bottom friction coefficient. Incorporation of the wave-current interaction BBIM dynamics in the simulation predicts currents significantly lower than observed. Sensitivity studies for Delia investigating vertical eddy viscosity magnitude and formulation and alternate representations of the vertical structure (from two dimensional vertically averaged to fully three dimensional with increasing numbers of Legendre polynomials) could not reconcile the difference between the model predictions (with the BBIM included) and the Observations. Using observed Delia currents at the surface measurement station and wave conditions as input, simulations were made with the BBIM to describe the current structure and bottom stress levels. Neutral and suspended sediment induced stratification cases were run. Comparison of these simulations to observations suggest that as the storm strength builds the near bottom, water column becomes stratified. When this occurs there is an effective decrease in the bottom stress level and an increase in currents compared to the neutral (non stratified) case. During later stages of the storm bed armoring appears to become active and the water column reverts to the neutral case. While this analysis cannot be confirmed from existing field observations from Delia, it appears highly plausible. It successfully explains the underprediction of the currents by the neutral BBIM included in the hydrodynamic model and the generally good agreement between the two dimensional vertically averaged simulations, using a low value of the bottom friction factor, and the observations. Simulations of the wind and wave fields for Delia, considering the simplicity of the model formulations and the limited data available for calibration, were generally quite good. The simple one dimensional parametric wave model appears adequate for describing the wind generated wave heights for high intensity small radius storm events in deep and intermediate water depths. In the nearshore area, the use of a shallow water wave model is recommended. The analyses performed on storm Delia suggest that stratification effects caused by variations in concentrations of resuspended bottom sediments must be carefully analyzed to determine their impact on the current profile. It is clear from the Delia BBIM simulations that stratification can substantially alter the predicted currept structure. Based on these observations, it is recommended that the simplified BBLM be modified to incorporate stratification effects. Once this is done, a detailed simulation of a storm for which a substantial amount of data exists, not only on currents but stratification as well, should be performed.
424
As a note on computational time, the full BBLM requires a factor of 50 more CPU time than the simplified model employed in this study.
For a simple
storm surge simulation with a square gridded domain of 19 (alongshore) by 5 (onshore
-
offshore), a resolution of 0.2S0, a constant depth of 100 m and
five Legendre polynomials requires a 0.0289 s of CPU time per cell per time step. With the simplified BBLM included, the CPU time increases by 0,0092 s per cell per time step. The computational time hence increases by 31.8% with the inclusion of the simplified BBLM.
Detailed specifications for this
simulation are included as the first test case in a model user’s manual (Isaji, 1985).
All CPU times are quoted for a PRIME 550 Model I1 computer
with 2 mb of main disk storage. CPU times may change substantially for modifications in the model domain size and configuration depending on paging requirements.
5 ACKNOWLEDGEMENTS This work was sponsored by the American Gas Association under Contract No, PR169-186 with G.L. Smith and Dr. D.T. Tsahalis serving in the role of technical contract monitors. ASA acknowledges the assistance of Dr. William G r a n t , W o o d s Hole Oceanographic I n s t i t u t e (WHOI) w h o s e r v e d as a subcontractor. Dr. Grant developed and verified the simplified bottom boundary layer model (BBLH), assisted in the task of interfacing the BBLM with the continental shelf hydrodynamics model, and performed the neutral and stratified BBLM simulations of Delia.
6 REFERENCES ASCE, Task Committee on Computational Hydraulics. 1980. Sources of Computer Programs in Hydraulics. ASCE Journal of the Hydraulics Division, Vol. 106, NO. H45, pp. 915-023. CERC, 1984. Shore Protection Manual, Volumes I and 11. Department of the Army, Coastal Engineering Research Center, Waterways Experiment Station, Vicksburg, Mississippi. Cooper, C.K. and B.R. Pearce, 1980. On the Forcing Mechanisms Affecting the Bottom Shear Stress in Coastal Waters. J. of Phys. Oceanog:, Vol. 10, No. 11, pp. 1870-1876. Cooper, C. and B. Pearce, 1982. Numerical Simulations of Hurricane Generated Currents. Journal of Physical Oceanography, Vo. 12, p. 1071-1091. Davis, A . M . , 1977a. The Numerical Solution of the Three-Dimensional Hydrodynamical Equations Using a B-Spline Representation of the Vertical Current Profile. In: Bottom Turbulence, Proceedings of the 8th Liege Colloquium on Ocean Hydrodynamics, J.C.J. Nihoul (ed), Elsevier, New York: 27-48. Davies, A.M., 1977b. Three-Dimensional Model with Depth-Varying Eddy Viscosity. In: Bottom Turbulence, Proceedings of the 8th Liege Colloquium on Ocean Hydrodynamics, J.C.J. Nihoul (ed.), Elsevier, New York: 27-48. EPA, 1982. Environmental Impact Statement (EIS) for the Galveston, Texas Dredged Material Disposal Site Designation. U.S. EPA Criteria and Standards Division WH-585).
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Estes, E.L. and R.J. Scrudato, 1977. Aquatic Disposal Field Investigations, Galveston, Texas Offshore Disposal Site. Appendix A Investigation of the Hydraulic Regime and Physical Nature of Sedimentation. Texas A6M University, Department of Marine Science, 134 pp. Forristall, G.Z., R.C. Hamilton, and V.J. Cardone, 1977. Continental Shelf Currents in Tropical Storm Delia: Observations and Theory. J. Phys. Oceanog., Vol. 7, No. 4, pp. 532-546. Garratt, J., 1977. Review of Drag Coefficients Over Oceans and Continents. Mon. Weather Rev., 105, 915-929. Glenn, S.M. and W. Grant, 1983. Continental Shelf Bottom Boundary Layer Model, Vol. 111-User's Manual. American Gas Association, Project No. PR153 - 126. Gordon, R. and M.L. Spaulding, 1974. A Bibliography of Numerical Models for Tidal Rivers, Estuaries and Coastal Water. Univ. of R.I., Marine Technical Report 32, 55 pp. Gordon, R., 1982. A Three-Dimensional Numerical Model of Circulation and Salinity Distribution for Estuarine Application. M.S. Thesis, Department of Ocean Engineering, University of Rhode Island, Kingston, R.I. 161 pp. Grant, W.D. and O.S. Madsen, 1979. Combined Wave and Current Interaction (C4), 1797Within a Rough Bottom. Journal of Geophysical Research, 1808. Grant, W.D. and O . S . Madsen, 1982. Moveable Bed Roughness in Unsteady Oscillatory Flow. Journal of Geophysical Research, fl, (Cl), 469-481. Grant, W.D. and S.M. Glenn, 1983. A Continental Shelf Bottom Boundary Layer Model, Volume I: Theoretical Development. Technical Report to the American Gas Association, May 1983. Bottom Estimates and Their Grant, W., A. Williams, and S. Glenn, 1984. Prediction on the Northern California Continental Shelf During CODE 1. The Importance of Wave-Current Interaction, Journal of Physical Oceanography, Vol. 14, p. 506-527. Harris, D.L., 1958, Meterological Aspects of Storm Surge Generation. Journal of Hydraulics Division, ASCE, paper 1859, 22 pp. Hasselman, K., D.B. Ross, P. Muller and W. Sell, 1976. A Parametric Wave Prediction Model. J. Phys. Oceanogr., 6 , 200-228. Hinwood, J.B. and I. Wallis, 1975. Review of Models of Tidal Water. ASCE J. of the Hydraulics Division, Vol. 101, No. HYll, pp. 1404-1421. Isaji, T., 1985. Design Flow Conditions Near Bottom Phase I1 Coupling of a Continental Shelf Hydrodynamics Model to a Bottom Boundary Layer Model. Vol. I1 - Computer Code and User's Manual, PR169-186, American Gas Association, Washington, D.C., pp. 184. Isaji, T. and M.L. Spaulding, 1984. A Model of the Tidally Induced Residual Circulation in the Gulf of Maine and Georges Bank. J. of Phys. Oceanogr., Vol. 14, NO. 6 , 1119-1126. Jaycor, Inc., 1979. Proceedings of the Continental Shelf Physical Oceanographic Model Evaluation Workshop. Document No. 5-79-310-001, Research Triangle Park, North Carolina, p. 175. Jelesnianski, C.P., 1974. SPLASH (Special Program to List Amplitudes of Surges from Hurricanes - 11. General Track and Variant Storm Coditions. NOAA Technical Memorandum NUS TDL-52. Kitaigorodskii, S.A., 1961. Applications of the Theory of Similarity of the Analysis of Wind-Generated Wave Motion as a Stochastic Process. Izv. Akad. Nauk SSSR, Ser. Geofiz., No. 1, 73-80. Large, W.G. and S. Pond, 1981. Open Ocean Momentum Flux Measurements in Moderate to Strong Winds. J. Phys. Oceanogr., 11, 324-336. Myers, V.A. and W. Malkin, 1961. Some Properties of Hurricane Wind Fields as Deduced from Trajectories. Report #49, U.S. Dept. of Commerce. Nowlin, W.D., 1971. Water Masses and General Circulation of the Gulf of Mexico. Oceanology 6(2), p. 28-33. Owen, A., 1980. A Three Dimensional Model of the Bristol Channel. J. Phys. Oceanogr. 10: 1290-1302. Pearce, B.R., C. Cooper and E. Doyle, 1979. Hurricane Generated Currents, Civil Engineering in the Oceans IV, ASCE/San Francisco, CA, September 1012.
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427
THREE-DIMENSIONAL MODEL OF CURRENTS I N THE BAY OF SEINE J.C. SALOMON, 6. THOUVENIN and P. LE H I R IFREMER, B.P. 337, 29273 BREST CEDEX
ABSTRACT Water movement and m i x i n g processes i n t h e Bay o f Seine a r e e s s e n t i a l l y i n duced by t i d e s ; winds and d e n s i t y d i f f e r e n c e s due t o r i v e r i n f l o w and temperat u r e g r a d i e n t s near t h e coast. To study t i d a l c u r r e n t s alone, i t i s s u f f i c i e n t t o use a two dimensional model. Howeveryfor wind and d e n s i t y e f f e c t s , t h e t h r e e s p a t i a l dimensions must be taken i n t o account. For t h a t purpose a three-dimensional model o f c u r r e n t s has been developed which solves t h e n o n l i n e a r Navier-Stokes equations i n a s t r a i g h t f o r w a r d manner through a f i n i t e d i f f e r e n c e technique. The model has been t e s t e d i n various schematic cases f o r which a n a l y t i c a l s o l u t i o n s o r r e s u l t s from previous models were a v a i l a b l e . It was then a p p l i e d t o t h e Bay o f Seine and proved t o be an e f f i c i e n t t o o l f o r coastal studies.
1 .REGIONAL SETTING The Bay o f Seine, as d e f i n e d by t h e c o a s t l i n e , i s shaped more o r l e s s l i k e a r e g u l a r q u a d r i l a t e r a l about one hundred k i l o m e t e r s long and f o r t y t o f i f t y kilometers wide, opening onto t h e E n g l i s h Channel ( F i g . 1).
The bay i s q u i t e
shallow (twenty f i v e meters deep) except f o r t h e a n c i e n t submerged v a l l e y o f the Seine which runs d i a g o n a l l y across t h e bay and reaches depths o f f o r t y met e r s i n t h e northwestern p a r t o f t h e bay. With an average flow o f 400 m 3 / s , t h e Seine i s a modest r i v e r . i t i s the o n l y s i g n i f i c a n t f l u v i a l discharge i n t h e Channel.
Nevertheless,
The estuary, which
used t o be upstream of t h e c i t y o f Le Havre, has now a s m a l l e r area, because o f the improvement o f t h e waterway. Over t h e l a s t hundred and f i f t y years, the estuarine area has decreased by a f a c t o r of two, and a t t h e same time t h e locat i o n has s h i f t e d towards t h e bay. downstream o f Le Havre.
Today t h e d e n s i t y nodal p o i n t i s s i t u a t e d
The c h a r a c t e r i s t i c s of t h e bay a r e q u i t e complicated. Indeed, t h e present s i t u a t i o n includes almost a l l t h e p o s s i b l e causes f o r water movements i n t h e coastal zone :
- The main p a r t of
t h e dynamics i s due t o t h e t i d e . The t i d a l range can exceed
e i g h t meters i n s p r i n g t i d e s .
The
c u r r e n t s increase r e g u l a r l y from e a s t t o
west, from one t o t h r e e knots. The t i d a l wave and t h e associated c u r r e n t s are s t r o n g l y d i s t o r d e d w h i l e propagating i n t h e shallow areas, so t h a t .the n o n l i near terms i n t h e hydrodynamicequations a r e i m p o r t a n t .
428
F i g . 1. Location map and bathymetry o f the Bay o f Seine.
-
The wind i s another d r i v i n g f o r c e o f water movements i n the bay, a l l the
more important as i t i s r a t h e r intense and as water depths are small.
-
As mentionned before, d e n s i t y c i r c u l a t i o n does e x i s t too, c l o s e l y related
t o the f l u v i a l r u n o f f i n the eastern p a r t o f the bay. It i s almost o f estuarine type (converging v e l o c i t i e s near the bottom, nodal p o i n t ) w i t h some geostrophic features ( r o t a t i o n o f surface v e l o c i t i e s t o the r i g h t ) .
-
F i n a l l y water movements i n the bay i s p a r t l y due t o the general c i r c u l a t i o n
i n the Channel.
2 MODEL SPECIFICATIONS The t i d a l components o f t h e c u r r e n t s i n t h e bay can be studied, f o r t h e i r essential p a r t , w i t h two-dimensional (2D) models, b u t those which are l i n k e d t o the d e n s i t y f i e l d o r t o the wind do r e q u i r e a three-dimensional ( 3 D ) s l c u l a t i o n .
429
A few years ago we made a f i r s t attempt a t modelling currents in three dimensions in t h i s area (Salomon and Le Hir, 1980). We proceeded i n a quite s i milar way t o t h a t used by Nihoul (1977) a t the time, by developing along the vertical coordinate the r e s u l t s of a 2D model a t specific points. The results revealed some interesting apects of the currentology, b u t such calculations are not really three dimensional :the f r i c t i o n a l s t r e s s i s considered to be in the same direction as the vertically averaged velocity. Moreover, the nonlinear advective terms are d i f f i c u l t to take into account, since the calculations are local. The technical specifications of the present 30 model are that the model should: - solve the complete nonlinear equations; take into account the precise form of the sea bed (very important in coastal problems) , which eliminates " s t a i r step" models; - have the same behaviour a t a l l times of the computation and over the whole bay. Due t o the depth differences in our zone, and due t o the tidal amplitude, we discarded the use of reduced coordinates. As different grids a c t l i k e d i f ferent media, waves and dynamic structures are partly reflected and distorted. On the other hand, the model could have modest performances as f a r as horizontal turbulent fluxes calculations are concerned because f o r the time scales we are interested in, these terms are small compared t o the advective terms. Simple considerations on the dynamics of the bay show t h a t due tx i t s restricted dimensions and the intensity of f r i c t i o n forces, i t s "memory" towards barotropic waves and even baroclinic ones i s of a few hours. I t s memory towards phenomena of transport of matter ( l i k e s a l t ) i s much higher. Calculations based on the lack of s a l t in the bay give a residence time of about a month. For financial reasons (among others) i t was impossible t o measure boundary conditions during such a long period (except f o r the t i d e ) and the project of such simulations was given u p . Model indications, in terms of density circulation, will essentially r e s u l t from the i n i t i a l s a l t and temperature distribution, and n o t from modifications of the boundary conditions d u r i n g the calculations. Barotropic movements, on the other hand, are driven through the boundary condi tioos.
-
3 MODEL CHARACTERISTICS 3.1 Equations The model solves the Navier-Stokes equations i n t h e i r usual form, using the hydrostatic approximati on , and assuming Fi cki an diffusion:
430
p = PO ( 1 t as)
as ~ t
L
as ax e t
v
as ay - t
asw az
as
a(KZ E) az
F, = o
: h o r i z o n t a l coordinates
: v e r t i c a l coordinate w) : v e l o c i t y components : surface e l e v a t i o n 4 H : t o t a l depth o f water : mass per u n i t volume P f : Coriolis factor : salinity S : Fx, FY h o r i z o n t a l d i f f u s i o n o f momentum Fs : horizontal diffusion o f s a l t Nx, NY : v e r t i c a l eddy v i s c o s i t y c o e f f i c i e n t s Kz : c o e f f i c i e n t f o r s a l t d i f f u s i o n along the v e r t i c a l
The h o r i z o n t a l d i f f u s i o n terms Fx, Fy, Fs are p h y s i c a l l y important i n vert i c a l l y i n t e g r a t e d models where they are c a l l e d dispersion because they include a t the same time the v e r t i c a l v e l o c i t y gradients and t h e v e r t i c a l 'diffusion. Here they are e s s e n t i a l l y r e q u i r e d t o damp small scale computational modes. It would be d i f f e r e n t i f long term
simulations were t o be done.
They are expres-
sed i n t h e f o l l o w i n g usual way : FX = NXX FY = NYX FS = KX
a2u
a2u + NXY 7 aY
a2v a2v v t NYY aY
a2s t ax.
7
KY
a2s
-2-
aY I n t h e present a p p l i c a t i o n t o the Bay o f Seine, which we know t o be rather
well-mixed, t h e pressure gradients are expressed i n the f o l l o w i n g simple form:
431 3.2 C o e f f i c i e n t s f o r v e r t i c a l exchange The model makes use of t h e concept o f t u r b u l e n t exchange c o e f f i c i e n t s parameterized i n a simple form.
The other, more sophisticated, p o s s i b i l i t y , which
would c o n s i s t s i n solving, a f t e r closure, the turbulence equations, has been rejected because: no i n i t i a l o r boundary c o n d i t i o n s are a v a i l a b l e w i t h enough accuracy,
-
i t has been established by d i f f e r e n t i n v e s t i g a t o r s t h a t i n the case o f t i d a l
currents i n shallow waters, a r e l a t i o n between the d i f f u s i o n c o e f f i c i e n t s and the v e l o c i t y gradients ( o r t h e concentration g r a d i e n t ) through a mixing length, gives reasonable r e s u l t s . point
No a d d i t i o n a l complication has been seeked on t h i s
.
The e f f e c t o f buoyancy forces, which h i g h l y modify t h e mixing length, i s introduced by means o f a global Richardson number:
AS: S a l i n i t y d i f f e r e n c e between bottom and surface U : v e r t i c a l l y averaged v e l o c i t y .
Several expression ( g e n e r a l l y empirical ones) have been t e s t e d i n t h e model. The f o l l o w i n g r e l a t i o n s appear t o g i v e the best r e s u l t s :
7 K, =
3.3.
t
5.10-2U log(O.l z 3 t 1 ) ( 1 t R i ) -
7
Method o f r e s o l u t i o n
The numerical procedure and t h e d i s c r e t i z a t i o n g r i d are extrapolated from the two dimensional v e r t i c a l model t h a t we already used i n several estuaries: Gironde, S t Lawrence (de Borne de Grandpre e t a l . , 1979, 1981), Seine (Salomon, 1981). It i s a mixed i m p l i c i t - e x p l i c i t f i n i t e d i f f e r e n c e technique. The o r i g i n a l i t y o f t h e method l i e s e s s e n t i a l l y i n the f a c t t h a t the locat i o n s o f the upper and lower f r o n t i e r s of t h e i n t e g r a t i o n area change during the c a l c u l a t i o n s . I n order t o achieve t h a t , use i s made o f v i r t u a l p o i n t s 10cated above t h e surface, and o t h e r s s i t u a t e d under t h e bottom (Hamilton, 1975). D e t a i l s o f t h i s technique have been exposed by Thouvenin and Salomon (1984).
432
HORIZONTAL
VERTICAL
j+l
j
j-1
Ax i-1
Ax
i
i +1
i -1
i
i +1
Fig. 2. Computational g r i d s B r i e f l y stated, a t ev.ery time step a f i n i t e d i f f e r e n c e method i s used t o go over the usual c a l c u l a t i o n sequence i n s i d e the area o f computations. The results are then e x t r a p o l a t e d t o external v i r t u a l p o i n t s by a n a l y t i c a l f u n c t i o n s which take i n t o account t h e upper and lower boundary conditions : -+ -+ a t the surface -+
av = k V1, Nz az -+
/Vim/
and
as
= o
a t t h e bottom
+ 1 ',
+
: t h e v e l o c i t y vector one meter above t h e bottom
T~
: wind s t r e s s
k
: f r i c t i o n coefficient
The f u n c t i o n s used a r e u s u a l l y polynomials o f t h i r d degree (splines), except f o r the v e l o c i t y near the bottom. To ensure a c o r r e c t v e l o c i t y p r o f i l e j u s t above t h e bottom where i t i s known t o be logarithmic, we use e i t h e r a l o g a r i t h m i c
curve o r an exponential one, obeying the c o n t i n u i t y o f
the v e l o c i t y curve and o f i t s f i r s t d e r i v a t i v e . 3.4 Numerical scheme The numerical scheme i s mixed i m p l i c i t / e x p l i c i t . A l l s p a t i a l d e r i v a t i v e s are centered except advective terms i n t h e s a l t equation. The pressure gradient term, p a r t i c u l a r l y important f o r the s t a b i l i t y o f t h e scheme, i s t r e a t e d as implicit.
433 Considering t h e equations i n t h e sequence described i n 3.1.,
i t i s possible t o
solve the complete system i n an e x p l i c i t way, i . e . w i t h o u t s o l v i n g any m a t r i x . The complete numerical scheme i s described and discussed i n Thouvenin (1983). 4. MODEL APPLICATIONS
4.1 Schematic t e s t s Before applying t h e model t o t h e Bay o f Seine, we t e s t e d i t s performances i n sane simple and r a t h e r schematic cases f o r which aspects o f t h e s o l u t i o n are known. I n p a r t i c u l a r , we have studied (Thouvenin, 1983):
- The propagation o f a sinusoidal wave i n a rectangular - The f r e e o s c i l l a t i o n s i n a closed basin. - The v e l o c i t y f i e l d i n a channel crossed by a trench. - The e f f e c t s o f t h e wind blowing over t h e open sea. - Convective movements along isopycnal surfaces.
basin.
Here, we s h a l l b r i e f l y comment on the l a s t two experiments.
N
WIND
Point 12 (170 Point 10 (130 m.) Point 6 ( 5 0 m . )
Fig.3. Water p a r t i c l e s t r a j e c t o r i e s under wind a c t i o n (Distance from the surface i n d i c a t e d i n brackets).
434
Fig. 3 shows the t r a j e c t o r i e s o f water p a r t i c l e s induced by a constant wind supposed t o blow during 33 hours from t h e south, and then t o stop.
According t o t h e known a n a l y t i c a l s o l u t i o n , we observe t h e superposition o f t h e Ekman d r i f t s p i r a l and o f i n e r t i a l o s c i l l a t i o n s .
When t h e wind stops t h e i n e r t i a l
r o t a t i o n , stronger near t h e surface. diminishes.
Some energy i s transferred
through viscous s t r e s s t o the lower l a y e r s which are thus accelerated. Comparing t h e thickness o f the Ekman l a y e r produced by t h e model w i t h the t h e o r e t i c a l s o l u t i o n , we f i n d f o r Nz a d i f f e r e n c e o f 20 %. This d i f f e r e n c e which sums up several numerical e r r o r s seems q u i t e reasonable. The movement associated t o a s a l t wedge i n a rectangular tank i s i n t e r e s t i n g Here, we observe the formation o f a d e n s i t y d r i v e n c i r c u l a as w e l l ( f i g . 4 ) . t i o n along the i n t e r f a c e and t h e appearance o f a b a r o t r o p i c wave propagating very q u i c k l y f a r from t h e s a l t wedge.
Also t o be noted i s
the polarisation
o f v e r t i c a l movements, more intense on t h e r i g h t side, due t o t h e C o r i o l i s f o r ce
.
Fig. 4 S a l t wedge c i r c u l a t i o n a) l a t e r a l views a f t e r 30 and 300 minutes
b) view from above a f t e r 5 hours
435 4.2 A p p l i c a t i o n t o the Bay o f Seine ( i ) I n i t i a l and boundary conditions.
Because o f computer l i m i t a t i o n s , t h e
application t o the Bay o f Seine i s done w i t h a r e l a t i v e l y coarse g r i d s i z e ( 4 km. 4 m).
It must be acknowledged t h a t due t o t h e l a c k o f i n f o r m a t i o n on
boundary conditions, a g r i d refinement would n o t have been completely j u s t i f i e d . I n i t i a l conditions f o r s a l i n i t y are taken from i n s i t u measurements made during the year 78. Boundary conditions along the open boundary (water l e v e l s and v e l o c i t i e s ) are taken from a 2D model (Salomon, 1985).
The v e r t i c a l s t r u c t u r e o f the h o r i -
zontal v e l o c i t i e s i s supposed t o obey a l o g a r i t h m i c p r o f i l e .
The v e r t i c a l com-
ponent o f the v e l o c i t y i s l i n e a r l y i n t e r p o l a t e d between t h e bottom and the surface where i t i s . c a l c u l a t e d through the f o l l o w i n g r e l a t i o n s : 'bottom
=
( i i ) T i d a l currents. The t i d a l component o f t h e currents i s the e s s e n t i a l part o f the dynamics b u t as i t seemed already w e l l known through measurements (Le H i r and l'Yavanc, 1985) o r 2D modeling (Salomon, 1985), l i t t l e a d d i t i o n a l information was expected. I n f a c t , we observe (see F i g . 5): important v e r t i c a l movements which, o f course, could n o t have been measured. They are t h e r e s u l t o f t h e bottom shape and h o r i z o n t a l convergences.
-
-
phase d i f f e r e n c e s and r o t a t i o n s along t h e v e r t i c a l which come from the d i f f e r e n t r a t i o o f the i n e r t i a l f o r c e and the d r i v i n g f o r c e (surface slope) as well as from c h a r a c t e r i s t i c s o f bottom Ekman l a y e r . Thus we are more aware o f the e r r o r we make i n a 2D model assuming t h e f r i c t i o n s t r e s s on the bottom t o be i n the same d i r e c t i o n as the average velocity. ( i i i ) Wind induced movements. The wind induced dynamics i s t y p i c a l l y threedimensional and cannot be approached by 2D simulations. This component o f the c i r c u l a t i o n i s l i n k e d t o others and e s p e c i a l l y t o the tide.
The r e s u l t s may be very complex and d i f f i c u l t t o understand from i n s i t u
measurements.
The model here i s o f g r e a t assistance, even i f i t s i n d i c a t i o n s
are n o t very accurate. Theoretically, c a l c u l a t i o n s should take i n t o account the response o f the e n t i r e Channel t o wind f o r c i n g and l a r g e scale atmospheric pressure gradients. However, some i n t e r e s t i n g i n f o r m a t i o n can be obtained from l o c a l simulations about the response o f t h e bay t o wind f o r c i n g , a t l e a s t q u a l i t a t i v e l y .
16
32 Km
-
\
-
8 -8
0
- 16
16
-40
96Km
32
m 1 16
80
-40
64
-24
48
-32
32
-32
16
-24
I
-32
- 24 48
64
16
32
Km
80
16
96Km
32
16
48
32
64
48
.--+--*__-
.\-&-&-b*D*Q*
BOTTOM
16
80
64
32Km
80
%Km
96Km
I
437
1 R
. . . .! . . . . . . . . . . . l
,
t
?
a
X
'
l
'
~
I . . .
. . . . . .
. . . . . I
R S , . r ? ! l p ? ?
I . . . . . . . . . . :ssnR.rcnnvnar
.'
F i g . 6. Wind induced v e l o c i t y f i e l d ( h o r i z o n t a l sections near the surface and the bottom ; N-S v e r t i c a l sections ; E-W v e r t i c a l s e c t i o n s ) . Wind blowing from the west.
438 So we have c a r r i e d o u t some s i m u l a t i o n s o f t h e d i r e c t a c t i o n o f t h e wind on t h e bay alone. The boundary c o n d i t i o n corresponds t o a h y p o t h e t i c a l steady regime.
I n the
dynamical equation along t h e n o r t h e r n boundary a balance i s assumed between t h e surface slope, f r i c t i o n stresses on t h e bottom and a t t h e surface, and the C o r i o l i s force. The r e s u l t s presented i n Figs. 6 and 7 are f o r u n i f o r m wind f i e l d s o f 15 m/s.
A permanent regime i s g e n e r a l l y observed a f t e r seven hours. The h o r i z o n t a l s e c t i o n s c l e a r l y show t h e s t e e r i n g o f t h e c u r r e n t s by bathym e t r i c features, p a r t i c u l a r l y near t h e bottom ( f i g . 6 ) . Cross c u r r e n t s a r e h a r d l y ever observed i n t h e Parfond ( a n c i e n t Seine valley) even close t o t h e surface, where c u r r e n t s a r e l e s s r e l a t e d t o t h e bottom shape. V e l o c i t i e s t h e r e o f t e n r o t a t e a few tens o f degrees.
A t t h e surface, t h e v e l o c i t i e s are l a r g e r i n t h e shallow zones, a consequence o f t h e f a c t t h a t t h e d r i v i n g f o r c e r e l a t e d t o t h e u n i t o f volume o r u n i t o f mass i s i n v e r s e l y p r o p o r t i o n a l t o t h e water depth.
This i s t h e basic reason
f o r t h e general dissymmetry between t h e eastern and western p a r t s o f t h e bay.
Fig. 7 H o r i z o n t a l v e l o c i t y f i e l d s f o r d i f f e r e n t wind d i r e c t i o n s .
439 For the same reason,the wind induced v e l o c i t i e s above t h e Parfond are weaker, When the conditions are such t h a t a d e f i c i t o f water appears i n t h e southern p a r t o f the bay, an adjustment c u r r e n t appears i n the Parfond, opposite t o the surface v e l o c i t i e s .
This i s a w e l l known f e a t u r e o f wind a c t i o n i n l i t t o r a l
zones and lakes : a back c u r r e n t appears i n the deepest parts, g e n e r a l l y the waterway.
I t i s a l s o i n close agreement w i t h measurements made during t h i s
study. The v e r t i c a l sections a l s o show very obvious v e r t i c a l s t r u c t u r e s corresponding t o v e r t i c a l movements near the banks and h e l i c o i d a l movements i n s i d e the trench o f the Parfond, which measurements would n o t have revealed. These simulations, as w e l l as the measurements, i n d i c a t e t h a t t h e wind i n duced v e l o c i t i e s are o f t e n o f an order o f magnitude o f 10 t o 30 an/s, which i s much higher than t i d a l r e s i d u a l currents.
This mechanism i s thus an essential
vector o f t h e long term movements o f waters i n t h e bay.
5 CONCLUSION . The model presented i n t h i s paper i s b u t one element o f a general study o f the bay. The study l a s t e d t h r e e years, during which a considerable e f f o r t has been made on measurements : t h r e e cruises o f a few months each have been c a r r i e d out, i n v o l v i n g d i f f e r e n t measurement l o c a t i o n s (up t o twelve), sometimes a t t h e surface, a t t h e bottom and a t mid-depth. Important human and technical resources have thus been involved.
Fig.8 Diagram showing t h e r e s i d u a l c i r c u l a t i o n i n t h e Bay o f Seine
440
The measurements produced a l o t o f i n t e r e s t i n q i n d i c a t i o n s but,finallv,when t came t o e l a b o r a t i n g a comprehensive synthesis o f thedynamical regime o f the bay, the main r o l e was devoted t o the model. This way we were able t o understand what occured and we could separate t h e proper e f f e c t s o f each o f the physical mechanisms involved : wind, t i d e , density, C o r i o l i s ... This way we understood what measurements where showing. The diagram f o r the residual c a l c u l a t i o n presented i n fig.8,
was e s s e n t i a l l y
deduced from model r e s u l t s . So, i t proved t o be an e f f i c i e n t t o o l f o r conducting coastal studies.
Some advantages o f t h i s model have been already mentioned, e s p e c i a l l y the p o s s i b i l i t y o f c o n c i l i a t i n g a f i x e d l o c a t i o n o f computational p o i n t s and a continuous representation o f the bottom and t h e surface, i n s p i t e o f a great t i d a l range. It i s h i g h l y p e r f e c t i b l e from the numerical p o i n t o f view as w e l l as f o r
the t u r b u l e n t exchange simulations, b u t before complicating i t , i t i s worth weighting t h e pros and cons. Our opinion i s t h a t f o r r e s t r i c t e d coastal s i t e s under t h e i n f l u e n c e o f the t i d e and the wind, preventing high s t r a t i f i c a t i o n t o occur, and f o r which i n i t i a l and boundary conditions are h a r d l y a v a i l a b l e , such a model i s an i n t e r e s t i n g compromise and an e f f i c i e n t t o o l . 6 REFERENCES De Borne de Grandpre, C., 1979. Modele bidimensionnel en temps r e e l de l a c i r c u l a t i o n v e r t i c a l e estuarienne. A p p l i c a t i o n I l a Gironde. Oecanologica Acta, 2, 1, 61-68. De Borne de Grandpre, C., E l Sabh, M . I . and Salomon, J.C., 1981. A twodimensional numerical model o f t h e v e r t i c a l c i r c u l a t i o n o f t i d e s i n the S t Lawrence Estuary. Estuarine, Coastal S h e l f Sci , 12, 375-387. Hamilton, P., 1975. A numerical model of t h e v e r t i c a l c i r c u l a t i o n o f t i d a l estuaries and i t s a p p l i c a t i o n t o t h e Rotterdam Waterway, Geophys. J.R. Astron. SOC., 40, 1-21. Le H i r , P. and l'Yavanc, J., 1985. Obssrvations de courant en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 7-14. 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 sea. Dynamics o f Atmosphere and Oceans, 2,29-47. Salomon, J.C. ana Le H i r , P., 1980. Etude de l ' e s t u a i r e de l a Seine. Modelisation numerique des phenomenes physiques. Rapport Scient. CNEXO/UBO, 286 p. Salomon, J.C., 1981. Modelling t u r b i d i t y maximum i n t h e Seine Estuary. I n : J.C.J. Nihoul ( e d i t o r ) , Ecohydrodynamics. Elsevier, Amsterdam, 285-317. Salomon, J.C., 1985. Courantologie calculee en b a i e de Seine. Rapport Scient. CNRS/Greco Manche, 15-22. Thouvenin, B. , 1983. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion pour des regions c b t i e r e s a maree. These 3e c y c l e U.B.O., 269 p. Thouvenin, B. and Salomon, J.C. , 1984. Modele tridimensionnel de c i r c u l a t i o n e t de dispersion en zone c b t i e r e 1 maree. Premiers essais : cas schematique e t baie de Seine. Oceanologica Acta, 7,4, 417-429.
44 1
TIDAL STREAMS IN SHALLOW WATER P.P.G. DYKE Department of Mathematics and Statistics, Plymouth Polytechnic, Drake Circus. Plymouth, Devon, PL4 8AA, U.K.
ABSTRACT The analysis of tides is particularly suited to perturbation methods, since there is normally a dominant oscillatory flow. In shallow water, a variety of non-linear effects can be justifiably considered. The advection terms, free surface terms and variations in vertical and horizontal momentum transfer are all important. This contribution examines, analytically, some of these effects, solves some simple problems and provides insight into how the steady circulation and higher harmonics can be thus generated. 1 INTRODUCTION
In the shallow continental shelf regions that surround most continents and which are particularly wide adjacent to North West Europe, barotropic tides are a very important source of water movements. Tides are predominantly oscillatory in character: however, non-linear interactions cause an asynaetry and give rise to tidal harmonics.
These non-linear interactions occur through
the advection terms, the free surface and in the representation of turbulent momentum exchange.
For example, if the primary tidal oscillations are plane
waves
v(x,y,t)
=
Vsin(kx + ly - ot)
with frequency o rad s-',
(2)
wave number vector (k.1) and tidal ellipse having
semi-major axis U. semi-ninor axis V: it is easily seen that a typical advective term that occurs in the equations of notion, say
u ax
=
3 k W (1 + cos2(kx + ly - ot))
gives rise to a steady term and a second harmonic.
(3)
This is simple.
However,
the solution of the full three dimensional partial differential equations which govern tidal dynamics is far more of a challenge.
Oceanographers are
still arguing about how best to represent turbulent momentum exchange.
Quite
early on in a typical computation, it is beneficial to concentrate the For example, in a series of papers
attention on a specific phenomenum.
Tee (1976, 1977, 1979, 1980, 1981, 1982, 1985) and Tang and Tee (1986) have addressed the steady problem which requires elimination of time-dependent effects.
This is most conveniently done by assuming all variables
proportional to eiot so that a/at is replaced by io (i occurs.
= 4 :)
wherever it
The fact that this can still remain a lifetime study is an indication
of the complexity of the tidal problem.
We shall follow this scheme in some
of the later sections of this paper. This paper is solely concerned with tides.
However, let us not lose sight
of the many and various other causes of coastal currents.
If an estimation of
the strength and direction of a current was required, then although tides may well provide the main input, from an engineering viewpoint it would be foolhardy indeed to ignore wind induced current, storm induced flows (including surges), density currents, and currents that arise from sea surface slopes outside the shelf region.
If such an estimate is required, the
analytical methods of this paper have to be abandoned and numerical techniques such as those of Davies (1986) come into their own. approach.
Davies favours a spectral
One could use finite differences in the vertical, this is done by,
for example, Backhaus (1983).
This paper is concerned more with fundamentals
and less with final applications. however it is hoped that sone of the ideas herein will later find their way into these applications. Tides are phenomena consisting of a dominant mode added to which are higher order but smaller effects.
An obvious candidate to choose from the
limited number of analytical techniques available is perturbation.
. This
requires all variables to be expanded in power series of some small parameter (eg. wave slope or ratio of boundary layer depth to total depth).
The
non-linear problem then transforms into a series or progressively more complicated, but linear. problems.
Perturbation methods are old,fashioned and
have received a bad press recently due to their overuse outside regions of applicability and validity.
The tidal problem is tailor made for perturbation
methods, so no excuse is required for using them here.
2 THE GOVERNING EQUATIONS
For a shallow, coastal sea region it is reasonable to assume that the Coriolis acceleration is constant and that variation in density can be ignored, at least in a preliminary investigation.
To include density
stratification would be to include internal tides which, admittedly, cause baroclinic residuals that can be important, but which would only confuse at
443
this early stage. A simple (the simplest) treatment of baroclinic residuals can be found in Dyke (1980). The conservation of momentum and mass equations take the form
ax
at +
a1
(1.0 )u --H -
+
w az +
1x
g
=
-g yHc
a
+
%
ax [.% ]
+
AHV@
,
(4)
and
ac ;;;:+YH.
r-
udz=O.
(5)
-h
In these equations, z , the coordinate axis which points vertically upward, has a zero value at mean sea level, and c(x.y.t) describes the elevation of the sea surface. The hydrostatic and Boussinesq approximations have been applied, 1
=
g(x,y,z)
=
(u,v) is the horizontal fluid velocity, w
=
w(x,y,z)
the vertical fluid velocity. PH is the horizontal gradient operator with x east and y north, N is a coefficient of eddy viscosity which may vary with position and/or velocity, z
=
-h(x,y) is the sea bed.
1
is twice the angular
velocity of the earth and AH is a constant horizontal diffusion coefficient. Since it is our wish to model tides which are mainly the result of gravitational forces dictated by the frequency of the earth-moon system, it seems a reasonable assumption to permit only oscillations of frequency o , the frequency of the dominant M, tide. This means that we adopt the scheme mentioned in the introduction of replacing a/at by io where o is the aforementioned frequency.
As
mentioned in section one, analytical progress is
facilitated by adopting a perturbation technique based upon expanding c , 1 and w as power series of a small parameter. The parameter we choose is slope which obeys
Accordingly ezc, +
..
EW, + € Z W 1 +
..
c = ec,
w =
+
The zero order equations are
e
the wave
444
and ioc, + (hu,)x + (hv,)y
(9)
= 0 ,
Implicit in deriving these last two equations is that the total depth is much larger than the tidal amplitude.
(Otherwise
LE
would cease to be small).
Tee (1986) has referred to this as the weakly non-linear case.
We later relax
this condition, ie. abandon the perturbation technique, temporarily. remaining non-constant yet to be expanded is the eddy viscosity N.
The only
To be as
general as possible, let us write
N
=
6N, + 6'N1
+
...
(10)
where 6 is another expansion parameter, the definition of which will depend on how N changes in various regimes
However, until section four of this
article, N will not be expanded. The first order equations wi 1 be
With adroit use of Taylor's series, subsequent order equations may be found, with the cross terms from previous order solutions providing the forcing. The boundary conditions are quite straightforward since we are 'concerned with tides which are very long waves (hydrostatic).
-u = 0
Thus
at coastlines
modified to
-u
. =~ 0 at coastlines
(14)
if AH, the horizontal diffusion coefficient of momentum is set to zero. the sea bed, there must be either no flow, or zero normal flow, again dependent upon whether friction is allowed, ie.
-u
=
0 at z
=
-h,
if N is non-zero at the sea bed
At
445
-u . h h
= 0
at z
=
if N = 0 close to the sea bed.
-h
P atmospheric at c(x,y,t) as a consequence of having assumed hydrostatic balance. (This
The sea surface boundary condition reduces to p z =
(16)
=
has already been subsumed in eliminating pressure from (4)). Now we shall make a few simplifying assumptions. AH
= 0.
First of all, set
This means we neglect horizontal boundary layers and demand only that
the component on velocity normal to the coast is zero. Next we suppose that a quadratic friction law applies at the sea bed
This is hardly a simplifying assumption, except that in tidal regimes it is possible to use Fourier Analysis to linearise (17) (see, for example Marchuk and Kagan (1984). ~ 2 4 4 ) . without significant loss of information.
Finally, we
will ignore surface stress, which rules out consideration of wind waves and tide-wind wave interaction. If we vertically integrate the zero order momentum equation ( a ) , we obtain, in Component form
where the bottom stress terms arise from undoing the stress/rate of strain (Newtonian) term in (8) and assuming that the bottom stress is the same as the stress in the sea itself at the sea bed.
go = (uo,vo) = Re((U,V)eiot)
The notation is as follows:
,
and
-h The continuity equation is
and elimination of
u
and
6
from (18) and (19) with substitution into (22)
leads to a second order elliptic partial differential equations for
u
and
c,
446 namely,
and (a’-f’) 6
=
Equation
reduces to the Helmholtz wave equation if h and the bottom
(25)
-fgCx
iagCy - f7bX/ph
+
+
iOTby/ph ,
stress are considered constant. The boundary conditions on c are
( i i . G ) . ~= o on coasts ,
(26)
n being a vector normal to the coast, and
6 and 6 being expressed in terms of
At open sea boundaries c is specified.
c through ( 2 3 ) and ( 2 4 ) .
Some special solutions to this problem are examined in section three. It is instructive to look at an alternative derivation of the zero order equations, this time retaining z dependence, and more importantly not explicitly demanding a small
6 .
All that is demanded is that non-linear terms
are in some sense small. 2.1
Direct Approach Consider again equation ( 4 ) with no non-linear terms, namely
Differentiate this with respect to z , which eliminates the first term on the right hand side. (Assuming AH constant). Using the suffix derivative notation t,!
+
1x2, =
Writing!
= (u,v)
ot + if@ = (Ne),,
+
AHV&
and o
=
uz
(28) +
iv,, equation
+ AH+@
H
If AH can be ignored, the transformation R
=
Neeift
(28)
is written in complex form
447
turns equation (29) into the standard diffusion equation Rt
=
NR,,
where N can be a general function of x,y,z, but not t. Alternatively, if N is not a function of x or y (or t) AH can be retained in equation (29), and the equation for R is
If N & time dependent, this can be catered for by the addition of a term -NtR/N on the left hand side of equation (31) or (32). or more naturally by dealing directly with equation (29). However, the beauty of deriving (31) and (32) is that it opens up all those analytical methods (Crank (1975)) or
numerical methods for solving the diffusion equation. The quantity o and its complex conjugate
0
are derivatives of the rotary
tidal current. These were first introduced in a slightly different form by Thorade in 1928 (Defant (1961), Soulsby (1983)). If N is constant, these rotary tidal currents resemble the Ekman spiral but with f replaced by and 0
+
o
o +
f
- f and with the two spirals spiralling very differently since
f >> o - f (and strange things happen at o = f). The equations that have been derived in this section are general for a
linear or quasi-linear formulation. If, however, the advection terms cannot be neglected even at zero order, then the problem needs re-examination. It has recently been suggested Davies (1985) that it is physically reasonable to let eddy viscosity vary as the current speed or its square. In either case, the previous formulation again fails due to the term
a ag -Naz az becoming fully non-linear.
It is still justifiable to ignore the advection
terms if horizontal variation of velocity is significantly slower than vertical variation. This will be the case far away from coasts and especially where there are vertical shear layers. Analytical progress is very difficult unless some kind of perturbation method is employed. Certainly, if eddy viscosity depends upon the square of velocity, then the vertical transfer of momentum becomes a term cubic in velocity in some general sense. If it is still assumed that the primary oscillation is o f frequency o , that is (u,v,c)proportional to eiot
(33)
then a cubic non-linearity will generate terms of time dependence eoiot.
In
the language of tides, an M, tide can in this way lead to the generation of M, tides, but with no M, tides.
Alan Davies (personal communciation) has
indicated that M, is present in some strength near North Sea coasts whereas M, is almost entirely absent.
This would support the above eddy viscosity
dependence and suggest that the energy of the tide is dissipated more through vertical transfer of momentum and less via horizontal advection (which would give rise to M, since these terms are velocity squared (or eliot) non-linearities). 3 RESIDUAL FLOW
Most analytical progress is possible when attention is restricted to time independent flow so that by time averaging over several tidal cycles, the complications caused by non linear generation of harmonics is eliminated. Referring back to the perturbation expansions of the last section; equations (11) and (12) (with A H = 0) have to be solved.
The complimentary function is
the same as in the zero order problem, with the extra terms consisting of (known) products of zero order terms.
The whole solution. complimentary
function plus particular integral, is then time averaged over several tidal periods.
Obviously, since there are non-linear interactions, the assumption
equivalent to (33) is technically false and a/at has to be carried explicitly in the equations. takes place.
This is of no consequence here since time averaging later
It is worth noting that no steady currents arise out of the
cubic non-linearities mentioned in the last section, but they would arise out of a friction term containing quadratic terms in velocity (or elevation).
The
solution for l1 will be of the form
where UE is independent of time, provided N is not of a form that generates even higher harmonics.
Upon time averaging, denoted by the time honoured
overbar,
and LIE is the Eulerian Residual Current, or Eulerian Tidal Residual.
The
Eulerian residual is what is observed by a stationary observer or current meter.
Particles which are moving with the fluid, or very nearly with the
fluid in the cases of pollution, fish eggs and the like, move with Lagrangian residual velocity.
Because the flow is primarily oscillatory, we can use the
well known, but often misused analysis in the introduction of Longuet-Higgins
449
(1953) which shows that
where
is the Lagrangian residual flow, and
&
is the Stokes' Drift, given
entirely in terms of zero order flow:
The question now arises as to the relative importance of UJ and&. and Dyke (1972).assumed that UJ was zero outside a boundary layer.
Johns
Moore
(1970) showed that, in the absence of friction if there are no closed geostrophic contours (depth contours for constant Coriolis parameter), then the total Lagrangian current
-
zero (unless 2s
=
UL is
zero, ie. YE
0, not the case here).
=
-US and
is certainly not
Examination of the Stokes' drift in
a frictionless sea thus gives information about Eulerian residuals provided there are no closed geostrophic contours. Tee (1980) calculates & and
&,
4~via
a numerical procedure and decides that both
in general, contribute to
with neither dominating.
interesting that Moore (1970) shows that
UL is,
It is
in general, indeterminate in a
frictionless sea and follows geostrophic contours.
The results of Tee (1980)
must, therefore, be heavily dependent on how frictional dissipation at the sea bed is parameterised. 3.1 Some Stokes' Drift Results From the results of the previous section, it is well worthwhile
calculating Stokes' drift because this is opposite in direction and equal in magnitude to the Eulerian residual current when there are no closed geostrophic contours. If it is assumed that
and the depth is allowed to vary, then geostrophic contours run north-south and are not closed.
The particularisation (38) is applicable to eastward
travelling long crested tidal waves.
With zero friction, equation (25)
reduces to the manageable (o'-f')<
+ g(hcX),
= 0
.
(39)
450 if we let c = A(X) sin ot ,
.
uo
=
U(x) cos ot
v0
=
v(x) sin ot ,
then U and V
=
=
f i A'(x)
43
.
&AI(x)
44 1
This analysis then becomes appropriate for standing waves with a straight north-south coastline.
&
= (US,VS)
where Us
vs
The Stokes' drift, from equation (36) is
=
= 0
-fg' 2(0*-fZ
) Z
A'(x) A"(x)
where, from (39), A(x) satisfies
(hA')' +
02-f A
= O .
At the coast, say x
=
(47)
x1
.
Now in this configuration. Eulerian residual.
U J
=
-
-%, an( a zero of
&
would give a zero of
We can perform some interesting analysis to obtain a
criterion for zeros of VS. By Rolle's theorem, provided (U.V) is not zero other than at the coast, the zero of Vs nearest the coast occurs when
(49) At this location, say x h'A' + h,X2A
= 0
= f ,
at x
= f
from equation (47)
(50)
451 where h, is some reference depth and
Integrating (47) with respect to x and using the mean value theorem, one obtains
hA' = h,A' 0 C C,C
(x1-x)A(C2)
x l (all x).
Hence, eliminating A' from (50) and (52),
where dh/dx is evaluated at x So far, (53) is exact.
= C.
the zero of A"(x).
However, if we assume that the ratio of the amplitude
on the right hand side is approximately unity, as is the case for tides, then 1 1 dh hdxa(x,-x)
(54)
at a location where VS is zero. Equation (54) is satisfied identically for a constant slope sea bed.
Dyke
(1973) examines particular cases and finds that reversals in Vs occur when the topography is convex upwards, and no reversals occur in VS when the topography is concave upwards.
4 FRICTIONAL EFFECTS There are many ways of introducing friction.
Through the perturbation
scheme, equations (7), ( 8 ) , (9) and (10). through methods such as those of section (2.1) and through dimensional arguments favoured by those working in turbulence (Monin and Yaglorn (1971)). The recent review by Soulsby (1983) is recommended reading.
Unfortunately, there is no best way to represent
friction. Returning to perturbation methods, if 6
= e
in equation (10). no friction
enters the equations at zero order, and the flow can be considered of boundary layer type with frictional effects restricted to sea-bed and sea-surface. Perhaps on continental shelves it is more reasonable to assume
N = N o + sN,
+
...
(55)
whence similar equations to Tee (1979, 1980) are obtained but with additional frictional terms at higher order.
Now, what are No, N,.
etc?
A good
question; let us see the results of assuming an expansion such as (55). Equation (8) remains unchanged save N is replaced by No.
If N o is constant or
linear in z (Soulsby (1983)). the result is Ekman type spirals or distorted versions thereof involving Kelvin functions (see Soulsby's article for details).
If No is a function of velocity. an idea increasing in popularity,
then the linear perturbation scheme will give rise to non-linear equations in
uo. vo and c o .
If time dependence is retained, then the occurrence of
non-linear terms as forcing terms on the right hand side of equation (11) enables one to see more easily how higher harmonics are generated.
ACKNOWLEGMENT The author is indebted to Kim Tee for access to unpublished material and to Alan Davies for useful discussions. REFERENCES Backhaus, J.O., 1985. A semi-implicit with an explicit scheme in a three-dimensional hydrodynamic model. Cont. Shelf Res. 2: 243-254. Crank, J., 1975. The Mathematics of Diffusion, Second Edition, Oxford University Press, 414 pp. Davies, A.M., 1985. On determining current profiles in oscillatory flows. Applied Math. Mod. 9. Davies, A.M., 1986. Numerical modelling of marine systems. In Numerical Modelling - Applications to Marine systems. J. Noye (ed) Elsevier North Holland (to appear). Defant. A., 1961. Physical Oceanography, Volume 2, Pergamon Press, Oxford, 590 pp. Dyke, P.P.G., 1973. Zeros of tidal mass transport under standing waves. Riv. Ital. Geof. 22: 353-358. Dyke, P.P.G., 1980. On the Stokes' drift induced by tidal motions' in a wide estuary. Est. Coast. Mar. Sci 11: 17-25. Johns, B. and Dyke, P.P.G., 1972. The structure of the residual flow in an offshore tidal stream. J. Phys. Oceanog. 2: 73-79. Longuet-Higgins, M.S., 1953. Mass transport in water waves. Phil. Trans. Roy. SOC. Lond. A245: 535-581. Marchuk, G.I. and Kagan, B.A.. 1984. Ocean Tides: Mathematical Models and Numerical Experiments. Perganon Press, Oxford, 292 pp. Monin, A.S. and Yaglom, A.M., 1971. Statistical Fluid Mechanics. Vol. I MIT Press, Cambridge Mass, 769 pp. Moore, D., 1970. The mass transport velocity induced by free oscillations at a single frequency. Geoph. Fluid Dyn. 1: 237-247. Soulsby, R.L., 1983. The bottom boundary layer of shelf seas. In Physical Oceanography of Coastal and Shelf Seas. B. Johns (ed). Ch.5. 189-266. Elsevier Science Publishers, Amsterdam. Tang, Y. and Tee, K.T., 1986. Effects of the mean and tidal current interaction on the tidally induced residual current. J. Physical Oceanography (submitted). Tee, K.T., 1976. Tide-induced residual current, a 2-D nonlinear numerical mode. J. Mar. Res. 34: 603-628. Tee, K.T., 1977. Tide-induced residual current - verification of a numerical model. J. Phys. Oceanog. 7: 396-402.
453
Tee, K.T., 1979. The structure of three-dimensional tide-generating currents. Part I Oscillating Currents. J. Phys. Oceanogr. 9: 930-944. Tee, K.T., 1980. The structure of three-dimensional tide-induced currents. Part I 1 Residual Currents. Tee, K.T., 1981. A three dimensional model for tidal and residual currents in bays. In Transport Models for Inland and Coastal Waters, 284-309. Tee. K.T., 1982. The structure of three-dimensional tide-generating currents: experimental verification of a theoretical model. Est. Coast and Shelf Sci. 14: 27-48. Tee. K.T., 1985. Depth-dependent studies of tidally induced residual currents on the sides of Georges Bank. J. Phys. Oceanogr. 15: 1818-1846.
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455
MODELLING AND OBSERVATIONS OF THE RESIDUAL CURRENT OFF SOUTHWEST NOVA SCOTIA
K.T. TEE and P.C. SMITH Bedford Institute of Oceanography, Department of Fisheries and Oceans, Dartmouth, Nova Scotia, B2Y 4A2 (Canada) D. LEFAIVRE Institut Maurice Lamontagne, 850 Route De La Mer, Ste-Flavie, Quebec, G5H 324 (Canada) ABSTRACT Using a depth-dependent weakly nonlinear tidal model, the threedimensional structure of the tidally induced residual current near Cape Sable off the Southwestern tip of Nova Scotia, is computed. The result is compared to 17 current meter observations located at seven stations. Examples of the depth-averaged residual current, and of the vertical structure of the current at all the experimental stations are presented. Using a multiple regression technique, the tidally induced residual currents are estimated from the current meter records. The estimated residual currents at two stations are well reproduced by the model. At one station, the predicted residual currents are smaller than the observed values. For the rest of the stations, the comparison is not conclusive because of large uncertainties in the estimated residual currents. 1 INTRODUCTION
Three-dimensional tidally induced residual currents have been computed in several nonlinear depth-dependent numerical tidal models (e.g., Leendertse and Liu, 1975). However, the results have not been examined carefully, and the structures of the current have not always been verified by observations. One of the reasons may be the lack of current meter data in the modelled area that indicate the signals (spring-neap oscillation) of the tidally induced residual currents. Another reason may be that there are problems in separating the tidally induced residual current from those induced by wind and freshwater runoff, especially when the latter forcings are dominant. The objectives of this paper are: ( 1 ) to compute the three-dimensional tidally induced residual current with a numerical model for the Cape Sable area, off southwestern Nova Scotia (Fig. 1 ) ; ( 2 ) to estimate the tidally induced residual current from 17 current meter data located at seven stations; and ( 3 ) to compare the model residual currents with those estimated from the data. The coastal waters off the southwestern tip of Nova Scotia (Cape Sable
456 68' 1
.
.
66'
.
,
. .. .. ..:
64-
46'
46'
440
440
42'
42'
Fig. 1. Location map for the Bay of Fundy and Gulf of Maine. area, Fig. 1) are among the most productive areas on the eastern coast of North America. It has been suggested that this is an area of upwelling (Lauzier, 1967; Garrett and Loucks, 1976; Smith, 1983). The modelled area is indicated by solid lines which extend across the Bay of Fundy and the Scotian Shelf in the north-eastern boundary, along the edge of the Scotian Shelf (-200 m isobath) at the south-eastern boundary, across the Gulf of Maine at
the south-western boundary, and along the New Brunswick coast at the northwestern boundary. There are 36 grid points at the north-western and southeastern boundaries, and 47 grid points at the north-eastern and south-western boundaries. The horizontal grid spacing is 7.047 km. There are two important topographic features near Cape Sable: one is the convex shape of isobaths around the Cape, and another is the Browns Bank, a shallow submarine bank approximatley 50 km long and 30 km wide. The crest of the Bank is "50 m
457
below the surface. The mooring array consists of a line of five sites (CO to C4) from Cape Sable across Browns Bank, and two more sites (C5, C6) on Scotian Shelf (Fig. 1). The depths of the stations are: 30 m for CO, 60 m for C1 and C5, and 110 m for the rest of the stations. The current measurements are available from CO between October 1981 and April 1983, and from C1 and C2 between April 1979 and April 1983, and from C3 to C6 between April 1979 and August 1980. A three-dimensional nonlinear numerical tidal model is described in section 2 and an analysis of the current meter data in section 3. In section 4 we discuss the computed residual current in the Cape Sable area and the comparison of the currents with the observations at stations CO-C6. 2 NUMERICAL MODEL
A simple method of calculating the three-dimensional tidal and residual currents has been described in detail by Tee (1979, 1980, 1986). Only those aspects that are related directly to the following solution and discussion are presented here. 2.1 The governing equations
For a Cartesian system with coordinates x, y, and z, where x and y are the horizontal coordinates and z is the vertical coordinate measured vertically upward from the mean sea level, the equations describing the tidal motion of a homogeneous fluid in shallow water are
-
-
au
au
+ a t
as
+ f x u = -gVs = az -+Waz....
* VU
U
-
5
+V
-D where
au
a N-+ -
u,
udz
-
az
(la)
-
AhV2u
0
is the horizontal velocity vector with components u and v in
coordinates x and y; w is the vertical velocity component; D is the depth of the bottom below mean sea level; 5 is the height of the water surface above mean sea level; f is the Coriolis parameter; N = N(x,y,z) is a vertical eddy Viscosity coefficient; Ah is a horizontal eddy viscosity coefficient; g is gravity; V is the horizontal gradient operator; and V2 is the horizontal Laplacian operator. By assuming that the advective term is small compared to the other terms in the momentum equation, a case we call the weakly nonlinear case, we obtain (Tee, 1979) the first order equations for the oscillating current au
-+ at
( ~ 1 )
au
-
f X u1
= -gVsl +
N
az + AhV2Yl
(2a)
458
and the second-order equations for the residual current ( i 2 ) au,
where the overbar denotes the time average over a tidal period, u is the -1 s value of u , ~at surface (z=O), and and U,z are the depth averaged values of
z1and u,~.
2.2 The boundary conditions The boundary conditions for a semi-enclosed basin are
-
aul
(a)
= az
2:a
0 and
-= az
3 s
-at
PN
z = 0, where (Tee, 1980)
and (b)
-
-
u 1 = u2 = 0 at z = -D and at the coastal boundary.
The open boundary conditions require the specification of the tidal elevation (5,) and the depth-averaged residual current (U,2). The tidal elevation is taken from Greenberg's (1983) large-scale model. Two boundary conditions on LIZ are tested: (1) 0 and ( 2 ) E2 specified from Greenberg's model. The result shows that the residual currents near Cape Sable, including the waters off the tip of the Cape and around Browns Bank, are generally the same for the two boundary conditions. The 'sensitivity of the residual circulation to the open boundary conditions is further examined by modelling the same area with a depth-averaged non-linear numerical tidal model. The linear version of this model is used to produce the depthaveraged tidal current for the 3-D tidal model. By assuming zero advection at the open boundary, we again find that the depth-averaged residual currents off the tip of Cape Sable, and around Browns Bank for the non-linear depthaveraged tidal model are consistent with those for the 3-D tidal model. The reason for the insensitivity of the residual current to the open boundary condition is that strong residual current near the Cape Sable area is mostly developed locally in the shallow water region, and not affected significantly
459
by the generally weak current at the open boundary. 2.3 The vertical eddy viscosity
The vertical eddy viscosity (N) is specified as:
1.4 x
where V o
m 2 r 1 is the molecular eddy viscosity, 6z is the
thickness of the transition layer between laminar and turbulent flow, and R2 is a parameter. This form of N has been applied by Johns and Dyke (1971, 1972), and Tee (1979, 1980, 1982, 1985). There are two adjustable parameters in (5): ( 1 ) 'the vertical eddy viscosity above the bottom transition layer, Nm; and (2) the thickness of the transition layer 6z. The value of Nm can be
estimated from the relationship between vertical eddy viscosity (N) and linear bottom friction coefficient (A) (Tee, 19801,
D -D -D
N
The bottom friction in the depth-dependent model will approximate that in the depth-independent model if (6) is satisfied. The value of X can be estimated from X(X,Y)
K
Yum 1 7
(7)
where KI is a constant taken to be one, y is the bottom friction coefficient taken to be 0.0021, and Um is the current amplitude. For given values of X,
(6) is used to compute Nm. The value of 6z is chosen by comparing the vertical variation of the observed and computed tidal current at station C1 (Fig. 1 ) . By using the observed tidal current at mid-depth (30 m), and estimating Nm from (61, the vertical variation of the tidal current can be Fig. 2 shows the comparison for the computed for various values of 6z. winter period. The observed tidal current is obtained by a 29-day tidal The water column at this analysis centered at day 56, 1980 (February 25). The best fit is obtained with 6z = 0.15 m, and is The comparison is also performed for the fall period when
period is homogeneous. shown in Fig. 2. the water surface to period ( = current is
column is stratified: at increased from 23.83 kg m-3 near the 25.13 near the bottom. For the same 6z as that used in the winter 0.15 m), the best fit between the computed and observed tidal found by reducing Nm to 20% of its value in winter period (0.0077
m2s-l). Figure 3 shows the comparison for the period centered at day 280, 1980 (October 7). Although the observed semi-major axis and the orientation
460
I
I
0-
-
-0.2
-
-04 D ,
N
-
-a6
-
-0.8
-
Fig. 2 . Vertical variation of the tidal current in homogeneous water comparison between theoretical and experimental results at station C1: (a) semi-major axis; (b) semi-minor axis (negative value indicates clockwise rotation); (c) phase lag; and (d) orientation of the semi-major axis. The best fit between the computed and observed results is for 6, = 0.15 m and N, computed from (6). The observation was taken between February and March, 1980. of this axis are well simulated by the model, the comparison of the semiminor axis and the phase lag are less satisfactory. More study is required to improve the comparison. In the following, only the parameters for the homogeneous case are applied.
461
0°-
a
-
-02.
-04.
Z/D -06-
-08-0 8.
-I 0
025
050
075
100
SEMI-MAJOR AXIS (ms-') 0.0-
I
C
-02.
-04-
Z/D -06.
PHASE LAG P)
/5
lo
-
Fig. 3. Vertical variation of the tidal current in stratified water comparison between theoretical and experimental results at station C1: (a) semi-major axis; (b) semi-minor axis (negative value indicates clockwise rotation); (c) phase lag; and (d) orientation of the semi-major axis. The best fit between the computed and observed results is for 6, = 0.15 m and N, = 0.2 x (value of N, for homogeneous water). The observation was taken between September and October, 1980.
2.4 Computational procedure The two-dimensional depth-averaged nonlinear tidal model described by Tee (1976) is first used to compute the depth-averaged tidal current. The linear friction coefficient, A , is then estimated from (71, where U, is the semimajor axis of the computed depth-averaged tidal current. The value of Nm can be computed from (6) using the estimated A. The depth-averaged tidal current is recomputed using the linearized model, which neglects the nonlinear terms and sets the bottom stress of the tidal current to be pAyi, where p is density and V_l is the depth-averaged tidal current. Knowing the values of V,i and N, and using the boundary condition (41, the vertical variation of the tidal current can be computed analytically from ( 2 ) (Tee, 1979). The accuracy of the computation can be improved by updating the bottom stress in the depth-averaged tidal model to be that computed from the vertical = pN ag,/az at z = -D). However, it was variation of the tidal current
(xbl)
(&,
462
found that the improvement was only 1-2$, and was neglected here.
The same
result was found by Tee (1979, 1980, 1985). Using the computed u , ~ ,and setting the bottom stress of the residual current (Lb2) equal to PA,! where E2 is the depth-averaged residual current, we compute U,2 by transforming the depth-averaged equations for residual current (depth-average of (3a)) into the vorticity equation, which is then solved numerically using a relaxation method (Tee, 1986).
From the known values of
U,J, the vertical variation of L I ~ can be solved numerically from (3a) (Tee, 1980). The accuracy of U,, is then improved by updating the bottom stress
(xb2)
in the depth-averaged momentum equation from X,,2 = PNau_,/az(. -D. The computation is iterated until is not significantly improved from the
a2
updated LI~. Here, it was found that five iterations were sufficient to obtain an accurate solution (i.e. to within .01$). 2.5 Spring-Neap computation The spring-neap variation of the tidal and residual currents is simulated by changing the tidal forcing at the open boundary by 9 5 % , which corresponds
*
*
approximately to the M2 S2 forcing on the continental shelf (see N2 or M2 Table 1 of Butman et al., 1983). It will be shown in section 3 that this spring-neap computation is important for comparing the computed and observed residual currents. Because the inertial acceleration of
u , ~(au_,/at)
is not included in the
computation, the above simulation of the spring-neap variation is valid only if A>>Au, where X is the linear friction coefficient (71, and AU = lU(M2)
-
u(N2,S2) 1 is the difference between M2 frequency and N2 or S2 frequency. In the Cape Sable area, the condition of b > h U is valid in most. of the area except in the deep water region near the south-western and north-eastern open boundaries. The values of X / A O for all the experimental stations are shown in Table 1. For the difference between M2 and N2 frequencies, the values of VAu are greater than three at most of the stations except C6 where X/AU
1.5.
-
AU = (U(S2) = 1.6 for C5 and T h s , neglecting the inertial acceleration in the computation of
The values of NlAul are also greater than three for
u(M2)) at most of the stations except C5 and C6 where V A U
0.8 for C6.
spring-neap variation of u,z is reasonable at most of the stations except the weak dissipative areas of C5 and C6. Note that because strong residual currents near Cape Sable are generated locally in the shallow water region, the problem of VAu not much larger than one in the deep water (weak residual current) area near the open boundaries is not expected to affect significantly the results of spring-neap variation in the shallow water region.
463 TABLE 1 Depths and ratios of the linear friction coefficient (1) to the monthly ( d M 2 -N2)) and fortnightly (u(S2-Mz)) frequencies for stations CO-C6. Stations
A/U(M2-N2)
qU(M2-S2)J
Depth (m)
co
38.3 19.2 6.8 6.2 6.2
20.3 10.2 3.6 3.3
3.1
1.6 0.8
30 60 110 110 110 60 110
c1
c2
c3 c4 c5 C6
3.3
1.5
3 ANALYSIS OF THE EXPERIMENTAL DATA The experimental data, measured by the Aanderaa RCM-5 current meters,
were sampled at 30-min intervals. The depths of the instruments, and the lengths of the record used for analyses, are given in Table 2. The time series of the current components were low-passed with a Cartwright filter (half-power at 31 h) and subsampled at 6 h intervals.
The low-passed data
TABLE 2 Depths of the current meters, periods of data used in the spectral analyses, values of a computed from_ (12), and comparison between the mean tidally induced residual current (t2) estimated from current meter observations and that computed from the numerical model. Estimated residual currents larger than 1.5 times the confidence limits are underlined.
a
Station Depth Period Name (m)
co c1
c2
c3 c4 c5
C6
15 22 25 15 30 40 50 55 57 50 90 100 15 50 15 50 100
Oct.81 Mar.82 Mar.82 OCt.79 Oct.81 Aug.80 Oct.81 Aug.80 Aug.80 Nova78 Aug.80 Oct.79 Aug.79 Apr.79 Apr.79 Apr.79 Apr.79
- Apr.83 - Apr.83 - Apr.83 - Aug.80 - Apr.83
-
-
2.2 2.1 2.1 3.0 2.7 Apr.83 2.4 Apr.83 2.1 Oct.81 2.0 O~t.82 2.0 Aug.80 2.1 Oct.81 2.0 Mar.82 1.9 Aug.80 1.6 Mar.80 1.7 Mar.80 1 .o Aug.80 1.1 Aug.80 2.9
Observed ii2 Magnitude Angle (m sec-1) (0)
Computed 5, Magnitude Angle (m sec-1) (0)
0.129 D .025 0.131 9.028 0.100 D.026 0.030 D .030 0.024D.021
0.099 0.093 0.091 0.045 0.042 0 037 0 035 0.031 0.030 0.012 0.015 0.014 0.042 0.028 0.012 0.010 0.005
257 f10 247 +11 247 f13 250 3 0 246 WO 239 3 1 226 k30 228 WO 0.026 i0.015 217P8 0.021 D.021 229 i.86 0.049 40.033 138a9 0.027 0.018 44 333 0.181 D.068 77 P 2 0.071 0.053 240 %5 0.152 D. 123 128 3 8 0.036 iO.030 93 a7 0.014%10.013 149 5 6
257 257 257 278 281 283 285 286 286
343 344 345 123
74 270 249 279
464
(g) can >e written as the sum of mean ( g ) and modulating u = u + u
-
6)components,
- . . -
(8)
For the tidally induced residual current, the modulating component contains u(N2)) and fortnightly {"14 day, AU = U(S2) monthly {"28 day, AU I u(M2) u(M2)) periods. Evidence of the tidally rectified residual current can be obtained from the coherent modulation in these two frequency bands between the low-passed data and the amplitude of the tidal signal (Tee, 1977; Butman
-
-
et al., 1983; Smith, 1983). Besides the tidal forcing, the wind forcing can also induce the low frequency modulations in the current meter data. Using these two forcings, the modulation at each frequency can be described by an optimum model u = aF + bE + cN + noise (9)
-.. -
-
where ,u is the low-passed current meter data with components u and v in the x- and y-directions, F, E and N are the modulation of the tidal forcing, and the east and north components of the wind velocity, and _a, ,b and ,c are complex vector coefficients with x and y components corresponding to u and v currents. The tidal forcing, F, is written as F = G (10) where Ua is the time-varying amplitude of the semi-major axis (Smith, 1983) and a is a spatially variable coefficient. Using moored current meters in the Minas Basin at the head of the Bay of Fundy, where only the tidal forcing is important for low frequency oscillations in the residual current, Tee (1977) found that the value of a varies between 1.0 and 1.4. The value of 1.0 was used by Smith (1983) to estimate the tidally induced residual current at stations C1 (50 m depth) and C3 (44 m depth). In Butman et al.'s (1983) study of the tidally induced residual current on the southern side of Georges Bank, the tidal forcing is taken to be the square of the low-passed tidal elevation. In these previous studies, wind forcing was not included in the spectral analysis of the tidally induced residual current. Here, a is estimated from the spring-neap computation described in section 2.5. Using (9) and (101, and neglecting the wind forcing, which is not included in the numerical model, a can be estimated from
-
t uz (spring) a = { Ua(spring)
- ...uz (neap)) /UZ ... (mean)
- Ua(neap))
/Ua(mean)
(11)
Theoretically,a is generally different for the x- and y-components of the residual currents. However, it was found that, unless the direction of the observed residual current is close to one of the axes, a in both the x-
and
465
y-components can be estimated from the magnitude of the residual current (Iu21)*
-
-
{ (u2(spring)J-lu2(neap)()/lu2(mean)l
a = {Ua(spring)
-
(12)
Ua(neap)}/Ua(mean)
The values of a, estimated from (12), for all the experimental stations are shown in Table 2. The values vary from a minimum of 1.0 at 15 m for station C5 to a maximum of 3.0 at the same depth for station C1. For given values of a, and the time series of u,, F, E and N, the values
-
of a, b_ and c, can be estimated from (9) using the spectral method described by Garrett and Toulany (1982). The least squares determination of the x component of a, b, and c, for each frequency ha is given by
ax@^^
+
bx @EF
ax $E
+
ax @FN
+
bx OEE
+
‘x #NF = QUF
+
‘x @NE = @UE
(13)
bx.8EN ‘x %N @UN where ax, bx and cx are the complex x-component of a_, b, and c_, #ij(Aa) is the cross-spectrum between variables i and j at frequency ha. Similar formulae can be used to obtain ay, by and cy, the y-component of the coefficients a_, b, and c,. +
The tidally induced residual current averaged over the period of is estimated from observation, 5,
where
Ea is the mean
value of Ua.
4 RESULTS AND DISCUSSIONS 4.1 Results of the numerical simulation Figure 4 shows the depth-averaged residual current induced by the mean (M2) tide. The basic features in the residual circulation near the Cape Sable area are: ( 1 ) the westward jet near the coast, which passes through stations CO and C1; (2) the clockwise circulation around Browns Bank, which passes through station C3; and ( 3 ) the eastward flow east of Browns Bank, which passes through station C4. Stations C2, C5 and C6 are situated in the weak current areas. Near the southern corner of the open boundary, there are strong eastward flows on the northern side of Georges Bank and north-western inflow in the deep channel. At the entrance to the Bay of Fundy (northThese western of Yarmouth, Fig. 41, there are irregular residual flows. irregularities, attributed to a resolution problem, have previously been observed in two depth-averaged tidal models which have similar horizontal
466
0.3 1.0 3.0 10. 430. -100 .
--
CM/S CM/S CM/S CM/S CM/S CM/S
50 u
0
NAUT I CAL M I LES
Fig. 4 . The computed depth-averaged residual current induced by the mean (M2) tide. grid spacings to those applied in this study (=7 km), but which includes the whole Gulf of Maine
-
Bay of Fundy
system
(Greenberg, 1983; Isaji and
467 Spaulding, 1984).
Here, we s t u d y only t h e r e s i d u a l c u r r e n t near Cape Sable
and around Browns Bank. Examples of t h e v e r t i c a l s t r u c t u r e of t h e r e s i d u a l c u r r e n t are shown i n Fig.
5 f o r t h e r e s u l t s a t a l l t h e experimental s t a t i o n s .
The magnitudes of
t h e r e s i d u a l c u r r e n t decrease with depth f o r a l l s t a t i o n s except C2 and C6.
The d i r e c t i o n s of t h e c u r r e n t , measured clockwise from t r u e North, w i t h depth a t s t a t i o n s CO, C 1 and C2, and decrease a t C 3 , C4 and C5.
increase
At
C6,
t h e d i r e c t i o n f i r s t i n c r e a s e s from t h e s u r f a c e t o a maximum a t 30 m , and then
Fig. 5. The v e r t i c a l s t r u c t u r e of t h e magnitudes ( 1 ~ ~ 1 , s o l i d line) and d i r e c t i o n s ( 0 , dashed l i n e ) of t h e t i d a l l y induced r e s i d u a l c u r r e n t a t s t a t i o n s CO-C6, computed from t h e numerical model.
468
decreases toward the bottom. The vertical variations of the direction are generally between 10 and 260, except at CO and C2 where the direction changes by 2.30 and 45O, respectively. 4.2 Comparison between computed and observed residual currents The predicted mean tidally induced residual current for the current meter The detailed vertical stations are shown in columns 7 and 8 of Table 2. structure of the current at these stations has been shown in Fig.
5.
The
i2
estimated (from 9 and 14) is shown in columns 5 and 6 of the same Table. The value of G2, estimated by using the monthly frequency, is more or less the same as that using the fortnightly frequency. The band-averaged result The 95% confidence limits for the currents are also is shown in Table 2. shown in the Table. Estimated residual currents that are larger than 1.5 times the confidence limits are underlined. The values of a used in the estimation are shown in column 4. The estimated is most reliable at station CO where the estimated
i2
i2
is
much larger than the confidence limit. There is also some reliability in estimating i2 at 40 m, 50 m and 57 m for station C 1 , and at 15 m for station C3, where the estimated i 2 is larger than 1.5 times the confidence limits. For the rest of the current meters, the values of i 2 are comparable to the confidence limit, and the reliability of the estimation is poor. From Table 2, we can see that the observed 1?_2 at station CO is reproduced
very well by the numerical model.
At station C1, the magnitudes of
i2
is
also reproduced well, but the directions of the current are off by about 40 to 50 degrees.
-
However, the uncertainties in the direction of the estimated
at this station are large, and the directions of the computed & change rapidly at nearby model grid points (Fig. 4). Thus, the disagreement in the computed and observed directions may not be significant. At station C3, the predicted i 2 is much lower than the observed value. One possible explanation of this disagreement is that, because the clockwise circulation around the bank is largely generated by topographic rectification of the tidal current {similar to the generating mechanism on Georges Bank and Norfolk Sandbank (Huthnance, 1973; Loder, 1980; Tee, 1985)}, the horizontal grid spacing of U,2
about 7 km used in the model may be too coarse to resolve the detailed topography on the sides of the Bank. The computed at stations C2 and C4 are about half of the observed values, and those at C5 and C6 are much smaller than the observed values.
i2
However, because the 95% confidence limits for the observed residual current less that 1.5 times the confidence at these stations are very large ( limits), the disagreement may not be conclusive.
k21
469
5 CONCLUSION The aim of this study is to compute the three-dimensional tidally induced residual current in the Cape Sable area, and to compare the model current with current meter observations at stations CO to C6 (Fig. 1). By using Tee's (1979, 1980, 1986) three-dimensional tidal model, the depth-averaged residual current and the vertical structure of the current at stations CO to C6 are shown in Figs. 2 and 3. Using a multiple regression technique, described by Garrett and Toulany (1982), and including both the tidal and wind forcings, the tidally induced residual currents are estimated for all the current meter records (Table 2). The estimated at station CO is reproduced very well by the model. At station C1, the magnitude of is also reproduced well. Although the directions of the current at this station are off by about 40 to 50 degrees, this may not be significant because of the large uncertainty in the estimation and the rapid variation of direction in the computed currents. At C3, on the northern side of Browns Bank, the estimated are much lower than the observed values, which may be because the horizontal grid spacing used in the model ( ~ 7 km) is not sufficient to resolve the topography in the area. For the rest of the stations (C2, C4-C6), the predicted is smaller than the estimated values. However, the result may not be conclusive because of large uncertainties in the estimated
iz
i2
i2
uz
i2 .
6 REFERENCES Butman, B., Noble, M., Chapman, D.C. and Beardsley, R.C. , 1983. An upper bound for the tidally rectified current at one location on the southern flank of Georges Bank. J. Phys. Oceanogr., 13: 1452-1460. Garrett, C.J.R. and Loucks, R.M., 1976. Upwelling along the Yarmouth Shore of Nova Scotia. J. Fish. Res. Board Can., 33: 116-117. Garrett, C. and Toulany, B., 1982. Sea level variability due to J. meteorological forcing in the northeast Gulf of St. Lawrence. Geophys. Res., 87: 1968-1978. Greenberg, D.A., 1983. Modelling the mean barotropic circulation in the Bay of Fundy and Gulf of Maine. J. Phys. Oceanogr., 13: 886-904. Huthnance, J.M., 1973. Tidal current asymmetries over Norfolk sandbanks. Estuarine Coast. Mar. Sci., 1: 89-99. Isaji, T. and Spaulding, M.L., 1984. A model of the tidally induced residual circulation in the Gulf of Maine and Georges Bank, J. Phys. Oceanogr., 14: 1119-1126. Johns, B., and Dyke, P., 1971. On the determination of the structure of an offshore tidal stream. Geophys. J.R. Astr. SOC. 23: 287-297. Johns, B. and Dyke, P., 1972. The structure of the residual flow in an offshore tidal stream. J. Phys. Oceanogr., 2: 73-79. Lauzier, L.M., 1967. Bottom residual drift on the continental shelf area of the Canadian Atlantic Coast, J. Fish. Res. Board Can., 24: 1845-1858. 1975. Modelling of three-dimensional flows Leendertse, J.J. and Liu, S.K., in estuaries, P-5461 , The Rand Corporation Publication, 18 pp. Loder, J.W., 1980. Topographic rectification of tidal currents on the sides of Georges Bank. J. Phys. Oceanogr., 10: 1399-1416.
470
Smith, P.C., 1983. The mean and seasonal circulation off southwest Nova Scotia, J. Phys. Oceanogr., 13: 1034-1054. Tee, K.T., 1976. Tide-induced residual current, a 2-D non-linear numerical tidal model. J. Mar. Res., 34: 603-628. Tee, K.T., 1977. Tide-induced residual current-verification of a numerical model. J. Phys. Oceanogr., 1: 396-402. Tee, K.T., 1979. The structure of three-dimensional tide-generating 930currents. Part I: Oscillating currents. J. Phys. Oceanogr., 9: 944.
Tee, K.T., 1980. The structure of three-dimensional tide-induced current. Part 11: Residual current. J. Phys. Oceanogr., 10: 2035-2057. Tee, K.T., 1982. The structure of three-dimensional tide-generating currents: Experimental verification of a theoretical model. Estuarine, Coastal and Shelf Sci., 14: 27-48. Tee, K.T., 1985. Depth-dependent studies of tidally induced residual currents on the sides of Georges Bank. J. Phys. Oceanogr., 15: 18181846.
Tee, K.T., 1986. Simple models to simulate three-dimensional tidal and residual currents. In: Three-dimensional shelf models, Meteor. Monogr., No. 5, Amer. Geophys. Union (in press).
471
A THREE-DIMENSIONAL WEAKLY NONLINEAR MODEL OF TIDE-INDUCED LAGRANGIAN RESIDUAL CURRENT AND MASS-TRANSPORT, WITH AN APPLICATION TO THE BOHAI SEA
SHIZUO FENG Institute of Physical Oceanography, Shandong College of Oceanology, 5 Yushan Rd., Qingdao, The People's Republic of China.
ABSTRACT A three-dimensional theory of the tide-induced Lagrangian residual currents and of the inter-tidal transport processes is proposed, based on a weakly nonlinear dynamical model of tides and intra-tidal transport processes. The first order Lagrangian residual velocity is the mass-transport velocity, that is the sum of the Eulerian residual velocity and the Stokes' drift velocity. The second order dynamics shows that the next contribution is a "Lagrangian drift velocity" which is a tidally periodic function of the tidal phase at which the marked water parcel is released. A set of field equations governing the mass-transport velocity is derived, which shows that the mean Lagrangian residual circulation is more relevant than the Eulerian residual circulation to the description of inter-tidal flow processes. A new inter-tidal transport equation for the tidal cycle average of the concentration of any conservative and passive tracer is derived. That equation correctly describes the Lagrangian nature of convective transport. The importance of the Lagrangian residual drifts for the dispersion or spreading of tracers in longterm transport processes is emphasized. Finally, a comparison between the present theory and observations in the Bohai Sea, China, shows good qualitative agreement.
1. INTRODUCTION In recent years, research on the hydrodynamics of coastal seas and tidal estuaries has focused on the inter-tidal processes which determine the long-term transport and distribution of important water properties such as salinity and temperature, and the concentration of pollutants or any tracer. While the dominant observable motion in coastal seas, such as the Bohai Sea and the East China Sea, is the rotary circulation associated with tides, the inter-tidal transport processes are mainly dependent on the residual circulation, but not on the tidal currents themselves, at least not in a direct way. It is becoming increasingly clear that the residual circulation should be described and studied by means of the Lagrangian residual velocity, since it is more relevant to use a Lagrangian mean velocity rather than a Eulerian mean velocity to determine the origin of water masses. The Lagrangian mean velocity of a marked water parcel leads to the concept of Lagrangian residual velocity, or Lagrangian residual current. It should be pointed out that the study of Lagrangian residual currents is a relatively recent topic, and thus any further investigation of Lagrangian residual currents may be a valuable contribution both from a theoretical and from a practical point of view.
472
As a first approximation of the Lagrangian residual velocity, the mass-transport velocity has been shown to be the sum of the Eulerian residual velocity and of the Stokes’ drift velocity (Longuet-Higgins. 1969). As is well known, the Eulerian residual velocity has been defined as the Eulerian mean velocity, i.e., as the velocity averaged at a fixed spatial point over one or several tidal cycles. The Lagrangian residual velocity, however, is related to the Lagrangian mean velocity of a marked water parcel, i.e., to the velocity averaged by following the marked water parcel over the tidal cycles. It is almost self-evident that, in contrast to the Eulerian residual velocity, the Lagrangian residual velocity is not only a function of the location where the parcel is released but also of the tidal phase at which that parcel is released. Indeed, the Lagrangian mean velocity depends upon the trajectory that the marked water parcel follows in the tidal field (Zirnmerman, 1979; Cheng and Casulli, 1982; Cheng, 1983). By considering second order dynamics, this dependence of the Lagrangian residual velocity on the tidal phase has been revealed in a depth-averaged two-dimensional model (Feng ef al., 1986a). It should be pointed out that a depth-averaged two-dimensional analysis of the Lagrangian residual velocity is, in general, questionable from the viewpoint of hydrodynamics because the interpretation of averaging a Lagrangian velocity over depth is not very clear unless the vertical water column is assumed to move in the tidal field as a rigid body (Stem, 1975; Alfrink and Vreugdenhil, 1981; Feng. 1986a). The dynamics of the Lagrangian residual current should be rigorously treated in a three-dimensional space. Therefore, a study of a three-dimensional theory of Lagrangian residual circulation in coastal seas is of much importance not only for practical applications, but also to satisfy the requirements of Lagrangian dynamics. In this paper, a weakly nonlinear three-dimensional model of Lagrangian residual current is introduced. The residual circulation in continental shelf seas, semi-enclosed shallow seas, gulfs or bays and tidal estuaries is, generally speaking, driven by tides, winds acting on the sea surface, variations of water density, and open boundary forces. While much effort has been applied to the study of residual circulation driven by winds and density variations, only recently has attention been drawn to the tide-induced residual circulation (Nihoul and Ronday, 1975; Heaps, 1978; Tee, 1976). In this paper, to avoid confusing the main issues, the effects due to surface wind and baroclinic variations are not included, only the dynamics of tide-induced Lagrangian residual circulation is discussed. The effects of wind and density variations on the Lagrangian residual circulation have been examined in previous papers (Feng, 1986b; Feng and Cheng, 1986). Here, the concept and the definition of Lagrangian residual velocity are reexamined. As stated above, the Eulerian residual velocity has been defined as the Eulerian mean velocity. Nevertheless, it would not seem proper to maintain that the Lagrangian residual velocity is synonymous with the Lagrangian mean velocity of a marked water parcel. Undoubtedly, the Lagrangian mean velocity of a marked water parcel in the tidal field is a Lagrangian velocity as far as inter-tidal processes are concerned. since it is proportional to the net displacement of the marked water parcel over the averaging period (several tidal cycles). On the other hand, it is natural to expect that the Lagrangian residual velocity could be used as a Eulerian field variable, and the aggregate of such local velocities may be specified as a Eulerian field of flow for
473
the inter-tidal processes. Thus it can be seen that the Lagrangian residual velocity cannot be unconditionally defined as the Lagrangian mean velocity of a marked water parcel unless the latter satisfies the continuity equation for an incompressible fluid. These conditions are discussed in sections 3 and 6 of this paper. Field equations governing the Eulerian residual circulation have been presented by Nihoul and Ronday (1975), who show that the introduction of nonlinear coupling between tides into the equations leads to the generation of a "tidal stress". Furthermore, a three-dimensional baroclinic model for the Eulerian residual circulation has also been proposed (Feng et al., 1984), in which there exists not only a "tidal stress" but also a "tidal surface source" due to the nonlinear interaction between tides. However, it is more relevant to the present problem to derive a set of field equations governing the mean circulation of the Lagrangian residual current. In this paper, a set of field equations for the tide-induced mean circulation is introduced. These equations govern the mean Lagrangian residual circulation and are the most appropriate for the description of inter-tidal processes. A long-term transport equation, namely, a convection-diffusion equation for the inter-tidal
processes describes the balance between convective and dispersive transports for the tidally averaged concentration of any conservative and passive tracer. As is well known, in the conventional long-term transport equation, the convection velocity is the Eulerian residual velocity and an assumption of "tidal dispersion" has to be made (Fischer ef al., 1979). As stated above, however, for the inter-tidal transport processes the convective transport should be characterized by the Lagrangian residual velocity. Furthermore, the physics and dynamics of the "tidal dispersion" are not well understood (Pritchard, 1954; Bowden, 1967; Fischer et af., 1979). A new inter-tidal transport equation has already been proposed, but it is a two-dimensional depthaveraged equation (Feng el af., 1986b). In this paper, we derive a three-dimensional inter-tidal transport equation which describes comedy the Lagrangian nature of convective transport without introducing the "tidal dispersion" terms.
In the last two sections of this paper, the Lagrangian dynamics of tracer spreading on long-term scales and a preliminary application of the proposed model to the Bohai Sea, China, are briefly described.
2. FORMULATION OF THE MODEL Based on a nonlinear three-dimensional tidal model (Feng, 1977), the generalized nondimensional dynamic problem proposed by Feng (1986a) is as follows : i) Field equations
v.u=o,
aU + K U . V U - fV = - az a aU ae ax a~ (" a ~ ) av + K U . V V + fU = - a +z-a(V z b) , ae ay aZ +
9
474
-as + + u . V S = e - (ak ~ ) as ;
aZ
ae
ii) Boundary conditions :
az ae ax + v -1, ay -au = - av = -as =o; aZ aZ aZ
az + K (u at z = K Z , w = and atz=-h,
u=O,
and
-as- - 0 ,
where
x = (x,y,z) =
aZ
:[
-1
, y' L'h,
'
e=to,
z = -Z* , ZC
a +e2 a + e3 a V =el ax
UC'K43L,
~,=KNL, = K N h, , f*
f=-,
0 V.
ay
aZ
9
475
Three nondimensional parameters, K, N, and E have been introduced. They are defined by ZC
K=-,
hC
N = 1 + (n-1)
K
,
(11)
In these equations, t* denotes time; (xI,yS,z+)form a Cartesian coordinate system on an fplane with corresponding unit vectors (el,e2,e3); (u+,v,,w*) are the velocity components in the (el,e2,e3)directions; Z, is the displacement of the free surface from the undisturbed sea surface; h, is the water depth measured from the undisturbed sea surface; g is the gravitational acceleration; f* is the Coriolis parameter, v* is the eddy viscosity; kr is the eddy diffusivity; S* is the concentration of any conservative and passive tracer; (k*,q&) are the Lagrangian displacements in the (el,e2,e3)directions of a marked water parcel at time t*, for which the initial condition is ((*,q&) = 0 when t, = to., and thus the position of the marked water parcel can be expressed as (x*,y*,z1) = (x,,.,yv,+) + (k*,q&); L and h, denote the horizontal and vertical scales, respectively; co-' is the time scale; the quantities with the subscript "c" indicate the characteristic value of the corresponding dimensional quantities; n is the number of tidal cycles over which the tidal current velocity is averaged to filter the periodic current velocity and to derive the residual velocity; KNis a measure of the relative scale of tidal excursion to L ; E ~ K is a measure of the relative effect on a concentration of the eddy diffusion compared to convection. Let us assume (Feng, 1986a) : O ( W = O(K)
(13)
o(K)C 1 O(dK) = o ( K ) .
(15)
These conditions imply weakly nonlinear dynamics : the nonlinearity due to the eddy viscosity and diffusion are excluded, and the transport mechanism is dominated by convection. Given the scale of tidal excursion, 5, = K N L, and conditions (13) and (14). the velocity of a marked water parcel can be expanded in a Taylor series expansion about q,or
476
where the notation ( )o indicates that the term is evaluated at xg. The displacement of the marked water parcel can be expressed by e N 5 = ju (xg + K N 5 , e‘) do’ eo where €lo = w
to
A perturbation technique with the small parameter K has been used to derive and solve the j*-order model (i = 0, 1, ...) of the dynamic problem proposed, (1)-(9) (Feng, 1977 and 1986a). In the present paper, we suppose that the solution for the tidal currents, and in particular the Eulerian residual currents and the Lagrangian displacements of marked water parcels, have been obtained. Thus, the basis to calculate the Lagrangian residual velocity and the concentration of any conservative and passive tracer has been laid. For clarity, a nonlinear M2 tidal system is used, instead of a complicated tidal system including several astronomical tides and associated shallow water constituents, to examine the Lagrangian residual current. It &.natural to select the circular frequency of M2 as the characteristic circular frequency, and thus the nondimensional circular frequency and period of Mz are 1 and 27t respectively. The zero-th order model, or M2 tide, can be expressed; in a nondimensional form, as ~0 = cose + sine , where the superscripts ’ and ” indicate the harmonic coefficients. As is well known, the first order model contains the M4 tide and the (first order) Eulerian residual velocity. The second order model contains the % tide and other harmonics of frequency equal to that of the M2 tide. Here, the harmonics of the order. of O(d) 0’ = 3,4, ...) are not considered.
4
<
3. LAGRANGIAN RESIDUAL CURRENT INDUCED BY AN M2-TIDAL SYSTEM
Let us use angular brackets < > to denote a tidal cycle average of any quantity A, or e, + 2m 1 = A do. where n = 1.2, ... . The Eulerian residual velocity, UER, defined as the 2xn eo Eulerian mean velocity (i.e., the velocity averaged at a fixed spatial point, say q.over n tidal cycles) is expressed as UER = < u (q, 8) >. The Lagrangian mean velocity, UL. of a marked water parcel released from the point xg at time 0, and moving over the n tidal cycles is defined
1
471
= .However, as pointed out in section 1. the Lagrangian residual velocity ULR cannot be unconditionally defined as the Lagrangian mean velocity UL unless UL satisfies the continuity equation for an incompressible fluid. In fact, in the proposed weakly nonlinear model, assumption (13) ensures that UL satisfies the continuity equation for an incompressible fluid. We shall prove that point at the end of this section. Thus, here the Lagrangian residual velocity can be expressed as uL,or as
UL
00
+ 2m
The substitution of (16) and (17) into (18) yields the Lagrangian residual velocity induced by an M2-tidal system, to be correct to the second order harmonics and normalized by urC = KU,, as follows ULR
=U
m
+ K ULD
(19)
where ULM
= UER + USD
(20)
and uLD = ut-,
coseo + uLD sineo
(21)
u;L) = u; . v u m - u m . vu; Urn=- u; . v u m + u m . vu; The 6rst order Lagrangian residual velocity expressed as the sum of the Eulerian residual velocity uER and the Stokes' drift velocity USD = < N b . (Vuo)~> 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 >
+-<(--
1 2
+-<-
a250
duo
a d y (v -)aZ
22110
+a d y (v
avo
+ 50
a2
duo
v ->
aZ
~ U O
a2
avo
a d y ( V -aZ- ) + 1 O a Z a Y ( V ~ ) >
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
- q - ( ua2 o
az2
,vo)>, 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
(30)
where Fo has been approximately expressed by
D0
rather than the Eulerian residual velocity. This further confirms that the mean Lagrangian residual circulation is appropriate to the description of inter-tidal flow processes. In fact, equation (30) is also valid when F is any conservative quantity in the flow field. The set of field equations governing the mass-transport velocity, (23)-(28),can in principle be solved if the forcing function (x1,x2), i.e., the tide-induced body force, is given and if the eddy viscosity v = v(x) is assumed to be known. A numerical model to solve the set of equations (23)-(28) was proposed by Song (1986). If the sea water is assumed to be an inviscid fluid, the tide-induced body force (x1,x2) is reduced to an irrotational force and the set of equations (23)-(28) describes some geostrophic motions. This implies that the careful and correct selection of the eddy viscosity, v (x), is of much importance for the numerical simulation of the mean Lagrangian residual circulation in coastal seas. 5. INTER-TIDAL TRANSPORT EQUATION The convection-diffusion equation for the tidal cycle average of the concentration of any conservative and passive tracer is also called the long-term transport equation, or the inter-tidal transport equation (Feng et al., 1986b). It can be derived by means of equation (5). Substitut~ + ~ ( dinto) equation (51,the zero-th order equation is aso I ae = 0, ing s = so + K S +
481
which says that the tidal cycle average of the concentration, <S>, can be approximately evaluated by So. The first order equation is as, / a0 + uo . VSo = 0, and the second order equation is as follows as2 ae +u,.
VSI +
where E = E /
K?,
~ 1 .VSo=
E-
a as0 aZ (k-)aZ
and, in view of (15), we have O(E) = 1.
Substituting the first order equation into equation (31) and using the zero-th order equation, a tidal cycle average of equation (31) yields the inter-tidal transport equation, or Urn.
a (6 a<s> V <S> = 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
<
0). the energy decreases rapidly, and falls to
zero when Ri = 0,4.
-2
Fig._ 1
-1
0
- Non-dimensional
1
2
Ri
turbulent kinetic energy
492
2.4 The numerical solution 2.4.1 Use of curvilinear coordinates In order to avoid evolution of the calculation field because of the free surface variations, a curvilinear coordinate
z*
is used to get a transformed
domain independant of time (fig 2) : where
*
-s
- ZF)
/ (S
-
-
z
S
is a horizontal reference level
S
(x, y, t)
zF
(x,y)
z
(x, y) vertical coordinate
(2
ZF)
water surface level
bed level
The equations written with the new coordinate are of the same form (see Benque et Al. 1982).
2 '
tz
TRANSFORMED DOMAIN
DOMAIN
Fig. 2
-
Curvilinear coordinates
2.4.2 The boundary conditions a
- Th_e-c!.!Efe"ts
- open boundaries Generally, at open sea boundaries, the mass-currents integrated.over depth can be obtained from field measurements or from a two-dimensional computation at a larger scale.
To deduce a velocity profile at the boundaries, the following Ekman type The open boundaries must be
relations are integrated (Benque et Al. 1982).
located in areas with regular bottom in order to allow to neglect horizontal gradients
a U =a at
az
av= a a t
a 2
I
ap
aZ
P
a x
($Za)-fu-
1
ap
aZ
P
a~
(3z?)+fv-
--
493
qZ : turbulent viscosity
where
f : coriolis parameter
- coastal boundaries Velocity components are zero or a slip condition is taken.
- free surface
. without wind, we assume : aU a,
-=
. with
0,
av az
-
-
0 and w = 0
-
wind, a surface shear tress T is generated, depending on the W wind speed and related to the flow velocity by the equation :
aii ”-=
-T w
az
-
sea bed
Two types of conditions have been tested :
. the velocity is zero. This condition imposes refined mesh grid near the
bottom, in order to have a good description of the gradients.
. a slip velocity condition is imposed by assuming that shear stress is a
quadratic function of the bed velocity (see CoEffe et al. 1982).
-
b The-te!?ee:ttu:e At the open boundaries or at the outlet, when the fluid flows into the domain the temperature repartition is imposed for the advection step, whereas when it flows out no condition is needed for this step (free outing condition). For the diffusion step the variation of temperature along the normal is set to zero at all the boundaries
.
2.4.3 The numerical scheme The governing equations are solved using a finite differences scheme based on a splitting of operators, with a rectilinear computational grid in the direction x, y, z*. A
completely
implicit solution of
the
equations would
lead
to a
prohibitive computation time. On the other hand, a completely explicit solution has some restriction upon the time step. It can be shown (CoEffe et Al. 1982) that the most restrictive conditions affect the determination of the free-surface wave and the vertical diffusion. Consequently these two steps are solved implicitly, the others being treated explicitly. The advection terms are solved by the characteristics method, and the vertical diffusion is treated with a double-sweep algorithm.
494
The pressure term and the continuity equation are solved together. The remaining equations averaged over the depth lead to 2D equations which relate the sea surface elevation and the two components of the fluxes. They are treated with a 2D iterative method using an alternating direction operator with coordination. Then the horizontal velocity profiles are modified with the new sea surface gradient and the vertical velocity is calculated by integrating the local continuity equation. 3
- APPLICATIONS Two applications were performed to test the validity of the assumptions and
to evaluate the predictive capability of the model ODYSSEE. The first chosen case is a quasi-two dimensional experimental flume with a stratified flow, for which extensive measurements were available (Gartrell 1979). The turbulence modelling could be discussed regarding the comparison of the measured temperature and velocity profiles and the numerical results.
In the second application, the dilution of the heated water plume of the nuclear power station of Gravelines was simulated and the results were compared with field surveys to test the reliability of the model. 3.1 Stratified flows in a flume 3.1.1 Description of the flume (fig 3 )
The data were taken from measurements made in a 40.m recirculating flume, with an average flow depth of 45 cm. Two pumps and a temperature control system allowed the production of a two-layered density-stratified flow. The lower layer had an initial depth of 30 cm while the upper layer had an initial depth of 15 cm. Profiles of the two velocity components and of the temperature were made at various points along the flume centerline.
3.1.2 Tests with smooth bottom Two similar tests were investigated ; only the initial conditions at the entrance of the model differed :
I l
mean velocity cmls upper layer U 1 lower layer U2 U 1 test 1 test 2
1
I
1;;;
I
’”
9,o
-
U2
mean temperature upper layer T1
1 i:iI(
34,l 30,9
I
I
OC
I
AT =
lower layer T2 T1 32,4 30.2
I
-
T2
1,7
0 ~ 7
495 In these two cases, the bottom of the physical model was smooth
so
that
little turbulence was generated from the bottom. A slip condition was imposed at the bottom and on the sides of the flume. The experimental temperature and velocity profiles were taken at the upstream boundary of the numerical flume. The computed results (fig 4 ) were very close to the measurements in the test 1 whereas some discrepancies could be noticed in the test 2 : although the shape of the velocity profile was somewhat preserved, it is staggered compared to measurements and the temperature gradient is too strong at the interface. The observed differences between measured and computed results in the second test could be explained by considering the equation of kinetic al - aii g z/T turbulent energy. In the test 2 , the Richardson number Ri was relatively large because of turbulent kinetic energy
I aiil
-
az( was small ; thus the local production
(that is
-
u 7 a
neglected in the turbulence model. In fact the shape of the velocity profile was not modelled badly because the dominant term in the Reynolds equation was the pressure gradient term, while the transport equation for T had no analogue for a heat source :
For these reasons, the term 3 T ”
/ az was modelled poorly in the test
2 ;
this induced a bad mixing of temperature at the interface of the two layers. On the contrary, in the test 1 the term of local production of kinetic energy prevailed with respect to the transport and the diffusion terms because the velocity and the temperature gradients are large at the interface. Therefore the turbulent mixing of temperature was well reproduced in that case.
-
note : The shape of the velocity profile was satisfying in the two tests but the value of the integrated velocity profile was smaller than the measured one. This was due to the fact that the experimental model was not really bidimensional because of friction on the flume sides. This induced greater velocities along the centerline in the physical model. 3.1.3 Effect of changes of the bottom roughness
This experiment was different from the others in the way that it was 4.5 m from rough isothermal and the roughness of the bed changed at x
-
496
(Darcy-Weisbach friction factor f
=
0,05) to smooth (f
= 0,017).
Thus, one was
able to measure the reaction of the model to sudden changes of the bed roughness. Figure 5 shows that the model behaved well in the first part of the smooth section but the agreement was not
so
good after x
-
2 0 meters. This was due to
the following reason : in the experiment, turbulent bursts generated in the roughened section continued to propagate in the smooth section, continuing the turbulent mixing. In the model, the "memory" of the various high turbulence levels was maintained only in the smooth section in the gradient of iT
. Hence,
the model reduced - u T too rapidly after the change in bed roughness and the mixing did not persist enough. 3 . 1 . 4 Summary of the results Though some small discrepancies could be displayed when the advection and
the diffusion terms in the turbulence model were not negligible, ODYSSEE did fairly well.
So
it was applied to a real case which is described in the
following paragraph. 3 . 2 Dilution of a heated water Plume under the action of tidal currents 3 . 2 . 1 Description The model ODYSSEE was used to
evaluate the temperature field
near
the coastal nuclear power plant of Gravelines located in the North of France, near Dunkirk. The power station is built on a sand mound encroaching upon the sea. To minimize recirculation between the intake and the outlet, cool waters are withdrawn inside the outer harbour of Dunkirk while heated waters are discharged on the shore out of the harbour by mean of a free surface channel (fig 6 ) . For the 2 units of 900 MWe considered in this applicatfon, the mean value of the discharge is 120 m3/s ; it induces velocities of 0 . 2 0 m/s at high water and 1 . 5 m/s at low water approximatively, at the end'of the outlet channel. For these conditions. the heated temperature elevation is 7OC. During mean spring tide, the tidal amplitude reaches 5 meters and the velocities exceed 1 m/s. The drift during a tidal period goes eastward and its value fluctuates between 2 km and 3 km. The bottom slope is gentle and the mean water depth (under lowest tide level) does not exceed 13 meters in the studied area except in the navigation channel of the harbour where it reaches 2 0 meters. 3.2.2
The numerical model
To test the predictive capability of the model, the thermal dilution of the discharged flume was computed. The studied domain covered an area of
497 5. x 3. km2 including the outlet works while the intake was not modelled ; only the mouth of the harbour was represented (fig 6). The computation grid had 32 640 nodes, the mesh-size was 75 meters long horizontaly and varied vertically, from 0.10 meter at the bottom and the surface, to 2 . 0 meters at mid-depth. The depth-integrated flows imposed at the open boundaries and at the harbour entrance were provided by interpolation of the results of a larger two-dimensional numerical model for mean spring tide conditions. Moreover the discharged flow imposed at the channel end is balanced by an additional flow distribution along corresponding to the intake withdrawal.
the
harbour
mouth
At the open boundaries, the temperature was zero when the water flowed into the domain and was derived from the advection step when the water flowed out. This means that we assumed that water which went out of the domain and came back after the slack period had a negligible temperature. The reliability of this assumption depended of course on the domain dimensions and a compromise had to be done here to avoid a prohibitive number of computational nodes. 3 . 2 . 3 The results
During the flood period
(fig 7a)
the outlet discharge was advected
eastwards against the harbour dyke and a strong dilution was observed at the work end because of the mixing in front of the harbour mouth (effects of the currents acceleration and the bottom irregularities).
A horizontal thermal
front could be noticed on the western limit of the plume. Moreover a vertical stratification due to the buoyancy effects could be noticed. When the tide was lowering (fig 7b). the current flowing out of the harbour deviated the plume northwards. During ebb-tide (fig 7c) the current flowed eastwards and the heated plume was transported along the coast. The plume was still stratified but the dilution in the ambient flow was not as effective as during the flood period : at mid ebb-tide the heat area exceeding 6 O C at the surface was around 0 . 2 2 km2 while it was only 0.09 km2 at mid flood. First the computed thermal impact at the water surface was compared with infrared teledetection (surface aerial thermographies). This comparison was in fact qualitative because the power of the station was not the same during the teledetection survey and in the computed case (6 MWe instead of 2 W e ) . The thermal front observed by teledetection (fig 6) is well reproduced by the
model
and
the
predicted
distribution of
temperature
is
similar.
Nerverthelesss the figure 9 illustrating the numerical and the measured distribution of temperature at mid flow shows that the model underestimated the dilution in the strong velocity shears occuring at the harbour pass, and that the computed plume was attached on the dyke while it was detached in
498
nature. That means that the modelling of the horizontal mixing is too simple when the velocity shear is strong. The vertical profiles of temperature were compared for the same conditions of outlet discharge. The figure 10 representing the profiles at various moments shows that the model reproduced correctly the vertical stratification. The measured temperature was generally higher than the computed one because it includes the far field heating, but the shapes are similar.
4
- CONCLUSIONS The three dimensional model ODYSSEE was succesfully tested in two cases
where stratification phenomena were important. It was shown that the modelling of horizontal mixing has to be improved in some particular situations where velocity shears are strong. Nevertheless the model ODYSSEE is a very interesting tool for the prediction of thermal impact. It can also be of use for other type of studies like dilution of all kind of pollutant or evaluation of sediment movement involving three-dimensional effects, and the knowledge of the
three-dimensional circulation is also
often needed
for simulating
ecological processes. 5
- ACKNOWLEDGEMENT The authors are very much indebted to the people who developed the model,
P. Dewagenaere. J.C.
Soliva and M.C.
Burg, under the supervision of J.P.
Benque. They appreciated the work done in cooperation with G. Gartrell, during a 9 months stay at the LNH. 6
- REFERENCES
Benque J.P.. Hauguel A. and Viollet P . L . , 1982. Engineering Application of Computational Hydraulics. Volume 2. Pitman. CoEffe Y., Warluzel A. and Burg M.C., 1982. Three-dimensional Numerical model for tidal and wind generated flow. Coastal Engineering, Cape Town. Darras M., Allen H., Cavelier D. and Caudron L. 1984. Synthese 'de mesures de temperature dans le rejet chaud d'une centrale en bord de mer. Rapport 1.5. 18e journees de l'hydraulique, Marseille, SociCtC hydraulique de France. Dewagenaere P.. 1979. Thesis, University Pierre et Marie Curie, Paris VI. Gartrell G. , 1979. Report KH-R-39 Keck Hydraulics Lab. California Institute of Technology, Pasadena CA. Gartrell G. and Pearson H . J . , 1982. "On the scaling of the vertical diffusivity in stably stratified flows". Submitted for publication. Gartrell G., 1983. Modelisation d'ecoulements stratifies B l'aide du modile ODYSSEE Rapport de stage LNH. Launder B.E., 1975. Journal of Fluid Mechanic 67. pp 569-581 Launder B.E.. 1972. Heat and mass transport topics in applied physics, 1 2 Springer-Verlag.
499
-
0.15
0.30
-
38.6 m
.Fig. 3
-
The experimental flume and the mesh g r i d
TEST 1
cm
I
32.0
J
0.0
4.0
u
12.0
16.0 cm/s
34.0
30.0
31 .O
30.5
f
x=24m
**
"C
f
cm/s
8.0
I
33.0
Computed Measured F i g . 4 : Tests i n a flume w i t h smooth bottom
OC
500
Rough
- -
Smooth bed
cm
40. 30.
20. 10.
U
0 o:
-
.*m..*.
0.0
16.0
U
cm/s
Computed Measured
F i g 5 : Test i n a flume w i t h v a r y i n g bottom roughness. V e l o c i t y p r o f i l e s i n t h e smooth p a r t .
16.0
cm/s
501
Fig. 6
-
The model lay-out for the thermal study of Gravelines nuclear power plant
502
a
-
mid f l o o d
b
-
s l a c k water
c
-
b e g i n n i n g of ebb t i d e
d
-
mid ebb t i d e
S t r a t i f i c a t i o n __
e
Fig. 7
- v e r t i c a l s e c t i o n s of t h e plume ( f l o o d p e r i o d ) - The c a l c u l a t e d plume e v o l u t i o n d u r i n g t i d a l c y c l e
503
@ IGN EOF 1986
a 9
1 =>25" 2 ~22-25" 3 =21-22"
4.20-21"
5 = 19-20' 6 = 18-19" 7.17-18" 8 =16-17" 9. < 16"
Fig. 8 - Observed temperature distribution drawn from an infrared teledetection of the plume HW + 3 h (to compare with fig. 7-b).
Fig. 9
- Temperature
elevation at the surface - m i d flood
504
measurements 0
2
4
6
*C
t
X
Fig.10-Temperature
profiles over the depth
505
ESTIMATION OF STORM-GENERATED CURRENTS
N . S. HEAPS and 3. E. JONES Institute of Oceanographic Merseyside, U.K.
Sciences,
Bidston
Observatory,
Birkenhead,
ABSTRACT Some r e l a t i v e l y s i m p l e c a l c u l a t i o n s are proposed f o r e s t i m a t i n g t h e v e r t i c a l s t r u c t u r e of storm-generated c u r r e n t s a t any p a r t i c u l a r l o c a t i o n on a c o n t i n e n t a l shelf. The water is supposed t o be homogeneous. H o r i z o n t a l gradients of storm-surge elevation from a two-dimensional v e r t i c a l l y - i n t e g r a t e d n u m e r i c a l model are i n t r o d u c e d i n t o a one-dimensional model f o r t h e v e r t i c a l c u r r e n t s t r u c t u r e a t t h e l o c a t i o n , a l o n g w i t h wind stress and a t m o s p h e r i c p r e s s u r e g r a d i e n t s . The c h a n g i n g d i s t r i b u t i o n of c u r r e n t s t h r o u g h t h e d e p t h is d e t e r m i n e d as t h e dynamic r e s p o n s e t o t h o s e forces.
1
INTRODUCTION I t is a f a c t of e x p e r i e n c e t h a t t h e c h a n g i n g d i s t r i b u t i o n of t h e e l e v a t i o n
s u r f a c e due t o s t o r m s u r g e s c a n b e w e l l p r e d i c t e d u s i n g two-dimensional v e r t i c a l l y - i n t e g r a t e d n u m e r i c a l models (Heaps 19831. T h i s
of
sea
the
assumes t h a t good meteorological forecasts, p e r t a i n i n g t o t h e s u r g e g e n e r a t i o n , are a v a i l a b l e . Thus, t h e e l e v a t i o n f i e l d may b e o b t a i n e d s a t i s f a c t o r i l y without studies this
by
Proudman
conclusion.
considering the (19541,
vertical
d i s t r i b u t i o n of c u r r e n t :
Roed (1979) and J o h n s e t a1 (1983) a l l reached
Specifically,
the
i n v e s t i g a t i o n by J o h n s and co-workers
found l i t t l e d i f f e r e n c e between t h e performance of a complex three-dimensional model and t h a t of a much s i m p l e r two-dimensional v e r t i c a l l y - i n t e g r a t e d model i n s i m u l a t i n g s t o r m s u r g e s i n t h e Bay of Bengal. Given t h a t a model is suitably
designed
and
properly
adjusted with
an
appropriate level
of
is two-dimensional o r three-dimensional
f r i c t i o n a l dissipation,
whether
appears to b e elevations.
i r r e l e v a n t t o the s a t i s f a c t o r y reproduction of surge
Taking gradients
the
of
largely
it
above facts i n t o a c c o u n t , t h e p r e s e n t p a p e r employs h o r i z o n t a l elevation
derived
from
a well-tested
vertically-integrated
n u m e r i c a l s t o m - s u r g e model t o d r i v e a one-dimensional model f o r t h e v e r t i c a l d i s t r i b u t i o n of s u r g e c u r r e n t a t any p a r t i c u l a r l o c a t i o n ( f i g u r e 1 ) . Wind
stress
and
atmospheric pressure
gradients
at
t h a t l o c a t i o n , d e r i v e d from
506
ID
SEA BED
Fig. 1 . Combination of a one-dimensional (1D) model t h r o u g h t h e v e r t i c a l , sea s u r f a c e t o sea bed, and a two-dimensional (2D) v e r t i c a l l y - i n t e g r a t e d model i n the horizontal. are also i n v o l v e d i n t h e d r i v i n g . The t h r e e - d i m e n s i o n a l problem of d e t e r m i n i n g c u r r e n t s t r u c t u r e t h r o u g h t h e v e r t i c a l is t h u s r e s o l v e d i n t o separate two-dimensional and one-dimensional c o m p u t a t i o n s , t h e f o r m e r
meteorology,
p r e c e d i n g t h e latter. Similar earlier
work on
the
determination
of
the
distribution
of
storm-surge c u r r e n t s through t h e v e r t i c a l a t a p o i n t , n o t a b l y t h a t done by F o r r i s t a l l (1974, 1 9 8 0 ) , c o n n e c t s t h e one-dimensional and two-dimensional models o f sea
bed
figure in
the
1 through bottom f r i c t i o n . The l a t t e r is e v a l u a t e d a t t h e one-dimensional model and i s e x p r e s s e d i n terms o f c e r t a i n
c o n v o l u t i o n i n t e g r a l s i n v o l v i n g time h i s t o r i e s of t h e wind stress and t h e storm-surge g r a d i e n t a t t h e p o i n t l o c a t i o n . S u r g e e l e v a t i o n . g r a d i e n t s from the
two-dimensional model a g a i n d r i v e t h e one-dimensional model t o o b t a i n t h e
changing
vertical
structure
of
current,
but
their
derivation
in
the
two-dimensional scheme i s based o n a d i f f i c u l t i n t e g r o - d i f f e r e n t i a l e q u a t i o n ( a r i s i n g from t h e t h r e e - d i m e n s i o n a l form of bottom f r i c t i o n assumed) which
t o a c o m p l i c a t e d a n a l y s i s . I n t h e p r e s e n t p a p e r i t is s u g g e s t e d t h a t t o t h e t h r e e - d i m e n s i o n a l i t y of t h e problem i n d e t e r m i n i n g s u r g e - e l e v a t i o n g r a d i e n t s i s l a r g e l y u n n e c e s s a r y , a s s e r t i n g rather t h a t a w e l l - a d j u s t e d two-dimensional v e r t i c a l l y - i n t e g r a t e d model of c o n v e n t i o n a l leads such
recourse
d e s i g n is s u f f i c i e n t t o p r o v i d e them. a s s e r t i o n is c o n s i d e r a b l e .
The s i m p l i f i c a t i o n r e s u l t i n g f r o m t h i s
507 2
BASIC EQUATIONS
The hydrodynamical e q u a t i o n s of m o t i o n are t a k e n i n t h e form
where
Boundary c o n d i t i o n s are, a t t h e sea s u r f a c e ,
a n d , a t t h e sea bed,
The a b o v e f o r m u l a t i o n a n d s u b s e q u e n t a n a l y s i s f o l l o w s t h a t o f Heaps ( 1 9 7 4 ) and t h e n o t a t i o n is
t
time C a r t e s i a n c o o r d i n a t e s f o r m i n g a l e f t - h a n d e d s e t i n which
X,Y,Z
x , y are measured i n t h e h o r i z o n t a l p l a n e o f t h e u n d i s t u r b e d
sea s u r f a c e and z is d e p t h below t h a t s u r f a c e , h
u n d i s t u r b e d d e p t h o f water,
5
e l e v a t i o n o f t h e water s u r f a c e , components o f c u r r e n t a t d e p t h z i n t h e d i r e c t i o n s o f
U9-J
i n c r e a s i n g x, y r e s p e c t i v e l y , a t m o s p h e r i c p r e s s u r e o n t h e sea s u r f a c e
Pa
components of wind stress o n t h e sea s u r f a c e i n t h e x , y
F,G
directions, P
c o e f f i c i e n t of v e r t i c a l e d d y v i s c o s i t y ,
k
c o e f f i c i e n t of f r i c t i o n i n a l i n e a r law of bottom stress,
P
d e n s i t y of t h e water, assumed c o n s t a n t (homogeneous sea)
Y
g e o s t r o p h i c c o e f f i c i e n t , also assumed c o n s t a n t , a c c e l e r a t i o n of t h e E a r t h ' s g r a v i t y .
g
Suffix
0
indicates
e v a l u a t i o n a t t h e s u r f a c e 2-0 and s u f f i x h e v a l u a t i o n a t
t h e sea bed z = h . By
a v e r t i c a l i n t e g r a t i o n , Heaps ( 1 9 7 4 ) showed t h a t e q u a t i o n s ( 1 ) and ( 2 ) ,
s u b j e c t t o c o n d i t i o n s ( 4 ) and ( 5 ) may b e t r a n s f o r m e d i n t o :
508
3 + irur
2+
-
yvr =
xrvr + yur =
-
ga P
F +-
(6)
- gar Q
G +Ph
(7)
r
Ph
f o r r = 1 , 2 , 3 ,...., where
k
ur =
0
Xr
The
,
fhufrdz
k
v
r (r=1,2,3, ...I
Jhvfrdz
,
ar =
0
fhfrdz 0
are t h e a s c e n d i n g e i g e n v a l u e s and t h e f r ( r = 1 , 2 , 3 ,
...)
t h e corresponding eigenfunctions through depth s a t i s f y i n g
f
= 1
(12)
Thus, s o l u t i o n s of ( 9 ) - ( 1 2 ) y i e l d
X =
xr , f
( r = 1,2,3,
= f
....
(13)
)
Eddy v i s c o s i t y p and f r i c t i o n a l c o e f f i c i e n t k are assumed t o b e i n d e p e n d e n t
of the
t
time
the
p=p(x,y,z),
and
hence
so
also are
k = k ( x , y ) and c o n s e q u e n t l y
positional
dependency,
depth
range
which
enables u
xr
therefore,
xr
and f r .
I n g e n e r a l , we have
xr(x,y), fr=fr(x,y,z).
A p a r t from
f r is a f u n c t i o n o f z d e f i n e d i n t h e
The . f u n c t i o n s f r form a n o r t h o g o n a l s e t i n t h a t r a n g e
Oszsh. and
v
t o b e e x p r e s s e d i n t e r m s of u r and vF i n t h e series
forms: m
u =
OD
1 Qrurfr
r=l
,
v =
z
Qrvrfr r=l
(14)
where
u and v may e a c h b e e x p r e s s e d , a t any time, as a a s e t o f c u r r e n t modes t h r o u g h t h e v e r t i c a l , namely f r ( z ) , r=1,2,3 S i n c e ur and v v a r y w i t h time, so also d o e s t h e p r o p o r t i o n of e a c h mode i n t h e t o t a l sum g i v i n g e i t h e r u o r v . C o n s i d e r i n g motion a t any p a r t i c u l a r p o s i t i o n , .the method of t h e p r e s e n t work is t o s o l v e (61, (7) t h r o u g h time f o r u r and v r w i t h F, G , P , Q as p r e s c r i b e d f o r c i n g f u n c t i o n s v a r y i n g w i t h time. The L a p l a c e t r a n s f o r m a t i o n Equation
linear
( 1 4 ) shows
combination
of
... .
that
509
method is employed t o do t h i s . Then ( 1 4 ) is u s e d t o deduce t h e v a r i a t i o n s i n h o r i z o n t a l c u r r e n t through t h e depth a t the position. Eddy v i s c o s i t y p is purposes
of
conditions
the
chosen
(see s e c t i o n
to
be
i n d e p e n d e n t of t h e d e p t h z f o r t h e
a n a l y s i s ; i t i s r e l a t e d however t o wind and t i d e
immediate 4).
T h i s c h o i c e of i n v a r i a b i l i t y through d e p t h
a b a s e - l i n e s o l u t i o n of e q u a t i o n s ( 9 ) - ( 1 2 ) . I t may b e j u s t i f i e d i f is m a i n l y d i r e c t e d towards t h e v e r t i c a l d i s t r i b u t i o n of c u r r e n t i n t h e main body of t h e v e r t i c a l water column, away from t h e s u r f a c e and bottom boundary l a y e r s . I n any case, f u r t h e r e l a b o r a t i o n t o a c c o u n t f o r a depth-dependent p is s p e c u l a t i v e a t t h e p r e s e n t time. I n t h e above c i r c u m s t a n c e s of a d e p t h dependent p, Heaps (1974) found provides
attention
A
= pa2/h2
f
= cosa r 5‘
a
= sinar/ar
(16)
(5
z/h)
(17) (18)
ar
= 2 / ( 1 + ar cosa )
(19)
where a
( r = 1,2,3
...)
(20)
d e n o t e t h e a s c e n d i n g non-negative roots of t h e e q u a t i o n (21 1
a t a n a = c ( c = kh/p)
and
F u r t h e r , t a k i n g t h e series i n ( 1 4 ) t o s a y M t e r m s ( a p r a c t i c a l n e c e s s i t y ) a d d i n g a c o r r e c t i o n term t o allow f o r t h e t r u n c a t i o n , l e a d s t o working
f o r m u l a e:
M
where
RM
(5)
(35‘
-
65 + 2 ) n2
M - 1 cosrnE r2 r=l
510 3
VERTICAL DISTRIBUTION OF CURRENT AT ANY PARTICULAR LOCATION
i s assumed t h a t F, G , P , Q are p r e s c r i b e d a t r e g u l a r i n t e r v a l s 6 t , from
It
time t = O , a t t h e p o s i t i o n where t h e v e r t i c a l d i s t r i b u t i o n of estimated. Wind stress components F, G and atmospheric pressure gradients spa/ ax, apa/ay come from m e t e o r o l o g i c a l a n a l y s e s w h i l e r e s i d u a l sea l e v e l g r a d i e n t s ar;/ax, a
some o r i g i n current
of
to
is
be
vertically-integrated
v a l u e s of 5 and pa t o appr oximate t h e s p a t i a l d e r i v a t i v e s (see V al ues are denot ed by:
grid-point section 5 ) .
F = F . , G = Gj, J
(j
Over
= 0,1,2,3
-
ime, we have:
Fj
indicates
2
t h e t y p e of v a r i a t i o n c o n s i d e r e d .
[jst,
+
Fj+l)
Average components of
( j + l ) 6 t l are: G. = J(Gj + Gj+l)
9
(26)
J
n
J ( Fj
I
}
Fj
wind stress o v e r n
(25)
- Gj J+1 (P. - P. J+1 J (a. - Q j ) (-t j 6 t ) / 6 t J+1
J P = P. + J 0 = Q. + J
Figure
J
time i n t e r v a l [ jst, ( j + l ) 6t1, assuming a l i n e a r v a r i a t i o n of
each v a r i a b l e w i t h
F = F. + (Fj+l J G I G. + (G.
Q = Q.; at t = j 6 t
P = P
,.... ) .is
typical
a
The P and Q are e v a l u a t e d from ( 3 ) u s i n g model
model.
(27)
j
and t h e a s s o c i a t e d a v e r a g e r e s u l t a n t wind stress is t h e r e f o r e
2,
=
On
($ j
the
(28)
+ B j 2 ) J
basis
of
(see s e c t i o n
p=p. J
coordinate,
t h e eddy v i s c o s i t y p f o r t h e i n t e r v a l may b e e s t i m a t e d : j' 4). It is assumed t h a t p is i n d e p e n d e n t of t h e depth
rem ai ns
i n v a r i a n t o v e r each i n t e r v a l , b u t g e n e r a l l y changes from
one i n t e r v a l t o t h e n e x t a c c o r d i n g t o t h e ch ange i n wind stress. Taking Laplace t r a n s f o r m s i n t h e e q u a t i o n s ( 6 ) and ( 7 1 , d e f i n i n g (Jaeger 1961) t h e t r a n s f o r m of a v a r i a b l e y ( T ) as co
(29)
./ e-PTy(T)dT ,
y(p) =
0
yields p'r
-
u
Pr'
-
vo + r .
+
Arcr -
Arvr
er =
-garP + F/ph
(30)
+ yiir = - g a p + F / p h
(31 1
511
(a)
F
Fig. 2. Diagrams i l l u s t r a t i n g t h e v a r i a t i o n i n t e r v a l [ j s t , [ j + l ) s t land ( b ) f o r a l l t.
Here, T
of F w i t h time t, ( a ) o v e r
is t h e time v a r i a b l e w h i l e u," and v o d en o t e t h e i n i t i a l (T=O) v a l u e s
o f ur and vr r e s p e c t i v e l y .
Using (30) and ( 3 1 ) t o s o l v e f o r
cr,
Fr g i v e s :
Laplace t r an s f o r m i n v e r s i o n f o r t h e i n t e r v a l [ j & t , ( j + l ) s t ] w i t h T = t-j6t may b e performed o n ( 3 2 ) and (33) u s i n g s t a n d a r d methods.
(26) t h a t
(34) I n t h i s , n o t e from
512
The result o f t h e i n v e r s i o n is -Arl ur = (u>osyr + v > i n y i ) e -Arl -A 1 + V.rSt(y ye cosy? Are sinyr) J -Arl -Arl + U.Gt(Ar Are cosy1 + ye sinyi)
-
-
-
J
+ (U.
J+1
v
-
J
-A
-
J
-
v + ve
-x
1
'cosyi 1
~ . ~ t ( yye
cosy1
cosy1
+ ye
-
-
-
V.)(Ari J
Ke
Uj)(yi
(36)
-
sinyi)
-Arl
sinyr)
Are
-
v + ve
-K
+ Ke
1
cosyr
-
-A Ke
1
sinyr)
-Arl
-AFT
(Uj+l
siny-r)
-x r 1
-A
+ (Vj+l
-
-Arl
- u>inyi)e
+ v . G ~ ( A ~ Are J
-
U.)(Ari
= (VOcosyi
-
-Arl
-AFT
cosyr + Ve
siny?)
where
with
ur
and
vr given by ( 3 6 ) and (371, t h e v e r t i c a l s t r u c t u r e of h o r i z o n t a l
a t t h e p o s i t i o n under c o n s i d e r a t i o n , f o r jrStSt<(j+l)rSt, is determined(see e q u a t i o n s ( 1 6 ) - ( 2 4 ) ) from
current
513 M
i n which
J where
...
a ( r = 1,2,3, M) d e n o t e , i n a s c e n d i n g o r d e r , t h e f i r s t M non-negative roots o f
a tan a = c in
As
( c = kh/p.)
(42) c may b e t a k e n as a known c o n s t a n t ,
J
previous
work
(Heaps
(41)
19741,
i m p l y i n g k = c p . / h ( v a r i a b l e w i t h p.). Then ar, f r , ar, or are unchanging b u t J J ir d i f f e r s from o n e i n t e r v a l t o t h e n e x t because of i t s dependence on p.. J
Alternatively, constant,
c v a r i e s between i n t e r v a l s , as would o b t a i n w i t h k a known
if
r , f r , ar, Or
t h e n so a l s o would
l e a d i n g c h a n g e s through time i n
t h e s t r u c t u r e of t h e v e r t i c a l modes. Currents
u,
[j&.,(j+l)6t] the one
Initially,
(39)
interval
the and j+l
depth
are e v a l u a t e d
z
over each i n t e r v a l
( 4 0 ) w i t h ur and v r from ( 3 6 ) and ( 3 7 ) .
The
follow d i r e c t l y a f t e r t h o s e f o r i n t e r v a l j .
c o m p u t a t i o n s advance i n a series o f s u c c e s s i v e s t e p s through time,
corresponding vr f o r
through
using
for
calculations Thus
v
respectively interval
it
are
to j = 0,1,2,3 taken
.... .
The f i n i s h i n g v a l u e s of up,
as t h e s t a r t i n g v a l u e s ,u:
is c o n v e n i e n t t o t a k e u:
= v:
v:for
t h e next.
= 0 corresponding to a state of
z e r o i n i t i a l motion.
4
EDDY VISCOSITY
by
Using a r e s u l t due t o Svensson (19791, and p u t t i n g t o g e t h e r arguments g i v e n Csanady (19761 and Heaps (19841, f o r wind-driven f l o w d u r i n g t h e i n t e r v a l
[ j & , ( j + l ) s t I we take
P kw where bw = 0.065hu,
= O.O26u,'/y
(43) hShw h>hw
514
and h
W
= 0.4u,/y
(45)
;
uI is t h e f r i c t i o n v e l o c i t y g i v e n by (46) h
and E
. I
t h e a v e r a g e wind stress o v e r t h e i n t e r v a l , d e f i n e d by ( 2 8 ) .
Turbulence
associated
with
the
tides
frictional
stress, Sb s a y , a t t h e sea bed.
associated
eddy
is generated
by
the
a c t i o n of
It is t h e r e f o r e s u p p o s e d t h a t t h e
v i s c o s i t y , pT s a y , may b e e s t i m a t e d from e q u a t i o n s
44)-(46)
r e p l a c i n g u,, by u , , ~where Urb
(47)
(8,/P)*
=
Here, 3b r e p r e s e n t s t h e bottom stress S a v e r a g e d o v e r a t i d a l c y c l e b a q u a d r a t i c law: Sb = Kp(u
'
i n which
K
m
Assume (48)
+ v '1
m
a f r i c t i o n a l c o e f f i c i e n t a n d u m , vm depth-mean components o f
is
t i d a l c u r r e n t f o r t h e mean t i d e g i v e n by u
m
= acoswt,
(49)
a , b are t h e semi-major and semi-minor a x e s of t h e t i d a l c u r r e n t e l l i p s e
where and
vm = b s i n w t
w
is t h e t i d a l f r e q u e n c y .
Then i n t e g r a t i o n o v e r a t i d a l c y c l e l e a d s t o
the result: h
Sb = t K p ( a ' + 0' )
(50)
Hence
u , , ~ = (K/2I3 (a' + b')'
(51
and s u b s t i t u t i o n i n ( 4 4 1 , ( 4 5 ) y i e l d s pT = 0.065h(K/2)
4 (a'
+b"*
= 0.013K (a' + b 2 ) / y
h6h
T h>hT
where hT 0 . 4 ( K / 2 ) 4 ( a 2 + b ' ) t /y Typically, K = 0.0026
i n which case pT = 0 . 0 0 2 3 4 h ( a 2 + b ' ) *
= 0.338 x
(a2 + b')/y
where hT = 0.0144 (a' + b') t /y
hSht h>hT
515 The form f o r pT, h
Using
obtained
( 4 4 ) and (551, f o r combined wind and t i d e we
from
take pJ. = pT f o r pT>pw
= 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, <-2, 5-1,
0.
C1,
C2, C3
< - v a l u e s a t t h e c e n t r e s of t h e g r i d boxes l o c a t e d s y m m e t r i c a l l y on
denote t h e either
two-dimensional n u m e r i c a l model ( f i g u r e 3 ) .
model, a t n e i g h b o u r i n g g r i d p o i n t s may b e employed t o compute aC/ax
side
of
standard
i n t h e x-direction.
0
These v a l u e s are marked i n f i g u r e 3.
f o r m u l a e f o r n u m e r i c a l d i f f e r e n t i a t i o n ( W h i t t a k e r and Robinson
1944; Comrie 1949, p.544) g i v e , g o i n g t o f i r s t o r d e r d i f f e r e n c e s ,
o r going t o t h i r d order,
or, to f i f t h order,
Here,
Ax
constant.
denotes
the
grid
spacing i n
t h e x-direction,
supposed t o b e a
S u f f i x 0 i n t h e above f o r m u l a e i n d i c a t e s e v a l u a t i o n a t g r i d p o i n t
0. The same e x p r e s s i o n s may b e u s e d i n t h e y - d i r e c t i o n t o g i v e ( 3 5
hu
replaced
associated
by with
Ay
-
the
grid
spacing i n
that
direction.
values
with
Values of 6
a p p r o p r i a t e n e i g h b o u r i n g g r i d p o i n t s , l y i n g s y m m e t r i c a l l y on
are t h e n employed. same g r i d p o i n t s e n a b l e s ( a p a / a x ) o and
e i t h e r s i d e of p o i n t 0 i n t h e y - d i r e c t i o n , Knowing
/%),,
of
p
at
the
(apa/ay)o t o be evaluated s i m i l a r l y . A l t e r n a t i v e l y , f o r convenience, P and Q may b e d e t e r m i n e d from
516
a
52
53
a
a
a
a
*AX* Fig. 3. G r i d p o i n t s a t w h i c h 5 is r e q u i r e d f o r t h e e v a l u a t i o n of ar;/ax and a r / a y a t p o i n t 0. V a l u e s of i n t h e x - d i r e c t i o n t h r o u g h 0 are marked. Each p o i n t l i e s a t t h e c e n t r e of a g r i d s q u a r e .
where
5
Then,
g r i d - p o i n t v a l u e s of
denotes
t h e h y d r o s t a t i c e l e v a t i o n a s s o c i a t e d w i t h t h e atmospheric
<-% may
be used t o o b t a i n t h e r e q u i r e d h o r i z o n t a l
g r a d i e n t s a t 0 f o r t h e e v a l u a t i o n of P a n d Q t h e r e .
617
Fig. 4. (0s).
6
Geographical
p o s i t i o n s of Stations G1, S t a t f j o r d ( S F ) and Oseberg
APPLICATIONS OF THE THEORY The
theory
of
currents measured of t h e North Sea.
this
paper
has
been
a p p l i e d t o p r e d i c t storm-generated
a t t h r e e s t a t i o n s l o c a t e d n e a r t h e s h e l f edge t o t h e n o r t h The s t a t i o n s , marked i n f i g u r e 4 , are: G1 (61°30.7'N,
0°2.5'E)
St a t f j o r d
SF
(6I019.54'N, I055.12'E)
Oseberg
0s
(60°33.9'N,
2O47.7'E)
meter p o s i t i o n s t h r o u g h t h e v e r t i c a l a t e a c h s t a t i o n are Thus a t G1 ( t o t a l d e p t h l g l m ) , r e c o r d s come f r o m 41m below t h e sea s u r f a c e and 25m above t h e sea bed; a t SF ( t o t a l d e p t h 15Om), from 30, 7Om below t h e s u r f a c e and 3m above t h e bed; and a t 0s ( t o t a l d e p t h 102m), f r o m 2 , 1 2 , 25, 50m below t h e s u r f a c e and 3 m above t h e bed. The
current
i l l u s t r a t e d i n f i g u r e 5.
meter, r e s i d u a l s were e x t r a c t e d from t h e raw d a t a by removal of t h e t i d a l s i g n a l , t h i s b e i n g d e t e r m i n e d c n t h e b a s i s o f a f u l l harmonic t i d a l However t h e s a m p l i n g i n t e r v a l o f t h e raw d a t a v a r i e d from meter to analysis. For
every
meter. the
top
A t G1 measurements were h a l f h o u r l y ; a t Oseberg t h e s e were h o u r l y f o r three
meters and e v e r y 1 0 m i n u t e s f o r t h e o t h e r s .
The S t a t f j o r d
518 STATFJORD
OSEBERG
150 m
102111
-
2. -
---- TOTAL DEPTH SEA SURFACE
10
50
I3
25 10
&
t
BED
Fig. 5. C u r r e n t meter p o s i t i o n s t h r o u g h t h e v e r t i c a l a t S t a t i o n s G1, S t a t f j o r d and Oseberg d u r i n g J a n u a r y 1983. D i s t a n c e s t h r o u g h t h e v e r t i c a l are marked i n metres.
measurements were
filter
and
hourly.
For convenience, by a combination of a low p a s s
c u b i c s p l i n e i n t e r p o l a t i o n t h e e x t r a c t e d r e s i d u a l s were converted
t o h o u r l y v a l u e s a t i n t e g e r clock hours.
a t G1
were p a r t of t h e C o n t i n e n t a l S l o p e Experiment (CONSLEX) d u r i n g t h e w i n t e r of 1982-3. The Oseberg data (raw measurements + r e s i d u a l s ) were p r o v i d e d by Norsk Hydro. Det Norske Veritas s u p p l i e d t h e S t a t f j o r d d a t a which was c o l l e c t e d u n d e r t h e a u s p i c e s of t h e Norwegian Meteorological I n s t i t u t e . Comparisons between t h e o r y and o b s e r v a t i o n have been made t h r o u g h o u t t h e The measurements
whole of J a n u a r y 1983.
Thus f o r each meter, h o u r l y time series of e a s t - g o i n g
519 TABLE 1
Parametric v a l u e s used i n a p p l y i n g t h e t h e o r y
Parameter
Units
h
G1
m -1
Y
191 1 .278xl 0-4 9.81 10.0 5.0 1 26.56 37.11 1.025 30 2
9 -
g
-1
a
ms- 1 ms hr
b
6t Ax AY
km
m g c100-3
P M C
and north-going
(u)
corresponding
of
range
periods
150 1 .278x1 0-4 9.81 4.5 4.5 1 26.56 37.11 1.025 30 2
102 1 . 2 7 0 ~ 10-4 9.81 8.1 3.6 1 26.56 37.11 1.025 30 2
( v ) c u r r e n t components have been p l o t t e d d i r e c t l y a g a i n s t
values
f r o m t h e theory.
obtained
Power s p e c t r a i n v o l v i n g t h e
e l u c i d a t e d p r o p e r t i e s o f coherence and phase spanning a
series have
various
0s
SF
from
the
Nyquist
frequency
o f two hours, t o beyond t h e
i n e r t i a l p e r i o d (approximately 14 h o u r s ) . relentless
A
feature
of
January
the
1983,
pattern
of
strong
weather
in
the
making
that
winds from a w e s t e r l y q u a r t e r was a key
area
area and
t o t h e n o r t h o f t h e North Sea d u r i n g period
suitable
f o r a s t u d y of
storm-induced c u r r e n t s . Parametric in
Table
evaluated In
At
each
s t a t i o n , c u r r e n t components u , v through depth were
a t h o u r l y i n t e r v a l s (6t 1 hour) u s i n g t h e e x p r e s s i o n s ( 3 6 ) - ( 4 2 ) . w e took c = 2 as i n p r e v i o u s work (Heaps 1974) and M=30 ( t h i s
latter,
the
being
v a l u e s used i n o u r p r e s e n t a p p l i c a t i o n of t h e t h e o r y are l i s t e d
1.
an
o v e r - p r e s c r i p t i o n o f t h e number o f nodes, t o e n s u r e accuracy w i t h i n
the l i m i t s of the theory). For and
were
each
Q,
obtained
(CXX) of
runs
w i t h a two-dimensional
v e r t i c a l l y - i n t e g r a t e d model
T h i s model was developed by Dr. R. A. F l a t h e r a t t h e I n s t i t u t e of
Oceanographic
Sciences,
Bidston.
1/2O l o n g i t u d e ( f i g u r e 6 ) .
effectively that
from
To t h i s end, h o u r l y s u r g e e l e v a t i o n s
t h e North-West European c o n t i n e n t a l s h e l f which extended beyond t h e
s h e l f edge. by
t h e computations of c u r r e n t r e q u i r e d h o u r l y v a l u e s of P
station
-[see e q u a t i o n s ( 3 ) and (6111.
gave h o u r l y C a t t h e g r i d p o i n t s of a r e c t a n g u l a r network such as
i n f i g u r e 3, w i t h
Meteorological coincidental
The model had a g r i d s i z e o f 1/3O l a t i t u d e
I n t h e v i c i n i t y of s t a t i o n s G 1 , SF and 0.3 t h i s
data points
Ax = 26.561an and Ay = 3 7 . 1 l h as e n t e r e d i n Table 1 .
yielded and
3-hourly v a l u e s of h y d r o s t a t i c e l e v a t i o n
times;
at
t h e s e were converted t o h o u r l y v a l u e s by a
520
O' E
I O"E
Fig. 6. SF, 0.5.
Two-dimensional model CXX, showing g r i d network and t h e p o s i t i o n s G,,
linear
interpolation.
station
Then, h o u r l y v a l u e s of P and Q were e v a l u a t e d a t each
using d i f f e r e n c e formula ( 5 9 ) w i t h each
of t h e e q u a t i o n r e p l a c e d by t h e a p p r o p r i a t e Also
the
a t e a c h s t a t i o n t h e c o m p u t a t i o n s of c u r r e n t r e q u i r e d h o u r l y v a l u e s o f
wind
provided
stress 3-hourly
interpolation. interval
from
term o n t h e r i g h t hand s i d e
(5-k).
the
Semi-major
to
components values,
F,
G.[see e q u a t i o n (2111.
which were
Meteorological d a t a
c o n v e r t e d t o h o u r l y v a l u e s by l i n e a r
Eddy v i s c o s i t y p ( d e p t h uniform and c h a n g i n g from one h o u r l y t h e n e x t ) , also r e q u i r e d i n computing t h e c u r r e n t s , was deduced
hourly and
wind stress components employing t h e f o r m u l a e of s e c t i o n 4 . semi-minor
a x e s of t h e t i d a l c u r r e n t e l l i p s e a t e a c h s t a t i o n
521
G (dyn cm-')
2,
-
Am
v
I
'cL/-w
v-
2
0 -2
2
0 -2
0 -2
2
0 L
- 2 ' 1
3
5
7
9
II
13
15'
171
19
'21
.
123
.
.. . .. 25'
.
27
.
.
29
.
.
'31
DAYS (JANUARY 1983)
Fig. 7. 1983.
V a r i a t i o n s of F, G,
were i n v o l v e d
a r / a x , a c / a y , P and Q a t G1 t h r o u g h o u t J a n u a r y
i n t h i s d e d u c t i o n b u t t i d a l effects o n p were i n s i g n i f i c a n t i n
t h e p r e s e n t c a l c u l a t i o n s ( a , b small, see T a b l e 1 ) .
7 PRESENTATION OF RESULTS The the
forcing
f u n c t i o n s of wind stress F, G and g r a d i e n t P, Q used t o d r i v e
model a t t h e p o i n t Gl are d e p i c t e d i n figure 7.
I t may b e n o t e d t h a t t h e
6th-10th J a n u a r y was a p e r i o d of h i g h wind stress f o l l o w e d by a q u i e t e r p e r i o d d u r i n g t h e m i d d l e of t h e month.
S u b s e q u e n t l y t h e 18th-22nd J a n u a r y was p e r i o d
of f a i r l y h i g h regular g r a d i e n t forcing. Throughout t h e whole month t h e as /ibc and P
and
Q
curves.
Thus
by
as / a y
values closely follow the
e q u a t i o n s ( 6 1 ) and (62) we may deduce t h a t t h e
40 20 0
cm s-'
40 v4 I
20
0
I
"I66
I
I
' I
3
51
.
.
7
.
' 9
I . .
/
II
.
.
13
.
. 15'
17
19
'21
DAYS (JANUARY 1983 v - c u r r e n t s a t G I , a t 41 and 166
23
25'
27
29
'31
Fig. 8. T h e o r e t i c a l p r e d i c t i o n s of u- and metres below t h e sea s u r f a c e , compared w i t h model t h e c o r r e s p o n d i n g c u r r e n t s measured a t those d e p t h s : f o r t h e whole of J a n u a r y 1983; - o b s e r v a t i o n , prediction. For t h e two five-day p e r i o d s d e l i n e a t e d , v e r t i c a l p r o f i l e s of c u r r e n t are drawn i n f i g u r e s ( 1 2 a ) and (12b).
-----
DAYS (JANUARY 1983 1
Fig. 9. (-----)
Components of depth-mean c u r r e n t u , v a t G , as d e r i v e d from t h e one-dimensional model through t h e v e r t i c a l and t h e two-dimensional model i n themhor'Pzontai ().
523
VI
W N
524
(01
Ue (model)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.
p.OS
I
I
I
24
0.)
0.1s
0 .1
0.I
0.16
.
I
14 (HOURS)
.
I
14 (HOURS)
0.4
cQh
(b)
24
0.I6
U ,. (observed )
0.46
0.6
525
( c l V,, (model)
V,, (observed 1
(d )
T
I
I
I
I
I I
I I
‘
I
I
I
I I I
I 1
1 . .
e.br .
I 24
I
..............................................
,
I
O!I
1 14 (HOURS)
0.L
o!,
0.L
o!.
0.:.
o!.
0.h
o!.
c ph
F i g . 10. Power s p e c t r a c o m p a r i s o n s f o r t h e e a s t - d i r e c t e d c u r r e n t s [ ( a ) model and ( b ) o b s e r v e d ] and n o r t h - d i r e c t e d c u r r e n t s [ ( c ) model and ( d ) o b s e r v e d ] f o r t h e t o p meter a t G,.
526 contribution
to
the
is
a
f o r c i n g from t h e a t m o s p h e r i c p r e s s u r e g r a d i e n t s must b e
small. Figure
8
predicted the
two
by
the
sets
of
or
period
comparison between t h e o b s e r v e d c u r r e n t s a t G I w i t h t h o s e T h e r e i s a n o b v i o u s m e a s u r e o f c o r r e l a t i o n between
model.
d a t a , p a r t i c u l a r l y so f o r o s c i l l a t i o n s w i t h a n e a r - d i u r n a l
greater.
However
there
are h i g h e r f r e q u e n c y components i n t h e I n b o t h cases c u r r e n t a m p l i t u d e s a t
o b s e r v a t i o n s n o t r e p r o d u c e d i n t h e model.
t h e t o p meter p o s i t i o n are greater t h a n a t t h e lower meter.
9
Figure
t h a t t h e o v e r a l l depth-mean c u r r e n t r e s p o n s e of t h e
illustrates
one-dimensional
is
model
calculated
by
frictional
dissipation
the
i n v e r y good a g r e e m e n t w i t h t h e depth-mean c u r r e n t
driving
two-dimensional
model.
i n b o t h models i s c o m p a r a b l e .
This indicates that the T h e r e are however some
o s c i l l a t i o n s i n t h e o n e - d i m e n s i o n a l model due t o t h e s t a r t - u p of t h e model from rest. T h i s g e n e r a t e s some t r a n s i e n t i n e r t i a l o s c i l l a t i o n s . initial The
power
noted
in
spectra
figures
displayed
i n f i g u r e 10 h i g h l i g h t t h e f e a t u r e s a l r e a d y
9.
T h e r e is a d i s t i n c t d i u r n a l peak i n b o t h t h e
and
8
and model s p e c t r a which are p r o b a b l y due t o c o n t i n e n t a l s h e l f waves.
observed
Cross s p e c t r a between t h e u- a n d v-components o f c u r r e n t y i e l d a 90° p h a s e l a g f o r b o t h o b s e r v e d and model d a t a , as m i g h t b e e x p e c t e d f o r s h e l f waves. Also the
of
amplitudes
reflect
general
the
the
u-
and
properties
v-components are a p p r o x i m a t e l y e q u a l and i n of
shelf
waves as h a v e been o b s e r v e d a t a
n e a r b y s t a t i o n ( S t e v e n s o n 6 I 0 3 0 ' N , OOO'E). The model s p e c t r a , l O ( a ) a n d I O ( c ) , show a d i s t i n c t peak a t 1 4 h o u r s due t o inertial The
oscillations
spectra
triggered
observations,
from
a
by
t h e i n i t i a l s h o c k a p p l i e d t o t h e model.
1 0 ( b ) a n d l O ( d 1 , show no s u c h p e a k , however
they
do
exhibit
Part
of
t h i s may b e t r a c e a b l e t o t h e m e t e o r o l o g i c a l i n p u t t o t h e model which
h i g h f r e q u e n c y c o n t r i b u t i o n a b s e n t i n t h e model s p e c t r a .
d e r i v e s from 3 - h o u r l y v a l u e s , i e less f r e q u e n t t h a n t h e h o u r l y o b s e r v a t i o n s .
To
compare
that
observed showed than
more
calculated and that the
c l o s e l y t h e observed c u r r e n t p r o f i l e through depth, with
by
the
model;
a cross c o r r e l a t i o n , between t h e r e s p e c t i v e
model s p e c t r a was performed.
a t G,
model,
F o r t h e l o w f r e q u e n c y regime t h i s
t h e o b s e r v a t i o n s e x h i b i t e d a greater s h e a r t h r o u g h d e p t h
for
b o t h t h e u- and v - c u r r e n t s ,
i e t h e model b e h a v i o u r was
somewhat more b a r o t r o p i c .
It is of i n t e r e s t t o compare t h e s u r f a c e wind s p e e d a n d t h e s u r f a c e c u r r e n t by t h e model. F i g u r e 11 shows t h a t a t G t h e maximum wind s p e e d
calculated just
exceeded
201n
s-'
1 and t h e c a l c u l a t e d s u r f a c e c u r r e n t r e a c h e d a maximum
i n e x c e s s of 40cm s - l . The r a t i o between s u r f a c e c u r r e n t and wind s p e e d h a s a mean of a b o u t 0.016 which i s low compared w i t h t h e u s u a l l y a c c e p t e d 0.03
just
or
3% v a l u e .
A s c a n b e s e e n t h e v a l u e of 0 . 0 3 is r e a c h e d o n l y i n f r e q u e n t l y .
Gmission of s u r f a c e boundary l a y e r e f f e c t s i e u s i n g a n e d d y v i s c o s i t y which is
527
SURFACE CURRENT (CMS-')
7
40
20
0
WIND SPEED (MS')
20
0
0.04 0.02
0
DAYS (JANUARY 1983)
Fig. 11. V a r i a t i o n s of s u r f a c e c u r r e n t , wind speed, t h e r a t i o of s u r f a c e c u r r e n t and wind s p e e d , hw and p . a t GI t h r o u g h o u t J a n u a r y 1983. J
528
c o n s t a n t through t h e v e r t i c a l , may b e t h e c a u s e of t h i s d i s c r e p a n c y . Figure
11
also d e p i c t s
g e n e r a l l y less
than
the
about
depth
of
f r i c t i o n a l i n f l u e n c e hw which is
w e l l w i t h i n t h e t o t a l d e p t h of 191111. The
lob,
c o e f f i c i e n t of eddy v i s c o s i t y p r e a c h e s v a l u e s of 3000 t o 4000 cm' s-l d u r i n g j
high
e p i s o d e s b u t d r o p s t o a b o u t 400 cmz s-l d u r i n g t h e q u i e t p e r i o d i n
wind
t h e m i d d l e of t h e month.
Figures
1 2 ( a ) and
through depth forcing
for
a
predominates.
12(b) provide
a comparison between c u r r e n t p r o f i l e s
p e r i o d of h i g h wind stress, and a p e r i o d when g r a d i e n t
to
Due
t h e p r e v a i l i n g w e s t e r l y wind d u r i n g J a n u a r y
east component of stress F h a s been s e l e c t e d as r e p r e s e n t a t i v e of t o t a l wind stress and h e a d s t h e c u r r e n t p r o f i l e s a t s i x - h o u r l y i n t e r v a l s . the
1983,
the
In
both
figures,
noted.
This
stress
current
is
a b a r o t r o p i c r e s p o n s e w i t h a 24-hour o s c i l l a t i o n may be
i n d i c a t i v e of s h e l f wave b e h a v i o u r .
O v e r a l l t h e h i g h wind
p r o f i l e s e x h i b i t a greater c u r r e n t a m p l i t u d e t h r o u g h o u t d e p t h
w i t h somewhat more s t r u c t u r e t h a n i n t h e lower wind case.
W e
can
now t u r n t o a comparison between model r e s u l t s and o b s e r v a t i o n s a t
t h e Oseberg p o s i t i o n , f i g u r e s 1 3 ( a ) and 1 3 ( b ) . Here, i n s h a l l o w e r water o b s e r v a t i o n s are a v a i l a b l e a t f i v e c u r r e n t meter p o s i t i o n s t h r o u g h depth. However d a t a from t h e t o p two meters are u n a v a i l a b l e from J a n u a r y 1 4 t h onwards there are also s h o r t b r e a k s i n t h e d a t a from t h e o t h e r three meters a b o u t
and
t h i s time.
In model
figure
1 3 ( a ) , considering t h e i n i t i a l 13 days, i t can be seen t h a t t h e
predictions
d i m i n i s h less r a p i d l y t h a n t h e o b s e r v a t i o n s a f t e r t h e peak
c u r r e n t s on t h e 7 t h J a n u a r y .
G e n e r a l l y t h e f i t between model and o b s e r v a t i o n s
i s p o o r e r t h a n f o r G, and once a g a i n i n e r t i a l o s c i l l a t i o n s are c l e a r l y v i s i b l e t h e model o u t p u t n e a r t h e s t a r t of computations. There also a p p e a r s t o be a mean e a s t e r l y c u r r e n t m i s s i n g from t h e model p r e d i c t i o n s f o r t h e bottom two meters. I t is p o s s i b l e t h a t t h e c a u s e of t h e d i f f e r e n c e i n t h e mean c u r r e n t s i s due t o t h e ' s l o p e c u r r e n t ' which c a n be t r a c e d i n many o b s e r v a t i o n s a l o n g t h e shelf edge and may b e d r i v e n by d e n s i t y effects n o t i n c l u d e d i n t h e s e models. However i n t h i s c o n t e x t i t is also p e r h a p s w o r t h n o t i n g t h a t t h e
in
l o w e r two meters a t Oseberg were of a d i f f e r e n t d e s i g n (Aanderaa.RCM-4) t o t h e t o p t h r e e meters (SIMRAD UCM V I ) . In
t h e model
results,
current
a m p l i t u d e s d e c r e a s e from t h e sea s u r f a c e
downwards. I n t h e o b s e r v a t i o n s there i s a c o r r e s p o n d i n g d e c r e a s e f o r t h e t o p t h r e e meters b u t i n c r e a s e d c u r r e n t a m p l i t u d e s are e v i d e n t f o r t h e lower two meters. Spectral
analysis
shows a
clear peak around a p e r i o d of 50 h o u r s i n t h e
model d a t a , b u t t h i s i s n o t p a r t i c u l a r l y d i s c e r n a b l e i n t h e o b s e r v a t i o n s . The c o h e r e n c e between t h e o b s e r v e d and model s p e c t r a l is greater than 0.60 for p e r i o d s greater t h a n 50 h o u r s .
2.93
8.44
11.22
9.75
6.82
5.83
3.23
3 35
F (dyn cm+)
365
U
b
4
8
I
12
16
20
6
4
8
I
6 JANUARY
12
16
20
7 JANUARY
6
I
4
8
HOURS
V
506
12
0.41
16
8 JANUARY
20
f
12 16
14.18
044
4
8
12 9 JANUARY
13.41
15.16
16
20
t
2 93
4
0.58
8
12
F (dyn ern+)
1.96
16
20
HOURS
10 JANUARY
Fig. 12a. Vertical2 p r o f i l e s of u- and v - c u r r e n t a t G,, 6-10 J a n u a r y 1983, shown a t two-hourly i n t e _ n ( a l s . Westerly stresses, F dyn cm , a t s i x - h o u r l y i n t e r v a l s , head t h e v a r i o u s p r o f i l e s . U n i t s of c u r r e n t : 10cm s (horizontal). Depths ( v e r t i c a l ) normalized t o f r a c t i o n s of u n i t y .
VI
8
cn 0
m I0 0
n C
8
StlnOH
91
02
StmOH
?
I
02
91
AMVllNVP 61 21
e
C
I
02
91
21
e
c
AMVllNVP 81
?
I
h
n 21
e
AYVIINVP 22
I
f
0
I
I
02
91
21 AMVIINVP I2
e
t
0
I
02
91
21
AtlVllNVP 02
Vertical p r o f i l e s of u- and v-current a t G , 18-22 January 1983, shown a t two-hourly i n t e f y a l s . Westerly Fig. 12b. stresses, F dyn a t six-hourly i n t e r v a l s , head the various p r o f i l e s . U n i t s of c u r r e n t : 1Ocm S (horizontall. Depths ( v e r t i c a l ) normalized to f r a c t i o n s of u n i t y .
40 20
0
.
. I
.
. 3
.
. 5
.
. 7
.
. 9
.
. II
.
.
. 13
.
.
.
.
.
.
15 17 19 DAYS ( J A N U A R Y 1983 1
' 21
I
23
25
I
27
29
31
F i g . 13a. T h e o r e t i c a l p r e d i c t i o n s of u - c u r r e n t s a t QS, a t 2 , 12, 25, 50, 99 metres below t h e sea s u r f a c e , compared w i t h t h e c o r r e s p o n d i n g c u r r e n t s measured a t t h e s e d e p t h s : f o r t h e whole of J a n u a r y 1983; -o b s e r v a t i o n , ----model p r e d i c t i o n .
532
13b. T h e o r e t i c a l p r e d i c t i o n s of v - c u r r e n t s a t OS, a t 2 , 1 2 , 25, 50, 99 metres below t h e sea s u r f a c e , compared t h e c o r r e s p o n d i n g c u r r e n t s measured a t t h e s e d e p t h s ; f o r t h e whole o f J a n u a r y 1983; -o b s e r v a t i o n , ----model p r e d i c t i o n .
Fig. with
4
.crl
c,Q)
.rl
W a , l
L >
32 c,
o x mc,
U h
P
3 0
-
h l m l R l
E l
.rl
-0
0
Q)
0u
am
.cl
h h 3a,
v)v)
m o
-a,
mL o
Fig. 14a. T h e o r e t i c a l p r e d i c t i o n s of u - c u r r e n t s a t SF, a t 30, 70 and 1 4 7 metres below t h e sea s u r f a c e , compared w i t h t h e c o r r e s p o n d i n g c u r r e n t s measured a t t h o s e d e p t h s : for t h e whole of J a n u a r y 1983; -o b s e r v a t i o n , ----- model prediction.
533
cn w
534
DAYS (JANUARY 1983)
14b. Theoretical p r e d i c t i o n s of v - c u r r e n t s a t SF, a t 30, 70 and 1 4 7 metres below t h e sea s u r f a c e , compared w i t h corresponding c u r r e n t s measured a t t h o s e depths: f o r t h e whole of January 1983; -o b s e r v a t i o n , ----- model prediction.
Fig. the
I
1
1
3
1
1
5
1
1
7
1
9
1
1
1
II
Fig. 15. Components of depth-mean c u r r e n t one-dimensional model through t h e v e r t i c a l (-----)
1
1
13
u
1
1
1
1
1
15 17 19 DAYS ( JANUARY 1983 I
1
1
21
I
I
23
I
I
25
I
I
27
I
I
29
I
I
I
31
and v m a t ( a ) Oseberg and ( b ) S t a t f j o r d , as d e r i v e d from t h e and t h e two-dimensional model i n t h e h o r i z o n t a l (-).
m
Figure
13(b)
discrepancy
at
displays
generally
the
same
features,
two l o w e s t meter p o s i t i o n s .
the
again
Here t h e r e a p p e a r s t o b e a
s o u t h e r l y c u r r e n t m i s s i n g from t h e model p r e d i c t i o n s .
mean
t h e r e is a
Also i n t h e model
s p e c t r a t h e 50 h o u r peak a p p e a r s o n l y i n t h e t o p two p o s i t i o n s .
a cross s p e c t r a l c a l c u l a t i o n is c a r r i e d o u t a t O s e b e r g , t h e t o p t h r e e
When
meters e x h i b i t similar b e h a v i o u r as n o t e d a t G ,
o b s e r v a t i o n s show a greater s h e a r t h r o u g h d e p t h t h a n t h e model r e s u l t s i n
the both
is
F o r t h e low f r e q u e n c y regime
the
F o r t h e lowest two meters however, t h i s t r e n d
u- a n d v-components.
so clear.
not
The model and o b s e r v e d s h e a r f o r b o t h u- and v-components
a p p e a r comparable. C o n s i d e r i n g t h e S t a t f j o r d p o s i t i o n , f i g u r e 1 4 ( a ) shows t h a t t h e f i t between I n p a r t i c u l a r an e a s t e r l y
model and o b s e r v e d d a t a is a g a i n p o o r e r t h a n a t G,. current data,
a
to be
appears
from t h e model p r e d i c t i o n s .
absent
t h e c u r r e n t amplitudes diminish through depth.
I n b o t h sets of
Spectral analysis gives
i n d i c a t i o n of a 36 h o u r p e r i o d i n b o t h model a n d o b s e r v e d d a t a .
clear
spectral
coherence
The
t h e model a n d o b s e r v a t i o n s is greater t h a n a b o u t
between
0.75 f o r p e r i o d s l o n g e r than t h i s . Figure
displays
14(b)
similar
features,
h e r e , t h e r e seems t o b e j u s t a
small s o u t h e r l y c u r r e n t a b s e n t from t h e model d a t a .
S p e c t r a o f b o t h model and
o b s e r v a t i o n s still e x h i b i t a peak a r o u n d 36 h o u r s b u t i t is less clear. Statfjord
At
the
cross-spectral
calculation
reveals
that,
i n t h e low
f r e q u e n c y regime b o t h o b s e r v e d a n d model u - c u r r e n t s e x h i b i t a c o m p a r a b l e s h e a r through
However f o r t h e v - c u r r e n t s t h e o b s e r v e d s h e a r a p p e a r s t o b e
depth.
somewhat greater t h a n i n t h e model. Finally
figure
15
compares
depth-mean
currents
derived
from
the
one-dimensional model w i t h t h o s e from t h e d r i v i n g two-dimensional model a t ( a ) Oseberg
and
(b)
Again t h e r e i s good a g r e e m e n t a p a r t from some
Statfjord.
i n e r t i a l o s c i l l a t i o n s a t t h e b e g i n n i n g of t h e model r e c o r d . 8 CONCLUDING REMARKS The model,
work
presented
driven
two-dimensional
by
here
shows
meteorological
model
can
t h a t a r e l a t i v e l y s i m p l e one-dimensional forcing
and
s e a - s u r f a c e g r a d i e n t s from a
a n o v e r a l l r e a s o n a b l e r e p r e s e n t a t i o n of low
give
frequency c u r r e n t behaviour through depth a t any s e l e c t e d point. However near-surface
the
present
and
is u n r e a l i s t i c .
model
is
i n a p p l i c a b l e for c u r r e n t s t r u c t u r e i n t h e
near-bottom boundary l a y e r s where a c o n s t a n t e d d y v i s c o s i t y
N e v e r t h e l e s s as c a n b e s e e n i n t h e O s e b e r g a n d S t a t f j o r d d a t a
t h e r e i s s t i l l r e a s o n a b l e a g r e e m e n t a t a p o s i t i o n o n l y 3 m above t h e sea bed.
It
is clear f r o m t h e o b s e r v a t i o n s t h a t a t t h e l o c a t i o n s c h o s e n t h e c u r r e n t
537
t h r o u g h d e p t h were n e a r - b a r o t r o p i c .
profiles
Although t h e model d i d show t h e
s h e l f waves t h i s was n o t a test of t h e model u n d e r h i g h s h e a r
barotropic-type as m i g h t b e e x p e c t e d u n d e r v e r y s t r o n g wind c o n d i t i o n s .
O f course
conditions
t h e model i s i n a p p l i c a b l e t o s t r a t i f i e d c o n d i t i o n s when h i g h s h e a r s c a n o c c u r . Nevertheless
the
Cross-spectral underestimate
observations
calculations the
current
did
showed shear
some
show
that
the
current
present
structure.
model
tended
to
d e p t h , t h o u g h i n c e r t a i n cases t h e
through
o b s e r v e d and model s h e a r were comparable. The
depth-mean
two-dlmensional
current
models
comparison
shows
that
between
the
the
frictional
one-dimensional dissipation
and
is well
represented.
is
It
deal
t h a t a more complex one-dimensional model may b e needed t o some o r a l l of t h e l i m i t a t i o n s of t h e p r e s e n t scheme. However t h e
obvious
with
measure
of s u c c e s s so f a r a c h i e v e d i n d i c a t e s t h a t f o r c e r t a i n a p p l i c a t i o n s a t
least,
t h e c o m p l e x i t y of a f u l l t h r e e - d i m e n s i o n a l model f o r c u r r e n t s t r u c t u r e
may n o t b e n e c e s s a r y . ACKNOWLEDGEMENTS The
authors
are g r a t e f u l
to
Norsk
s u p p l y i n g d a t a from t h e Oseberg p o s i t i o n . Aune
of
Det
Norske
Meteorologiske
Hydro,
especially
L.
I. Eide f o r
W e are also v e r y g r a t e f u l t o D r . B.
Institutt
for
permission to use t h e i r
S t a t f j o r d d a t a and t o F. E. Dahl of Det Norske Veritas f o r s u p p l y i n g i t .
Dr. She
Judith
also
(N.S.H.)
Wolf
presented
k i n d l y p r o v i d e d d a t a from t h e CXX two-dimensional model. this
paper
at
Liege
on
b e h a l f of o n e of t h e a u t h o r s
and h e r h e l p i n a d v i s i n g t h e o t h e r a u t h o r (J.E.J.) on c o m p l e t i n g t h e
m a n u s c r i p t i s much a p p r e c i a t e d . Thanks programs,
are
also
due
t o Dr. J. M. Vassie f o r p r o v i d i n g s p e c t r a l a n a l y s i s
Mr. R. A. Smith f o r p r e p a r i n g and p h o t o g r a p h i n g t h e diagrams and to
Mrs. J. Huxley for t y p i n g t h e m a n u s c r i p t .
REFERENCES Bowden, K.F., 1953. Note o n wind d r i f t i n a c h a n n e l i n t h e p r e s e n c e o f t i d a l c u r r e n t s . Proc. A. Soc., A, 219: 426-446. Comrie, L.J., 1949. Chambers's S i x - F i g u r e Mathematical T a b l e s , Vol. 2: N a t u r a l Values. W. & R. Chambers Ltd.. Edinburnh. Csanady, G.T., 1976. Mean c i r c u l a t i o n i n . s h a l l o w seas. J. Geophys. Res., 81 : 5389-5399. F o r r i s t a l l , G.Z., 1974. Three-dimensional s t r u c t u r e of s t o r m - g e n e r a t e d c u r r e n t s . J. Geophys. Res., 79: 2721-2729. F o r r i s t a l l , G.Z., 1980. A t w o - l a y e r model f o r h u r r i c a n e - d r i v e n c u r r e n t s o n a n i r r e g u l a r g r i d . J . Phys. Oceanogr., 10: 1417-1438. Heaps, N.S., 1974. Development of a t h r e e - d i m e n s i o n a l model of t h e I r i s h Sea. Rapp. P.-v. Reun. Cons. perm. i n t . Explor. Mer., 167: 147-1 62.
538 Heaps, N.S., 1983. Storm s u r g e s , 1967-1982. Geophys. J . R . a s t r . SOC., 74: 331 -376. Heaps, N.S., 1984. Vertical s t r u c t u r e of c u r r e n t i n homogeneous and s t r a t i f i e d waters. pp. 153-207 i n , Hydrodynamics of Lakes, e d . K. H u t t e r , S p r i n g e r - V e r l a g , 341 pp. Jaeger, J . C . , 1961. An I n t r o d u c t i o n t o t h e L a p l a c e T r a n s f o r m a t i o n . Methuen & Co. L t d . , London. J o h n s , B . , S i n h a , P.C., Dube, S.K., Mohanty, U.C. and Rao, A . D . , 1983. S i m u l a t i o n of storm s u r g e s u s i n g a t h r e e - d i m e n s i o n a l n u m e r i c a l model: a n a p p l i c a t i o n t o t h e 1977 Andhra c y c l o n e . Q. J1. R . met. SOC., 109: 21 1-224. Proudman, J . , 1954. Note o n t h e d y n a m i c a l t h e o r y of s t o r m s u r g e s . Arch. Met. Geophys. Bioklim, A , 7: 344-351. Roed, L.P., 1979. Storm s u r g e s i n s t r a t i f i e d seas. T e l l u s , 31: 330-339. S v e n s s o n , V . , 1979. The s t r u c t u r e o f t h e t u r b u l e n t Ekman l a y e r . T e l l u s , 31: 340-350. W h i t t a k e r , E.T. and Robinson, G . , 1944. The C a l c u l u s of O b s e r v a t i o n s . B l a c k i e and Son L t d . , London.
539
A COUPLED 2-D/3-D MODELLING SYSTEM FOR COMPUTATION OF TIDAL CURRENTS
Ah?)
WIND-INDUCED
J.M. USSEGLIO-POLATERA AND P. SAUVAGET SOGREAH Consulting Engineers, BP 172 X, 38042 GRENOBLE Cedex (FRANCE)
ABSTRACT Although computer costs are rapidly decreasing, it is not yet common to carry out 3-D modelling all over large areas where detailed results are required. The authors present the development of a promising code allowing the definition of a 3-D block in any 2-D computational field. 3-D computation is limited to the area where it is absolutely necessary while the feed back of 3-D behaviour towards the outer area is accurately simulated. The algorithm based upon fractional steps and the solution provided for the coupling problems are described in detail. 1
INTRODUCTION
Mathematical modelling for coastal engineering or coastal environment problems displays typical feetures that may significantly influence the trend of code developments. Since most of the time water is very shallow and there is no significant stratification, two-dimensional (2-D) models are well adapted, but sometimes the conventional 2-D hypotheses are locally inadequate (sharp depth variations or strong wind forcing) and three-dimensional (3-D) models are recommended. Indeed, it is exactly where the flow field displays 3-D behaviour that reliable results are often required (sediment transport, ship manoeuvring). On the other hand, detailed results are needed locally (for instance. around new structures) and the grid size must be very fine (sometimes lower than 20 m) whereas the tidal excursions of water particles may be very long (several kilometers), hence the area involved in the flow computation may be very large. Consequently, a large number of computational points is needed despite the most advanced techniques of grid refinement (Chenin-Mordojovich et el, 1985). Advanced and reliable 3-D codes exist but up to now, only a few people are ready to pay for 3-D simulations on such a dense grid for zny local coastal engineering project. This is mainly a cost efficiency concern but the inveotigation of any method able to provide an effective, even partial solution to this problem, does involve a series of interesting theoritical problems.
u l 0 P
-
EQUAL DEPTH CURVES
FIG.l : SCHEMATIC EXAMPLE OF A REAL APPLICATION REQUIRING A COUPLED 2-D / 3-D MODELLING SYSTEM
541
Starting from a schematic but typical example, the authors will show the content and the first applications of a new code coupling 2-D and 3-D equations. 2
PRESENTATION OF THE PROBLEM USING A SCHEMATIC EXAMPLE Consider the schematic example shown on Fig.la. A new harbour is planned
in a coastal area. The shoreline is nearly linear, the bathymetry very uneven, (sudden depth but extensive shallows off-shore).
The main access to the harbour
is a natural channel crossing the shallows but the channel position and direction vary significantly in time due to major sedimentological changes. A significant tidal range induces alternating tidal currents parallel to the shoreline. It is assumed that the problem concerns simulation of the tidal flow in order to study the sedimentological processes in the channel area. 3-D modelling is particularly recommended because the velocities near the
channel vary considerably in direction and intensity throughout the water layer. Since the tidal range is wide, the tidal currents are strong. Thus, there is a large area involved in the generation of the flow field around the channel. 3-D modelling over the whole area is possible but it would be very expensive. On the other hand, in most of the area, 2-D modelling is adequate and even recommended near the shoreline because flooding and drying of tidal flats has not yet been reliably simulated in 3-D models (as far as we know). A solution with nested (imbedded) models would be possible : a 2-D model would give boundary conditions for a local 3-D model (fig.lb) provided a reasonable assumption for the vertical distribution of horizontal velocities on the 2-D/3-D boundary line. But the main drawback of the nested models is obvious : the simulation will provide no feed back of the internal information towards the surrounding area and if the inner structure changes significantly. for instance because of civil engineering works, feed back will be needed. Clearly, a coupling method is necessary.
3
A NEW COUPLED 2-D/3-D MODELLING SYSTEM A new modelling system has been developed in order to provide an effective
solution to the above-mentioned problem. This new code couples as far as possible 2-D and 3-D equations so that, in any given computational area, a 3-D block may be defined. Therefore, the computation is 3-D where it is absolutely necessary, and 2-D elsewhere. For the case of the above example, the boundaries of the 3-D block could be the same as the boundaries of the internal model.
542 3.1
Preamble For clarity,the local velocity and the depth-averaged velocity are first
defined. The local (horizontal) velocity which varies throughout the water layer is a typical 3-D modelling parameter. The depth averaged (horizontal) velocity is obtained by integrating the local velocity over the water layer. 3.2
Governing 2-D and 3-D equations Consider now both the 2-D and 3-D governing shallow water wave equations.
In the momentum equations, the different terms may be related to the various physical processes occurring simultaneously, namely advection, diffusion, wave propagation, wind forcing, Coriolis deviation and bed friction. In table 1 . these equations are presented using this split operator approach which is the basis of fractional step resolutions. Now consider the formulation of each of these processes.
For advection, horizontal momentum diffusion and wave propagation, integration over the water layer of the 3-D
formulation gives the 2-D
formulation. Therefore, 2-D and 3-D formulations are consistent. This is the same for Coriolis deviation although this is responsible for most of the typical 3-D behaviour when wind is involved. Owing to the assumption of hydrostatic pressure distribution, the vertical velocity remains very low and vertical advection is of very little importance with respect to horizontal advection. Wind forcing and bed friction are surface stresses. In two dimensions, their formulation depends on typical 2-D parameters: bed friction depends on the depth-averaged velocity according to Manning-Strickler formulation and wind effects directly influence the depth-averaged velocity as well. The 3-D formulation is very different because bed and wind stress can be held as actual surface stresses and their propagation throughout the water layer, is modelled through the formulation of vertical diffusion processes. 3.3
Coupling problems This
analysis
of
the
governing
equations
emphasizes
obvious
inconsistencies in the formulation inducing the main coupling problems, Consequently, it seems necessary to find a coupling algorithm displaying the following properties :
.
the physical processes having a similar formulation in 2-D and 3-0
must be isolated in order to use similar algorithms for these 2-D and 3-D operators ;
the algorithm must provide some solution to maintain the coherence in the results for the processes which are a priori inconsistent. In addition, at each time step, boundary conditions are needed for the local velocities on the 2-D/3-D boundary line where only the depth-averaged velocities are known ( 3.4
(I
-
coordinates are used in the 3-D block).
Algorithm of the coupled code The coupling algorithm is based on the splitting-up techniques developed
by Yanenko (1968). The governing equations are split into several fractional steps corresponding to the various physical processes involved. For each time increment. the various steps are solved successively using a numerical approach adapted to their mathematical properties. The theory of fractional step techniques shows that the governing equations are approximated to first-order accuracy in time. This technique is already applied in existing 2-D and 3-D codes developed by Sogreah Consulting Engineers and the French Electricity Board (EDF/LNH) (Benque et al, 1982, Burg et al, 1982). The coupling algorithm in which nine fractional steps are involved is presented first. The solution to the above-mentioned problem of surface stress representation will be described subsequently (3.5). The nine successive fractional steps are the following :
.
1st step (2-D advection): advection of the depth-averaged velocities
is computed using the 2-D characteristics method throughout the computational field except in the 3-D block. The spatial interpolation at the foot of the characteristics line is calculated using linear or cubic interpolation.
.
2nd step (horizontal 3-D advection): horizontal advection of the
local velocities is calculated using the same method of characteristics as mentioned for the 3-D block.
.
3rd step (vertical 3-D advection): vertical advection of the local
velocities is computed within the 3-D block, using a 1-D characteristic method, a vertical integration of the horizontal local velocities then yields the advected depth-averaged velocities in the 3-D block.
.
4th step (2-D diffusion): diffusion of the depth-averaged velocities
is performed throughout the computational area by splitting the operator in each horizontal direction and using the implicit double sweep method; the computation is not necessary within the 3-D block but simplifies the programming effort while involving a negligible increase in cost. Moreover, it allows for continuity across the 2-D/3-D boundary line since the
advected
depth-averaged velocities have been evaluated consistently in the 1st and 3rd steps.
VI P
TABLE 1. A/
2D SHALLOW WATER WAVE EQUATIONS SOLVED I N CYTHERE E S I USING THE: SPLIT-OPERATOR APPROACH
Bf
E= E I N G THE SPLIT-OPERATOR
APPROAM
HORIZONTAL
VERTICAL ADVECTION
1 with: -
g+gh,+-+-= aZ TSX P
1 -3:
+au + ax
av = aY
0
I
TBX P
-
(K
HORIZONTAL
&)-3 aY
(K &)- Fv = 0
aY
(mass conservation)
x, y , z and t = the independant space and time variables
DIFFUSION
U, V = the unit-width discharges through the depth i n the xand y- directions; C, 5 = the corresponding depth-averaged
au
velocities
at
u, v. w = the local velocities in the x-, respectively Z = the free surface elevation
Ig+wE=o
+
y- and z- directions
/ ' h = the flow depth
+ g
PROPAGATION
az,o
Rax
az
aU + &! +
= 0
*
= o (mass conservation) a Y az 7BX, T~~ = the bed friction shear stresses / 7sx, 7sy = the surface wind shear stresses / P = the density of the water F = the Coriolis acceleration parameter / g = the acceleration due to gravity
K , u z = the coefficients of horizontal and vertical diffusion
ax
545
.
5th stee (horizontal 3-D diffusion): horizontal diffusion of the
local velocities is computed within the 3-D block using the same implicit double sweep method; a vertical integration of the horizontal local velocities then yields the new depth-averaged velocities in the 3-D block.
.
6th step (vertical 3-D diffusion): vertical diffusion of the local
velocities is computed inside the 3-D block using the implicit double-sweep method. Up to now, turbulence effects have been modelled according to Prandtl's mixing length hypothesis and the mixing length 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 (Burg et al, 1982). A vertical integration of the new local velocities enables an evaluation of the momentum sink due to wind action, bed friction and turbulent diffusion to be made (see 3.5 for details).
.
7th step (free surface elevations):
the wave propagation step
consists in'an equation relating depth averaged velocities and surface elevations which are typical 2-D parameters. Accordingly, in the coupled code, throughout the computational field, free surface elevations are computed using a 2-D algorithm, applied to the equations of the 2-D propagation step of table 1: an iterative method based on a special variant of the conjugate gradient
approach is carried out through an alternating direction operator with co-ordination in space (BenquC et el. 1982). Inside the 3-D block, new friction coefficients are evaluated in order to obtain, with 2-D formulation of bed shear stress, the head loss computed in the previous step (see 3.5 for details).
Once surface elevations are known, depth averaged velocities can be
evaluated over the whole computational field.
.
8th step (boundary velocity profile):
at this time, the vertical
distribution of horizontal velocities on the 2-D/3-D boundary line is needed.
An Ekman-type relation is currently used (Burg et al, 1982). This relation is certainly not the most appropriate relation for all problems but it is particularly suitable when wind is involved, work is at present under way to improve this formulation.
.
9th step (local velocities after propagation): the horizontal local
velocities are corrected according to the variations of the free surface elevations. They do not need to be integrated since the actual depth averaged velocities have already been computed with the same numerical scheme as in the 7th step. Finally, the vertical local velocities are evaluated.
546
3.5
A solution to the problems of surface stress
(i) bed friction
--.
In 2-D, the momentum sink due to bed friction is expressed as a bed stress
related to the Chezy or Strickler head loss formulae, as follows (for tB' instance, with Chezy's approach) : 4
ZB
'Pg
i-Il-al -
ChZ hZ
= the density of the water,
where: p
= the acceleration due to gravity,
g 4
U
= the unit-width discharge vector,
IIFll
= the modulus of the unit-width discharge vector,
h
= the flow depth,
Ch
= Chezy's coefficient.
The friction head loss therefore only depends on the depth averaged velocity. In 3-D, a slip velocity condition is imposed on the sea bed, assuming that the shear stress is a quadratic function of the bed velocity (Burg et al., 1982) :
where : 4
u
= the bed velocity vector (no vertical component),
c
= a constant,
'2
k A2
= the eddy viscosity (or vertical diffusion coefficient). = the Von Karman constant, i.e.
-
0.41,
the distance between the bed and the point where the condition is applied, which must be within the logarithmic boundary layer.
547
In this case, the friction head loss depends on the local velocity and is transferred to the depth averaged velocity by mean of vertical turbulent diffusion. Clearly, the friction formulation in the two systems is very different and a relation between Ch, the Chezy coefficient, and c is not immediately apparent. Assuming unidimensional flow, a real relationship exists; consistency is excellent as far as steady flow is concerned but becomes much less so in the case of transient flows. In the coupled algorithm, the free surface elevations are calculated all over the computational field using the 2-D algorithm (7th step). However, the momentum sink due to bed friction is already knom in the 3-D block (6th step). Therefore, a new coefficient Ch
is evaluated for each computational point
within the 3-D block in order to obtain with the 2-D formulation the head loss computed by the 3-D
formulation. As things stand at present, Che is given by
the following equation : A
AU At
- I
'
4
u
I I A I I
IIUII Che2 h
(4)
where :
-
=
momentum sink computed in the vertical diffusion step,
=
depth-averaged velocity at the end of the 2-D diffusion step.
At 4
u
Studies are at present in progress to investigate a similar formulation for an implicit expression of bed friction. This formulation is. however, far more complex and considerable programming investments will be required. (ii)
wind
In order to allow for wind effects correctly, the 2-D bed friction formula has been somewhat modified in order t o take into account certain 3-D aspects. Kith Chezy's approach :
This formulation corresponds to ( 1 ) but an additional discharge, called "q",
has been introduced. "q"
depends on the Elanan depth, the wind
speed, Coriolis deviation, the water depth and on the current velocity. More details on this formulation are available (see Hamm et al. (1985)).
This
formulation eliminates some well-known drawbacks of the conventional 2-D formulation of wind effects: the poor representation of the vertical profile of horizontal velocities on the one hand, and the zero depth averaged velocities leading to zero bed friction, on the other hand. (This is the case when a steady wind is blowing over a flat bottom basin).
Consequently, good
consistency is obtained for wind effects. 4
APPLICATION EXAMPLE In order to illustrate the present state of these developments, a simple
but interesting test case is described below. The computational area is a rectangular basin closed on three sides, 3.800 m long and 1.100 m wide (Fig.2).
Its average depth is 4 m. An S-shaped channel
crosses the basin in its middle portion and the average water depth in this channel is 12 m. Chezy's friction coefficient is 40. The boundary condition on the open side is a significant tidal range (6 m). The computation was performed over half a tidal cycle, starting from high water (ebb). Three runs were performed. The first one with a 2-D code, CYTHERE ES1 (Benque et al., 1982). and the last one with the coupled 2-D/3-D code, CYTHERE 3-D.
The computational grid size was 100 m hence the number of
computational points in the horizontal plane is 468 (39 x 12). A 3-D block was defined covering an area of 1.600 x 900 m. The smallest distance between the 2-D/3-D boundary line and the channel is 300 m. The following results have been sketched : the depth averaged velocities obtained with the two codes (Fig.3) and the horizontal velocities in the 3-D block at various fractions of the depth (Fig.4 and Fig.5). It is clear that the coupling scheme introduces no singularity near the 2-D/3-D boundary lines. The particular flow field observed with the results of the 2-D/3-D calculation near the western entrance to the channel exists as well as with the results of the 2-D calculation. Accordingly, it emphasizes the efficiency of the 2-D code itself and the reliability of such a tool with regard to coupling purposes.
549
E 4 4 UPTH
I
TIDcy_ RPJUGE
wsm a 2-0
1.39 x
4.00 M
cwr*a U P T H
I
6.00 M
Bm FRICTD-4
32 (STRIMLER)
I
T I K STEP
FIG.2
:
12.00 M
mwls
CWUTATIN
3-0BLaX 100 M 120. SEC. (CFL > 10)
12
KFUCNTk GRID SIZE
I
I
17
x
10
x
18
I
APPLICATION EXAMPLE :
RECTANGULAR BASIN CLOSED ON
TmEE SIDES
Going into detail, discrepancies are apparent and further investigations have shown that this is because bed friction is slightly under-estimated in the channel in 2-D,
due to the significant depth variations (4 m versus 12 m).
Clearly, the feed back of the internal information to the surrounding area is satisfying. Finally, the coupled 2-D/3-D
code provides an interesting estimate of the
variations in direction and intensity of the horizontal velocities around and within the channel. These results are directly usable as far as sedimentation processes or ship manoeuvring problems are concerned. The cost efficiency of the coupled code has been evaluated, and is approximately proportional to the number of computational points, i.e. (b)
468 (- 39 x 12) 8 424 (- 39 x 12 x 18)
(c)
3 528 (= ( 1 7 x 10 x 18)
(a)
for full 2-D computation. for full 3-D computation,
+
(39 x 1 2 ) ) for the coupled code.
The ratio between (b) and (c) will be significantly higher in most real applications because in this schematic case the 2-D area is limited to the minimum necessary around the 3-D block.
5
CONCLUSIONS An original tool has been developed. The results obtained on regular
geometries and reasonably even bathymetries are very promising. Work is at present under way to test the code on real applications displaying complex natural layouts provided that the basic hypotheses are respected. 6
REFERENCES
Benque, J.P.,
Cunge, J.A.,
Feuillet, J., Hauguel, A. and Holly, F.M.,
1982. New
method for tidal current computation. Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 108 WW3: 396-417. Burg, M.C.,
Coeffe, Y. and Warluzel, A., 1982. Tridimensional numerical model
for tidal and wind generated flow. Proc. 18th Coastal Engineering Conf., ASCE, Capetown, South-Africa, 635-651. Chenin-Mordojovich, M.I. and Hauguel. A.,
1985. The internal refined grid in
particular areas inside a 2-D mathematical model. Proc. 21st Congress. IAHR, Melbourne, Australia.
Hamm, L., Quetin. B. and Usseglio-Polatera, J.M..
1985. Two dimensional
modelling of wind-induced currents in coastal and harbour areas. Proc. Int. Conf. on Numerical and Hydraulic Modelling of Ports and Harbours, Birmingham, U.K., Usseglio-Polatera, J.M.
13-22. and Chenin-Mordojovich. M.I.,
1985. Numerical problems
in coupling two- and three-dimensional models: the CYTHERE 3-D system. Proc. Int. Symp. on Refined Flow Modelling and Turbulence Measurements, Iowa City, USA. Yanenko, N.N, 1968. Mhthodes B pas fractionnaires. Annand Colin, Paris. 203 pp.
55 I
FIG.3
: DEPTH AVERAGED VELOCITIES COMPUTED WITH THE 2-D CODE AND THE COUPLED 2-D / 3-D CODE, 4 HOURS AFTER HIGH
WATER AND A T LOW WATER (APPLICATION EXAMPLE OF FIG.2)
552
I '
I
1600
Moo
2400
J
1600
Moo
2400
> 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.
This Page Intentionally Left Blank
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 &<0<1,
they
are
i n terms of sigma
defined
r a t h e r t h a n o v e r t h e i n t e r v a l @k
The t r a n s f o r m a t i o n t o sigma c o o r d i n a t e s is a c c o m p l i s h e d u s i n g O=z/h Expanding
t h e two components o f v e l o c i t y u , v i n terms o f m f u n c t i o n s f r ( a )
and h o r i z o n t a l - s p a c e a n d time d e p e n d e n t c o e f f i c i e n t s A r ( X , ( P gives
(6)
, t ) and Br(X,(P , t )
576
Using
t h e G a l e r k i n method i n t h e v e r t i c a l s p a c e domain, e q u a t i o n s ( l ) , ( 2 )
(3) are
and
first transformed t o sigma c o o r d i n a t e s u s i n g e q u a t i o n ( 6 ) .
sima form of
these
equations
is
then
multiplied
by
The
e a c h of t h e b a s i s
f u n c t i o n s f ( o ) and i n t e g r a t e d w i t h r e s p e c t t o o o v e r t h e i n t e r v a l 0 to 1 . By k integrating the t e r m i n v o l v i n g t h e v e r t i c a l eddy v i s c o s i t y , boundary c o n d i t i o n s ( 4 a , b ) and ( 5 a , b ) i n s i g m a c o o r d i n a t e s c a n b e i n c l u d e d , (see D a v i e s 1980,
F i n a l l y , e x p a n s i o n s ( 7 ) a n d ( 8 ) are s u b s t i t u t e d
for d e t a i l s ) .
1983a,b
i n t o t h e r e s u l t i n g equations, giving,
m
m
1 =r.f0frfkdo aA
-y
r-1
1 B r . fol fr f k d o = Rcoscp r=l
lofrdU
where k:l , 2 , .
..,m
With a similar e q u a t i o n t o ( 1 0 ) d e r i v e d from t h e V e q u a t i o n of m o t i o n ( 3 ) . 2 . 3 C a l c u l a t i o n of v e r t i c a l modes For
an
corresponding the
b a s i s set of f u n c t i o n s , t h e m e q u a t i o n s i n ( 1 0 ) a n d t h e
arbitrary
equations,
V
viscosity
removed,
with
Stephens
1983),
are c o u p l e d t o g e t h e r t h r o u g h bottom f r i c t i o n and
However t h e c o u p l i n g t h r o u g h t h e v i s c o s i t y term c a n be
term.
a s s o c i a t e d s i g n i f i c a n t s a v i n g i n computer time' ( D a v i e s and
an
by u s i n g a b a s i s set of modes which are e i g e n f u n c t i o n s 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 eddy v i s c o s i t y , of t h e form d
with
- and f E
Writing
df
[PZI =
-
(11)
-Ef
denoting r e s p e c t i v e l y eigenvalue and eigenfunction. in
terms
of
a
time-dependent
and h o r i z o n t a l l y v a r y i n g p a r t
a( X, cp, t ) and v e r t i c a l l y v a r y i n g f u n c t i o n @( a ) , g i v e s
511
is a f u n c t i o n of X , c p , t ,
cx
Since
i t o n l y a f f e c t s t h e e i g e n v a l u e s , and t h e
e i g e n f u n c t i o n s a r e d e t e r m i n e d from
with
-
E=aE
For
a r b i t r a r y v e r t i c a l v a r i a t i o n of
an
readily
computed
Runge-Kutta
using
approach
either
(Davies
G a l e r k i n method ( D a v i e s 1 9 8 3 a , b ) o r t h e
the and
0 , t h e e i g e n v a l u e s of ( 1 3 ) c a n b e
Furnes
1984).
E x t e n s i v e d e t a i l s of t h e
method are g i v e n i n t h e s e p a p e r s a n d w i l l n o t b e r e p e a t e d h e r e . In
order
consistent
t o s o l v e e q u a t i o n (131, s u r f a c e and sea bed boundary c o n d i t i o n s ,
w i t h t h o s e g i v e n i n e q u a t i o n s ( 4 a , b ) and ( 5 a , b ) must b e s p e c i f i e d .
I t c a n b e r e a d i l y shown ( D a v i e s 1 9 8 3 a , b ) t h a t b o u n d a r y c o n d i t i o n s ,
are a p p r o p r i a t e and g i v e r i s e t o r e a l e i g e n v a l u e s and e i g e n v e c t o r s of ( 1 3 ) . For c o n v e n i e n c e t h e e i g e n f u n c t i o n s are n o r m a l i z e d by r e q u i r i n g t h a t f (0) = 1
r=l,2,.
.., m
(15)
The e i g e n f u n c t i o n s of ( 1 3 ) are o r t h o g o n a l , and c o n s e q u e n t l y 1
Jofrfkdo
0
rfk
(16)
Also w i t h b o u n d a r y c o n d i t i o n ( 1 4 ) , we have, 1
Jofrdo = 0
r=2,3,
...,m
(17)
2 . 4 S p e c t r a l form of t h e hydrodynamic e q u a t i o n s With
a
basis
set of e i g e n f u n c t i o n s ( m o d e s ) , e q u a t i o n s ( 9 ) , ( l o ) , and t h e
c o r r e s p o n d i n g V e q u a t i o n c a n b e s i m p l i f i e d (see D a v i e s 1 9 8 3 a , b f o r d e t a i l s ) by u s i n g ( 1 3 ) , ( 1 5 ) , ( 1 6 ) and (171, t o g i v e ,
578
where
m
m (21a , b )
(181, (19) and (20) are t h e w o r k i n g e q u a t i o n s w h i c h have t o b e
Equations solved
to
find
5, A r and Br o v e r t h e sea area, s u b j e c t t o
t h e v a r i a t i o n of
i n i t i a l and boundary c o n d i t i o n s .
C u r r e n t s a t any d e p t h c a n t h e n be c a l c u l a t e d
and Br u s i n g e x p a n s i o n s ( 7 ) and (8).
f r o m the A
Any a r b i t r a r y v e r t i c a l v a r i a t i o n of e d d y v i s c o s i t y , v a r y i n g w i t h h o r i z o n t a l p o s i t i o n and time c a n b e i n c o r p o r a t e d i n t h e model. In
to
order
discretize
in
accomplished using
equations
horizontal
a
using
at
evaluated
solve the
(18), (19) and
and
w i t h time.
s t a g g e r e d f i n i t e d i f f e r e n t g r i d , i n which
time-stepping
5 , U, V are
The time d i s c r e t i z a t i o n is a c c o m p l i s h e d
d i f f e r e n t g r i d points.
a forward
(20) i t is n e c e s s a r y t o
H o r i z o n t a l d i s c r e t i z a t i o n is
method.
Details
of
t h e s e t e c h n i q u e s are
s t a n d a r d and w i l l n o t b e p r e s e n t e d h e r e .
a s t a t e of z e r o d i s p l a c e m e n t a n d m o t i o n . Along a c l o s e d boundary t h e normal component of c u r r e n t is s e t t o z e r o . A t t h e o p e n b o u n d a r i e s of t h e model, M2 and S 2 t i d a l i n p u t d e t e r m i n e d from a Solutions
are g e n e r a t e d
from
t h r e e d i m e n s i o n a l model ( D a v i e s and F u r n e s 1980), a n d o b s e r v a t i o n s ( C a r t w r i g h t 1976) t o g e t h e r (Flather,
with
Proctor
data
and
from a two d i m e n s i o n a l model of t h e N.E. A t l a n t i c
Wolf p e r s o n a l c o m m u n i c a t i o n ) were s p e c i f i e d .
A three
dimensional
r a d i a t i o n c o n d i t i o n was employed a l o n g t h e o p e n b o u n d a r y t o allow
disturbances
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 . ' Details of
these
conditions
can
be
found
i n D a v i e s and F u r n e s (1980) a n d w i l l n o t b e
given here. An
appropriate
parameterization
of
vertical
model i s d i f f i c u l t t o d e t e r m i n e .
three-dimensional
eddy v i s c o s i t y to u s e i n a D a v i e s and F u r n e s (1980),
r e l a t e d e d d y v i s c o s i t y t o d e p t h mean c u r r e n t s iI and 7, by, p = K(ii2+T2)/U
w i t h 6 a f r e q u e n c y of l . O ~ l O - ~ s -and ~, K a dimensionless c o e f f i c i e n t having a value
K=2.0x10-5.
This
formulation
and
t h e s e p a r a m e t e r s are u s e d i n t h e
present calculation. The
coefficient
comparable
with
of bottom f r i c t i o n k , u s e d i n ( 5 a , b ) , s h o u l d h a v e a v a l u e
o b s e r v e d v a l u e s of
Cleo,
w h e r e C l o o i s t h e c o e f f i c i e n t which
579
60- I
50' I
I1 V
F i g . 1 . F i n i t e - d i f f e r e n c e g r i d of t h e t h r e e - d i m e n s i o n a l c o n t i n e n t a l s h e l f model, showing p o s i t i o n of some t i d e g a u g e s u s e d i n comparisons.
F i g . 2. (
---- )
M
2
c o - t i d a l c h a r t showing a m p l i t u d e i n cm (-)
and p h a s e i n d e g r e e s
580 bottom stress t o t h e c u r r e n t 100cm above t h e sea bed.
relates
r a n g e from 4 . 3 0 ~ 10-3 Wolf ( 1980 In
the
present
to 5.29 a
calculations
f 1 .7x1
value
5 . 0 ~ 1 0 - ~was
of
V a l u e s of Cloo
0-3 (Bowden a n d Ferguson 1 9 8 0 ) . employed; a v a l u e
c o n s i s t e n t with t h i s range o f observed values. 3
COMPARISON OF OBSERVED AND COMPUTED ELEVATIONS AND CURRENTS
3.1 E l e v a t i o n s Co-tidal
charts
harmonically
the
of
analysing
M2
output
and
S2
from
components
the
o f t h e t i d e computed by
model, u s i n g s t a n d a r d methods, are
g i v e n i n F i g u r e s 2 and 3 a , b .
to
Referring (Lennon
1961)
F i g u r e 2 , i t is a p p a r e n t t h a t t h e model r e p r o d u c e s t h e known three
three
amphidromes
shown
in
M2 amphidromes i n t h e N o r t h Sea.
and
the
distribution
of c o - a m p l i t u d e and co-phase l i n e s
2 is i n e x c e l l e n t a g r e e m e n t w i t h M
Figure
The p o s i t i o n o f t h e s e
2
co-tidal c h a r t s derived
r e s o l u t i o n North sea m o d e l s ( D a v i e s 1 9 7 6 , P r a n d l e 1980, M a t h i s e n
from
higher
and
Johansen
1983,
Davies,
S a u v e l and Evans 1 9 8 4 ) and a l s o c o - t i d a l c h a r t s
d e r i v e d from o b s e r v a t i o n s (Lennon 1 9 6 1 ) . In
order
to
tidal
input
along
examine the
t h e s e n s i t i v i t y o f t h e S2 t i d e t o c h a n g e s i n t h e S 2 open b o u n d a r y , r e s u l t s f r o m two c a l c u l a t i o n s (namely
A and B ) w i l l now b e d e s c r i b e d . I n t h e first c a l c u l a t i o n ( C a l c n .
calculations
t o t h e r a d i a t i o n c o n d i t i o n ( D a v i e s and F u r n e s 2 1 9 8 0 ) was i n t e r p o l a t e d from a two d i m e n s i o n a l N.E. A t l a n t i c model of F l a t h e r ,
A ) , the S
Proctor
t i d a l i n p u t ( S T , qT
and
radiation The M2
Wolf
and the
communication).
The
M2
tidal
input to the
was i d e n t i c a l t o t h a t u s e d by D a v i e s a n d F u r n e s ( 1 9 8 0 ) .
t i d e s were s e p a r a t e d by a harmonic a n a l y s i s of t h e t i d a l time
S2
series computed although
(personal
conditions by
M2
t h e model w i t h t h i s boundary f o r c i n g .
tidal
I t was f o u n d t h a t
e l e v a t i o n , a m p l i t u d e and p h a s e was i n good agreement
with
o b s e r v a t i o n s d e r i v e d from o f f - s h o r e t i d e g a u g e s , i n t h e v i c i n i t y o f t h e
open
boundary
too
high
(see T a b l e 11, t h e S2 a m p l i t u d e was of t h e o r d e r ’ o f u p t o 10cm
(approximately 20%),
(see g a u g e s 5 a n d 6 i n T a b l e 1 ) .
This error
r i s e t o a n a m p l i t u d e e r r o r of t h e o r d e r of 6cm, a t o f f s h o r e t i d e g a u g e s
gave
H , 5 4 , 55, 5 6 , 5 7 which i n e s s e n c e mark t h e n o r t h e r n e n t r a n c e t o t h e N o r t h Sea
(see T a b l e 2 ) .
( T h e s e t i d e g a u g e s l i e a p p r o x i m a t e l y a l o n g a l i n e from Wick t o
Bergen, see D a v i e s , S a u v e l a n d Evans ( 1 9 8 4 ) ) . I d e a l l y an i t e r a t i v e process should then be used ( F l a t h e r 1976) to d e r i v e a
cT ,
new an
qT f o r t h e S t i d a l i n p u t t o t h e r a d i a t i o n c o n d i t i o n . However s u c h 2 p r o c e s s is c o m p u t a t i o n a l l y e x p e n s i v e w i t h a t h r e e d i m e n s i o n a l
iterative
model
since
associated
each
harmonic
iteration
involves
a
a n a l y s i s to s e p a r a t e M
full 2
3D t i d a l s i m u l a t i o n w i t h t h e
and S2.
I n p r a c t i c e a number o f
581
were
calculations
performed,
order
of
tidal
amplitude
exceeded
the
was
obtained
in
(Calcn.
B)
agreement
with
observations
b o u n d a r y (see T a b l e 1 ) ;
56,
57
model
in
cT , qT
f o r t h e S2 t i d e , r e d u c e d by t h e
observed.
t h i s means a s e c o n d s o l u t i o n
By
which t h e S2 a m p l i t u d e a n d p h a s e was i n b e t t e r
at
off-shore
tide
g a u g e s n e a r t h e model open
i n t h e Celtic Sea and p a r t i c u l a r l y a t rigs H , 5 4 , 55,
n o r t h e r n N o r t h S e a (see T a b l e 2 ) .
the
The M2 t i d a l i n p u t t o t h e
i n b o t h C a l c n s . A a n d B was i d e n t i c a l t o t h a t u s e d by D a v i e s and F u r n e s
I t is e v i d e n t from T a b l e s 1 a n d 2, t h a t w i t h t h i s i n p u t t h e M2 t i d e
(1980).
is
with
15% i n t h e r e g i o n of t h e o p e n boundary where t h e S2
approximately
in
agreement w i t h o b s e r v a t i o n s a t o f f s h o r e t i d e gauges.
good
I t i s also
t h a t t h e M2 t i d e a t t h e s e g a u g e s is n o t s i g n i f i c a n t l y i n f l u e n c e d ( o f
apparent
o r d e r lcm a m p l i t u d e and 1 d e g r e e p h a s e ) by c h a n g e s i n S2 t i d a l i n p u t . TABLE 1 OSTG Boundary
M Ogs
h
go
(cm) C1
C2 C3 C4
C5 C6 C7
129 111 116 109 112 112 117
102 108 107 109 116 121 144
Calcn. A h go (cm)
129 128 120 121 124 123 113
103 104 103 108 113 121 144
Calcn. B h go
(cm)
Calcn.
S
ogs h
A go
h
go
(cml
(cm)
Calcn. B h go
(cm)
129 128 120 121 124 123 113
103 104 103 108 113 121 144
45 36 39 37 38 37 41
137 138 140 142 150 154 179
50 45 44 44 45 45 46
140 142 144 146 148 150 179
41 43 42 43 44 44 42
137 138 135 139 146 153 178
111 109 103 101 91 83 74 58 53 53 59 50 52
164 168 179 190 206 225 259 256 257 282 291 289 279
42 41 40 39 35 31 26 20 21 20 23 19 18
196 202 21 2 224 238 261 287 298 293 31 6 333 328 320
46 46 42 43 39 38 37 30 28 25 29 23 22
189 191 21 2 225 246 270 305 302 299 323 334 335 328
42 40 37 36 33 30 27 21 19 19 21 17 18
202 207 21 6 227 243 261 297 294 294 320 329 327 31 7
West C o a s t / N o r t h e r n N o r t h Sea Boundary
14 13 1 2 3 4 5 6 7 8 9 10 11
112 110 108 104 93 84 70 54 56 55 66 51 52
163 167 179 190 205 228 252 264 256 282 302 282 285
111 109 103 101 91 83 74 58 53 53 59 50 52
164 168 179 190 206 225 259 256 257 282 291 289 279
Table 1 . Comparison of computed a n d o b s e r v e d a m p l i t u d e h ( c m ) and p h a s e g ( d e g r e e s 1 f o r t h e M and S components of t h e t i d e a t o f f - s h o r e g a u g e s i n t h e 2 v i c i n i t y of t h e s h e l ? e d g e .
289
582
S
c o - t i d a l c h a r t showing a m p l i t u d e i n cm (-) d e r i v e d f'rom c a l c u l a t i o n A.
and p h a s e i n
S
c o - t i d a l c h a r t showing a m p l i t u d e i n cm (-1 d e r i v e d fron c a l c u l a t i o n B.
and p h a s e i n
F i g . 3a. degrees
(-2-1,
F i g . 3b. degrees
(-2-1,
583
C o t i d a l c h a r t s f o r t h e S2 t i d e , d e r i v e d from c a l c u l a t i o n s A and B are shown in
Figs
3a,b.
S2
tide
i n t h e n o r t h e r n N o r t h S e a , computed i n c a l c u l a t i o n A (see F i g 3a and
I t i s a p p a r e n t from t h e s e F i g u r e s , t h a t t h e a m p l i t u d e of t h e
2 a n d 31, is s i g n i f i c a n t l y h i g h e r t h a n t h a t i n c a l c u l a t i o n B , which i s
Tables in
reasonable
S2
tide
agreement
derived
with observations.
The s p a t i a l d i s t r i b u t i o n of t h e
c a l c u l a t i o n B is i n good a g r e e m e n t w i t h t h a t g i v e n by
from
M a t h i s e n a n d J o h a n s e n (1983),from a h i g h r e s o l u t i o n two-dimensional North S e a model. TABLE 2 Calcn. A
M
C e l t i c Sea/ I r i s h Sea
06s
h go (cm)
h go (cm)
123 112 138 122 156 136 163 144 189 142 144 154 112 184 138 309
109 116 129 134 134 144 00 172 34 305
28 39 55 60 81 34
Calcn B
.
S
Calcn A h go
Calcn. B h go (cm)
h go (cm)
06s h go (cm)
108 116 129 160 134 182 134 134 144 101 173 135 305
41 145 45 158 52 178 55 190 66 187 49 202 44 230 44 344
46 148 48 156 49 174 49 184 59 187 41 210 44 237 39 327
44 47 50 51 59 42 41 39
140 151 168 175 177 196 224 328
19 24 18 13
25 348 33 6 24 2 16 357 12 346
19 25 18 12
341 358 355 349 338
128 139 155
(cm)
N o r t h Sea H
50 67 50 32 25
54 55 56 57
54 71 51 33 25
309 323 318 309 286
301 316 314 307 291
54 72 52 33 25
301 315 314 307 290
338 360 354 345 10 328
9
T a b l e 2. Comparison of computed a n d o b s e r v e d a m p l i t u d e h (cm) and p h a s e g ( d e g r e e s ) f o r t h e M and S components of t h e t i d e a t o f f - s h o r e g a u g e s .
2
Considering It
is
2
t h e d i s t r i b u t i o n of t h e S2 t i d e o v e r t h e N o r t h Sea.
initially
evident
from
Tables
2
and
3, t h a t
i n t h e n o r t h e r n North Sea, i n
c a l c u l a t i o n A , t h e S2 t i d e computed w i t h t h e model h a s a s i g n i f i c a n t error, i n that
t h e computed a m p l i t u d e is t h e o r d e r of +10cm t o +15cm above t h e o b s e r v e d
(see f o r example Wick a n d A b e r d e e n ) . boundary other
error
hand,
off-shore
small
both
of
B,
gauges
of
error,
particularly average
( o f o r d e r 5cm) as t h e t i d e p r o p a g a t e s i n t o t h e model.
solution
tide
examination
T h i s reflects a n i n t e n s i f i c a t i o n of t h e
order the
along
which
in
was
optimised
to
On t h e
s i m u l a t e t h e S2 t i d e a t
n o r t h e r n N o r t h Sea (see T a b l e 21, c o n t a i n s o n l y a
-4cm a t Aberdeen; -5cm a t N o r t h S h i e l d s .
However a n
S2 t i d a l s o l u t i o n i n t h e S o u t h e r n B i g h t of t h e N o r t h S e a the
solutions
Dutch,
German
underestimate
and the
D a n i s h coasts, r e v e a l e d t h a t o n amplitude
of
t h e S2 t i d e ; w i t h
584 B, g i v i n g a s i g n i f i c a n t l y lower a m p l i t u d e t h a n c a l c u l a t i o n A (see
calculation Table
This
3).
energy
within
suggests
that
w i t h i n t h e model, t h e d i s s i p a t i o n of t i d a l
t h e s h a l l o w r e g i o n s of t h e N o r t h S e a is larger t h a n t h a t w h i c h
would o c c u r i n n a t u r e . I t i s p a r t i c u l a r l y i n t e r e s t i n g t o n o t e t h e c h a n g e i n a m p l i t u d e and p h a s e of
t h e M2
to
the
e l e v a t i o n a t coastal g a u g e s , d u e t o t h e d i f f e r e n c e s i n S2 i n p u t
tidal
Although T a b l e s 1 a n d 2 , showed t h i s t o b e small a t o f f - s h o r e
model.
g a u g e s , i t is e v i d e n t from T a b l e 3, t h a t c h a n g e s i n S2 do s i g n i f i c a n t l y a f f e c t the
a m p l i t u d e , a l t h o u g h n o t t h e p h a s e of t h e M2 t i d e .
For example t h e c h a n g e
of t h e o r d e r of 10cm i n t h e a m p l i t u d e of M2 a t S h i v e r i n g S a n d s between c a l c n s . A and B (see T a b l e 3 ) .
TABLE 3
M2 Obs
h go (cm) Wick Invergordon Aberdeen North S h i e l d s Scarborough Immingham I n n e r Dowsing S h e l l AD Lowestof t S h i v e r i n g Sands Ostende Helgol and Cuxhaven Esb j erg
101 136 130 158 171 225 197 75 71 182 176 109 134 66
Calcn. A
h go (cm)
322 109 337 151 23 127 8 9 154 111 175 163 191 162 1 8 3 74 194 66 258 346 130 5 158 76 312 344 101 36 51
Calcn. B h go (cm)
309 112 326 154 1 3 129 81 1 5 8 117 181 154 202 175 194 194 77 246 69 336 140 4 169 318 81 316 109 26 55
308 326 13 82 117 155 175 194 248 337 5 319 318 27
Calcn. A h go h go (cm) (cm) S 06s
35 48 45 53 58 74 68 29 22 54 52 29 34 16
360 15 61 130 153 214 209 242 297 40
58 20 54 98
48 68 54 65 73 82 80 34 26 42 45 23 32 15
3 20 68 142 181 232 256 275 316 71 96 86 85 142
Calcn. B h go
(cm) 35 354 50 11 41 59 4 8 133 54 172 60 222 57 246 25 265 1 9 303 29 62 33 86 16 81 23 80 10 136
Table 3. Comparison of computed a n d o b s e r v e d a m p l i t u d e h ( c m ) a n d p h a s e g ( d e g r e e s ) f o r t h e M and S components of t h e t i d e a t some coastal g a u g e s i n 2 2 t h e North Sea. The e f f e c t of t h e S2 t i d e upon t h e M2 t i d e , i s t h r o u g h t h e n o n - l i n e a r t e r m s in
t h e equations.
through
I n p a r t i c u l a r its i n f l u e n c e upon e n e r g y d i s s i p a t i o n arises
c h a n g e s i n b o t t o m stress t h r o u g h two mechanisms.
The f i r s t mechanism
i s d i r e c t , i n t h a t bottom stress d e p e n d s i n a non l i n e a r manner upon t h e total current The
square A
at
t h e sea bed, which i s i n c r e a s e d when t h e S c u r r e n t is i n c r e a s e d . 2 i n d i r e c t , i n t h a t v e r t i c a l eddy v i s c o s i t y is r e l a t e d to t h e
is
second
of
t h e t o t a l c u r r e n t , a n d h e n c e i n c r e a s e s w i t h i n c r e a s i n g S2 c u r r e n t .
consequence
of
an
increase
in
e d d y v i s c o s i t y ; is a d e c r e a s e i n c u r r e n t
v a r i a t i o n t h r o u g h t h e v e r t i c a l , and i n t h e l i m i t as t h e e d d y v i s c o s i t y goes to infinity
the
current
will
have no v e r t i c a l s t r u c t u r e .
I n t h i s case bottom
585
M, tlde calcn. A 40
30
2
P
-25-20-15
-10 -5
0
5
10 I5
20
p
10
4=
20 25
S, tide calcn. P 40
30
20
10
-25 -20 -I5 -10 -5
0
5
10
I5 20 25
-25-20 -15 -10 -8
o
6
10 15
-
P
t
5
z
w za
Phase
hplllude
F i g . 4a. Histogram of t h e d i s t r i b u t i o n of errors i n c a l c u l a t i o n A , f o r t h e M and S components of t h e t i d e . 2 2 M, tlde calcn. B 40
30
2
H 20
; 0
- 1 5 - 2 0 -I5 -10 -5
0
5
10 IS PO 25
-25 -20-15 -10 -5
0
5
10
I5
w
25
S, tide calcn. B
so 4a L
=x6 W j
z
10
-21 -20 -15 -10 -5
o
5
Amplllud*
m
15 PO 21
-28-20 -I5 -I0 -5
0
5
K)
I5 20 25
PhllU
F i g . 4 b . Histogram of t h e d i s t r i b u t i o n of errors i n c a l c u l a t i o n B, f o r t h e M and S2 components of t h e t i d e . 2
current
and
depth
stress w i l l vertically
be
mean
c u r r e n t w i l l have t h e same v a l u e .
related
integrated
to
the
depth
mean
current
vertically
integrated
situation i n a
I n s u c h c i r c u m s t a n c e s t h e u s e of a f r i c t i o n a l
model).
c o e f f i c i e n t k = 0.005 w i l l b e i n a p p r o p r i a t e ( a v a l u e of k
a
Hence t h e bottom
(the
model
Flather
0.0025 is u s u a l i n
( 1 9 7 6 ) ) and w i l l p r o d u c e e x c e s s i v e
damping i n t h e model. The the
high
of damping of t h e M2 and S2 t i d e i n t h e S o u t h e r n B i g h t of
level
S e a , s u g g e s t s t h a t t h e v a l u e of bottom f r i c t i o n c o e f f i c i e n t k, and
North
v i s c o s i t y c o e f f i c i e n t K ( e q u a t i o n 22) a l t h o u g h a p p r o p r i a t e f o r a s i m u l a t i o n o f the
M2 t i d e a l o n e , ( D a v i e s and F u r n e s 1 9 8 0 ) may r e q u i r e some s l i g h t r e d u c t i o n
and S t i d e are s i m u l a t e d t o g e t h e r . 2 H i s t o g r a m s o f e r r o r s , b a s e d upon c o m p a r i s o n s a t o n e hundred and t w e n t y n i n e
when t h e M
coastal these
2
and o f f - s h o r e t i d e g a u g e s are g i v e n i n F i g s 4 a , b . figures
significantly
that
both
the
amplitude
and
o v e r - e s t i m a t e d i n c a l c u l a t i o n A.
phase
I t is e v i d e n t from
of
the
S2
tide
is
I n c a l c u l a t i o n B however, t h e
a m p l i t u d e o f t h e S2 component agrees t o w i t h i n 5cm of t h e o b s e r v e d a t o v e r 54% of
the
tide
gauges. evident
w i t h p h a s e a g r e e i n g t o w i t h i n 1 0 d e g r e e s a t 67% o f t h e
gauges,
s l i g h t b i a s t o w a r d s a n u n d e r e s t i m a t i o n o f a m p l i t u d e and p h a s e i s
A
The g r i d of t h e model however i s v e r y
c a l c u l a t i o n B, see F i g 4b.
in
i n t h e S o u t h e r n B i g h t of t h e N o r t h Sea, a n d w i t h s u c h a g r i d i t may b e
coarse
to
unrealistic
a high
expect
of
level
accuracy
from
t h e model, due t o
l i m i t a t i o n s of g r i d r e s o l u t i o n . 3.2 C u r r e n t s
The c r u c i a l t e s t of a n y t h r e e d i m e n s i o n a l model i s i t s a b i l i t y t o r e p r o d u c e the
vertical
Comparisons
are
tide,
However,
the
presently
subsequently and
variation of
.
some
of
form in
tidal presented
progress
indication
at
current
from
sea s u r f a c e
to
sea bed.
by D a v i e s a n d F u r n e s ( 1 9 8 0 ) , f o r t h e M2 for
the
S2
tide,
and w i l l b e r e p o r t e d
o f t h e s p a t i a l and v e r t i c a l v a r i a t i o n of t h e M
S2
tidal
currents
maximum
tidal
c u r r e n t o n t h e s h e l f w i l l o c c u r a t s p r i n g t i d e when t h e M
S2
tidal
current
currents
are
in
2 The
s p r i n g t i d e c a n b e o b t a i n e d from F i g s 5 a , b . 2
and
phase.
F i g s 5 a , b , show c o n t o u r s of s p r i n g t i d a l
a t sea s u r f a c e a n d sea bed.
Comparing F i g 5a w i t h n e a r s u r f a c e t i d a l
c u r r e n t a m p l i t u d e s d e r i v e d from o b s e r v a t i o n s (see F i g 4.15 i n Howarth and Pugh 1983) spring
i t is e v i d e n t t h a t t h e model r e p r o d u c e s t h e s p a t i a l d i s t r i b u t i o n of t h e t i d a l c u r r e n t s t o w i t h i n a n a c c e p t a b l e l e v e l of a c c u r a c y . I n g e n e r a l
surface
spring
t i d a l c u r r e n t s computed w i t h t h e model are t h e o r d e r of a few
percent
higher
t h a n t h o s e g i v e n by Howarth a n d Pugh ( 1 9 8 3 ) .
T h i s may b e due
587
Fig. 5a.
Fig. 5b.
Computed c o n t o u r s of t h e major a x i s (M + S ) a t t h e sea s u r f a c e . 2
2
588
to
t h a t F i g u r e s 5 a , b are b a s e d upon C a l c u l a t i o n A , i n w h i c h , as w e
fact
the
have
seen,
observed.
i n p u t t o t h e model was of t h e o r d e r of 10% h i g h e r t h a n t h e 2 A l t h o u g h i n g e n e r a l , b o t h c a l c u l a t i o n s y i e l d e d a lower a m p l i t u d e of the S
i n t h e s h a l l o w r e g i o n s . The o t h e r p o s s i b i l i t y i s t h e d i f f i c u l t y of d r a w i n g 2 accurate distribution of surface t i d a l c u r r e n t s from p o i n t c u r r e n t
S
o b s e r v a t i o n s t a k e n a t d e p t h (Howarth 1982, Howarth a n d Pugh 1 9 8 3 ) .
is e v i d e n t from F i g u r e s 5 a , b , t h a t s p r i n g c u r r e n t s n e a r t h e sea bed are
It
s i g n i f i c a n t l y lower t h a n t h e s u r f a c e c u r r e n t s . CONCLUDING REMARKS
4
t h r e e d i m e n s i o n a l n u m e r i c a l t i d a l model of t h e N o r t h West European S h e l f
A
has
been
bottom
horizontal method
The model i n c o r p o r a t e s a q u a d r a t i c law of
developed i n t h i s paper.
friction,
eddy
position
a
with
accurate
method
equations
has
through
and
the
time i n a n a r b i t r a r y manner.
and
By u s i n g t h e G a l e r k i n
s e t of e i g e n f u n c t i o n s , a c o m p u t a t i o n a l l y economic and
basis
(Davies been
v i s c o s i t y c a n v a r y t h r o u g h t h e v e r t i c a l and w i t h
and
Stephens
presented.
(see f o r
vertical
19831
This
method
of
solving
t h e hydrodynamic
u n l i k e t h e u s e of g r i d b o x e s
example D a v i e s and S t e p h e n s 19831, y i e l d s a
c o n t i n u o u s c u r r e n t p r o f i l e from sea s u r f a c e t o sea bed. Computed
M2
are
shelf
S2
t i d a l e l e v a t i o n s a n d c u r r e n t s o v e r t h e r e g i o n of t h e
A r i g o r o u s c o m p a r i s o n of o b s e r v e d and computed t i d a l c u r r e n t s a t
area.
the
and
i n g e n e r a l i n r e a s o n a b l y good a g r e e m e n t w i t h o b s e r v a t i o n s t a k e n i n
different
points
in
the
v e r t i c a l water column is p r e s e n t l y i n p r o g r e s s and
r e s u l t s w i l l be reported subsequently. The u s e of a q u a d r a t i c law of b o t t o m f r i c t i o n , r a t h e r t h a n t h e a p p l i c a t i o n
a n o - s l i p c o n d i t i o n a t t h e sea b e d d o e s however mean t h a t t h e h e i g h t above
of
sea
the
bed
a t w h i c h t h e b o t t o m c u r r e n t s are computed c a n n o t b e d e t e r m i n e d .
Consequently
important
spatial
temporal
and
particular
variations
importance
to
information of
the near
o i l i n d u s t r y , on t h e c u r r e n t s and stresses (of
off-shore bed
i n d e t e r m i n i n g s e d i m e n t t r a n s p o r t and p o s s i b l e e r o s i o n
of t h e sea bed i n t h e v i c i n i t y of o f f - s h o r e s t r u c t u r e s ) c a n n o t b e d e t e r m i n e d . The method d e v e l o p e d h e r e h a s r e c e n t l y b e e n e x t e n d e d t o d e a l w i t h a n o - s l i p bottom can
boundary
be
determined.
analysed. significant
I n t h i s case n e a r bed c u r r e n t s and bed stresses
condition.
This
Results
extension
practical
from
these
importance
c a l c u l a t i o n s are p r e s e n t l y b e i n g
s c i e n t i f i c a l l y w o r t h p u r s u i n g a n d is of
appears
to
the
oil
industry,
where
near
bed
measurement programmes are a l r e a d y i n p r o g r e s s . The eddy
sensitivity viscosity
is
of
tidal
also of
c u r r e n t s t o c h a n g e s i n t h e v e r t i c a l p r o f i l e of scientific
importance,
and
could
yield
an
589 improvement
in
accuracy
of
computed
tidal current profiles.
The u s e o f a
eddy v i s c o s i t y from sea s u r f a c e to sea bed is q u e s t i o n a b l e f o l l o w i n g
constant
r e s u l t s from r e c e n t f i e l d measurements (Wolf 1 9 8 0 ) which show a n e a r p a r a b o l i c variation below
from
t o bed, w i t h v i s c o s i t y r e a c h i n g a maximum a t a p o i n t
surface
The model d e v e l o p e d h e r e c o u l d i n c o r p o r a t e s u c h a v a r i a t i o n
mid-depth.
i t would b e of s i g n i f i c a n t s c i e n t i f i c and p r a c t i c a l i m p o r t a n c e t o examine
and
t h e e f f e c t of s u c h a p r o f i l e upon computed t i d a l c u r r e n t s . It
is
extended
apparent to
that
the
model
a
number
of
examine
developed important
i n t h i s p a p e r c a n be r e a d i l y scientific
problems i n t i d a l
p h y s i c s , and s e d i m e n t t r a n s p o r t , w i t h t h e a s s o c i a t e d b e n e f i t s t o t h e o f f - s h o r e
o i l i n d u s t r y , and a p p l i c a t i o n s t o v a r i o u s p o l l u t i o n problems.
5
ACKNOWLEDGEMENTS
The
author
i s i n d e b t e d t o D r . N. S. Heaps f o r a number of u s e f u l comments
and s u g g e s t i o n s i n c o n n e c t i o n w i t h t h i s p a p e r .
Data made a v a i l a b l e by p a r t i c i p a n t s i n t h e JONSDAP e x p e r i m e n t and by Mr. M. J . Howarth from a C e l t i c sea e x p e r i m e n t , w i t h o u t which t h e comparisons made i n
t h i s p a p e r would n o t b e p o s s i b l e , are v e r y much a p p r e c i a t e d .
Numerical
r e s u l t s s u p p l i e d by D r s . F l a t h e r , P r o c t o r and Wolf from a
model
two-dimensional North-East A t l a n t i c model are g r a t e f u l l y acknowledged. The
numerical
Consortium
modelling
consisting
of
work the
described
Natural
in
this
p a p e r was funded by a
Environment R e s e a r c h C o u n c i l and t h e
Department o f Energy. 6
REFERENCES
Bowden, K.F. and Ferguson, S.R., 1980. V a r i a t i o n s w i t h h e i g h t of t h e t u r b u l e n c e i n a t i d a l l y - i n d u c e d bottom boundary l a y e r . Marine T u r b u l e n c e . P r o c e e d i n g s o f t h e 1 1 t h Liege Colloquium' on Ocean Hydrodynamics, J . C . J . N i h o u l , Ed. E l s e v i e r , Oceanography Series Vole 28, 259-286. C a r t w r i g h t , D.E., 1976. S h e l f boundary t i d a l measurements between I r e l a n d and Norway. Mem.Soc.R.Sci. L i e g e , 1 0 , 133-1 39. D a v i e s , A.M., 1976. A n u m e r i c a l model of t h e North S e a and i t s u s e i n c h o o s i n g l o c a t i o n s f o r t h e deployment of o f f s h o r e t i d e g a u g e s i n t h e JONSDAP ' 7 6 o c e a n o g r a p h i c e x p e r i m e n t . D t s c h . Hydrogr. 2. 29, 11-24. D a v i e s , A.M., 1980. A p p l i c a t i o n of t h e G a l e r k i n method t o t h e f o r m u l a t i o n o f a t h r e e - d i m e n s i o n a l n o n - l i n e a r hydrodynamic n u m e r i c a l sea model, A p p l i e d Mathematical M o d e l l i n g , 4 , 245-256. D a v i e s , A.M., 1983a. F o r m u l a t i o n o f a l i n e a r t h r e e - d i m e n s i o n a l hydrodynamic sea model u s i n g a G a l e r k i n - E i g e n f u n c t i o n method. I n t e r n a t i o n a l J o u r n a l of Numerical methods i n F l u i d s , 3 , 33-60. D a v i e s , A . M . , l983b. A p p l i c a t i o n of a G a l e r k i n - E i g e n f u n c t i o n method t o computing c u r r e n t s i n homogeneous and s t r a t i f i e d seas. pg. 287-301 i n Numerical Methods f o r F l u i d Dynamics e d K.W. Morton and M.J. B a i n e s , Academic P r e s s (London).
590 D a v i e s , A.M., 1986. A t h r e e - d i m e n s i o n a l model o f t h e N o r t h w e s t European C o n t i n e n t a l S h e l f , w i t h a p p l i c a t i o n t o t h e M 4 t i d e . J . Phys. Oceanogr., 1 6 , 797-813. D a v i e s , A.M. a n d Owen, A . , 1979. T h r e e - d i m e n s i o n a l n u m e r i c a l sea model u s i n g t h e G a l e r k i n method w i t h a p o l y n o m i a l b a s i s s e t . Appl. Math. M o d e l l i n g , 3, 421-428. D a v i e s , A.M. a n d F u r n e s , G . K . , 1980. Observed a n d Computed M t i d a l c u r r e n t s i n t h e North Sea. J . Phys. Oceanogr., 1 0 , 237-255. D a v i e s , A.M. and F u r n e s G.K., 1984. On t h e d e t e r m i n a t i o n of v e r t i c a l s t r u c t u r e f u n c t i o n s f o r time d e p e n d e n t f l o w problems ( t o a p p e a r i n T e l l u s ) . D a v i e s , A.M. and S t e p h e n s , C.V., 1983. Comparison of t h e f i n i t e d i f f e r e n c e a n d G a l e r k i n methods as a p p l i e d t o t h e s o l u t i o n of t h e hydrodynamic e q u a t i o n s . A p p l i e d M a t h e m a t i c a l M o d e l l i n g , 7 , 226-240. D a v i e s , A.M., S a u v e l , J . , and Evans, J . , 1984. Computing n e a r coastal t i d a l dynamics from o b s e r v a t i o n s and a n u m e r i c a l model. C o n t i n e n t a l S h e l f R e s e a r c h , 4 , 341-366. F l a t h e r , R . A . , 1976. A t i d a l model of t h e n o r t h west European c o n t i n e n t a l s h e l f . Mem. SOC.Roy.Sci. Liege, 1 0 , 141-1 64. Heaps, N.S. and J o n e s , J . E . , 1981. T h r e e - d i m e n s i o n a l model f o r t i d e s a n d s u r g e s w i t h v e r t i c a l eddy v i s c o s i t y p r e s c r i b e d i n two l a y e r s - 11. I r i s h S e a w i t h bed f r i c t i o n l a y e r . G e o p h y s i c a l J o u r n a l Royal A s t r o n o m i c a l S o c i e t y , 6 4 , 303-320. Howarth, M.J., 1982. T i d a l c u r r e n t s of t h e c o n t i n e n t a l s h e l f . I n : A.H. S t r i d e ( E d i t o r ) O f f s h o r e T i d a l Sands. Chapman and Hall, London pp. 10-26. Howarth, M.J. and Pugh, D.T., 1983. O b s e r v a t i o n o f t i d e s o v e r t h e C o n t i n e n t a l S h e l f of NorthlWest Europe, pg. 135-188 i n P h y s i c a l Oceanography o f Coastal a n d S h e l f Seas e d B. J o h n s , p u b l i s h e d E l s e v i e r Oceanography S e r i e s No. 35. Lennon, G.W., 1961. The d e v i a t i o n o f t h e v e r t i c a l a t B i d s t o n i n r e s p o n s e t o t h e a t t r a c t i o n o f o c e a n t i d e s . Geophys. J . R . A s t r o n . s o c . , 6 , 64-84. M a t h i s e n , J . P . , J o h a n s e n , .O., 1983. A n u m e r i c a l T i d a l a n d Storm S u r g e Model o f t h e North S e a . Marine Geodesy, 6 , 267-291. Owen, A . , 1980. A t h r e e - d i m e n s i o n a l model of t h e B r i s t o l Channel. J o u r n a l o f P h y s i c a l Oceanography, 1 0 , 1290-1302. P r a n d l e , D . , 1980. C o - t i d a l c h a r t s f o r t h e S o u t h e r n N o r t h S e a . D e u t s c h e H y d r o g r a p h i s c h e Z e i t s c h r i f t , 3 3 , 68-81. Wolf, J . , 1980. E s t i m a t i o n o f s h e a r i n g stresses i n a t i d a l c u r r e n t w i t h a p p l i c a t i o n t o t h e I r i s h sea, i n , Marine T u r b u l e n c e , e d , J . C . J . N i h o u l , E l s e v i e r S c i e n t i f i c P u b l i s h i n g Company, Amsterdam, pp 319-344.
591
A GENERAL THREE-DIMENSIONAL EDDY-RESOLVING MODEL FOR STRATIFIED SEAS
I . D . JAMES I n s t i t u t e o f Oceanographic Sciences, Bidston Observatory, B i r k e n h e a d , L 4 3 7RA, England.
ABSTRACT Models w h i c h p r e d i c t t h e development of b a r o c l i n i c f e a t u r e s , s u c h a s f r o n t s and e d d i e s , i n l i m i t e d sea areas, p a r t i c u l a r l y i n t h e s h e l f seas, are r e q u i r e d f o r s e v e r a l p u r p o s e s . F o r example, t h e c i r c u l a t i o n i n t h e s e f e a t u r e s a f f e c t s b i o l o g i c a l a c t i v i t y a n d c a n d e t e r m i n e t h e p a t h s o f p o l l u t a n t s . Also, t h e e n h a n c e d c u r r e n t s i n e d d i e s c a n a f f e c t o f f s h o r e e n g i n e e r i n g , a s i n ti? Norwegian Coastal C u r r e n t , where n e a r - s u r f a c e v e l o c i t i e s of o v e r 1 . 5 m s have been recorded. Fundamental problems w i t h m o d e l s of t h i s k i n d are t h e i n c l u s i o n of b o t h b a r o t r o p i c a n d b a r o c l i n i c flows, w i t h t h e i r d i f f e r e n t time s c a l e s , t h e need t o r e t a i n s h a r p f r o n t a l g r a d i . e n t s w i t h o u t r i p p l i n g o r e x c e s s i v e s m e a r i n g and t h e The model d e s c r i b e d h e r e u s e s a t r e a t m e n t of open boundary c o n d i t i o n s . f i n i t e - d i f f e r e n c e s i g m a c o o r d i n a t e g r i d box scheme, a s p l i t i n t o b a r o t r o p i c and b a r o c l i n i c modes w i t h a s e m i - i m p l i c i t scheme f o r t h e b a r o t r o p i c p a r t , a h y b r i d a d v e c t i o n scheme t o r e t a i n p o s i t i v i t y a n d s h a r p n e s s a t f r o n t s and partly prescribed, partly radiation boundary c o n d i t i o n s . Arbitrary f o r m u l a t i o n s of v e r t i c a l eddy v i s c o s i t y a n d d i f f u s i v i t y , v a r y i n g i n s p a c e and time, may be used. The c o d i n g is d e s i g n e d f o r e f f i c i e n t u s e o n t h e Cray-1s c o m p u t e r . The r e s u l t s show t h e a b i l i t y of t h e model t o s i m u l a t e t h e e v o l u t i o n of e d d i e s i n a c o a s t a l c u r r e n t f r o n t . 1
INTRODUCTION A f u n d a m e n t a l l e n g t h s c a l e f o r m o t i o n s i n s t r a t i f i e d seas i s t h e b a r o c l i n i c 1
where g' is t h e r e d u c e d g r a v i t y
Rossby r a d i u s of d e f o r m a t i o n ( g ' h ' ) ' / f , (=
gAp/po f o r a two l a y e r f l u i d , w h e r e Ap is t h e d e n s i t y d i f f e r e n c e and p
r e f e r e n c e d e n s i t y ) , h ' i s a d e p t h scale upper
layer
This
length
to
speed
the
energetic implying
thickness
scale
parameter,
eddies
for
a n e d d y - r e s o l v i n g model.
and
stir
atmosphere.
seas
shelf
coasts. but
water
the
hl)/H for a two-layer f l u i d ,
t o t a l d e p t h H), and f is t h e C o r i o l i s p a r a m e t e r .
hl,
seen
:
is
and in
d i a m e t e r s from 5 t o 50 km. the
-
is t h e r a t i o of t h e ( f i r s t mode) l o n g i n t e r n a l wave p h a s e
Coriolis
baroclinic
( = h,(H
the
satellite
typical
s c a l e of t h e most
images
of t h e s h e l f seas,
This determines the resolution necessary
S h a r p g r a d i e n t s are formed a s t h e e d d i e s a d v e c t
this
process
is
analogous to frontogenesis i n t h e
I t c a n e n h a n c e a l r e a d y large h o r i z o n t a l g r a d i e n t s produced i n t h e by
variations
in
tidal
mixing
and f r e s h water o u t f l o w a t t h e
T h e r e f o r e t h e i d e a l model n o t o n l y r e s o l v e s t h e Rossby r a d i u s scale
also allows s h a r p g r a d i e n t s t o d e v e l o p a n d p r o p a g a t e r e a l i s t i c a l l y .
This
problem
shared
is
with
prediction
weather
models, w h i c h , a l t h o u g h e a s i l y
r e s o l v i n g t h e major d e p r e s s i o n s a n d a n t i c y c l o n e s , b e n e f i t i n t h e n e i g h b o u r h o o d
of
fronts
from
the
limits
sub-regions boundary
increased
feasibility the
of
resolution. of
the
The r e q u i r e m e n t of h i g h r e s o l u t i o n with
models,
present-day
computers,
to
s h e l f , s o t h e s o l u t i o n w i l l depend o n t h e choice o f open
conditions.
r e l a t i v e l a c k of m o n i t o r i n g of t h e sea (compared
The
t h e a t m o s p h e r e ) limits t h e d e v e l o p m e n t of a f o r e c a s t i n g model, b u t i t i s
with
to
possible
compare
t h e model r e s u l t s q u a l i t a t i v e l y w i t h w h a t is known from
o b s e r v a t i o n s , a i d e d p a r t i c u l a r l y by remote s e n s i n g , and l a b o r a t o r y m o d e l s . are
There
seas.
many
potential
Transport
stirring
by
concentration while
the
d i s p e r s a l of p o l l u t a n t s are l i k e l y t o be d e p e n d e n t on
and
c o n v e r g e n t flow.
by
of
instability
transfer,
which
i n many r e g i o n s , w h i l e f r o n t s are p r o b a b l e areas of
eddies
the
a p p l i c a t i o n s of e d d y - r e s o l v i n g m o d e l s of s h e l f
may
There is r a p i d t r a n s p o r t i n f r o n t a l jets,
fronts
determine
provides
a
mechanism
for c r o s s - f r o n t a l
q u i c k l y p o l l u t a n t s are d i s p e r s e d o u t o f
how
coastal r e g i o n s i n t o t h e o p e n sea. I t is n o t p o s s i b l e t o p a r a m e t e r i z e a l l t h e s e e f f e c t s i n a c o a r s e - g r i d model. F r o n t a l z o n e s are known t o be biologically regions an
show
cause
as
be
mixed o r u p w e l l e d i n t o
a d e q u a t e l i g h t for phytoplankton growth.
Pingree e t a l . (1979)
productive
with
of
example
cross-frontal
v e r t i c a l motions.
here
can
p r o d u c t i o n i n a f r o n t a l eddy.
enhanced transfer
nutrients
of
The e d d i e s may
n u t r i e n t s , a n d may be e x p e c t e d t o i n c r e a s e
Enhanced h o r i z o n t a l v e l o c i t i e s i n e d d i e s c a n c a u s e problems
to
o f f s h o r e e n g i n e e r s : n e a r - s u r f a c e flows h a v e been m e a s u r e d a t o v e r 1.5ms-1
in
e d d i e s i n t h e Norwegian Coastal C u r r e n t i n areas of o i l a n d gas p r o d u c t i o n
( J a m e s and the
McClimans
1983,
et
Carstens
al. 1984).
The c u r r e n t i n c r e a s e s
( o v e r a few h o u r s ) t o t h e s e v a l u e s as t h e e d g e of t h e e d d y moves p a s t
rapidly
site,
measuring
so
h e r e t h e r e is a c l e a r a p p l i c a t i o n f o r a f o r e c a s t i n g
model. 2
MODEL EQUATIONS The
model
to
is
based
on t h e p r i m i t i v e equations w i t h t h e Boussinesq, incompressible,
It
hydrostatic
and
for d e n s i t y
p
eddy
f-plane
approximations.
and
A
mixing.
diffusivity Mellor
As
diffusion
should
not
horizontal
eddy
scale
be
fronts.
and
required
dictated
t o p o g r a p h y ) , as is t h e case h e r e . out
They i n c l u d e a p r o g n o s t i c e q u a t i o n
i n t h e form of bucyancy b = g ( p
viscosity
horizontal
be d e s c r i b e d h e r e i s a d e v e l o p m e n t of t h a t of James ( 1 9 8 6 ) .
by
are
K
the
p)/po.
Arbitrary vertical
assumed, b u t t h e r e i s no imposed
Blumberg
for
-
models Rossby
(1985) point out, horizontal that
resolve
t h e dominant
r a d i u s ( o r v a r i a b l e bottom
An imposed h o r i z o n t a l d i f f u s i o n would smear
T h e r e d o r e m a i n p o s s i b l e n u m e r i c a l p r o b l e m s s u c h as a t e n d e n c y
593
for
nonlinear
sharp
terms to d r i v e e n e r g y t o w a r d s t h e g r i d scale a n d r i p p l i n g n e a r
gradients,
term.
which
can
b e masked by i n t r o d u c i n g a h o r i z o n t a l d i f f u s i o n
However, t h e aim h e r e is t o u s e a n u m e r i c a l scheme w h ic h t a k e s care of
t h e s e p r o b l e m s w i t h , i f p o s s i b l e , a minimum of n u m e r i c a l d i f f u s i o n . The
equations
in
a r e c t a n g u l a r ( x , y , z , t ) coordinate system, right-handed
w i t h z u p w a r d s , are t h e n
L(1)
= 0,
(5)
w h e r e t h e o p e r a t o r L is d e f i n e d by
A
transformation
surface points depth
z in
to
sigma c o o r d i n a t e s
allows
the
i n c l u s i o n of a free
= c ( x , y , t ) a n d b a t h y m e t r y z = - h ( x , y ) w i t h t h e same number of g r i d the
vertical
at
each p o s i t i o n , a r b i t r a r i l y spaced.
becomes a constant i n t e r v a l i n 0
, with
The v a r y i n g
= 0 on t h e surface a n d
= -1
o n t h e sea be d . The transformation is t o (a,B,u,~) c o o r d i n a t e s , w i t h
where H ( x , y , t ) = h + c i s t h e t o t a l d e p t h . The t r a n s f o r m e d e q u a t i o n s are:
(10)
(11)
594
(13)
@ = Pa/Po + g 5 where P
-
H
,E
b do
(14)
i s a t m o s p h e r i c p r e s s u r e and t h e o p e r a t o r
is d e f i n e d by
I\
The v e r t i c a l v e l o c i t y w i n t h e o r i g i n a l c o o r d i n a t e s is
3
THE G R I D SCHEME AND TIME STEPPING
The
Arakawa
corners
of
a
(1986)
for
a
this
B-grid,
which
u
and v d e f i n e d a t common p o i n t s a t t h e
is u s e d t o d e f i n e t h e f i n i t e - d i f f e r e n c e e q u a t i o n s i n
diagram)
model.
has
s q u a r e , w i t h b , h and 5 d e f i n e d a t t h e c e n t r e ( s e e James
grid
was
This
found
t o be much more s a t i s f a c t o r y t h a n t h e C-grid
( w h i c h h a s u and v d e f i n e d a t t h e c e n t r e s of a d j a c e n t s i d e s of t h e s q u a r e ) f o r maintaining a sharp f r o n t i n approximate geostrophic balance. (1987,
this
C-grid, This
in
colloquium) near
J a m a r t and Ozer
a similar problem of s p u r i o u s flows w i t h t h e
n e x t t o a r i g i d boundary i n a r e c t a n g u l a r sea model.
case
their
trouble
found
boundaries,
whether
rigid
or
f r o n t a l , arises from t h e
a v e r a g i n g n e c e s s a r y i n t h e C o r i o l i s term o n t h e C - g r i d . The
v a r i a b l e s have p o s i t i o n s d e n o t e d by a s i n g l e i n d e x r u n n i n g ' t h r o u g h t h e
three-dimensional time-stepping Points
grid.
provides
long
inner
do-loops
in
the
of t h e c a l c u l a t i o n which are v e c t o r i z a b l e o n t h e Cray-1s.
part
outside
This
the
calculation
area
are
t h e n d i s c a r d e d ( p u t t o z e r o ) and
boundary p o i n t s m o d i f i e d b e f o r e t h e n e x t time s t e p . leapfrog
A
the
scheme
unconditionally
terms.
The
is u s e d f o r t h e time d i f f e r e n c i n g .
T h i s allows use o f
for
t h e v e r t i c a l mixing
stable
separation
l e a p f r o g methods
is
of
DuFort-Frankel solutions
suppressed
on
method
odd and e v e n time s t e p s found i n
by t h e time f i l t e r of A s s e l i n (19721, which
r e p l a c e s v a l u e an a t time l e v e l n by t h e v a l u e n n+l a n = an + 4 (al"-' - 2 a + a (17) 1 n+ 1 a f t e r t h e new v a l u e a h a s been c a l c u l a t e d . A s s e l i n ( 1 9 7 2 ) showed t h a t any W
>
0
damps t h e unwanted c o m p u t a t i o n a l mode and t h a t v
s h o u l d b e t a k e n less
595
than
1.
The
comparison
higher
values
w
of
damp t h e h i g h e r f r e q u e n c i e s more.
of t h e r e s u l t s w i t h w = 0.1 and
I,
A
= 0.8 f o r t h e problem shown h e r e
confirms t h i s , t h e changes being very s l i g h t . 4
SEMI-IMPLICIT TREATMENT OF THE BAROTROPIC TERMS The
is
extremely
James
(1986)
part,
and
the
(CFL) c o n d i t i o n o n t h e time s t e p f o r e x p l i c i t
Courant-Friedrichs-Levy
methods
problem
to
seconds
the
the
restrictive
equations
for
f i n e g r i d s and b a r o t r o p i c waves.
In
s p l i t i n t o a gravity-wave ( o r b a r o t r o p i c )
were
r e m a i n d e r , which were s o l v e d u s i n g d i f f e r e n t ' t i m e s t e p s .
time
steps
For
had t o b e as l i t t l e as t e n
shown,
the
barotropic
Satisfy
the
CFL c o n d i t i o n , w h i l e t h e CFL c o n d i t i o n b a s e d on t h e
baroclinic
( i n t e r n a l ) wave p h a s e s p e e d a l l o w e d t h e l o n g e r time s t e p t o b e s i x
minutes.
A s i g n i f i c a n t improvement i n c o m p u t a t i o n time h a s been a c h i e v e d by
solving the
the
b a r o t r o p i c p a r t by a s e m i - i m p l i c i t method.
advantage
that
the
scheme
T h i s method a l s o h a s
can be chosen t o s u p p r e s s two-grid-interval
n o i s e , which c a n b e t r o u b l e s o m e o n t h e B-grid. The v e l o c i t y is d i v i d e d as follows:
and
a(Hur)/aT
= Hfvr +
+ H - ~ a ( A aur/au)/aa
(20) - F S + F B' w h e r e A M c o n t a i n s n o n l i n e a r and b a r o c l i n i c t e r m s and FS and FB are s u r f a c e and bottom stress. T h e r e are similar e q u a t i o n s f o r V a n d v r' g The s e m i - i m p l i c i t scheme is similar t o t h a t o f Kwizak and R o b e r t ( 1 9 7 1 ) . The v a r i a b l e s
ug' -vg and
5 are f o u n d a t time l e v e l n+l from
596 have b e e n n e g l e c t e d , 6 t is t h e b a r o t r o p i c time s t e p and
Here, v a r i a t i o n s i n P
-
-
‘t-9
‘r
are
integrals
and
u
of
v
r
over
the
depth
with respect to
U.
S u b s t i t u t i o n of ( 2 1 ) a n d ( 2 2 ) i n t o ( 2 3 ) l e a d s t o a n i m p l i c i t e q u a t i o n f o r g n + l , which c a n be s o l v e d by r e l a x a t i o n .
To allow v e c t o r i z a t i o n , s i m u l t a n e o u s n+ 1 r e l a x a t i o n is u s e d , w i t h g n as a f i r s t g u e s s f o r 5 Boundary v a l u e s of n+ 1 are r e q u i r e d : a n e x p l i c i t boundary c o n d i t i o n which f i x e s t h i s v a l u e throughout
relaxation
the
process
.
speeds
convergence.
Over-relaxation is
a l l o w e d , b u t t h e optimum p a r a m e t e r i s n o t o b v i o u s . The e q u a t i o n f o r g i n v o l v e s terms of t h e form
a (Hag/aa)/aa
+
a (sa
(24)
which c a n b e f i n i t e - d i f f e r e n c e d i n s e v e r a l ways.
The v a l u e a t a p a r t i c u l a r
< - p o i n t i n v o l v e s v a l u e s of 5 a t some o r a l l of t h e e i g h t s u r r o u n d i n g < - p o i n t s . The
two-grid
false
a
from
of
two
these
is
(24)
representation points)
n o i s e , w i t h its chess-board appearance, arises
interval
of s o l u t i o n s o n two s u b s e t s of t h e g r i d .
separation
used
solutions
is
used
interval
is
suppressed.
(
wave
( t h e c e n t r e p o i n t and t h e f u r t h e s t c o r n e r
remain unconnected.
I f , however, a n i n e - p o i n t
are
c o n n e c t e d and t h e t w o - g r i d
two
representation
the
If a f i v e - p o i n t
solutions This
is
s i m i l a r t o t h e scheme of M e s i n g e r
1973). T h i s s e m i - i m p l i c i t scheme may s t i l l i n v o l v e a d i f f e r e n t time s t e p from t h a t t h e b a r o c l i n i c e q u a t i o n (201, b u t f o r t h e r e s u l t s shown h e r e t h e same time
of
s t e p ( s i x m i n u t e s ) was u s e d .
T h i s n e c e s s i t a t e d a c h a n g e i n t h e a r r a n g e m e n t of
time l e v e l s shown i n James ( 1 9 8 6 ) s o t h a t t h e b a r o t r o p i c a n d b a r o c l i n i c l e v e l s 3 were c o n c u r r e n t .
5
H Y B R I D ADVECTION SCHEME
The h y b r i d a d v e c t i o n scheme u s e d h e r e was d e s c r i b e d by James ( 1 9 8 6 ) .
It is
i n t e n d e d t o g i v e some of t h e a d v a n t a g e s of t h e f l u x - c o r r e c t i o n scheme of B o r i s and
Book
(1976)
scheme
would
leading
to
diffusion.
with
much
less com put atio n al effort.
The i d e a l a d v e c t i o n
combine
p o s i t i v i t y ( s o there would b e no u n r e a l i s t i c r i p p l i n g ,
obviously
wrong d e n s i t y v a l u e s , n e a r a f r o n t ) w i t h low n u m e r i c a l
The
upwind
scheme h a s p o s i t i v i t y b u t h i g h n u m e r i c a l d i f f u s i o n .
Both
t h e f l u x - c o r r e c t i o n scheme a n d t h e h y b r i d scheme c a l c u l a t e t h e a d v e c t i v e
flux
as
upwind
a w e i g h t e d a v e r a g e of t h a t from a l o w - o r d e r , differencing
and
a
higher
p o s i t i v e scheme s u c h as
o r d e r scheme, h e r e c e n t r e d d i f f e r e n c i n g .
Hence, F l u x = C x ( f l u x from upwind s c h e m e )
+ (1-C) x ( f l u x from c e n t r e d s c h e m e ) The
(25)
d i f f e r e n c e - b e t w e e n t h e f l u x - c o r r e c t i o n a n d t h e h y b r i d scheme l i e s i n t h e
method of c h o o s i n g C.
Here, C is s i m p l y d e f i n e d from t h e b f i e l d from a t e s t ,
597
to
due
Harten
(1978),
the
for
e d g e of f r o n t s , where t h e d e n s i t y g r a d i e n t
The v a l u e of C d e f i n e d a t t h e b - p o i n t l a b e l l e d i , f o r t h e a - d i r e c t e d
changes.
f l u x , i s g i v e n by
>
for S A.
1+;
i+l
-
b = b.
1+1
being
as
defined
where
+ the
of
point
next
to
i
i n t h e +a d i r e c t i o n .
The
E
allow some f u r t h e r c o n t r o l of t h e s h a r p n e s s of t h e s w i t c h
Values
of C a t p o i n t s r e q u i r e d f o r t h e f l u x c a l c u l a t i o n s c a n be
maxima of
definition front,
label and
p
C.
E,
b . and S =
the
parameters
factor
= 0 for S 5
a n d Cai
E
where
those at the appropriate neighbouring b-points.
of
ensures
C
the
that
gradient
of
This
t h e upwind scheme is u s e d n e a r t h e e d g e of a b
suddenly
changes
from
a
low v a l u e , s o
t h e c e n t r e d scheme is u s e d when d e n s i t y g r a d i e n t s
preventing
ripples,
are
James ( 1 9 8 6 ) showed t h a t t h i s h y b r i d scheme h a s s i g n i f i c a n t l y l e s s
low.
numerical
while
diffusion
adjustment
p
of
than
can
the
upwind
scheme,
w h i l e l a t e r i t is shown t h a t
c o n t r o l t h e amount of n u m e r i c a l d i f f u s i o n .
The h y b r i d
scheme is u s e d i n t h i s model f o r t h e a d v e c t i v e terms f o r a l l v a r i a b l e s , i n a l l three directions. 6
OPEN BOUNDARY CONDITIONS Open
eddies
boundary
conditions
generated
within
the
f o r a l i m i t e d area model s h o u l d allow waves a n d model t o p a s s o u t t h r o u g h t h e boundary w i t h o u t
r e f l e c t i o n , w h i l e p r e s e r v i n g t h e i n f l u e n c e of known e x t e r n a l v a l u e s , s a y those
a
area.
obtained
from
boundary
condition
model.
R a d i a t i o n c o n d i t i o n s , w h i c h allow i n f l u e n c e s t o p a s s o u t t h r o u g h t h e
coarse-grid is
not
model
of
the
surrounding
This ideal
known f o r a g e n e r a l t h r e e - d i m e n s i o n a l s t r a t i f i e d
were p r o p o s e d by O r l a n s k i ( 1 9 7 6 ) . They are b a s e d o n t h e Sommerfeld r a d i a t i o n c o n d i t i o n , w h i c h f o r v a r i a b l e a a t a n a = c o n s t a n t boundary is
boundary,
a a l a t + caalaa = 0 (27) p h a s e s p e e d ( o r a d v e c t i o n v e l o c i t y ) c i s c a l c u l a t e d from i n t e r i o r p o i n t s
The
next
to
the
approaching
boundary, u s i n g ( 2 7 ) . the
I n e f f e c t , t h i s is l o o k i n g f o r a p a t t e r n
b o u n d a r y and a s s u m i n g t h a t i t c o n t i n u e s across t h e boundary. 1
If
for
t h e problem i m p l i e s a p h a s e v e l o c i t y known from t h e o r y ( f o r example, ( g h ) ' barotropic
Kelvin
w a v e s ) t h i s may b e s u b s t i t u t e d for c , b u t t h i s is n o t
t h e case h e r e . Hence
if
the
unknown b o u n d a r y v a l u e a t time l e v e l n+l h a s l a b e l i , i t is
598
related
interior
to
i-1 a n d i - 2 ( f o r t h e example of a n a = c o n s t a n
points
boundary w i t h a i n c r e a s i n g o u t of t h e model a r e a ) by n+ 1 n-1 = [(1-p)ai + 2p an. I / (1+p)
a.
( 28
1- 1
n-1 where p = ( a1-1 .
-
n+l aiml
n+l
n-1 + a1-1 .
/ (ai-l
-
2ani-2),
u n l e s s t h i s p i s n e g a t i v e , i n which case there is i n f l o w and p i s p u t e q u a l t o z e r o , o r t h i s p > 1 , i n w h i c h case p is p u t e q u a l to 1 . T h i s is t h e i m p l i c i t n+l v e r s i o n of t h e O r l a n s k i b o u n d a r y c o n d i t i o n , which is u s a b l e i f ai-l is a l r e a d y n+ 1 calculated before a. is needed. The e x p l i c i t form c a l c u l a t e s p from o n e
earlier.
level
time
interpolation point.
Chapman
barotropic overall
Whatever
formula
model
(with
a
value
the
of p, ( 2 8 ) t a k e s t h e form of a n
i n v o l v i n g v a l u e s a t e a r l i e r times and t h e n e i g h b o u r i n g
(1985) and
has
found
sponge
v a r i o u s open boundary c o n d i t i o n s f o r a
tested this
added
Orlanski
c o n d i t i o n t h e most s a t i s f a c t o r y
f o r a waves o n l y c a s e ) .
Blumberg and Kantha
1
a r a d i a t i o n c o n d i t i o n w i t h c = ( g h ) ' and found a s t e a d y emptying
(1985)
used
of t h e
model b a s i n .
to
right
the
case,
To p r e v e n t t h i s , t h e y i n t r o d u c e d a n e x t r a term ( a - a ) / T f
hand
surface
s i d e of ( 2 7 1 , t o b r i n g t h e b o u n d a r y v a l u e of a ( i n t h e i r
elevation)
back
to
a g i v e n v a l u e ao, w i t h time c o n s t a n t Tf '
T h i s may b e t e r m e d a " l o o s e l y clamped" b o u n d a r y , where "clamped" refers t o t h e
=
case Tf equation
0
=
a
(so
(27)
and " r a d i a t i o n a l " t o t h e case Tf i n f i n i t e , when
ao)
Chapman ( 1 9 8 5 ) presumed t h e b a s i n e m p t y i n g to b e t h e
holds.
1
r e s u l t of ( g h ) ' b e i n g a n o v e r e s t i m a t e of c. A
made
further by
development
Raymond
direction
rather
possibility,
for
the
boundary,
and
then
and
Kuo
than
of t h e O r l a n s k i r a d i a t i o n a l boundary c o n d i t i o n was
(1984),
simply
the B-grid,
who
normal
allowed
to
for r a d i a t i o n i n a general
t h e b o u n d a r y as i n ( 2 7 ) .
is t o l i n e a r i s e t h e ur a n d v
Another
equations next to
d e t e r m i n e 5 and b o n t h e b o u n d a r y from t h e b o u n d a r y c o n d i t i o n s
calculate
t h e l i n e a r i s e d u a n d v.
However, i t h a s been f o u n d more
s a t i s f a c t o r y h e r e t o a p p l y t h e o p e n b o u n d a r y c o n d i t i o n s t o v e l o c i t y as well as 5 and
b,
as
avoids
u n r e a l i s t i c spurious v e l o c i t i e s running along the
C l e a r l y , t h e r e are numerous p o s s i b l e o p e n b o u n d a r y c o n d i t i o n s , t h e
boundary. best
this
of which is l i k e l y t o depend o n t h e problem i n h a n d , a n d t h e q u e s t i o n i s
still
a n a c t i v e t o p i c of r e s e a r c h .
current
with
F o r t h e r e s u l t s shown h e r e , f o r a coastal
a g i v e n i n f l o w , s e v e r a l d i f f e r e n t o u t f l o w c o n d i t i o n s were t a k e n
t o test t h e s e n s i t i v i t y of t h e model t o them. 7
RESULTS OF A COASTAL CURRENT MODEL
This dense
model
fluid
was into
applied
a
to t h e problem of t h e release of a p a t c h of less
coastal c u r r e n t , a n d i t s s u b s e q u e n t d e v e l o p m e n t i n t o a n
599
200 m
Basic configuration of the model region.
Fig.1.
- - - - -O ;5I
100.
O"""""""...
. . .
I
600
150
I m/sec 100
50
1 0
~~
I
:
. .
;
W
'
I
l
l
'
1
1
1
'
I
1
1
1
;
, l l , . . . . . . . . . . . . .
, l l , . . . . . . . . . " . ' ' l , , . ' . ' . . . . . . . . .
~
I
50
100
I
150
200 km
Fig.2(b).
Surface current vectors.
Fig.2(c).
Perspective plot of the surface elevation, viewed from offshore.
601 eddy.
The
to
2(a)
shown i n F i g . 1 , are similar t o those of t h e
conditions,
C o a s t a l C u r r e n t i n t h e Norwegian T r e n c h , w i t h a bottom s l o p i n g from
Norwegian 200
basic
m
300
shows
and a d e n s i t y c h a n g e a c r o s s t h e c u r r e n t of o n e u t u n i t .
d e n s i t y c o n t o u r s w i t h a n i n t r o d u c t i o n of less d e n s e
surface
the
w a t e r i n a p a t c h n e a r t h e coast. model
Fig.
T h i s is taken a s t h e i n i t i a l d e n s i t y for t h e
The i n i t i a l c u r r e n t s ( t h e s u r f a c e v a l u e s are shown i n F i g . 2 ( b ) )
run.
and t h e i n i t i a l s u r f a c e e l e v a t i o n ( F i g . 2 ( c ) ) are c a l c u l a t e d o n t h e a s s u m p t i o n of g e o s t r o p h i c flow w i t h z e r o v e l o c i t y a t t h e sea bed.
start to t h e calculation.
T h i s p r o v i d e s a smooth
The i n i t i a l c h a n g e i n l e v e l across t h e f r o n t a t t h e
i n f l o w ( o r o u t f l o w ) boundary was 1 0 . 3 cm. boundary c o n d i t i o n s used i n o b t a i n i n g t h e r e s u l t s shown i n F i g . 3 ( t h e
The surface days)
density,
surface
and
bottom c u r r e n t s and s u r f a c e e l e v a t i o n a f t e r 6
appropriate t o a c o a s t a l c u r r e n t c o n t r o l l e d at t h e inflow: inflow
were
and d e n s i t y p r e s c r i b e d , z e r o g r a d i e n t of e l e v a t i o n a t t h e i n f l o w and
velocity Orlanski
radiation vertical
little
coefficient
at
(A=K=20 cm2
0.0025,
of
two r e m a i n i n g open b o u n d a r i e s .
the
mixing
= 10
E
quadratic
These r u n s had a bottom
friction
time f i l t e r w i t h V = 0.8, and h y b r i d s w i t c h -4 The g r i d had 3 2 x 3 2 ( h o r i z o n t a l ) x 1 2
Asselin
p a r a m e t e r s p = 1 and
factor
a
s-'1,
.
( v e r t i c a l ) p o i n t s and 5 km h o r i z o n t a l r e s o l u t i o n . These coast
results
Current
eddies
T h i s a g r e e s w i t h t h e s u r f a c e prominence of a n t i c y c l o n i c
( C a r s t e n s e t a l . 1 9 8 4 ) and t h e l a b o r a t o r y o b s e r v a t i o n s of G r i f f i t h s (1981)
anticyclonic, gives
pair
t a k e n t o be t h a t f o r 60°N). I t h a s d e v e l o p e d i n t o an
is
(f
velocities.
Linden
and
of t h e o r i g i n a l d i s t u r b a n c e a l o n g t h e
t o g e t h e r w i t h a c y c l o n i c eddy which is more e v i d e n t i n t h e
"whirls" and
progression
eddy
anticyclonic bottom
a
show
a r a t e of t h e o r d e r of 10 km d - l , as o b s e r v e d f o r Norwegian C o a s t a l
at
the
structure
different boundary present
scheme
these
in
the
upper
"backwards-breaking"
seen
also
numerical
motion
in
based
the
primarily
on
appearance.
left
This vortex
model of James ( 1 9 8 4 ) , which u s e s a a
rigid-lid
a s s u m p t i o n and p e r i o d i c
and a s i n u s o i d a l i n i t i a l d i s t u r b a n c e .
results,
is
layer
i n t h e lower l a y e r e n t r a i n s u p p e r l a y e r f l u i d
motion
characteristic was
conditions, in
while
that
cyclonic
behind
at
the
position
O t h e r e d d i e s are of t h e o r i g i n a l
d i s t u r b a n c e , and a h e a d of t h e main v o r t e x p a i r .
latter
The boundary initial
value
solution
eddy
condition. except
(Fig. very
may
be
thought
However,
with
to
b e rather d e p e n d e n t on t h e o u t f l o w
the
o u t f l o w b o u n d a r i e s clamped a t t h e
4 ) t h e r e is v e r y l i t t l e d i f f e r e n c e from t h e r a d i a t i o n a l near
the
boundary.
T h i s r e s u l t , of n e a r - i d e n t i c a l
s o l u t i o n s , was f o u n d a l s o f o r t h e l o o s e l y - c l a m p e d c o n d i t i o n , w i t h Tf between 1 and these
5
hours. boundary
The
c o n c l u s i o n is t h a t t h e model is i n s e n s i t i v e t o which of
conditions
is
taken,
a t l e a s t up t o t h e times shown.
c o n c l u s i o n may c h a n g e as t h e main d i s t u r b a n c e r e a c h e s t h e boundary.
This
602
1
1
150--
100-.
50 -.
O !
'
.
.
.
0 Fig.3.
i
.
.
.
A f t e r 6 days.
. . . . . . . . . . . . ..
. . . . . . . . . . . . ..
.
. i
.
100
50
.
.
i
150
' ' ' ' 1
200 km
( a ) Surface density.
............................ ............................ ............................ ............................ ............................ ............................ ............................ ............................ ............................ ............................ ............................ ............................ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
1501r7 l 100
Im/sec
.............................. ..............................
.............................. ....................,._....... .... ,_.._ _------- _ - .. . . .... . . . . ---, -,,,// .. I
,
....
------_--_. . . . .----.,,
/ / /
#
I
I
# .
. . . . . ,*..,,............. .........,,................... ...._____.._-----C . . . . - _ - - - - -
0 Fig.3(b).
50
Surface currents.
100
150
200 km
603
I50
100
.............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. .............................. ..............................
Im/sec'
.............................. ............................... .............................. . . . . . . . . . . . . . . . . _ _ . . . ......... . . . . - - - - ,,,,,--.\\ . . I ,_.... ... I .....,//,... ,\\,..\\,.-... -. . . . . . . . . . , \ , r , I , . . \ \ \ \ . . - . 50 ........ \\\.... . . . . . . . r , . . , , , . . - . . r r . . . , \ \ . - , . . . . . . . . .- - , - ____ , .. .. ., , , ... .*-. . . . . . . . . _ _ _ . . ,,. . ....-.. . .........,,,,.....,/.......... ........., . ..............................
__
I
.
I
,
,
-
-
.
.
.
.
I
1
.
,
,
.
- . . - - - * - - . , i . .
0
,
L
I
,
.
Fig.3(c).
Bottom currents.
Fig.3(d).
Surface elevation.
-
-
-
-
.
,
%
\
-
-
.
I
I
,
.
.
I
604
-
50
0 Fig.4.
.
.
.
.
.
.
.
150
100
After 6 days with the clamped boundary condition.
I50 . . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . .
............................. ............................. ............................. ............................. .............................
.............................. ............................. .............................. ............................. . . . . . . . . _ . . . . . . . . . . . . ......... .... - -. .-----,---. . . . . . . . . ,, . . - ---///-.. . . a
,
/-.. ....
d
~
~
~
.
.
’
,
200 km ( a ) Surface density
605
150
100 L
50
0 50
0 Fig.5.
100
150
A f t e r 6 d a y s , f o r t h e f r i c t i o n l e s s case w i t h p = 4 .
.............................. 150-. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ............................... .............................. ............................... .............................. ............................... .............................. . . .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. lo()-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.............................. ............................... .............................. . . .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. . . . . . . . . . . . .. .. .. .. . . . . .-.- ..... . - ... I //..&--. . . . I
I
4
c..
/
-
-
-
.
.
.
.
.
.
200 km ( a ) Surface density.
Im/sec
606
150 --
Im/sec
too
50 -.
Fig.5(c).
Bottom c u r r e n t s .
Fig.5(d).
Surface e l e v a t i o n .
607
The the
different
mean
level
Blumberg
and
o u t f l o w b o u n d a r y c o n d i t i o n s d o p r o d u c e small d i f f e r e n c e s i n (of
the
Kantha
o f a c e n t i m e t r e , much smaller t h a n t h o s e of
order
(1985)).
Such d i f f e r e n c e s i n l e v e l c a n be p r o d u c e d by
small d i f f e r e n c e s i n t h e c u r r e n t s a t t h e b o u n d a r i e s .
The
region
the
from
coast where d e n s i t y is u n i f o r m h a s b a r o t r o p i c
T h i s r e m a i n s much smaller t h a n t h e flow i n t h e coastal c u r r e n t .
motion o n l y . This
away
model
is
able
to
run
and b o t t o m stresses z e r o ) .
surface
h a p p i l y w i t h z e r o imposed f r i c t i o n ( A , K and The r e m a i n i n g damping i s d u e t o n u m e r i c a l T h i s c a n t h e n be a d j u s t e d t o as low
d i f f u s i o n and t h e A s s e l i n time f i l t e r i n g .
a
level
possible.
as
S u c h a f r i c t i o n l e s s case, w i t h
shown i n F i g . 5, w i t h r a d i a t i o n a l o p e n b o u n d a r i e s . shows
directed
a greater a n g l e t o t h e c o a s t .
at
Compared w i t h F i g . 3, t h i s
s h a r p e r r e s u l t , w i t h l a r g e r b o t t o m c u r r e n t s , and a v o r t e x p a i r
much
a
v = 0.1 and p = 4 , is
A s i g n i f i c a n t p a r t of t h i s c h a n g e
d u e t o t h e i n c r e a s e i n t h e h y b r i d s w i t c h p a r a m e t e r p.
is
This underlines the
i m p o r t a n c e of t h e a d v e c t i v e terms i n b a r o c l i n i c e d d y - r e s o l v i n g models. 8
CONCLUSIONS For
This of
problem, t h e model r a n i n a b o u t 40 s e c o n d s p e r day on a Cray-1s.
this
shows t h e f e a s i b i l i t y , w i t h p r e s e n t a n d f o r t h c o m i n g c o m p u t e r f a c i l i t i e s ,
eddy-resolving
shelf
and
again
been
primitive
equation
baroclinic
m o d e l s f o r r e g i o n s of t h e
f o r time p e r i o d s from s e v e r a l d a y s up t o weeks.
A vortex p a i r has
f o u n d t o r e s u l t from t h e d i s t u r b a n c e , and s u b s e q u e n t i n s t a b i l i t y ,
of a f r o n t .
Open
boundary
conditions
and
a d v e c t i o n schemes r e m a i n areas f o r f u r t h e r
i n v e s t i g a t i o n : h e r e , r a d i a t i o n a l o p e n b o u n d a r i e s and a h y b r i d a d v e c t i o n scheme have been found u s e f u l . 9
REFERENCES
A s s e l i n , R., 1972. F r e q u e n c y f i l t e r f o r time i n t e g r a t i o n s . Mon.Weath.Rev., 100: 487-490. Blumberg, A.F. and K a n t h a , L.H., 1985. Open boundary c o n d i t i o n f o r c i r c u l a t i o n m o d e l s . J.Hydraul.Engng., ASCE, 1 1 1 : 237-255. B o r i s , J . P . and Book, D.L., 1976. S o l u t i o n of c o n t i n u i t y e q u a t i o n s by t h e method of f l u x - c o r r e c t e d t r a n s p o r t . Methods Comput.Phys., 16: 85-129. C a r s t e n s , T., McClimans, T.A. and N i l s e n , J . H . , 1984. S a t e l l i t e i m a g e r y of b o u n d a r y c u r r e n t s . I n : J.C.J. N i h o u l ( E d i t o r ) , Remote S e n s i n g of S h e l f S e a Hydrodynamics. E l s e v i e r Oceanography Series, 38. E l s e v i e r , Amsterdam/Oxford/New York/Tokyo, pp. 235-256. Chapman, D . C . , 1985. N u m e r i c a l t r e a t m e n t of c r o s s - s h e l f open b o u n d a r i e s i n a b a r o t r o p i c c o a s t a l o c e a n model. J.Phys.Oceanogr., 15: 1060-1075. G r i f f i t h s , R.W. and L i n d e n , P.F., 1981. The s t a b i l i t y of buoyancy d r i v e n coastal c u r r e n t s . Dynam.Atmos.Oceans, 5: 281-306.
608
H a r t e n , A . , 1978. The a r t i f i c i a l c o m p r e s s i o n method f o r c o m p u t a t i o n of s h o c k s a n d c o n t a c t d i s c o n t i n u i t i e s : I11 - S e l f - a d j u s t i n g h y b r i d schemes. Math.Comput., 32: 363-389. J a m a r t , B.M. and O z e r , J . , 1987. Real and s p u r i o u s b o u n d a r y l a y e r e f f e c t s i n t h r e e - d i m e n s i o n a l hydrodynamic m o d e l s . I n : J . C . J . N i h o u l ( E d i t o r ) , T h r e e - d i m e n s i o n a l Models of M a r i n e and E s t u a r i n e Dynamics. E l s e v i e r Oceanography S e r i e s . E l s e v i e r , Amsterdam/Oxford/New York/Tokyo ( t h i s volume) James, I . D . , 1984. A t h r e e - d i m e n s i o n a l n u m e r i c a l s h e l f - s e a f r o n t model w i t h v a r i a b l e eddy v i s c o s i t y and d i f f u s i v i t y . C o n t . S h e l f Res., 3: 69-98. James, I . D . , 1986. A f r o n t - r e s o l v i n g s i g m a c o o r d i n a t e sea model w i t h a s i m p l e h y b r i d a d v e c t i o n scheme. Appl.Math.Modelling, 10: 87-92. James, I . D . and McClimans, T.A., 1983. Coastal c u r r e n t w h i r l s i n l a b o r a t o r y and n u m e r i c a l models. Ocean M o d e l l i n g , 53: 1-3 + f i g s . ( u n p u b l i s h e d manuscript). Kwizak, M. and R o b e r t , A . J . , 1971. A s e m i - i m p l i c i t scheme f o r g r i d p o i n t a t m o s p h e r i c m o d e l s of t h e p r i m i t i v e e q u a t i o n s . Mon.Weath.Rev., 99: 32-36. Mellor, G.L. and Blumberg, A.F., 1985. M o d e l l i n g v e r t i c a l a n d h o r i z o n t a l d i f f u s i v i t i e s w i t h t h e s i g m a c o o r d i n a t e s y s t e m . Mon.Weath.Rev., 113: 1379-1383. M e s i n g e r , F., 1973. A method f o r c o n s t r u c t i o n of s e c o n d - o r d e r a c c u r a c y d i f f e r e n c e schemes p e r m i t t i n g n o f a l s e t w o - g r i d - i n t e r v a l wave i n t h e h e i g h t f i e l d . T e l l u s , 25: 444-458. O r l a n s k i , I . , 1976. A s i m p l e boundary c o n d i t i o n f o r unbounded h y p e r b o l i c f l o w s . J.Comput.Phys., 21: 251-269. P i n g r e e , R.D., H o l l i g a n , P.M. and M a r d e l l , G.T., 1979. P h y t o p l a n k t o n g r o w t h and c y c l o n i c e d d i e s . N a t u r e , 278: 245-247. Raymond, W.H. and Kuo, H.L., 1984. A r a d i a t i o n boundary c o n d i t i o n f o r m u l t i - d i m e n s i o n a l flows. Quart.J.R.Met.Soc., 110: 535-551.
.
609
A 3-D MODEL OF THE SEVERN ESTUARY
J WOLF I n s t i t u t e of Oceanographic S c i e n c e s , B i d s t o n O b s e r v a t o r y , B i r k e n h e a d , M e r s e y s i d e L43 7RA, U.K.
ABSTRACT
The S e v e r n e s t u a r y is a high-energy t i d a l regime w i t h s t r o n g r e s i d u a l s The p r e s e n t knowledge o f t h e t i d a l and g e n e r a t e d by n o n l i n e a r effects. r e s i d u a l flow i s reviewed. Some measurements o f c u r r e n t p r o f i l e s were used t o c a l c u l a t e t h e M2 t i d a l p a r a m e t e r s and a r e s i d u a l flow and also t h e f r i c t i o n stress t h r o u g h d e p t h a t a p a r t i c u l a r l o c a t i o n . A 3-dimensional ( 3 - D ) model o f t h e i n n e r B r i s t o l Channel and S e v e r n E s t u a r y was u s e d t o g e n e r a t e t i d a l and r e s i d u a l c u r r e n t s i n t h e area and t h e s e are compared w i t h t h e a v a i l a b l e observations. 1 INTRODUCTION The
Severn
see F i g . 1 .
England,
lower
the
l i e s e n c l o s e d between t h e coasts of S. Wales and S.W.
Estuary
I n t h i s paper, discussion w i l l be l i m i t e d mainly t o ( i )
S e v e r n E s t u a r y , between t h e S h o o t s ( u p s t r e a m of Avonmouth) and t h e
Holms
I s l a n d s , and ( i i ) t h e i n n e r Bristol Channel between Swansea Bay and t h e
Holms
Islands.
to
less
The mean water d e p t h r a n g e s f r o m o v e r 5Chn a t t h e seaward end
than
10m n e a r
Avonmouth.
I n t h e S e v e r n E s t u a r y t h e deep water is
c o n f i n e d t o r a t h e r narrow c h a n n e l s w i t h e x t e n s i v e s h a l l o w f l a t s . The world
normal and
semi-diurnal
all
the
maximum
the and
way
tidal
largest up
in
r a n g e i n t h i s area is t h e t h i r d largest i n t h e the
British
Isles.
The t i d e is predominantly
t i d a l wave e n t e r s from t h e Celtic S e a and is a m p l i f i e d
the
t h e Bristol Channel (Heaps, 1 9 6 7 ) .
The a m p l i f i c a t i o n stems
from e n e r g y p r o p a g a t i n g i n t o a n i n c r e a s i n g l y narrow area ( P r a n d l e and Rahman, 1 9 8 0 ) and p a r t l y from a broad-band r e s o n a n c e w i t h c e n t r a l p e r i o d partly
between
11
and
12
hours
Severn
(Fong
and
strong
tidal
currents
Bristol
Channel
in
tidal
the
Heaps,
of
between t h e s h e l f e d g e and t h e t i d a l l i m i t of t h e These h i g h t i d a l a m p l i t u d e s l e a d to v e r y
1978). over
4
and S e v e r n E s t u a r y . range.
knots
i n t h e d e e p c h a n n e l s of t h e i n n e r
T h e r e is also a large s p r i n g - n e a p c y c l e
The mean s p r i n g r a n g e a t Avonmouth i s 12.3111 w h i l e t h e
mean n e a p r a n g e i s 6.5m, which r e p r e s e n t s a v a r i a t i o n i n e n e r g y l e v e l s o f 4 : 1 ,
610
Fig. 1 . The B r i s t o l Channel and S e v e r n E s t u a r y , showing t h e l o c a t i o n o f o b s e r v a t i o n s from M.V. G a r d l i n e Locater, September 1978.
since quite
e n e r g y is 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 s q u a r e d . The t i d e is a l s o n o n l i n e a r w i t h t h e e b b flow sometimes l a s t i n g 2 h o u r s l o n g e r t h a n t h e the
flood. Due
to
bottom
friction
the
tidal
wave
is
p a r t i a l l y p r o g r e s s i v e i.e.
c u r r e n t s are i n advance o f e l e v a t i o n s by less t h a n 90° ( H u n t , 1 9 6 4 ) .
There is
a n e t e n e r g y f l u x i n t o t h e Channel which is d i s s i p a t e d i n t h e shallow r e g i o n s where f r i c t i o n i s s t r o n g e s t . Gwen ( 1 9 8 0 ) c a l c u l a t e s t h e e n e r g y f l u x and shows t h a t t h e e n e r g y d i s s i p a t i o n reaches a maximum i n t h e area between Swansea Bay and Lavernock P o i n t . The p r o g r e s s i v e n a t u r e of t h e M2 t i d e a l s o r e s u l t s i n a net residual t r a n s p o r t o f water o r Stokes' d r i f t , i n t o t h e e s t u a r y (Longuet-Higgins,
1969;
Prandle,
1978).
By
continuity
of
mass t h i s is
compensated by a seaward-flowing E u l e r i a n r e s i d u a l c u r r e n t ( U n c l e s and J o r d a n , 1980). O t h e r components of r e s i d u a l flow may b e produced by winds, d e n s i t y gradients,
fresh-water
flow,
topography and other t i d a l n o n l i n e a r i t i e s e.g.
t h e a d v e c t i o n of momentum and n o n l i n e a r f r i c t i o n . relative compared
importance
Uncles ( 1 9 8 2 a ) examined t h e
of t h e s e effects i n a depth-averaged n u m e r i c a l model and He found t h a t a d v e c t i o n is t h e most
h i s findings with observations.
important 1978;
mechanism
d r i v i n g many
1 9 7 6 ) ) followed
Tee,
g y r e s , e s p e c i a l l y off h e a d l a n d s ( P i n g r e e ,
by
b a l a n c i n g t h e Stokes’ d r i f t .
the
seaward-flowing
continuity
residual
Depth-averaged d e n s i t y c u r r e n t s were found t o be
i m p o r t a n t i n t h e e a s t e r n Bristol Channel b u t small i n t h e S e v e r n Estuary. observations
and
observations
of
model
currents
are
residuals
His
g a v e r e a s o n a b l e q u a l i t a t i v e agreement, b u t quite
sparse.
Observations
show
the
down-channel r e s i d u a l flow w i t h some i n d i c a t i o n s of g y r e s n e a r coast. However t h e o b s e r v e d r e s i d u a l s are n o t uniform through t h e water column. The d i r e c t i o n of t h e r e s i d u a l s t e n d s t o e x h i b i t v e e r i n g w i t h i n c r e a s i n g d e p t h below t h e s u r f a c e , w h i c h is e x p l a i n e d p a r t l y by d i f f e r e n t i a l flow due to d e n s i t y g r a d i e n t s and p o s s i b l y also by f r i c t i o n . Neither o f t h e p r edominantly the
l a t t e r effects c a n b e p r o p e r l y m odel l ed i n a depth-averaged model.
The no
with
S e v e r n E s t u a r y was o n c e c o n s i d e r e d t o be a t y p i c a l well-mixed e s t u a r y s i g n i f i c a n t v e r t i c a l d i f f e r e n c e s of s a l i n i t y . However Bowden ( 1 967)
p o i n t s o u t t h a t v e r t i c a l s a l i n i t y d i f f e r e n c e s of up t o 0.5% have been observed l o n g i t u d i n a l d e n s i t y g r a d i e n t r e q u i r e s some v e r t i c a l c u r r e n t t o b a l a n c e i t s i n c e i t pr oduces a d e p t h - v a r y i n g p r e s s u r e g r a d i e n t . Also he q u o t e s v a l u e s of t h e e f f e c t i v e l o n g i t u d i n a l d i s p e r s i o n from o b s e r v a t i o n t h a t are 10-100 times larger t h a n those p r e d i c t e d from well-mixed and
a
that
structure
theory.
This
suggests
vertical
and
transverse
gradients
is
not
negligible.
However,
the
an d t r a n s v e r s e s a l i n i t y g r a d i e n t s i n t h e Bristol Channel are n o t
longitudinal and
large,
t h a t t h e d i s p e r s i o n due t o a d v e c t i o n o f these small
at
most
r e s i d u a l s of a few c e n t i m e t r e s p e r second.
pr oduce
No
t h e r m a l s t r a t i f i c a t i o n o c c u r s e x c e p t l o c a l l y i n calm sunny weather ( P a r k e r and K i rby, 1981 1. i m p o r t a n t factor i n r e s i d u a l flow w h i c h may also affect t h e
potentially
A
oscillatory
tidal
sediment.
To
flow
date
is
this
density
the
has
not
s t r a t i f i c a t i o n c a u s e d by suspended
been i n c l u d e d i n numerical models.
The
dynamics of t h e s e d i m e n t b a l a n c e are c o m p l i c a t e d and n o t y e t f u l l y understood. There
is
fairly
recent
evidence
that
the
suspended s e d i m e n t c a n produce
d e n s i t y s t r a t i f i c a t i o n a t l e a s t of t h e same o r d e r as t h e s a l i n i t y d i s t r i b u t i o n and
locally
extensive su s pended
p e r h a p s much
larger.
P a r k e r and Kirby ( 1 9 8 1 ) have c a r r i e d o u t
s u r v e y s of suspended s e d i m e n t and i n p a r t i c u l a r found a marked s e d i m e n t f r o n t a l i g n e d l o n g i t u d i n a l l y i n t h e Severn E s t u a r y (Kirby
and P a r k e r , 1 9 8 2 ) . Sediments relationship higher
size Severn
i n t h e Bristol Channel / Sever n E s t u a r y do n o t show t h e classical of g r a i n s i z e w i t h e n e r g y l e v e l s (Hamilton, 1979). Normally t h e
energy than
that
Estuary.
r egi m es of
c o i n c i d e w i t h a s e d i m e n t p o p u l a t i o n of coarser g r a i n
low e n e r g y regimes b u t t h e r e v e r s e is t r u e i n t h e upper Hamilton s u g g e s t s t h a t t h i s is due t o t h e c o m p l e x i t y and
of s o r t i n g i n t h e h i g h l y e n e r g e t i c o s c i l l a t o r y t i d a l c u r r e n t s .
intensity
The
S e v e r n E s t u a r y h a s t h e h i g h e s t e n e r g y l e v e l s of any B r i t i s h e s t u a r y , b u t there is
no
source of
of
supply
coarse s a n d s and g r a v e l s h e n c e t h e o n l y m o b i l e These are v e r y m o b i l e d u e t o t h e
s e d i m e n t s are g e n e r a l l y f i n e s a n d s and muds. exceptionally
strong
flocculating into slack water.
springs
i n and o u t o f s u s p e n s i o n between maximum t i d a l stream t o
It
e s t i m a t e d t h a t 70% of t h e s e d i m e n t which is m o b i l e a t
is
settled
is
p a r t i c l e s which t h e n have enhanced s e t t l i n g rates.
larger
s e d i m e n t moves
The
The f i n e c l a y s e d i m e n t h a s t h e p r o p e r t y of
currents.
neaps
at
( P a r k e r and Kirby, 1 9 8 1 ) .
s e t t l e d mud are i n Bridgewater Bay and Newport Deep.
The main b o d i e s of
The B r i s t o l Channel and
t h e deep c h a n n e l s of t h e S e v e r n E s t u a r y have m a i n l y r o c k y floors w i t h p e r h a p s a t h i n l a y e r o f s e d i m e n t . A t maximum f l o o d and e b b t h e suspended s e d i m e n t t e n d s t o be well-mixed t h r o u g h t h e d e p t h . Towards s l a c k water t h i s s e d i m e n t
settles o u t , gradients
of
sediment
may
forming c h a r a c t e r i s t i c density have
layers
with
( P a r k e r and Kirby, 1 9 8 1 ) .
locally
can
vertical
a s t r o n g i n f l u e n c e o n t h e d e n s i t y g r a d i e n t s which i n t u r n
affect t h e r e s i d u a l flow and p o s s i b l y e v e n t h e t i d a l flow.
gradients
large
These f e a t u r e s of suspended
produce
residuals
while
vertical
Horizontal density
density gradients inhibit
v e r t i c a l mixing and c o n s e q u e n t l y t h e exchange of b o t h mass and momentum. T h i s h i g h e n e r g y t i d a l regime h a s t h e effect of e n h a n c i n g v a r i o u s n o n l i n e a r effects w h i c h
may
without
careful
current
profiles
elsewhere
tidal
and
These s h o u l d n o t b e o m i t t e d
consideration. The n e x t s e c t i o n d i s c u s s e s some o b s e r v e d from a s t a t i o n i n t h e S e v e r n E s t u a r y and t h e c a l c u l a t i o n of
f r i c t i o n a l shear stresses. the
be i n s i g n i f i c a n t .
residual
S e c t i o n 3 d e s c r i b e s t h e r e s u l t s of a 3-D model f o r c u r r e n t s i n t h i s area.
These are compared w i t h t h e
observed c u r r e n t s t r u c t u r e . 2 OBSERVED CURRENT PROFILES The measurements were made from M.V. I.O.S.
G a r d l i n e L o c a t e r i n September 1978 by
Taunton as p a r t of t h e i r c o h e s i v e s e d i m e n t s p r o j e c t .
obtained
o n t h e 12-13/9/78 and 22-24/9/78.
from 07.41
GMT
on
O b s e r v a t i o n s were
The l o n g e s t c o n t i n u o u s s e r i e s was
1 2 / 9 / 7 8 u n t i l 05.00 GMT o n 13/9/78.
The l o c a t i o n of t h e
is shown i n F i g . 1 . The s h i p was a n c h o r e d o n s t a t i o n u s i n g 4 t o r e d u c e any movement which m i g h t c o n t a m i n a t e t h e measurements. Two a r r a y s of c u r r e n t meters were d e p l o y e d , b o t h c o n t i n u o u s l y m o n i t o r e d from t h e
observations anchors
ship. bottom
The first a r r a y of 5 c u r r e n t meters was l o w e r e d u n t i l i t j u s t touched t h e i n s t r u m e n t s were a t f i x e d h e i g h t s above t h e sea-bed. The
i.e.
second a r r a y of 3 c u r r e n t meters was p o s i t i o n e d a f i x e d d i s t a n c e below t h e sea surface. amount.
During
the
tidal
cycle
the
two a r r a y s o v e r l a p p e d by a v a r y i n g
Some of t h e c u r r e n t meters measured flow o n l y , w h i l e others measured
613
l+zfh t-10
.oo
-RO
RO
cmfs
ltzlh 11,-
,~
major a x i s -RO
0 C
X
l+z/h
1 t-1R.15
-RO
I + zminor /h acxm 80 i sf s
; i
-RO
0 ma J o r a x80 is
cmls
Fig. 2. Major and minor a x i s c u r r e n t p r o f i l e s through a t i d a l c y c l e . 12-13/9/78. T i m e , t , is i n G.M.T. Mean d e p t h = 21.4m. T i d a l range = 8.75m. T i m e of HW = 14.52. LW = 21.15.
614 direction
as
P r e s s u r e s e n s o r s were i n c l u d e d t o check t h e
speed.
as
well
d e p t h s of t h e meters. The
at
and
into
speed
north
calculated
e l l i p s e was
and
and
amplitude
M2
the
current
the
for
each
meter
was
o b t a i n e d by l i n e a r
e x t r a p o l a t i o n from t h e three d i r e c t i o n a l o b s e r v a t i o n s made
The
time.
each
converted obtain
of
direction
interpolation
direction
east
and
time
components. phase
series f r o m each l e v e l were
A F o u r i e r a n a l y s i s was used t o
o f e a c h component.
Then a n M2 c u r r e n t
and t h e o r i e n t a t i o n o f t h e major a x i s was determined
f o r each h e i g h t above t h e sea-bed.
The dept h -averaged v a l u e o f t h e m a j o r a x i s as a more r e p r e s e n t a t i v e d i r e c t i o n a l o n g which t o T h i s d i r e c t i o n was a b o u t 027O ( M a g n e t i c ) . r e s o l v e t h e c u r r e n t components. The n o r t h and east components of t h e t o t a l c u r r e n t were t h e n c o n v e r t e d to components a l o n g t h e m aj or (now l a b e l l e d x ) and minor ( y l a x i s d i r e c t i o n s of
orientation
tidal current.
M2
the
chosen
was
Some c u r r e n t p r o f i l e s are shown a t 2-hourly i n t e r v a l s
through one t i d a l c y c l e i n Fi g. 2. It c a n be s e e n t h a t t h e v-component o f c u r r e n t (minor a x i s ) changes s i g n T h i s r e f l e c t s t h e r o t a t i o n of t h e major a x i s of t h e e l l i p s e d e p t h below t h e s u r f a c e . I n f a c t t h e M 2 c u r r e n t s are a l m o s t
through d e p t h . clockwise w i t h
r e c t i l i n e a r a t e v e r y d e p t h . The u-component o f c u r r e n t (major a x i s ) i s more interesting. I t shows a marked r e d u c t i o n i n c u r r e n t s p e e d n e a r t h e sea s u r f a c e w h i c h shows up i n t h e t o p three c u r r e n t meter r e c o r d s and therefore is not
just
an
instrument malfunction.
It d o e s n o t seem t o b e c o n f i n e d t o t h e
f l o a t i n g a r r a y , s i n c e a t low t i d e t h e two a r r a y s o v e r l a p and t h e t o p meter of the
bottom
not
a
a r r a y is above t h e lowest meter of t h e s u r f a c e a r r a y .
I t is a l s o
f u n c t i o n of t h e c h o i c e of t h e d i r e c t i o n s f o r r e s o l u t i o n of t h e c u r r e n t
components as i t c a n be s e e n i n t h e o r i g i n a l c u r r e n t s p e e d p r o f i l e s . The
current
time
series were
used
t o c a l c u l a t e f r i c t i o n a l stress time
series u s i n g l i n e a r i s e d e q u a t i o n s as i n a p r e v i o u s p a p e r , Wolf ( 7 9 8 0 ) . The s u r f a c e and bottom stress must b e p r e s c r i b e d . The s u r f a c e stress was set t o zero.
Winds were
generally
light
(less t h a n 1 6 k n o t s , g i v i n g a maximum
s u r f a c e stress of less t h a n 1 dyne/ cm 2) , so t h i s a p p r o x i m a t i o n was r e a s o n a b l e . The bottom stress was related t o t h e depth-mean c u r r e n t by a q u a d r a t i c f r i c t i o n law. The a m p l i t u d e and phase of t h e M2 t i d a l c u r r e n t components and of t h e M2 stress components t h r o u g h d e p t h are p l o t t e d i n F i g . 3 . The major a x i s is dominant and t h e c a l c u l a t i o n s more r e l i a b l e . The c u r r e n t phase is almost
constant
t hr ough
decreases
almost
classical
eddy
p r oduce negative
d e p t h as i s t h e stress phase.
linearly with
viscosity
height
above
the
The stress a m p l i t u d e
sea-bed.
Obtaining a
by r e l a t i n g stress t o t h e v e l o c i t y g r a d i e n t , would
n e g a t i v e v a l u e s n e a r t h e s u r f a c e , s i n c e t h e v e l o c i t y g r a d i e n t becomes there. T h i s s u g g e s t s t h a t t h e s i m p l e model used does not f u l l y
615
I+r/h
I+z/h Ma velocity amplitude
Fig. 3. M2 c u r r e n t and stress a m p l i t u d e and phase as a f u n c t i o n o f d e p t h .
t h e f r i c t i o n a l stresses.
predict
as
are
advection
important
in
I t seems l i k e l y t h a t n o n l i n e a r e f f e c t s s u c h
this
area.
The n e x t s e c t i o n d e s c r i b e s t h e
of a f u l l y n o n l i n e a r 3-D model which shows t h e c o m p l e x i t y of t h e flow
results
i n t h e area. 3 THE 3-D MODEL o r d e r t o examine t h e s t r u c t u r e of t i d a l and r e s i d u a l flow i n t h e Severn
In
E s t u a r y a three-dimensional model was r e q u i r e d which s h o u l d b e of s u f f i c i e n t l y fine
r e s o l u t i o n t o r e s o l v e t h e complex topography of t h e area i n some d e t a i l .
This
includes
Channel square
has by
sufficient grid-boxes the
time
finite Estuary
deep
c h a n n e l s , s a n d banks and i n t e r - t i d a l f l a t s .
The Bristol
been modelled i n t h r e e dimensions o n a g r i d of a p p r o x i m a t e l y 4km Owen detail
(1980b) in
the
and S t e p h e n s ( 1 9 8 3 ) . Severn
Estuary
T h i s r e s o l u t i o n does n o t g i v e
amounting
to only about t h r e e
across t h e c h a n n e l . I n o r d e r t o u s e a f i n e r g r i d w i t h o u t r e d u c i n g s t e p as i n a n e x p l i c i t scheme a s e m i - i m p l i c i t a l t e r n a t i n g d i r e c t i o n
d i f f e r e n c e scheme was a p p l i e d t o a model of t h e Bristol Channel/Severn which had p r e v i o u s l y o n l y been modelled i n 2-D
(Owen, 1980a; Proctor,
616 1981b) w i t h described
of
grid-size
a
about
Bristol Channel and S e v e r n E s t u a r y .
The f i n i t e d i f f e r e n c e scheme is
The M2 c o t i d a l c h a r t is w e l l - r e p r o d u c e d .
v e r t i c a l eddy v i s c o s i t y u s e d i n t h e model was t a k e n from Owen ( 1 9 8 0 b ) .
The
means
t h a t t h e eddy a t mid-depth and v a r i e s local v e l o c i t y and d e p t h . r e s u l t s of Wolf (1980) and
This
lkm.
Wolf ( 1 9 8 3 ) , w i t h t h e r e s u l t s f o r t h e M2 t i d a l e l e v a t i o n i n t h e
in
v i s c o s i t y is p a r a b o l i c t h r o u g h d e p t h w i t h a maximum i n s p a c e and time d e p e n d i n g on t h e magnitude of t h e T h i s form of v i s c o s i t y is s u g g e s t e d i n p a r t by t h e
also by t h e work of Bowden and Hamilton ( 1 9 7 5 ) .
The model was u s e d t o o b t a i n t h r e e - d i m e n s i o n a l t i d a l r e s i d u a l s . Figs. 4 ( a ) and ( b ) shows t h e r e s i d u a l s o b t a i n e d by a v e r a g i n g o v e r a n M2 t i d a l c y c l e f o r t h e s u r f a c e and near-bottom c u r r e n t s i n t h e model. The a d v e c t i v e t e r m s have been i n c l u d e d and these p r o v i d e t h e main s o u r c e o f t h e t i d a l r e s i d u a l s . I n p a r t i c u l a r , t h e y d r i v e t h e g y r e s w h i c h may b e s e e n a t a l l d e p t h s n e a r h e a d l a n d s and a l l c o r n e r p o i n t s ( t h e l a t t e r p r o d u c e g r i d - s c a l e e d d i e s which are u n r e a l i s t i c ) . Two v e r y pronounced g y r e s may b e s e e n on e i t h e r s i d e of Lavernock P o i n t . The c l o c k w i s e g y r e is t h e r e s u l t of a lee eddy o n t h e ebb
t i d e and t h e a n t i c l o c k w i s e o n e o n t h e f l o o d . The p r e d o m i n a n t l y seaward flow due to compensation f o r S t o k e s , d r i f t is also e v i d e n t i n t h e c e n t r a l c h a n n e l . Other
eddies
corresponding
to
sandbanks
a p p e a r o u t s i d e Swansea Bay (Owen,
1980a). The o v e r a l l p a t t e r n is v e r y similar t o t h a t o b t a i n e d by Owen ( 1 9 8 0 a ) and c o n f i r m e d by Uncles ( 1 9 8 2 a ) , who shows t h e main c a u s e s of r e s i d u a l flow t o be a d v e c t i o n and c o n t i n u i t y .
is
This
first
the
attempt
to
examine
r e s i d u a l s in t h i s area in three
d i m e n s i o n s o n s u c h a f i n e scale. In g e n e r a l , t h e s u r f a c e and bottom r e s i d u a l s are
very
similar
areas
certain
e x c e p t t h a t t h e s u r f a c e c u r r e n t s are much s t r o n g e r , b u t i n
there
i s a marked change i n d i r e c t i o n w i t h d e p t h .
T h e r e is a
tendency f o r a v e e r i n g ( i . e . clockwise r o t a t i o n ) of l o o t o 20° i n t h e c u r r e n t d i r e c t i o n f r o m t h e s u r f a c e t o t h e bottom. However, i n some areas e.g. t h e SW open boundary and east of Lavernock P o i n t , i t I s i n t h e i . e . a n t i c l o c k w i s e . T h i s characteristic i s q u a l i t a t i v e l y i n agreement w i t h t h e o b s e r v a t i o n s shown by Uncles ( 1 9 8 2 a ) f o r t h e few s t a t i o n s a v a i l a b l e i n t h e i n n e r Bristol Channel. In t h e S e v e r n E s t u a r y , t h e r e s i d u a l s show more v a r i a b i l i t y , b e i n g i n o p p o s i t e d i r e c t i o n s a t s u r f a c e and bottom a t some g r i d p o i n t s and c h a n g i n g d i r e c t i o n markedly from o n e g r i d p o i n t t o t h e n e x t . It would t h e r e f o r e b e corner
near
opposite
the
sense
d i f f i c u l t t o compare q u a n t i t a t i v e l y w i t h o b s e r v a t i o n s as t h e r e s o l u t i o n of t h e model may be i n s u f f i c i e n t here. Some of t h e v a r i a b i l i t y may b e u n r e a l i s t i c , due t o grid-scale eddies. Uncles stations
and in
Jordan the
( 1 9 7 9 ) look
in
detail
at
r e s i d u a l c u r r e n t s a t two
S e v e r n E s t u a r y , s t a t i o n A b e i n g j u s t seaward of S t e e p Holme
Fig. 4 ( a ) . Surface r e s i d u a l s . 617
618
\
\
"
....
....,.\l.........l.
,,,,,-..
I . . . . . . ,
,.>
, , , . ,
;,
I
.
1
' 1
1
\
-,
\
-
\
- \ \ \ \ , \ \ \ \ \
.
- \ \ . \
.
.
L
I
,
,
.
,
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.
i
;
.
1 .
.,\-.:;! . . ( . . \ . . ........ . .
,
'
.
'
.
.
a
.
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.
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.
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, . . \ : .,
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6
,
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.
.,... .
. .
n - .
\ , - - I
.
.
-
....-....
a a - , , . l ' d , ' a , . . . . . . 3 \ - , . . - - - - - - - ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
F i g . 4 ( b ) . Bottom residuals.
619 and
station
b e i n g a b o u t 17km upst r eam . Both s t a t i o n s l i e a p p r o x i m a t e l y i n
B
mid-channel. about
station
At
5cm/s
(half
the
t h e model g i v e s a s u r f a c e r e s i d u a l seawards of
A
observed
residual
but i n the right d i r e c t i o n ) , the
r e s i d u a l s have t h e r i g h t m agni t ude o f 4cm/s b u t i n t h e o p p o s i t e d i r e c t i o n to t h e o b s e r v e d r e s i d u a l , t h e model r e s i d u a l s rotate c l o c k w i s e w i t h
bottom
increasing
depth
anticlockwise.
below
surface
the
whereas
t h e observed r e s i d u a l s r o t a t e
A t s t a t i o n B t h e o b s e r v e d r o t a t i o n is a g a i n a n t i c l o c k w i s e w i t h
increasing
depth except for a reversal i n d i r e c t i o n very near to the surface.
The
gives
model
direction direction magnitude. flow
to
an a n t i c l o c k w i s e r o t a t i o n and t h e near-bed r e s i d u a l c u r r e n t
in
is
good
agr eem ent b u t t h e s u r f a c e r e s i d u a l is i n t h e o p p o s i t e
t o , t h a t o b s e r v e d . The s p e e d s of 5-l0cm/s are t h e correct o r d e r of These d i s a g r e e m e n t s between t h e model and o b s e r v a t i o n s may b e due d r i v e n by t h e l o n g i t u d i n a l s a l i n i t y g r a d i e n t a l t h o u g h t h i s h a s been
shown t o b e small i n t h e Sever n E s t u a r y ( S t e p h e n s , 1983) and g e n e r a l l y i n t h e same d i r e c t i o n from s u r f a c e t o bottom. The d i s c r e p a n c i e s may also b e due t o insufficient (b)
resolution
illustrate
how
of
t h e model i n t h e S e v e r n E s t u a r y .
F i g s . 4 ( a ) and
much v a r i a b i l i t y there is i n t h e r e s i d u a l s from o n e g r i d
box t o t h e n e x t . The t i d a l c u r r e n t s and r e s i d u a l s f r o m t h e model are examined i n more d e t a i l
over
a
c r o s s - s e c t i o n t h r o u g h t h e Locater p o s i t i o n , shown i n F i g . 1.
In this
s e c t i o n t h e r e is a d e e p c h a n n e l w i t h a complex system of banks around it.
M2
and
residual
The
c u r r e n t s were r e s o l v e d i n t h e NE- and NW-going d i r e c t i o n s ,
across t h e main c h a n n e l . The M2 c u r r e n t a m p l i t u d e s are shown i n F i g s . 5 ( a ) and ( b ) . The t i d a l c u r r e n t s are almost rectilinear. The maximum c u r r e n t s of a b o u t lOOcm/s are found i n t h e d e e p e s t p a r t of t h e s e c t i o n , f l o w i n g NE o r SW. T h e r e i s no s i g n of a r e d u c t i o n i n c u r r e n t a m p l i t u d e n e a r t h e s u r f a c e . The r e s i d u a l s , shown i n Figs. 6 ( a ) and ( b ) , seem t o flow NE o n t h e s o u t h ( E n g l i s h ) s i d e of t h e s e c t i o n and SW on t h e n o r t h (Welsh) s i d e , which agrees w i t h t h e g e n e r a l p i c t u r e d e s c r i b e d by P a r k e r a nd K irby ( 1 9 8 1 ) . The t r a n s v e r s e r e s i d u a l s are d i r e c t e d NW almost everywhere and are small e x c e p t n e a r t h e E n g l i s h coast where there is q u i t e a s t r o n g offshore c u r r e n t f e d b y a l o n g s h o r e c u r r e n t s c o n v e r g i n g o n t h e headland of Sand Point. C o n t i n u i t y is s a t i s f i e d i n t h e h o r i z o n t a l by r e t u r n flow a t other approximately
aligned
sections. These unfortunately. The
tidal
and
along
and
features residual
cannot
be
checked
against
observations
c u r r e n t s a t t h e Locater p o s i t i o n are compared i n
The t i d a l c u r r e n t a m p l i t u d e i s i n good agreement e x c e p t f o r t h e F i g . 7. n e a r - s u r f a c e r e d u c t i o n i n a m p l i t u d e of t h e l o n g i t u d i n a l component i n t h e
observations. however t h i s
The p h a s e i s i n good agr eem ent f o r t h e l o n g i t u d i n a l component, n o t t h e case f o r t h e t r a n s v e r s e component. There is a
is
620
CARDIFF SAND POINT
Fig. 5 ( a ) . M
2
I
j0 CARDIFF
c u r r e n t a m p l i t u d e . NE-going component. cm/s.
I
j-j F i g . 5 ( b ) . M2 c u r r e n t a m p l i t u d e . NW-going component. cm/s.
SAND POINT
62I
CARDIFF
SAND POINT
F i g . 6 ( a ) . NE-going r e s i d u a l c u r r e n t . c d s .
CMDIFF SAND POINT
1 m.
1 1 km.
F i g . 6 ( b ) . NW-going r e s i d u a l c u r r e n t . cm/s.
622
Rcsid". I..
F i g . 7. Comparison of o b s e r v e d and model c u r r e n t p r o f i l e s . L o n g i t u d i n a l a x i s is a l i g n e d 027°/2070M. T r a n s v e r s e a x i s is a l i g n e d 117°/2970M. -model ---observed.
of
discrepancy
as
surface,
the
given
of
by
the
the
current
model
observed
transverse ellipse
c u r r e n t p h a s e c h a n g e s t h r o u g h 180° a t
major a x i s .
The d i r e c t i o n of major a x i s
is 3 5 O n e a r t h e bed, r o t a t i n g clockwise t o 37O n e a r t h e
t o rotate i n t h e o p p o s i t e d i r e c t i o n , b e i n g surface. The d i r e c t i o n of r o t a t i o n of t h e
major
axis
starts
N-S
near
the
almost
aligned
is a n t i c l o c k w i s e n e a r t h e bed i n b o t h model and o b s e r v a t i o n s b u t t h e
currents model
observed
n e a r t h e bed and t h i s becomes a b o u t 150° n e a r t h e
which is v e r y c l o s e to t h e o b s e r v e d d i r e c t i o n up t o mid-depth, where
surface, the
45O
T h i s p h a s e r e v e r s a l c o r r e s p o n d s t o a change i n t h e d i r e c t i o n of
mid-depth. alignment
about
t h e n e x h i b i t s c l o c k w i s e r o t a t i o n t h r o u g h o u t t h e u p p e r p a r t of t h e water
column
w h i l e t h e o b s e r v e d c u r r e n t s r e m a i n a n t i c l o c k w i s e u n t i l above mid-depth
when t h e y become c l o c k w i s e . The
is
show some q u a l i t a t i v e agreement.
residuals
directed
observed transverse
NE
magnitude residuals
and
are
The l o n g i t u d i n a l r e s i d u a l
t h e d e p t h b u t t h e model g i v e s less t h a n h a l f t h e
throughout does
not
directed
exhibit
NW
a
through
a t mid-depth. The t h e main p a r t of t h e water
maximum
623 column, model and o b s e r v a t i o n s b e i n g f a i r l y close, b u t t h e model d o e s n o t g i v e t h e r e v e r s e flow a t s u r f a c e and bottom s e e n i n t h e o b s e r v a t i o n s . 4 DISCUSSION The
model
gives
quite
of
tidal
and
magnitude
agreement
flow
with
t h e o b s e r v e d d i r e c t i o n and
but
does n o t reproduce e x a c t l y t h e
t h e v e r t i c a l p r o f i l e s of c u r r e n t .
It demonstrates t h e importance
details
of
of
advective
the
good residual
t e r m s b u t i t d o e s n o t i n c l u d e d e n s i t y effects which may b e
The g r i d - s i z e of a p p r o x i m a t e l y Ikm may n o t be s u f f i c i e n t l y small
important.
t o r e s o l v e a l l t h e t o p o g r a p h i c f e a t u r e s and t h e smallest e d d i e s . The dominant forces i n t h i s area are a b a l a n c e between t h e s u r f a c e elevation since
pressure
this
gradient
a very
.is
are
Nonlinearities
a n d t h e bottom f r i c t i o n , o n a t i d a l time scale,
energetic
tidal
regime
in
rather
i m p o r t a n t i n g e n e r a t i n g r e s i d u a l flows.
s h a l l o w water. I t may b e t h a t a
more s o p h i s t i c a t e d model f o r f r i c t i o n is n e c e s s a r y i n s u c h a n area. From p r e v i o u s s t u d i e s t h e s a l i n i t y i s v e r y well-mixed i n t h e S e v e r n E s t u a r y and
t o make much d i f f e r e n c e to t h e c u r r e n t s t r u c t u r e .
unlikely
density
stratification
important.
strong
A
induced
by
transverse
suspended gradient
sediment
of
may
be
However t h e relatively
suspended s e d i m e n t h a s been
o b s e r v e d i n t h e S e v e r n E s t u a r y which may have effects o n t h e r e s i d u a l c u r r e n t s such
as
a t o t h e r f r o n t s d u e t o t e m p e r a t u r e and s a l i n i t y .
occur
I t may also
a f f e c t t h e t i d a l c u r r e n t s t h r o u g h n o n l i n e a r effects, i . e . t h e a d v e c t i v e t e r m s , and
by
The
pattern
explained
the by
r e d u c t i o n i n t u r b u l e n c e and v i s c o s i t y d u e t o t h e s t r a t i f i c a t i o n . of the
t i d a l r e s i d u a l s s u g g e s t t h a t t h e d e n s i t y f r o n t may b e p a r t l y suspended
sediment
being
trapped
i n t h e gyres near its
o r i g i n s i n B r i d g e w a t e r Bay.
5 REFERENCES Bowden, K.F, 1967. C i r c u l a t i o n and d i f f u s i o n . I n : G.H. Lauff ( E d i t o r ) , P u b l i c a t i o n no. 83, Washington, pp. 13-36. E s t u a r i e s . A.A.A.S. Bowden, K.F. and Hamilton, P., 1975. Some e x p e r i m e n t s w i t h a n u m e r i c a l model of c i r c u l a t i o n and m i x i n g i n a t i d a l e s t u a r y . E s t . C s t l . Mar. Sci, 3: 281-301. Fong, W.W. and Heaps, N.S., 1978. A n o t e o n quarter-wave t i d a l r e s o n a n c e i n t h e Bristol Channel. I.O.S. R e p o r t no. 63 ( u n p u b l . ) , 11 pp + f i g s . Hamilton, D., 1979. The h i g h e n e r g y , s a n d and mud regime of t h e S e v e r n E s t u a r y , S.W. B r i t a i n . I n : R.T. S e v e r n , D. D i n e l e y , L.E. Hawker ( E d i t o r s ) , T i d a l power and e s t u a r y management. C o l s t o n p a p e r s no. 30. S c i e n t e c h n i c a , B r i s t o l , pp. 1 62-1 72. Heaps, N.S., 1967. Storm S u r g e s . I n : H. B a r n e s ( E d i t o r ) , Oceanography and m a r i n e b i o l o g y , a n n u a l review. A l l e n and Unwin, London, pp. 11-47. Hunt, J . N . , 1964. T i d a l o s c i l l a t i o n s i n E s t u a r i e s . Geophys. J . R. astr. SOC., 8: 440-455. Kirby, R. and P a r k e r , W.R., 1982. A suspended s e d i m e n t f r o n t i n t h e Severn E s t u a r y . N a t u r e , 195: 396-399.
624
Longuet-Higgins, M.S., 1969. On t h e t r a n s p o r t of m a s s by time v a r y i n g o c e a n c u r r e n t s . Deep-sea Res., 16: 431-447. Gwen, A., 1980a. The t i d a l regime of t h e Bristol Channel: a n u m e r i c a l m o d e l l i n g approach. Geophys. J . R . astr. Soc., 61: 59-75. Gwen, A . , 1980b. A three-dimensional model of t h e Bristol Channel. J.P.O., 10 : 1290-1 302. P a r k e r , W.R. and Kirby, R., 1981. The b e h a v i o u r of c o h e s i v e s e d i m e n t i n t h e i n n e r Bristol Channel and S e v e r n E s t u a r y i n r e l a t i o n to c o n s t r u c t i o n of t h e Severn E s t u a r y . I.O.S. Report no. 117 ( u n p u b l . ) , 54 pp. P i n g r e e , R.D., 1978. The f o r m a t i o n of t h e Shambles and o t h e r banks by t i d a l s t i r r i n g of t h e seas. J. Mar. Biol. Assoc. U.K., 58: 211-226. P r a n d l e , D., 1978. R e s i d u a l flows and e l e v a t i o n s i n t h e s o u t h e r n North Sea. Proc. R. SOC. Lond., A, 359: 189-228. P r a n d l e , D. and Rahman, M . , 1979. T i d a l r e s p o n s e i n e s t u a r i e s . J.P.O., 10: 1552-1 573. P r o c t o r , R . , 1981. Mathematical m o d e l l i n g of t i d a l power schemes i n t h e Bristol Channel. I n : Proc. 2nd I n t . Symp. on Wave and T i d a l Energy, Cambridge, September 1981. BHRA F l u i d E n g i n e e r i n g , pp. 33-51. S t e p h e n s , C.V., 1983. Hydrodynamic m o d e l l i n g developments f o r t h e west coast of t h e B r i t i s h Isles. L i v e r p o o l U n i v e r s i t y , Ph.D t h e s i s , 213 pp. Tee, K.-T., 1976. Tide-induced r e s i d u a l c u r r e n t , a two-dimensional n o n - l i n e a r n u m e r i c a l t i d a l model. J. Mar. Res., 34: 603-629. Uncles, R . J . , 1982. Computed and o b s e r v e d r e s i d u a l c u r r e n t s i n t h e Bristol Channel. Ocean. Acta, 5: 11-20. Uncles, R . J . and J o r d a n , M.B., 1979. R e s i d u a l f l u x e s of water and s a l t a t two s t a t i o n s i n t h e S e v e r n E s t u a r y . E s t . C s t l . Mar. Sci., 9: 287-302. Uncles, R . J . and J o r d a n , M.B., 1980. A one-dimensional r e p r e s e n t a t i o n of r e s i d u a l c u r r e n t s i n t h e S e v e r n E s t u a r y and a s s o c i a t e d o b s e r v a t i o n s . E s t . C s t l . Mar. Sci., 10: 39-60. Wolf, J . , 1980. E s t i m a t i o n of s h e a r i n g stresses i n a t i d a l c u r r e n t w i t h a p p l i c a t i o n t o t h e I r i s h Sea. I n : J . C . J . Nihoul ( E d i t o r ) Marine Turbulence. Proc. 1 1 t h I n t . Liege Colloq. o n Ocean Hydrodynamics, 1979. E l s e v i e r Amsterdam, pp. 319-344. Wolf, J . , 1983. A comparison of a s e m i - i m p l i c i t w i t h a n e x p l i c i t scheme f o r a t h r e e - d i m e n s i o n a l hydrodynamic model. Cont. S h e l f Res., 2: 215-229.
625
THE VARIATIONAL INVERSE METHOD REVISITED* C. PROVOST L.O.D.Y.C., Universitt P. et M. Curie, Tour 14,2E,4 place Jussieu. 75252 Paris Cedex 05, France.
ABSTRACT The variational inverse method addresses the classical problem of estimating the three dimensional field .of velocity from hydrographic data. It seeks the smoothest velocity field (in the sense of an arbitrarily defined norm) wich is consistent with the data and selected prescribed misfits. These misfits should not be zero: they represent errors in the observations and in the approximate dynamical constraints. The admission of errors necessitates the choice of weights. The choice is made by prior estimation of the relative error variances. The solutions are produced in one single step by a global optimization by demanding overall consistency in the whole domain. By varying the misfits relatively to one another in their respective admissible range, we explore the full envelope of physically plausible estimates of the average geostrophic flow. We discuss the choice of weights, the choice of smoothing criteria, and problems such as resolution of the solutions, compatibility between constraints and parameter estimations.
*
Paper presented at the Colloquium and submitted too late for publication in this volume.
This Page Intentionally Left Blank
627
THE BRANCHING OF THE GULF STREAM REVISITED USING THE VARIATIONAL INVERSE METHOD* F. MARTEL and C. PROVOST L.O.D.Y.C., Universitk P. et M. Curie, Tour 14,2E,4 place Jussieu, 75252 Paris Cedex 05, France.
ABSTRACT We investigate the circulation, and the mass and heat transports in the region between the Grand Banks and fhe Mid-Atlantic Ridge. We use the variational inverse method developed by Provost and Salmon (J. Mar. Res., 44, 1-34,1986) to analyse the dense data set described by Clarke et al. (J. Phys. Oceanogr., 10, 25-65,1980). Using only two constraints (geostrophy and total transport dynamics), we can conclude that the data does not support Worthington's hypothesis of two independent gyres. All the flows obtained from our solutions show a branching of the Gulf Stream. However, the solutions are not well resolved in areas with steep bottom topography, and in regions of strong shear. More constraints (data and dynamics) are needed to understand clearly the behavior of the flow, and to establish firmly the "average dynamical balances" in this complex region.
*
Paper presented at the Colloquium and submitted too late for publication in this volume.
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629
ABOUT A DIAGNOSTIC ANALYSIS OF THE HISTORICAL HYDROGRAPHIC DATA IN THE TROPICAL ATLANTIC* C. PROVOST and M. S. SUK L.O.D.Y.C., Universid P. et M. Curie, Tour 14,2E, 4 place Jussieu, 75252 Paris Cedex 05, France.
ABSTRACT We presentan application of the variational inverse method (Provost and Salmon, 1986, J. Mar. Res., 44, 1-34) to the historical hydrographic data of the tropical Atlantic. The idea is to develop a method for analyzing hydrographic data in the tropics and to evaluate the present data set. This paper discusses only the use of temperature and salinity. We had to face three main difficulties. First, the geostrophic constraint that controls efficiently the vertical shear of the horizontal velocity in mid-latitudes is not valid in the equatonal band. We modify the traditional geostrophic relation with the explicit inclusion of vertical eddy diffusion near the equator. Second, the tropical Atlantic exhibits a strong seasonal variability. We invert the data season by season. Finally, the area is large (20ON-20°S), and the original method has to be adapted to handle large domains and large data sets. We define the averaging spatial and time scales and estimate the aliasing errors in the data with respect to that definition. The misfits on the data constraint are adjusted in order to filter the aliasing present in the data, and the misfits on the dynamical constraints are adjusted in relation to the estimated accuracy of the corresponding dynamical equation. We explore the domain of solutions for each season with a special attention to where, how, and at which rates water masses cross the equator and to heat transport estimates. Only spring season has a sufficient data coverage for a small domain of solutions. However, even in spring, uncertainty on longitudinally averaged heat transport estimates is large. The inadequacy of the historical data is mainly due to the lack of deep hydrographic measurements and of meridionally closely spaced stations across the equator. Estimated heat transport at depth accounts for a large part of the estimated total transport.
*
Papex presented at the Colloquium and submitted w late for publication in this volume.
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![Three-dimensional models of marine and estuarine dynamics: [proceedings of the 18th International LieРњР‚ge Colloquium on ocean hydrodynamics, 1986]](https://epdf.tips/img/300x300/three-dimensional-models-of-marine-and-estuarine-d_5b5518e4b7d7bc5877d6508b.jpg)
