synoptic coherent structures in geophysical turbulence: proceedings of the 20th International Liege Colloquium on Ocean Hydrodynamics

MESOSCALE/SYNOPTIC COHERENT STRUCTURES IN GEOPHYSlCAL TURBULENCE FURTHER TITLESI N wIs SERIES 1 J L MERO 38 J C J NIH...

Author: Jacques C. J. Nihoul | B. M. Jamart

9 downloads 430 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

MESOSCALE/SYNOPTIC COHERENT STRUCTURES IN GEOPHYSlCAL TURBULENCE

FURTHER TITLESI N wIs SERIES 1 J L MERO 38 J C J NIHOUL (Editor) REMOTE SENSING OF SHELF SEA HYDRODYNAMICS THE MINERAL RESOURCES OF THE SEA 2 L M FOMIN 39 T ICHlYElEditor) OCEAN HYDRODYNAMICS OF THE JAPAN AN0 EAST CHINA SEAS THE DYNAMIC METHOD IN OCEANOGRAPHY 3 E J F WOOD 40 J C J NIHOUL (Editor1 COUPLED OCEAN-ATMOSPHERE MODELS MICROBIOLOGYOF OCEANS AND ESTUARIES 4 G NEUMANN 41 H KUNZEDORF (Editor) MARINE MINERAL EXPLORATION OCEAN CURRENTS 5 N G JERLOV 42 J C J NlHOULIEdctor) OPTICAL OCEANOGRAPHY MARINE INTERFACESECOHYDRODYNAMICS 43 P LASSERRE and J M MARTIN (Ednors) 6 V VACOUIER BIOGEOCHEMICAL PROCESSES AT THE LAN0 SEA BOUNDARY GEOMAGNETISMIN MARINE GEOLOGY 7 W J WALLACE 44 I P MARTINI (Editor) CANADIAN INLAND SEAS THE DEVELOPMENTS OF THE CHLORlNlTYf 45 J C J NlHOUL and B M JAMART (Editors) SALINITY CONCEPT IN OCEANDGRAPHY THREE DIMENSIONAL MODELS OF MARINE AND ESTUARINE DYNAMICS 8 E LlSlTZlN 46 J C J NIHOUL and B M JAMART (Edftors) SEA LEVELCHANGES SMALL SCALE TURBULENCE AN0 MIXING IN THE OCEAN 9 R H PARKER 47 M R LANDRY and B M HICKEY (Editors) THE STUDY OF BENTHIC COMMUNITIES COASTAL OCEANOGRAPHY OF WASHINGTON AND OREGON 10 J C J NIHOUL (Editor) 48 SR MASSEL MODELLINGOF MARINE SYSTEMS HYDRODYNAMICS OF COASTAL ZONES 1 1 01 MAMAYEV 49 V C LAKHAN and A S TRENHAILE (Editors) TEMPERATURE SALINITY ANALYSIS OF WORLD OCEAN WATERS APPLICATIONS IN COASTAL MODELING 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 SUBMERSIBLESAND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINEERING 18 J W CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS 19 J C J NIHOUL (Editor) BOTTOM TURBULENCE 20 P H LEBLONDand L A MYSAK WAVES IN THE OCEAN 21 C C VON DER BORCH (Editor) SYNTHESIS OF DEEP SEA DRILLING RESULTS IN THE INDIAN OCEAN 22 P DEHLINGER MARINE GRAVITY 23 J C J NIHOUL (Editor) HYDRODYNAMICS OF ESTUARIES AN0 FJORDS 24 F T BANNER M B COLLINS and K S MASSIE (Editors) THE NORTH WEST EUROPEAN SHELF SEAS THE SEA BE0 AND THE SEA IN MOTION 25 J C J NIHOUL (Editor) MARINE FORECASTING 26 H G RAMMING and Z KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R A GEYER (Edirorl MARINE ENVIRONMENTALPOLLUTION 28 J C J NIHOUL (Editor) MARINE TURBULENCE 29 M M WALDICHUK G B KULLENBERG and M J ORREN (Editors) 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 (Editor1 ECOHYDRODYNAMICS 33 R HEKlNlAN PETROLOGY OF THE OCEAN FLOOR 34 J C J NIHOUL [Editor) HYDRODYNAMICS OF SEMI ENCLOSED SEAS 35 B JOHNS(Edi1or) PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS 36 J C J NIHOUL IEditorl HYDRODYNAMICS OF THE EOUATORIAL OCEAN 37 W LANGERAAR SURVEYING AND CHARTING OF THE SEAS

Elsevier Oceanography Series, 50

MESOSCALE/SYNOPTIC COHERENTSTRUCTURES IN GEOPHYSICAL TURBULENCE PROCEEDINGS OF THE 20TH INTERNATIONAL LIEGE COLLOQUIUM ON OCEAN HYDRODYNAMICS

Edited by

J.C.J. NIHOUL University of Liege, 65 Sart Tilman, 6-4000 LiGge, Belgium and

B.M. JAMART MUMM, Institute of Mathematics, 15 Avenue des Tilleuls, 6-4000 Liege, Belgium

ELSEVIER Amsterdam - Oxford -New

York -Tokyo

1989

ELSEVIER SCIENCE PUBLISHERSB.V. Sara Burgerhartstraat25 P.O. Box 2 1 1, 1000 AE Amsterdam, The Netherlands

Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 655, Avenue of the Americas New York, NY 10010, U S A .

ISBN 0-444-87470-4 (Vol. 50) ISBN 0-444-4 1623-4 (Series)

0 Elsevier Science PublishersB.V., 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./ Physical Sciences & EngineeringDivision, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulationsfor readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper.

Printed in The Netherlands

V

FOREWORD

The International Liege 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 Litge, the Scientific Committee on Oceanographic Research (SCOR), the Intergovernmental Oceanographic Commission, the Division of Marine Sciences (UNESCO), and the Office of Naval Research for their most valuable support. We dedicate this volume to the memory of Professor Konstantin N. FEDOROV, of the Academy of Sciences of the USSR. Professor Fedorov was one of the world leading researchers in the field of turbulence. His many contributions cover a wide range of the spectrum, with particular emphasis, in recent years, on the small- and mesoscale phenomena. A hallmark of Fedorov’s approach to physics is his open-minded insistence on combining all possible sources of information, from laboratory data to theoretical generalizations through interpretation of satellite observations: Professor Fedorov was an integrator. In addition, Professor Fedorov was a driving force in international scientific communications. Konstantin has attended many of the Li&ge Colloquia on Ocean Hydrodynamics: all who met him have the fondest memories.

Jacques C. J. Nihoul

Bruno M. Jamart

IN MEMORIAM

Professor Konstantin N . Fedorov

TABLE OF CONTENTS MUSHROOM-LIKE CURRENTS (VORTEX DIPOLES): ONE OF THE MOST WIDESPREAD FORMS OF NON-STATIONARY COHERENT MOTIONS IN THE OCEAN K.N. Fedorov and A.I. Ginsburg

....................................................................................................

1

MODELLING OF "MUSHROOM-LIKE" CURRENTS (VORTEX DIPOLES) IN A LABORATORY TANK WITH ROTATING HOMOGENEOUS AND STRATIFED FLUIDS K. N. Fedorov, A.I. Ginsburg and A.G. Kostianoy

........................................................................

ON THE MULTITUDE OF FORMS OF COHERENT MOTIONS IN MARGINAL ICE ZONES (MIZ) A.I. Ginsburg and K.N. Fedorov ....................................................................................................

15

25

DIFFERENTIAL ROTATION (BETA-EFFECT) AS AN ORGANIZING FACTOR IN MESOSCALE DYNAMICS V.D. Larichev

..................................................................................................................................

41

GEOSTROPHIC REGIMES AND GEOSTROPHIC TURBULENCE BEYOND THE RADIUS OF DEFORMATION B. Cushman-Roisin and Bcnyang Tang

..........................................................................................

51

THE EVOLUTION OF COOLING RINGS William K. Dewar

...........................................................................................................................

75

VORTICITY FRONTOGENESIS Melvin E. Stem

...............................................................................................................................

95

WEAKLY NON-LOCAL SOLITARY WAVES

J.P. Boyd

.........................................................................................................................................

103

NONLINEAR INTRUSIONS D. Nof

.............................................................................................................................................

113

THE DECAY OF MESOSCALE VORTICES Richard P. Mied

..............................................................................................................................

EKMAN DISSIPATION OF A BAROTROPIC MODON Gordon E. Swaters and Glenn R. Flier1 ..........................................................................................

135

149

VlII

ON THE STABILITY OF OCEAN VORTICES P. Ripa

.............................................................................................................................................

167

INFLUENCE OF TOPOGRAPHY ON MODON PROPAGATION AND SURVIVAL G.F. Camevale, R. Purini, M. Briscolini and G.K.Vallis

..............................................................

181

MESOSCALE STRUCTURES ON DENSITY DRIVEN BOUNDARY CURRENTS

Scott A. Condie

...............................................................................................................................

197

FACTORS INFLUENCING ASYMMETRY AND SELF ADVE(JT1ON IN OCEAN EDDIES David C. Smith IV and Arlene A. Bird

..........................................................................................

211

BAROTROPIC AND BAROCLIMC INSTABILITIES OF AXISYMMETRIC VORTICES IN A QUASIGEOSTROPHIC MODEL X.J. Canon an J.C. McWiLliams

.....................................................................................................

225

EDDY-GENESIS AND THE RELATED HEAT TRANSPORT: A PARAMETER STUDY

S.S. DrijIhout

..................................................................................................................................

245

EDDY GENERATION BY INSTABILITY OF A HIGHLY AGEOSTROPHIC FRONT:MEAN FLOW INTERACTIONS AND POTENTIAL VORTICITY DYNAMICS Richard A. Wood

............................................................................................................................

265

EDDY-CURRENT INTERACTIONS USING A T W S L A Y E R QUASI-GEOSTROPHIC MODEL M. Ikeda and K. Lygre

...................................................................................................................

277

SIMULATION OF OCEAN TOMOGRAPHY IN A QG MODEL Fabienne Gaillard

............................................................................................................................

293

SIMULATION EXPERIMENTS OF THE EVOLUTION OF MESOSCALE CIRCULATION FEATURES IN THE NORWEGIAN COASTAL CURRENT P.M. Haugan, J.A. Johannessen, K. Lygre, S. Sandven and O.M. Johannessen

............................

303

NUMERICAL MODELING OF AGHULAS RETROFLECTION AND RING FORMATION WITH ISOPYCNAL OUTCROPPING D.B. Boudra, K.A. Maillet and E.P. Chassignet

.............................................................................

315

EVOLUTION OF RINGS IN NUMERICAL MODELS AND OBSERVATIONS E.P. Chassignet, D.B. Olson and D.B. Boudra

...............................................................................

337

IX

THE ROLE OF MESOSCALE TURBULENCE IN THE AGULHAS CURRENT SYSTEM J.R.E. Lutjeharms

............................................................................................................................

357

MODELLING THE VARIABILITY IN THE SOMALI CURRENT Mark E. Luther and James J. O'Brien

............................................................................................

373

GENERAL CIRCULATION OF THE MID-LATITUDE OCEAN: COUPLED EFFECTS OF VARIABLE WIND FORCINGS AND BOTTOM TOPOGRAPHY ROUGHNESS ON THE MEAN AND EDDY CIRCULATION B. Bamicr and C. Le Provost

..........................................................................................................

387

ASYMMETRICAL WIND FORCING DRIVING SOME NUMERICAL EDDY-RESOLVING GENERAL CIRCULATION EXPERIMENTS

J. V e m n and C. Le Provost

...........................................................................................................

407

ON THE RESPONSE OF THE BLACK SEA EDDY FIELD TO SEASONAL FORCING E.V. Stanev

.....................................................................................................................................

423

THE DYNAMICAL BALANCE OF THE ANTARCTIC CIRCUMPOLAR CURRENT STUDIED WITH AN EDDY RESOLVING QUASIGEOSTROPHIC MODEL J.-0. Wolff and D.J. OlberS

............................................................................................................

435

A LIMITED-AREA PRIMITIVE EQUATION MODEL OF THE GULF STREAM: RESULTS IN STATISTICAL EQUILIBRIUM

J. Dana Thompson and W. J. Schmitz, Jr.

......................................................................................

459

A SYNOPSIS OF MESOSCALE EDDIES IN THE GULF OF MEXICO A.W. Indest, A.D. Kirwan, Jr., J.K. Lewis and P. Reinersman

......................................................

485

MESOSCALE EDDIES AND SUBMESOSCALE, COHERENT VORTICES: THEIR EXISTENCE NEAR AND INTERACTIONS WITH THE GULF STREAM J.M. Bane, L.M. O'Keefe and D.R. Watts

....................................................................................

501

A SUMMARY OF THE OPTOMA PROGRAM'S MESOSCALE OCEAN PREDICTION STUDIES IN THE CALIFORNIA CURRENT SYSTEM

....................................................................

519

GEOMETRY-FORCED COHERENT STRUCTURES AS A MODEL OF THE KUROSHIO LARGE MEANDER T. Yamagata and S . Umatani ..........................................................................................................

549

Michele M. Rienecker and Christopher N.K. Mooers

X

THE BEHAVIOR OF KUROSHIO WARM CORE RINGS NEAR THE EASTERN COAST OF JAPAN

T. Matsuura and M. Kamachi

.......................................

..............

561

ADVECTIVE SURFACE VELOCITIES DERIVED FROM SEQUENTIAL IMAGES OF EDDY FIELDS M. Kamachi

.....................................................................................................................................

577

LABORATORY EXPERIMENTS ON DIPOLE STRUCTURES IN A STRATIFIED FLUID G.J.F. van Heijst and J.B. Flor

...............................................................................................

59 1

ON TRIPOLAR VORTICES R.C. Kloostelziel and G.J.F. van Heijst

..........................................................................................

609

LABORATORY STUDIES OF ISOLATED EDDIES IN A ROTATING FLUID J.A. Whitchead

...................................................................

............................................

.

627

LABORATORY STUDIES OF PSEUDO-PERIODIC FORCING DUE TO VORTEX SHEDDING FROM AN ISOLATED SOLID OBSTACLE IN A HOMOGENEOUS ROTATING FLUID G . Chabcrt d’Hieres, P.A. Davies and H. Didelle

....................................................................

639

TIME-DEPENDENT ROTATING STRATIFIED FLOW PAST ISOLATED TOPOGRAPHY

Don L. Boyer, X. Zhang and P.A. Davies

..............................................

.........

655

FLAT VORTEX STRUCTURES IN A STRATIFIED FLUID S.I. Voropayev ................................................................................................................................

67 1

LABORATORY EXPERIMENTS WITH BAROCLINIC VORTICES IN A ROTATING FLUID A.G. Kostianoy and A.G. Zatsepin .................................................................................................

69 1

LONG-LIVED SOLITARY ANTICYCLONES IN THE PLANETARY ATMOSPHERES AND OCEANS, IN LABORATORY EXPERIMENTS AND IN THEORY M.V. Nezlin and G.G. Sutyrin

...........................................

..............................................

701

NUMERICAL MODELLING OF THE FORMATION, EVOLUTION, INTERACTION AND DECAY OF ISOLATED VORTICES

G.G.Sutyrin and I.G. Yushina

.....

.....

72 1

EDDY-RESOLVING MODEL OF IDEALIZED AND REAL OCEAN CIRCULATION D.G. Seidov, A.D. Marushkevich and D.A. Nechaev

.....................................................................

737

XI

ON THE EVOLUTION OF INTENSIVE CYCLONIC-ANTICYCLONIC VORTEX G.I. Shapiro and V.N. Konshin

.......................................................................................................

757

FORECAST OF INTENSE VORTEX MOTION WITH AN AZIMUTHAL MODES MODEL G.G. Sutyrin

....................................................................................................................................

771

ON THE DYNAMICS OF LENSLIKE EDDIES G.1. Shapiro

.....................................................................................................................................

783

SYNERGETICS OF THE OCEAN CIRCULATION D.G. Seidov

.....................................................................................................................................

A SURVEY OF OBSERVATIONS ON INTRATHERMOCLINE EDDIES OCEAN A.G. Kostianoy and I.M. Bclkin

797

IN THE WORLD

.....................................................................................................

821

This Page Intentionally Left Blank

LIST OF PARTICIPANTS ARMI L., Prof. Dr., Scripps Institution of Oceanography, La Jolla, California, USA. BANE J.M., Prof. Dr, University of North Carolina, Chapel Hill, North Carolina, USA. BALBI J.H., Prof. Dr., Universit6 de Corse Pascal Paoli, Corte, Corse. BARNIER B., Dr.,Institut de M6canique de Grenoble, Saint Martin d’Hdres, France. BECKERS J.M., Eng., Universit6 de Lidge, Liege, Belgium. BOISSIER Chr., Eng., Centre National de la Recherche Mttkorologique, Toulouse, France. BOUDRA D.B., Prof. Dr., University of Miami, Miami, Florida, USA. BOUKORTT R., Eng., Institut des Sciences de la Mer et de I’Ambnagement du Littoral, Alger, AlgCrie. BOUQUEGNEAU J.M., Dr., University of Litge, Libge, Belgium. BOYD J.P., Prof. Dr., University of Michigan, Ann Arbor, Michigan, USA. BOYER D.L., Dr., University of Wyoming, Laramie, Wyoming, USA. CARNEVALE G.F., Dr., Scripps Institution of Oceanography, La Jolla, California, USA. CARTON X.J., Dr., Ecole Normale SupCrieure, Paris, France. CHABERT D’HIERES G.,Dr., Institut de MCcanique de Grenoble, Saint Martin d’Htres, France. CHASSIGNET E.P., Dr., University of Miami, Miami, Florida, USA. CONDIE S.A., Dr., Australian National University, Canberra, Australia. CUSHMAN-ROISIN B., Prof. Dr., Florida State University, Tallahassee, Florida, USA. DAUBY P., Dr., University of Lidge, Litge, Belgium. DAVIES P.A., Dr., The University, Dundee, UK. DELCOURT D., Eng., University of Litge, Libge, Belgium. DELEERSNUDER E., Eng., University of Litge, Litge, Belgium. DEWAR W.K., Prof. Dr., Florida State University, Tallahassee, Florida, USA. DIRKS R., Dr., University of Utrecht, Utrecht, The Netherlands. DJENIDI S . , Dr., University of Lidge, Lidge, Belgium. DIKE P.P.G., Prof. Dr., Plymouth Polytechnic, Plymouth, UK. DRUFHOUT S.S., Dr., Royal Netherlands Meteorological Institute, De Bilt, The Netherlands. EVERBECQ E., Eng., University of Litge, Libge, Belgium. FEDOROV K.N., Prof. Dr., Academy of Sciences USSR, Moscow, USSR. FU JIA, Prof. Dr., Academia Sinica, Beijing, People’s Republic of China. GAILLARD F., Dr., IFREMER, Plowant, France. GARCIA E., Mr., Instituto de Ciencias del Mar, Barcelona, Spain. GOFFART A., Miss, University of Litge, Litge, Belgium. GOFFART P., Eng., University of Litge, Lidge, Belgium. GREGORIS Y., Eng., Centre National de la Recherche MCtCorologique, Toulouse, France. GRIFFITHS C., Miss, Exeter University, Exeter, UK.

XIV

GUGLIELMACCI D,, Mr., UniversitL de Corse. Ajaccio, Corse. HAPPEL J.J., Eng., University of Litge, LiZge, Belgium. HAUGAN P.M., Mr., Nansen Remote Sensing Center, Bergen, Norway. HEBURN G.W., Dr., NORDA, NSTL Station, Mississipi, USA. HECQ J.H., Dr., University of Litge, Likge, Belgium. HUA B.L., Dr., IFREMER, Brest, France. IKEDA M., Dr., Bedford Institute of Oceanography, Dartmouth, Canada. JAMART B.M., Dr., Management Unit of the Mathematical Models of the North Sea and the Scheldt Estuary (MUMM), Litge, Belgium. KAMACHI M., Dr., Kyushu University, Kasuga, Japan. KARAFISTAN-DENIS A., Dr., University of Likge, Litge, Belgium. KELLY F.J., Dr., Texas A&M, College Station, Texas, USA. KINDER T.H., Dr., Office of Naval Research, Arlington, Virginia, USA. KIRWAN A.D., Prof. Dr., Old Dominion University, Norfolk, Virginia, USA. KLOOSTERZIEL R.C., Dr., University of Utrecht, Utrecht, The Netherlands. KOSTIANOY A., Dr., Academy of Sciences USSR, Moscow, USSR. KRUSE F., Dr., Alfred-Wegener-Institut fur Polar und Meeresforchung, Bremerhaven, Federal Republic of Germany. LARICHEV V.D., Prof. Dr., Academy of Sciences USSR, Moscow, USSR. LEBON G . , Prof. Dr., University of Liege, Litge, Belgium. LENSU M., Mr., Finnish Institute of Marine Kesearch, Helsinki, Finland. LINDEN P.F., Dr., University of Cambridge, Cambridge, UK. LIU Y . , Prof. Dr., Institute of Mechanics, Beijing, People’s Republic of China. LUTHER M.E., Dr., Florida State University, Tallahassee, Florida, USA. LUTJEHARMS J.R.E., Dr., NRIO/CSIR, Stellenbosch, South Africa. LYGRE K., Mr., Nansen Remote Sensing Center, Bergen, Norway. McCLIMANS Th. A., Prof. Dr., Norwegian Institut of Technology, Trondheim, Norway. McWILLIAMS J., Dr., NCAR, Boulder, Colorado, USA. MIED R.P., Dr., Naval Research Laboratory, Washington, DC, USA. MILLOT C., Dr., Antenne du Centre d’OcCanologie de Marseille, La Seyne, France. MOOERS C.N.K., Prof. Dr., Institute for Naval Oceanography, NSTL Station, Mississipi, USA. MOUCHET A., Miss, University of Litge, Litge, Belgium. NIHOUL J.C.J., Prof. Dr., University of Likge, Libge, Belgium. NISHIMURA T., Dr., Science University of Tokyo, Noda City, Japan. NOF D., Prof. Dr., Florida State University, Tallahassee, Florida, USA. NORRO A., Mr., University of Libge, LiZge, Belgium. ONYANGO H., Mr., Kenya Marine and Fisheries Institute, Mombasa, Kenya. PINARDI N., Dr., IMGA-CNR, Modena, Italy. REES J.M., Mr., Fisheries Laboratory, MAFF, Lowestoft, UK.

xv RIPA P., Dr., CICESE, Ensenada, Mexico. ROBINSON A.R., Prof. Dr., Harvard University, Cambridge, Massachusetts, USA. ROED L.P., Prof. Dr., Veritas Offshore Technology and Services, Veritec, Hovik, Norway. RONDAY F.C., Dr., University of Li&e, Libge, Belgium. SALUSTI S.E., Prof. Dr., Universita La Sapienza, Roma, Italy. SMETS E., Dr., Ministerie Van Openbare Werken, Borgerhout, Belgium. SMITH D.C. IV, Prof. Dr., Naval Postgraduate School, Monterey, California, USA. SMITZ J., Eng., University of Litge, Libge, Belgium. SPITZ Y., Miss, Florida State University, Tallahassee, Florida, USA. STANEV E., Dr., University of Sofia, Sofia, Bulgary. STERN M.E., Prof. Dr., Florida State University, Tallahassee, Florida, USA. SUTYRIN G.G., Dr., Academy of Sciences USSR, Moscow, USSR. SWATERS G.E., Prof. Dr., University of Alberta, Edmonton, Canada. THOMPSON J.D., Dr., Naval Ocean Research and Development Activity, NSTL Station, Mississipi, USA. VAN HELTST G.F., Dr., University of Utrecht, Utrecht, The Netherlands. VERRON J., Dr., Institut de Micanique de Grenoble, Saint Martin d’Hkres, France. VOROPAYEV S.I., Dr., Academy of Sciences USSR, Moscow, USSR. WHITEHEAD J.A., Dr., Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA. WOLFF F.O., Dr., University of Hamburg, Hamburg, Federal Republic of Germany. WOOD R.A., Dr., University of Southampton, Southampton, UK. YAMAGATA T., Prof. Dr., Kyushu University, Kasuga, Japan. ZORKANI M., Dr., Ecole Hassania des Travaux Publics et des Communications, Casablanca, Maw.

ACKNOWLEDGMENTS The following grants and contracts are gratefully acknowledged: FNRS contract C 31/5-MO. 24360,572-569; - IOCNNESCO contract SC/RP 267006.8; - ONR grant Nr. 0001487-J-1124. -


1

MUSHROOM-LIKE CURRENTS (VORTEX DIPOLES): ONE OF THE MOST WIDESPREAD FORMS OF NONSTATIONARY COHERENT MOTIONS IN THE OCEAN K.N. FEDOROV and A.I. GINSBURG Institute of Oceanology, Acad. Sci. USSR, 23 Krasikova Street, 117218, Moscow, USSR

ABSTRACT The space-time characteristics of mushroom-like currents (vortex dipoles) in the ocean are discussed. Some consistent properties in the development of such currents and some features of their kinematics are noticeable. It is shown that such currents are both non-stationary and nonlinear motions exhibiting strong deformation fields and relatively low turbulent Reynolds numbers (Re* order of 100) on the scale of a single mushroom-like structure. Satellite images allow us to illustrate some interesting peculiarities of the local dynamics of near-surface waters as related to mushroom-like current formation in two different sets of local conditions: a) in a semi-enclosed region of the Subarctic Frontal Zone (142'-146' E) as a result of the Oyashio interaction with a Kuroshio frontal eddy, b) in the Oyashio proper (152'-160'E) most likely under the influence of a non-homogeneous atmospheric forcing. There are two powerful factors in the Ocean bringing order into turbulent motions which occur on a variety of scales due to a multitude of stochastic forcings. These are the Earth's rotation and density stratification. Since stratification is particularly characteristic of the nearsurface Ocean layer, and since the latter is permanently the subject of numerous multi-scale forcings (including atmospheric ones), it is in the near-surface layer of the Ocean that it is most logical to expect a systematic generation of a variety of orderly (organized) and long existing motions from the chaotic field of initial disturbances. Numerous satellite IR and visible images give us a conclusive proof that there is a multitude of forms of such motions which are so frequently observed in dynamically active ocean areas, such as frontal and marginal ice zones (see a companion paper by Ginsburg and Fedorov in this volume). One of the most typical and widespread forms of such orderly non-stationary oceanic motions is represented by quasi-symmemc currents on scales of 10 to 200 km,which redistribute passive scalars (floating ice, suspended matter, near-surface temperature, etc) in such a way as to produce characteristic patterns which remind of a champignon cross-section (Figs. 1 and 2). The "cap" of such currents corresponds to the vortical portion of the motion (a vortex dipole), which consists of two vortices of opposite sign (cyclonic and anticyclonic), while the "stem" of the mushroom-like pattern corresponds to the inflowing jet. In most cases, the jet length and the size of the vortical portion are of the same order of magnitude, while the jet width does not exceed 10-25% of its length.

2

Fig. 1 : Mushroom-like structure in the south-west part of the Sea of Okhotsk (ice is the tracer). Fragment of the visible image obtained 13 March 1987 from Meteor-30 satellite, medium resolution scanner, 0.5-0.7 pm band.

3

The mushroom-like currents have no preferential spatial orientation (see Figs. 1 and 2, and also satellite images reproduced in Ginsburg and Fedorov, 1984a, b, c; Fedorov and Ginsburg, 1986, 1988). Such currents are often observed in areas where there are no marked bottom topography features, and their typical lifetime is of the order of 1 to about 10 days. During this period of time, the mushroom-like currents grow in size. Their vortical portion (vortex dipole) increases due to the entrainment in spiral motion of water from the outside. This means that the dipole is not driven by P-dynamics, and hence is not restricted to move only in a zonal direction according to the theoretical predictions of Larichev and Reznik (1976) for Rossby solitons on a rotating sphere. The observed currents are also not the cyclone-anticyclone pairs predicted as a theoretical solution of the problem of stationary interaction between uniform flow and bottom topography features (Kamenchovich et al., 1982). In addition, the available sequential series of satellite images (some of them of several days duration) demonstrate clearly that such mushroom-like currents do not originate from a by-chance encounter of two independent eddies of opposite sign, but represent a fully selfcontained type of organized water motion on its own. The mushroom-like currents are evidently formed in a relatively thin near-surface ocean layer (some meters or few tens of meters thick; see below) under the action of a locally applied, short-time, impulse-like forcing. Such local impulsive forcings which act on the Ocean surface or the near-surface layer may be, in the open ocean: localized wind jets, local unbalanced sea level or atmospheric pressure differences, local frontal instabilities; in coastal areas: river discharge or water exchange through straits or lagoon entrances (in particular, when the openings are protected by jetties assuring a directional discharge), ice melting, local jet-like wind forcing focused by coastal geometry or coastal mountain topography, as well as some peculiarities of current-eddy interaction in enclosed and semi-enclosed basins (such interactions apparently involve frontal instabilities, some of which may be similar to those of the open ocean). This has been confirmed in many instances by satellite images presented by Ginsburg and Fedorov (1984a, b, c, 1985, 1986) and Fedorov and Ginsburg (1986, 1988). Since various sharp spatial and temporal inhomogeneities of forcing (including the atmospheric one) are very typical for the ocean, the mushroom-like currents may be a widespread universal form of nonstationary horizontal motion of the near-surface water of the Ocean (Ginsburg and Fedorov, 1984c; Fedorov, 1984; Fedorov and Ginsburg, 1986, 1988). The thickness of the layer involved in the vortex dipole motion cannot possibly be determined from the satellite data alone. It has been hypothesized, on the basis of energy arguments, that such currents are formed in a relatively thin near-surface ocean layer, and that, correspondingly, the near-surface density stratification plays an important role in limiting the depth of their penetration (Ginsburg and Fedorov, 1984a, b, c). For an impulse-type forcing (e.g., a local wind of 10m s-' with a tangential stress of 0.4 Pa acting for 10 hours), the water layer set in motion should not be thicker than 30-40 m to ensure the observed high current velocities (up to = 1 knot) during the initial stages.

4

Fig. 2: Surface manifestations of water circulation in the south-west part of the Sea of Okhotsk as seen in the melting ice distribution. Fragment of the visible image obtained 17 April 1987 from Meteor-30 satellite, medium resolution scanner, 0 . 5 4 . 7 pm band.

5

The seasonal conditions typical of the periods of time when mushroom-like currents have been observed (e.g., near-surface low salinity or high temperature layer), or the local peculiarities of larger scale circulations (particularly near fronts), could usually support the above arguments. A final confirmation of those arguments was found in laboratory experiments modelling vortex dipoles (mushroom-like currents) in a stratified fluid (Ginsburg et al., 1987) and in the results of some recent observations in the Baltic Sea (Victorov, 1987). The latter observations, conducted in 1985, showed that the depth of the layer in which a mushroom-like current was observed did not exceed 10 m, while the current itself existed for about 2 days. At the beginning of their formation, mushroom-like currents involve a very shallow layer and are relatively small in size. At later stages of their development, however, their spatial scales L and typical velocities u correspond to Rossby numbers of the order of lo-', which implies a considerable degree of geostrophic adjustment. Therefore, in those cases when the lifetime exceeds several days, one may expect the same effects as have been observed in laboratory experiments (Ginsburg et al., 1987), i.c, the appearance on the lower boundary of the initial current of a density interface topography typical for geostrophic eddies (i.e., an upward dome under the cyclonic vortex of the dipole, and a downward curvature under the anticyclonic one). If the forcing is sufficiently intense, the vortical motion may not remain limited to the near-surface layer but spread downward through the geostrophic adjustment of the density field as observed in the laboratory experiments (Ginsburg et al., 1987). The non-dimensional parameter

@, which U

determines the importance of the P-effect

relative to the nonlinear terms in the vorticity equation, is estimated to be smaller than 0.1 for the oceanic mushroom-like currents (Fedorov and Ginsburg, 1986). Therefore, there are reasons to believe that the nonlinear effects play a more important role than the p-effect in the genesis and the evolution of the mushroom-like currents. A similar conclusion was reached by Gorbunov et al. (1987) on the basis of some pertinent measurements in the ocean. Thus it is evident that mushroom-like currents are both non-stationary and nonlinear vortical motions. Mushroom-like currents are accompanied by a very strong deformation field. Estimates of the local deformation rate on the basis of satellite data give values up to 5.104 s-l, which is 2 to 3 orders of magnitude larger than the rate of deformation typical for climatic frontal zones of the Ocean (Fedorov, 1983). Consequently, a considerable redistribution of scalars (or temperature) occurs in the near-surface layer under the influence of the deformation fields associated with mushroom-like currents. When strong gradients of such scalars or temperature exist in the area, the sharpest concentration or thermal fronts appear on the leading edge of the "cap" of the growing "mushroom" and on the lateral edges of the jet. Therefore, mushroom-like currents which form in frontal zones (e.g., due to upwelling or of climatic origin) must create some particularly strong frontal contrasts of colour, brightness or/and temperature, sharply visible on satellite images in the visible or infra-red parts of the spectrum (see Fig. on p. 22 in NOAA, Dept. of Commerce, 1979; Fig. 6 in Vastano and Bernstein, 1984; Fig. 1 in Ginsburg and Fedorov, 1986). In such cases, images with enhanced brightness or colour contrasts are a much

6

more convenient way to visualize the coherent structures than, say, isotherms or isoconcentration lines. (An example can be found through comparison of Fig. 3a and 3b in Amone and La Violette, 1986.) It is not excluded that internal viscous friction at the sharp frontal boundaries associated with mushroom-like currents is in fact instrumental in maintaining their coherent (organized) character, ensuring relatively low values (order of 100) of the turbulent Reynolds number Rer on the scale of an entire current. There should exist in this case a spectral separation of scales between the small-scale turbulence of frontal nature and the next most energetic scale of the current itself. For a thermal front of width Bf = 100 m to be in an equilibrium state (Fedorov, 1983) under a deformation rate D, = 5.10-4

8 , the horizontal turbulent heat diffusivity K,

should be K,= 2Bf D, = 10 m2 s-'. If the momentum diffusivity has the same order of magnitude, then with a mean velocity value ii= 0.1 m s-l and a typical width d of the jet portion equal to 10 km = lo4 m, we obtain Re, = lid / K1 = 100. Under a weaker deformation rate (order of gives K,

s-'), more diffuse fronts are observed (Bf: 1 km) (see Fedorov, 1983), which 20 m2 s-' and ensures Re* = 100 for mushroom-like currents of a larger size.

The kinematics of mushroom-like currents has a number of peculiarities.

The vortex

dipole at the end of the jet may not appear at once. It is possible (and it occurs often) that only the anticyclonic vortex is formed at the beginning, with the cyclonic counterpart developing somewhat later (Figs. 7 and 8 in Solomon and Ahlnls, 1978; see also discussion in Ginsburg and Fedorov, 1984a). Sometimes, one of the vortices of the pair or both of them are practically undeveloped, so that the whole structure evokes a hammer or the letter T (e.g., structures in the California upwelling zone; see Fig. on page 22 in NOAA, Dept. of Commerce, 1979). A dipole asymmetry, i.e., the dominance of one of the two vortices, may be of any sign. More often the asymmetry is anticyclonic, but a cyclonic asymmetry is also possible and has been observed in the ocean (see, for example, Fig. on p. 49 in Horstmann, 1983) and in the laboratory tank (Fedorov and Ginsburg, 1986, 1988; Ginsburg et al., 1987). The sign of the asymmetry may be related to the background local shear affecting the dipole during its formation (see companion paper by Ginsberg and Fedorov in this volume). The jet portion of a mushroom-like current may be either straight or curved, the curvature corresponding to the dipole vortical asymmetry. Sometimes, one of the vortices of the dipole or even both vortices give birth to new mushroom-like currents, which gives structures of the type represented in Fig. 3. Typical examples of such structures are, from our point of view, some elements of instability of the Algerian and of the Leeuwin Currents (see, respectively, Fig. 3 in Millot, 1985 and Fig. 1 in Griffiths and Pearce, 1985a). Frequently mushroom-like currents form rather tightly "packed" patterns in which it is not unusual for neighbouring dipoles to share one common vortex (Fig. 4). Such tightly "packed" patterns appear either in complex situations where several localized forcings act in different directions (see Fig. 5 in Fedorov and Ginsburg, 1986) or when compensating motions of a secondary nature develop due to local pressure gradients generated by the strong primary

7

u

t

Fig. 3 : Schematic representation of two mushroom-like currents newly formed from an initial vortex dipole. disturbance (see Figs. 3 and 4 in Fedorov and Ginsburg, 1986; Fig. 3a in Amone and La Violette, 1986). One of the dynamically active areas where mushroom-like currents form very frequently and are particularly clearly visible on IR and visible images is the Subarctic Polar Frontal Zone east of Honshu Island. A complicated horizontal circulation occurs in this zone where waters of the Oyashio, Kurushio and Tsugaru Currents meet and interact. This circulation is associated with numerous eddies (cyclonic and anticyclonic) and jets spreading northward and southward from the main streams of the Kuroshio and the Oyashio (Bulatov, 1980a, b; Vastano and Bemstein, 1984; Ginsburg and Fedorov, 1986). A typical situation observed here is that the formation or the movement of one eddy brings about as a consequence the formation or strengthening of a whole series of jets (meridional or zonal) and associated eddies at distances up to 300 km from the initial disturbance. Examples are the following two satellite images: a visible band image obtained from the Meteor-30 satellite on 19 May 1984 (Fig. 5) and an IRimage obtained from NOAA-6 satellite on 20 May 1981 (Fig. 6, adapted from Vastano and Bemstein, 1984). It is easy to see that in spite of some substantial differences in detail, the positions and configurations of the three major circulation features: an eddy A(A’), a mushroom-like structure l(1’) and a jet 2(2’) are almost identical in both cases (Figs. 5 and 6). The jet portion of the

8

I

II

Fig. 4 : Schematic representation of two types (I and 11) of compact packing of mushroom-like currents in the ocean. I: with anti-parallel jet portions; 11: with mutually perpendicular jet portions. mushroom-like current l(1’) is some 20-35km wide in both instances and it is tangential to the eddy A(A’) at a point on its western periphery. In this case, the direction of the jet is opposite to the water rotation on this side of the eddy (A,A’), the situation being different from the usual one when the eddy motion entrains jets of surrounding water with their characteristic T,S properties (see e.g., Fedorov, 1983). The lengths (L)and the size (H)of the dipole portion of the mushroom-like structures (l,l’), measured on the two images, are. as follows: L, = 140 km,LIP= 220 km, HI = 80 km and HI, = 150 km. In both cases, the jets (2,2’) which are observed near the “caps” of the mushroom-like structures are directed westward and have the following lengths : L, = 250 km and L,. = 215 km. A series of IR-images reproduced by Vastano and Bemstein (1984) allows us to determine the sequence of events which led to the formation of the three circulation elements 1’ to 3’. The surface waters in the eddy A’ started to form a spiral between 28 April and 4 May 1981, the appearance of the jet current 1’ is noticeable on 13 May, while the mushroom-like structures 2’ to 4’ appear on the satellite image of 20 May when the already formed mushroom-like current 1’ reached the northern edge of the Kuroshio frontal system. It is likely that in May 1984 (Fig. 5) the time sequence of events was analogous.

9

144 " E

I

139"E

149"E

144 E

149"E

Fig. 5 : Satellite image of the Subarctic Polar Frontal Zone (a), and its interpretational scheme (b). The image was obtained on 19 May 1984 from the Meteor-30 satellite, low resolution scanner, 0.5-0.6 p band. Arrows on the scheme show directions of water movements. Letters and numbers denote eddies and jets streams, respectively.

10

144 " E 'A/

149" E

. . . .. ..

40"N

40" N

144O E

149" E

Fig. 6 : Interpretational scheme of an IR-image obtained on 20 May 1981 from the NOAA-6 satellite (published by Vastano and Bernstein, 1984): 1 - cold subarctic waters; 2 - transformed cold waters; 3 - warm waters; 4 - transformed warm waters. The letters A' and B' designate eddies, while the numbers 1' to 4' show jet streams.

The most likely cause for the formation of this typical pattern of circulation is the entrainment of Oyashio water from the east along the southern periphery of the anticyclone A(A'). As a result, an intensive westward jet is formed. Its velocity on 20 May 1981 was estimated by Vastano and Borders (1984) at 75 cm s-'. The inevitable slowing down of this jet on approaching the western boundary of the region (the coast of Honshu), in conjunction with an impulse from the north-west to the south-east (mushroom-like current 4') from the Tsugaru Strait in a semi-enclosed area (which is the region under consideration), most likely led to the

11

formation of a gigantic mushroom-like structure. Its right anticyclonic branch embraced the anticyclone A(A’), while its left cyclonic extension degenerated into a new mushroom-like current l(1’). When reaching the Kuroshio frontal system, this current itself produced the jet 2(2’), which in turn influenced the Oyashio Stream along the Honshu coast. The rate of growth of the mushroom-like Structures 2’ and 3’ was estimated by us from the sequence of IR-images for 20 and 21 May (as reproduced by Vastano and Bemstein, 1984) to be approximately equal to 25-30 cm s-’. Correspondingly, current velocities in their jet parts should be much higher (Ginsburg and Fedorov, 1984~). Thus, in the semi-enclosed area between Hokkaido (to the North), the Kuroshio frontal system (to the south) and Honshu (to the west), the initial impulse corresponding to the entrainment of Oyashio water along the southern periphery of the anticyclone A(A’) led to the formation of several interrelated jets of zonal and meridional directions with associated vortex dipoles at their extremities. An additional argument in favour of this hypothesis is the obviously passive character of the heat and plankton redistribution by the newly formed jets and eddies 1-2 (1’-2’). The passive character of the heat redistribution is also clearly evident in the mushroomlike current pattern shown in Fig. 7, where a graphic interpretation of the NOAA-9 IR-image for 20 September 1987 is presented. This figure illustrates an extremely interesting dynamical situation in the Oyashio zone between 147’ and 155’ E. In this zone which is limited to the north by the Kuril Islands and to the south by the Subarctic Front, one can distinctly see at least three large mushroom-like structures (marked with roman numerals). For the smallest one, the width of the vortex pair (the “cap”) is about 180 km. Structures I and In, which contain cold water (shaded) spreading from the north, have their jet portions oriented along the Oyashio axis. Structure 11, with somewhat warmer water, forms with structure I a very typical tightly packed pattern of mushroom-like currents. It is not clear whether structure IV reflects a real mushroom-like current or whether its shape is a mere consequence of currents II and III.

As seen in Fig. 7, the cold Oyashio waters in structures I and HI are separated by warmer waters from the south. Since it is rather difficult to expect even a short period interruption of the Oyashio current over its entire thickness, it is logical to assume that the circulation features seen on Fig. 7 are related mostly to the near-surface layer of the ocean. In the absence of relevant hydrographic and meteorological information, and since no consecutive satellite images are available for the preceding days, it is practically impossible to establish the causes of this unusual non-stationary situation. One can only suggest that this was due to some strong atmospheric forcing or some deep instability of the Oyashio Stream, and that it does not seem like having been caused by the bottom topography of the Kuril-Kamchatka Trench (whose contours are shown in Fig. 7 by a dash-dot line). What is important in this example is the complete decoupling of the near-surface dynamics (which is entirely transient) from the deeper water dynamics associated with a permanent or quasi-permanent current such as the Oyashio.

12

150"E

145"E

155" E

45" N

45" N 155"E

/40°N

40" N

145"E

150.

E

Fig. 7 : Interpretational scheme of an IR-image of the north-westem part of the Pacific Ocean obtained on 20 September 1987 from the NOAA-9 satellite. Thick solid lines are used to show boundaries of mushroom-like currents (marked with roman numerals). The thin line shows the position of the Subarctic Front. The broken lines indicate the boundary between waters of different temperature. The dash-dot line shows the 6000 m contour line of the KurilKamchatka Trench. Arrows show the directions of water movements. Areas of cold water spreading from the north are shaded. The probability that a strong atmospheric influence is at the origin of the above case (Fig. 7) seems quite high to us. Bulatov's (1980b) observations show that the appearance of warm surface jets spreading northward from the Kuroshio towards the Subarctic Frontal Zone is often related to the strengthening of wind of favourable direction. In our case (Fig. 7), such a forcing could have been produced by a south-easterly wind producing the surface flow IV which

13

cut across the Oyashio in the near-surface layer. The inevitable slowing down of this current upon approaching the Kuril Islands and the local pressure gradients thus created by the flow IV or by a total change of the wind field could have produced the mushroom-like currents I-III and the interlocking pattern 1-11. One thing may be stated in a definitive way: this non-stationary situation in the area of the Oyashio current is not associated with an instability of its density front. Mushroom-like structures produced by frontal instabilities have been modelled in laboratory experiments (Griffiths and Linden, 1981) and have been observed often enough in the Ocean (Ginsburg and Fedorov, 1984a, b, c; Griffiths and Pearce, 1985b; Fedorov and Ginsburg, 1986, 1988) to tell the difference. Similar cases of vortex dipole formation which are not related to frontal instabilities have also been observed in other locations, e.g., in the Irminger current (see Ginsburg and Fedorov, 1984a) and in the GuIf Stream (see ERTS-1 image of 4 July 1973 in Sawyer and Apel, 1976 and its discussion by Fedorov and Ginsburg (1986, 1988)). Analyzing such situations along with the relevant meteorological and hydrographic data will broaden our conceptual ideas on the near-surface layer dynamics and will help to understand which mechanisms in the Ocean effect the transfer, redistribution and accumulation of momentum supplied by a variety of localized short-period atmospheric forcings.

REFERENCES h o n e . R.A. and La Violette. P.E.. 1986. Satellite definition of the biooDtical and thermal vhation of coastal eddies associated with the African current. J. Geoihys. Res., 91 C2: 2351-2364. Bulatov, N.V., 1980a. Vortical structure of the Subarctic front in the northwestern Pacific Ocean. Uchebnye zapiski LGU. Sputnikovaya okeanologia, 403, 2: 61-72 (in Russian). Bulatov, N.V., 1980b. On the structure and dynamics of warm water streams north of the Subarctic front in the Pacific Ocean. Izvestia Tihookeanskogo Nauchno-Issledovatelskogo Instituta Rybnogo Hoziaystva i Okeanografii (TINRO), 104: 50-57 (in Russian). Fedorov, K.N., 1983. The Physical Nature and Structure of Oceanic Fronts. Gidrometeoizdat, Leningrad, 296 pp. (in Russian). Fedorov, K.N., 1984. Satellite technique and development of modem concepts in Ocean dynamics. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 4: 3-13. Fedorov, K.N. and Ginsburg, A.I., 1986. "Mushroom-like" currents (vortex dipoles) in the ocean and in a laboratory tank. Annales Geophysicae, B 5: 507-516. Fedorov, K.N. and Ginsburg, A.I., 1988. Near-surface Layer of the Ocean. Gidrometeoizdat, Leningrad, 303 pp. (in Russian). Ginsburg, A.I. and Fedorov, K.N., 1984a. Mushroom-like currents in the Ocean (based on the analysis of satellite images). Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 3: 18-26. Ginsburg, A.I. and Fedorov, K.N., 1984b. Evolution of mushroom-like currents in the ocean. Dokl. Acad. Sci. USSR, 276, 2: 481-484. Ginsburg, A.I. and Fedorov, K.N., 1984c. Some consistencies in the development of mushroom-like currents in the Ocean revealed by analysis of space imagery. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 6: 3-13. Ginsburg, A.I. and Fedorov, K.N., 1985. Systems of transverse jets in coastal upwelling: satellite information and physical hypotheses. Issled. Zemli iz Kosmosa (Earth Res. from

14

Space, in Russian), 5: 3-10. Ginsburg, A.I. and Fedorov, K.N., 1986. Near-surface water circulation in the Subarctic Frontal Zone from satellite data. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 1: 8-13. Ginsburg, A.I., Kostianoy, A.G., Pavlov, A.M. and Fedorov, K.N., 1987. Laboratory reproduction of mushroom-like currents (vortex dipoles) under rotation and stratification conditions. Izv. Acad. Sci. USSR, Ser. Atmos. and Ocean Physics (in Russian), 2: 170-178. Gorbunov, Yu.A., Eremeev, V.N., Ivanov, L.M., Losev, S.M. and Smelianskiy, V.I., 1987. Some dynamic characteristics of mushroom-like current. Dokl. Acad. Sci. Ukr. SSR, B, 3: 5-8 (in Russian). Griffiths, R.W. and Linden, P.F., 1981. The stability of vortices in rotating, stratified fluid. J. Fluid Mech., 105: 283-316. Griffiths, R.W. and Pearce, A.F., 1985a. Satellite images of an unstable warm eddy derived from the Leeuwin current. Deep Sea Res.. 11: 1371-1380. Griffiths, R.W. and Pearce, A.F., 1985b. Instability and eddy pairs on the Leeuwin current south of Australia. Deep Sea Res., 12: 1511-1534. Horstmann, U., 1983. Distribution patterns of temperature and water colour in the Baltic Sea as recorded in satellite images: indicators for phytoplankton growth. Bench. Inst. Meereskunde. Christian-Albrechts-Universittit(Kiel), 106, 147 pp. Kamenkovich, V.M., Koshlyakov, M.N. and Monin, A.S., 1982. Synoptic Eddies in the Ocean. Gidrometeoizdat, Leningrad, 264 pp. (in Russian). Larichev, V.D. and Reznik, G.M., 1976. On the two-dimensional solitary Rossby waves. Dokl. Acad. Sci. USSR, 231, 5: 1077-1079 (in Russian). Millot, C., 1985. Some features of the Algerian current. J. Geophys. Res., 90 C4: 7169-7176. NOAA, US Dept. of Commerce, 1979. Oceanic and Related Atmospheric Phenomena as Viewed from Environmental Satellites. The Walter A. Bohan Company, Illinois, 43 pp. Sawyer, G. and Apel, J.R., 1976. Satellite images of ocean internal wave signatures (atlas). NOAA, Atlant. Oceanogr., and Meteorolog. Lab., Miami, 17 pp. Solomon, H. and Ahlntis, K., 1978. Eddies in the Kamchatka current. Deep Sea Res., 4: 403410. Vastano, A.C. and Bernstein, R.L., 1984. Mesoscale features along the first Oyashio intrusion. J. Geophys. Res., 89 C1: 587-596. Vastano, A.C. and Borders, S.E., 1984. Sea surface motion over an anticyclonic eddy on the Oyashio front. Remote Sens. Envir., 1: 87-90. Victorov, S.V., 1987. Complex investigation of the Baltic Sea on the basis of satellite information. Abstracts, 3th Congress of Soviet Oceanographers. Ser. "Physics and Chemistry of the Ocean: Polar and Regional Oceanography". Gidrometeoizdat, Leningrad, pp. 49-51 (in Russian).

15

MODELLING OF "MUSHROOM-LIKE" CURRENTS (VORTEX DIPOLES) IN A LABORATORY TANK WITH ROTATING HOMOGENEOUS AND STRATIFIED FLUIDS K.N. FEDOROV, A.I. GINSBURG and A.G. KOSTIANOY Institute of Oceanology, Acad. Sci. USSR, 23 Krasikova Street, 117218, Moscow, USSR

ABSTRACT The dynamics of vortex dipoles in homogeneous and stratified fluids has been studied on a rotating platform in a laboratory. It is shown that, unlike what happens in non-rotating conditions, a disturbance produced by a localized impulse in a solidly rotating homogeneous fluid spreads instantly through the whole column in accordance with the Proudman-Taylor principle, independently of the water depth or of the level at which the disturbance is applied. In a twolayer system, a weak forcing causes an eddy disturbance limited to the upper layer only. Under a stronger impulse the motion is transmitted to the lower layer, the density interface being lower under the anticyclonic part of the eddy pair and higher under the cyclonic one. The characteristic "width" D and "length" L of the vortex dipole are found to be functions of time, D - ta and L - tp, the values of a and p increasing with an increase of the effective impulse (a from 0.13 to 0.30, and p from 0.19 to 0.47). The propagation speed of dipoles is higher during some initial stage and considerably lower at later stages, which matches the observations of oceanic vortex dipoles from satellites.

1. INTRODUCTION "Mushroom-like" currents (vortex dipoles) were discovered several years ago as a result of a systematic analysis of numerous visible band and IR satellite images of the ocean (Ginsburg and Fedorov, 1984; Fedorov and Ginsburg, 1986, 1988; see also a paper by the same authors in this volume). Until recently, satellite data have been practically the only way to study the evolution of such mushroom-like currents. It is impossible, however, to answer a number of important specific questions using the satellite data only. Among these questions are the following: 1. What are the roles of the Earth's rotation and of the p-effect in the processes of formation

and evolution of mushroom-like currents ? 2. What is the influence of stratification on the downward penetration of the vortex motion ?

3. Which factors govern the horizontal and vertical dipole scales, and how do these scales vary with time ? 4. What is the cause of a dipole vortical asymmetry (anticyclonic or cyclonic), both of these types being observed in the Ocean ?

16

To answer these questions, we undertook a series of laboratory experiments with vortex dipoles artificially generated in rotating homogeneous and stratified (two-layer) fluids (Ginsburg et al., 1987). The results of the experiments have been analyzed and compared with those obtained by other investigators in a rotating homogeneous fluid (Flierl et al., 1983) and in nonrotating homogeneous and stratified fluids (Voropayev and Filippov, 1985; Afanasyev et al., 1987). 2. LABORATORY EXPERIMENTS

Our experiments were performed in a cylindrical glass tank 40 cm in diameter and 10 cm in height, placed on a turntable. The tank was filled either with fresh water (layer thickness H = 2.5 - 8 cm) or with a two-layer fluid. In the latter case, the lower layer contains salt water ( S = 5 or 10 o/oo and has a thickness denoted H,; the upper layer contains fresh water and its thickness is denoted H,. We used H, = H2 = 4 cm. The tank was turned clockwise with an angular speed of rotation R = 1.05 rad s-* (one revolution per 6 s), i.e., the Coriolis parameter f = 2Q was equal to 2.1 s-'. The experiments always started after solid body rotation of the fluid had been achieved. This was verified by dye track verticality and immobility in the rotating frame. In the case of homogeneous fluid, the achievement of solid body rotation was accelerated by the use of a removable radial partition installed in the tank before switching on the rotation. The partition was either removed after solid body rotation had been achieved, or it was kept in place during some experiments to simulate the effects of coastal features. The experimental configuration corresponded to f-plane conditions, since the p-effect due to the parabolic curvature of the fluid surface was very small indeed. A soluble blue dye was used to visualize the development of eddy disturbances. For the generation of vortex dipoles, different localized impulse-type modes of forcing were used, including some which permitted to vary locally the sign and the intensity of the net relative angular momentum introduced into the rotating fluid a) a pulsating air jet directed tangentially to the surface from a glass tube fixed in the rotating frame (pulse duration 1 s, jet maximum velocity = 10 cm s-I); the relative intensity of the forcing was measured by the number "n" of pulses (from 1 to 10); b) a submerged axisymmemc jet of dyed water analogous to that used in the experiment of Flierl, Stem and Whitehead (1983);

c) short directional pushes by the end of a glass rod, submerged some 0.5

- 1 cm;

d) falling drops of water, e) breakdown of the flow near the partition; f) a local spinning up of the fluid near the free surface during relatively short periods of time

(3 - 12 s) with the aid of a small (1 cm) shallowly submerged blade rotating cyclonically or anticyclonically from an electrical motor fixed in the rotating frame.

One of the major outcomes of all these experiments is that a vortex dipole (or a system of dipoles) represents a universal reaction of a solidly rotating fluid system to any kind of localized, impulse-type disturbance which i n d u c e s into the system finite quantities of positive and negative relative angular momentum (Ginsburg et al., 1987). The vertical structm of the dipoles in a rotating fluid differs strongly from that in a nonrotating fluid. In a non-rotating stratified fluid, the plane flow set in motion by an air or a submerged water jet forcing is limited to a thin layer (a few millimeters) on a particular density surface. The forcing of a homogeneous non-rotating fluid by a submerged water jet leads to the formation of a toroidal spherical vortex, like the well-known vortex rings, growing on the jet front propagation (Voropayev and Filippov, 1985). In a rotating fluid, following the Proudman-Taylor principle, the motion quickly spreads throughout the whole column of water independently of its depth or of the level at which the impulse has been applied. This was c o n h e d by the dye tracers. In the case of a relatively weak air jet forcing applied to the free surface of a two-layer stratified fluid, the motion was limited to the upper layer only; under a stronger forcing, the vortex motion spread also to the lower layer through geosmphic adjustment of the interface. After 2 - 6 revolutions (depending on the intensity of the forcing), the density boundary caved in downward under the anticyclonic vortex and upward under the cyclonic one. The magnitude of the flexure (i.e., the vertical deformation), as predicted by Orlanski and Polinsky (1983), was directly proportional to the strength and duration of the forcing and inversely proportional to the density difference between layers. For example, for AS = 5 HI = 4 cm and n = 5 , the flexure under the anticyclone reached 0.25 HI after 5 revolutions; for n = 10, it was 0.33 HI after 1 - 2 revolutions; for n = 16, it attained 0.50 HI. The main features of vortex dipoles behaviour in both homogeneous and stratified rotating fluids are basically the same and may be summarized as follows (Ginsburg et al., 1987): 1. Symmetric vortex dipoles move in any direction determined by the direction of the forcing.

While propagating, they transport water from the region of the original disturbance and they entrain surrounding water.

2. Dipoles may have any vortical asymmetry (cyclonic or anticyclonic) (Figs. 1, 2) in contrast with the experiments by Flierl, Stem and Whitehead (1983). For an axisymmetric forcing, an anticyclonic dipole asymmetry is the most typical one. However, the local shear (e.g., due to the interaction of two dipoles) may result in a cyclonic dipole asymmetry. The behaviour of asymmemc vortex dipoles is similar to that predicted by Lamb (1932) for point vortex pairs in an ideal fluid, i.e., they propagate along circular trajectories, rotating around the stronger vortex of the pair. 3. The interaction of two dipoles or that of a dipole with a tank wall may result in the loss of one vortex of the pair with subsequent regeneration of the missed vortex in another location. Different types of compact packing of dipoles may also arise, and are often observed in the ocean (Fedorov and Ginsburg, 1986, 1988). When approaching each other, vortices of the same sign weaken and vortices of opposite signs intensify. This leads either to a

18

Fig. 1: Dipole with an anticyclonic asymmetry generated by an axisymmetric air jet applied H, = H, = 3.5 cm, n = 10, tangentially to the surface of a two-layer fluid system: AS = 10,,"/ t = 56 s.

Fig. 2: Dipole with a cyclonic asymmetry generated by an axisymmetric air jet applied tangentially to the surface of a two-layer fluid system: AS = 5"/,, HI = H2 = 4 cm, n = 5, t = 18 s. The cyclonic asymmetry is a result of the interaction between the newly born dipole, and the weak residual anticyclonic motion left by the vortex dipole which had evolved for more than 200 s in the previous experiment.

19

change of each dipole trajectory or to a change of their rotation direction and velocity. The reflection of dipoles from the tank wall is usually in accordance with the principle: "the angle of reflection is equal to the angle of incidence".

4. The dipole lifetime depends on the intensity of the initial forcing and it can reach up to 5 min (some 50 or so revolutions of the system). This was verified by using labeled particles rotating in the dipole vortices. The dipole final adjustment to the state of solid body rotation occurs through a slow elongation of the vortices and their transformation into vertical vortex sheets. It is obviously of interest to investigate whether such dipoles evolve in a "consistent fashion", i.e., whether rules can be formulated to describe their behavior. Let us consider the evolution of dipoles generated by an air jet forcing on the surface of a rotating homogeneous fluid. Subsequent photographs of the dipole can be used to estimate the characteristics scales (transverse D and longitudinal L) of the dipole as a function of time, starting from the moment the forcing is applied. The values of L were defined as the distance from the point of forcing to the front of the dipole. The functions D(t) and L(t) are shown in Figs. 3 and 4 in logarithmic coordinates. The main parameters and results of the experiments are summarized in Table 1, where the values of the layer depth H, the number of pulses n, as well as their values H* and n* relative to a benchmark case (exp. 1) are presented. TABLE 1 Main parameters and results of the experiments.

NN

f(s-')

H(cm)

H*

n

n*

n*/H*

a(D-ta)

P(L-tP)

1 2 3 4

2.1 2.1 2.1 2.1 2.1

4 6 7 7 7

1. 1.5 1.75 1.75 1.75

3 6 7 6 6

1. 2. 2.3 2. 2.

1. 1.33 1.33 1.14 1.14

0.13 0.30 0.25 0.22 0.19

0.19 0.47 0.25 0.21 0.23

5

Because the applied impulse produces motion in the whole column of fluid, it makes sense to consider the "effective" impulse n*/H*. However, this parameter only varies from 1. to 1.33 over the five experiments reported here, so that the resulting statistics are rather scanty. The results shown in Figs. 3 and 4 enable us to draw the following conclusions. First of all, it is obvious that D(t) and L(t) have power dependences on time (D - ta and L - tD). However, the powers a and p are not constant, in contrast to the empirical relation D = 0.5L - t1'3 obtained in a non-rotating stratified fluid forced by an axisymmetric water jet (Afanasyev et al., 1987). These powers depend on the "effective" intensity of the forcing

20

20,

I

I

I

I

-

51

I

I

1

Fig. 3: The dipole transverse scale, D, as a function of time. The sequential numbers of the experiments correspond to those of Table 1.

5

10

20

t [seq

50

100

Fig. 4: The dipole longitudinal scale, L, as a function of time. The sequential numbers of the experiments correspond to those of Table 1.

21

impulse, i.e., on the value n*/H*. As n * / d increases by a factor 1.33, the values of ct and also increase.

p

The formation of dipoles usually takes about 1 - 2 revolutions (6 - 12 s). At this initial stage, the transverse scale of the symmetrically shaped dipoles is equal to 7 - 8 cm, independently of the forcing conditions. The latter influence only the subsequent speed of the dipole growth due to the entrainment of surrounding water. The resulting increase of the volume V of the dipole is proportional to the increase of the horizontal section B of the dipole: V = BH - D2 - t2”, where 2 a = 0.26 - 0.60, which is a quite significant value. The propagation speed of the dipole at the initial formation stage (1 to 2 revolutions) is considerably higher than that at all subsequent stages (Fig. 4). Such kinematics of a dipole is very similar to that of ocean “mushroom-like” currents (Fedorov and Ginsburg, 1986, 1988). We can also see the following tendency: for higher values of n*/H*, the dipoles are formed closer to the source of disturbance and the speed of their subsequent motion is greater. This observation may be explained as follows: the jet produced by a stronger effective impulse meets with a stronger resistance, leading to the formation of a more intense dipole much closer to the source. Such a dipole is capable of moving faster at later stages than a less intense one. The same properties were also observed in the experiments performed in a two-layer stratified fluid. It was found that the transverse scale D is not limited by the Rossby radius of where g’ is the reduced gravity (8’ = 4 cm s-2 and R, = 2 cm deformation RN = (g’ HI)’” f’, in the experiments). The scale D of the initial dipole was 7 - 9 cm, which corresponds to 3.5 - 4.5 times RN, and it increases with time to 7 - 8 times RN in accordance with the power law: D - t‘, where a = 0.24 - 0.37. This result is in good agreement with those obtained for a homogeneous fluid. In contrast with the axisymmetric jet forcing, a localized input to the water of additional angular momentum of any sign produced a multitude of dipole structures. Cyclonic spinning with an angular velocity of 2 - 3 revolutions per second during 6 - 24 s produced an intensive cyclone with a weak anticyclone (or a series of anticyclones) at its periphery. Anticyclonic spinning produced a cloud of small-scale turbulence, that became organized into a compact system of two to three dipoles after 2 - 3 revolutions (Fig. 5). Such a compact system of three dipoles with one common anticyclone and three cyclones placed at 120’ from each other (Fig. 5) has not yet been observed in the ocean. A noteworthy detail of this system is that the cyclonic part of each dipole is in fact a new independent dipole. Such a situation has often been observed in the experiments with axisymmetric jet forcing as well as in the Ocean (Fedorov and Ginsburg, 1988). Finally, it was often observed that the instability of an anticyclonically spinning region produced a pair of dipoles moving away from each other in opposite directions, with a small anticyclone remaining at the original point of spinning (Fig. 6). The same feature has been obtained both in the numerical modelling of the instability of geostrophic eddies in a barotropic ocean by the method of contour dynamics (Kozlov and Makarov, 1985) and in laboratory

22

Fig. 5: A complicated combination of three dipoles, generated by the instability of the anticyclonically spun region during 6 s (H = 4 cm, t = 60 s).

Fig. 6: A system of two dipoles, generated by the instability of the anticyclonically spun region during 6 s (H = 4 cm, t = 32 s). The local spinning in this case is stronger than in Fig. 5.

23

modelling of baroclinic instability of vortices in a rotating fluid (Griffiths and Linden, 1981). It may be suggested in this connection that this structure may be common to both types of instability. This question calls for further theoretical and experimental investigations.

3. CONCLUSIONS The results obtained so far enable us to answer a number of questions concerning the origin and the evolution of "mushroom-like'' currents in the ocean. The Earth's rotation and the stratification of the upper layer of the ocean must influence significantly both the vertical structure and the process of dipole formation. The stratification determines the penetration limit of the vortex motion into the deeper layers of the ocean in accordance with the Proudman-Taylor principle. The Earth's rotation together with the stratification and the energy input define, apparently, the dynamics and the kinematics of the dipoles. For a fixed stratification, the horizontal and vertical scales of the dipoles are mainly determined by the strength of the forcing impulse. The almost complete analogy of dipoles behaviour in the ocean and in the laboratory experiments on an f-plane gives reasons to suppose that the P-effect is not important for the formation and the evolution of dipoles in the ocean. This conclusion is confirmed by the estimates of the parameter PL2/u (u being a typical velocity), which quantitatively expresses the relative importance of the p-effect as compared to the nonlinear terms of the vorticity equation: for the ocean and for the laboratory experiments, pL2/u a 1 (Fedorov and Ginsburg, 1986; Ginsburg et al., 1987). The anticyclonic asymmetry of the dipoles is, apparently, more typical in the real ocean than the cyclonic one. The latter is observed less frequently, and its appearance is related to the background vorticity of the flow field in the region. 4. REFERENCES Afanasyev, Ya.D, Voropayev, S.1 and Filippov, LA., 1987. A model of coherent structures of "mushroom-like" form in the ocean. Abstracts of the III Congress of soviet oceanologists, "Currents, synoptic and mesoscale eddies", Leningrad, pp. 16-1 8 (in Russian). Fedorov, K.N. and Ginsburg, A.I., 1986. "Mushroom-like" currents (vortex dipoles) in the ocean and in a laboratory tank. Annales Geophysicae, 4B: 507-516. Fedorov, K.N. and Ginsburg, A.I., 1988. The Nearsurface Layer of the Ocean. Leningrad, Gidrometeoizdat, 303 pp. (in Russian). Flier], G.R., Stem, M.E. and Whitehead, J.A., 1983. The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atm. and Oceans. 7:

233-263. Ginsburg, A.I. and Fedorov, K.N., 1984. "Mushroom-like'' currents in the ocean sed on the analysis of satellite images). Issled. Zemli iz Kosmosa (Earth Res. form Space), 3: 18-26 (in Russian). Ginsburg, A.I., Kostianoy, A.G., Pavlov, A.M. and Fedorov, K.N., 1987. Laboratory reproduction of "mushroom-like'' currents (vortex dipoles) under rotation and stratification conditions. Izv. Acad. Sci. USSR, ser. Physics of Atmos. and Ocean, 23: 170-178 (in Russian).

24

Griffiths, R.W. and Linden, P.F., 1981, The stability of vortices in a rotating stratified fluid. J. Fluid Mech., 105: 283-316. Kozlov, V.F. and Makmv, V.G., 1985. Hydrodynamic model of the formation of "mushroom-lie" currents in the ocean. Dokl. Acad. Sci., USSR, 281: 1213-1215 (in Russian). Lamb, H., 1932. Hydrodynamics, 6-th Ed., pp. 22G222. Orlanski, J. and Polinsky, L.J., 1983. Ocean response to mesoscale atmospheric forcing. Tellus, 35A: 296-323. Voropayev, S.I. and Filippov, I.A., 1985. Development of a horizontal jet in homogeneous and stratified fluids: a laboratory experiment. Izv. Acad. Sci. USSR. ser. Physics of Atmos. and Ocean, 21: 964-972 (in Russian).

25

ON THE MULTITUDE OF FORMS OF COHERENT MOTIONS IN MARGINAL ICE ZONES (MIZ) A.I. GINSBURG and K.N. FEDOROV Institute of Oceanology, Acad. Sci. USSR, 23 Krasikova Street, 117218, Moscow, USSR

ABSTRACT A variety of forms of orderly, non-stationary motions in marginal ice zones is illustrated using examples from satellite images. Basic elements of such motions are jets, vortices, and vortex dipoles. The spatial scales of these orderly motions range from a few km to approximately 100 km, while their life-time is in most cases limited to a few days. The mechanisms generating such motions are considered to be: frontal instability; upwelling in the marginal ice zones; local horizontal shear; uneven ice melting; local water circulation caused by irregularities of the ice edge. Various aspects of transverse jets formation and of their origin are considered in greater detail. It is well known (Vize, 1944; Nickolayev, 1973; Muench, 1983; Nickolayev et al., 1984; Johannessen et al., 1983) that the structure of waters in marginal ice zones (ME) is clearly of frontal character due to a number of natural contrasts in physical conditions between ice and open water. Air-sea interaction parameters undergo here an abrupt change, while the partial shielding of the ocean surface by ice cover from the direct wind forcing and a considerable share of solar irradiance is the cause of some interesting dynamical and thermal effects both in marginal ice zones and in polynyas and openings. This peculiarity of dynamical conditions in marginal ice zones became already evident by the end of the ~ O ’ S when , the first ”ice vortices” were discovered through the analysis of a series of Soviet meteorological satellite images (Preobrazhenskaya, 1971; Gayevskaya, 1971; Aizatullin and Nazirov, 1972). During the next 15 years, the remote sensing data (satellite visible band images, aerial photographs, side-scanning radar images) remained practically the only means of studying the kinematics and the space-time characteristics of the “ice vortices” (Johannessen et al., 1983; Bushuev et al., 1979; Gorbunov and Losev, 1978, 1979; Nazirov, 1982; Kuz’mina and Sklyarov, 1984; Ginsburg, 1988; Fedorov and Ginsburg, 1988). It is only during the latest years that purposeful projects with specialized equipment for both conventional and remote sensing measurements started to be organized (e.g., project MIZEX-84, see Johannessen et al., 1987; Shuchman et al., 1987) in connection with the rapidly increasing interest of scientists in studying the M E . The analysis of information available to date from various remote sensing instruments shows that a number of specific organized forms of local non-stationary currents are frequently observed in MIZ’s, in particular during the period of intensive ice melting when strong near-

26

surface stratification occurs. The major elements of these organized currents are jets, vortices and vortex pairs (dipoles). They become visible because of the accumulation of floating ice in convergence zones against a background of sparse ice or clear water, or as a result of openings in divergence zones against a background of denser floating ice. Different types of such organized motions are considered below, and hypotheses as to their origin and nature are put forward. One typical MIZ form of organized currents is represented by relatively narrow transverse jets up to 100 km long directed almost normally to the ice edge towards the open sea. Frequently they terminate in a vortex, either cyclonic (C) or anticyclonic (AC), or in a vortex dipole (Figs. 1 and 2). It seems that the first mention in the literature of stsuctures of this type was made by Bushuev et al. (1979). In that paper, the jets without a terminal vortex were called "mow-like tongues" (Fig. 4f in Bushuev et al., 1979), while jets with vortices were treated simply as marginal eddies (see Fig. 4d, ibid.). Similar structures were also treated as eddies by Johannessen et al. (1983) and by Wadhams and Squire (1983). It is our opinion that the jet character of such off ice-edge flows must be stressed, independently of the existence or not of a vortex or of a vortex dipole at their outer end. The characteristics of such structures (length L and width d of the jet, vortex sign) and also the regions of their sighting are given in Table 1. Starting with the available satellite information and also with some scanty meteorological data, one can compose the following picture of the phenomenon under consideration: 1. Transverse MI2 jets are typical of both stationary ice cover edges (East-Greenland Polar Front, the region NW of Spitzbergen (see Johannessen et al., 1983, and Wadhams and Squires, 1983), and of coastal ice or big floating ice field edges which exist only during winter and spring (Tartar Strait, Sea of Okhotsk, see Figs. 1 and 2). Hence, one can consider that this phenomenon is general for different MIZ conditions and related to some peculiar physical processes developing near ice edges. 2. Practically all sightings of such jets took place during periods of slight winds in the spring or summer seasons, i.e., during intensive ice melting. In the case reported by Johannessen et al. (1983), the appearance of transverse jets was preceded by a strong wind.

3. The jets may have a rectilinear or a slightly curved form. Vortex sign or dipole asymmetry (C or AC) at the end of a jet may be of any type (see Table 1).

4. Jets have been found near the edge of a very dense ice cover (e.g., Bushuev et al., 1979) as well as near the edges of sparse floating ice areas (Fig. 2).

5. Current velocity in jets (obtained from the analysis of satellite images) is on the order of 40-50 cm s-'. The life-time of the jets is several days (7-10 days as reported by Bushuev et al., 1979).

6. The jet structure in the area of the East-Greenland Polar Front (Wadhams and Squire, 1983) did practically not change its position during the 4 days of observations.

21

TABLE 1 Summary of observations of transverse jets in the MIZ Approximate dimensions (km)

Region of sighting

L

Description of the associated vortex or vortex dipole

Source of information

d

NW of Spitzbergen 82" OO'N, 10" 10'E

6

1

Structure E l on Dipole with AC asymmetry Fig. 16 in Johannessen et al., (*) 1983

8 I 50'N, 7" OO'E

15

2

C-vortex or dipole with a strong C-asymmetry

Ibid., structure E2 on Fig. 16

Greenland Sea : Fram Strait

5 100

5-10

No vortex

Fig. 4f in Bushuev et al., 1979

Near Shannon Island

5 100

5-10

C and AC vortices

Ibid., Fig. 4d

East-Greenland Polar Front : 79" 20'N, 0" 38'E

50

15

Almost symmetric vortex dipole

Wadhams and Squire, 1983

Tartar Strait, near Cape Peschany

40

6

AC vortex

Fig. 1 (this paper)

Dipole with a strong AC asymmetry

Structure 1 on Fig. 2 (this paper)

O

SW part of the Sea of Okthotsk

I

I3O

I

l s 0 l 7

l3

Dipole with AC asymmetry Structure 2 on Fig. 2 (this paper)

(*) AC or C asymmetry means here the predominance in intensity of either the AC or the C vortex in the dipole (Fedorov and Ginsburg, 1986, 1988). Note: the conclusion as to the character of vortex dipoles found on the images reproduced by Johannessen et al. (1983) and Wadhams and Squire (1983) was made by the authors of the present paper on the basis of the information contained in these images, while the authors of the original information treated these dipoles simply as cyclones.

28

Fig. 1 : Transverse jet with AC vortex in the Tartar Strait coastal ice. Fragment of the visible image obtained on 4 April 1979 by Meteor-29 satellite, medium resolution scanner, 0.5-0.7 pm band.

29

Fig. 2 : Organized structures in the south-west part of the Sea of Okhotsk. Fragment of the visible image obtained on 10 April 1984 by Meteor-30 satellite, medium resolution scanner, 0.5-0.7 pm band. Numbers refer to specific structures (see text and Table 1). 7. Sometimes, systems of transverse jets are observed. Such was the case of 4 jets 5 to 15 km long separated by 25-40 km from each other NW of Spitzbergen (Johannessen et al., 1983).

8. The length (L) of a jet is often comparable to the width (H) of the vortex part, but relates differently to the local baroclinic radius of deformation (RN)in various observed cases (L ranged from 1.5 RN to 10 RN).

30

Taking into account the general picture presented above, let us consider various possible mechanisms of transverse-jet generation in the MIZ. The following four hypotheses have been advanced in the literature to explain the origin of the observed structures: 1) Baroclinic or barotropic frontal instability (Johannessen et al., 1983; Wadhams and Squire,

1983); 2) Water entrainment by topographic eddies located off the ice edge (Smith et al., 1984);

3) Eddy motion of water and ice within the ice covered area leading to the squirting of a jet into the M E (Bushuev et al., 1979); 4) Ice edge upwelling (Ginsburg, 1988; Fedorov and Ginsburg, 1988). Jets with a vortex dipole at their extremity (“mushroom-likecurrents”) are indeed typical for frontal instability (Ginsburg and Fedorov, 1984; Fedorov and Ginsburg, 1986, 1988). However, in our opinion, weak fronts with a strong thermoclinicity are more likely to produce vortex dipoles as a result of their instability, than strong baroclinic fronts associated with intensive along-front geostrophic jet currents. The latter, when unstable, are likely to produce meanders and to shed rings. Barotropic instability of strongly baroclinic fronts, as shown by the Gulf Stream example, takes the form of elongated frontal cyclonic spin-off eddies whose shape is greatly distorted by the horizontal current shear at the front. These shallow eddies are usually transported along the front with the velocity of the frontal geostrophic jet. Therefore, barotropic instability may hardly be the cause of transverse jets in the case of the frontal boundary between the colder, fresher M E water and the saltier, warmer water of the open sea NW of Spitzbergen, as surmised by Johannessen et al. (1983) on the ground of the Griffiths and Linden criteria (1981). Such a front should be strongly thermoclinic, and the authors indeed recognize its weak character and the absence of a “substantial horizontal velocity shear“. Besides, it is well known (e.g., Vize, 1944) that the salinity fronts caused by ice melting are usually located up to some tens of miles off the ice edge, while the transverse jets, according to satellite data, always begin at the very edge of the ice. Wadhams and Squire (1983) attribute the origin of the MIZ eddy they observed (79’ 20’N, 0’ 38’E) to an instability (this time - baroclinic, according to the same criteria) of the East Greenland Polar Front, triggered by the proximity of the Molloy Deep. If so, however, the observed eddy disturbance should be due to the barotropic component of the total current which cannot be related to the local dynamics of the M E . One should also note that this topographic eddy could not be the direct cause of the disturbance in this particular case because it is located at a great distance from the ice edge (see Fig. 1 in Smith et al., 1984). Moreover, similar transverse jets are generated in regions where topographic features like the Molloy Deep are totally absent. Vortices and eddies which can entrain MIZ water and floating ice into their orbital nearsurface motion should not necessarily be of topographic origin. In principle, this can be any type of oceanic eddies including intrathermocline ones (ITE) (Bellcin et al., 1986) which happen

31

to pass near the ice edge, and also any eddy or vortex under the ice cover, the existence of which is presupposed by Bushuev et al. (1979). However, in order to generate transverse jets with velocities of the order of 40 cm s-l in the MIZ, the orbital velocities of such eddies or vortices should be of the same magnitude. This is not true of the ITE or of the eddies usually found under the ice cover. Their near-surface velocities, as a rule, do not exceed a few cm s-' (Nickolayev et al., 1984). Also, it is difficult to expect that each case of transverse jet formation in the MI2 be related to the passage of an open ocean synoptic eddy near the ice edge. It is possible, however, that a mechanism of transverse jet formation is connected to the ice-edge upwelling. The latter may be accompanied by the formation of one or several fronts near the ice edge (Buckley et al., 1979). Instabilities of such fronts may be instrumental in causing the occurrence of the phenomenon under consideration. Theoretical analysis (Hakkinen, 1986) shows that the upwelling-producing winds may be of various directions depending upon the character of the ice edge movement. In principle, for the upwelling to occur it is sufficient to have only a tangential stress due to ice moving over water, with no wind at all. Thus, there may be a variety of conditions favorable for ice-edge upwelling. It is not excluded that the local bottom topography and the ice edge configuration may be reinforcing factors for the M E upwelling, in much the same way as for coastal upwelling. If the transverse jet formation is indeed due to upwelling at the ice edge, then there is an inevitable question concerning the T, S-characteristics of the water transported by the jet. This question is applicable here exactly as in the case of coastal upwelling jets (Ginsburg and Fedorov, 1985). To be transported tens of kilometers off the ice edge, the upwelled water must be less dense than the water immediately below it. The only source of additional buoyancy for the warmer but saltier upwelled water may be the fresh water produced by ice melting which dilutes the upwelled water in the process of its transformation. This may explain why most transverse jet sightings took place in spring and summer, although the phenomenon of ice-edge upwelling itself is not a seasonal one. As to the sign of the terminal vortex or the asymmetry of the terminal dipole, the rules are probably the same as for oceanic "mushroom-like" currents (Fedorov and Ginsburg, 1986, 1988; see also a paper by the same authors in this volume). In the absence of a background cyclonic shear, an AC-asymmetry is more typical for dipoles. A background vorticity of one or the other sign may either weaken or reinforce the AC-asymmetry, and even lead to a C-asymmetry. It is worth pointing out that in many cases the fact that only one vortex (cyclonic or anticyclonic) is visible on a satellite image (e.g., at the end of a transverse jet) does not preclude the existence of a dipole. In a number of cases, it may happen that the floating ice which serves as a tracer has, for some reasons, been entrained in only one vortex of the dipole. In such a case, there is no means to make an unambiguous judgement as to the nature of the observed phenomenon.

32

Fig. 3 : Cyclonic vortices near the ice edge east of Sakhalin. Fragment of the visible image obtained on 21 May 1985 by Kosmos-1602 satellite, medium resolution scanner, 0.7-1.1 pm band.

33

Let us now list briefly some other forms of orderly motions observed in the M E . Chains of cyclonic vortices are often observed near the ice edge when there is a strong horizontal velocity shear (Fig. 3; features 3 on Fig. 2, and the same features 5 days later on Fig. 4; Fig. 5). In some cases (Figs. 3 and 4), their formation is probably related to the ice-edge front which, as all oceanic fronts, may exhibit cyclonic instability elements due to high frontal shear. In other cases, the observed shear is evidently unrelated to M E dynamics (the chain of 3 vortices in the upper right comer of Fig. 5). These vortices are some 30 to 50 km in diameter, and their pattern enters deeply into the rather extended area of sparsely distributed floating ice, the whole picture being of a less marginal character than in other cases. The distribution of current (or wind) velocity which was the cause of the horizontal cyclonic velocity shear in this case is unknown. Judging from the position of an opening in the ice relative to a big ice floe (marked A on Fig. 5), it may reasonably be supposed that the ice drifted in the general east-west direction north of the vortex chain. It should be noted that chains of vortices developing in shear zones are well known in hydrodynamics (e.g., Brown and Roshko, 1974; Winant and Browand, 1974). Such chains have also been observed in many areas in the ocean (Onishi, 1984; Flament, 1986; Washburn and Armi, 1988) under conditions of strong frontal or tidal shear. However, for the ice vortex chains observed in the Sea of Okhotsk (Fig. 5 ) and in the East-Siberian Sea (Gorbunov and Losev, 1978), permanent local sources of horizontal velocity shear are unknown. Gorbunov and Losev (1978) hypothesised that the observed cyclonic vortices, which have a diameter of 7 km, were caused by the reorganization or the variability of the wind field. Atmospheric forcing may also be the cause of the vortex chain seen in Fig. 5. The vortex lifetime in both cases was of the order of several days. When the horizontal velocity shear in the MIZ is weak, various local inhomogeneities of forcing or buoyancy distribution, such as differences in heat exchange with the atmosphere or uneven ice melting, produce strong local sea-level and pressure gradients. As a consequence, numerous "mushroom-like" currents (vortex dipoles) are formed which often occupy a considerable space and, being of different orientations, those dipoles sometimes interlock with one another, sharing common eddies and giving the impression of being densely packed. An example of this situation may be seen in the pattern of vortex dipoles produced during ice melting in the Barents Sea (Fig. 6). Another example is given in Fig. 4, where densely packed "mushroom-like'' currents are generated as a result of the complete melting of a large rectangular ice floe (see Fig. 2 for its original shape). It can also be construed that vortices or eddies (this time, anticyclonic) may be formed as a result of the spin-up (under the influence of the Earth's rotation) of large surface lenses of low salinity melt water in the process of their radial spreading (collapse) due to outwardly directed sea level (pressure) gradients. Although such hypotheses have been advanced by several authors (Kuz'mina and Sklyarov, 1984; Nickolayev et al., 1984), no observational proof of such a mechanism at work has yet been produced.

34

Fig. 4 : Packing of vortex dipoles and cyclonic vortices in the south-west part of the Sea of Okhotsk. Fragment of the visible image obtained on 15 April 1984 by Meteor-30 satellite, medium resolution scanner, 0.5-0-7 pm band.

35

Fig. 5 : Vortices in the area of sparsely distributed floating ice in the south part of the Sea of Okhotsk. Fragment of the visible image obtained on 23 March 1987 by Meteor-30 satellite, medium resolution scanner, 0.7-1.1 pm band.

36

Fig. 6 : Vortices in the melt ice zone in the Barents Sea. Enlarged fragment of the visible image obtained on 25 May 1984 by Meteor-30 satellite, low resolution scanner, 0.7-1.1 pm band. The last but not the least among the factors which may affect or even cause the development of vortices or vortex dipoles in the MIZ is the ice-edge configuration. It may influence the flow by distorting its path, it may provide boundaries for accumulating melt water, or it may help focusing wind forcing on the water surface. The result in all cases would be in the form of a close fitting of the vortex pattern in the open water space bounded by the specific form of the ice edge. As an example, we can supply our own interpretation (Fig. 7) of the peculiar combination of "mushroom-like" currents in a semi-enclosed MIZ area in the Fram Strait as seen on a satellite IR-image reproduced by Johannessen et al. (1987, their Fig. 2; in OUT Fig. 7, we kept the original designation of vortices by the numbers 1 to 5). Instead of the

37

Fig. 7 : Interpretation sketch of the IR-image obtained on 4 July 1984 by NOAA-7 satellite (published by Johannessen et al., 1987). The numbers denote vortices as in Johannessen et al., 1987; the letters denote mushroom-like currents. The arrows show the directions of motion in the jet portion of mushroom-like currents. two eddies identified as 4 and 5 by Johannessen et al. (1987), we see in fact four "mushroomlike" structures (A, B, C and D). They are combined in a pattern where both types I and I1 of dipole packing (see Fedorov and Ginsburg, 1986, 1988 and also their companion paper in this volume) are present with eddies 4 and 5 being shared by the pairs of dipoles A-B and C-D, respectively. Although we cannot say what initial disturbance triggered the development of this pattern, or whether this disturbance was in some way related to the MIZ dynamics, it seems likely that the ice configuration in this case determined both the pattern of the dipoles and their horizontal scales. In conclusion we should like to point out that the observed variety of foms of coherent motions of surface water in marginal ice zones can be properly understood if M E ' S are treated

38

as dynamically active frontal zones whose properties are very close in certain ways to those of coastal upwelling frontal zones. The near-surface salinity stratification is apparently as important here as in the Oregon and California coastal upwellings (Ginsburg and Fedorov, 1985; Fedorov and Ginsburg, 1988), and the observed temporal and spatial scales of the similar forms of coherent motions are of the same order of magnitude in all these cases. Owing to the continuous presence of stratification (in summer) and of a very convenient tracer (floating ice), MIZ's are extremely suitable places for studying the dynamics and kinematics of coherent structures in natural geophysical flows.

REFERENCES Aizatullin, T.A. and Nazirov, M., 1972. Ice vortices on the sea surface. Priroda, 9: 101-102 (in Russian). Belkin, I.M., Emelyanov, M.V., Kostianoy, A.G. and Fedorov, K.N., 1986. Thermohaline structure of intermediate waters of the ocean and intrathermocline eddies. In: Intrathermocline Eddies in the Ocean. Moscow, Academy of Sciences of the USSR, P.P. Shirshov Institute of Oceanology, pp. 8-34 (in Russian). Brown, G.L. and Roshko, A., 1974. On density effects and large structure in turbulent mixing layers. J. Fluid Mech., 4: 775-816. Buckley, J.R., Gammelsrod, T., Johannessen, J.A., Johannessen, O.M. and Roed, J.A., 1979. Upwelling: oceanic structure at the edge of the Arctic ice pack in winter. Science, 4376: 165- 167. Bushuev, A.V., Bichenkov, Yu.D. and Provorkin, A.V., 1979. Distribution and dynamics of ice in the Greenland Sea in March-July 1976 based on space-borne information. In: POLEXSEVER-76 (Gidrometeoizdat), Leningrad, 1: 115-128 (in Russian). Fedorov, K.N. and Ginsburg, A.I., 1986. "Mushroom-like'' currents (vortex dipoles) in the Ocean and in a laboratory tank. Annales Geophysicae, B5: 507-516. Fedorov, K.N. and Ginsburg, A.I., 1988. The Nearsurface Layer of the Ocean. Gidrometeoizdat, Leningrad, 303 pp. (in Russian). Flament, P., 1986. Subduction and finestructure associated with upwelling filaments. Ph.D. Dissertation, Scripps Institution of Oceanography, University of California, San Diego, 123 PP. Gayevskaya, O.V., 1971. Television images of clouds over ice obtained from Meteor satellite. Antarktika. Doklady Komissii, 1969. Nauka, Moscow, pp. 122-128 (in Russian). Ginsburg, A.I., 1988. On the nature of transverse jets in marginal ice zones observed on satellite images. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 3: 3-10. Ginsburg, A.I. and Fedorov, K.N., 1984. Mushroom-like currents in the ocean (based on the analysis of satellite images). Issled. Zemli iz Kosmosa (Earth Res. form Space, in Russian), 3: 18-26. Ginsburg, A.I. and Fedorov, K.N., 1985. Systems of transverse jets in coastal upwelling: satellite information and physical hypotheses. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 5: 3-10. Gorbunov, Yu.A. and Losev, S.M., 1978. Vortex disturbances in drifting ice field. Trudy AANII, 354 52-57 (in Russian). Gorbunov, Yu.A. and Losev, S.M., 1979. Some peculiarities in local sea ice distribution in connection with its drift. Trudy AANII, 364: 64-69 (in Russian). Griffiths, R.W. and Linden, P.F., 1981. The stability of vortices in a rotating stratified fluid. J. Fluid Mech., 105: 283-316. Hakkinen, S., 1986. Coupled ice-ocean dynamics in the marginal ice zones: upwellingdownwelling and eddy generation. J. Geophys. Res., 91, C1: 819-832.

39

Johannessen, O.M. and Johannessen, J.A., 1983. Oceanographic conditions in the marginal ice zone north of Svalbard in early fall 1979 with an emphasis on mesoscale processes. J. Geophys. Res., 88, C5: 2755-2769. Johannessen, O.M., Johannessen, J.A., Svendsen, E., Schuchman, R.A., Campbell, W.J. and Josberger, E., 1987. Ice-edge eddies in the Fram strait marginal ice zone. Science, 4800: 427-429. Kuz’mina, N.P. and Sklyarov, V.E., 1984. Circulation research using drifting ice as a tracer of underlaying sea water. Issled. Zemli iz Kosmosa (Earth Res. from Space, in Russian), 1: 16-25. Muench, R.D., 1983. The marginal ice zone experiment. Oceanus, 2: 55-60. Nazirov, M., 1982. Ice and suspended matter as hydrothemodynamic tracers. Gidrorneteoizdat, Leningrad, 165 pp. (in Russian). Nickolayev, S.G., 1973. The experience of oceanographic research organization in the Chukchi sea marginal zone. Problemi Arktiki i Antarktiki, pp. 31-36 (in Russian). Nickolayev, Yu.V., Makshtas, A.P. and Ivanov, B.V., 1984. Physical processes in marginal zones of drifting sea ice. Meteorologia i Gidrologia, 11: 73-80 (in Russian). Onishi, S., 1984. Study of vortex structure in water surface jets by means of remote sensing. In: Remote Sensing of Shelf Sea Hydrodynamics, J.C.J. Nihoul, Ed., Elsevier Oceanography Series, 38: 107-132. Preobrazhenskaya, T.I., 1971. Mesoscale vortices in the East Atlantic coastal regions. Antarctica. DoMady Komissii, 1968. Nauka, Moscow, pp. 135-137 (in Russian). Shuchman, R.A., Bums, B.A., Johannessen, O.M., Josberger, E.G., Campbell, W.J., Manley, T.O. and Lahnelogue, N., 1987. Remote sensing of the Fram strait marginal ice zone. Science, 4800: 429-431. Smith, N,D.C., Morison, J.N., Johannessen, J.A. and Untersteiner, N., 1984. Topographic generation of an eddy at the edge of the East Greenland current. J. Geophys. Res., 89, C5: 8205-8208. Vize, V.Yu., 1944. Hydrometeorological conditions in marginal ice zones of Arctic Seas. Trudy AANII, 184: 122-151. Wadhams, P. and Squire, V.A., 1983. An ice-water vortex at the edge of the East Greenland current. J. Geophys. Res., 88, C5: 2770-2780. Washburn, L. and Armi, L., 1988. Observations of frontal instabilities on an upwelling filament. Submitted to J. Phys. Oceanogr. Winant, C.D. and Browand, K., 1974. Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech., 63: 237-255.


41

DIFFERENTIAL ROTATION (BETA-EFFECT) AS AN ORGANIZING FACTOR IN MESOSCALE DYNAMICS V.D. LARICHEV P.P. Shirshov Institute of Ocemology, USSR Academy of Sciences, Krasikova 23, Moscow 117218, USSR

ABSTRACT There are two ke factors in mesoscale dynamics: wave radiation and nonlinearity. The organizing role of the &effect is analyzed in two contrary but mutually supplementing cases: a) the case of a statistically homogeneous field with an essential influence of advection processes (the problem of quasi-geostrophic turbulence); b) the case of high spatial inhomogeneity of the eddy field (isolated eddies, for example) where radiation of Rossby waves is particularly important. In the problem of quasi-geostrophic turbulence, the theory of self-organization by the peffect is discussed. The results of numerical modelling experiments which describe the selforganization on the P-plane as a phenomenon of recurrent spatial localization and intensification of eddies are shown. As typical for the second case, the free (decaying) nonlinear evolution of an isolated eddy is considered (within the limits of the barotropic vorticity equation on a p-plane). The cardinal difference between the linear and nonlinear evolutions lies in the trapping of wave radiation by the strongly nonlinear field of the eddy, which leads to the formation of non-radiating organized states; in particular, two-dimensional Rossby solitons can be formed.

1. INTRODUCTION There exists a number of arguments which show that the concepts of order and selforganization may be fruitful in mesoscale dynamics. I shall name some of them:

1. The existence of long-lived oceanic eddies. For example, "rings" live for several years, which is significantly longer than typical dynamical times. Furthermore, "rings" essentially maintain their shape during the evolution, so that they can be characterized as structured formations. But are "rings" exceptions ? Formally, yes, since there are mesoscale eddies of different type (like "eddies of open ocean") which live only a short time and significantly change their form. Nevertheless, the point is that all eddies have a common dynamical nature: they are governed by fundamentally the same equation, i.e., the vorticity equation. Thus it is reasonable to ask the following question: how much of the order do the shortlived eddies contain? So, in this consideration the problem of order arises as a general one in mesoscale dynamics. 2. The very existence of Rossby solitons gives an example of structure in the model related to mesoscale eddies.

42

In considering the dynamics of oceanic eddies, one can distinguish two limiting cases: a) a statistically homogeneous case, when the eddies are closely packed; this is the problem of geostrophic turbulence; b) the case of a strong spatial inhomogeneity of the mesoscale field (for example, a single isolated eddy). 2. GEOSTROPHIC TURBULENCE ON A P-PLANE

2.1. Phenomenological description Let us begin with the geostrophic turbulence, taking the simplest model of 2D-motion on a P-plane:

Rhines (1975) gives the following account of the p-influence on the evolution of 2Dturbulence: the spectral energy flux to large scales, going over the boundary k = (j3/U)1h between linear and nonlinear domains, decreases significantly while there appears a substantial angular energy transport resulting in dominant quasi-one-dimensional motions (a / ax = 0). It should be emphasized that this account lies completely within a turbulence concept. There exists another mechanism of p-influence on 2D-turbulence, which shall be described below in short. Since equation (1) expresses Lagrangian conservation of the potential vorticity Ayr + Py = co = const. in a fluid parcel. The vorticity A y of a given fluid parcel may be measured by its displacement from the so-called "latitude of rest", = / p, where the relative vorticity of the parcel or particle is equal to 0. So Ayr = - p 6y, where 6y = y -9. Therefore, the total enstrophy

c = Ayr + py , we have

co

__

E=

(Ayr)' dS

(2S)-' 2

s

can be interpreted as a value of mean squared displacements of the fluid particles from their "latitudes of rest":

In a turbulent regime, the diffusion leads particles away from their initial positions unrestrictedly: Turbulent regime Hence, from their "latitudes of rest",

jdiffusion

(y(t)

- ~ ( 0 )+ ) ~m

43

The point is that on a 0-plane the enstrophy is conserved (for doubly periodic boundary conditions) or bounded from above (in a closed domain with no-flow across the boundary). According to the relation (2), this means that the mean squared particle displacements should be bounded as well:

So unlimited departures of the particles, inherent to a turbulent regime, become impossible due to condition (3). As a result, the turbulent regime has to make way for another nonlinear regime that one may call coherent, in which motions are by necessity regular in their spacetime structure. However, this phenomenological theory, described in more detail elsewhere (Larichev, 1985a, b) does not provide an explicit description of such a coherent regime. Therefore, we now turn to the results of numerical experiments aimed at studying the basic features of the coherent regime.

2.2. Numerical experiments and their results* The initial value problem for the vorticity equation

ao + J at

(tp)

a = - 1AAw ; w = Aw ; Re = UL3 +2 ax Re V

has been solved in a square domain with doubly periodic boundary conditions. The spatial grid was uniform with 129x129 points so that lc- = 1, = 64. A finite-difference scheme of

Lm

second-order accuracy was used. The viscous term was added to organize a sink for enstrophy near the maximum wave number. The initial isotropic stream function spectrum was non-zero within the band of wave numbers from 3.5 to 6.5 (more details are given by Fedotov and Larichev, 1988). The Fourier amplitudes and phases were prescribed by random number generators of normal and uniform distributions, respectively. The initial energy level corresponds to a Rhines' scale (p/U)''2 = 4.32. It should be emphasized that the choice of such a stochastic initial field was dictated by the necessity to have initially the most chaotic ("turbulent") conditions so that the following evolution could reveal the organizing role of the 0-effect. A remark concerning the coincidence of the initial energy spectrum position and of the Rhines' scale is in order. In a sense, this coincidence was accidental since we first derived the spectral estimate for a transition to coherent regime (Larichev, 1985a, b), and then tried to determine the initial conditions in such a way that this transition would occur during the

* This part of the study was performed with A.B. Fedotov (Fedotov

and Larichev, 1988).

44

computation time. Recurrent spatial localization of the vorticity and velocity fields is one of the basic features of the temporal evolution of the system under consideration. So it was natural to choose the kurtosis of the vorticity field Qo =

-

1

, where, by definition, (I = - $ dS (2XY

as a diagnostic characteristic of the vorticity field localization. The temporal variability of the kurtosis is shown in Fig. 1. The initial value of about 3 corresponds to the statistically homogeneous initial vorticity field. The subsequent increase of Q, indicates the appearance of spatial localization, with the main peak at N = 360. It is very important to note that in two additional runs with different (random) sets of ini-

tial phases but the same spectrum, the time series of kurtosis did not change by more than 10%. Therefore, the observed behavior of Q, is not a coincidence resulting from a particular specification of the phases, and there is a physical phenomenon behind the scenes.

Fig. 1. Evolution of the kurtosis, Qo,as a function of time. N is the number of timesteps, and At = 0.226.

A time series of the mean squared y-displacement of 4096 passive tracers (particles whose initial distribution was spatially uniform) is shown in Fig. 2. A tendency for the ydisplacements to saturate is clearly seen here in agreement with the theoretical predictions of section 2.1.

45

2

-1

-

-2-

I

I

N

-

Fig. 2. Time evolution of the average of the squared y-displacements of 4096 passive tracers. As in Fig. 1, N is the number of timesteps. A visual analysis of time series of maps of stream function, kinetic energy and vorticity (not shown here but given in Fedotov and Larichev, 1988) allows us to establish two facts: 1. The existence of recurrent (as Fig. 1 indicates for vorticity) spatial localization of energy and vorticity into a cluster of a few intensive vortices (usually 2 or 3). This cluster is also clearly seen in the stream function field, but here there is no noticeable localization. 2. An intensification of the vortices induced by the localization of the fields. The stream function, energy and enstrophy amplitudes of the most intensive vortices increase approximately by a factor two at the major peak of vorticity kurtosis (N = 360, Fig. 1) as compared to the initial fields.

The appearance of a few strong vortices on comparatively small-amplitude background at the periods of Q, maxima means that at these times the system is characterized by a relatively small number of effective degrees of freedom. More formally, one can see that from a calculation of the informational entropy, S = -

-

f(o) In f(w) do, where f(o) is the distribution func-

tion of vorticity values in the spatial domain at a given time t. Hence, S = S(t). A time series of the informational entropy is given in Fig. 3. Strictly speaking, this is only the conservative part of the entropy, since, because of the openness of the system due to viscosity effects, there is another component of entropy changes which shows a monotonous decrease of entropy. But one has to single out the conservative part because it is that part which describes the structural transformation of the flow. The main feature of this diagram is

46

that the major minima of entropy coincide exactly with the maxima of vorticity kurtosis.

Fig. 3. Time series of the conservative part of the entropy,

x = XG = 1.419 corresponds to a Gaussian distribution.

x

(solid line); the dashed line at

Hence, one can conclude that when the vorticity kurtosis increases, an organization of the nonlinear flow on the P-plane takes place. Since external influences are absent, this is a selforganization which is accomplished through spatial localization and intensification of vortices. An important question is the role of the low-amplitude background. At first, we supposed that the background works on the intensive vortices like an external disturbance making their lifetime finite. To verify this assumption, we have performed the following numerical experiment. At one of the kurtosis peaks, when a nice isolated dipole couple had been formed, we erased the background outside the couple and used a field similar to the initial one for the subsequent integration. To our surprise the couple decayed much more quickly than in the basic run. In parallel, we have studied spectral nonlinear interactions and we have obtained results which may be interpreted as an indication of strong interaction between the vortices and the background at those times when Q, is maximum. Finally, some estimates of the enstrophy cascade, done separately fnr the intensive vortices and for the background, show synchronous changes of the cascade rates. All these results together indicate that at the stage of self-organization the low-amplitude background plays an active role, participating in the generation and maintenance of the intensive vortices. In other words, self-organization is spatially global.

47

2.3. Physical description of self-organization In section 2.1, we have discussed the reasons why the P-effect should suppress turbulence transferring the flow from a chaotic (turbulent) regime to a more ordered (coherent) one. Theoretical considerations based on Boltzman statistics (Fedotov and Larichev, 1988) show that the ordering of the flow requires its spatial localization and an intensification of vortices (this was clearly seen in the numerical experiments). But localization doesn't go very far. It is limited due to two factors: The finiteness of the total energy E and enstrophy E. There is a strict estimate from above for a particular form of the vorticity kurtosis Q, which limits its growth: & S const . (E/E) (Fedotov and Larichev, 1988). More importantly, the existence of waves in the medium, which tend on the contrary to make the fields more or less spatially homogeneous. An increase of spatial localization simultaneously initiates a radiation of Rossby waves from the intensive vortices. This radiation carries the energy and enstrophy of the vortices away or, in spectral language, it destroys the phase correlations between spectral harmonics which have provided the specific structure of the intensive vortices. As a consequence, the vortices decay resulting in approximately statistically homogeneous fields. Eventually, one arrives at a state of strongly nonlinear motion with relatively weak phase correlations or, in other words, a turbulent motion which is exactly the point of departure for our analysis. Hence, the process under consideration should repeat itself. It is worthwhile to emphasize the dual role of the p-effect in the phenomenon. On one hand, the p-effect leads to a coherent state in an essentially nonlinear process (the Lagrangian conservation of absolute vorticity which, together with the boundedness of the enstrophy restrains the diffusion of fluid particles, is a nonlinear constraint). On the other hand, the &effect destroys this coherent state due to radiation, a process whose nature is linear. This understanding of the physics of the phenomenon allows us to estimate theoretically a lifetime of the coherent state:

where E and E are the total enstrophy and energy (Fedotov and Larichev, 1988). Comparing this value with the "numerical" estimate taken as the time interval between two adjoining minima of vorticity kurtosis, one obtains a quite good correspondence (in the numerical experiment, T,, = 6.10 in the above theory, T,, = 4.10). It is interesting to apply this estimate to real geophysical phenomena such as mesoscale eddies in the ocean and blocking situations in the atmosphere. For a typical size of a mesoscale eddy in mid-latitudes, the above mentioned lifetime is about 300 days which, being a Lagrangian time scale, is significantly larger than the period of mesoscale motions in, for example, the POLYMODE experiment (= 50 days). A more instructive comparison comes from the

48

application to the blocking phenomenon since observations give a lifetime of this atmospheric event. For blocking, the theoretical estimate is about 15 days which corresponds well to the observed duration of the phenomenon(Rex, 1950). These are two examples which show a possible area of application of the self-organization phenomenon. A final remark is in order. Everything we have discussed in this section indicates that the formation of strong eddies is a manifestation of primarily internal properties of nonlinear 2Ddynamics under the effect of differential rotation (p-effect). This explains the wide geographical ubiquity of mesoscale eddies in the ocean since they do not require a vast area for their energy sources. It is worthwhile to note that self-organization is also possible without the p-effect (McEwan, 1976; Fomberg, 1977; McWilliams, 1984). However, that mechanism seems to be quite different from the one discussed here (Fedotov and Larichev, 1988).

3. THE CASE OF STRONG SPATIAL INHOMOGENEITY Let us consider for the sake of simplicity an isolated eddy in a horizontally unbounded ocean. On the basis of the model (l), what is the outcome of the eddy evolution? Since the eddy can radiate Rossby waves, it is reasonable to suppose that the eddy will gradually decay up to its total collapse when all its energy will have been converted to wave energy. In terms of vorticity A y , the total eddy collapse means that A y I + -+ 0 everywhere. To attain that state, every fluid particle has to depart to its "latitude of rest" because of Lagrangian conservation of the potential vorticity (section 2.1). But this is not always possible. For example, consider an ocean bounded from the north by a zonal boundary in which there exists an isolated vortex of positive vorticity, such that the "latitudes of rest" of the vortex fluid particles lie above the boundary. It is obvious that in spite of the possibility for the vortex to radiate Rossby waves, the total collapse of the vortex ( A y I + + 0) is impossible. Not only can ocean boundaries secure a halt of radiation from a vortex, but a particular spatial vortex structure (at the initial time) can also prevent its total collapse. All these cases are encompassed by the following statement: the property of nonradiation from a localized vortex is determined by a particular topological feature of the potential vorticity field 6, namely the presence of closed isolines 6 = const. (Larichev, 1983a). However, the analysis carried out in the paper just cited is local, i.e., it maintains the existence of large (and non-decaying) vorticity values at some spatial points. But how many such points are there, or, speaking mathematically, what is the measure of the set of such points? The solution to this problem given by Larichev (1983b) shows that the whole domain of closed isolines 6 (the vortex "core") consists of these points. More strictly, it was demonstrated that the evolution of strongly nonlinear (due to presence of closed isolines 4LR being already sufficient). This second approach leads to frontal geostrophic dynamics (Cushman-Roisin, 1986). Here, both possibilities will be retained in a unified fonnulation. The beta effect yields a second dimensionless number

(7) f0

LP

which measures the relative variation of the Coriolis parameter over one eddy scale. Another dimensionless number, which emerges from the scaled continuity equation, is the Froude number

55

Since p and s are two measures of the same eddy length scale, L, they are not independent. It follows from definition ( 5 ) that sp2 = a2. Also, since 6H is bounded above by H (if 6H were to exceed H, H can always be redefined to be SH), there is an upper bound on the value of E:

(9)

E<S.

Finally, the time derivatives call for a time scale, T, which will later be determined by the dynamics. A last dimensionless number arises:

which compares the evolution time scale of the geostrophic turbulence to the inertial period.

2.2 A generalized geostrophic eauation With the scales L for x and y, T for time, U for u and v, and SH for q, the governing equations (1-3) become mut +E(UU, + w ) - v - pyv = -qx Y Wvt

+ E(UV, + vvy) + u 4- pyu = -11Y

According to our objective of restricting the attention to geostrophic flows, we impose w\ \

-2.0

0.

1.

3.

2.

4.

5.

dRd

F i g . 5 . A comparison o f t h e v e l o c i t y f i e l d o f t h e i n i t i a l and f i n a l r i n g s t a t e s . Note t h a t t h e f i n a l p r o f i l e i n d i c a t e s t h a t t h e s w i r l v e l o c i t y has g e n e r a l l y decreased i n magnitude. The f l o w has become b a r o t r o p i c , however, and t h e l o w e r - l a y e r flow has been a c c e l e r a t e d . average r a d i a l v e l o c i t i e s a s s o c i a t e d w l t h these displacements a r e q u i t e small.

Assuming f o r example t h a t t o t a l v e n t i l a t i o n c o u l d o c c u r o v e r 300 days

(most l i k e l y an u n d e r e s t i m a t e ) , t h e average r a d i a l v e l o c i t i e s a s s o c i a t e d w i t h an O(60 km) displacement would be .2 cm/sec and t h u s k i n e m a t i c a l l y unimportant.

On t h e o t h e r hand, these r a t h e r s l i g h t v e l o c i t i e s induce m a j o r

changes i n t h e a z i m u t h a l r i n g flow, and can e f f i c i e n t l y a c c e l e r a t e t h e deep fluid. One aspect o f t h i s problem which I s n o t very r e a l i s t i c i s t h e complete removal o f t h e r i n g h e a t anomaly, a f e a t u r e n o t observed i n t h e f i e l d .

Rather

I t I s t h e case t h a t warm r i n g s s u r v i v e i n t h e Slope Water f o r about s i x months

and a r e reabsorbed by t h e G u l f Stream ( F i t z g e r a l d and Chamberlin, 1983). What i s t h e r e f o r e observed i s t h a t o v e r w i n t e r e d r i n g s l o s e o n l y a f r a c t i o n o f t h e i r h e a t anomaly and t h a t p a r t o f t h e h e a t i s l o s t t h r o u g h s h o r t - t e r m , i n t e n s e c o o l i n g events (Joyce and S t a l c u p , 1985). F u r t h e r , t h e two l a y e r model used here p r o b a b l y overemphasizes t h e r o l e o f t h e bottom. on these p o i n t s i s c o n s i d e r e d below. 3.

A model which improves

A TWO AND ONE-HALF LAYER MODEL WITH PARTIAL VENTILATION

Consider t h e two and o n e - h a l f l a y e r model i n F i g . 6.a.

The e q u a t i o n s

g o v e r n i n g t h e m o t i o n i n each l a y e r a r e e s s e n t i a l l y ( 1 ) w i t h a v e r t i c a l l y integrated form of c o n t i n u i t y :

84

a

(Ti) + - { t ar

1

(i+l)

uidz }

(-1)

=

s

(6)

where the subscript ' i ' obtains the values 1 and 2, and denotes the upper and lower layer respectively: ui denotes radial velocity and Ti denotes thickness. The third layer in this model will be assumed t o be stagnant. The two density defects, associated with the two interfaces, will be assigned identical values and denoted by g'. A s discussed below, S can be related t o the surface heat fluxes. 3.1 Fluid Thermodvnamics In order t o use a layered formalism t o examine the effects of heat loss on a warm ring, it is necessary t o model the effects of cooling in a manner consistent with a layered model. The layered constraint will be maintained in the present problem by requiring the fluid t o exist only in the three initial density states. Heat losses from the upper layer will be balanced entirely by converting upper layer warm fluid t o cold second layer fluid. The volume of fluid converted will be that necessary t o balance the heat budget. This fluid response t o cooling is reflected in the continuity equation ( 6 ) by the inclusion of the cross-interfacial velocity denoted by S.

a 0

1

2

3

4

L

/ / /

v I

-'.O

,/'

I V

v,=o

v,=

0

pa

Ill

Fig. 6. (a) A two and one-half layer model. Upper layer variables are denoted by the subscript '1' and lower layer variables by the subscript ' 2 ' . hl is the depth of the upper interface and h2 the depth of the second interface. The upper bowl is characterized by zero angular momentum (u=O),and the lower layer is initially at rest. The deepest layer i s always at rest. (b) The lower layer of a cooled ring divides naturally into the four indicated regions. Region I contains zero angular momentum fluid and extends t o radius r (r, 2 / 2 = P),. Region I1 extends from ,r t o the upper layer outcrop. Region IPI extends from the outcrop t o the location of the fluid column which was initially at 2J2 Rd. This position is denoted Pc (Pc = rc */2). Region I V extends from Pc t o -.

85

Assuming z e r o a n g u l a r momentum i n t h e upper l a y e r , t h e i n i t i a l r i n g s t r u c t u r e i s t h e same as t h a t i n s e c t i o n 2 .

The i n i t i a l v a l u e f o r t h e d e p t h

of t h e second i n t e r f a c e i s assumed t o be c o n s t a n t , and t h e second l a y e r i s assumed t o be i n i t i a l l y m o t i o n l e s s , i . e . , h2

(7a)

hc,

=

v2 = 0

(7b)

where h, i s t h e d e p t h of t h e second i n t e r f a c e and h C 2 i s t h e c o n s t a n t second For convenience, we w i l l assume hcl = hc2. T h i s s t r u c t u r e i s

l a y e r depth.

shown I n F i g . 6.a. 3.2 A d j u s t ment C a l c u l at l o n s

Now suppose t h e r i n g i n F i g . 6.a l o s e s h e a t t o t h e atmosphere.

A f t e r the

c o o l i n g has stopped, t h e r i n g w i l l e v e n t u a l l y s e t t l e i n t o a new steady s t a t e . The d e n s i t y f i e l d w l l l have been a l t e r e d by t h e c o o l i n g and t h e new v e l o c i t y and p r e s s u r e f i e l d s w i l l r e f l e c t t h i s .

The remainder o f t h i s s e c t i o n w i l l be

devoted t o computing these f i e l d s as a f u n c t i o n o f t h e n e t h e a t l o s s from t h e ring.

C o o l i n g i n t h i s model c o n v e r t s warm water t o cool water, so t h e n e t

h e a t loss w i l l r e s u l t i n a r e d u c t i o n o f upper l a y e r volume.

The volume o f

c o n v e r t e d warm water w i l l be denoted by 6V and w i l l be used as a measure o f h e a t loss. A f t e r c o o l i n g , t h e steady r i n g s t r u c t u r e w i l l be governed by t h e c y c l o s t r o p h i c balance: 2

V 1

- +

fv

r

=

g'(h + h ) 1

1

z r

2

V 2 -

r

+

fv2

(8b)

= 'Ih2r

The upper l a y e r i n t h e f i n a l s t a t e w i l l c o n s i s t o f z e r o a n g u l a r momentum f l u i d ; t h e r e f o r e , v1 = - f / 2 r and (8a) can be i n t e g r a t e d t o y i e l d : 2

h + h =1

2

-f r

89'

2

+c

where C i s a c o n s t a n t t o be determined

(9)

86

The s o l u t i o n o f (8b) i s more c o m p l i c a t e d and proceeds i n a s e r i e s o f s t e p s . F i r s t , n o t e i t i s necessary t o c o n s i d e r f o u r r e g i o n s i n t h e second l a y e r o f t h e a d j u s t e d s t a t e (see F i g . 6.b). The innermost r e g i o n ( I ) i s The f l u i d i n t h i s c h a r a c t e r i z e d by a = 0 and extends f r o m r = 0 t o r = r., r e g i o n was l o c a t e d i n i t i a l l y i n t h e upper l a y e r and r e l e a s e d t o t h e lower l a y e r by c o o l i n g . Reversals i n t h e r a d i a l g r a d i e n t s o f a n g u l a r momentum i n a symmetric v o r t e x a r e d y n a m i c a l l y u n s t a b l e (see Charney (1973) for a d i s c u s s i o n ) ; t h e r e f o r e , t h e z e r o a n g u l a r momentum f l u i d must c o l l e c t a t r i n g c e n t e r I f t h e r e s u l t i n g p r o f i l e i s t o be s t a b l e (and hence s t e a d y ) . Region I1 c o n s i s t s o f lower l a y e r f l u i d which has been d i s p l a c e d r a d i a l l y

t o ra, where r a i s t h e o u t c r o p r a d i u s o f h, ( i . e . , h,(ra) = 0). Region I11 c o n s i s t s o f second l a y e r f l u i d which was i n i t i a l l y under t h e r i n g b u t which has s u r f a c e d i n t h e f i n a l s t a t e . Region I11 extends from r a t o r c , where r c i s t h e f i n a l l o c a t i o n o f t h e f l u i d column l o c a t e d i n i t i a l l y a t 242 Rd, Region I V c o n s i s t s o f lower l a y e r f l u i d which was i n i t i a l l y o u t s i d e o f t h e r i n g . T h i s r e g i o n

outward by t h e f o r m a t i o n o f r e g i o n I .

Region I 1 extends from r,

extends f r o m rc t o 00. The s t r u c t u r e s o l u t i o n s i n a l l f o u r r e g i o n s can be o b t a i n e d i n much t h e same way as was done i n s e c t i o n 2 (Dewar, 1988a).

Mass and a n g u l a r momentum

c o n s e r v a t i o n i s used t o w r i t e a g o v e r n i n g e q u a t i o n i n each r e g i o n , and s o l u t i o n s of those e q u a t i o n s a r e e x t r a c t e d . The r e s u l t i n g s o l u t i o n s c o n t a i n 1 1 unknowns. A p p l y i n g boundary c o n d i t i o n s which r e f l e c t smooth v e l o c i t y and p r e s s u r e f i e l d s a t t h e r e g i o n boundaries r e s u l t s i n a s e t of 1 1 coupled, n o n l t n e a r a l g e b r a i c e q u a t i o n s s u b j e c t t o t h e f r e e parameter SV. S o l u t i o n s o f t h i s system f o r a r b i t r a r y 6V can be e x t r a c t e d n u m e r i c a l l y , and p e r t u r b a t i o n s o l u t i o n s can be o b t a t n e d a n a l y t i c a l l y f o r s m a l l v a l u e s o f 6V. more u s e f u l small 6V s o l u t i o n s a r e : r,

p

h

(sv)~

(10a)

5 - (6V)% 16

(lob)

(1.8)x

= 1 2

where r,

+

Two o f t h e

denotes t h e boundary o f r e g i o n I .

A s k e t c h o f t h e a d j u s t e d r i n g s t r u c t u r e f o r 6V = .16 i s g i v e n i n F i g .

6.b.

Note t h a t t h e two upper l a y e r isotherms, which were i n i t i a l l y t o u c h i n g

a t r = 0, have everywhere separated by a f i n l t e d i s t a n c e . F u r t h e r , t h e p e r t u r b a t i o n s t o t h e t h e r m o c l i n e s t r u c t u r e decay q u i c k l y away from r i n g center.

The azimuthal v e l o c i t y p r o f i l e i n t h e l o w e r l a y e r c o n s i s t s o f u = 0

f l u i d w i t h v = - f r 1 2 o u t t o r,

f o l l o w e d by a r a p i d decay o f v t o z e r o .

87

3.3 Comparisons with Observations Joyce and Stalcup (1985) show that the 'bowl of warm water in warm ring 821 is approximately 200 m thick. This value wi 1 therefore be used as hcl and hC2 in order t o compare t h e model results wi h observations. T h e Initial model ring radius Is thus 2J2 Rd -40 km, which is comparable t o the -30 k m radius of the observed velocity maximum in 821. Joyce and Stalcup estimate a net heat loss from ring center t o the atmosphere of . 3 5 ~ 1 0 ~ J mbecause -~ of the passage of t w o cold air outbreaks. Assuming this heat loss falls off linearly t o the ring boundary (ZJZRd), a net heat loss over the ring of 5.8x101'J is realized. It I s admittedly not obvious how this heat loss should be converted t o a measure of volume loss for comparison with the analytical results, but here the model upper layer is equated with the heat anomaly of the ring. Examining the X B T data presented in Joyce and Stalcup (their Fig. 7 ) . suggests that 8 2 1 is - 3 O C warmer than its surroundings. This gives 821 a heat anomaly of 5.7x10lnJ. It therefore appears that O(lO%) of the heat anomaly in 821 was lost as a result of the t w o cooling events. Using this estimate t o determine the volume loss parameter yields a value of SV = .4 (the nondimensional inltial ring volume is 4.). From lob, a net depression of

Sh

=

5 Y, (SV) h 16 1c

-

=

40 m

for the thermocline under ring center is computed. Joyce and Stalcup (1985) argue that the thermocline under 821 deepened at an average rate of 1 m/day during the period o f intense cooling. This represents a net downward thermocline displacement o f 12 m. Given the inherent uncertainty In the calculation of SV and other model assumptions, the comparison of these observations with the model prediction of 40 in is encouraging. The theory suggests that f o r reasonable heat extractions, the net effect a t ring center will be 10's of meters in magnitude. [As further evidence of this order of magnitude, Olson et al. (1985) found that the 10°C isotherm under warm ring 8 2 8 experienced a downward displacement of 2 5 m f r o m March t o April. It Is also clear that this was a period o f active ventilation in 828.1 Joyce and Stalcup (1985) also discuss the outward movement of an anomalous water mass a t a depth of 300-350 m in 821. ThIs lateral displacement moved the anomaly from their 0-20 km averaging bin t o their 20-40 km radial bin, and is suggestive of an O(10 km) displacement. According t o (lOa), outward displacements under 821 of

88

Again t h e model p r e d i c t i o n s agree i n magnitude w i t h t h e

are predicted.

suggestions f r o m data. Given model shortcomings ( i . e , ,

2-1/2 l a y e r s , hcl = hc,,

a = 0, c o o l i n g

p a r a m e t e r i z a t i o n ) , t h e u s e f u l aspect of t h e s e p r e d i c t i o n s I s t h e i r o r d e r of 10 km, 6h 10 m) r a t h e r t h a n t h e i r a b s o l u t e v a l u e . T h i s magnitude ( 6 r

-

-

o r d e r o f magnitude agreement between p r e d i c t i o n s and o b s e r v a t i o n s suggests t h a t Rossby a d j u s t m e n t - l i k e mechanisms i n t h e presence o f c o o l i n g , which a r e a t t h e h e a r t o f t h e p r e s e n t c a l c u l a t i o n s , a r e a reasonable model o f t h e p h y s i c s i n a v e n t i l a t i n g r i n g . The c a l c u l a t i o n s a l s o emphasize t h a t t h e e f f e c t s o f v e n t i l a t i o n a r e a m p l i f i e d a t r i n g c e n t e r b o t h i n v e l o c i t y and t h e m o d i f i c a t i o n of the thermocline.

F u t u r e o b s e r v a t i o n a l programs m i g h t be w e l l

a d v i s e d t o m o n i t o r t h e e v o l u t i o n o f r i n g s near t h e i r c e n t e r . 4.

BETA-PLANE MODELS

The p r e v i o u s two s e c t i o n s have c o n s i d e r e d f - p l a n e models o f r i n g s s u b j e c t t o s h o r t - t e r m c o o l i n g . The nondimensional s t r e n g t h o f t h e c o o l i n g , g i v e n i n ( 2 ) , s u p p o r t s t h e a p p l i c a t i o n o f such models t o r i n g s a f f e c t e d by c o l d

storms.

On t h e o t h e r hand, t h e o b s e r v a t i o n s a l s o show t h a t r i n g s a r e a f f e c t e d

by t h e weaker c o o l i n g a s s o c i a t e d w i t h t h e average w i n t e r t i m e Slope Water atmosphere.

F u r t h e r , t h i s c o o l i n g can o p e r a t e f o r l o n g p e r i o d s o f t i m e .

If

( 2 ) I s e v a l u a t e d u s i n g these weaker f l u x e s , i t r e t u r n s a nondimensional

measure o f t h e s t r e n g t h o f t h e c o o l i n g , SL -

CphU(AT)

.04,

=

which i s comparable t o t h e nondimensional measure o f t h e b e t a e f f e c t . I t t h u s appears necessary t o p e r f o r m a beta-plane a n a l y s i s o f r i n g dynamics t o examine t h e e f f e c t s o f weaker c o o l i n g .

Such an a n a l y s i s based on t h e reduced

g r a v i t y s h a l l o w water e q u a t i o n s i s c o n s i d e r e d i n t h i s s e c t i o n . The nondimensional e q u a t i o n s g o v e r n i n g such a system a r e But + BVt

+

-

uUX

+

V U ~

uvX

+

vvY + ( 1

(1

+

BY)V = -hx

(lla)

+ BY)U = -h Y

(llb)

where x and y have been n o n d i m e n s i o n a l i z e d by t h e d e f o r m a t i o n r a d i u s , I3 = BoRd/fo where 3I, I s t h e m e r i d i o n a l g r a d i e n t o f f , t i m e has been n o n d i m e n s i o n a l i z e d by (BoRd)-l, hc i s t h e maximum i n i t i a l r i n g t h i c k n e s s ,

89

BS,

=

S+/f,hc

s a scale estimate o f heat f l u x .

and S*

The j o i n t e f f e c t o f

The system weak cool ing and b e t a w i l l be s t u d i e d by s e t t i n g P < < l and So-O(l). S u b s t l t u t i n g p e r t u r b a t i o n expansions i n powers o f B i s depicted i n Fig. 7 for u, v and h y i e l d s t h e steady f - p l a n e e q u a t i o n s a t l o w e s t o r d e r .

u,

UOX

+ v, uoy - v o

=

-hox

( 1 2a)

u,

VOX

+ vo voy - u,

=

-h,y

(12b)

(u,

h,)x

+ (v, hO ) Y

(12c)

= 0

A s was done I n t h e p r e v i o u s two s e c t i o n s , we w i l l cons d e r z e r o a n g u l a r

momentum s o l u t i o n s t o these e q u a t i o n s . The t r a n s l a t i o n speed o f t h e l e n s i s determined a t t h e n e x t o r d e r i n t h e expansion ( f o l l o w i n g F l i e r l , 1984).

A f t e r manipulating the

O(8) e q u a t i o n s ,

one o b t a i n s t h e i n t e g r a l balance: h,

dA +

X-

/I

dt

h, dA

+ Q[

dA = -Zi,[/S

dA - So//vo

SdA

(13)

where :

i s t h e eddy zonal c e n t e r o f mass speed and

Q

i s a s t r e a m f u n c t i o n d e f i n e d by:

ox

= voho oy = -Uoh,

b

F f g . 7. Schematic o f a v e n t t l a t i n g r i n g on a b e t a p l a n e . A c r o s s s e c t i o n i s shown i n ( a ) and a p e r s p e c t i v e view i s shown i n ( b ) . The r i n g i s l o s i n g h e a t t o t h e atmosphere, which produces a c r o s s i n t e r f a c e flow. T h i s i s I n d i c a t e d by t h e I S ' . The r i n g i s p r o p a g a t i n g t o t h e west because o f b e t a . The lower l a y e r i s assumed t o be v e r y deep r e l a t i v e t o h c , t h e l e n s t h i c k n e s s , and motions I n t h a t l a y e r a r e i g n o r e d .

90

Assuming t h a t S i s a t most a f u n c t i o n o f r ( w h i c h seems r e a s o n a b l e ) , t h e quanti t y : SSV,S

dA = 0, i s an odd f u n c t i o n o f x .

s i n c e v,

I n t e g r a t i n g c o n t i n u i t y o v e r t h e eddy y i e l d s : (14)

dt and t h u s : c=-

-SS9 dA SSh, dA

(15)

upon s u b s t i t u t i o n o f (14) i n t o (13).

The above e q u a t i o n f o r c i s i d e n t i c a l t o

t h a t f o u n d by Nof (1981) and F l i e r 1 (1984) i n t h e l r s t u d i e s o f a d i a b a t i c The new r e s u l t h e r e i s t h a t t h e same f o r m u l a h o l d s i n t h e presence o f

lenses. cool ing

.

The above i n t e g r a l c o n s t r a i n t s can be e v a l u a t e d u s i n g t h e z e r o p o t e n t i a l v o r t i c i t y l o w e s t o r d e r s o l u t i o n s , and y i e l d : C(t)

=

-213 hc.

and h c ( t ) = ,h, where hc, yields :

(161

- SS,t

(17)

i s t h e maximum eddy t h i c k n e s s p r i o r t o c o o l i n g .

J o i n i n g these

Note t h a t c i s i n h e r e n t l y t i m e dependent due t o t h e presence o f c o o l i n g . F u r t h e r , by t a k i n g a d e r i v a t i v e o f (18) w i t h r e s p e c t t o t i m e , t h e m a j o r p o i n t

of t h i s s e c t i o n i s o b t a i n e d , 1.e.: d

-~ dt

( t=) 213 SS,

> 0

Note t h a t a c c o r d i n g t o

(19)

(la),

c ( t ) < 0, so (19) demonstrates t h a t t h e r i n g

d r i f t r a t e decreases i n magnitude towards z e r o as c o o l i n g proceeds. The mechanism b e h i n d t h i s l e n s " d e c e l e r a t i o n " i n v o l v e s t h e a d j u s t m e n t o f

91

t h e r i n g . C o o l i n g induces low p r e s s u r e p e r t u r b a t i o n s a t l e n s c e n t e r . The r i n g responds by drawing r a d i a l l y i n w a r d and, i n t u r n , l o s e s some o f i t s c i r c u l a t i o n ( t h i s was demonstrated i n s e c t i o n s 2 and 3). Propagat ion r a t e i s p r o p o r t i o n a l t o n e t c i r c u l a t i o n ; t h u s , t h e l e ns slows down. Equation (19) suggests t h a t t h e n e t d e c e l e r a t i o n o f a v e n t i l a t i n g l e n s i s p r o p o r t i o n a l t o i t s n e t h e a t l o s s . More q u a n t i t a t i v e l y , a warm l e n s has a C h a r a c t e r i s t i c he a t anomaly o f : H = poCp V AT

where V i s t h e l e n s volume, AT t h e temperature d i f f e r e n t i a l between t h e l a y e r s , Cp t h e s p e c i f i c h e a t o f water and po a r e f e r e n c e d e n s i t y . From (14) one sees t h a t t h e n e t warm water volume change i s p r o p o r t i o n a l t o t h e i n t e g r a t e d heat l o s s and, f u r t h e r , u s i n g t h e z e r o angular momentum s o l u t i o n s : d

-V dt

= 4~

d

- hc2 dt

Thus, t h e r a t i o o f t h e n e t h e a t l o s s from t h e r i n g t o i t s i n i t i a l h e a t anomaly

is: SH

_ = _

Shc

hc where Shc denotes t h e n e t t h i c k n e s s change a t r i n g c e n t e r . But, upon u s i n g (16), t h e f r a c t i o n a l h e a t l o s s can be r e l a t e d t o f r a c t i o n a l l e n s d e c e l e r a t i o n . SH Sc _ _ H c

Warm r i n g 82-8 was c o o l e d f o r 40 days a t a r a t e o f 400 W/m2. Assuming a de f ormat io n r a d i u s o f 22 km ( g ' = 1 cm/sec2, hc = 500 m) and u s i n g t h e z e r o 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 a l s o l u t i o n s y i e l d s a n e t h e a t loss o f : SH = FSt A = 400

W

- 40x10

5

9 2

19

sec 8x10 m = 1 . 2 8 ~ 1 0 J

m2 w h i l e t h e i n i t i a l warm water anomaly was r o u g h l y : lgm 4.25

H = pocpATV =

-- 5°C cm3 gm"C

12

3

19

3x10 cm = 6 . 5 ~ 1 0 J

A c h a r a c t e r i s t i c temperature anomaly f o r 82-8 o f 5°C was used i n t h e above

est imat e s .

92

Thus the scaling from the present analysis suggests that over the course of a winter, a typical Gulf Stream ring might experience a net 20% reduction in its rate of beta driven propagation. Although this is a reasonably significant effect, it i s probably not measurable. A 20% loss of speed for most rings translates t o a change of -1 cm/sec. The best current means o f tracking rings i s by satellite and the uncertainties involved in determining propagation rate from such observations are generally of this magnitude (Hooker and Olson, 1984). 5. SUMMARY

Some simple models of the evolution of warm rlngs subject t o atmospheric cooling have been reviewed. The solutions demonstrate that both ring structure and propagation are affected by the observed rates of cooling. The most significant alteration forced by cooling is in ring circulation; indeed, this i s the effect responsible for the ring deceleration noted in section 4. One of the more surprising results involves how efficiently the lower layers are spun up, suggesting that future ring observational programs might focus on deep layers near ring center. There are several obvious shortcomings t o the models presented here; however, one slightly more subtle shortcoming merits explicit mention. A reduced gravity model was used in the beta plane examination of warm core rings. Such a model allows no interesting lower layer flows t o develop and is asymptotically equivalent t o a thin ring over a much deeper lower layer. Rings are, however, not very thin, and one wonders i f reduced gravity models miss important processes associated with the lower layer evolution. One possibility is the development of a Taylor column under the ring. This shortcoming (i.e., the neglect o f the lower layer) plagues all ring studies based o n the reduced gravity equations. In particular here, one worries that if the lower layer were allowed to adjust in response t o cooling, the additional circulation developed under the ring might remain with the ring, and thus affect the integral momentum balances. Although propagation speed should still be altered, the rates and tendencies could differ significantly from those computed using the reduced gravity model. 6. ACKNOWLEDGEMENTS

The author gratefully recognizes the support of the National Science Foundation, through grant OCE-8711030, and the Offfce of Naval Research, through contract N00014-87-6-0209, wfthout which thls research could not have been conducted. Many more people have contributed t o this work than can be

93

named; however, John Bane, Glenn Flierl and Doron Nof merit particular mention. I apologize t o my other colleagues, whose assistance has been only less in quantity and not in quality. This i s also a welcome opportunity t o thank Prof. Jacques Nlhoul and the other members of the Colloquium Steering Committee, Profs. A.R. Robinson, B. Cushman-Roisin and K.N. Fedorov, for organizing and hosting a most exciting research conference. Finally, It is a pleasure t o recognize Ms. Sheila Heseltine for her continued successful efforts without which I should find life much more difficult indeed. 7. REFERENCES Charney, J., 1973. Planetary fluid dynamics. In: P. Morel (Editor), Dynamic Meteorology. Reidel, pp. 77-352. Dewar, W.K., 1986. Mixed layers in Gulf Stream rings. Dyn. Atmos. Oceans, 10: 1-29. Dewar, W.K., 1987. Ventilating warm rings: theory and energetics, J. Phys. Oceanogr., 17: 2219-2231. Dewar, W.K., 1988a. Ventilating warm rings: structure and model evaluation, J. Phys. Oceanogr., 18: 552-564. Dewar, W.K., 1988b. Ventilating beta plane lenses, J. Phys. Oceanogr., 18: 1193-1201. Evans, R.H., Baker, K.S., Brown, 0.6. and Smith, R.C., 1985. Chronology of warm-core ring 828. J. Geophys. Res., 90: 8803-8812. Fitzgerald, J. and Chamberlin, J.L., 1983. Anticyclonic warm core Gulf Stream rings off the northeastern United States during 1980. Annual. Biol., 37: 41 -47. Flierl, G.R., 1984. Rossby wave radiation from a strongly nonlinear warm eddy. J. Phys. Oceanogr., 14: 47-58. Flierl, G.R.. 1979. A simple model for the structure of warm and cold core rings. J. Geophys. Res., 84: 781-785. Hooker, S. and Olson, D., 1984. Center of mass estimation in closed vortices: a verification in principle and practice. J. Atmos. Ocean. Tech., 1: 247-255. Joyce, T.M., 1985. Gulf Stream warm-core ring collection: An introduction. J. Geophys: Res., 90: 8801-8802. Joyce, T.M. and Stalcup, M.C., 1985. Wintertime convection in a Gulf Stream warm-core ring. J. Phys. Oceanogr., 15: 1032-1042. Nof, D., 1981. On the beta-induced movements of isolated baroclinic eddies. J. Phys. Oceanogr., 1 1 : 1662-1672. Nof, D., 1983. On the mtgratlon of isolated eddies with application t o Gulf Stream rings. J. Mar. Res., 41: 399-425. Olson, D., Schmitt, R., Kennelly, M. and Joyce, T., 1985. A two-layer diagnostic model of the long-term physical evolution of warm-core ring 828. J. Geophys. Res., 90: 8813-8822. Schmitt, R.W. and Olson, D.B., 1985. Wintertime convection in warm-core rings: thermocline ventilation and the formation of mesoscale lenses. J. Geophys. Res., 90: 8823-8838.


95

VORTICITY FRONTOGENESIS MELVIN E. STERN Department of Oceanography, Florida State University, Tallahassee, Florida, U.S.A.

ABSTRACT

The temporal evolution of a weak downstream velocity convergence in a barotropic jet leads to the merging of vorticity isopleths (“frontogenesis”) accompanied by large offshore velocities. This effect may be important for the formation of temperature/salinity or density fronts in a baroclinic ocean. INTRODUCTION Vorticity (or potential vorticity) frontogenesis is the process by which a field with weak lateral gradients evolves into one with strong gradients. Since temperature and salinity as well as potential vorticity are conserved in a large scale flow, the process of vorticity frontogenesis in a barotropic fluid should be relevent to the formation of temperature/salinity fronts in a baroclinic fluid. These are important sights for initiating the heat/salt flux between different water masses (e.g., at coasts and in the mid-ocean). The flux depends on the frontal gradients, rather on the weaker (geostrophic) gradients from which they evolve, and thus the question arises as to what limits the width of the frontal region (when we abandon the artificial eddy diffusion co-efficients which are conventionally introduced in large scale dynamics). Although our discussion is limited to a highly specific barotropic coastal model, such as is amenable to a relatively transparent calculation, a generalizable physical idea emerges.

2 QUALITATIVE CONSIDERATIONS Figure 1 is a schematic diagram of a semi-infinite barotropic coastal jet intruding into otherwise resting water (at x = - co), and the curves represent vorticity isopleths. Far upstream (x = + co) the profile of mean horizontal velocity ii (y) is assumed to be laminar, and initially (t = 0) the downstream variation of u(x,y) is small compared to the cross stream variation. Thus there is a large horizontal spacing between the anticyclonic vorticity isopleths, and also between the underlying cyclonic isopleths. As t increases the “nose” of these isopleths will move downstream, displacing irrotational fluid up and over the anti-cyclonic isopleths.

96

t=0

X

Figure 1. Schematic diagram of a jet (8(y) < 0) at x = 00 intruding along a wall and into an otherwise resting (x = -00) fluid. The isopleths of vorticity at t = 0 (top) are widely spaced in z , but later (1 >> 0) a vorticity front (arrowheads) forms at the nose of the intruding jet.

Because of the downstream convergence

(e)

the vorticity isopleths will tend to

merge, and upward velocities at the anti-cyclonic nose should occur for reasons of mass continuity. Thus particles in this region should move upwards and clockwise as shown by the anti-cyclonic isopleths at t >> 0. The motion near the nose of the cyclonic isopleths, on the other hand, is mainly determined by the proximate cyclonic vortices, whose net effect is to turn this nose downwards and counterclockwise. These inferences will be confirmed by calculations for an inviscid model having piecewise uniform vorticity, with particular attention directed to the minimum distance between isopleths as time increases (cf the frontal region indicated by the arrows in Figure 1). Before turning to this calculation it is useful to refer to a laboratory experiment illus trating vorticity frontogenesis. A dyed axi-symmetric jet was forced [Stern and Vorapayev 19841 out of a round nozzle and into a large open tank containing water of the same density, the Reynolds number being such that the flow was dominantly inertial but still stable. Aftcr a steady flow was established the control valve of the nozzle was opened further, thereby causing a faster flow to emerge from the nozzle. This proceeded to “catch-up’’ to the old-slow flow further downstream, but the initial downstream velocity convergence was so small (compared to the cross stream variation of velocity) that no visible manifestation of the increased discharge appeared on the dyed jet boundary. As time increased an axi-symmetric interfacial bulge in the dyed region developed, and its propagation speed

97

was measured. This bulge was interpreted as a vorticity front which formed as the fast flow converged on the old slow flow. The formation of the front is dependent on the existence of a “potential” vorticity invariant which, in the case of an axi-symmetric jet, is based on the radius of (material) annular rings. In a strictly two dimensional version (Stern and Pratt, 1985) of this problem vorticity is conserved, the axis of symmetry is replaced by a slippery wall, and a piecewise uniform vorticity jet is used. The configuration then consists of a (half) jet with uniform vorticity lying beneath a single free interface, which separates the jet from the irrotational “ocean.” This interface intersects the coast at a “nose” point, and becomes parallel to the coast at large upstream distances. The inviscid evolution of this interface is computed using having a small interfacial slope contour dynamics. It was found that a weak initial at the nose, will be amplified with time causing the nose slope to increase (and causing the nose speed to increase to slightly less than half the maximum upstream velocity). At this stage the leading edge of the intruding jet develops a bulbous nose, on the rear side of which filaments of irrotational fluid are entrained. A qualitatively similar evolution was found [Stern (1986)] for a quasi-geostrophic potential vorticity intrusion in an equivalent barotropic model, in which the vertical density difference and the Rossby deformation radius are important. Although the evolution of the steeply sloping nose in the previous problem is suggestive of the frontogenetical mechanism (Fig. l),a more convincing calculation requires at least two interfaces separating three regions, each one having uniform vorticity. In such a model we may relate the temporal decrease of the minimum separation between the interfaces to the reciprocal of the maximum vorticity gradient in the continuous case (Fig. 1).

3 DISCUSSION OF CALCULATIONS The reader is referred to Stern (1988) for the details of the calculation which are discussed here in a different context. Consider a region of vorticity C bounded by y = 0 and a lower interface y = L(x,t). Above this, a region of vorticity -1 is bounded by L(x,t) and an upper interface R(x,t). Above this, the fluid is irrotational. In the non-dimensionalization, the length unit is taken as the uniform y-separation between the interfaces at x = 00. For the velocity unit, we take the maximum speed at x = 00, and for the initial interfacial shapes we take:

L(z,O) = Hw

1 - exp-(x-l)/B

,

I > 1 x < l

98

If C = -1, this reduces to the aforementioned one-interface problem. The contour dynamical method will now be applied to the two-interface problem, and the numerical results will be presented in a coordinate system whose origin (x = 0) is fixed to the R = 0 nosepoint.

7 0R

1-

I

-X

I

Figure 2. Vorticity frontogenesis (see text). The symbol “1OL” indicates the lower interface a t time 10, the symbol “ 1 0 R indicates the upper interface at t = 10, the symbol “1OLR” indicates the merged segments at t = 10, and similar conventions are used in all following figures. H , = 2 5 , B = 1.5, C = -2. The lower interface converges on the upper interface in a coordinate system moving with the nosepoint (x = 0) of R. The velocity profile a t x = 00 consists of a wall jet with irrotational fluid a t y = 00, and also a t x = -00.

For C = -2 and a rather large B (Fig. 2), the separation of the L-nosepoint from the R-nosepoint decreases from unity to zero at t = 5. At t = 10, entire segments of the two nose regions have merged, i.e., the minimum thickness between the two interfaces has decreased to zero. Thus at t = 10, we have a nose region across which the jump in vorticity i s two, whereas at t = 0 we have two interfaces each with a vorticity jump of unity. This extreme frontogenesis occurs for a smaller B (Fig. 3), and also for a larger H , (Fig. 4). Fig. 5 shows a similar evolution of the nose shape when ( = 0, but this case cannot be called “frontogenetical” since irrotational fluid exists on both sides of the nose and the

99

jump in vorticity across the merged intcrfaces does not increase. When the sign of the vorticity of the lower layer is changed (Fig. 6) then there is no frontogenesis because the nose of the lower layer fluid is forced downward by the underlying cyclonic vortices. The L-nose is therefore unable to merge with the R- nose, along which particles move upwards due to the proximate anti-cyclonic vortices. The implication is that strong frontogenesis can only occur in regions where the vorticity has the same sign, i.e., in the upper half of Fig. 1.

R

I

I

-X

Figure 3. Same as Figure 2 except that B = .5. Note the coincidence of the two fronts in the nose region at t = 9.

100

R

3.75R

1

1

I

--X

I

I Figure 4. Same as Figure 3 except for a larger H , = .5. Extreme frontogenesis in the nose is again indicated.

1 Figure 5. Vorticity frontogenesis in the halfjet

C = 0, H ,

= .25, B = 1.5. At t = 12, the nose speed is-0.33.

101

1 Figure 6. An extreme case C = +4, H , = .25, B = 1.5 chosen so that u(m,O,t) = 0 At t = 16, the leading nose velocity is -.44 and the trailing nose velocityis -.35.

4 CONCLUSION Weak downstream convergences in a coastal jet are amplified in time, leading to very large horizontal gradients of (potential) vorticity whose dynamical effect produces strong offshore velocities. Note that our barotropic calculation is formally valid in a fluid where the temperature and salinity are exactly (density) compensating, and such fronts exist in the ocean in addition to density fronts. As a n approach t o the latter problem it would be interesting to demonstrate potential vorticity frontogenesis in a quasi-geostrophic baroclinic model. As we have shown, one advantage of the contour dynamical method is its ability to fully resolve the large gradients implied by the merging of vorticity isopleths. 5 REFERENCES Stern, M.E. and Pratt, L.J., 1985. Dynamics of vorticity fronts. J. Fluid Mech. 161: 513-532. Stern, M.E., 19S6. On the amplification of coastal currents and the formation of “squirts.” J. Mar. Res., 44: 403-421. Stern, M.E., 19SS. Evolution of locally unstable shear flow near a wall or coast. J. Fluid Mech., in press. Stern, M.E. and Vorapayev S.I., 19S4. Formation of vorticity fronts in shear flow, Phys. Fluids, 27: 848-855.


103

WEAKLY NON-LOCAL SOLITARY WAVES J. P. BOYD Department of Atmospheric, Oceanic and Space Sciences and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109

ABSTRACT Williams and Wilson (1988) have shown that in their numerical model, the n = 1 equatorial Rossby solitons predicted by Boyd (1980) are long-lived, easily generated, and agree well with perturbation theory. However, solitary waves of higher latitudinal mode number are long-lived only for very small amplitude. To quote Williams and Wilson: “When an anticyclone of order n is overly strong, it stabilizes itself by shedding an anticyclone of order (n - 2) - something the n = 1 mode cannot do.” The first goal of this work is to understand this radiative decay of higher mode equatorial solitary waves. It turns out that the n = 3 and higher modes are representatives of a new and as yet poorly understood class of nonlinear waves: “weakly non-local solitons.” The second goal of this study is a deeper understanding of these generalizations of ordinary solitary waves. INTRODUCTION Boyd (1980) analyzed the nonlinear shallow water wave equations on the equatorial betaplane. The main result, derived via the method of multiple scales, is the prediction of Rossby solitary waves. The latitudinal structure of these solitons is (to lowest order) identical with that of linear equatorial Rossby waves in the limit of infinitely large zonal scale. The zonal-and-temporal dependence of each field is either a solution of the Korteweg-deVries (KdV) equation [zonal velocity, pressure] or the x-derivative of a KdV solution [latitudinal velocity]. The solitons form a one-parameter family parameterized by the “pseudowavenumber” B. The perturbation theory requires that B > 1 (with the soliton peak at z = 0) where the solitary wave has decayed to such small amplitude that its dynamics becomes linear. THEOREM 1 (FAR FIELD ANALYSIS): If the phase speed of a nonlinear coherent SEUCture lies outside the range of the linear phase speeds of all waves in the system, then it is possible that the soliton decays exponentially as 1x1 -+ 00. If c is equal to the speed of some linear wave mode for some real wavenumber k, then the nonlinear coherent structure will be oscillatory rather than decaying in the “far field” and cannot be a classical, localized soliton. The concept of “far field” analysis has been applied, correctly but informally, by many authors. A proof of the theorem is given in Boyd (1989d). Unfortunately, the page limits of this article require omitting the proofs of this and other theorems. The relevance of this very general theorem to the particular case of equatorial waves is that the n = 1 soliton passes the test: c is outside the range of any linear equatorial wave. All the higher mode Rossby waves flunk this test and consequently oscillate in the “far field.” Fig. 1 illustrates this. The heavy solid curves are the linear dispersion functions, c(k), for the three n = 1 equatorial modes. For any c in the range [-1/3,0], there is some real value of k such that the n = 1Rossby mode has this phase speed. Similarly, c varies over the interval [ l , co] for the n = 1 eastward-travelling gravity while e ranges from c = -co (for k + 0) to c -1 (as k --t co) for the n = 1 gravity wave which propagates westward. The dispersion functions for all other equatorial modes of odd mode number are contained within the ranges of these n = 1 waves, leaving two gaps. (Because the soliton is symmetric about the equator, only the symmetric modes (odd mode number n) are relevant.) The phase speed of the n = 1 Rossby soIiton lies on the gap c E [-1, -1/3] as may be seen by recalling the perturbative prediction of Boyd (1980):

-

c

-

1/3 - 0.395B2

( B 0),

VZ11/,

< 0 and V 4 $o > 0 near the center of the ring. As the ring

> 0), the vortex stretching term assumes the opposite sign (a$,,/aT 0 (i.e. eastward-travelling). It will be assumed that throughout the dissipation process a = a(T) and

IE

= K(T) such that the modon dispersion relationship (2.9) remains

continuously satisfied. The modon dispersion relationship forms a single constraint on the evolution of a, c and

K.

Consequently, two additional transport equations are required

in order to uniquely determine the evolution of the modon parameters. Linear Liapunov stability for the westward-travellingmodon has been proved by Laedke and Spatschek (1985) using a variational argument closely related to the energy-Casimir methods of Holm et al. (1985). For the eastward-travelling modon there is no proof of linear stability. However, Swaters (1986a) and Flierl (1987) have established sufieient conditions which characterize the neutral modes for the linear normal-mode stability problem.

153

ylo -2

L~ I -2

0

I

I 2

Fig. 1. Contour plot of the relative vorticity for the barotropic modon with a = c = 1.0 and IC = 3.9226. The zero contour occurs just slightly radially-outward from the modon radius. The maximum and minimum are about +16.1 and -16.1, respectively. There is no known proof of nonlinear stability for either the westward or eastward-travelling modon. Numerical experiments by McWilliams et al. (1981) demonstrate, however, that the eastward-travelling modon is stable for a large class of finite-amplitude perturbations. Swenson (1987) showed that modon-with-rider solutions were unstable. 2.2 Derivation of the transport equations

Following Swaters (1986b) the transport equations describing the slow time evolution of the modon will be obtained from averaged conservation laws. An alternate viewpoint, described by Kodama and Ablowitz (1981) and Ablowitz and Segur (1981, Sec. 3.8), is to obtain the transport equations for slowly-varying solitary waves aa the consequence of orthogonality conditions derived from examining the kernel of the adjoint operator associated with the first order equations in a perturbation theory with an asymptotically small damping coefficient. Both these approaches are, of course, equivalent. The asymptotic

154 methods developed here are two-dimensional extensions of the theory that has been developed for perturbed one-dimensional solitary wave equations [e.g. Grimshaw (1979a,b; 1981)]. Unfortunately, the transport equations cannot be derived from an averaged Lagrangian since an variational principle for (2.1) in Eulerian variables is unknown. For the inoiscid linear version of (2.1) a variational principle is well known, e.g. Seliger and Whitham

(1968). For the inuiscid nonlinear version of (2.1), Virasoro (1981) has found a variational principle, but unfortunately the action density is written in form which makes it not particularly convenient to use in problems of the kind discussed here. As well, it is unknown whether or not the unperturbed (2.1) (i.e. with e = 0 ) is integrable in the sense of a multidimensional inverse scattering theory (IST).If it were, the powerful analytical machinery developed over the last decade by, among others, Karpman (1977), Karpman and Maslov (1978), Kaup and Newell (1978) and Knickerbocker and Newell (1981) for perturbed soliton equations based on the IST formalism would be available. Thus, as of this time, it seems that any analytical progress that can be made in describing the evolution of perturbed modons must be based on a direct singular perturbation theory. The conservation laws exploited to determine the leading order evolution of the dissipating modon will be the energy and enstrophy. Conservation equations for (2.1) can be put into the general form Et

+V

7 = EH.

where E, 7 and

(2.10)

H are the particular density, flux and “source” terms, respectively.

For the energy we have

E = V p . Vp,

(2.1 l a )

(2.11b) (2.1lc) for the enstrophy we have

E = (A(P)~,

S t r i c t l y s p e a k i n g exprcsaions of t h e form ( 2 . 1 0 ) are n o t really conservation s t a t e m e n t s b e c a u s e of t h e p r e s e n c e of t h e s o u r c e t e r m s . It w o u l d b e m o r e correct t o refer t o ( 2 . 1 0 ) as a balance l a w .

155

H = -2(Ap)', where

(2.12c)

and

dl

are the unit basis vectors in the zonal and vertical directions, respec-

83

t ively. When an asymptotic expansion of the form (2.2) and (2.5) is substituted into (2.10) expressions of the form -C[E(O)

+ cE(1)(I + €EP) + v

*

[7@) + c7(')] = €If@)

+ O(€'),

(2.13)

are obtained. Define the fast variable averaging operator

((*I) = J-", :--j

dEdY(*).

(2.14)

If (2.14) is averaged, it follows to O(c) , that (E(O))T= ( H " ) ,

(2.15)

where it has been assumed p(') + 0 as r ---t 00 and that V p ( l ) is continuous a t r = a. Calculation of (2.15) for the energy and enstrophy terms (2.11) and (2.12) yields, respectively, [ a 2 c 2 E 1= ] ~-2a2c2E1,

(2.16a)

[c'E~IT= -2c2E2,

(2.16b)

where El

Ez

+ v2)(1+ 4R/7)/(2u2), 'ya (7' + u 2 )R2/2, T2(7'

and where R

= Kz(y)/K1(7). Physically, the quantities

(2.16~) (2.16d) a2c2E1 and c2E2 correspond

to the kinetic energy and rigid-lid relative enstrophy, respectively. The transport equations (2.16) can be integrated immediately to yield a2c2E1= [a'c2E1]~=0exp(-2T),

(2.17a)

c2Ez = ( c ' E z ] r , ~exp(-ST).

(2.17b)

The three "algebraic" relations (2.9), (2.17a) and (2.17b) are sufficient to determine the slow time evolution of the modon radius, speed and wavenumber. We will compare the

156

predictions of the theory with a numerical solution of (2.1) for a modon initial condition in Section 3. Before moving on to the next Section we want to show that the transport equations (2.17) are in fact solvability conditions for the asymptotic expansion (2.5). The O(c)

problem can be put into the form (2.18)

where X = 1/c in r

> a and X = -n2 in r < a , respectively. It is easy to show that

the homogeneous adjoint problem associated with (2.18) can be put into the form (2.19)

for which there are two immediately obvious independent solutions: (2.20) (2.21)

If (2.18) is multiplied on the left by q given in (2.20) or (2.21) and the product integrated by parts repeatly, the resulting integrals on the left-hand-side eventually vanish and the right-hand-side can be shown to yield (2.17a) on (2.17b), respectively. In other words, (2.17) expresses the requirement that the inhomogeneity in (2.18) must be orthogonal

to the kernel of the adjoint operator associated with (2.18). It is the geometric functionspace viewpoint which Ablowitr and Segur (1981, Sec. 3.8) adopt to derive the appropriate transport equations in their treatment of perturbed one-dimensional solitary waves. Flier1 (1984) has used this latter procedure to generate the appropriate transport equations in a study of the Rossby wave field generated by a strongly nonlinear warm-core eddy in a

two-layer model. The error in Swaters (1985) can now be clearly seen. The appropriate integration domain over which the "inner-products" must be taken is the entire two-dimensional plane. Swaters (1985) took the "inner-products" over the exterior and interior regions separately. This would be only acceptable if the boundary integrals on the modon radius that arise during the above integration by parts were identically zero. However, as is easily seen this is not the case. It turns out, interestingly enough, that the single most important difference between the qualitative predictions of the two approaches is that the Swaters

157

(1985)calculation implied that modon radius monotonically decreased during the dissipation process, whereas the theory presented here will predict a monotonic increase in the modon radius during the dissipation process. The numerical simulations of McWilliams

et al. (1981)showed an increasing modon radius during the decay. Finally, we would like to comment that we have been able to solve (2.18)to find solutions for the perturbation pressure field, The solution procedure is complicated and has been presented elsewhere (Swaters, 1988).

3. COMPARISON BETWEEN THE THEORY AND A NUMERICAL

SIMULATION In this Section a comparison between the prediction of the leading-order theory developed in this paper and a numerical solution for (2.1) assuming a barotropic modon with a = c = 1.0 and

K

= 3.9226 as the initial condition is presented. The numerical scheme

used was a pseudo-spectral code for solving (2.1)in a doubly-periodic (64 x 64) domain provided by D. Haidvogel. The value of the Ekman damping coefficient was chosen to be identical to that in McWilliams et al. (1981)so that

o.oo

I

I

1

I

I

2

I

E

= 0.2.

I

I

3

I

4

I

I

5

Nondimensional Fast Time Fig. 2. Decay in the normalized kinetic energy for an Ekman-dissipating barotropic modon. The solid curve is our theoretical prediction and the open circles represent the results of a numerical solution.

158

Figure 2 is a plot of the globally-integrated barotropic energy (normalized by the initial value) versus the nondimensional fast time. The normalized integrated energy is given by

Figure 3 is a plot of the globally-integrated relative enstrophy (normalized by the initial value) versus the nondimensional fast time. The normalized relative enstrophy is defined as

'

0.0 0

I

I

I

I

I

2

I

I

I

3

I

4

I

I 5

Nondimensional Fast Time Fig. 3. Decay in the normalized relative enstrophy for an Ekman-dissipating barotropic modon. The solid curve and open circles are as described in Figure 2. In addition to accurately reproducing the dissipation rates in global properties such as the area-averaged energy and enstrophy, the theory is able to accurately reproduce "local" properties such as the decay in the zonal translation speed, zonal position and vorticity and streamfunction extrema. Figure 4 depicts the comparison between the zonal position of the dissipating modon as computed from our theory, i.e.

159

and the inferred position of the decaying modon as determined from the numerical solutions. C

0 U

8

w

C .c

(d Cl 2 .u) .-(I)

n + 0 C

.-0 .4-

a

(d C

0

N

n K "V

I

0

I

1

I

I

2

I

I

I

3

I

4

I

I

5

Nondimensional Fast Time Fig. 4. Zonal position of an Ekman-dissipating barotropic modon. The solid curve and open circles are as described in Figure 2. In Figure 5 we compare the decay of the streamfunction maximum as computed with

our theory and the numerically computed maximums. Figure 6 illustrates the comparison between the theoretical decay computed for the maximum in the relative vorticity and the numerically computed maxima. The agreement between the theory and the numerical simulations is striking.

160

1.6,

$ 3 c .-

aE"

1.4

1.2

C

.-0 .I-

0

5

1.0

.c

E m

s?

0.8

9)

.I-

-3

$ 0.6 a

a E .-

0.4

X

2 0.2

0

I

0

I

1.o

I

I

2.0

I

I

3.0

I

I

4.0

I

I

5.0

Nondimensional Fast Time

Fig. 5 . The decay in the streamfunction extrema for an Ekman-dissipating barotropic modon. The solid curve and open circles are as described in Figure 2.

161

16.0

14.0 Q,

T3

3

g

12.0

E“

a

2.

c 10.0

.-0 r

0

>

-3

’z

8.0

s E

6.0

.-

X

3

4.0

2.0

C

I

I

1.o

I

I

2.0

I

I

3.0

I

I

4.0

I

I

5.0

Nondimensional Fast Time

Fig. 6. The decay in the relative vorticity extrema for an Ekman-dissipating barotropic modon. The solid curve and open circles are as described in Figure 2. The McWilliams et al. (1981) numerical simulation of a dissipating modon indicated that the effective radius of the modon increased throughout the dissipation process. Our numerical simulations also had this qualitative trend but we were unable to compute with any confidence what exactly the radius was as a function of time due to “noise” at the modon boundary. However, we were able to compute the radial coordinate of the extrema in the stream function and vorticity fields. Figure 7 illustrates the comparison between the numerically inferred radial positions and the predictions of our theory.

162

t

Nondimensional Fast Time Fig. 7. The radial coordinate of the extrema in the stream function and vorticity fields throughout the decay. The open boxes and circles correspond to the numerically determined coordinate of the vorticity and stream function extrema, respectively. The solid lines are the theoretical predictions.

4. SUMMARY AND CONCLUSIONS

A multiple-scales perturbation theory was developed to describe the Ekman dissipation of barotropic modons. The modon evolves so as to satisfy leading-order globally-integrated energy and enstrophy balances. These transport equations were shown to be properly formulated solvability conditions for an asymptotic expansion assuming a relatively small damping coefficient. The theory predicts that the globally-integrated energy and enstrophy will decrease to zero exponentially rapidly. As well, the stream function and vorticity extrema will decrease to zero. The zonal translation speed will decrease monotonically to zero. Throughout the decay process the modon dipole will dilate.

A comparison between the predictions of our theory and a numerical solution was presented. The results are in very good agreement. Global properties such as the decay in the area-integrated energy and enstrophy are accurately reproduced. As well, local

163

properties such as the zonal position, and the decay and position of the streamfunction and vorticity extrema are also accurately reproduced. Finally, we would like to mention that we have extended the above analysis to higherorder friction laws (e.g. horizontal and bi-harmonic damping) and to baroclinic modons. The results of this analysis and comparisons with numerical simulations will be published in a forthcoming paper. ACKNOWLEDGEMENT This project was initiated while G.E.S. was a post-doctoral fellow supported by National Science Foundation grants awarded to G.R.F. Final preparation of the manuscript was supported in part by an Operating Research Grant awarded to G.E.S. by the Natural Sciences and Engineering Research Council of Canada, and by a Science Subvention from the Atmospheric Environment Service of Canada.

REFERENCES Ablowitz, M.J. and Segur, H., 1981. Solitons and the inverse scattering transformation. SIAM Studies in Applied Mathematics, 425 pp. Carnevale, G.F., Vallis, G.K., Purini, R. and Briscolini, M., 1988. Propagation of barotropic Modons over topography. Geophys. Astrophys. Fluid Dynamics, 41: 45101. Flierl, G.R., 1984. Rossby wave radiation from a strongly nonlinear warm eddy. J. Phys. Oceanogr., 14: 47-58. Flierl, G.R., 1987. Isolated eddy models in geophysics. Ann. Rev. Fluid Mech., 19: 493-530. Flierl, G.R., Larichev, V.D., McWilliams, J.C. and Reznik, G.M.,1980. The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans, 5: 1-41. Flierl, G.R., Stern, M.E. and Whitehead, Jr., J.A., 1983. The physical significance of Modons: Laboratory experiments and general integral constraints. Dyn. Atmos. Oceans, 7: 233-263. Flierl, G.R. and Wroblewski, J.S., 1985. The possible influence of warm core Gulf Stream rings upon shelf water larval fish distribution. Fishery Bulletin, 83(3): 313-330. Grimshaw, R.H.J., 1979a. Slowly varying solitary waves, I. Korteweg-de Vries equation. Proc. R. SOC.Lond., A368: 359-375. Grimshaw, R.H.J., 1979b. Slowly varying solitary waves, 11. Nonlinear Schroedinger equation. Proc. R. SOC.Lond., A368: 377-388. Grimshaw, R.H.J., 1981. Slowly varying solitary waves in deep fluids. Proc. R. SOC. Lond., A376: 319-332. Haines, K. and Marshall, J., 1987. Eddy-forced coherent structures as a prototype of atmospheric blocking. Q.J.R. Meteorol. SOC.,113: 681-704.

164

Holm, D.D., Marsden, J.E., Ratiu, T. and Weinstein, A., 1985. Nonlinear stability of fluid and plasma equilibria. Physics Reports, 123(1,2): 1-116. Karpman, V.I., 1977. A perturbation theory for the Korteweg-de Vries equation. Phys. Lett., A60: 307-308. Karpman, V.I. and Maslov, E.M., 1978. Structure of tails produced under the action of perturbations on solitons. Sov. Phys. JETP, 48: 252-259. Kaup, D.J. and Newell, A.C., 1978. Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. R. SOC.Lond., A361: 413-446. Knickerbocker, C.J. and Newell, A.C., 1980. Shelves and the Korteweg-de Vries equation. J. Fluid Mech., 98: 803-818. Kodama, Y. and Ablowitz, M.J., 1981. Perturbations of solitons and solitary waves. Studies in Appl. Math., 64: 225-245. Laedke, E.W. and Spatschek, K.H., 1986. Two-dimensional drift vortices and their stability. Phys. Fluids, 29(1): 133-142. Lai, D.Y. and Richardson, P.L., 1977. Distribution and movement of Gulf Stream rings. J. Phys. Oceanogr., 7: 670-683. Larichev, V.D. and Reznik, G.M., 1976. Two-dimensional Rossby soliton: an exact solution. Rep. USSR Acad. Sci., 231: 1077-1079. McWilliams, J.C., 1980. An application of equivalent Modons to atmospheric blocking. Dyn. Atmos. Oceans, 5: 43-66. McWilliams, J.C., Flied, G.R., Larichev, V.D. and Reznik, G.M., 1981. Numerical studies of barotropic modons. Dyn. Atmos. Oceans, 5: 219-238. Mysak, L.A., Hsieh, W.W. and Parsons, T.R., 1982. On the relationship between interannual baroclinic waves and fish populations in the northeast Pacific. Bio. Oceanogr., 2(1): 63-103.

Pedlosky, J., 1987. Geophysical Fluid Dynamics, 2nd Edition, Springer-Verlag, 710pp. Richardson, P.L., 1983. Eddy kinetic energy in the North Atlantic from surface drifters. J. Geophys. Res.,88: 4355-4367. Seliger, R.L. and Whitham, G.B., 1968. Variational principles in continuum mechanics. Proc. Roy. SOC.Lond., A305: 1-25. Stern, M., 1975. Minimal properties of planetary eddies. J. Mar. Res., 33: 1-13. Swaters, G.E., 1985. Ekman layer dissipation in an eastward-travelling modon. J. Phys. Oceanogr., 15(9): 1212-1216. Swaters, G.E., 1986a. Stability conditions and a priori estimates for equivalent-barotropic Modons. Phys. Fluids, 29(5): 1419-1422. Swaters, G.E., 1986b. Barotropic Modon propagation over slowly varying topography. Geophys. Astrophys. Fluid Dynamics, 36: 85-113. Swaters, G.E., 1988. Propagation of two-dimensional solitary drift-vortices in a viscous rotating fluid or plasma. To appear in Continuum Mechanics and its Applications, ed. G.A.C. Graham, Springer-Verlag. Swenson, M., 1986. Equivalent Modons in simple shear. J. Atmos. Sci., 43(24): 31773185.

Swenson, M., 1987. Instability of equivalent-barotropic riders. J. Phys. Oceanogr., 17(4): 492-506.

165

Verkley, W.T.M., 1987. Stationary barotropic Modons in westerly background flows. J. Atmas. Sci., 44(17): 2383-2398. Virasoro, M.A., 1981. Variational principle for two-dimensional incompressible hydrodynamics and quasigeostrophic flows. Phys. Rev., 47(17): 1181-1183. Wyrtki, K., Magaard, L. and Hagar, J., 1976. Eddy energy in the oceans. J. Geophys. Res., 81: 2641-2646.


167

ON THE STABILITY OF OCEAN VORTICES' P. R P A Centro de Investigacibn Cientifica y de EducaciQ Superior de Ensenada, BC. Ave. Espinoza 843, 22800 Ensenada, BC. (Mtxico), and Pacific Marine Environmental Laboratory, 7600 Sand Point Way NE, Seattle, WA 98115-0070 (USA)

ABSTRACT The conservation laws of an f-plane model are used to study the stability of steady nonlinear solutions to finite amplitude perturbations. The method can only be applied to symmetric flows; circular vortices are therefore chosen for this work. Sufficient stability conditions are found for a general N-layer model, with either a "reduced gravity" or "flat bottom" lower boundary condition. Low values of N are explicitly presented and the difficulty with the continuous stratification limit, N --f 00, is discussed. There might exist states with a constant or monotonic potential vorticity horizontal dependence, but which are, nevertheless, unstable. In this case, the basic flow must violate a kind of subcriticality condition and a growing perturbation must have negative or zero energy. INTRODUCTION Mesoscale eddies are coherent structures that wander around in the ocean, transporting and mixing water with particular physical and biological properties. Internally, they experience a vigorous rotation in addition to radial and vertical motions. The longevity of some vortices is remarkable, given the turbulent environment in which they are embedded. Others, though, do not last long. One important ingredient for the explanation of the long life expectancy of these eddies is their stability to small perturbations. More complicated non-linear phenomena, like collisions with other energetic eddies or interactions with narrow currents, are doubtless also important, but in this paper attention is restricted to the problem of the stability of an isolated vortex. More precisely, the problem discussed here is that of its stability to small perturbations of arbitrary shape, viz., in the Lyapunov sense (Hahn, 1980; Holm eral., 1985; Abarbanel etal., 1986; McIntyre and Shepherd, 1987). Examples are given of models with increasing vertical resolution. A relevant metaquestion is how general (and useful) the sufficient conditions for stability can be. Thus, one serious handicap of the method is that it is restricted to cases with a finite vertical resolution; the continuum stratification limit cannot be reached. Moreover, making use of the relationship of the integrals of motion with the symmetries of the system, it can be shown (Andrews, 1984; Ripa, 1987, 1988) that the method can only be used with flows that have one of the symmetries of the system; hence, the choice of circular vortices for the f-plane model without topography or coasts.

'

PMEL Contribution No. 1067

168

The stability of non-symmetrical eddies, then, is studied only in the h e a r sense - i.e., to infinitesimal perturbations - or by means of numerical methods. The first one is usually not an easy task because the normal mode equations are non-separable (e.g., Cushman-Roisin, 1986; Ripa, 1987, for the case of an elliptical vortex in a one-layer model). Numerical modeling is not trivial either, because unstable eddies have a tendency to form very thin filaments (Ripa & Jimenez, 1988; Dritschel, 1988b). One important exception to the statements in the last paragraph are the cases of two-dimensional, non-divergent flow and that of quasigeostrophic models (for which horizontal divergence is kinematically not important). In these cases, the vorticity field uniquely determines the rest of the variables, a nice property that is extensively used in the study of the stability of non-symmetric vortices (e.g., see Love, 1893, and Tang, 1985, for the cases of elliptical eddies). This paper, then, addresses the problems for which horizontal divergence is important. The equations of motion and conservation laws for an N-layer model, with either "flat bottom" or "reduced gravity" lower boundary condition, are presented in Section 2. Section 3 deals with the derivation and discussion of the stability conditions; this is the heart of this paper. Finally, the concluding remarks are in Section 4. Polar coordinates are used in the horizontal plane; the same analysis in Cartesian coordinates (and including the p effect) is presented in Ripa (1988).

2 THE N-LAYER MODEL 2.1 Equations of motion Consider a f-plane model with N active layers of constant density pj (i = 1 through N, from top to bottom), thickness hj(x,t), horizontal velocity Kj(z,t) (with radial and azimuthal components uj and vj, respectively), and kinematic pressure pj(x,t); denotes the polar coordinates (r,O). For simplicity, the layer subscript will be omitted, unless there is danger of confusion (e.g., in expressions involving more than one layer). The evolution of this system is governed by the momentum and incompressibility equations

x

--[f+;]v++O, Du Dt Dt

+ [f + ;] u +

a

;g =o,

where

and the relationship between pj and hj. The latter is obtained making the hydrostatic, Boussinesq

169

and rigid lid approximations', and choosing between either "reduced gravity" or "flat bottom" as the lower boundary condition. Reduced gravity means that the active fluid is on top of an inert, motionless, layer, with density pN+l;the other possibility represents a rigid horizontal bottom. Denoting by the vertical displacement of the interface between layers "j" and '?+I", it is

cj

co

where is the mean value of hj and = 0 in accordance with the rigid lid approximation. The hydrostatic approximation gives p,(z,t) = grl(z,t) and j-1

(i = 2, ...,N), where q(E,t) is the sea surface elevation, and

is the buoyancy of layer "k+l" relative to layer "k. The reduced gravity case requires that pN+l= 0, i.e., grl(z,t) = -g; c,(z,t) - ... -gk c&,t), which implies pj(x,t) = -gicj(z,t) - ... -g&cN(z,t),for any value of j, between 1 and N. In the flat bottom case, on the other hand, one simply has = 0, i.e., there is one less vertical degree of freedom for the density field, due to the invariance of the total depth: h,(z,t) + ...+ h&t) = constant. Recall that Lo(= 0) is not equal to the sea-surface elevation; q is the diagnostic variable, calculated so as to assure that the total transport is non-divergent. The contribution of 9 to the potential energy is also neglected, to be consistent with the rigid lid approximation.

cN

2.2 Conservation laws From the equations of motion, (1) and (4), it is straightforward to derive the following conservation laws. First, the total volume of each layer, jjhjrdrde, is constant and the potential vorticity of an elementary fluid column in any layer,

is conserved, viz. Dqj / Dt = 0. From these conservation laws it is easy to derive N

'[YE

h.F.(q.) J J J rdrde =constant,

(7)

j=1

The rigid lid approximation is based on neglecting the relative change of potential density, 0(10-3)for the ocean, whereas the Boussinesq approximation neglects changes of the in situ density, O( lo-'). The hydrostatic approximation is valid for periods much larger than the buoyancy one, O(d(h/g')), where h and g' are typical values of layer thickness and buoyancy jump between layers.

170

where the functions Fj are arbitrary. Total energy is also conserved, viz.

where N : reducedgravity N-1 : flat bottom Finally, total linear momenta are conserved, and so is the total angular momentum N AI

Jjj=l hj (r v, + If 3 ) rdrde = constant.

3 SUFFICIENT STABILITY CONDITIONS The set of integrals of motion of a certain fluid system, such as those derived in the last section, can be used to derive stability conditions, following the procedure developed by Amol'd (1965, 1966). The conditions are only sufficient for stability (there could exist stable flows that simply do not satisfy them). The converse, then, is necessary for instability: an unstable flow must violate at least one of those conditions. These conditions are not only quite easy to derive, but also they apply to perturbations with small but finite amplitude, i.e., not just to infinitesimal ones as the normal modes (or spectral) method does. However, their use is restricted to basic flows that have one of the symmetries of the system: there cannot exist non-symmetric flows that satisfy them (Andrews, 1984; Ripa, 1987, 1988). Therefore, for an f-plane model, and without topography or coasts to break the symmetries, integrals of motion can be used to derive stability conditions only for flows which are either axisymmetric,

Uj= 0, V, = Vj(r), Hj = H,(r),

(11)

or parallel (the latter possibility is considered in Ripa, 1988). In order for this to be an exact steady solution of the equations of motion, (1) and (4),it must satisfy the gradient wind balance: dPj/dr = (f + Vj/r)Vj. Moreover, Pj(r) and Hj(r) must related as in equation (4) andfS. Combining equations (7),(8) and (10) it follows that for any solution [~&t),~(z,t)] of the governing equations (1) and (4),

Sll,&l E - oA - I[F] =constant,

(12)

where B is an arbitrary constant (with dimensions of angular velocity). If it is possible to prove that

171

is positive definite for any non-vanishing perturbation (ui = uj - Uj, h: = hj - Hj), sufficiently small3, then the basic flow [E,HJ is stable. This is based on the fact that 6s can, in that case, be considered a time-invariant4 measure of the perturbation (see, for instance McIntyre and Shepherd, 1987). The procedure to find conditions under which 6s > 0 for small non-vanishing perturbations proceeds as follows: Expanding the integrand of S in & ’![,I’ one first chooses the arbitrary functions Fj(qj) so that the linear terms, 6‘”S, are canceled. Then, by examining the integrand of 6%, one sees which conditions the basic flow (2,g)must satisfy in order for this integral, the dominant part of 6S,to be positive definite. In the axisymmemc case, 6‘”s vanishes if F(Q) is chosen as the solution of F - Q dF/dQ = iVz - o(Vr - if?‘)+ P, in each laye?; N.B. Q = (f + V/r + dV/dr)/H. With this choice of F(q),

where 5 = v’/r + W/& - r-’au’/& - Qh‘, is related to the perturbation of potential vorticity in each layer (N.B. q = Q + 0). From equation (14) it follows easily that a fluid in solid rotation (Vj(r) = a)is stable, since using o = C2 makes 6“’S positive definite. But for a flow with radial and/or vertical shear it is not immediately apparent whether or not 6% is positive definite, because of the two terms between curly brackets. The f i s t one comes from the h’uz part of the kinetic energy; this term, responsible for the possibility of negative perturbation energies, is not present in a two-dimensional, non-divergent model or in a quasi-geostrophic one. The second term can be shown (using potential vorticity conservation) to be proportional to the square of the radial particle displacement. Replacing Hv” + 2(V-m)v‘h’ by H[v’ + (v-~r)h’/H]~ - (V-m)zh‘2/H in (14), making h,’ = - and requiring 6”’s to be positive for non-vanishing perturbations, one arrives to the following sufficient conditions for stability:

F, cj,

As [u’,h’] + 0 it is 6s = O(u’,h’)’, but there might also be cubic or higher order terms in 6s: “sufficiently small” then means “%u& that the quadratic terms dominate”.

Recall S[x+!’,

E+k’]is a constant of motion by construction, and S[E,HJ is but a number.

The last term, P, comes from the potential energy integral after some algebraic manipulation. For that purpose, it is better to write down that part of the integral as a function of the thickness perturbations h;, instead of the displacement perturbations C; (Ripa, 1988).

172

If there exists any non-vanishing6 value of o such that vj -0T

dQ/& 2 + (VZ -

g; HI

. T h e r e s u l t s s h o w a w i d e range of m o d e l behavior. W e find t h a t a t w o - l a y e r m o d e l w i t h a horizontal r e s o l u t i o n of half t h e R o s s b y - r a d i u s of d e f o r m a t i o n is sufficient t o m o d e l t h e physics involved in eddy-genesis. The most realistic solution d i s p l a y s a t i m e m e a n heat t r a n s p o r t , averaged o v e r t h e m e r i d i o n a l e x t e n s i o n of t h e m o d e l (1000 km> of 0 . 0 7 p e t a w a t t f o r a zonal s e c t i o n of 320 k m length. 1 INTRODUCTION

Our

present

results

climate

from

the

energy

radiative

r e c e i v e d f r o m t h e s u n and t h e r e d i s t r i b u t i o n of t h e absorbed p a r t b y the a t m o s p h e r e / o c e a n

s y s t e m . T h e r e d i s t r i b u t i o n is a c c o m p l i s h e d

t h r o u g h a p o l e w a r d heat t r a n s p o r t caused by t h e a t m o s p h e r e and t h e and Vonder H a a r

ocean circulations. Oort

show

that

the

in

atmosphere

oceans the

a

play

poleward

comparable

heat

transport.

estimates

( C a r i s s i r n o e t al., 1985)

are

large

still

transport

in

the

uncertainties ocean,

as

stations d i r e c t l y m e a s u r i n g

w e r e

role

in the

confirm

local

ocean

temperature

this

to

that

recent

Although

lacking

and c u r r e n t

of a

to

the

of

conclusion,

estimates is

first

the

there

the

heat

network

velocity

of

as a

f u n c t i o n of depth.

O n e of t h e concerns of

c l i m a t e s t u d i e s is t h e question of

whether

246 oceail do

eddies

plav

a similar

svnoptic-scale

exploration of advanced

in

estimate

the

being,

role

disturbances

the

mesoscale

recent

transport

experiments

data

properties

with

the

global

the

an

heat

transport

atmosphere.

variability

the

years,

in in

in

base

of

the

ocean

eddy-resolving

ocean

has

too

still

is

Although

eddies. numerical

greatlv

sparse

to

time

the

For

as the

ocean

model

seem t h e m o s t a d e q u a t e way of e s t i m a t i n g eddv t r a n s p o r t s . Two

eddy-resolving

explicitly

with

numerical

the

eddy

Cox, 1985>. The

and

circulation displays transport

The

North-Atlantic. between

0

latitude

The

of

the an

is

and 0.5 p e t a w a t t . The

heat

eddy-resolving

have

dealt

simulates

This

since

heat

heat

the

basin

Mintz

eddy

heat

the

mean

by

approximation

transport

in

this

transport

as

a

and

the

the

simulation

the

transport

idealized

poleward

for

net

and

transport

exactly Cox

ocean

(Semtner and Mintz, 1977,

North- Atlantic.

heat

almost

model

an

Semtner

of

net

of

transport

western

any

cancels

currents.

study

the

of

hardly

models

heat

of

the

study

is

function

of

non eddy-resolving

s o l u t i o n s are q u i t e s i m i l a r . However,

both

studies

a

use

resolution

of

the

order

of

which s e e m s n o t a d e q u a t e t o r e s o l v e m e s o s c a l e e d d i e s of of

the

of

Rossby-radius

deformation.

Moreover,

the

40 km,

t h e order

authors

of

t h o s e s t u d i e s do n o t examine, i n o r d e r t o a s c e r t , a i n t h e g e n e r a l i t y of

the

used

results,

in

the

effects

the

models.

A

of

changing

parameter

study

the

various

exploring

parameters

the

dynamical

p r o c e s s e s r e l a t e d t o eddv t r a n s p o r t s would be needed f o r t h i s . A t

present

eddies

a variety

and

Holland

their

and

parameter

of

influence

Lin, 1975,

Schmitz, 1985>. The

eddy

on

s t u d i e s have the

McWilliams formation

general

e t al., 1978, process

with

ocean

circulation

dealt

Ce.g.,

and

was

Holland

thoroughly

and

studied

by Ikeda C1981> and by Ikeda and Ape1 C1981>. The p u r p o s e of heat

in

t h i s p a p e r is t o s t u d y t h e meridional t r a n s p o r t

relation

hydrodynamic

with

instability

primitive equations

eddy-genesis an

of

model.

The

that

eastward phvsical

results

flowing

from

jet.

parameters

using

and

of

the a

initial

conditions are chosen t o s i m u l a t e t h e Qulf S t r e a m .

By m e a n s of a p a r a m e t e r s t u d y w e want t o e s t i m a t e t h e dependence of

the

most

results on

appropriate

the

model

configuration

parameters f o r

simulating

and

to

the

eddy-genesis

determine

a n u n s t a b l e ocean c u r r e n t and t h e s u b s e q u e n t h e a t flux.

the from

247 2 DESCRIPTION O F THE MODEL. use

We

diabatic

a

following

the

principles

equations

are

cast

simplifications. and

the

and

in

,e-plane

Laplacian

equation

friction

the

by

Cartesian form

more

of

model

The

model

Bryan .

< x , v , z > . The

hydrostatic,

the

the

introduced.

selective

usual

Boussinesq

incompressibility

are

state

scale

primitive equations

coordinates the

of

approximations,

linearized

a

in

the

multi-laver outlined

condition

In

biharmonic

stead

of

friction

is

used. The governing e q u a t i o n s of t h e model are

dun

+ u au n + w n + fu '>=

at

J

ax

1 aP

- - --

BmV4Un

P axn

az

The e q u a t i o n of s t a t e is l i n e a r i z e d around a r e f e r e n c e t e m p e r a t u r e giving:

p = p < l - n < T - T >

The c o n t i n u i t y e q u a t i o n r e t a i n s t h e free s u r f a c e :

The CarLesian n

tensor

= i,2; m = 1,2;

j

velocity

vector

vertical

velocity,

the

vertically

basin only.

depth.

u

notation

= 1.2; u

are

with

summation

the

horizontal

< u , u > ,m e q u a l s 3 pressure.

i n t e g r a t e d velocity, The

V

-

11.

w. p .

t e m p e r a t u r e and

operator

h

the

applies

convention

T and p

density. U

surface to

is

components

the

represent represents

elevation,

horizontal

used,

of

H

the

coordinates

248 For

the

numerical

technical

formulation

report

c o n s e r v a t i o n of

of

of

the

momentum,

mass,

we

equations

S e m t n e r (1974).

The

refer

formulation

the

to

guarantees

t e m p e r a t u r e and energy. The only

deviation f r o m t h e Bryan model is t h a t w e do n o t u s e t h e rigid-lid but

retain

variable.

We

use

a

equations

for

the

assumption. prognostic to

solve

within

the

domain

a

the

split-explicit

method

barotropic

320 km x 1000 km

of

as

free-surface elevation h

and

with

(Madala, 1981>

baroclinic

a

a

modes,

bottom

flat

of

4200 m depth.

Boundary

are

conditions

northern

and

southern

we

boundaries

use

periodic

east-west

the

boundaries

no-slip

a

in

closed.

are

condition

and

direction, these

A t

there

no

is

the

closed

heat

and

m a s s flux o u t of t h e domain. No e x t e r n a l f o r c i n g is used i n any of t h e experiments.

We jet

have

used

between

an

initial

= U/e2>.

a n i n v e r s e Rossby number based on t h e (3-parameter,

where f =

/3y, t h e

the

fo upper-layer

+

ratio

of

the

lower-laver

jet

velocity

to

249 velocity

jet

and

the

ratio

the

of

far-field

velocity

to

the

jet

velocity. Flier1

estimated

these

parameters

for

best-guess,

a

two-layer approximation of t h e Qulf S t r e a m t o be:

fo

=

8.7

HZ

=

3440 m

Le

= 85.8

, (3

s-i

km

=

1.8

lo-''

rn-ls-'

, Hi =

760 m

,Ui =

90.3 c m s-'

, Uz =

19.7 c m s-'

, Rd =

29.2 km

, u0 =

o

0.34

,c

0.67

c m s.'

g' = 1.02 cm s-'

giving: c

=

0.22

, c

=

c

=

0.22

, c

= o

=

These values have been used i n o u r s t u d y t o determine t h e i n i t i a l conditions f o r t h e basic experiment. F u r t h e r model p a r a m e t e r s are

Bm =

1.0

The

potential

loie

cm4s-',

- are

and Mintz kinetic

and

B t = 3.0 10" kinetic

determined

C1977>, where

energy,

eddy

energy

by

,a =

cm4s-'

the

transfers

following

Km, K e , Pm, Pe kinetic

lo-*

resulting

equations,

are

energy,

2.0

respectively

mean

from

see

available

eqs.

Semtner the

mean

potential

energy and eddy available p o t e n t i a l energy. W e define x =

the

zonally

x

+ x' , where x =

averaged

value

l/D

of

SE x dX;

any

quantity

x.

The

numerical

formulations are given in Semtner C1974>.

-

Km =

uz/2

+ vz/2

C8>

c10>

250

-

Froni e q u a t i o n s 1

5 t h e e n e r g y c o n v e r s i o n s can be dei.ived,

Cwhere

< K m , K e ) d e n o t e s a c o n v e r s i o n f r o m Km t o Ke, e t c . 3

= u

>

naz

a - C P m , P e > = a g T -

C13>

ax

= -ocgT’w’

= -agTw

3 BASIC EXPERIMENT In t h i s experiment equal

to

the

t h e b a s i n width

wavelength

of

the

(320

km> is c h o s e n t o be

growing

fastest

wave.

First

we

examine t h e evolution of t h e c o n t o u r s of p r e s s u r e , t e m p e r a t u r e and

vorticity

potential with

10-day

growing

in

time.

intervals

meander

from

during

the

non-linear

e f f e c t s becoming

see

pinch-off

the

day

results 30

first

shown

are

until

60. W e

day

30 d a y s

of

in

the

cyclonic

eddy

and

figure observe

simulation

p e r c e p t i b l e after day 20. A t

a

of

The

around

1

a

with

day 40 w e

50

dav

an

anticyclonic eddy is formed. The eddy

diameter

After

is

about

100 km,

the

eddy p r o p a g a t i o n

velocity

its s w i r l v e l o c i t y 1 m/s.

a b o u t 6 cm/s,

t h e jet

day 50 t h e flow o f

becomes more zonally o r i e n t a t e d .

This r e s t o r a t i o n t o z o n a l i t y is b r o u g h t

about

by

transfer

a

from

eddy k i n e t i c t o mean k i n e t i c e n e r g y a f t e r t h e eddy d e t a c h m e n t . The potential

vorticity

contours reveal

that

the

eddies

contain

water

from a c r o s s t h e f r o n t .

The t i m e evolution of t h e e n e r g y c o n v e r s i o n s In

the

first

barotropic day 14,

the

s t a g e

and

baroclinic

barotropic

of

the

instability conversion

is shown i n f i g u r e 2.

instability

mechanisms

term

process,

both

active.

After

are

starts

to

diminish

become n e g a t i v e after day 20. After day 36, t h e r e s t o r a t i o n of zonal

flow

begins.

The

baroclinic

conversion

clearly

to the

starts

251

Fig. la. Flow p a t t e r n s i n t h e upper layer for case 1, above day 30, below day 40. From l e f t t o r i g h t : P r e s s u r e , t e r n erature and Ip potential v o r t l c i t y . Contour d i f f e r e n c e s are 0.667 m /s2 for t h e p r e s s u r e , 0.4 C for t h e temperature and 1.5 10-5’s-ffor the potential v o r t i c i t y .

252

Fig. lb. F l o w p a t t e r n s in t h e upper layer for case 1, above day 50, below day 60. From left t o r i g h t : Pressure, t e m p e r a t u r e and potential vorticity.

253 decreasing and t h e b a r o t r o p i c t r a n s f e r rapidly becomes n e g a t i v e . From

this

completely barotropic linear

figure

we

dominated

by

the

instability

phase

of

that

see

entire

baroclinic effective

is

the

the

eddy-genesis

instability

instability only

and

during

process.

process

Hence

p o t e n t i a l e n e r g y is t h e main source for t h e growing

that

the the

is

the

initial, available

meanders

and

t h e f o r m a t i o n of eddies.

64.

- 6-

36

-10

48

60

t (days)

Fig. 2 . Time v a r i a t i o n of t h e Fig.3. Vertically i n t e g r a t e d v e r t i c a l l y i n t e g r a t e d zonal t r a n s p o r t s of p o t e n t i a l mean k i n e t i c and p o t e n t i a l v o r t i c i t y , m o m e n t u m and h e a t e n e r g y and t h e v e r t i c a l l y as a f u n c t i o n of t i m e for case i n t e g r a t e d eddy k i n e t i c and 1. Values of Q V are given i n potential energy with r e s p e c t 10-5mZs-2, uv i n m3s-', VT i n t o t h e initial energy 1oi3watt. d i s t r i b u t i o n for case 1. For symbols see eqs_.aC8>
. Values are in 10 m's-'. Fig 2 . below, t h e t i m e v a r i a t i o n of the vertically integrated e n e r g y c o n v e r s i o n s f o r case 1. PKE d e n o t e s -, PK d e n o t e s -$P d e n o t e s (Pm,Pe>. see e q s . -ffm,Km), KK d e n o t e s CKmKe:, .The v a l u e s are i n m s .

-

Figure over

the

3 shows

basin

the

length

meridional and

width,

transport

as

a

of

heat,

function

of

integrated

time.

The

254 comparison of f i g u r e s 2 and 3 r e v e a l s a s t r o n g c o r r e l a t i o n between the

transfer

transport. petawatt

from

The

mean

heat

and h a s

eddy

to

transport

potential

a

reaches

a mean value of

energy

and

maximum

.031 p e t a w a t t

the

value

during

heat

of

the

.08

whole

eddy g e n e s i s p r o c e s s . The f o r m e r f i g u r e s confirm t h a t i n t h i s model s t u d y 60 days is a reasonable

measure

-

eddy shedding

for

eddy-genesis.

Moreover,

over

implies

60 days

the

process

of

r e s t o r a t i o n t o zonality averaging

the

-

meander g r o w t h that

statistics

about

f o r m a t i o n of 6 eddy p a i r s

characterizes eddy

genesis

a y e a r , which

is n o t u n r e a l i s t i c .

4 PARAMETER STUDY

W e have c a r r i e d o u t a number meridional

transports

of

heat,

of

e x p e r i m e n t s t o examine

momentum

and

potential

the

vorticity

t h a t a p p e a r during t h e eddy-genesis p r o c e s s in our model ocean. The

horizontal

density been

and

difference

varied

to

vertical

between

examine

the

the

resolutions, layers,

sensitivity

the

viscosity,

the

velocity

have

and

the

jet

of

the

model

results

to

various p a r a m e t e r s . The experiments are l i s t e d in t a b l e I.

TABLE I. Dimensional p a r a m e t e r s for the experiments study. Blanks indicate no change from case 1.

discussed

this

in

~

Case

no. I 2 3 4 5 6

Bm

5’ -2 i~*’ cms 4

1

-L

Resolution l a y e r s km

Rd

U

km

m/s

basin width

Lm 1.02

7

10 8 16 20 32 40 5

2

29

8 9

10 11 12 13 14 15 16 17 18

U m/s

0.9

0.2

0.45 1.35 0.9

0.1 0.3 0.0

320

0.32 3.2 0.45 2.30

19 39

12.5 12.5

400 400

4 8.0

40

0.63

0.15

255 Experiments 1 Experiments 8

-

7 i n v e s t i g a t e t h e dependence o n g r i d size. 10 i n v e s t i g a t e t h e e f f e c t o f t h e j e t velocitv.

Experiments 11 and 12 explore t h e e f f e c t s of v i s c o s i t y . Experiments

13

-

study

15

the

role

the

of

Rossby

of

radius

deformation. C a s e 16 i s p e r f o r m e d with a f o u r - l a y e r model. C a s e 17 is a r u n t a k i n g s a l i n i t y i n t o a c c o u n t .

Experiment

18 is a

run w i t h p a r a m e t e r s

resembling

those

used

in

t h e s t u d y of Cox . A s u m m a r y of t h e r e s u l t s is p r e s e n t e d i n Table 11.

TABLE I1 Meridional t r a n s p o r t s , various experiments. ~

and

energy

transfers

the

for

~~~

case h e a t no. transp.

10'' I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

energy levels

momentumpotential transp. vorticity transp.

lo-*-2

3 -2

watt

m s

0.031 0.032 0.015 0.016 0.011 0.011 0.028 0.002 0.042 0.041 0.022 0.023 0.025 0.009 0.022 0.023 0.065 0.004

-5.0 -2.1 -1.3 -3.1 1.2 -1.6 1.5 -1.0 -5.0 5.0 -4.1 -7.2 4.8 -4.2 -6.0 -3.6 -4.9 -0.5

First w e

eddy potential energy level 102-4-2

m s

-1.9 0.3 -2.0 -2.0 1.9 0.8 -0.8 -0.2 -4.6 14.9 4.4

244 229 124 107 88 94 172 28 322 296 156 195 280 77 217 225 244 36

84 86 51 48 37 32 71 10 166 134 78 58 153 35 69 79 84 I1

116 121 68 78 6U 57 102 11 211 311 86 80 214 36 101 119 117 17

shows c o n t o u r

. W e have been formed i n cases 1 and 4, a n extended

lobe

of

2

m s

3.6 -0.2 2.4 3.4 -1.3 0.1

the

increasing

that

at

day

2

-3

the

p l o t s for cases

see

Pe,Ke

10-l6

2 :

m s

0.0

4a

1 0 ;

Pm,Pe

ms

w i l l examine t h e e f f e c t of

g r i d spacing. Figure

eddy kinetic energy level

m s

-3

49 56 35 34 20 19 50 4 124 105 47 26 81 14 36 47 49 6

horizontal I,4

50

and

6

eddies

w h e r e a s i n case 6 w e only see

meander. A t

day 60,

we

see

that

only

in

case 1 a n eddy containing c r o s s f r o n t a l w a t e r remains. From t h e h e a t

transport

time

series ( f i g u r e

4b>, w e

observe

that

256

Fig. 4a. Pressure field in t h e upper layer for ( f r o m the right) cases I, 4 and 6, above day 50, b e l o w day 60.

the

left

to

257

200,

801

t(days1

Fig. 4b. H e a t . transport for cases 1, 4 and 6

( f r o m above to below>.

t(days)

Fig.5. Heat transport for cases 1, 8 , 9, 10 ( f r o m above t o b e l o w > .

258 t h e e d d y d e t a c h m e n t o c c u r s earlier

i n case 4 t h a n

maximum

two-thirds

transport

heat

about

is

of

i n case

The

1.

maximum

the

in

case 1.

In case

transport, entirely.

to

Increasing

distinct

a

with

associated

resolution

the

lower

is still a

there

6,

meander

km,

80

the

horizontal

the

in

maximum

in

the

growth-decay

resolution

5

to

heat

If

cycle.

cycle

growth-decay

the

we

disappears

km

did

not

of

the

jet

change m u c h w i t h respect t o case 1 w i t h 10 km.

E x p e r i m e n t s 8-10 stream.

In

all

In

formation.

explore

velocity of

the

8

30 c m / s .

of

experiments

these

case

effect

the

Also

eddy

model

the

much

is

strength

the

displays

weaker,

t h e e d d y does n o t

with

a

eddy

surface

really b e c o m e

detached

f r o m t h e jet.

In case 9 eddy f o r m a t i o n h a s a l r e a d y occurred a f t e r 30 d a y s , a f t e r which the

flow

is v i r t u a l l y

remains

vigorously

in

meandering. A t

From

eddies.

figure

d a y 60 t h e

note

5 we

flow,

that

the

t r a n s p o r t in case 8 is v e r y w e a k (about 5% of t h e v a l u e s i n

heat

case l>, whereas by

up

split

a

factor

in case 9, t h e h e a t

2.5. Figure

5

transport

shows

also

that

m a x i m u m is larger

the

formation

eddy

period h a s decreased t o 30 d a y s . Modelling

Qulf

the

(case 10) enhances

Figure

5

shows

Stream the

two

with

no

flow

thermocline

the

beneath

h e a t t r a n s p o r t and t h e i n s t a b i l i t y

maxima

associated

with

two

process. formation

eddy

events.

w e l o o k a t t h e e f f e c t of

Next,

t h e d e n s i t y difference b e t w e e n t h e

two layers.

In

case

results

13

the

smaller

in a R o s s b y

case 1. Figure

6

density

radius

shows that

which

difference is

a t day

between

two-thirds 40

the

the

of

layers

radius

in

is detached. T h e

a cyclone

jet r e m a i n s m e a n d e r i n g .

In case 14 t h e R o s s b y r a d i u s is doubled w i t h respect t o t h e f o r m e r case.

Maximum

eddy-genesis

growth

occurs

at

develops m u c h s l o w e r ,

a

wavelength

of

and only a w e a k

400

km.

The

eddy is f o r m e d

a t day 50. In case 13 t h e heat t r a n s p o r t m a x i m u m is l o w e r t h a n i n e x p e r i m e n t 1, b u t

due

to

t h e g r o w t h cycle does n o t s t o p after the

augmented

available

potential

one e d d y d e t a c h m e n t ,

energy.

In

case

14

the

259

F i g . 6. Pressure f i e l d in t h e upper layer for < f r o m t h e t h e right> cases 13, I,and 14, above day 40, below day 50.

left

to

260 h e a t t r a n s p o r t is o n e - t h i r d of t h e value i n case 1. We

conclude

that

an

overestimation

of

the

Rossby-radius

affects

t h e instability process m o r e seriously t h a n a n underestimation.

Case

12

with

behavior energy

as

viscosity the

7/10

basic

displays

experiment,

are

exchanges

higher

but

t h e s m a l l scales c o n t a i n m o r e

smaller.

wave

amplitude,

rate

growth

11,

with

e n e r g y and

the

same

and

the

viscosity

7/10

they

the

affect

o c c u r s a t a n earlier s t a g e and

v o r t i c i t y balance. The eddy c u t - o f f at a smaller

the

case

For

smaller.

qualitatively

producing

weaker

eddies

than

in

the

b a s i c experiment. 16

Case

yields

performed

is

a

with

statistics

eddy

four-layer

to

similar

model.

of

those

This

the

experiment

two-layer

model,

e x c e p t t h a t t h e h e a t t r a n s p o r t is somewhat less < t h e maximum value is t h e s a m e , b u t t h e s h a p e of t h e peak is n a r r o w e r > . C a s e 17 is a run wherein s a l i n i t y is t a k e n i n t o a c c o u n t . If

a

salinity

of

difference

0.9

permill

in

the

upper

w e use

layer,

and

double t h e t e m p e r a t u r e s t e p across t h e f r o n t . H o w e v e r .

Qulf S t r e a m

is

This value is of t h e s a m e o r d e r of by

I n all o u r

Newton.

is

eddy-genesis

experiments

positive

during

the

the

values

mean

heat

.2 p e t a w a t t

in

found

in

the

transport

at

experiment

17.

magnitude as t h e o n e e s t i m a t e d the

heat

the

whole

transport process,

related with

to the

h i g h e s t v a l u e s o c c u r r i n g during t h e eddy d e t a c h m e n t phase.

To

illustrate

discussed

the

values

comparable

about

40

experiment

sensitivity

of

eddy-genesis

to

the

parameters

i n t h i s s t u d y w e have r u n a n e x p e r i m e n t w i t h p a r a m e t e r

km,

the

with

the

viscosity

study 8

and a r e l a t i v e l y weak

of

times

Cox

(1985):

higher

a

than

grid in

size

our

of

basic

Qulf S t r e a m a s s o c i a t e d w i t h eddy

261

Fig. 7. P r e s s u r e f i l e d In the upper Layer for case 18. Contour differences are 0.5 m z s - 2 . C f r o m the upper left t o the l o w e r right: Day 30 day 8 0 )

-

262 kinetic energy levels of 0.1 mZ/s2 (case 18>.

7 shows t h a t

Figure the

basic

there

we

experiment

is no see

eddy

much

detachment. Comparing

wave

smaller

with

amplitudes.

The

energy exchange and t h e h e a t t r a n s p o r t are a b o u t 15% o f t h e values obtained in t h e basic experiment. We

therefore

with

conclude

quantitative

description

of

the

conclusions o f

a

that

errors eddy

representation

may

lead

behavior.

Cox and

to

Therefore, Bryan

of

of

the

qualitative

in

Uulf

errors

our

Stream in

the

opinion,

(1986) cencerning

the

the eddy

h e a t t r a n s p o r t are questionable, and a s t u d y w i e h more a p p r o p r i a t e p a r a m e t e r s is needed t o

draw

more d e f i n i t e conclusions a b o u t

the

global e f f e c t of t h e eddy h e a t t r a n s p o r t .

Finallv

we

present

will

some

general

conclusions

from

this

study. numerical

1. A

calibrated f o r a

horizontal

radius

of

model

with

vertical

a

problem a t

the

resolution

deformation

with is

hand

a

resolution

according t o

grid

spacing

sufficient

to

of

two

layers,

Flier1 , and

of

half

adequately

the

Rossby

simulate

the

h e a t t r a n s p o r t and energy exchange r e l a t e d t o eddy-genesis. 2.

The

higher to

estimation order

various

of

the

meridional

s t a t i s t i c s related physical

to

parameters.

heat

transport

eddy-genesis The

s e n s i t i v e t o t h e r a t i o Rossby r a d i u s o f

very

is

results

are

and

other

sensitive especially

deformation/current

width,

t h e c u r r e n t s t r e n g t h and its v e r t i c a l r e p r e s e n t a t i o n .

3. The basin-averaged heat t r a n s p o r t a s s o c i a t e d with t h e f o r m a t i o n of

s i x eddy p a i r s p e r

year

in a periodic

channel of

320 km l e n g t h

and 1000 km width is 0.07 p e t a w a t t . 4.

An

ocean-wide

simulation

with

carefully

chosen

parameters

needed t o t e s t t h e hypothesis t h a t t h e eddy h e a t t r a n s p o r t to

eddy-genesis

is

not

negligible

for

the

global

heat

is

related

budget

of

t h e ocean.

References. Bryan, K., 1969. A numerical method for the study of the circulation o f t h e world ocean. J. Comput. Phys., 4: 347-376. -, 1986. Poleward buoyancy t r a n s p o r t i n t h e ocean and mesoscale eddies. J. Phys. Oceanogr., 16: 927-933. Carissimo, B.C.. Oort, A.H. and Vonder Haar, T.H., 1985. Estimating t h e meridional energy t r a n s p o r t s i n t h e a t m o s p h e r e and ocean. J. Phys. Oceanogr., 15: 82-91.

263 Cox. M.D., 1985. An eddy-resolving numerical model of the v e n t i l a t e d t h e r m o c l i n e . J. P h y s . O c e a n o g r . , 15: 1312-1324. Flierl, Q.R., 1975. G u l f S t r e a m meandering, ring formation and r i n g p r o p a g a t i o n . Ph.D. d i s s e r t a t i o n , H a r v a r d U n i v e r s i t y . -, 1978.Models of vertical structure and the calibration of t w o - l a y e r m o d e l s . Dyn. A t m o s . O c e a n s , 2: 341-381. H o l l a n d , W.R. and Haidvogel. D.B., 1980. A p a r a m e t e r s t u d y of t h e mixed instability of idealized ocean currents. Dyn. Atmos. O c e a n s , 4: 185-215. -, and L i n , L.B., 1975. O n t h e generation of m e s o s c a l e eddies and t h e i r contribution t o t h e oceanic general circulation. 11. A p a r a m e t e r s t u d y . J. P h y s . O c e a n o g r . , 5: 658-699.9. -, and Schmitz, W.R., 1985. Zonal penetration scale of m i d l a t i t u d e j e t s . J. Phys. O c e a n o g r . , 15: 1859-1875. I k e d a , M., 1981. Meanders and detached eddies of a s t r o n g e a s t w a r d f l o w i n g j e t using a t w o - l a y e r quasi-geostrophic m o d e l . J. P h y s . O c e a n o g r . , 11: 526-540. , and Apel, J.R., 1981. Mesoscale eddies detached from s p a t i a l l y g r o w i n g m e a n d e r s i n a n e a s t w a r d . f l o w i n g oceanic jet. using 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 m o d e l . J. P h y s . O c e a n o g r . , _ .

11: 1638-1661.

Madala, R.V., 1981. Efficient time integration schemes for atmosphere and ocean models. In: D.L. Book <Editor>, Finite-difference techniques for vectorized fluid dynamics calculations. Springer-Verlag, p p 56-74. McWilliams, J.C., Holland, W.R. and Chow. J.H.S., 1978. A description of n u m e r i c a l A n t a r c t i c C i r c u m p o l a r c u r r e n t s . Dyn. A t m o s . O c e a n s , 2: 213-291. N e w t o n , C.W., 1961. E s t i m a t e s of v e r t i c a l m o t i o n s and m e r i d i o n a l heat exchange i n aulf S t r e a m eddies and a c o m p a r i s o n w i t h a t m o s p h e r i c disturbances. J. U e o p h y s . R e s . , 66: 853-870. O o r t , A.H. and Vonder H a a r , T.H., 1976. O n t h e o b s e r v e d annual c y c l e in t h e o c e a n - a t m o s p h e r e h e a t balance o v e r t h e N o r t h e r n h e m i s p h e r e . J. P h y s . O c e a n o g r . , 6: 781-800. S e m t n e r , A.J., 1974. A n oceanic general circulation m o d e l with bottom topography. Technical Report NO.^., Dept. of M e t e o r o l o g y , U n i v e r s . of C a 1 i f o r n i a . a . and Mintz, Y., 1977. N u m e r i c a l s i m u l a t i o n of t h e S e m t n e r , A.J. Uulf Stream and mid-ocean eddies. J. Phys. Oceanogr., 7: 208-230.


265

EDDY GENERATION BY INSTABILITY OF A HIGHLY AGEOSTROPHIC FRONT: MEAN FLOW INTERACTIONS AND POTENTIAL VORTICITY DYNAMICS RICHARD A. WOOD Department of Mathematics, University of Southampton, Southampton SO9 5NH, U.K.

ABSTRACT A primitive equation numerical model is used to study the mean flow interactions and potential vorticity (PV) fluxes associated with the meandering of an unstable front, which may be taken as a simple model of the Gulf Stream. The mean flows produced show a closer agreement with Gulf Stream observations than in previous models; in particular, the deep eastward jet which is produced in this model is displaced to the south of the original frontal position, as observed in the ocean. Cross-frontal PV fluxes are achieved by the formation of coherent 'blobs' of low and high PV water in the lower layer. The low PV blob moves in such a way as to flux PV down the basic PV gradient. On a 8-plane, the basic state has a PV maximum which limits the southward propagation of the low PV blob and hence determines the final southward displacement of the eastward jet. Diagnostic studies show that the dynamics of the lower layer are dominated by quasigeostrophic processes, but that in the upper layer ageostrophic processes enhance the production of relative vorticity, leading to stronger eddies.

1 INTRODUCTION There is now considerable evidence that the transient meander/eddy

field

near the Gulf Stream plays an important r61e in driving the deep mean

flows

there (see Hogg, 1983 and references therein). The results of eddy-resolving general circulation models ( e . g . Holland, 1978, Holland and Rhines, 1980) also suggest strong interactions between the eddy field and the mean flow. One way to think of the eddies is as large-scale redistributors of potential vorticity, and a feature of several theories of the

large-scale mean

circulation (Rhines and Young, 1982, Marshall and Nurser, 1986, Greatbatch, 19871 is the assumption that

the eddy field tends to homogenise the

potential vorticity in regions of the ocean not exposed to vorticity forcing (for example by wind stress curl). Wood (1988a, hereafter referred to as W88, and 1988b) describes a simple, primitive equation model of Gulf Stream meanders and eddies. In the present paper we summarise some of the results of W88 concerning eddyhean

flow

266

interactions ($31, and then proceed to examine the potential

vorticity

dynamics of the meander process. We consider two questions: 1. What is the process by which the eddies mix potential vorticity across

the front? ($4) 2. How well do the quasigeostrophic (QG) equations, which have been widely

used to study eddies, model the dynamics of regions such as the Gulf Stream, where the existence of a strong front means that QG theory strictly does not apply? (65). We begin in section 2 by giving a brief description of the model. 2 THE MODEL

The numerical model used in this study is described

in W88, and

reader is referred to that paper for full details; here we

give a

the

brief

summary. Fig. 1 shows the initial state of the model. A two-layer fluid is contained in a channel, with rigid walls to north and south and a periodic boundary condition in the east-west direction. For consistency with W88 have retained the somewhat nonstandard co-ordinate system used

we

there, with

the x-axis pointing south and the y-axis pointing east; thus the Coriolis parameter f varies with x as f=f -fix. However, for ease of viewing we

have

oriented the diagrams in this paper so that north is to the top of the page. The interface profile upper layer flow

;(XI

IX(:

intersects the surface in a front, and

is prescribed in geostrophic balance with

layer is initially at rest. The profile

h is chosen to

profile used in Stommel’s (1966,p.109) model of

be

h.

an

The lower

the exponential

the Gulf Stream; on an

f-plane this gives uniform potential vorticity in the upper initiate a meander a small, sinusoidal perturbation frontal position X in the initial state, as shown by

is

layer. To

imposed on the

the dotted

Fig. 1. The wavelength of this disturbance, and the channel

line

in

length M, are

chosen to be equal to the wavelentgh of the fastest-growing wave according to the linear theory of Killworth et a l . (1984). A three dimensional, primitive equation numerical model

is used to study

the system shown in Fig. 1. The initial density field in the numerical model is set up as two regions of constant density separated by

a narrow

pycnocline region, and the middle isopycnal of the pycnocline is identified with the interface in Fig. 1. Further details can be found in W88, and Wood, 1988b contains a report on an extensive series of approach. The results will be presented here system.

tests to justify this

in terms of

the

two-layer

267

FIG.l

The initial state of the model. A two layer fluid is contained in a

periodic channel of depth D, with rigid walls to north and south

(note the

nonstandard coordinate system in which the negative x direction represents north). The interface intersects the surface in a front at "X0,

and in the

initial state the front is slightly perturbed by making X, vary sinusoidally with y (dotted line). Initially the lower layer is at

rest, and an upper

layer velocity v is prescribed to give a geostrophic balance.

Table 1 gives parameter values for the three experiments discussed

in

this paper. In all cases f =10-4s-1, ho, the interface depth at the southern

wall, is 600m and the Rossby radius is 30km, intervals of

6-9km.

Parameters f o r

with finite difference grid

experiment DB axe chosen to

be

representative of the Gulf Stream between Cape Hatteras and the Grand Banks. For fuller details the reader is again referred to WEE.

TABLE 1. Parameter values for the experiments reported in this paper.

Desi gnation

Description

D(m)

@(m-'s-')

SF

Shallow f-plane

1200

0

SB

Shallow 8-plane

1200

1.50

Deep p-p 1ane

4000

DB

Length of Integration M(km) (days)

,

180

28

180

28

270

79

x10-'O

1.57 x10-"

268

In much of what follows we shall be concerned with interactions between the eddy field and the mean flow. For any quantity #

we shall define

the

‘mean’ component of $ as a zonal average:

3 EDDY/MEAN FLOW INTERACTIONS

The development of the initial meander shown in Fig. 1

W88 and consists of a linear growth phase, followed at

is described

in

large amplitude by

backward breaking and the formation of cutoff eddies. At

large amplitude

strong Reynolds stress convergences generate mean flows in the lower layer, and in this section we summarise and discuss some results from W88 concerning this interaction between the eddy field and the mean flow.

-

Fig. 2a shows the evolution of the mean zonal velocity v

in the

lower

layer, as a function of x and t , for experiment SF. We see the development of a double jet, with a westward flow directly under the original position of the front and a somewhat more intense eastward flow displaced

to

the

south. The eastward flow migrates southward with time, apparently without limit. When 6

is introduced (experiment SB, Fig. 2b)

the double

jet

structure remains, but now the eastward jet appears to find an equilibrium position somewhat to the south of the original position of the front. In W88 it was suggested that this difference between the f-plane and @-plane

cases

could be interpreted in terms of the potential vorticity fluxes associated with the meanders and eddies, and we shall take up this point in section 4. Observations of the deep mean flow under the Gulf Stream between Cape Hatteras and the Grand Banks (Schmitz, 1977, Hogg, 1983) show a

westward

flow directly below the Gulf Stream, with a stronger, eastward flow further south. Hogg argues that the westward flow is distinct from the deep western boundary current in this region, and Schmitz’s observations suggest that

it

is eddy-driven. However, as Hogg

in

notes,

eddy-resolving general circulation models

the

deep

velocities

(EGCMs) show an eastward flow

directly below the eastward surface jet, with recirculations to the north and south. Our results are closer to what is observed in the ocean than are are those of the EGCMs, and with Gulf Stream-like parameters (Expt. DB) the positions, breadths and relative strengths of the two jets show a

striking

similarity to Schmitz’s observations (see W88). The reason for the discrepancy between the EGCM results and the observed deep flows is not known. However, one possible explanation is that with the comparatively coarse resolution of most

EGCM studies (typically 20-40km,

compared with 6-9km for the present model) horizontal temperature or

layer

269

0

0 0

80

160 240 320 400 480

0

80

* (km)

160 240 320 400 480

x (km)

FIC.2 Contours of the mean lower layer zonal velocity vs. ( x , t l . North is to the left. Contour interval 0.15 ms-’. Dashed contours denote negative ( i . e . westward] velocity. a.

Experiment SF

b . Experiment

SB

thickness gradients cannot grow sharp enough to represent adequately the highly asymmetric frontal structure of the Gulf Stream (the length scale of cross-stream variations of the thermocline depth near the Gulf Stream is about 30km); with a weaker, more symmetric front the dynamical processes responsible for shifting the eastward jet to the south may be weakened. Some support for this conjecture can be found by comparing our results with those of Kielmann and Kase (1987). In their model of a weaker front than the Gulf Stream (still with fine resolution: 10 km) the meandering of the front leads to downwards penetration of

the eastward frontal jet, with a

southward shift of the jet (see their Fig. 161,

but

the shift

slight is much

smaller than in the present model. 4 POTENTIAL VORTICITY DYNAMICS The basic state described in 52 is particularly convenient for studying the potential vorticity dynamics of the meander/eddy

process, because the

uniform (or on the 6-plane, near-uniform) potential vorticity

(PV1

upper layer allows us to concentrate our attention on the

lower layer.

in the

270

Fig. 3 shows the initial profiles of the q 2 = ( < + f o - f 3 x ) / ( D - h ) (where

, ...->,. 360 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ........... ... ... ... ... ... ... ... ... ... ... ... ...........

.

0

60

120 180

0

90

180

Y (km) FIC.5 Experiment SF after 5.8 days: a.

Contours of lower layer potential vorticity q, vs. (x,y) Contours at 9 . 5 ~ 1 0 -1.25~10-~ ~, and 1 . 3 5 ~ 1 0 rn-ls-l. ~ The shaded region denotes low PV (

......I,

LAYER I

L L . . r . K . , , T .

.. ,. ,. .. ., /*: C

w-.

C

*

L

*

+

C

. ‘ .. .. .. .. ....

t... C

1 .

: - . < : . : . :

r

.

.

.

a

SOUTH AFRICA/

L

/

LAYER 2

3465 DAYS

SOUTH AFRICA/ AGULHAS BANK

/

Figure 4: Typical layer 1(a) and layer 2 (b) velocity and interface depth fields for experiment FCTB. The contours and barbs are as in Fig. 2. Grid points represented by dots indicate where the layer heiow is outcropped.

330

from 4 to 5 , but the total volume of leakage remains about the same.

1:

I

i):

: I\

AGULHAS RETROFLECTION.

., b)

.I

Figure 5 : Instantaneous layer 1 (a) and layer 2 (b) velocity and interface depth fields for experiment FCT3 very close to an event of ring cutoff. The contours and barbs are as in Fig. 2 and the grid points represented by dots indicate where the layer below has outcropped.

Also evident in the Figure is a near-surface tongue of subpolar ggre fluid which is wedged between the developing and the previously formed ring just southwest of Africa’s tip. It is quite common for the two water masses to interact in this fashion during ring formation

331

events in experiments FCT2-4 and in the real Agulhas Retroflection region (Lutjeharms, 1988, personal communication).

To encourage additional isopycnal outcropping, the mean thicknesses of the upper two layers are changed from 150 db each to 75 d b and 175 db in the four-layer experiment FCT3. The data in Table 2 suggests that the impact which this has on the structure and behavior of the rings is a slight reduction in maximum swirl velocities to 0.5-0.75 m s-'. Also, most of the rings are now west of Africa's tip before they cut off from the main Aylhas. From Table 1 we note that there is no mean retroflection in layer one, and the vertically integrated mass transport streamfunction (not shown) suggsts that the mean retroflection intensity is slightly weaker than in FCT2. The individual layer streamfunctions (also not shown) suggest that there is no significant change in the rate of leakage from the Indian into the Atlantic. The rings and the mean circulation, then, South of Africa's tip are less energetic as resolution is concentrated toward the upper surface, suggesting again that the energy input by the wind is dissipated more effectively. An illustration exhibiting the FCT3 preference for a more westerly ring formation position is given in Fig. 5, where instantaneous mass and flow fields for layers l and 2 are depicted just prior to ring cutoff. Note that a portion of the Agulhas turns back toward the Indian Ocean near the southeast tip of the shelf break geometry, but another significant portion continues around the outer edge of the developing ring circulation. Between the two is an intrusion of subpolar gyre fluid as much as 240 km north of the wind curl zero latitude. This intrusion, strongly evidenced in the 2nd interface depth (Fig. 5b) contours, is associated with and is likely playing an important role in the ring formation event. The subtropical front is sharply inclined in general, but especially here where the depth of the 2nd interface increases from less than 50 db to 400 d b over a distance of N 60 km. 4.3

Five Layers (FCT4)

The rings of FCT4, in which two upper layers now have mean thicknesses of 75 db, the third 100 db, the intermediate layer 950 db, and the bottom layer 3800 db (as in all experiments), are even less energetic than in FCT3. Horizontal plots of total and eddy kinetic energy (not shown) reveal that the retroflection region is 20-30% less energetic than in FCTS. From column 3 of Table 2 we note that the interface depth change from the ring edge to center is typically only 150 to 200 m, and this is true for all three of the upper ocean interfaces. The maximum swirl velocities are generally little more than 0.5 m s-l. Again, this further increase in upper ocean resolution has enhanced the transfer of energy input by the wind to the intermediate layer, as well as allowed for more efficient dissipation through release of instabilities. Also, as pointed out a t the end of Section 3, the energetic and dynamical character of the retroflection region may be significantly influenced in FCT3-4 by the fact that layer 1 fluid tends to accumulate in the South Atlantic subtropical gyre, finding it somewhat difficult to return to the Indian Ocean in the short distance south of Africa to its outcrop. This configuration is an artifact of the lack of thermohaline forcing

332

in the model. In actuality, some of the warm fluid leaking from the Indian into the Atlantic returns to the Indian Ocean as modified denser water after circulating through the South Atlantic. An accounting for such conversion must be made to determine the effect of the non-thermohaline forced circumstances on the retroflection dynamics. We will return to this issue in the closing remarks. Of particular significance in the experiments with more upper ocean resolution is the realistic tendency for rings to cut away from the Agulhas farther south and west than in E l l and FCT1. This is perhaps the factor most responsible for increased interaction with the fluid from the sultropical/polar front in FCT2-4. Such volatile intcraction in FCT4 is illustrated in Fig. 6, where layer 3 flow and interface 3 depth are depicted at two instants 25 days apart. As in Fig. 5, a pronounced intrusion of subpolar fluid is involved in the ring formation event near Day 3440 (Fig. Ga). In the subsequent snapshot (Fig. Gb), this intrusion has shifted eastward where it is influencing the formation of the next ring. This interaction in the model is especially significant because satellite imagery commonly displays filaments of cold surface water from the northern edge of the ACC between the Retroflection and a newly formed ring (Lutjeharms, 1968, personal communication). 5

SUMMARY AND DISCUSSION

The sensitivity of Agulhas retroflection and ring formation in an idealized model of the South Atlantic-Indian Ocean to increases in upper ocean vertical resolution and isopycnal outcropping has bcen esplored. Here we have described a first series of experiments with a pure-isopycnic coordinate model which handles the outcropping of coordinate surfaces in a robust fashion, replacing the quasi-isopycnic coordinate model used in previously reported work. To encourage isopycnal outcropping, first, the relative strengths of the interface density jumps in the 3-layer framework are reversed. The uppcrmost interface g’ is reduced from 0.02 to 0.01 m s - ~ ,but a major impact is the more rapid transfcr of energy from layer 1 to 2, as well as more efficient dissipation of energy input by the wind. Thus, isopycnal outcropping is not a predominant feature of the experiment. The retroflection is slightly weakened and the rings formed are smaller and less energetic than in the case with the stronger upper interface g’, but they have more intermediate layer expression. When the number of layers is increased first to 4 and then 5 , the strength of the retroflection and the energetic magnitudes of the retroflection activity continues to decrease. The rings tend to separate from the main Agulhas more often and at a more southwesterly position, however, which is a trend more toward the position where real Agulhas rings form with respect to the Agulhas Bank. Likely because of this more southerly location, intrusions of subpolar origin fluid are quite often fouiid playing an active role in the ring formation events, as is known to happen in the Retroflection area. A preliminary conclusion from these results is that reduced static stability and improved vertical resolution near the model ocean surface both lead to more efficient dissipation of

333 ,.** If-!, . .i, ;,f\ . * &

. . ..........

AGULHASRETROFLECTION,

. , . a .

AGULHAS RETROFLECTION. LAYER 3 3465.DAYS EXP FCT4

. b) /-*< ,. ,I

2

f

c

/ /

SOUTH AFRICA/

Figure 6: Instantaneous layer 3 flow patterns and interface 3 depth fields 25 days apart in FCTC Note the intrusion of subpolar fluid associated with development of two rings. The contours and bar3s are as in Fig. 2 and the grid points represented by dots indicate where the layer belcw has outcropped.

334

the energy input by the wind at the surface. In the first case, the upper ocean flows become unstable more rapidly, and in the second, the release of instability is apparently better resolved. A thorough energetics analysis of the experiments would be required to verify the second assertion. Chassignet and Boudra (1988) found coherent peaks in basinaveraged barotropic and baroclinic eneergy conversion to coincide with events of Agulhas ring cutoff in the quasi-isopycnic experiment E l l , and they concluded that a weak mixed barotropic/baroclinic instability is associated with ring formation in the experiment. Due to the relative weakness of the rings generated in the experiments FCT2 to FCT4 here, it seems that peaks in the basin-averaged energetics would be less prominent and that dynamical instabilities play a less significant role in the formation process. What is mainly apparent here is that the onset of instabilities occurs at a lower level of basin available potential energy, preventing the circulation from reaching the mean energetic intensity of Ell. In order to verify or refute the above preliminary conclusion, additional experimentation is also suggested. In particular, a series of pure-isopycnic coordinate experiments in which the mean upper ocean static stability is similar to that in E l l , and the resolution is increased as in the series here FCT1-FCT4, would be helpful. Secondly, some representation of thermohaline forcing must be incorporated into the experiments with very thin upper layers. Without this, layer 1 fluid tends to become trapped in the South Atlantic subtropical gyre. In the real South Atlantic/Indian ocean, pools of lightest fluid are generally isolated from each other in the southwest portions of the basins. Small parcels of the Indian pool occasionally break away into the southeast Atlantic through the retroflection region. Such light fluid is then modified on its journey which will eventually carry it across the equator or around the South Atlantic subtropical gyre and back into the Indian Ocean south of Africa. In the latter case, the fluid is denser than when it originally drifted into the Atlantic. Models of the type discussed here must ultimately incorporate buoyancy gain/loss. In the near future, we propose to develop the capability in the model to convert layer 1 fluid in the South Atlantic to intermediate layer fluid, so that it may freely return to the Indian Ocean south of Africa, where it will return to the upper layer near the eastern boundary. The conversion rate may be varied to determine the impact of thermohaline circulation strength on the retroflection dynamics. Finally, once more computer power becomes available, the sensitivity of the results to increased horizontal resolution should be tested. As more vertical resolution is introduced, the radii of deformation associated with the new vertical modes become progressively smaller. In order to resolve the baroclinic processes associated with these modes, finer horizontal grid spacing should be used. At the current stage of computer technology, such sensitivity studies with 5 or 10 km grid spacing are feasible only with filtered models' which allow use of a much larger time step than primitive equations models. 'Such as those based

01the

balance equation and the quasi-geostrophic approximation

335

ACKNOWLEDGEMENTS This work has been supported by the Office of Naval Research Grant No. N0001487-GO116. The calculations have been performed on the Cray computers at the Naval Research Laboratory and at the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science Foundation.

G

7

REFERENCES

Bleck, R. and Boudra, D.B., 1981. Initial testing of a numerical ocean circulation model using a hybrid (quasi-isopycnic) vertical coordinate. J. Phys. Oceanogr., 11: 755-770. Bleck, R. and Boudra, D.B., 1986. Wind-driven spin-up in eddy-resolving ocean models formulated in isobaric and isopycnic coordinates. J. Geophys. Res., 91(C6): 76117621. Boris, J.P. and D.L. Book, 1973. Flux-corrected transport, I, SHASTA, a fluid transport algorithm that works. J. Computat. Phys., 11: 38-69. Boudra, D.B. and De Ruijter, W.P.M., 1986. The wind-driven circulation of the South Atlantic-Indian Ocean. 11: Expcriments using a multi-layer numerical model. DeepSea Res., 33: 447-482. Boudra, D.B. and Chassignet, E.P., 1988. Dynamics of Agulhas Retroflection and ring formation in a numerical model. I: The vorticity balance. J. Phys. Oceanogr., 18: 280-303. Charney, J.G., 1955. The Gulf Stream as an inertial boundary layer. Proc. Natl. Acad. Sci. U.S.A., 41: 731-740. Chassignet, E.P. and Boudra, D.B., 1988. Dynamics of Agulhas Retroflection and ring formation in a numerical model. 11: Energetics and ring formation. J. Phys. Oceanogr., 18: 304-319. Chassignet, E.P., Olson, D.B., and Boudra, D.B., 1989. Evolution of rings in numerical models and observations. In Mesoscale Synoptic Coherent Structures in Geophysical Turbulence. J.C.J. Nihoul and B.M. Jamart, Eds. Elsevier, Amsterdam. Submitted. De Ruijter, W., 1982. Asymptotic analysis of the Agulhas and Brazil Current system. J. Phys. Oceanogr., 12: 361-373. De Ruijter, W.P.M. and Boudra, D.B., 1985. The wind-driven circulation of the South Atlantic-Indian Ocean. I: Expcriments in a one-layer model. Deep-sea Res., 32: 557-574. Gordon, A.L., 1985. Indian-Atlantic transfer of thermocline water at the Agulhas Retroflection. Science, 227: 1030-1033. Gordon, A.L., 1986. Interocean exchange of thermocline water. J. Geophys. Res., 73: 531-534. Hellerman, S. and Rosenstein, M., 1983. Normal monthly wind stress over the world ocean with error estimates. J. Phys. Oceanogr., 13: 1093-1104. Lutjeharms, J.R.E., 1981. Spatial scales and intensities of circulation of the ocean areas adjacent to South Africa. Deep-sea Res., 28A: 1289-1302. Lutjeharms, J.R.E. and van Ballegooyen, R.C., 1984. Topographic control in the Agulhas Current System. Deep-sea Res., 31: 1321-1337. Lutjeharms, J.R.E. and Gordon, A.L., 1987. Shedding of an Agulhas ring observed at sea. Nature, 325: 138-140. Moore, D.W. and Niiler, P.P., 1974. A two-layer model for the separation of inertial boundary currents. J. Mar. Res., 32: 457-484. Nof, D., 1983. On the migration of isolated eddies with application to Gulf Stream rings. J. Mar. Res., 41: 399-425. Olson, D.B. and Evans, R.H., 1986. Rings of the Agulhas. Deep-sea Res., 33: 27-42. Ou, H.W. and De Ruijter, W.P.M., 1986: Separation of an inertial boundary current from a curved coastline. 16: 280-280. Parsons, A.T., 1969. A two-layer model of Gulf Stream separation. J. Fluid Mech., 39: 511-528. Veronis, G., 1973. Model of world ocean circulation: I. Wind-driven, two layer. J. Mar. Res., 31: 228-288. Zalesak, S.T., 1979: Fully multi-dimensional flux-corrected transport algorithm for fluids. J. Computat. Phys., 31: 335-362.


337

EVOLUTION OF RINGS IN NUMERICAL MODELS AND OBSERVATIONS E.P. CHASSIGNET', D.B. OLSON AND D.B. BOUDRA Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami FL 33149-1098, (USA)

ABSTRACT Observed properties of rings are compared to rings produced in a two-gyre wind driven circulation model and in a model of the Agulhas retroflection. Their temporal evolution is discussed in terms of structure and translation rate. In both observations and numerical model results, propagation speeds 2 to 5 times faster than of an equivalent isolated eddy (which is of the order of the long Rossby wave speed) were observed. The decay rate of model rings with a lateral viscosity of 330aZs-' is found to be faster than in observations. Furthermore, it is observed that the model rings have a coherent structure all the way to the bottom and it seems likely that this may also b e the case in real oceanic rings. In the specific case of the model Agulhas ring, the factors influencing its motion and evolution are isolated in a series of subsidiary experiments. It is found that as the ring rounds the tip of Africa, there is only a small influence of the large scale flows on the ring propagation. On the other hand, the presence of the African continent provides an additional westward movement in addition to P. As soon as the ring drifts into the South Atlantic tropical gyre, advection by the large scale flows dominates the ring motion. 1

INTRODUCTION The temporal evolution of rings has been described in models where isolated vortices

are placed on a P-plane (McWilliams and Flierl, 1979; Meid and Lindemann, 1979) and in several time series field observations (Cheney and Richardson, 1976; Olson, 1980; Olson et a]., 1985). There have not been any detailed intercomparisons, however, between these observed rings and the model results, nor have rings produced by gyre scale simulations of the ocean been examined in relation to isolated model rings or observations. The early models used amplitudes of ring layer depth and velocities characteristic of very weak observed rings. Recent primitive equation model experiments have included more realistic ring simulations and thus prompted the comparison made here. In the following, the properties of model rings created in a square basin and in an idealized South Atlantic/Indian Ocean are compared to rings observed from various western boundary currents. The observations use a diagnostic two-layer model described in full in Olson e t al. (1985) and Olson and Evans (1986). The diagnostic model, which makes use of the topography of observed thermocline depths, is a close match to the multi-layer quasi-isopycnic coordinate numerical model developed by Bleck and Boudra (1981), which is used in the simulations. Comparisons will be made in terms of ring structure, ring translation, ring parameters (Rossby number, Burger number, etc.) and ring energetics. In addition, a number of subsidiary 'Present address: NCAR, P.O. Box 3000, Boulder CO 80307-3000, (USA)

338

experiments help to identify and quantify the primary factors influencing the translation speed and direction of the rings. The final discussion considers both 1) the general similarity of the observed and model rings and 2) the temporal decay in the models and observations in relation to the decay mechanisms at work and the frictional parameterization in the model. 2

DESCRIPTION OF THE MODEL AND EXPERIMENTS

The model used in this paper is the Bleck and Boudra (1981) quasi-isopycnic coordinate, primitive equation model. The behavior of rings formed in a double-gyre square flat bottom basin experiment (referred to as 2G) and in an idealized South Atlantic-Indian Ocean flat bottom basin (referred to as E l l ) are studied and compared to some observations. Both model configurations are driven by a steady zonal wind stress. The parameters for the experiments are presented in Table 1.

Experiment

Number of layers

Thickness

Bottom

of the layers (m)

g’

drag

( m ~ - ~ ) (s-l)

3800

1 For both experiments, Ax = 20 km.

Basin size

km x km

1280

Table 1: Parameters of experiments 2G and E l l . Blanks indicate no change from the previous experiment. 2.1

E x p e r i m e n t 2G The applied wind stress is 7== --r,

cos(-)2SY

L

where 7, = 1 x 10-4m2s-2 and L = 2000 km. The resulting circulation pattern is nearly symmetric with a counterclockwise gyre north of the wind stress curl zero and clockwise south. The upper layer mean mass transport stream function is presented in Figure 1. The boundary conditions are free-slip everywhere. Meanders develop within the free jet and grow in time resulting eventually in eddy detachment. 2.2

Experiment E l l The applied wind stress is 7* = 7 ,

7r

cos(-(y - a ) )

L

(2)

where 7, = 2 x 10-4m2s-a,L = 1000 km and a = 280 km. The resulting circulation pattern includes anticyclonic subtropical gyres in the Atlantic and Indian sectors, respectively, and an elongated cyclonic subpolar gyre south of the wind curl zero. Africa is represented by a

339

Figure 1: Upper layer mean mass transport stream function of 2G. The contour interval is 5 sv. triangular shape, approximating that of the Agulhas Bank shelf break. The upper layer mean mass transport stream function is shown in Figure 2. The boundary conditions are no-slip on meridional boundaries and free-slip on zonal boundaries. The intense boundary current along the eastern coast of Africa constitutes the Agulhas Current and the major part of it retroflects south of the tip of Africa, returning eastward toward the Indian Ocean. In turning back, several times a year the Agulhas intercepts itself and forms a ring which translates into the Atlantic. For more details on this experiment, the reader is referred to Boudra and Chassignet (1988) and Chassignet and Boudra (1988).

_- __ -- - - - - - - - - - \

I

- - _ _- - _

- --

-

_.__

Figure 2: Upper layer mean mass transport stream function of Ell. The contour interval is 5 sv. 3

3.1

FUNG MOTION AND EVOLUTION

Description of the model rings 2 6 1 and 2 6 2 At day 6160 of the 7200 day double-gyre simulation (Figure 3a), a ring which we call 2G1

340

Figure 3: Time evolution of the upper layer mass stream function of 2G from day 6160 to day 6240. The contour interval is 5 x 1 O E r n 3 ~ - ' .

341

has just separated from the free jet and a second ring (2G2) is forming. The time evolution of both rings is illustrated in Figure 3 for a period of 90 days. At day 6160, the shape of 2G1 is quite elongated and it is only at day 6180 (Figure 3b) that the ring becomes more symmetric. 2G1 keeps this quasi circular shape until day 6210 (Figure 3c) when the ring starts to interact with the western boundary of the basin. The propagation speed of 2G1 until day 6210 is approximately 9 cm 8 - l in the westward direction with a very small northward propagation. The ring 2G2 (Figure 3b) separates around day 6180 with a fairly circular shape and interacts with the free jet until day 6210. 2G2 moves westward at approximately the same speed as 2G1 until day 6240 (Figure 3d) where it encounters the now-weakened 2G1. The diameters of the rings 2G1 and 2G2 just after formation are of approximately the same magnitude (250 to 300 km), their maximum velocities of the order of 60-80 cms-' (2G2 and 2G1, respectively) at a radius of about 70 km and a maximum interface displacement at the center of 250 to 300m. In 50 days, the diameters and interface displacements at the center remain almost unchanged. There is first an increase in the maximum velocities to 100 - 120 cm s-l (2G2 and 2G1, respectively) followed by a decrease to 60 - 80 cm 8 - l . Decay becomes important as soon as the rings reach the western boundary. 3.2

Description of the m o d e l ring RE11

The chosen ring RE11 formed around day 2950 (beginning of year 9 from a 10-year simulation) and the time evolution of its upper depth interface anomaly field is illustrated in Figure 4, for a period of 90 days. The ring formation process is described in detail in Chassignet and Boudra (1988). At day 2955 (Figure 4a), the ring has just separated from the Agulhas proper and moves in a southwestward direction. The shape is quite symmetric, but on its northern side some deformation occurs because of the presence of the African continent. As the ring rounds the tip of Africa, it is compressed between the continent and the return flow from the Atlantic basin. The ring's shape becomes elliptic and undergoes strong deformation as it propagates westward (Figure 4b,c,d). At day 3035 (Figure 4e), once the ring has escaped from the influence of Africa, it regains a more circular shape and starts to move toward the west-northwest. It then gets slowly absorbed by the South Atlantic subtropical gyre (Figure 4f). The propagation speed of the ring until day 3025 (when moving westward) is approximately 4.3cms-'. The speed then increases to 6.2cms-' in the northwestward direction (5.5 c m d toward the west and 2.8cms-' toward the north). At day 2955, the ring diameter is of the order of 300 km, the maximum velocities of 120 cm 8-l at a radius of about 80 km and the interface displacement at center of 300 m. In 50 days, the diameter remains approximately unchanged, but the magnitude of the velocities has dropped by about 45% to 75 em a - l and the interface displacement at the center to 260 m. 3.3

Comparisons with observations

(i) Ring structures. The physical structure of rings from various boundary currents around the world can be compared in a uniform manner through the use of a simple two-layer approximation of their first baroclinic structure. This approach is fully described in Olson et

342

DAY 2955

DAY 2975

DAY 2995

DAY 3015

DAY 3035

DAY 3065

Figure 4: Time evolution of the upper depth interface anomaly field of RE11 from day 2955 to day 3065. The contour interval is 20 m.

343

al. (1985) who use as the interface the 10°C surface, thought to be a good approximation

of the main thermocline of the ring. The model stratifications in the present study are chosen such that the upper layer thickness at rest gives a good representation of the mean thermocline depth in the domain (Table 1). Therefore, in order to make comparisons, the radial distribution of the depth of the first interface from the rings 2G1, 2G2 and R E l l and of the 10°C isotherm from Gulf Stream warm core ring 82B (Olson et a[., 1985) and two Agulhas rings surveyed in 1983 (Olson and Evans, 1986) are presented in Figure 5 and 6. The first three profiles of 82B (March-June 1982) were taken when the ring was not interacting with the Gulf Stream, the last three when the ring was in contact (Figure 5a). Both 2G1 and 2G2 have a smaller interface displacement at the center (of the order of 50m), but the major differences appear in ring diameter ( w 160 km for 82B, 200 to 300 km for 2G1 and 2G2) (Figure 5) and in volume (Table 2). 82B has velocity maxima between 30 and 70cms-' (between 60 and 120crns-' for 2G1 and 2G2) at a radius of about 50 km (70km for 2Gl and 2G2) (Olson et al., 1985). Differences between R E l l at day 2955 (just after formation when most vigorous) and the observed Agulhas rings (Figure 6) are a smaller interface depth of the model ring of 100 to 150m and less volume (Table 2 ) . A better agreement between the model and observations might be obtained with a choice of a thicker upper layer (400m at rest instead of 300m as in 2G) (Table 1) or of a shallower isotherm than 10°C. Both observed rings have velocity maxima between 110 and 130 km from center (80 km for RE11). The newer, southern ring (referred to as the Retrofection eddy in Table 2 ) is more intense, with maximum velocities of nearly 9Ocms-' (between 75 and 120cms-' for RE11) as compared to approximately 60 cms-' in the northern ring (referred to as the Cape Town eddy) (Olson and Evans, 1986). (ii) Ring translation speeds. Gulf Stream rings were found by Brown e t al. (1986) to have translation rates up to ten times faster than those expected for an equivalent upper layer isolated eddy from the theory of Nof (1981) and Flierl (1984).' Advection by the larger scale mean circulation and interactions with the Gulf Stream were suggested to account for the difference. Furthermore, this behavior is not restricted to Gulf Stream rings. Olson and Evans (1986) compared the translation speed of two Agulhas rings surveyed in Nov.-Dec. 1983 to Nof and Flied's estimate. They found the observed translation rate to be approximately 2 to 5 times faster. This was attributed to advection by the larger scale flow. The model rings (2G1, 2G2 and RE11) also exhibit this more rapid motion. Their propagation speeds are approximately of the same magnitude as their observed counterparts (9cms-' for 2G1 and 2G2 to 6.5cms-' from long-lived Gulf Stream rings (Brown et al., 1986); 6.2 cm 8-l from R E l l to 4.8 em 8-l and 8.5 cm 8 - l from observed Agulhas rings (Olson and Evans, 1986)). Some of the factors affecting the rings propagation will be discussed in more detail in Section 4. lThese estimates are based on momentum integral over the ring

(= c

imately equal to the long Rossby wave speed (Nof,1981; Flierl, 1984).

s) and are approx-

344

-.(219-221) ..--.August August (228-231) Olson eta/. (1985)

100

200

300 400

500

0

20

40

60

80

100

120

140

160

r (km) Figure 5: Radial distributions of the depth (a) of the 10°C isotherm from Gulf Stream warm core ring 82B (from Olson et al., 1985) and of the first interface from (b) 2G1 (day 6160..6210) and (c) 2G2 (day 6180-6240).

345

200

-

300

-

400

-

500

-

8000

'

I

'

'

I

'

'

'

I

20

40

80

80

100

120

140

160

180

'

'

200 220

(b)

lo:[

. .

200

I

800 0

,

' , . . . , a ,

20

40

80

80

,

~

100 120

, 145

,

,

,

,

I60 780 200 220

Figure 6: Radial distribution of the depth of the first interface from RE11 at day 2955 (dashed line) and of the 10°C isotherm (solid line) in (a) the northern and (b) the southern observed Agulhas rings (from Olson and Evans, 1986).

346

P x 1 0 1 5 ~ K x 1015.7 voi. ~ 1 0 % ~

Ring 2G1 2G2 82B

13.4 8.8 4.2 11.2 51.4

RE1 1 Retroflection eddy Cape Town eddy

30.5

8.1 4.9 1.0

8.3 6.0 3.9

8.5 8.7 6.2

19.2 15.2

8.3

Table 2: Available potential energy, kinetic energy and volume for 2G1 a t day 6160,2G2 at day 6210, Gulf Stream warm core ring 82B (Olson e t al., 1985), RE11 a t day 2955 and two Agulhas rings (Olson and Evans, 1986).

(iii) R i n g APE a n d KE. The depth of the 10°C isotherm is also used within the context of the two-layer model of the observed ring’s structure (Olson et aL, 1985) to compute kinetic and available potential energy. The same calculation can be performed with the model rings by using the first interface depth (which corresponds approximately to that of the 10°C isotherm as shown previously). Upper layer kinetic energy ( K l ) , available potential energy associated with the first interface (PI) and corresponding volume are calculated assuming the ring to be symmetric, which is not always the case as shown in section 3.1. Both 2Gl and 2G2 are more energetic than 82B (Table 2). On the other hand, the model Agulhas rings are smaller and less energetic than the two counterparts observed in Nov.-Dec. 1983 (Table 2). Top layer decay time scales ( r p , r ~for ) a certain period of time can be calculated by dividing the energy content of the ring by the time rate of change of each quantity over the time period. Model ring time decays (- 100 days for both r p and r ~are) found to be much faster than observed ones. In the case of the Gulf Stream ring 82B (Olson el a l , 1985), when the ring is outside the Gulf Stream and topographic influences, the ~p decay scale is 500 days (compared to 600 and 520 days for cyclonic rings in the Sargasso sea (Olson, 1980)). (iv) G r a d i e n t balance state. Once formed, the ring should evolve in time and reach an approximate gradient balance state

-

Va

-

+ f v = g‘-ah a,T

(3)

where v is the azimuthal velocity and the radial derivative of the interface depth h. Equation (3)is derived for a symmetric ring from the equations of motion for an inviscid fluid in a cylindrical coordinate system associated with the ring:

a. -+.-

at av -+ at

au - v 2 + -v a u + w -a. - f v = _--1 aP aT T Tae aZ p aT’ VU av + -vav + w -av +f. = --1 aP -+.T aT rae az pT ae’ aw aw vaw aw lap at +u-+--+w-=----g aT T ae az paz

347

where u, v and w are the radial ( r ) , azimuthal (0) and vertical

(2) velocities,

respectively;

p the pressure and p the density. If the ring is perfectly symmetric, u = 0 and then (3) is

satisfied. If the ring is strongly out of balance, then the extra terms in (4) and (5) could start to play a significant role. Common practice for the scaling of rings is to consider the length scale L as the radius of maximum velocity and the velocity scale V as the maximum velocity V,,, (Olson, 1980). If the ring is considered as consisting of two layers of fluid with different densities (Olson et al., 1985), we can define a Burger number B' = g'* for the change in interface height across the ring and the scaling of (3) leads to

R:

+ R, = B',

(7)

5

where Ro = is the Rossby number for the gradient flow. This scaling allows intercomparisons between different rings. R, versus B' is represented in Figure 7. The dots in Figure 7a represent a compilation of 30 observed rings from the world ocean. The time evolutions of 2G1, 2G2, 82B and RE11 in the Rossby-Burger plane are represented in Figure 7b,c. 2G1, 2G2 and R E l l are well located among the majority of the world anticyclonic rings. On the other hand, Gulf Stream warm core ring 82B, 2G1 and 2G2 do not show a very organized pattern (Figure 7b), probably because of the presence of the near-by jet. At day 2955, RE11 is slightly off balance, but then adjusts and follows closely in time the scaled gradient balance curve (Figure 7c). Olson et al. (1985) found that the direct surface velocity measurements of observed ring were consistently higher than those predicted by the gradient balance relation of (3) in a two-layer configuration, assuming the lower layer to be at rest. They suggested that the difference is due either to surface effects or to coherent barotropic modes. A similar result is obtained for R E l l even when the contribution of the second interface displacement is taken into account, the lower layer also being at rest. In this latter case, the upper layer gradient velocities are derived from v1" -

a + fv1 = -(g:hl ar

+gih2),

(8)

where gi, hl, g; and ht correspond respectively to the reduced gravity and depth at the first and second interface and v1 to the upper layer velocity. The corresponding velocity profile (model, gradient and geostrophic) of the upper layer of RE11 for day 2955 and day 3005 are displayed in Figure 8. The upper layer geostrophic velocity is computed as follows,

One can therefore ask to what extent is the approximation of a lower layer at rest valid. Equation (3) can be rewritten as VZ

-

+ fv = f v , ,

-

(10)

-

where vg is the total geostrophic velocity. For example, at day 2955 (Figure 8a), the observed maximum velocity in the upper layer is v 1 . 2 7 7 ~ ~ - 'and from (lo), this leads to vg

348

0.5

I’

-22

0.4

0.3 0.2

0.1

-0.3 -0.2

B’

-0.1

0.1

0.2

0.3

0.4

0.5

-0.1 -0.2

-0.3 -0.4

-0.5

tRo

(c)

-0.2

/

I

I I I I

I

/

TRE1l

t

-Om2

Figure 7: Representation of the Rossby number (Ro) for the gradient flow versus the Burger number (B’)for the change in interface height acrosn the ring. (a) Compilation of 30 world oceanic rings; (b) Evolution of Gulf Stream ring 82B, 2G1 and 2G2; (c) Evolution of RE11.

349

DAY 2955

100 u

2

)

. .

Radius in km

Figure 8: R E l l upper layer velocity profiles for day 2955 and day 3005 (Solid line for model velocities, dashed for gradient and dotted for geostrophic).

-

I n s - ' . The derived geostrophic velocity from Equation (9) is 0.85ms-' (Figure 8) which implies velocities of the order of 0.15ma-' in the lower layer to maintain the balance. This is effectively the case right underneath the ring. The same reasoning is valid for the other days. A similar calculation can be made for the two-layer model. The difference between the gradient velocities calculated assuming the lower layer at rest and the model velocities is the barotropic component of the flow. If the lower layer is considered to be at rest, the upper layer velocities of the ring are underestimated by about 10%. The model rings, therefore, have a coherent structure to the bottom. This may also be the case for observed rings since surface

velocities are found to be higher than those predicted by the gradient balance assuming the lower layer at rest (Olson et al., 1985; Olson and Evans, 1986). In fact, in the Agulhas case,

+

vertically coherent structures (99%) of the flow (retroflection rings) were recently found in two years of measurements with an array of current meter moorings south of Africa (Luyten, 1988, personal communication). 4

FACTORS AFFECTING THE RING MOTION AND EVOLUTION

In this section, only the specific case of R E l l is considered and the influence of several external factors on its propagation are investigated. In subsection 4.1, the model is first configured in the same basin as in section 2, but no wind stress is applied. The initial conditions are defined with the interface displacements and velocities of RE11. Then, in subsection 4.2, the African continent is removed to investigate the influence of its proximity on the ring propagation. The parameters used in each experiment are given in Table 3. 4.1

Ring R1

- Advection by the larger scale flows

In order to study the influence of the larger scale flows, first, the ring RE11 is extracted from E l l a t day 2965. The position of the center was estimated and u,v and h in each layer within a radius of 150 Icm (distance from the center to the tip of Africa) were saved. The model

350

is then initialized with R E l l only and no wind forcing is applied. The initial conditions in the ring, referred now to as R1, are interface displacements and velocities of R E l l in each layer. The time evolution of R1 for a period of 200 days is illustrated in Figure 9. The ring is initially in contact with the African continent and generates a Kelvin wave along the eastern coast of Africa, leaving a strip of current behind it. This Kelvin wave propagates cyclonically around the basin (southern hemisphere). The measured wave speed is of the order of 2ms-', which is between the first and second baroclinic mode Kelvin wave propagation speeds (2.7ms-1 and 1.7ms-l, respectively, with ~ ~ = f R ~d ) . 3 As l this ~ pressure i disturbance ~ reaches the eastern boundary, it excites a wave whose western edge progresses at a speed of approximately 1.5cms-', which is of the order of the long Rossby wave speed PR;,as expected (1.5 cms-' and 0 . 5 m s - ' for the first and second baroclinic modes, re~pectively).~Nof (1988) found that quasi-geostrophic eddies (i.e., vortices with small amplitudes and weak circulation) on an f-plane, when in contact with a wall, leak fluid in a fashion similar to R1. Curiously, he also found that the leakage was completely blocked once the initialized eddy was at least weakly nonlinear. Comparisons are limited by the fact that Nof considers only upper-layer eddies with a lower layer at rest (reduced gravity model) and does not include the p-effect. Because of the small basin size, the Kelvin wave can propagate around the entire basin and form a closed circulation (Figure 9c). The excitation of a Rossby wave at the eastern boundary, as a consequence of this Kelvin wave propagation, might also be linked to the generation of basin mode resonance. This was found to be important in the ring formation in a two-layer experiment with a rectangular Africa (Chassignet and Boudra, 1988). The ring R1 initially moves westward until day 40 and northwestward thereafter, leaving behind it a Rossby wave wake (Flied, 1977, 1984). It is only after day 150 (Figure 9f) that the ring detaches from the African continent. The propagation of R1 is almost directly westward during the first 40 days (Figure 9h) at a speed of 3 . 6 ~ ~ ~ 3(Table -' 4). This is close to the initial 4.3 cm s-' of RE11, nevertheless suggesting a small influence of the external flows on the propagation of R E l l during its passage from the Indian to the Atlantic basin. This

-

'The internal Rossby radii of deformation ( R d ) are computed following Lighthill (1969) and assuming the at-rest stratification. Equivalent depth values of 74.2 and 27.4 em are obtained for the two baroclinic modes, corresponding to radii ( R d ) of 27 and 16 km, respectively.

Number of

Ring

layers 3

Rl

R2

1

I

Thickness of the g' layers (m) (ms-') 300 900 3800

Bottom drag

.02 .005

(s-')

African continent (Yes or No)

Basin km x km

Yes

2520

size

x

Initial conditions

Ring extracted from E l l , no wind forcing.

1280

No

As R1,but no Africa

For all experiments, Ax = 20 km.

Table 3: Parameters of the experiments. Blanks indicate no change from the previous experiment.

351

Figure 9: Upper layer interface displacement anomaly of contour interval is 10 m.

R1 for a period of 200 days. The

352

Ring

cwest (cms-1)

RE11

R1 R2

4.3 3.6 2.7

5.5 1.9 2.6

cLorth

(cms-1) 0 0.4 2.1

2.8

CTotal

(cms-1) 4.3

6.2

1.1 3.8 2.2 1.0 3.4 2.7

Table 4: Ring propagation speed. The first number in each column corresponds to the propagation speed averaged over the first 40 days, the second number to the propagation speed after the f i s t 40 days. is not surprising since there is almost no mean flow between the two basins. It is only when the ring rounds the tip of Africa that some leakage occurs between the ring and the African continent. After 40 days, R1 decelerates and moves northwestward at a speed 2.2cms-' (Table 4). This propagation speed is much slower than that of RE11 ( 6 . 2 m s - ' ) , suggesting a strong influence of the South Atlantic subtropical gyre on the movement of the latter.

-

4.2

R i n g R2

- Influence of the African continent

In order to investigate the influence of the African continent, an experiment without Africa, R2, otherwise similar to R1, was run. Its time evolution is presented in Figure 10. The ring propagates in a northwestward direction, as expected from previous numerical studies (McWilliams and Flierl, 1979; Mied and Lindemann, 1979; Flierl, 1984) and also leaves behind a Rossby wave wake. This wake is of the same order of magnitude in size and intensity as for

R1. During the first 40 days, R2 propagates toward the west at a speed of the order of 2.7 cm s-' (Table 4), suggesting an influence from the African continent on the westward propagation speed of R1 of the order of l c m s - l . The total speed of R2 for this period is 3.4cms-' (which is approximately equal to that of Rl), with a northward component of 2.1cms-'. This direction of motion (north) was not allowed in R1. It is apparent that the presence of Africa in R1 influences the ring propagation and enhances the westward movement over that due to p. Sommerfeld (1950) showed that the interaction of a point vortex with a lateral boundary is analogous to the interaction of neighboring vortices of opposite sign. The analytical solutions which describe the resulting motion of two point vortices are well known and are a function of the circulation and the separation distance (Basset, 1888; Lamb, 1932; Sommerfeld, 1950; Batchelor, 1967). The problem becomes considerably more complex if the point sources are replaced by finite area vortices which allow the vortices to exchange mass (Hooker, 1987). As described by Sommerfeld, the vortex is pushed forward by its virtual image obtained by reflection in the wall. This theory implies maximum velocities at the boundary and therefore is applicable in numerical models only when a free-slip boundary condition is used. When a no-slip condition is prescribed, this mirror image theory breaks down since the velocities

-

353

Figure 10: As in Figure 9 for R2.

354

have to be identically zero at the boundary. However, propagation along the wall in the same direction as for the effect of a virtual imge has been observed in numerical experiments using no-slip boundary conditions (Cox, 1979; Smith, 1986). Cox suggested that, by virtue of the no-slip condition, a narrow but intense band of vorticity of sign opposite to that of the eddy is present between the wall and the eddy and, considering the interactive effects of neighboring vortices (Sommerfeld, 1950; Hooker, 1987), this results in an alongshore movement of the eddy. It is surmised here that such an interaction may account for the differences between the propagation speeds of R1 and R2. The westward propagation speed of R2 remains the same after the first 40 days, but the northward component decreases sharply, as if the initial period was one of adjustment to the environment (Table 4). Initially, most of the fluid is at rest and the only active force is due to ,B. In the absence of compensating pressure gradient forces, the stronger Coriolis force on the poleward versus the equatorward side will produce an equatorward force on an anticyclone, referred to as Rossby drift (Rossby, 1948; Holland, 1983). The ring will therefore experience an acceleration to the north. As compensating pressure forces develop, the motion is then turned to the west and the ring drifts westward in a circular analogy to the classical Rossby wave. Also, transient Rossby waves are generated which further influence the translation of the eddy, i.e. slow northward drift (e.g., see Flied, 1984). 5

SUMMARY AND DISCUSSION

Rings produced in a two-gyre wind driven circulation model (experiment 2G) and in a model of the South Atlantic/Indian Ocean (experiment Ell)have been compared to observed Gulf Stream and Agulhas rings in a consistent fashion through the use of a diagnostic twolayer model. At this stage, especially the model vertical structure is still highly simplified with respect to reality as might be required to correctly simulate world ocean rings. Some attempt is made to begin to address this issue elsewhere in this volume (Boudra e t aL, 1989). One also has to keep in mind that these comparisons have been carried out with only a limited number of rings, both observed and modeled. However, a good agreement is obtained between the observed rings' 10°Csurface and the first interface depths of the modeled rings when the model mean upper layer thickness (approximation of the mean thermocline depth in the domain) is chosen to be 400 m. Therefore, some qualitative comparisons with reality can be made. The model and observed rings exhibit some substantial similarity in terms of thermocline depth, ring size, swirl velocities, and translation speeds, in addition to parameters such as the Rossby and Burger numbers. Direct surface velocity measurements in observed rings were found to be consistently higher than those predicted by the gradient balance assuming the lower layer to be at rest, and the same result was obtained for the model rings. In the model, it was demonstrated that this difference is due to the barotropic component of the flow, suggesting that the model rings have a coherent structure to the bottom. It seems likely that this is also the case for real oceanic rings. One of the major differences between observed and model rings is in their decay rates. The decay rates of the model rings were found to be 4 to 6 times faster than in observed rings,

355

and are apparently strongly influenced by the lateral viscosity. McWilliams and Flierl (1979), in their study of quasigeostrophic isolated vortices, stated that the vortex amplitude decay rate, in the limit of strong nonlinearity, is governed by the frictional coefficient rather than dispersion. The question arises as to what extent the decay is due to Rossby wave radiation (horizontal and vertical) in the above rings. In the case of an inviscid upper layer lens in the same parameter range as one of those studied here (REll), Flierl (1984) found a decay of the order of 7 years (- 2500 days) due to Rossby wave radiation in the lower layer. This is at least one order of magnitude slower than the decay rates obtained in the model. Thus, it seems most likely that, as in McWilliams and Flierl (1979), viscous effects are the predominant energy sink. It is felt that the use of a lateral viscosity of 50 to 100 mas-1 (instead of the current 330 m'8-l) might bring about comparable decay times between the modeled and observed rings. In the framework of this numerical model, such small viscosities could be employed only with a reduction in grid spacing to perhaps one-half the current 20 km. A successful intercomparison between observed and modeled rings provides the opportunity and justification to isolate the factors in the model influencing the ring motion and evolution, which is not possible with observations alone. This has been carried out by examining in some detail the motion and evolution of one E l l ring and comparing with the behavior of a similar ring in each of several subsidiary experiments. It is found that the presence of Africa provides a westward motion in addition to that due to p, and it seems that this owes to interaction between the ring and a high vortidty band along the no-slip boundary. This topic deserves further investigation in a study focused on eddy-wall interaction. There is apparently only a small influence from the external flows on the propagation of the ring during its passage from the Indian to the Atlantic basin. There is no continual flow between the two basins. It is only when the ring rounds the tip of Africa that a leakage is observed between the ring and the African continent. The advection by the large scale flows is found to dominate the motion once the ring drifts into the South Atlantic subtropical gyre. On the one hand, the relative simplicity of the model allowed the possibility to analyze some of the physical mechanisms behind ring propagation, such as advection by the mean flow or boundary influence. On the other, the effects of bottom topography/realistic coastline are also felt to be of importance, and the model's simplicity has, thus far, inhibited their determination. It is, therefore, expected that more insight into ring propagation and evolution can continue to be gained as the realism/complexity of the models is further increased. 6 ACKNOWLEDGEMENTS This work was supported by NSF grants OCE-8502126 and OCE-8600593 and by the Office of Naval Research grant No. N00014-87-G0116. Computations were carried out using the CRAY computers at the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science Foundation. 7 REFERENCES Basset, A.B., 1888: A treatise on hydrodynamics. Dover Publications, 2: 328 pp. Batchelor, G.K., 1967: An introduction to fluid dynamics. Cambridge University Press, 615 pp. Bleck, R. and D.B. Boudra, 1981: Initial testing of a numerical model ocean circulation

356 model using a Hybrid (quasi-Isopycnic) vertical coordinate. J . Phy. Oceanogr., 11, 755-770. Boudra, D.B. and E.P. Chassignet, 1988: Dynamics of Agulhas retroflection and ring formation in a numerical model. Part I. The vorticity balance. J. Phys. Oceanogr., 18, 280-303. Boudra, D.B., K.A. Maillet, and E.P. Chassignet, 1989: Numerical modeling of Agulhas retroflection and ring formation with isopycnal outcropping. In Mesoscale Synoptic Coherent Structures in Geophysical Turbulence. J.C.J. Nihoul and B.M. Jamart, Eds. Elsevier, Amsterdam. Submitted. Brown, O.B., P.C. Cornillon, S.R. Emmerson and A.M. Carle, 1986: Gulf Stream warm rings: A statistical study of their behavior. Deep-sea Res., 53, 1459-1473. Chassignet, E.P. and D.B. Boudra, 1988: Dynamics of Agulhas retroflection and ring formation in a numerical model. Part 11. Energetics and ring- formation. J . Phys. Oceanogr., 18,304-319. Cheney, R.E. and P.L. Richardson, 1976: Observed decay of a cyclonic Gulf Stream ring. Deeo-Sea Res.. 23. 143-155. COX, M.D., 1979: 'A numerical study of Somali Current eddies. J. Phys. Oceanogr., 9, 312-326. Flierl, G.R., 1977: The application of linear quasigeostrophic dynamics to Gulf Stream rings. J. Phys. Oceanogr., 7,365-379. Flierl, G.R., 1984: Rossby wave radiation from a strongly nonlinear warm eddy. J . Phys. Oceanogr., 14,47-58. Holland, G.J., 1983: Tropical cyclone motion: Environmental interaction plus a beta effect. J. Atmos. Sci., 40,328-342. Hooker, S.B., 1987: Mesoscale eddy dynamics by the method ofpoint vortices. Ph.D. Thesis, University of Miami, 158 pp. Lamb, H., 1932. Hydrodynamics. 6th ed. Cambridge University press, 738 pp. Lighthill, M.J., 1969: Linear theory of long waves in a horizontally stratified ocean of uniform depth (Appendix). Philos. Trans. R. Soe. London, 265, 85-92. McWilliams, J.C. amd G.R. Flierl, 1979: On the evolution of isolated non-linear vortices, with application to Gulf Stream rings. J. Phys. Oceanogr., 9,1155-1182. Mied, R.P. and G.J. Lindemann, 1979: The propagation and evolution of cyclonic Gulf Stream rings. J. Phys. Oceanogr., 9,1183-1206. Nof, D., 1981: On the P-induced movement of isolated baroclinic eddies. J . Phys. Oceanogr., 11, 1662-1672. Nof, D., 1988: Eddy-wall interactions. J. Mar. Res., 46,527-555. Olson, D.B., 1980: The physical oceanography of two rings observed by the cyclonic ring experiment. 11. Dynamics. J . Phys. Oceanogr., 10,514-528. Olson, D.B., R.W. Schmitt, M. Kennelly and T.M. Joyce, 1985: A two-layer diagnostic model of the long-term physical evolution of warm-core ring 82B. J . Geophys. Res., 90, 8813-8822. Olson, D.B. and R.H. Evans, 1986: Rings of the Agulhas. Deep-sea Res., SS, 27-42. Rossby, C.G., 1948: On displacements and intensity changes of atmospheric vortices. J. Mar. Res, 7 , 175-187. Sommerfeld, A., 1950: Mechanics of defomable bodies. Academic Press, New York, 396 pp. Smith, D.C, IV, 1986: A numerical study of Loop Current eddy interaction with topography in the western Gulf of Mexico. J . Phys. Oceanogr., 16,1260-1272.

357

T H E ROLE OF MESOSCALE TURBULENCE IN T H E AGULHAS CURRENT SYSTEM J.R.E. LUTJEHARMS National Research Institute for Oceanology, CSIR, P.O. Box 320, Stellenbosch 7600, South Africa Abstract

The terminal region of the Agulhas Current has for some time been recognised Jor iis particularly intense mesoscale turbulence field. This has been evident f r o m statisticai analyses of historic hydrographic data, satellite altimetric data as well eddy kinetic energy as measured by free-drifting

as the geographic distribution of

weather buoys.

This area of higher

ihan

average mesoscale variability is geographically extensive. Investigations in the area, making use of salellite remote sensing as well as dedicated research cruises. have identified a number of regional mechanisms involved in the generation of mesoscale vortices. Amongst these are instabilities in ihe Agulhas relrofleciiori which lead to the budding off of Agulhas rings.

These rings have been observed to drijt into the south-easi

Atlantic Ocean and may dominate the heat and momentum f l u x of a large part of ihis ocean. Perturbations in the this front.

Subtropical Convergence have been observed to cause eddy formation ai

Fluctuations in the Rossby waves

oii

the

Agulhas

Return Current

have

been

identified as being responsible for the shedding of both warm and cold eddies. This eddy spawning across the Subtropical Convergence may be held responsible f o r substantial meridioiial heat f l u x into the Southern Ocean. I t is further conjectured that the Subaritarciic Front south o f Africa may in fact be maintained by a balance in ihe meridional f l u x of such warn7 SUhtI'(Jpicd and cold Antarctic eddies. Recent observational results have suggested that circulation features in the sub-mesoscale range may act as triggering mechanisms for some of the larger fluctuaiions in the f l o w patterns. implying that modelling of the system will have to take cognizance of features on these finer scales. INTRODUCTION Sometimes the fabric of science is so well-prepared for a new advance that its general acceptance occurs more or less unconsciously.

Such perhaps has been the case with the

establishment of the distinct and persistent geographic patterns in the distribution of mesoscale turbulence in the ocean. With the advent of serious studies on mesoscale eddies (e.g.

The MODE Group, 1978;

Robinson, 1983) Wyrtki e t al. (1976) made use of all available historic ships' drift data in an attempt to determine the distribution of energy due to the mean flow and the kinetic energy due to fluctuations, which they interpreted as the eddy kinetic energy. They were the first to show that the distribution of this variable revealed distinct maxima in geographic areas

358 consistent with the location of the major western boundary currents such as the Gulf Stream, the Kuroshio, the Agulhas Current and the Brazil/Falkland confluence.

South of 40" S the

dearth of data prevented statistically reliable analysis. This global pattern has subsequently been confirmed by a number of studies, in particular altimetric satellite data (Figure I).

The

consistency of the results, notwithstanding the entirely different data base, confirms this portrayal as a persistent one.

This discovery is an important milestone in global ocean

hydrodynamics which deserves greater recognition that it has perhaps received.

Figure 1 Global distribution of intensity of mesoscale turbulence as observed in the varratice of alfimefer. mensuremetils accorditig 10 data f r o m lhe Srasal salellite (according to Cheney el al.. 1983).

In both these results the intensity of mesoscale variability in the region of the Agulhas Current termination stands out as particularly intense. This has subsequently been confirmed by a number of studies using historic hydrographic data (Lutjeharms and Baker, 1980), satellite thermal infrared radiances (Lutjeharms and

van Ballegooyen,

1984), satellite altimetric

measurements (Gordon et al., 1983; Cheney et al., 1983) and the drift records of free-drifting weather buoys forming part of the First Global C A R P Experiment of 1978-79 (e.g. Piola e t al.,

1987).

In all cases where spatial resolution was sufficient

Agulhas Current is noted for high variability.

the total southern area of the

A tongue of high variability is also seen to

extend from here eastward into the southern Indian Ocean, along the mean path of the Agulhas Return Current, or the Subtropical Convergence (Lutjeharms and Valentine, 1984). The core,

359 and most intense part of the variability, seems always to lie just north of 40" S and between 15" and 2 0 " E , i.e. at the location of the Agulhas Current retroflection (Lutjeharms and van Ballegooyen, 1988a). This extensive region of intense mesoscale activity south of Africa (Griindlingh, 1983b) may present a very propitious and ideal region for studying the interaction between major ocean circulation systems and the generation of mesoscale turbulence. Because of the extreme horizontal thermal gradients exhibited at the sea surface (Lutjeharms and Valentine, 1984) the disposition of

circulation

elements

is

easily

variability in this area may furthermore mesoscale coherent structures.

discerned.

single it

The

extreme intensity of the

out as particularly suitable for studying

A general description of

the

characteristic circulation

elements of the area are probably called for at this point. Two outstanding circulation features of the area are the

Agulhas Current

and the

Subtropical Convergence. The Agulhas Current, flowing poleward along Africa's east coast reaches

its full stature at about 30" S although there may be intermittent

inflow from

the

east due to the Mozambique Ridge Current (Grundlingh, 1984, 1985a) downstream from here. U p to the latitude of

(35" S )

Port Elizabeth

the Current tends to

slope closely (Griindlingh, 1983a). Downstream of here,

adjacent to

follow the

the

wide

continental continental

shelf of the Agulhas Bank, lateral meanders in the course of the Current occur (Lutjeharms, 1981a) with

dimensions

southernmost tip of the

120 km

up to continental

(Lutjeharms,

boundary, the

1981b).

Agulhas

On overshooting the

Current retroflects sharply to

flow back into the South Indian Ocean as the so-called Agulhas Return Current (Harris, 1970). For the most part this return current closely follows the Subtropical Convergence in its eastward

not all the water is returned to the Indian

progression. However,

Ocean.

A

substantial proportion may be lost to the South Atlantic Ocean by the formation of Agulhas rings (Gordon et al., 1987). On

flowing eastward, the Agulhas Current tranverses a number

topographic

features such

as

the

of substantial bottom

(at 40" S , 26" E)

Agulhas Plateau

and the Mozambique

Ridge extension at 40" S, 3 8 " E, which cause Rossby waves in the current (Harris, 1970). All the circulation features described here have been observed in hydrographic

data

(Bang, 1970), in the drift patterns of weather buoys (Grundlingh, 1977, 1978) and in satellite imagery (Lutjeharms, 1981a). In combining these results (Griindlingh and Lutjeharms, it becomes clear that most of

them

1979)

form a generic part of the circulation system and are

present for most of the time.

AGULHAS CURRENT RETROFLECTION REGION TURBULENCE The coastal, and therefore more accessible, part of the been quite intensively investigated (e.g. Pearce, stable (Grundlingh,

1983a)

while fluctuations in

velocity (Pearce and Grundlingh, the data presently available.

1977).

northern Agulhas

Current

has

Its course has been shown to be

its

transport (Grundlingh,

1980) and

1982) were shown to be statistically insignificant, based on

It has, however, been suggested that on occasion there may

be a major current flowing parallel

to

the usual

location

of the Agulhas Current

proper,

360 the Mozambique Ridge Current (Griindlingh, 1985a). As its name implies, as

flowing

poleward

along

the

1985b,

Mozambique

1988b;

Ridge,

Griindlingh

shedding

(Griindlingh,

1984,

(Griindlingh,

1988a) at its suggested termination at about 32" S.

it

is conceived

mesoscale

eddies

and Pearce, 1984) and eddy-pairs The drift directions and

contribution of these eddies to the general mesoscale environment is not yet known. Agulhas Current exhibits

of Port Elizabeth the landward border of the

Downstream

meanders with a range of dimensions. (Lutjeharms, 1981a).

In general their amplitudes increase downstream

These are strictly shear edge perturbations.

shedding such as is observed in the Gulf

No looping

or ring-

Stream has been observed. Analysis of long

records of satellite imagery for the area (Lutjeharms et al., 1988) suggests that there are two preferred geographic areas along the continental shelf the meanders, form. The

where eddies, associated

with

mesoscale structures on the landward edge of the Current have two

possible effects on water overlying the adjacent shelf. First, the spread of warm surface water may enhance

by plumes associated with the meanders (Swart and

Largier, 1987).

the seasonal thermocline from above

Second, seasonal injection of cold water into the

bottom layers

Orren, 1985) and may be due to the

over the Agulhas Bank has been observed (Eagle and

upwelling action at the cores of the shear edge eddies associated with the meanders. The meanders continue downstream where they probably have a significant influence on the Agulhas Current retroflection.

This sharp

turnabout of

the Current,

first

hydrographically measured and described in detail by Bang (1970), seems by all accounts to be unstable.

On

occasion

(Lutjeharms, 1981a;

it will coalesce and

Lutjeharms and

bud

Gordon,

off

1987).

an independently These

circulating ring

rings are

of

the

most

energetic observed in the world ocean (Olson and Evans, 1986) and may carry substantial amounts of

Indian Ocean water into the Atlantic (Gordon, 1985; Gordon e t al., 1987). In

his perceptual model identified

this

of the

global

flux

of

ring shedding process at

intermediate

the

Agulhas

depth

water

Gordon

Current retroflection

(1986)

as

a key

component of water mass balance of the world ocean. Subsequent studies have shown that the Agulhas retroflection loop exhibits sequence of progradations into

the

retroflection loop

at

to again

progradation (Lutjeharms

lie

and van

South Atlantic, about

20" E)

Ballegooyen,

and

1988a).

a consistent

ring shedding (which causes the subsequent

renewed

These events

westward

occur on average

about 9 times per year, suggesting that on average 9 rings per year may be shed. Not all rings, however, may be of the same extent and intensity as those described by Olson and Evans (1986).

Modelling of

Chassignet,

the

retroflection,

and particularly

of

ring

shedding (Boudra

and

1988; Boudra and de Ruijter, 1988; Chassignet and Boudra, 1988) has become

increasingly realistic and closer to that observed in nature. An early symptom of incipient ring-shedding is the formation of a wedge subantarctic surface water Ballegooyen,

1988a).

at

the

On satellite

Subtropical Convergence

(Lutjeharms

imagery this wedge is observed to penetrate

during the shedding of an Agulhas ring.

It is believed that this wedge

is

the

of

cold

and

van

the loop result of

361

opposing currents within the Agulhas Retroflection loop.

Such opposing currents may in

turn be triggered by incipient instabilities forming the initial stages of ring separation. Important

to

a better understanding of the role

Agulhas

Ocean, and thus in global ocean climate, is an estimate of

rings play in the Atlantic

their lifetime and

their

final

Based on estimates of the life times of similar Gulf Stream rings (The Ring

distribution.

Group, 1981; Lai and Richardson, 1977) a duration of 2 years would not seem unrealistic. At the Gulf Stream,

worm-core rings have been observed to drift in a south-westerly

and

merge with the Gulf Stream

to

again

somewhere

termination of rings' existence in this manner has Current.

along

its length.

direction The early

not, as yet, been observed in the Agulhas

Whereas the full length of the Gulf Stream is potentially available for such mergers,

only a small part of the Agulhas retroflection loop protruding from between the southern tip of

Africa

and

the Subtropical Convergence is available for reabsorption of

thus unlikely that such mergers would

occur.

One

may assume that

rings.

It

is

in general Agulhas

rings spin down by mixing with their general environments. The recent observations of what might

have been the remnants of an Agulhas ring at 23" S

McCartney, personal communication) suggests

a substantial

in

the South Atlantic (M.

lifetime and a wide range of

distribution for such features. Recent analyses (Lutjeharms, 1988) show that the recently produced rings may

penetrate as

far

west

Agulhas retroflection and its most The distribution of

as 8 " E.

warm

Agulhas rings distinguishable by their surface thermal expression is portrayed in Figure 2. It shows the relatively invariant location of Current,

the

the landward

border of the northern Agulhas

preferred location of the Agulhas Return Current/Subtropical Convergence as

well as two seemingly preferred locations for the eastward edge of the Agulhas retroflection loop, one at 20" E and the other at 15" E. Surrounding the retroflection region is a wide border of rings and eddies.

It is clear that the Agulhas retroflection

mesoscale features radiating in a range of directions. in

this area

averages

400

W/m

(Walker and Mey,

assume that most features of this nature would expression rapidly and

is the origin of a range of

Since the heat loss to the atmosphere 1988)

one

may

with confidence

loose their distinguishing surface thermal

that therefore the area of influence of these features as described by

satellite infrared imagery (Figure 2) is a gross underestimate.

During a recent cruise in the

area (Lutjeharms, 1987b) a number of circular mesoscale features in the area were traversed and delineated by hydrographic measurements.

It is hoped that analysis of these results

will help in determining the mixing rates for characteristic mesoscale features of this kind and thus establish their estimated lifetimes. A

close scrutiny of detailed satellite infrared imagery

demonstrated that Agulhas rings are probably present 30% of

the

of

the area

has

in fact

south west of Cape Town at least

time (Lutjeharms and Valentine, 1988~). It was only feasible to

make this

estimate by the recurrent presence in the specific area of Agulhas shear edge filaments which encircle subsurface Agulhas rings. These rings would otherwise be indistinguishable from their surroundings at the sea surface having lost their characteristic warmsurface expression. This process is very similar to that described by Nof (1986) for rings colliding with the Gulf Stream.

362 Due to the fortuitous, tell-tale presence of Agulhas filaments in this particular area one may arrive at estimates of the prevalence of rings here.

It may

possibly be representative of all

other areas equidistant from the Agulhas retroflection locus where such revealing surface features are not in evidence.

Figure 2 The distributiori of superimposed sea surface temperature fronts i n the southerri Agulhas Currelit system for the period December 1984 to December 1985. Data is from the radiometer 011 board the satellite METEOSAT I I . ( A f t e r Lutjeharms and vari Ballegooyen. 1988a)

SUBTROPICAL CONVERGENCE COHERENTSTRUCTURES In the distribution of eddy kinetic energy, of sea level variance and of surface temperature variability, discussed above, the area of

very high values at the Agulhas

retroflection trails a long plume of raised values stretching into the South Indian Ocean (e.g. Patterson 1985; Daniault, 1985; Piola et al., 1987). This plume follows the mean location of the Subtropical Convergence in this area as observed in surface readings (Lutjeharms and Valentine, 1984) in expendable bathythermography (Lutjeharms, 1985a) and in the drift of free-drifting weather buoys (Hoffman, 1985). One of the distinguishing characteristics of the Subtropical Convergence is an extreme surface thermal gradient exceeding, at times, 0,15" C/km (Lutjeharms and Valentine, 1984). This is reflected to a lesser degree in density and dynamic topography gradients.

Lateral

meanders of this front will therefore be portrayed in, for example, altimetry analysis as high mesoscale variability. The course of the Agulhas Return Current on the whole closely follows that of the Subtropical Convergence (Grundlingh, 1978; Lutjeharms and Valentine, 1984). The

363 fact that the tongue of very high variability ends about one third of the way across the South Indian Ocean suggests that the interaction between the Agulhas Return Current and the Subtropical Convergence is important for generating the high level of mesoscale dynamic variability. A large number of hydrographic sections taken across the Subtropical Convergence south of Africa have, however, shown that not only lateral meandering is the cause of variability but that intense eddies are also formed here (Lutjeharms and Valentine, 1988a).

The formation

mechanisms of such frontal eddies has been described in detail for the Antarctic Polar Front (Joyce and Patterson, 1977; Petersen e t al., 1982).

Eddies observed adjacent the Subtropical

Convergence may be either cold- or warm-core.

No preference has been observed.

The

distribution of such eddies relative to the location envelope for the Convergence south of Africa is portrayed in Figure 3.

Observations on such features have heretofar been entirely random

and dependent on tracks of ships of opportunity.

Because of their very distinctive initial

surface expressions, eddies of this type are ideal subjects for study by thermal infrared remote sensing.

Figure 3 Distribution of four distrricl e d d y types. categorised according to lateral Subtropical Convergence south o f Africa. Circles are Agulhas Retroflectioti retort-shaped frontal eddies. Open triangles are cold eddies associated filled-in lriarigles are cold frontal eddies associated wilh perturbalions Convergence. ( A frer Lutjeharms and Valentine, 19886)

morphology. at the eddies. dots warni, with retort-eddies: in the Subtropical

Careful scrutiny of satellite imagery for the area extending over a period of ten years has shown that eddies are preferentially formed in certain distinct geographic areas (Lutjeharms and Valentine, 1988b). Eddies for each area exhibit very characteristic lateral morphologies i.e. on

364 satellite imagery they are clearly distinguishable by their shape as belonging to a class (Figure

3). The first class consists of eddies formed at the Agulhas retroflection and advected southward across the Subtropical Convergence (see e.g. Lutjeharms, 1987a). These eddies may possibly represent most of the features observed south of the retroflection in Figure 2. They are usually round or slightly elliptical and are, as far as can be ascertained by sequential satellite images, not formed as rings from occluded Agulhas retroflection loops. The few that have been observed for extended periods did not exhibit rapid drift in any particular direction (Lutjeharms, 1987).

Coming from a warm oceanic environment into the Subantarctic with a

much lower ambient temperature these eddies represent a substantial meridional heat transport. Downstream of the retroflection, at the Agulhas Plateau and immediately to the east of this relatively shallow area, a dense clustering of warm eddies is observed (Figure 3). A Rossby wave is formed over the Agulhas Plateau as the Agulhas Return Current moves across it (Harris, 1970, Griindlingh and Lutjeharms, 1979). Upstream and in the lee of the wave, pools of warm water are occasionally enclosed by protrusions of cold Subantarctic Surface Water. These eddies then exhibit a very characteristic retort shape. As they drift southward the connecting band of cold water becomes increasingly longer until it extends over about 150-200 km with an estimated width of only about 10 km (Lutjeharms and Valentine, 1988a. 1988b).

No eddies

observed south of the Agulhas retroflection, nor cold eddies yet to be discussed, have this unusual retort shape.

Very few have been observed for more than a short period.

No

statistically reliable evidence is therefore available on their subsequent drift or disposition. Two perturbations in the Subtropical Convergence bud off cold mesoscale features (Figure

3). The one is the large planetary wave over the Agulhas Plateau (Lutjeharms and Valentine, 1988 a,b), the other the smaller eastward protrusion which encloses the warm eddies mentioned previously. Each occasionally lengthens and forms a cold coherent feature.

These also loose

their distinguishing surface expressions within weeks (Lutjeharms and Valentine, 1988a). The extent of their geographic distribution depicted in Figure 3 is therefore probably an underestimate since it represents the surface expression of the features only as seen in thermal infrared satellite imagery. The geographic distributions of each of the type of eddies associated with the Subtropical Convergence is, according to present data, quite distinct. Their surface morphologies are very characteristic and closely tied to the specific areas of distribution of each. It may therefore be assumed with some degree of confidence that the controlling dynamics is area specific. This requires further and more detailed investigation. SUBANTARCTIC FRONT SUSTENANCE The lifetime of the distinctive surface expressions of warm eddies cast off from the Subtropical Convergence are not in general sufficient to follow individuals from satellite thermal infrared and to establish either their drift rates, drift patterns o r distribution range. A number of warm eddies have, however, been found far south of the Subtropical Convergence (e.g. Lutjeharms and McQuaid, 1986; Lutjeharms 1985b). One may assume that such eddies with a

365 southerly drift component represent a significant proportion of the eddies shed at the Convergence.

It is conceivable that they may eventually even reach the other major front of

the area, the Antarctic Polar Front (Deacon, 1933, 1937). South of Africa the locations of these two fronts is now fairly well established (Hoffman, 1985; Lutjeharms, 1985a).

It has also been noted that an additional front, the Subantarctic

Front, is found here and has much the same characteristics as its counterpart observed in the Drake Passage (Sievers and Emery, 1978). Whereas the dynamics of the more classical fronts such as the Subtropical Convergence and the Antarctic Polar Front are fairly well understood (Deacon, 1933, 1937; Baker et al., 1977; Whitworth, 1980) that of the Subantarctic Front is not. Hoffman (1985) has surmised that it may be caused by a horizontal flow convergence.

This

may be due to a minimum in the average zonal wind stress. The results presented thus far are suggestive but not entirely convincing. South of Africa the Subantarctic Front exhibits a very characteristic and unusual surface thermal expression.

I t has been noted that in about all instances during which measurements

were made crossing the front a sharp, but localised, increase in temperature was observed just north of the front and a similar but inverse decrease in temperature was noted just south of the front (Lutjeharms and Valentine, 1984).

It has, in fact, been demonstrated (H.R. Valentine,

personal communication) that surface thermograph traces are sufficient to unequivocably establish the location of the Subantarctic Front as confirmed by depth readings, at least south of Africa. The causative mechanism for this curious surface expression has not been established but the presence and possible drift patterns of mesoscale eddies suggest a possible one. The temperature section to about 800111 depth portrayed in Figure 4 was obtained by expendable bathythermographs south of Africa. It shows the Subantarctic Front, being the most vertically oriented isotherm within a subsurface gradient between 3 " and 5" C at about 46" S. O f particular interest is the eddy pair adjacent to the front itself, namely a warm eddy to the north and a cold eddy to the south made evident by the sharp bulgings in the isotherms.

Although

not observed in this distinct fashion on all crossings of the front, finding single eddies or eddy pairs adjacent to the front occurs sufficiently often to strongly raise the suspicion that they may be involved in establishing and maintaining the front itself. The hypothesis being put forward now is that that proportion of the warm eddies formed at the Subtropical Convergence that exhibit a southward drift component as well as that proportion of the cold eddies formed at the Antarctic Polar Front (Joyce and Patterson, 1977) which exhibit a net northward drift, establish and sustain the Subantarctic Front. The location of the front may therefore represent the line at which a balance is maintained between warm and cold mesoscale turbulent flux.

The distances of this front from the Subtropical

Convergence and the Artarctic Polar Front respectively may, based on this supposition, conceivably be a measure of the relative intensity of warm and cold eddy fluxes. Significant temporal shifts in the front's location may represent climatic changes in these fluxes. This is but one example of mesoscale turbulence sustaining larger and more enduring hydrodynamic features.

366

-Om

-100

-200

-300

-400

-500

I

Subontarctlc Front

Figure 4 Thermal section across the Suhartiarctic Froni south of Africa. The location 01 the Subaniarcilc Frortl is indicated. A warm edd.v lies to the rrorth and a cold eddy to the south 01 [he J r ~ r i t . (Ajter Lutjeharms. 1986)

SUBMESOSCALE TRIGGERING MECHANISMS The question arises whether most oceanic circulation systems can be adequately described and sufficiently modelled on the scales of mesoscale coherent features such as have been described above for the Agulhas Current and for the Subtropical Convergence.

Experience

suggests that submesoscale features may play crucial roles in an important part of the Agulhas dynamics. A case in point is the role of the Natal pulse. The Natal pulse is the collective name given to extreme, solitary meanders that occur on the southern reaches of the Agulhas Current (Harris et al., 1978; Grundlingh , 1979). Their genesis usually takes place in the Natal Bight, a small offset in the coastline north of Durban (Lutjeharms and Roberts, 1988) whence they progress downstream growing to features in excess of 250 km and thus forcing the Agulhas Current far offshore and probably triggering or dampening ring shedding at the Agulhas retroflection.

It is believed that small lee eddies

known to form in the Natal Bight (Grundlingh, 1974; Malan and Schumann, 1979), similar to these formed in the Delagoa Bight (Lutjeharms and Jorge da Silva, 1988), are dislodged by a

367

process of vortex shedding and thus initiate the Natal pulse. As these eddies move downstream on the landward side of the current their lateral dimensions increase considerably. The causes and mechanisms of vortex shedding in the Natal bight is unknown.

Lutjeharms and Roberts

(1988) have suggested that the collision of offshore eddies with the Agulhas Current at the Natal

bight might trigger the dislodging of an eddy. An extreme meander, similar to a Natal pulse, has been described by Griindlingh (1986). A feature of this meander was the presence of an attendant mesoscale eddy seaward of the meander.

Figure 5 Conceptual portrayal o f mesoscale triggering mechanisms in the southern Agulhas Current circulation, such as the Natal Pulse and possible feedback mechanisnis from the Agulhas Return Current. Numbered features are: I , Nalal Bight lee eddy: 2. Nalal pulse: 3. cold eddies rhed f r o m the Subiropical Convergeiice ai 4 , the planeiary Rossby wave over the Agulhas PLaieau: 5 . cyclonic eddies of the Mozambique Ridge Current. ( A f t e r Boden. Duricombe Rae atld Lutjeharms. 1988 and Griindlingh, 19886)

368 The origin of such offshore eddies might be varied.

Mozambique Ridge Current eddies

(Griindlingh, 1984; Grundlingh, 1985b; Grundlingh and Pearce, 1984) or cold eddies originating at the Subtropical Convergence are likely candidates.

If shedding of the latter was precipitated

by a Natal pulse travelling down the Agulhas and Agulhas Return Current, one would have an unusual feedback mechanism with Natal pulses generating Subtropical Convergence eddies and advecting Subtropical Convergence eddies triggering Natal pulses (Lutjeharms, 198 la).

A

conceptual portrayal of such a mechanism is given in Figure 5. The possible influence of Natal pulses on ring shedding has as yet not been established. A different dramatic influence on the whole circulation of the southern Agulhas Current system has. It has been noted (Lutjeharms and van Ballegooyen, 1988b) that on occasion a significant diversion of Agulhas Current water occurs just south of Port Etizabeth. This current bifucation at times resembles an early retroflection.

This location corresponds to a gap in the bottom

topography between the continental shelf and the shallow Agulhas Plateau offshore.

Modelling

the Agulhas Current as an inertial jet stream has been shown to be an effective way of simulating the influence of bottom topography on current behaviour (Darbyshire,

1972;

Grundlingh, 1978, Lutjeharms and van Ballegooyen, 1984). Doing this for an Agulhas Current that is forced 250 km offshore by a Natal pulse at Port Elizabeth causes upstream retroflection due to the presence of the Agulhas Plateau.

This early retroflection, triggered by the Natal

pulse, may cause a dramatically reduced flow downstream, fewer ring shedding events, or alternatively, smaller rings and as a result less leakage of Indian Ocean water into the Atlantic Ocean.

CONCLUSIONS From the above example of the influence of the Natal pulse on the Agulhas Current system it is clear that submesoscale constituents of the circulation system may critically determine the system as a whole, at least on specific occasions.

This has important

consequences for modelling of this and, most probably, other similar systems. Modelling of the Agulhas Current (De Ruijter, 1982; De Ruijter and Boudra, 1985; Ou and de Ruijter, 1986; Boudra and Chassignet, 1988; Chassignet and Boudra, 1988; Boudra and de Ruijter, 1988) is becoming increasingly more sophisticated and detailed.

Many of the

features observed in nature are now accurately simulated in models. The final aim of modelling is prediction.

If submesoscale triggering mechanisms have to be included in at least the

initialisation of models this implies a much finer grid scale than presently being used. It would be surprising if other systems are not influenced by similar submesoscale perturbations.

I t is,

for instance, known that the extent and location of eddies on the Norwegian Coastal Current are critically determined by submesoscale wind stress changes at the Skaggerak (McClimans, 1986).

ACKNOWLEDGEMENTS 1 thank Drs Chassignet and Boudra for useful conversations on aspects of the above work

and Dr Grundlingh for constructive criticism of the manuscript. This paper presents a synthesis of work over several years by Mr. Henry R. Valentine, Mr. Roy C. van Ballegooyen, Dr. Marten

369 L. Grundlingh and the author.

Funding from the South African National Committee for

Oceanographic Research (SANCOR) and the South African Scientific Committee for Antarctic Research

(SASCAR)

through

the

Foundation

of

Research

Development

is

gratefully

acknowledged.

REFERENCES

Baker, D.J., Nowlin, W.D., Pillsbury, R.D. and Bryden, H.L. 1977. Antarctic Circumpolar Current : Space and time fluctuations in the Drake Passage. Nature 268 696-699. Bang, N.D. 1970. Dynamic interpretations of a detailed surface temperature chart of the Agulhas Current retroflection and fragmentation area. S. Afr. Geogr. J . 52 : 67-76. Boden, B.P., Duncombe Rae, C.M. and Lutjeharms, J.R.E. 1988. The distribution of the diatoms of the S.W. Indian Ocean waters between Cape Town and the Prince Edward Island archipelago. S. Afr. J . Sci. 84 : 811-818. Boudra, D.B. and Chassignet, E.P. 1988. Dynamics of Agulhas retroflection and ring formation in a numerical model. 1. The vorticity balance. J. Phys. Oceanogr. 18 : 280-303. Boudra, D.B. and de Ruijter, W.P.M. 1986. The wind-driven circulation in the South Atlantic-Indian Ocean - 11. Experiments using a multi-layer numerical model. Deep-sea Res. 33 : 447-482. Chassignet, E.P. and Boudra, D.B. 1988. Dynamics of Agulhas retroflection and ring formation in a numerical model. 11. Energetics and ring formation. J . Phys. Oceanogr. 18 : 304-319. Cheney, R.A., Marsh, J.G. and Beckley, B.D. 1983. Global mesoscale variability from colinear tracks of SEASAT altimeter data. J. Geophys. Res. 88 : 4343-4354. Daniault, N. and Menard, Y. 1985. Eddy kinetic energy distribution in the Southern Ocean from altimetry and F G C E drifting buoys. J. Geophys. Res. 90 : 1187711889. Darbyshire, J. 1972. The effect of bottom topography on the Agulhas Current. Pure Appl. Geoph. 101 : 208-220. Deacon, G.E.R. 1933. A general account of the hydrology of the South Atlantic Ocean. Discovery Rep. 7 : 171-238. Deacon, G.E.R. 1937. The hydrology of the Southern Ocean. Discovery Rep. 15: 124 PP. De Ruijter, W. 1982. Asymptotic analysis of the Agulhas and Brazil Current systems. J . Phys. Oceanogr. 12 : 361-373. De Ruijter, W.P.M. and Boudra, D.B. 1985. The wind-driven circulation in the South Atlantic - Indian Ocean. - I. Numerical experiments in a one-layer model. DeepSea Res. 32 : 557-574. Eagle, G.A. and Orren, M.J. 1985. A seasonal investigation of the nutrients and dissolved oxygen in the water column along two lines of stations south and west of South Africa. Res. Rep. S. Afr. Coun. Sci. Ind. Res. 567 : 52 pp. Gordon, A.L. 1985. Indian-Atlantic transfer of thermocline water at the Agulhas retroflection. Science 227 : 1030-1033. Gordon, A.L. 1986. Inter-ocean exchange of thermocline water. J . Geophys. Res. 91 : 5037-5046. Gordon, A.L., Horai, K.-I. and Donn, M. 1983. Southern hemisphere western boundary current variability revealed by GEOS 3 altimeter. J . Ceophys. Res. 88 : 775-762. Gordon, A.L., Lutjeharms, J.R.E. and Grundlingh, M.L. 1987. Stratification and circulation at the Agulhas Retroflection. Deep-sea Res., 34(4): 565-599. Grundlingh, M.L. 1974. A description of inshore current reversals off Richards Bay based on airborne radiation thermometry. Deep-sea Res. 21 : 47-55. GrOndlingh, M.L. 1977. Drift observations from Nimbus VI satellite-tracked buoys i n the southwestern Indian Ocean. Deep-sea Res. 24 : 903-913. Grundlingh, M.L. 1978. Drift of a satellite-tracked buoy in the southern Agulhas Current and Agulhas Return Current. Deep-sea Res. 25 : 1209-1224.

370

Boudra, D.B. and de Ruijter, W.P.M. 1986. The wind-driven circulation in the South Atlantic-Indian Ocean - 11. Experiments using a multi-layer numerical model. Deep-sea Res. 33 : 447-482. Chassignet, E.P. and Boudra, D.B. 1988. Dynamics of Agulhas retroflection and ring formation in a numerical model. 11. Energetics and ring formation. J. Phys. Oceanogr. 18 : 304-319. Cheney, R.A., Marsh, J.G. and Beckley, B.D. 1983. Global mesoscale variability from colinear tracks of SEASAT altimeter data. J. Geophys. Res. 88 : 4343-4354. Daniault, N. and Menard, Y. 1985. Eddy kinetic energy distribution in the Southern Ocean from altimetry and FGGE drifting buoys. J. Geophys. Res. 90 : 1187711889. Darbyshire, J. 1972. The effect of bottom topography on the Agulhas Current. Pure Appl. Geoph. 101 : 208-220. Deacon, G.E.R. 1933. A general account of the hydrology of the South Atlantic Ocean. Discovery Rep. 7 : 171-238. Deacon, G.E.R. 1937. The hydrology of the Southern Ocean. Discovery Rep. 15: 124 PP: . De Ruijter, W. 1982. Asymptotic analysis of the Agulhas and Brazil Current systems. J. Phys. Oceanogr. 12 : 361-373. De Ruijter, W.P.M. and Boudra, D.B. 1985. The wind-driven circulation in the South Atlantic - Indian Ocean. - 1. Numerical experiments in a one-layer model. DeepSea Res. 32 : 557-574. Eagle, G.A. and Orren, M.J. 1985. A seasonal investigation of the nutrients and dissolved oxygen in the water column along two lines of stations south and west of South Africa. Res. Rep. S . Afr. Coun. Sci. Ind. Res. 567 : 52 pp. Gordon, A.L. 1985. Indian-Atlantic transfer of thermocline water at the Agulhas retroflection. Science 227 : 1030-1033. Gordon, A.L. 1986. Inter-ocean exchange of thermocline water. J. Geophys. Res. 91 : 5037-5046. Gordon, A.L., Horai, K.-I. and Donn, M. 1983. Southern hemisphere western boundary current variability revealed by GEOS 3 altimeter. J. Geophys. Res. 88 : 775-762. Gordon, A.L., Lutjeharms, J.R.E. and Griindlingh, M.L. 1987. Stratification and circulation at the Agulhas Retroflection. Deep-sea Res., 34(4): 565-599. Griindlingh, M.L. 1974. A description of inshore current reversals off Richards Bay based on airborne radiation thermometry. Deep-sea Res. 21 : 47-55. Grundlingh, M.L.' 1977. Drift observations from Nimbus VI satellite-tracked buoys in the southwestern Indian Ocean. Deep-sea Res. 24 : 903-913. Grundlingh, M.L. 1978. Drift of a satellite-tracked buoy in the southern Agulhas Current and Agulhas Return Current. Deep-sea Res. 25 : 1209-1224. Grundlingh, M.L. 1979. Observation of a large meander in the Agulhas Current. J. Geophys. Res. 84 : 3776-3778. Grundlingh, M.L. 1980. On the volume transport of the Agulhas Current. Deep-sea Res. 27 : 557-563. Grundlingh, M.L. 1983a. On the course of the Agulhas Current. S. Afr. Geogr. J. 65 : 49-57. Griindlingh, M.L. 1983b. Eddies in the southern Indian Ocean and Agulhas Current. In: A.R. Robinson (Editor), Eddies in Marine Science. Springer-Verlag, Berlin, pp. 246-264. Grundlingh, M.L. 1984. An eddy over the northern Mozambique Ridge. S . Afr. J. Sci. 80 : 324-329. Grundlingh, M.L. 1985a. Features of the circulation in the Mozambique Basin in 1981. J. Mar. Res. 43 : 779-792. Grundlingh, M.L. 1985b. An intense cyclonic eddy east of the Mozambique Ridge. J . Geophys. Res. 90 : 7163-7167. Grundlingh, M.L. 1986. Features of the northern Agulhas Current in spring, 1983. S. Afr. J. Sci. 82 : 18-20. Grundlingh, M.L. 1988a. Two contra-rotating eddies of the Mozambique Ridge Current. Deep-sea Res., submitted. Griindlingh, M.L. 1988b. Review of cyclonic eddies of the Mozambique Ridge Current. S . Afr. J . Mar. Sci. 6 193-206.

371 Griindlingh, M.L. 1979. Observation of a large meander in the Agulhas Current. J . Geophys. Res. 84 : 3776-3778. Griindlingh, M.L. 1980. On the volume transport of the Agulhas Current. Deep-sea Res. 27 : 557-563. Griindlingh, M.L. 1983a. On the course of the Agulhas Current. S. Afr. Geogr. J. 65 : 49-57. Grundlingh, M.L. 1983b. Eddies in the southern Indian Ocean and Agulhas Current. In: Springer-Verlag, Berlin, pp. A.R. Robinson (Editor), Eddies in Marine Science. 246-264. Griindlingh, M.L. 1984. An eddy over the northern Mozambique Ridge. S. Afr. J . Sci. 80 : 324-329. Grundlingh, M.L. 1985a. Features of the circulation in the Mozambique Basiii in 1981. J. Mar. Res. 43 : 779-792. Griindlingh, M.L. 1985b. An intense cyclonic eddy east of the Mozambique Ridge. J . Geophys. Res. 90 : 7163-7167. Grundlingh, M.L. 1986. Features of the northern Agulhas Current in spring, 1983. S. Afr. J. Sci. 82 : 18-20. Griindlingh, M.L. 1988a. Two contra-rotating eddies of the Mozambique Ridge Current. Deep-sea Res., submitted. Grundlingh, M.L. 1988b. Review of cyclonic eddies of the Mozambique Ridge Current. S. Afr. J. Mar. Sci. 6: 193-206. Griindlingh, M.L. and Lutjeharms, J.R.E. 1979. Large scale flow patterns of the Agulhas Current system. S. Afr. J. Sci. 75: 269-270. Grundlingh, M.L. and Pearce, A.F. 1984. Large vortices in the northern Agulhas Current. Deep-sea Res. 31 : 1149-1 156. Harris, T.F.W. 1970. Planetary-type waves in the South West Indian Ocean. Nature 227 : 1043-1044. Harris, T.F.W., Legeckis, R. and van Foreest, D. 1978. Satellite infra-red images in the Agulhas Current system. Deep-sea Res. 25 : 543-548. Hofmann, E.E. 1985. The large-scale horizontal structure of the Antarctic Circumpolar Current from FGGE drifters. J. Geophys. Res. 90 : 7087-7097. Joyce, T.M. and Patterson, S.L. 1977. Cyclonic ring formation at the polar front in the Drake Passage. Nature 365: 131-133. Lai, D.Y. and Richardson, P.L. 1977. Distribution and movement of Gulf Stream rings. J. Phys. Oceanogr. 7 : 670-683. Lutjeharms, J.R.E. 1981a. Features of the southern Agulhas Current circulation from satellite remote sensing. S. Afr. J. Sci. 77 : 231-236. Lutjeharms, J.R.E. 1981b. Spatial scales and intensities of circulation in the ocean areas adjacent to South Africa. Deep-sea Res. 28 : 1289-1302. ' Lutjeharms, J.R.E. 1985a. Location of frontal systems between Africa and Antarctica : some preliminery results. Deep-sea Res. 32 : 1499-1509. Lutjeharms, J.R.E. 1985b. The South African SIBEX I cruise to the Prydz Bay region, 1984 : 111. Detail of the upper thermal structure of the Southern Ocean between South Africa and Prydz Bay during March - May 1984. S. Afr. J. Antarc. Res. I5 : 12-18. Lutjeharms, I.R.E. 1986. Die wjsselwerking tussen die Suidelike Oseaan en subtropiese sirkulasiekolke grensend aan Suider-Afrika: 'n Oorsig. S. Afr. T. Antarkt. 16 : 2-8. Lutjeharms, J.R.E. 1987a. Meridional heat transport across the Sub-Tropical Convergence by a warm eddy. Nature 331: 251- 253. Lutjeharms, J.R.E. 1987b. Die Subtropiese Konvergensie en Agulhasretrofleksievaart (SCARC). S. Afr. J. Sci. 83: 454-456. Lutjeharms, J.R.E. 1988. Examples of extreme circulation events of the Agulhas Retroflection. S. Afr. J. Sci. 84 : 584-586. Lutjeharms, J.R.E. and Baker, D.J. 1980. A statistical analysis of the meso-scale dynamics of the Southern Ocean. Deep-sea Res. 27 : 145-159. Lutjeharms, J.R.E. and Gordon, A.L. 1987. Shedding of an Agulhas Ring observed at sea. Nature 325 : 138-140. Lutjeharms, J.R.E. and Jorge da Silva, A. 1988. The Delagoa Bight eddy. Deep-sea Res. 35 : 619-634. Lutjeharms, J.R.E. and McQuaid, L.H. 1986. Changes in the structure of thermal ocean fronts south of Africa over a three-month period. S. Afr. J . Sci. 82 : 470-476.

372 Lutjeharms, J.R.E. and Roberts, H.R. 1988. The Natal pulse; an extreme transient on the Agulhas Current. J. Geophys. Res. 93 : 631-645. Lutjeharms, J.R.E. and Valentine, H.R. 1984. Southern Ocean thermal fronts south of Africa. Deep-sea Res. 31: 1461-1476. Lutjeharms, J.R.E. and Valentine, H.R. 1988a. On mesoscale ocean eddies at the Agulhas Plateau. S. Afr. J. Sci., 84: 194-200. Lutjeharms, J.R.E. and Valentine, H.R. 1988b. The formation of eddies at the SubTropical Convergence south of Africa. J. Phys. Oceanogr., 18: 761-774. Lutjeharms, J.R.E. and Valentine, H.R. 1 9 8 8 ~ . Evidence for persistent Agulhas rings southwest of Cape Town. S. Afr. J. Sci. 84 : 781-783. Lutjeharms, J.R.E. and van Ballegooyen, R.C. 1984. Topographic control in the Agulhas Current system. Deep-sea Res. 31 : 1321-1337. Lutjeharms, J.R.E. van Ballegooyen, R.C. 1988a. The Agulhas Current retroflection. J. Phys. Oceanogr., 18 : 1570-1583. Lutjeharms, J.R.E. and van Ballegooyen, R.C. 1988b. An anomalous upstream retroflection in the Agulhas Current. Science 240 : 1770-1772. Lutjeharms, J.R.E., Catzel, R. and Valentine, H.R. 1988. Border phenomena of the Agulhas Current. Cont. Shelf Res., in press. Malan, O.G. and Schumann, E.H. 1979. Natal shelf circulation revealed by Landsat imagery. S. Afr. J. Sci. 75 : 136-137. McClimans, T.A. 1986. Laboratory modelling of dynamic processes in fjords and shelf waters. In: S. Skreslet (Editor), The Role of Freshwater Outflow in Coastal Marine Ecosystems. Springer-Verlag, Berlin, pp. 67-84. Nof, D. 1986. The collision between the Gulf Stream and warm-core rings. Deep-sea Res. 33 : 359-378. Olson, D.B. and Evans, R.H. 1986. Rings of the Agulhas Current. Deep-sea Res. 33 : 27-42. Ou, H.W. and De Ruijter, W.P.M. 1986. Separation of an inertial boundary current from a curved coastline. J. Phys. Oceanogr. 16 : 280-289. Patterson, S.L. 1985. Surface circulation and kinetic energy distributions in the.southern hemisphere oceans from F G G E drifting buoys. J. Phys. Oceanogr. 15 : 865-884. Pearce, A.F. 1977. Some features of the upper 500m of the Agulhas Current. J. Mar. Res. 35 : 731-753. Pearce, A.F. and Grundlingh, M.L. 1982. Is there a seasonal variation in the Agulhas Current? J. Mar. Res. 40 : 177-184. Petersen, R.G., Nowlin, W.D. and Whitworth, T. 1982. Generation and evolution of a cyclonic ring Bt Drake Passage in early 1979. J. Phys. Oceanogr. 12 : 712-719. Piola, A.R., Figueroa, H.A. and Bianchi, A.A. 1987. Some aspects of the surface circulation south of 2 0 " s revealed by First C A R P Global Experiment drifters. J Geophys. Res. 92 : 5101-5114. Robinson, A.R. (ed.) 1983. Eddies in Marine Science. Springer-Verlag, Berlin, 609 pp. Sievers, H.A. and W.J. Emery 1978. Variability of the Antarctic Polar Frontal Zone in the Drake Passage - summer 1976-1977. J. Geophys. Res. 83 : 3010-3022. Swart, V.P. and Largier, J.L. 1987. Thermal structure of Agulhas Bank water. In: A.I.L. Payne, J.A. Gulland and K.H. Brink (Editors), The Benguela and Comparable Ecosystems. S. Afr. J. Mar. Sci. 5: 243-253. The Mode Group 1978. The Mid-Ocean Dynamics Experiment. Deep-sea Res. 25 : 859-910. The Ring Group, 1981. Gulf Stream cold-core rings : their physics, chemistry and biology. Science 212 : 1091-1100. Walker, N.D. and Mey, R.D. 1988. Ocean/air fluxes within the Agulhas Retroflection region. J. Geophys. Res., in press. Whitworth, T. 1980. Zonation of the Antarctic Circumpolar Current at Drake Passage. Deep-sea Res. 27 : 497-507. Wyrtki, K., Magaard, L. and Hagar, J. 1976. Eddy energy in the oceans. J. Geophys. Res. 18 : 2641-2646.

373

MODELLING THE VARIABILITY J NTHE SOMALI C U R R E " Mark E. Luther and James J. O'Brien The Florida State University Mesoscale Air-Sea Interaction Group, Tallahassee, FL 32306-3041

ABSTRACT A numerical model of the wind driven circulation in the Indian Ocean is used to study the variability of the circulation on seasonal and interannual time scales. The model is a nonlinear reduced gravity model driven by observed winds. Model simulations use a monthly mean climatology of ships' winds a s forcing and the 23 year long monthly mean Cadet and Diehl winds a s forcing, The model is very successful in simulating the observed features of the circulation in this region, such a s the formation and decay of the two-gyre system in the Somali Current during the southwest monsoon and the formation of the eddies off the coasts of Oman and Yemen. Examination of model statistics from many years of simulation using climatological monthly mean winds shows that the model fields are exactly repeating from one year to the next over most of the basin, even in the highly nonlinear eddies like the great whirl. Exceptions occur in the smaller scale eddies that form in the strong shear zones around the great whirl and in the southern gyre recirculation region, where the flow field exhibits a more chaotic nature, but even these features are nearly repeating from one year to the next. When observed, interannually varying winds are used to drive the model, the variability from year to year increases dramatically. This indicates that interannual variability in the model fields is due solely to variability in the winds and not due to inherent variability in the model physics, a s is seen in mid-latitude models of the oceanic general circulation.

WCKGROUND Numerous modelling efforts have sought to explain the observed flows in the tropical Indian Ocean, with particular attention given to the semi-annual reversals in the Somali Current along the east coast of Africa (e. g. Cox, 1970, 1976, 1979; Hurlburt and Thompson, 1976; Lin and Hurlburt, 1981; Luther and O'Brien, 1985; Luther et al., 1985; see Luther (1987) or Knox (1987) for a review). While the Somali Current is similar to mid-latitude western boundary currents in some respects, it is unique in many others. The most striking feature of this current is its reversal with the changing monsoon winds (see Schott, 1983; Knox, 1987). During boreal summer, the boundary current flows toward the north and northeast from the coast of Mozambique (11's)to the island of Socotra at 12"N, driven by the southwest monsoon winds. A "two-gyre" system is often observed in this current (Swallow and Fieux, 19821, with a southern gyre that straddles the equator, flowing offshore a t 3-4"N, and a northern gyre, called the great whirl, between 5 ' N and 9"N. Wedgeshaped areas of cold, upwelled water are found along the coast t o the north of these gyres. Late in the summer, the southern gyre and its cold wedge migrate rapidly

374

northward and coalesce with the great whirl (Brown et al., 1980; Evans and Brown, 1981; Swallow et al., 1983). This appears to be the case in most years; however, there is evidence of some years when the two-gyre system does not form (Swallow and Fieux, 1982). The summer Somali Current to the north of 2-3"s gradually breaks up during the fall transition period, and is replaced by the southwestward winter Somali Current with the onset of the northeast monsoon in December. The boundary current to the south of 3"sflows to the north throughout the year and is called the East African Coastal Current (EACC). During the summer, the EACC feeds the northward Somali Current; during winter, i t meets the southwestward Somali Current and both flow offshore into the South Equatorial Counter Current (SECC). Woodberry et al. (1989) show that this region of the EACC is a tropical analog to a mid-latitude western boundary current recirculation region that is strongly modified by the presence of the equatorial wave guide and by the seasonal reversals in the wind. It closes the circulation in a tropical Sverdrup-like gyre in the southern hemisphere, consisting of the eastward SECC that meanders between the equator and 8"s and the westward South Equatorial Current (SEC) between 10 and 20"s. We use "Sverdrup-like" to describe this gyre because it is far from steady state; indeed, it has large seasonal variability (Schott et al., 1988; Swallow et al., 1988; Woodberryet al., 1989). In this paper, we will show that the model response to observed winds is largely deterministic, with only limited regions where the response appears to be chaotic. It then follows that interannual variability in ocean fields seen in the model is due to variability in the wind fields, rather than to inherent variability contained in the dynamics of the ocean itself. We begin by briefly describing the model and then summarizing some results from recent calculations. We describe the interannual variability in the summer Somali Current from a 23 year simulation and its relationship to Indian monsoon rainfall. Next, we describe results from a n 18 year simulation using climatological monthly mean winds. We then present statistical fields from these two simulations and identify regions where the ocean's response is largely deterministic and where it exhibits a chaotic nature. 2 "HEMODEL

The model used here is that of Luther and O'Brien (1985). It is a nonlinear reduced gravity model forced by observed winds. The advantage to such a model is ite inherent simplicity. As demonstrated by Lin and HurIburt (1981),this is the simplest model that contains the necessary physics to reproduce the observed eddy patterns in the Somali Current. Luther and O'Brien (1985) and Luther et al. (1985) show that the model faithfully reproduces most of the observed features of the

375

seasonal cycle of the northwest Indian Ocean circulation, such as the formation and coalescence of the two-gyre system during the southwest monsoon, the formation and decay of the energetic eddy field off the Arabian Peninsula during the fall transition and the formation of the southwestward Somali Current with the onset of the northeast monsoon. Simmons et al. (1988) show that the model can accurately simulate the features observed in a particular year by comparing model fields driven by the observed winds for 1985 with extensive observations taken off the coasts of northern Somalia and the Arabian Peninsula during the fall of that year. Because the model physics are well known and the model fields are more easily analyzed, and because it reproduces many of the important features of the observed flows in this region, this model is ideal for use in long term, multi-year integrations, to assess the importance of interannual and seasonal variability in the region. Results from two versions of the model will be discussed here. The first is a limited area version that covers the northwest portion of the Indian Ocean from 10"sto 26"N and from 40"E to 74"E at a resolution of 118 degree zonally and 114 degree meridionally. The second version covers the entire Indian Ocean basin from 35"E t o 120"E and from 25"s to 26"N at a resolution of 0.1 degree in both directions (Fig. 1). We will call these the limited area model and the full basin model, respectively. Both versions employ open boundaries along the south and along a portion of the east boundaries. The free parameters in the model are the initial upper layer thickness, Ho, the wind stress drag coefficient, Cd, and the

Fig. 1: Model Geometry. The land boundaries are closed, no-slip boundaries. The southern boundary and a portion of the eastern boundary are open boundaries. The shallow banks located along the Seychelles-Mauritius Ridge, around Socotra, the Maldives and the Chagos Archipelago that are less than 40m deep are land boundaries in the model.

376

Laplacian eddy viscosity coefficient, A,. For the results presented here, the limited area model uses Ho = 200 m, cd = 1.25 x 10-3, and A, = 750 m2s-1; the full basin model uses HO= 200 m, Cd = 1.5 x 10-3, and A, = 750 m2s-I. These models have been run for over 200 years of integration using various wind data sets as forcing, including interannual simulations using ship observed winds from 1954 to 1986 and long, multi-year simulations using climatological winds. Earlier model simulations have used a monthly mean climatology of ships' winds as forcing (Luther and O'Brien, 1985) and a monthly mean of the FGGE 1000 mbar winds as forcing (Luther et al., 1985). Our longest continuous interannual simulation covers the period 1954 through 1976 using the ship winds analyzed by Cadet and Diehl(1984).

3.1 Interannual Variability in the Somali Cumnt A great deal of interannual variability is seen in the model fields from the 23 year simulation using the Cadet and Diehl (1984) ships' winds as forcing. This variability can be attributed directly to variability in the wind fields. The two-gyre system is clearly present in all but two of the 23 years (Fig. 2), but the strength, location and timing of formation and collapse of the gyres vanes from year to year. In the two years when the two-gyre system is apparently absent ('72 and '731, the northern gyre, or great whirl, is present but the southern gyre is absent or at least very weak. Of the 21 years when there was clearly a two-gyre pattern, the northern and southern gyres coalesced in July-August in 14 years ('55, '56, '58, '60, '61, '62, '63, '64, '66, '68, '69, '70, '71, '74). In the other 7 years, a blocking flow formed to the north of the southern gyre a t about 2"N and prevented it from migrating northward. In these years, the along-shore winds near the equator were anomalously strong. In 5 of these years ('54, '57, '59, '65, '75) smaller eddies formed between the southern gyre and the great whirl and then coalesced with the great whirl. In 1967 and 1968 the great whirl was anomalously weak while the southern gyre was anomalously strong, and no coalescence was observed in the model fields. 3.2 Indian Monsoon Rainfall In a cooperative study with S. K. Dube of the Indian Institute of Technology, Delhi, we investigated the relationship between the interannual Variability in the model fields and variability in Indian monsoon rainfall (Dube et al., 1989). We found that the period 1954 to 1966 is a period during which the southern gyre of the Somali Current is generally stronger, the cross-equatorial winds are stronger and the upper layer is thinner (the thermocline is shallower) in the central Arabian

377

Upper Layer Velocity

Upper Layer Thickness

10 N

ION

5N

5N

EQ

EQ

August 16. 1958

a

4; E

5iE

5SE

45 E

Upper Layer Thickness

WE

55E

Upper Layer Velocity

August 16. 1972

b

I

rJ

I

I

I

45 E

WE

55E

Fig. 2: Two very different circulation patterns for mid-August from the Somali Current region of the model. (a) In 1958, there was a pronounced two-gyre system, that collapsed in late August. In this image, the southern gyre is just beginning to coalesce with the great whirl, while another clockwise eddy has formed a t the equator. This was also a year of very good Indian monsoon rainfall. (b) In contrast, the circulation pattern for the same period in 1972 shows no evidence for a southern gyre, although there is some smaller scale eddy activity to the south of the great whirl. This year was one of the worst Indian monsoons in recent decades.

378

Sea, implying that sea surface temperatures are lower. This period was identified by Cadet and Diehl (1984) as one of generally higher Indian monsoon rainfall. In contrast, the period 1967 to 1975 was one of generally lower Indian monsoon rainfall and also was a period during which the southern gyre was weaker, the cross-equatorial winds were weaker and the model upper layer was thicker (deeper thermocline) implying higher SST. A pronounced quasi-biennial oscillation (QBO) was found in the model fields and in the wind stress curl fields during the period of higher rainfall but was absent during the period of lower rainfall. The significance of the presence o r absence of this QBO signal is not yet clear, but is certainly suggestive.

3.3 Southern Hemisphere Circulation The full basin model is integrated for 20 years using the Hellerman and Rosenstein (1983) monthly mean climatological winds. A steady seasonal cycle is achieved throughout the basin by the tenth year of integration (Fig. 3). The southern hemisphere circulation from the 10th year of this simulation has been investigated i n detail (Woodberry et al., 1989). The primary feature of this circulation is a basin wide tropical gyre between the equator and approximately 20"S, consisting of the South Equatorial Current (SEC) in the south, the South Equatorial Counter Current (SECC) t o the north and the East African Coastal Current (EACC) closing the circulation in the west. A strong recirculation region exists in the EACC between 10"s and 2% during most of the year, with numerous eddies forming and being reabsorbed into the EACC. During the boreal summer, the separation point of the EACC moves northward across the equator, and the recirculation region becomes the southern gyre of the two gyre system in the Somali Current. Late in the summer monsoon, a n eddy separates from this region and migrates northward to coalesce with the great whirl. Due to the annual cycle in the wind stress curl, the Sverdrup-like interior consists of a series of first baroclinic mode nonlinear Rossby waves that propagate slowly westward. The required poleward Sverdrup flow from the SECC t o the SEC thus occurs in meridional bands in the lee of the minima in model upper layer thickness associated with these nonlinear waves. The model reproduces the SEC and its branches a s it splits around the Nazareth Bank and Cargados Carajos Shoals along the Seychelles-Mauritius Ridge at 60°E,14%, and again a t the east coast of Madagascar. The transport in the current around the northern tip of Madagascar shows a prominent 40 to 50 day oscillation during the months of February through April (Fig. 4), even though there is no forcing in that period band in the winds, but shows little annual cycle, as observed by Schott et al. (1988) and Swallow et al. (1988). This is due to the blocking effect of the very shallow banks

379

Fig. 3: Model upper layer thickness (ULT) and velocity from the standard case (with islands) for August after 10 years of spin-up with the Hellerman winds. Colors denote ULT, with red being deeper ULT and higher heat content of the upper ocean, and blue being shallower ULT and lower heat content. Brown shading denotes land points. Arrows indicate upper layer velocity, with arrows shown only once per 1.6 degree in each direction. No arrows are shown for velocities less than 5 em s-l and velocities greater than 1m are truncated for clarity of display.

Fig. 4: Time-longitude contours of meridional transport across 12"S,from 50"E to 60"E. Westward propagation of the 40-50 day waves from the Seychelles-Mauritius Ridge toward Madagascar can be seen in February - April as sloped contours . These waves are excited by an annual Rossby wave that is blocked by the shallow Saya de Malha and Nazareth banks along the Ridge at 60"E. Units are in Sverdrups (1 Sv = 106 m3s-l). Contour interval is 0.2 Sv.

380

along the Seychelles-Mauritius Ridge a t 60"E (see below). The annual Rossby wave that is generated in the interior farther to the east is trapped on the east side of these banks where it then sheds smaller scale, higher frequency waves through the gap between these banks. These higher frequency waves are the source of the 40 to 50 day oscillations to the north of Madagascar. 3.4 Equatorial Waves

The model also reproduces the 26 to 28 day waves seen along the equator in current meter records (O'Neill, 1984;Luyten and Roemmich, 1982)and in similar ocean models (Kindle and Thompson, 1989). These waves are identified as mixed Rossby-gravity waves, and are generated a t the western boundary when strongly nonlinear eddies in the East African Coastal Current or the Somali Current cross the equator. These waves have a n eastward group speed, but a westward phase speed, as seen in Fig. 5 . The existence of these waves constitutes the primary difference between this region and a mid-latitude western boundary current. The presence of the equatorial wave guide allows energy to be carried away from the western boundary a s either mixed Rossby-gravity waves or a s equatorial Kelvin waves. A t midlatitudes, there are no long waves available to carry energy eastward.

Fig. 5: Time-longitude contours of meridional transport along the equator between the coast of Africa and Gan (43-73"E). Solid (dashed) contours show northward (southward) transport. East of 50"E the presence of mixed Rossby-gravity waves, with eastward group speed and westward phase propagation, is apparent in the sloping contour lines. These waves are generated at the western boundary several times a year by the abrupt changes that occur in the western boundary current, seen in the convoluted contours west of 50'E.

381

3.6 Island Effects The standard version of the model considers as land boundaries any bathymetry less than 200 m deep. This results in large islands along the Seychelles-Mauritius Ridge (around the Seychelles, the Saya de Malha Bank, the Nazareth Bank and the Cargados Carajos Shoals), along the Maldives, around Socotra and around the Chagos Archipelago, as well as numerous smaller islands. These land boundaries are shown in dark shading in Fig. 1. This is a realistic geometry, as these banks are extremely shallow, typically 10 to 40 m deep, are dotted with tiny islands and reefs, and therefore present substantial barriers to flow. Models of the Indian Ocean circulation presently in use by others omit these islands in their model geometry. Fig. 6 shows the flow field in the model for August after 10 years of integration using the Hellerman climatological winds as in Fig. 3; however, in Fig. 6, all the islands have been removed at the beginning of the integration. There are substantial differences between the two cases, particularly in the South Equatorial Current (SEC) between Madagascar and 60"E, in the equatorial wave guide and in the Somali Current. In the standard case (Fig. 3), the great whirl of the summer Somali Current is blocked t o the south of Socotra in August. The SEC splits at the Ridge at 60"E, with the northern branch flowing around the tip of Madagascar at about 12"s and the southern branch splitting again at the coast of Madagascar at 18%. There is almost no annual variation in the flow around the northern tip of Madagascar, which is consistent with observations of Swallow et al. (1988) and Schott el al. (1988). In the no islands case (Fig. 6) the great whirl continues t o migrate northward in the absence of the blocking effects of Socotra. The SEC impinges on the coast of Madagascar as a single broad current at about 15"S, with a pronounced annual signal, due to the absence of the blocking effect of the Ridge on the annual Rossby wave generated by the wind stress curl in the east. There is higher eddy energy in the East African Coastal Current (EACC) in the no islands case, again due to the absence of blocking of Rossby wave 'energy from the interior by the islands. The inclusion of these islands therefore appears to be crucial for an accurate simulation of the Indian Ocean circulation. 4 INHERENT vs. lN!CJ3RA"uAL VARIABILITY

From the last ten years of the 20 year full basin simulation, we compute the mean and standard deviation of the model upper layer thickness (ULT) field for the 16th of each month of the year, i.e., we compute the mean and standard deviation of ULT at each grid point from all January 16ths, from all February 16th's, etc. The standard deviation fields, expressed as meters of deviation about the mean, are thus a measure of the inherent variability in the model physics, since the wind

382

Fig. 6: Same as in Fig. 3, for the no islands case. Flow in the SEC, in the EACC and in the great whirl are changed appreciably.

Fig. 7: ULT standard deviation for all August 16th'~from years 11 through 20 of model integration using the Hellerman climatological monthly mean winds.

383

cycle is exactly repeating from year to year. These fields are shown in Fig. 7 for representative months. Over most of the basin, on a particular day of the year, ULT varies by less than l m from one year t o the next; for instance, the ULT field for August 16, year 15, is within l m of the value it had on August '16, year 14, over most of the basin, indicating that the model solution is a nonlinear periodic response t o the seasonal winds. Exceptions to this occur only in limited regions such as the East African Coastal Current recirculation region, which is a tropical analog of a mid-latitude western boundary current recirculation region. High values of ULT standard deviation are also found in the intense shear zone around the great whirl during the summer monsoon, but this is confined to very small scale motions around the periphery of the great whirl; the center of the great whirl is found in the same position with the same intensity from one year to the next. The highest values of ULT standard deviation in these regions are in the range of 12-15m, indicating a more chaotic nature in the model response. In the EACC recirculation region and around the great whirl, the chaotic nature of the flow stems from horizontal shearing (barotropic) instabilities; still, the solution over much of these regions is repeating to a high degree. The situation is much different when real-time, interannual winds are used to drive the model. Fig. 8 shows the same standard deviation calculation from a version of the model driven by 23 years of real-time, interannually varying ship winds from Cadet and Diehl (1984). Fig. 8a shows the ULT standard deviation, expressed in meters of deviation about the mean, from 10 years of simulation driven by the climatological monthly mean of the 23 years of winds, while Fig. 8b shows the the same quantity from the 23 year simulation using the interannually varying winds. The ULT standard deviation for the interannual case is everywhere an order of magnitude larger than for the climatological case, except in the highly nonlinear shear zones around the great whirl, indicating that interannual variability in the model response is solely due to variability in the winds, rather than to inherent variability contained in the model physics. 5 CONCLUSION

The variability seen in this model is in stark contrast to that seen in midlatitude ocean models, where the flow field approaches chaotic motion and agreement between model fields and observations is found only in a statistical sense (eg. Schmitz and Holland, 1982; Holland and Schmitz, 1985). One reason for

384

Fig. 8: (a) ULT standard deviation, expressed as departure from the mean in meters, for all August 1 6 t h ’ ~from a 10 year integration driven by the climatological monthly mean winds ~ compiled from ship observations from 1954 through 1976. 6)Same for all August 1 6 t h from the 23 year simulation using the interannual winds from 1954 through 1976.

385

this disparity is that baroclinic instability is more important in mid-latitudes than in the tropics, giving rise to more random eddy production. This mechanism is absent in the reduced gravity formulation; however, a t low latitudes, barotropic instability, which is contained in our model physics, is the dominant instability mechanism. A more likely reason is that the mid-latitude models use steady, idealized winds in an idealized basin, while our model uses a realistic basin geometry with observed winds as forcing. It may be possible that a more deterministic solution would arise from the use of realistic, time varying winds with realistic basin geometry in mid-latitude oceans as well.

6 ACKNOWLEIIGE3lENTS This research was supported by the Office of Naval Research through the Secretary of the Navy Chair in Oceanography, by the Naval Ocean Research and Development Activity and by the Institute for Naval Oceanography. Additional support was provided by the Florida State University through time granted on their Cyber 205 supercomputer. This is contribution number 272 of the Geophysical Fluid Dynamics Institute and contribution number 89-09 of the Supercomputer Computations Research Institute at Florida State University.

Brown, 0. B., J. G. Bruce and R. H.Evans, 1980. Evolution of sea surface temperature in the Somali Basin during the southwest monsoon of 1979. Science, 209:595-597. Cadet, D. L. and B. C. Diehl, 1984. Inter-annual variability of surface fields over the Indian Ocean in recent decades. Mon. Wea. Rev., 112: 1921-1935. Cox, M. D., 1970. A mathematical model of the Indian Ocean. Deep-sea Res., 17:47-75. Cox, M. D.,1976. Equatorially trapped waves and the generation of the Somali Current. Deep-sea Res., 23: 1139-1152. Cox, M. D., 1979. A numerical study of Somali Current eddies. J. Phys. Oceamgr., 9:311-326. Dube, S. K, M. E. Luther and J. J. OBrien, 1989. Relationships between interannual variability in the Arabian Sea and Indian monsoon rainfall. J. Geophys. Res. - Oceans, 94 (to appear). Evans, R. H. and 0. B. Brown, 1981. Propagation of thermal fronts in the Somali Current system. Deep-sea Res., 28: 521-527. Holland, W. R. and W. J. Schmitz, 1985. Zonal penetration scale of model midlatitude jets. J. Phys. Oceanogr., 15: 1859-1875. Hurlburt, H. E. and J. D. Thompson, 1976. A numerical model of the Somali Current. J. Phys. Oceanogr.,6: 646-664. Kindle, J. C. and J. D. Thompson, 1989. The 26 and 50 day oscillations in the western Indian Ocean: Model results. J. Geophys. Res. - Oceans, 94 (in press).

386 Knox, R. A,, 1987. The Indian Ocean: Interaction with the monsoon. In: J. S. Fein and P. S. Stephens (Editors), Monsoons, John Wiley, New York, pp. 365-397. Lin, L. B. and H. E. Hurlburt, 1981. Maximum simplification of nonlinear Somali Current dynamics. In: M. J. Lighthill and R. P. Pearce (Editors), Monsoon Dynamics, Cambridge University Press, pp. 541-555. Luther, M. E., 1987. Indian Ocean modelling. In: E. Katz and J. Witte, (Editors), Further Progress in Equatorial Oceanography, Nova Univ. Press, Dania, FL, pp. 303-316. Luther, M. E., and J. J. OBrien, 1985. A model of the seasonal circulation in the Arabian Sea forced by observed winds. Prog. Oceanogr., 14:353-385. Luther, M. E., J. J. O'Brien and A H. Meng, 1985. Morphology of the Somali Current System during the southwest monsoon. In: J. C. J. Nihoul (Editor), Coupled Ocean-Atmosphere Models, Elsevier, Amsterdam, pp. 405-437. Luyten, J. R., and D. H. Roemmich, 1982. Equatorial currents at semi-annual period in the Indian Ocean. J. Phys. Oceanogr., 12: 406-413. O'Neill, K., 1984. Equatorial velocity profiles: I. Meridional component. J. Phys. Oceanogr., 14: 1829-1841. Schmitz, W. J. and W. R. Holland, 1982. A preliminary comparison of selected numerical eddyresolving general circulation experiments with observations. J. Mar. Res., 40: 75-117. Schott, F., 1983. Monsoon response of the Somali Current and associated upwelling. Prog. Oceanogr., 12:357-382. Schott, F., M. Fieux, J. Kindle, J. Swallow and R. Zantopp, 1988. The boundary currents east and north of Madagascar, 2, Direct measurements and model comparisons. J. Geophys. Res.Oceans, 93:4963-4974. Simmons, R. C., M. E. Luther, J. J. O'Brien and D. M. Legler, 1988. Verification of a numerical model of the Arabian Sea. J. Geophys. Res. - Oceans, 93:15 437-15455. Swallow, J. C., and M. Fieux, 1982. Historical evidence for two gyres in the Somali Current. J. Mar. Res., 4O(Suppl.): 747-755. Swallow J. C., M. Fieux and F. Schott, 1988. The boundary currents east and north of Madagascar, 1, Geostrophic currents and transports. J. Geophys. Res.-Oceans, 93:4951-4962. Woodberry, K. E., M. E. Luther and J. J. O'Brien, 1989. The wind driven seasonal circulation in the southern tropical Indian Ocean. J. Geophys. Res.-Oceans, 94 (in press).

387

GENERAL CIRCULATION OF THE MID-LATITUDE OCEAN: COUPLED EFFECTS OF VARIABLE WIND FORCINGS AND BOTTOM TOPOGRAPHY ROUGHNESS ON THE MEAN AND EDDY CIRCULATION.

B. BARNIER and C. LE PROVOST lnstitut de Mkanique de Grenoble, BP 53,38402 Saint Martin dHhres, France.

ABSTRACT The aim of the present study is to compare a set of numerical experiments which simulate the general circulation of a subtropical gyre within the same overall conditions. The basic simulation is a double-gyre winddriven experiment conducted with a multi-layer EGCM-QG model. The different process studies include a stochastic wind forcing superimposedon theclassicalantisymmetricdouble gyre wind, and a mesoscale random bottom topography. The results identify the effects of the bottom roughness and the role of a variable wind on the mesoscale eddies and the mean circulation. INTRODUCTION The roleof the bottom topography is recognizedto be important in the dynamics of theturbulent flows which govern theoceaniccirculations.Onemajor effect is thedevelopment of steadycurrents lockedtothetopography, as demonstrated by Bretherton and Haidvogel (1976), Herring (1977), and Holloway (1978). Another major effect, demonstrated by Rhines (1977) in free-decaying turbulence experiments, is the tendency of the topography to act against the nonlinear energy cascade toward large barotropic scales, by transferring energy toward smaller horizontal and vertical scales. This process seems important inthedynamicsof the MODEeddies (Owens and Bretherton, 1978). However, a recent study of Treguier and Hua (1988) shows that, in the case of quasi-geostrophic oceanic turbulence forced by large scale surface wind stress fluctuations, the transfer from large to small scales occurs mostly for the barotropic mode, and that little energy exchange can be observed between vertical modes. Therefore, it seems that in the presenceof bottom topography, the nature of the forcing of the oceanic turbulence is an important factor to consider when studying the dynamics of the flow. In theocean, a major sourceof eddy variability is thetransferof energy from the mean flow and stratification fields to mesoscaletransient motions by baroclinic and barotropic instability processes. The role of these eddies in the general circulation of the Ocean has been intensively studied over the past ten years (Hollandet al., 1983, Evansetal., 1987).HoweverlittlehasbeenreportedontheeffectofbottomtopcgraphyinEddy-resolvedGeneral Circulation Model (EGCM) simulations. Schmitz and Holland (1982) pointed out smaller abyssal energies in an idealized double-gyre EGCM experiment with random small scale topography. Verron et al. (1987) in a similar numerical experiment, and Barnier (1984 and 1988) through analytical and numerical approach, have demonstrated the strong constraint of a large scale, ridge-like, bottom topography, which acts like a polarising barrier to the energy radiated from the intense eddying jet regions, or from wind induced Rossby waves. Very recently, Boning(1988)has reexamined the double gyre case with small scale roughness in a primitive equation model study. One of his conclusions is that such a random bottom topography blocks the energy radiating in the lower layers, leading to a much more depth dependent structure of the flows, especially in areas of weak flows. The goal of the present study is twofold. First, it aims to investigate the influence of a mesoscale random bottom topography on the dynamics of the jet stream itself. To that effect, two classical double-gyre,quasi-

388

(a) Bathymetry

(b) spectrum

Fig. 1. The random bottom topography. (a) The model basin and the bathymetry. The mean depth is 5000 m. Dashed (full) lines indicate levels below (equal or above) the mean depth. Contour interval is 100 m. (b) The wavenumber spectrum of the topography. Wavenumbers are dimensionless, such that k=l corresponds to a 4000 km wavelength. At high wavenumbers the slope is k”.’.

389 geostrophic, EGCM simulations, one with a flat bottom and one with a mesoscale random bottom topography are compared. Second, this study attempts to see if, in the presenceof the topography, the eddy flows generated by the dynamic of the jet are sensitive to transient wind forcings. An experiment similar to the previous one is conducted with a stochastic wind forcing added to the antisymmetric double gyre forcing, and results are compared. 2 THE NUMERICAL EXPERIMENTS 2.1 The experiments

Three experiments are considered. The first is a reference experiment (referredto as LR64). in which the modelisina basicconfigurationwithaflatbottomandasteadyforcing.Thesecondisatopographicalexperiment (referredtoasTRW), inwhichthemodelisinaconfigurationsimilartoLR64, butamesoscalerandomtopography is introduced. In the third experiment (referredto as TWR64) the model is configured like in the topographical case, but a stochastic wind is addedtothe steady forcing. Each experiment requires 10 years of integration (with a 4 hour time step) to reach a state of statistical equilibrium. Then for each case, a further 1500 day run is conductedfromwhichthestatisticsoftheexperimentsarecalculated.Therefore,inthe following, meanquantities

are time-averaged over 1500 days. The results of the experiments are cornpared in section 3: the influence of the bottom topography and of the variable wind on the gyre circulation can be identified since the three experiments have been performed in the same overall conditions. 2.2 Basic configuration of the model

In its basic configuration (experiment LRW), the ocean model is a six-layer, quasi-geostrophic model, with a rigid top and a flat bottom, driven by a steady wind. Details of the model formulation, the geometry, the wind forcing and the nondimensionalparameters that govern the flow can be found in Schmitz and Holland (1986). The basin is rectangular (Lx = 3600 km, Ly = 3200 km), and the forcing is a steady sinusoidal wind stress (0.75 dyn cm-*)which drives a two-gyre ocean. Linear bottom friction and high order biharmonic lateralfrictionprovide thedissipativemechanisms. Thedepthofthesix layers, the jumpindensityatevery interface, thecorresponding internal radii of deformation and all physical and numerical parameters which define the numerical experiments of the present study are shown in Table 1. (The values of the reducedgravity are such that the main thermocline is the third interface, at 1050 m depth). The governing equations are the quasi-geostrophic, nonlinear potential vorticity equations for every layer, coupled by the continuity equation applied at every interface. The model is an eddy-resolvedgeneralcirculation model in the sense that the geometry is basin-size and that the horizontal resolution is fine (20 km). The model isthus ablelodevelopstrong instabilities,and togiveaturbulent pictureoftheoceancirculation.Inconsequence, calculations must be performed over long periods of time, to let the model reach a statistically steady state in which mean and eddy flows are in mutual balance. 2.3 Mesoscale random bottom topography

In the topographical experiment (TRW), a variable bottom topography is used instead of the flat bottom. whereas all the other parameters remain identical to those of the basic experiment (LR64). Statistical characteristics of the sea floor topography have been presented by Bell (1975). The significant features of the wavenumber spectrum are a k-2slope at high wavenumbers (larger than 2d100 km’), and a flattening at lower wavenumbers indicating a lack of 100-km and larger scales.

390

(a) Wavenumber spectrum

€

(b) Frequency spectrum

(c)ECMWF frequency spectrum

Fig. 2. The stochastic wind forcing. (a) The wavenumber spectrum (white, with a cut-off wavenumber of 2dl50 krn.'). As in Fig. 1, wavenurnbers are dimensionless. (b) The frequency spectrum at the center of the basin produced by the Markovianprocess. (c) The frequency spectrum in the middle North Atlantic from the ECMWF data (Mac Veigh et al., 1986).

391 The bottomtopography used in experiment TR64 aims to represent a random mesoscale bottomroughness Fig. 1(b). It roughlyhasthesame (Fig.l(a)). It hasbeenrandomlygeneratedfromtheisotropicspectrumshownin spectral features as pointedout by Bell (1975). The spectrum is flat for wavenumbers (k) rangingfrom 261000 kml to 2d250 kml, and has a k 1 slope for k rangingfrom 26250 to 2d120 km-l.There is no variability at scales largerthan1000kmbecausewe believethat inthis rangeof wavenumbers, thebottomtopography is best defined by deterministic patterns (like the mid-oceanridge), rather than by random features. The short wavelength cut-

off is 120 km. It is larger than for the real sea floor (Bell 1975), but is imposed by our computational grid size of 20 km. Thus we have six grid points to resolve the smallest topographic scale. The rms topographic height is 120 m, which is less than the observations given by Bell (1975), (around 200 m or more), but is required to be consistent with the quasi-geostrophic approximation. TABLE 1 Model parameters. Parameters are common to all experiments, except for those relative to the bottom topography and the time dependent wind forcing. (Symbols are similar to Schmitz and Holland, 1986). Grid scale: Ax = 20 km Zonal Lx = 3600 km Meridional Ly = 3200 km Basin size: p = 2x10-l1m 1sec-’ Bottom friction: E = set' Coriolis parameter: f, = 9 . 3 ~O1 5 secl Steady sinusoidal wind forcing: T, = 0 . 7 5 ~ 1 0m2 ~ secz Biharmonic Friction: A, = 4x10lo m4secl Stratification (Layer number): Layer depth (m): Reduced gravity ( l o 3m sec2): Radius of deformation (km):

3 400

2 350

1

300 12 38.8

8.08 18.7

4 500

5.24 12.6

5 6 1350 2100 4.99 1.17 10.2 9.2

~

Rms topographic height (TR64 and TWR64): 120 m

Rms wind stress curt (TWR64): Z X ~ OPa - ~m-l

2.4 Stochastic wind forcing In experiments LR64 and TR64, the forcing is the curl of a steady sinusoidal wind stress. Inthe experiment with variable wind forcing (TWRM), we add a time dependent curl to the mean curl. All the other paremeters are thoseofthetopographicalcaseTR64,inorderto investigatethecombined effectsof a random bottomtopography

and an unsteady wind forcing on the gyre circulation. The variable wind stress curl is generated with a Markovian process, from an isotropic white wavenumber spectrum (Treguier and Hua, 1988). The spectral features of the variable wind stress curl are shown in Fig. 2. The cut-off wavelength is 150 km. The integral time scale of the Markovian process is 10 days, which yields a white frequency spectrum at periods longer than 30 days, and a w2decay at shorter periods. The wind stress curl rms, determined after Mac Veigh et al. (1986), is 2x10’ Pa m1and gives a spectral levelat low wavenumbers which is quite comparable to that shown by these authors with the ECMWF data (see Fig. 2(c)).

3 MAIN RESULTS OF THE COMPARATIVE EXPERIMENTS 3.1 Instantaneous flow This section comments the main features of the time dependent circulation in the three experiments from instantaneous maps of the upper and lower layer streamfunctions. -0(

.Theupperlayercirculation(Fig.B(a))showstheclassicaldouble

gyre. The jet stream penetrates more than 2000 km in the interior ocean. It shows meanders whose lengthscale

392

LR64

SIRERHFUFICIIBN LAYER 6

Fig.3. Instantaneousmapsof thesurface(a) and bottom(b) layer streamfunctionsforthe flat bottomexperiment (LR64). Contour interval is 2x10' m*sec' for the surface layer, and 6000 m2sec-'for the bottom layer. Dashed contours indicate negative values.

393

(4 TRM

STRERMFUNCTIBN LRYER 6

Fig. 4. Instantaneous maps of the surface (a) and bottom (b) layer streamfunctions for the topographical experiment (TR64). Contour intervalis 2x104m2sec1forthe surface layer, and 6000 rn2sec"forthe bottomlayer. Dashed contours indicate negative values.

394

(a! TWR64

SIRERHFLJNCTIEN LRYER 6

Fig. 5. Instantaneousmaps of the surface (a) and bottom (b) layer streamfunctionsforthe case with the variable wind forcing (MRM). Contour interval is 2x1 O4 m2sec"for the surface layer, and 6000 mzseci for the bottom layer. Dashed contours indicate negative values.

395 is around 500 km and amplitude increases towards the end of the jet. Large eddies are present in the vicinity of the jet. The instantaneous upper layer transport in the jet reaches 54 sv at 500 km off the western boundary. Inthe lower layer (Fig. 3(b)),the jet stream is much weakerand can be identifiedonly over its first 1200 km. Elsewhere, it looks like a turbulent field, with mesoscale eddies and Rossby waves filling the whole basin. However, the eddies seems to be spatially positively correlated with the upper layer eddies, indicating a strong coupling between the eddy flow at all levels. Notice the significant signal in the eastern part of the basin, due to the radiation of barotropic Rossby waves generated by the instabilities of the end of the jet. (ii) p

e lTR6Q . The upper layer circulation (Fig. 4(a)) shows drastic

changes with respect to thr reference experiment. The eastward penetration of the jet is considerably reduced. The jet stream itself is well defined over its first 600 km, then it looks rather chaotic, showing series of mesoscale features, like spliting, eddies and rings, in a much larger number than in LR64. The instantaneous upper layer transport in the jet, near the western boundary, is not significantly changed (50sv). The lower layer circulation (Fig. 4(b)) is also quite different. Significant mesoscale features are now limited to the region of the jet.The correlation with the upper layer eddies is not striking anymore, thus the topography would tend to decouple the upper and lower layers (baroclinisationof the eddy flow). In the eastern basin, the signal is at least three time weakerthan in LR64, which indicates that the random topography is a barrier to the radiation of barotropic waves. It must benotedthatthetopographicscalesseemtobeverymuchpresent inthetimedependentcirculation,

even in the upper layer. (iii)

(TWR64). The upper and lower layer streamfunctions (Fig.

5(a),(b)) are not significantly different lrom the case TR64. The surface jet is still short and chaotic. Concerning the far field (the eastward basinand the regions nearthezonal boundaries), we expect the wind induced currents to modify the local circulation. But this does not appear on the plots presented here, and we need local statistical investigation of the mean and eddy currents to derive quantitative results. 3.2 The mean fields The analysis of the mean fields (streamfunction, mean and eddy energetics, etc...), pointed out several features which illustrate the effectsof the random bottom topography and the stochastic wind forcing on the gyre circulation. In the following we focus on three dynamical features which we believe are most important in the comparison of these three experiments. They are the penetrationof the jet, the baroclinisation or barotropisation of the eddy and mean flows, and the evolution of the far field.

. The upper layer mean streamfunctions for cases LR64 and TR64 are 0) shown in Fig. 6(a) and Fig. 7(a). They illustrate the fact that the bottom topography drastically reduces the eastward penetration of the jet. In the flat bottom case (LR64), the mean jet is straight and reaches the middle

of the basin (Fig. 6(a)), its maximum transport is 52 sv at 500 km off the western boundary, and is still 24 sv 1000 km further. In the topographical case (TR64). the jet is shorter (Fig. 7(a)), and its intensity, as well as the

inertialrecirculation are greatly reduced. The transport at 500 and 1500 km are 24 sv and 10.5 sv, respectively; the maximum transport (33 sv) occurs at 200 km off the western boundary. Notice in Fig. 7(a) the mesoscale meanders inthe mean current, signatureof lower layer mean circulation (Fig. 7(b)).The comparison of the lower layer Streamfunctions for cases LR64 (Fig. 6(b)) and TR64 (Fig. 7(b)) also stresses drastic differences. In the case of a flat bottom, the deep jet is coupled to the surface jet. In the topographic case, the deep circulation is controlled by the mesoscale features of the bottom topography (anticyclonic above hills and cyclonic above

396

Fig. 6. Surface (a) and bottom (b) mean streamfunction forthe flat bottom experiment (LRM). Contour interval is 1O4 m2SeC-'for the surface layer, and 3000 m2sec" for the bottom layer.

397

Fig. 7. Surface (a) and bottom (b) mean streamfunction for the topographical experiment (TR64).Contour interval is 10' m*secl lor the surface layer, and 3000 m2sec1lor the bottom layer.

398

TWR64

lUR64

HERN STRERMFUNCTIBN

MEAN SIRERMFUNCTIBN

LAYER NLMBER I

LAYER NUMBER 6

Fig.8. Surface(a)andbottom(b) meanstreamfunctionforthecasewithvariablewindforcing(TWR64). Contour interval is 10' rn2Sec1lor the surface layer, and 3000 m2sec-'for the bottom layer.

399 hollows), and feeds back the meanders of the surface jet. The analysis of global and local energetics is necessary to understand the new equilibrium reached by the jet in the presence of bottom topography. Global (spatially averaged over the basin) energy budgets are presented in Fig. 9. Comparedto the flat bottom case, the topographic case presents the following discrepancies which may explain the shortening of the jet. For the surface layers (1 to 3), the energy input by the wind in the mean circulation, the kinetic and potential energy of the mean flow (MKE and MPE) are weaker, but the mean kinetic energy of the eddy flow (EKE) is aboutthe same. (In fact, EKE is larger in the jet incaseTR64, but because the jet is shorter, the area of large values of EKE is smaller, and yields an integralvalue of the same order as for case LR64). Furthermore,the energy transfer rate related to the barotropic instability (EKE t MKE) is larger in all three surface layers. Therefore, the jet is problably reduced because it transfers more of its mean kinetic energy to the eddies, via horizontal shear (barotropic) instability. To complete this picture, we need to look at the energetics of the lower layers for case TR64. As shown in the plot of the mean streamfunction of layer 6 (Fig. 7(b)), the bottom flow is controlled by the topography. The eddies generated at the surface by barotropic instability propagate energy downward. They interact with the topography, generating rectified mean mesoscale currents (in Fig. 9 rectification is the transfer MKE, t EKE,). Thus the mean energy is (in case TR64) scattered at smaller scales than in the flat bottom case, and is more efficiently removed by lateral friction (note in Fig. 9that the dissipation of the mean energy in the bottom layers is always largerinthetopographiccase).Theupwardtransferofmeankineticenergyfromthelowerlayerstoward the surface (MKE, t MKE, t MKE, in Fig. 9). is peculiar to the topographical experiment and indicates that the rectified deep currents have a strong barotropic component which constrains the surface flow (the meanders of Fig. 4(a)). We believe that it is the interaction of the bottom flow (controlled by the mesoscale topography), with the surfaceflow (drivenby a basin scale wind forcing), which favours the barotropic instability of the jet stream. The main consequence is a weakening of the jet and of its recirculation, but the level of eddy energy remains high. The case with stochastic wind forcing (TWR64) presents no drastic discrepancy compared to the case with the topography alone. Upper and lower layer mean streamfunctions (Fig. 8(a),(b))are quite similar, differences appearingonly locally. Thevariable wind problably amplifies, near majortopographic features, the eddy currents, and thus enhances the rectification of the deep currents, and therefore modifies the upper mean flow. But those changes are small (the maximum transport in the jet is 34 sv in case TWR64, an increase of 1 sv compared to case TR64). (ii) v

. . n of the &. The diagrams of Fig. 10 and Fig. 11 show the variations

with depth of the kinetic energy of the mean flow (MKE) and of the eddy flow (EKE). Fig. lO(a) sketches local values of MKE at one point located in the most active part of the jet, for each of the six layers. In the flat bottom case, the jet is strongly baroclinic, most of the mean energy being above the thermocline (MKE, is 1200 cm2/sec2 and MKE, is 200 cm2/sec2).In the topographic case, we observe a barotropisation of the jet, in the sense that the mean energy is significantly reduced at the surface and is increased at the bottom (MKE, and MKE, have comparable values, around 500 cm2/sec2).When the variable wind is added, the mean energy increases only in the surface layers. Thus the variable wind seems to reinforce the baroclinic character of the jet stream. Fig.lO(b) is similar to Fig. 1O(a). but for EKE. No drastic modification in the vertical distribution of EKE in the jet is noted between the three cases, except for an increase in the upper layers in the cases with topography. The physical processes responsible for the features of Fig. 10 have been commented in the latest subsection.

400

WINO

2.57

2-05

EKE, 5.04 4.87

0.20

zo 0.50

106.14

se.12

Fig.9. Theglobalenergy diagram for the flat bottomexperiment (italic) andthe topographicalexperiment (bold). Units are in rn(nPsec-*) for the energies (boxes) and lo8 rn(m2sw2)hecfor the energy transfer rates (arrows).Transfersbelow lod are noted 0.00. Negative numbers indicate transfers opposite to the direction of thearrows. Inthat diagram, all quantities are basin-averaged,mean (averagedover 1500days) quantities.MKE and MPE stand for the kinetic and potential energy of the mean flow, and EKE and EPE for the kinetic and potential energy of the eddy flow.

401

Thelargervaluesof MKE inthebottom layers incasesTR64andTWR64areduetotherectifcationofthebottom flow by the topography. The reduction of MKE and the increase of

EKE in the surface layers in cases with

topography are explained by the same phenomenomas for the shorter penetration of the jet. The mean current is reduced becauseit transfers more of its energy to the eddies, via horizontal shear instability, resulting in larger values of EKE and smaller values of MKE. The diagrams of Fig . 11 illustrate the variation with depth of the basin-averaged values of MKE and EKE (denotedJMKEand IEKE in Fig. 11). They must be interpreted with caution, since they average quantities which are not homogeneously distributed in space. These plots confirm that, compared to the flat bottom case, the topographical cases present MKE values which are larger in the bottom layers and lower in the surface layers (Fig. 11(a)). An important feature of Fig. 11(b) is the reduction of the eddy kinetic energy in the bottom layers in the topographical cases. Since this feature is not present in Fig. 10(b),we deduce it is characteristic of the flow outside the region of the jet. Therefore we can say that outside the jet itself, the topography produces a baroclinisation of the eddy flow, in the sense that the eddy activity is reduced under the thermocline. (iii) Evolution of the

.

. The far field is the part of the basin which is far away from the jet and its

recirculation. Basically it covers the eastern basin and the regions adjacent to the southern and northern boundaries. In the flat bottom case (LR64), the variability of the eastern basin is too weak to be realistic; This is a well known failure of the type of model used in this study. As noted in section 3.1, the main effect of the topography on the far field is to block the eastward energy radiation due to the barotropic Rossby waves generated at the extremity of the jet (see Fig. 3 and Fig. 4). Therefore, the level of EKE of the eastern basin is lower in case TR64 than in case LR64, which shows that the random topographic roughness alone cannot compensate for the lack of variance in that region. When the variable wind is added to the random topography (case TWR64). the eddy kinetic energy of the surface layers reaches a level higher than in case LR64, but still too low comparedto observations of the North-Eastem Atlantic. However, complete local energy diagnostics of

caseTWR64are presently undertakenin ordertoquantifythe importanceof thevariable wind and the mesoscale topography in generating mesoscale variance in the eastern basin. Fig. Sshowedthat in the topographical case (TR64).theglobal rateof the baroclinic instability is considerably

reduced (transfer EPE t MPE). Infact, whereasincaseLR64thetransferbetweenthepdentialenergiesis from mean to eddy (indicating dynamics dominated by the baroclinic instability), it is from eddy to mean at every interface, except for the first one in TR64 (indicatingeddy driven dynamics). Howeverthese results are averaged over the basin, and significant discrepanciescan be observed dependingon the area we consider. In experiment

LR64,the regionsof the far field near the zonal boundariesare regionswhere the baroclinic instability is the main source of eddyvariability. The introductionof the randomtopography reducesthe rate of instability. It seems that the topography scatters at mesoscale the available MPE, making the development of the instabilities more difficult, since a growing perturbation might randomly meet favorable or unfavorable conditions of growth over a characteristic topographic length scale (a few 100 km). However this process remains the dominant source of eddy variability in that region. Again, detailed energetics of that region are presently undertaken in order to precisely describe the effect of the mesoscaletopography and transient wind forcings on thedevelopment of the baroclinic instability.

402

Fig. 10. Comparison (in the three experiments)of the vertical distribution of, (a)the mean kinetic energy (MKE), and (b)the mean eddy kinetic energy (EKE), at a point located in the axis of the jet at a distance X of the western boundary. To make the comparison meaningful X has been chosen in each experiment, such that the kinetic energy is maximum in the upper layer. Dashed horizontal lines indicate the interfaces between the layers of the model.

403

0

30

60

JMKE (cm/sec)2 90 120

150

180

150

180

1000-

D 2000E P T 3000H

e

0 TR64

(m)4000- D)

+ TWR64

500G

(4

J EKE 30

0

60

(cm/sec) 2 90 120

1000

D 2000 E P

*

Q

t

0

T 3000H (m)4000.

0 TR64

+ TWR64

5000

Fig. 11. Comparison (in the threeexperiments)oftheverticaldistributionofthe basin-averaged, (a) mean kinetic energy (IMKE), and (b) mean eddy kinetic energy (IEKE). Dashed horizontal lines indicate the interfaces between the layers of the model.

404

4 CONCLUSION

Theresultsofthe present study identify several aspectsofthedrasticinfluenceofa mesoscaletopography roughness on the dynamics of a subtropical ocean gyre model driven by a steady wind. It appears that, underthe thermocline, the flow is controlled by the interaction between the eddies and the

mesoscalefeaturesof the bottomtopography (generatingsignificant mean rectified currents), whereas theupper flow is driven by the large scale wind forcing and the eddies generated by internal instabilities. We believe that the interaction betweenthe surface and the deep flows favours the barotropic (horizontal shear) instability of the upper layer jet stream. In consequence, the eastward penetration of the jet is considerably reduced,the intensity of the jet and its recirculation are weakened, but the level of eddy energy above the thermocline remains high. Therefore, we notice a barotropisation of the jet itself, in the sense that the mean flow is reduced (by shear instability) at the surface and is increased (by rectification) in the bottom layers. Outside the region of the jet itself, an effect of the mesoscale bottom roughness is to reduce the eddy energy inthedeep layers, resulting in a baroclinisationof theeddy flow. This is another indication (afterwunsch, 1981 ; Schmitz and Holland, 1982; and Boning, 1988) of the possible contribution of the bottom topography to

the surface-intensified and weak abyssal eddy energies observed in the interior of the subtropical Ocean gyres. A component of the baroclinisation of the flow far away from the region of the jet (the far field), is the blocking, which occurs in the lower layers, of the energy radiating away from the jet (eddies and Rossby waves). This is another indication that the low-frequencyvariabiliies in the western and the eastern basins of the North-Atlantic Ocean do not have the same origin (Wunsch, 1981). The additiin of a transient stochastic wind forcing to the double-gyrewind pattern reveals several features on the combined effects of a mesoscale bottom roughness and unsteady winds on the gyre circulation. In the present study, only preliminary results have been presented, and a more thorough investigation of the experiments remains to be done. The variable wind does not drastically modify the circulation. It amplifies, near major topographic features, the barotropic rectified mesoscale currents. It also slightly increases the kinetic energy of the upper layer mean flow, acting against the topography (which tends to render the flow more barotropic), in reinforcing the baroclinic character of the jet. In the far field, the variable wind increases the eddy energy of the upper layers, but this is not enough to compare favorably with observations of the eastern NorthAtlantic. However, this is an indication of the possible role of the combined effects of the variable wind forcing and the bottom topography for the surface-intensified nature of the eddy field in the eastern North-Atlantic. ACKNOWLEDGMENTS Support for computations was provided by the Conseil Scientifique du Centre de Calcul Vectoriel pour la Recherche in Palaiseau (France). Fundings came from the CNRS and IFREMER support to the lnstitut de Mkanique de Grenoble through the Programme dEtude de la Dynamiquedu Climat. REFERENCES Barnier. B., 1984. Energy transmission by barotropic Rossby waves across large-scale topography. J. Phys. Oceanogr., 14,438-447. Barnier. B.,1988. A numericalstudy on the influence of the Mid-Atlantic ridge on non linear first mode baroclinic Rossby waves generated by seasonal winds. J. Phys. Oceanogr.. 18,417-433. Bell, T. H.. 1975. Statistical features of sea-floor topography. Deep-sea Res., 22, 883492.

405

Boning, C., 1988. Influence of a rough bottom topography on flow kinematics in an eddy-resolving circulation model. J. Phys. Oceanogr., submitted. Bretherton, F. P.. and D. B. Haidvogel, 1976. Two dimensional turbulence above topography. J. Fluid Mech., 78,129-154. Evans, J. C., D. B. Haidvogel, and W. R. Holland,1987. A review of numerical ocean modeling (1983-1986): Midlatitude mesoscale and gyre scale. Rev. of Geophysics, 25,235-244. Herring, J. R., 1977. Onthestatisticaltheoryoftwodimensionaltopographicturbulence.J.Atmos. Sci., 34,17311750. Holland, W. R., D. E. Harrison and A. J. Semtner, 1983. Eddy-resolving numerical models of large scale circulation. In Eddies in Marine Science (A. R. Robinson, Ed.). Springer Verlag. Holloway, G. 1978. A spectral theory of non linear barotropic motion above irregular topography. J. Phys. Oceanogr.. 8,414-427. MacVeigh, J. P., 8. Barnier, and C. Le Provost, 1986. Spectral and EOFanalysisof fouryearsof ECMWF wind stress curl over the North Atlantic ocean. J. Geophys. Res., 92,13141-13152. Owens, W. R., and F. P. Bretherton,l978. A numerical study of midlatitude mesoscale eddies. Deep-sea Res., 225, 1-14. Rhines, P. B., 1977. The dynamics of unsteady currents. The Sea, Vol. 6, Marine Modelling, E. D. Goldberg, 1. N. Mc Cane, J. J. O’Brien, and J. H. Steele Editors - Wiley, 189-318. Schmitz, W. J., and W. R. Holland, 1982. Apreliminarycomparison of selected numericaleddy resolvinggeneral circulation experiments with observations. J. Mar. Res.. 40, 75-117. Schmitz, W. J., and W. R. Holland, 1986. Observed and modeled mesoscale variability near the Gulf Stream and Kuroshio extension. J. Geophys. Res.. 91,9624-9638. Treguier, A. M., and B. L. Hua, 1988. Influenceof bottomtopography on stratified quasigeostrophic turbulence in the wean. J. Phys. Oceanogr., submitted. Verron, J., C. Le Provost, and W. R. Holland, 1987. On the effects of a mid-ocean ridge on the general circulation: Numerical simulation with an eddy resolving ocean model. J. Phys. Oceanogr., 17, 301-312. Wunsch, C., 1981. Low frequency variability of the sea. Evolution of Physical Oceanography, MIT Press, 342-374.


407

ASYMMETRICAL WIND FORCING DRIVING SOME NUMERICAL EDDY-RESOLVING GENERAL CIRCULATION EXPERIMENTS .l. VERRON and C. LE PROVOST Institut de MBcanique de Grenohle, BP 68, 38402 Saint Martin d’H6res CBdex, France

ABSTRACT The classical double-gyre wind-driven mid-latitude ocean circulation simulated by means of eddy-resolving quasi-geostrophic numerical models is reconsidered by introducing some asymmetr in the forcing configura.tion. The main objective is to explore the changes in the basin scaL dynamics due to the wind asymmetry : jet profile and penetration, eddy energy radiation, downward energy pro agation, homogenization, rinq formation and gyre exchange, etc... Some rationalization o f t h e dynamical control of the jet detachment from the coast is also attempted from a survey of several asymmetrical QG experiments : mult,ilayer unsteady configuration as well as more simple steady barotropic ocean. 1. INTRODUCTION

In EGCM studies, the large scale structure of the wind stress at mid-latitudes which, for a part, drives the subtropical gyre general circulation has often been schematized by the now classical (anti-) symmetrical double-gyre pattern (Holland, 1978). Assuming appropriate tuning of other model parameters, this approximation has encountered some success to predict some realistic features of the intense mid-latitude jet stream and the associated variability fields (Schmitz and Holland, 1982, 1986). Actually, the wind structure is a much more complex combination of different time and space scales. Our motivation in this paper is to investigate how the simple departure from symmetry may alter the classical double-gyre Holland’s type ocean model circulation and to identify the possible modifications to the circulation : jet penetration and general circulation characteristics, eddy variability geography and intensity, gyre exchange, etc...An other important issue is to try clarifying what controls the latitude the jet departs from the coast, which supposes clearly a better understanding of the Western Boundary Current (WBC) dynamics. 2. NUMERICAL SIMULATIONS

Several experiments were realised with a quasi-geostrophic multi-layered model of the type which has been first presented by Holland (1978). The model is used along the paper in several configurations : one, two or three layers with either slip or no-slip conditions. Its formulation may be found for example in Verron et al (1987). The rotation parameters

408

are common in all simulations : j 0 = 9.3x i05s-', p = 2 x 10-"m-'s-'. Biharmonic dissipation is employed as lateral friction mechanism everywhere. Other specific parameters are indicated on Table 1. The asymmetrical wind pattern driving the model is a simple extension of the classical double-gyre symmetrical wind pattern. The same rectangular basic shape for the wind pattern is kept but different tuning is allowed for each gyre (Figure 1) concerning :

-

the meridional extension L y

-

the wind stress intensity

-

the zonal extension L X

70

LYN

L LY s

4

Lxs Figure 1. Sketch of the horizontal pattern of the wind stress forcing used in the numerical experiments. Subscript N (S) are respectively used for the Northern (Southern) gyre. Because the wind pattern may be limited in its zonal extension we have imposed a sinusoidal ponderation in the zonal direction to smoothly reach zero at the limits. But the vorticity fluxes injected by the wind are kept equivalent to their symmetrical counterparts in order to make possible a comparison of the induced circulation. Consequently this vorticity flux in each gyre will depend only on the wind stress amplitude 7o and not of the pattern area. The wind stress curl is expressed as :

409

and curlrs = - . -, TOR

?r

P

Lys

-.L r

2Lxs

. ry

sin-.

LYS

,

sin-

i?z

Lxs

Outside of these ranges the wind stress is set to zero. In each gyre the total injected vorticity flux per unit of mass is :

which is exactly the value for each symmetrical gyre having a zonal extension given by L and a meridional extension given by L / 2 . This wind pattern schematization is a first order representation (the zero order would be the symmetrical double-gyre case) of the real mean wind stress-curl over a mid-latitude ocean. Figure 2 shows an example of such mean wind stress-curl field over the North Atlantic from a 4 year average of ECMW'F data (Mac Veigh et al, 1987). The northern wind cell is smaller than the southern : its extension is restricted in the meridional direction at the latitude of about, 35" N ( L y parameter). It, is also limited in the zonal direction by the abrupt northward shift of the wind stress at the longitude of about 50" W ( L x parameter).

MERN CURL

Figure 2. Distribution of the mean wind stress curl over the North Atlantic. Units are in Pa / 1000 km. The contour interval is 40 (After Mac Veigh et al, 1987).

410

Figure 3. Mean streamfunction fields for the two layers of experiment 1. Contour intervals are 3000 m2/s for the surface layer (a) and 1000 m 2 / s for the deep layer (b). Statistical average is performed over 2000 days.

411

Moreover the wind intensity is clearly different in each cell, generally northern wind stress curl is stronger (7" parameter), However because of the southern wind pattern area the total vorticity flux is significantly larger in the southern gyre. 3. ELEMENTARY EFFECTS

Before considering a complete asymmetrical experiment let us look at the inferences of varying L v , r0 and Lx in isolation: 3.1 Shifting the zero wind stress curl line [ZWL).

Because of the wind configuration, shifting the ZWL has no consequence on the vorticity input by the wind which stays the same for both gyre as far as ro is unchanged. Figure 3 shows the mean streamfunction of a two-layer unsteady experiment (Experiment 1) in which the ZWL has been shifted 200 km northward. The consequences of this shift may be divide in two parts :

-

in the WBC, it modifies the latitude the jet departs from the coast. The jet detachment latitude (JDL) is now about 120 km south of the ZWL.

-

in the open ocean : the jet starts oscillating around the ZWL and quickly restores along this line.

3.2 Changing the w r e vorticitv fluxes

Changing the gyre vorticity fluxes has a more dramatic consequence especially in the deep flow (Experiment 2, Figure 4). The JDL is ako shifted and the surface jet is restoring besides the ZWL. But at the bottom the familiar symmetrical configuration with a well defined mid-latitude deep jet is no longer present. The cyclonic ceIl is filling out all along the western boundary current (WBC) and now there is a countercurrent in the deep ocean right under the mid-latitudes. 3.3 Limiting the zonal extension of the wind pattern

Varying the zonal extension of the wind pattern has little influence on the circulation for moderate change of L x . The jet axis is hardly deviated as well as the JDL or its transport characteristics. Nevertheless, in normally stable conditions, reducing strongly this dimension L x assuming the flow is inertial enough, may lead to a two regime unsteady flow instead of a unique steady flow. The general circulation is then regularly shifting from a straight jet regime to a meandering regime. A controlling parameter is deduced to be :

In the range of experiments we have done a critical value was about 1.8 , i.e. : Rx > 1.8 , straight jet and steady regime Rx < 1.8 , meandering regime.

412

Figure 4. Mean streamfunction fields for the two layers of experiment 2. Contour intervals are 6000 m 2 / s for the surface layer (a) and 1000 m2/s for the deep layer (b). Statistical average is performed over 2000 days.

413

Experiments L

(km)

1

2

3

4000

4000

4000

1800

2000

1800

1. 1. 2000 2000

1.33 1.

1.25 1. 2000

Wind pattern

TON

--

~ n c

m2/s2 m2/s2

2: [k]

2000 2000

2000

Vertical structure 2

2

3

1000

(m)

1000 4000

4000,

300 700

(m/s2)

0.02

0.02 0.0162

Layers

4000 9’

0.0357 Lateral friction A,

(m4/s)

8. lo9

8 . lo9

8. lo9

10-7

10-7

10-7

Bottom friction €

(8-1)

TABLE 1: Parameters of the numerical experiments

4. A COMBINED EXPERIMENT

The present asymmetrical three-layered experiment (Experiment 3) is obtained by both modifying the parameters ro and L r : the ZWL is shifted 200 km north from the mid-basin and ?ON is set to 1.25 while ros stays 1. 4.1 Mean flow

Figure 5 shows the mean streamfunction fields averaged over 2000 days of statistically steady state. The flow above the main thermocline has similarities with the symmetrical one. In particular the jet profile is similar and the jet penetration is about the same : a little less than the 2000 km. But the mean jet axis line is now meandering with about 700 km wavelength. The jet detachs from the coast at a latitude which is about 300 km south of the ZWL. So there is an overshooting inertial circulation due to the northern wind pattern features : reduction of the meridional dimension and stronger wind stress. When leaving the coast the jet is restoring firstly northward then oscillating around the ZWL. Moreover the mean jet axis is slightly tilted also in the northward direction. A time series analysis shows that the surface jet. is quickly moving besides its mean detachment latitude with a clear variability a t long period of a year or so. The surface layer mean transport is almost unchanged and each recirculation cell carries about 48 Sv.

414

415

F'igure 5 . Mean streamfunction fields for the three layers of experiment 3. Contour intervals are 10000 m 2 / s for the surface layer (a), 7000 m 2 / s for the intermediate layer (b) and 3000 m 2 / s for the deep layer (c). Statistical average is performed over 2000 days. Under the thermocline (i.e. in the deepest layer) the vort,ex-street-like mean structure is enhanced although the standing meanders have a strong barotropic signature in the recirculation area. A cyclonic cell is spreading from the south resulting in a countercurrent running southward under the surface WBC which itself flows northward in this area. The boundary current is no longer barotropic because of the presence of a countercurrent under the area the surface jet leaves the coast. This would suggest that the particular deep flow configuration obtained in the symmetrical case might be partly an artefact of the forcing symmetry. The deep layer mean transport, much more concentrated, is even larger: about 113 Sv compared to 91 Sv in the equivalent symmetrical case. 4.2 Eddies

In looking at the instantaneous streamfunction field of Figure 6 a mechanism emerges for the excess of vorticity originated from the northern gyre to transfer southward and act, on average, to shift the jet detachment southward. It seems mostly due to the pinching of a cyclonic meander along the wall because the open ocean jet perturbations are travelling westwards and have tendencies to restore their mean latitude to the mean jet axis. The coastally trapped eddy is then travelling southward, between the wall and the WBC, then disappears after one month or so.

416

Figure 6. An example of the instantaneous surface layer streamfunction field in experiment 3. Countour interval is 20000 mz/s.

On Figure 5 the kinetic energy fields are presented. Because of the previous mechanism the kinetic energy field is now spreading a little more along the WBC compared to the symmetrical case. In a general matter the variability is stronger especially a t the bottom and extends more uniformly. This would be clear from an energy diagram : the asymmetry is resulting in more eddy kinetic energy in each layer, although mean potential energy and kinetic energy are almost unchanged. Because of the northern wind intensification the total input of wind energy is larger than in the symmetrical case. It is remarkable that the excess of energy is to be transferred to eddy activity. Saying shortly, the effect of asymmetry is mostly going to eddies. Looking a t the potential vorticity field on Figure 8 one can see the pool of Q homogenization in the intermediate layer has a wider latitudinal extension. Moreover due to the eddy activity reaching the western wall this pool extends to the central part. of the WBC. 4.3

Rinas

Regarding the jet stability arid eddy or ring formation the wind asymmetry raises the question of the possible asymmetry of the jet profile and the ring formation. Does the new jet stability conditions account for an asymmetrical cold- and warm-core ring formation

417

418

Figure 7 Mean kinetic energy fields for the three layers of experiment 3. Maximum values are 8278 m Z / s 2for the surface layer (a), 1020 mz/sz for the intermediate layer (b) and 299 m2/sZfor the deep layer (c). Maximum contours are 2216 m2/sZfor the surface layer, 985 m2/sZfor the intermediate layer and 292 mz/sz for the deep layer. The contour are decreasing in power of 1.5. rate. We have already said that the instantaneous velocity profile in the jet is relatively symmetrical as will be also the mean jet profile. But the ring formation seems to be altered. During the 2000 days of the experiment 3 a census of ring formation will result in :

-

7 anticyclonic rings (WCR, warm core ring) forming at north,

-

14 cyclonic rings (CCR, cold core ring) forming at south except one forming at north,

-

1 vortex pair forming at south.

This formation rate of about 4 rings per year is somewhat low compared to the numbers mentioned by some observational works. Let us remark that ring formation must be distinguished from eddy formation. In the model we found more eddies in the northern gyre than in the southern but more rings in the southern than in the northern. When the transcription of that result is done to the real ocean where the vorticity input by the wind is larger in the southern basin than in the northern (the opposite of the model configuration) the following conclusion for the ocean would be reached : in the north, less eddies but more WCR, in the south, more eddies but less CCR.

419

Figure 8. Mean potential vorticity field in the intermediate layer of experiment 3. Contour interval is 5 x ~ O - ' S - ~ .

6. WHAT CONTROLS THE JET DETACHMENT ?

In the case of EGCM simulations with slip boundary conditions there is a rather clear decoupling between the WBC and the jet dynamics. The open ocean jet dynamics integrates the basin-wide fluxes (including the WBC ...) and the main jet axis stays alined with the ZWL. Reversely the JDL position and the local jet behavior seem to be determined from a local response t o the WBC dynamical constraints. For a given dynamics, the JDL will be controlled by the 3 parameters ro, L y and LX which set the wind configuration in each north and south cell. Because L X has little effect on the JDL in the range of realistic values for the North At,lantic we have momentarily discarded this parameter effect. A number of numerical experiments have been ran to attempt rationalizing the JDL controlling mechanism. These experiments are not described here, only their final results are presented. Assuming the JDL is determined by a latitude

YJDL

one can introduce :

Ls = Y J D L LN = L - Y J D L it is convenient to build the numbers R L , RLY and R, such as :

420

LYN- LYS LYN - LYS L LYN + LYS R, = TON - 705

RLY=

~

TON

+ TDS

which content all the ingredients of the problem. When RL varies in the range - 1 < RL < + 1 , it gives a measure of the departure of the JDL besides the mid-latitude (RL = 0). RL is negative (positive) for a northward (southward) shifting of the JDL. RLY and R, measure the dissymmetry of the wind pattern. Before considering complex turbulent, multi-layered models, let us go back to some fondumentals. As was noted by Boning (1986)the horizontal structure of the transport in such EGCM experiment, characterized by the existence of intense recirculation cells besides the jet transport, has more analogy with steady (slip boundary conditions) barotropic circulation damped by lateral friction, than Stommel-type circulation damped by bottom friction even though bottom friction is the dominant dissipation mechanism in the EGCM. The dissipative control through eddies at the bottom draining energy from the surface is still unclear given the complexity of such turbulent flows. Let us assume a simple linear ocean where dissipation is either lateral friction ('Munk ocean') or bottom friction ('Stommel ocean'). An asymmetrical wind forcing the Munk ocean will result in the relationship : RL = RLY independantly of R,. Similarily, solving numerically the Stommel flow configuration provide us for a relation such as :

Introducing non-linearity in the Stommel ocean configuration (i.e. Veronis-like solutions) will result in two possible regimes as determined from our numerical simulations :

-

a frictional regime which fits quite well with the Stommel solution :

-

an inertial regime which has similar dependency for RLY but the wind intensity effect is strictly proportional : RL = 0.6.RLY+ 1.0.R,

At the moment it is unclear t o explain the origin for the 0.G coefficient of R L which ~ is a feature of bottom stress damped ocean. Nevertheless a 0.5 coefficient is straightforward to explain by involving the only p-effect. 0 . 5 will correspond to the linear response to the restoring effect of ,8 which equally balances the jet between the symmetrical and the new asymmetrical ZWL positions.

421

Considering the fully unsteady experiments with one, two or three layers will result in an approximate relationship of the type : RL

-

0.5. RLY + 1.0.R,

This relation is satisfied with a certain level of approximation because of residual unsteadiness in the statistics (in each experiment, statistics has been performed for 2000 days). But the main tendency is indisputable when data is drawn in a (RL,RLy:R,) diagram. Moreover, it seems more effective when the number of layers is increased. Then the JDL position would be controlled mostly by :

-

the ,+-effect for the response to a change in L y ,

-

directly from the wind intensity similarly to the Veronis inertial (steady) regime.

We think this may be interpreted by identifying the main terms in the balance equation in the surface WBC. The complete equation writes :

If we assume on one hand that the WBC is not mostly eddy-driven, and on the other hand that the wind input as well as the stretching may be neglected because of the narrowness of the WBC region; if, in addition, we neglect biharmonic dissipation in this region, the equation reduces to :

___

a?i aZ

J(!hi,A$i)+B.-=O

that is a vorticity equiIibrium controlled by @-effectand mean advection. This is consistent with the Harrison and Holland (1981) study of vorticity budgets over a basin in a similar two-layer simulation. From a simple integration of the equation using V >> U for the horizontal velocity, one obtain that the WBC vorticity is equal to the vorticity input from the wind modulated by @.Then integrating along the western boundary up t o JDL, within each gyre, one recover the previous relationship for RL. 6. CONCLUSIONS The classical double-gyre wind-driven mid-latitude ocean circulation has been revisited after adding some wind forcing asymmetry. The basic dynamics of the surface flow is not much affected i.e. jet penetration and intensity, eddy energy level and geography, ... The deep flow is rather strongly modified especially a deep jet stream is not longer present. and the WBC has a marked baroclinic structure in the mid-latitudes. Gyre exchange enhanced by a possible unbalance of vorticity input between gyre seems to be realised owing tlo asymmetrical ring formation but also eddy pinching along the WBC. The latitude the jet detachs form the west coast is tentatively explained by a local WBC equilibrium for the potential vorticity where the main balance is between mean advection and beta effects.

422

7. REFERENCES Boning, 1986: On the influence of frictional parameterization in wind-driven ocean circulation models. Dyn. Atmos. and OceansJO, 63-92. Holland, W.R., 1978: The role of mesoscale eddies in the general circulation of the ocean - Numerical experiments using a wind-driven quasigeostrophic model. 1.Phys. Oceanogr., 8, 363-392. Harrison D.E. and W.R., Holland, W.R., 1981: Regional eddy vorticity transport and the equilibrium vorticity budgets of a numerical model ocean circu1ation.J. Phys. Oceanogr., 11>190-208. Mac Veigh J.P., Barnier B and C. Le Provost, 1987: Spectral and empirical ortho onal function analysis of four years of European Center for Medium range Weather orecast wind stress curl over the North Atlantic ocean. J. of Geophys. Res.,92,1314113152. Schmitz, W.J., and W.R. Holland, 1982: A preliminary comparison of selected numerical eddy-resolving general circulation experiments with observations. J . Mar. Res., 40, 75-117. Schmitz, W.J., and W.R. Holland, 1986: Observed and modeled mesoscale variability near the Gulf Stream and Kuroshio extension. J . Geophys. Res., 9 1 , 9624-9638. Verron J. C. Le Provost, et W.R. Holland, 1987: On the effects of a mid-ocean ridge on the general circulation - Numerical simulations with an eddy-resolved model. J . Phys. Oceanogr., 17 (3),301-312.

%

423

ON THE RESPONSE OF THE BLACK SEA EDDY FIELD TO SEASONAL FORCING

E.V. S T A N E V ~ , ~ Department of Meteorology and Geophysics, University of Sofia, 5 Anton Ivanov Blvd., 1126 Sofia (Bulgaria) 2 Institute of Oceanography, University of Hambur Troplowitzstr. 7, 2000 Hamburg 54 (F.R. Germany? 1

ABSTRACT

The numerical results of a coarse resolution Black Sea circulation model with horizontal grid intervals A A = 1 deg and A $ = 0.5 deg and 12 levels are used to initialize the model with a fine resolution: A?, = 20 min and A @ = 10 min on the same vertical levels. The boundary conditions include wind forcing, heat and salt fluxes on the sea surface and inflows through the Strait of Bosphorus and river inflow as well. Bryan's (1969) numerical model with Laplacian horizontal mixing is used to simulate the circulation. Seasonality in the model forcing contributes to some fundamental changes in eddy processes which can be better observed below the halocline. The results show a trend of better organization of the eddy energy patterns when the model is driven by seasonally variable boundary conditions. Comparison of the model results with some experimental ones is done.

1 INTRODUCTION The Black Sea is a semienclosed sea with a restricted water exchange with the Mediterranean Sea and with a large river discharge into its nortwestern part. The volume averaged salinity is about 22 salinity units. The sharp halocline separates the upper layer (salinity about 18) from the deep water (salinity about 23). The seasonal atmospheric signal does not penetrate below the halocline, and the water in deep layers is stagnant. The oxygen supply is also very small and the Black Sea now accumulates a substantial part of the hydrogen sulphide which exists in the World Ocean. Geological observations indicate that the Black Sea salt content and its stratification are very sensitive to changes in the water and salt budget. During the Pleistocene glaciation, this sea was well aerated and aerobic life existed down to the bottom. One of the main problems in the numerical modelling of the Black Sea circulation i s to take account for the real buoyancy fluxes. The models should simulate realistically the vertical stratification and the vertical exchange which are the background for the general circulation processes. In Friedrich and Stanev (1988) (hereafter FS) the physical mechanisms governing the vertical stratification are analysed.

424

Recently t h e mesoscale v a r i a b i l i t y i n t h e Black Sea has been considered i n numerous studies.

The e x i s t i n g measurements o f Black Sea c u r r e n t s show a

strong eddy v a r i a b i l i t y . Due t o t h e b a r o c l i n i c i n s t a b i l i t y t h e core o f t h e main Black Sea c u r r e n t changes i t s p o s i t i o n , as can be seen i n F i g . l a . Also, t h e t i m e changes o f temperature and s a l i n i t y a r e more pronounced between t h e main c u r r e n t and t h e coast than i n t h e c e n t r a l area ( F i g . 1 b,c).

......... 153,

---

a

-.- IS34 1552

111

C

Fig. 1. P o s i t i o n s o f t h e core o f t h e main Black Sea c u r r e n t a t t h e deDth o f 100 m (a); r.m.s. of temperature ( b ) and s a l i n i t y ( c ) i n August ( f r o m B l a t o v e t a l . 1984).

425

In order to simulate realistically the processes of eddy variability, one has to develop a numerical model having fine enough horizontal resolution. According to Blatov et al. (19841, the baroclinic radius of deformation in the Black Sea is about 30 km. It is obvious that the coarse resolution model discussed in FS with Ay=50km cannot simulate such processes as frontal variability and eddy formation. I n the present work, in order to improve the resolution, we use a grid interval three times finer than in the coarse model. Every coarse grid element is divided into nine fine grid elements A X = 20' and A @ = 10'. I n the vertical direction, the same grid levels as in the coarse resolution model are used.This resolution is still not fine enough to resolve a wide spectrum of mesoscale motions, but as the results indicate, provides good information on the role of the eddies in the Black Sea circu1 at i on. The aim of the present work is to study the response of the Black Sea eddy field to seasonal atmosheric forcing. As the experimental and numerical model results show, the main pycnocline changes its depth throughout the year mainly due to seasonal variations in wind stress. The increase in wind forcing in winter results in increase in the pycnocline slope. The available potential energy increases too. In the transition period between winter and sumner part of the available potential energy at the depth of the pycnocline is released. The meandering of the main Black Sea current can be partly regarded as a transition process between two circulation regimes (winter and summer ones). The seasonal changes i n atmospheric forcing provide substantial variations in the frontal area, and one can expect that the eddy field will also respond strongly to seasonality. Our task in this work is to analyse the establishment of the eddy field in the Black Sea as a result of the seasonal changes in the atmospheric forcing. 2 NUMERICAL EXPERIMENTS Bryan's (1969) oceanic general circulation model and Semter's (19741 code have been used in this study. A detailed description of the physics of the model and of the finite difference scheme may be found in the references given above. The bottom topography was obtained from a bathymetric map, and the depths are resolved approximately by using different numbers of levels at different locations. In the real Black Sea, the surface and volume are 423 000 km' and 537 OOOkm', and in the fine resolution model they are 413 000km' and 543 0o0km3, respectively. The surface boundary conditions are considered in Stanev et. a1 (1988) and in FS. They were calculated in the grid points after interpolation of the climatic wind data of Hellerman and Rosenstein (19831, heat flux data of Makerov

426

(1961) and of precipitation and evaporation data from the Climatic Reference Book of the Black Sea (Sorkina, 1974). All boundary conditions data were determined for the four months: January, April, July, October. The actual values for the boundary conditions at each time step were calculated by linear interpolation. Water balance and the parameterization of strait's and river's inflows are considered in FS. In the horizontal direction every third fine resolution grid point coincides with one grid point of the coarse resolution model considered in FS. The initial values at the points where no information exists were calculated by linear interpolation of the coarse resolution model data from the FS experiment, driven by annual mean forcing. The experiments were carried out in several stages. In the first stage (experiment F1) the model is forced by annual mean boundary conditions and the coefficients of the horizontal turbulent exchange and diffusivity are the same as in the coarse resolution model (AMH = AHH = 4.10' m' s-1). This experiment was run for 5 years. In the first month of integration a splash of energy can be observed due to the change of the grid, to the noise produced by the interpolation of the coarse resolution model data and also to the change in bottom topography. With increasing time, a small increase in energy can be observed, compared to the coarse resolution model. After the second year of integration no trend of energy exists. In the second stage of integration (F2) we decrease the coefficients of horizontal turbulent exchange and diffusion to AMH = 5.101 and AHH = lo2 'm s-l respectively. This stage was integrated for 8 years. In the first 5 years the mean kinetic energy increases strongly due to the decrease of eddy mixing coefficients. In the third stage of the integration (F3) the annual mean boundary conditions are replaced by seasonally varying boundary conditions. Kinetic energy increases in winter, resulting mainly from the increase in wind speed. However the annual mean energy does not change and the model tends to a quasiperiodic state.The process of establishing the steady state is examined also by comparing the northward transport of heat and salt, derived from the boundary conditions and from model data. The first stage F1 of our numerical experiment corresponds to Semtner and Mintz' (1977) spin up experiment after replacing the coarse grid with a fine one. As in Serntner and Mintz, our model results show that in this stage no mesoscale eddies arise. The second stage (F2) corresponds to the Laplacian experiment o f Semtner and Mintz. As in the quoted study, in our experiment the circulation also evolved very quickly toward a state with mesoscale eddies. Unlike Semtner and Mintz we do not carry out biharmonic experiment. In-

427

stead o f t h i s i n t h e t h i r d stage o f i n t e g r a t i o n (F3) we involve time v a r i a t i o n i n boundary conditions. No increase i n the mean k i n e t i c enegy can be observed. However t h e permanent change o f f o r c i n g generates more intense eddy processes than i n t h e stage F2. Below we r e f e r t o F1 as general c i r c u l a t i o n model (GCM) anf t o F2 and F3 as eddy general c i r c u l a t i o n model (EGCM).

3 CIRCULATION The t o t a l stream f u n c t i o n corresponding t o t h e stages F1 and F2 i s shown i n Fig. 2a,b. The decrease o f the c o e f f i c i e n t s o f eddy mixing and eddy d i f f u s i v i t y leads t o a substantial increase i n the t o t a l transport. C i r c u l a t i o n i n t e n s i f i e s e s p e c i a l l y i n the western Black Sea, where the t o t a l t r a n s p o r t obtained from the EGCM exceeds t w i c e t h a t corresponding t o the GCM. I n t h e shallow northern p a r t o f the Black Sea no substantial differences can be observed between the t o t a l mass t r a n s p o r t simulated i n both experiments. Obv i o u s l y i n t h i s area the t r a n s p o r t by surface currents i s compensated by subsurface current transport. The sea surface currents p a t t e r n s corresponding t o the GCM and t o t h e EGCM are c y c l o n i c and q u a l i t a t i v e l y s i m i l a r . However, i n the easternmost Black Sea t h e currents i n the GCM a r e small and mainly wind driven. Examinat i o n o f the model data shows t h a t the surface currents increase t w i c e i n t h e EGCM compared t o ones i n t h e GCM. Below 500 my and e s p e c i a l l y i n the eastern Black Sea, c y c l o n i c c i r c u l a t i o n weakens and some a n t i c y c l o n i c gyres are f o r -

med. The reversal o f the c i r c u l a t i o n explains why the t o t a l t r a n s p o r t decreases i n the eastern Black Sea. The c i r c u l a t i o n i n the western Black Sea r e mains cyclonic down t o the bottom and, as a r e s u l t , t h e t o t a l t r a n s p o r t i s higher than i n the eastern Black Sea. 4 EDDY VARIABILITY The time averaged k i n e t i c energy q = (1 /

TIT:

q dt

E = p o v2 / 2

, where

,

can be divided i n t o two p a r t s : k i n e t i c energy o f t h e time averaged motion Em = p o v / 2 and eddy k i n e t i c energy E, =P v" / 2 .Thus,

428

DEG N

us.1

a

4u.u

u3.u

92.4

u1.u

u0.u 1

2

.28.2

,30.2

,32.2

,%.2

.36.2

,38.2

.M.2

OEG E

DEG N

b

u5.u

uu.4

u3.u

u2. u

41.4

uo.9 ,26.2

,2a.2

,30.2

,32.2

,3u.2

.36.2

,38.2

,110.2

OEG L

Fig. 2. Total transport stream function (in 106 m3 s-1) (a) stage F1, (b) stage F2.

In the following we will analyse the eddy kinetic energy to obtain some insight into the Black Sea eddy processes. After reaching steady state the EGCM is integrated for 8 additional years and the contribution of the seasonal variability to eddy processes is analysed. The time averaging is carried out for a period of four years.

429

4.1 Experiments driven with annual mean forcing Mean kinetic energy in the EGCM (not shown) increases in the western Black Sea and along the south and north branches o f the gyre. In the northern Black Sea the mean energy patterns are more irregular. In the eastern Black Sea energy reaches its minimum.

DEG N

a

u5.u

u4.u

u3.u

u2.u

J=

4I.P

uo. 9 ,26.2

,28.2

,30.2

,32.2

.3u.2

,36.2

.38.2

,90.2

OEG E

,28.2

,30.2

,32.2

.W.Z

,36.2

,3e.t

,40.2

OEG E

)EG N 15.1

19.4

13.4

12. u

11.4

10.4

,26.2

Fig. 3 . Eddy kinetic energy in the experiment driven by annual mean forcing. The dashed line indicates the maximum o f Ee (a) 10 m, ( b ) 210 m

430

Eddy energy at 10 m and 210 m is shown in Fig. 3. Like the mean energy, the eddy energy increases substantially in the western part of the basin, and its patterns are much more irregular than the patterns of the mean kinetic energy. As can be seen from Fig. 3a, in the northern part of the sea the eddy energy increases from East to West along the north branch of the current. The patterns indicate that the fluctuations of the velocity field are advected to the West. The area of strong variability broadens after the current reaches the Crimean Peninsula. I n the northern shallow area the processes are highly transient and the eddy energy increases. The patterns of the eddy energy in surface layers mark a southward distribution o f eddy energy by the mean current along the western coast. However, this energy decreases to the South, which indicates that in the western Black Sea the eddy kinetic energy dissipates. Thus, according to the model results, eddy energy in the surface layers is generated in the northern Black Sea, transported to the west, and dissipated in the western region. The level 210 m corresponds to the Black Sea halocline. It domes strongly in the western Black Sea, where the eddy energy is substantially greater than the eddy energy in the eastern Black Sea. As can be seen from Fig. 3b, the transient processes are intensified after the north branch of the main current reaches the Crimean Peninsula and over the slope separating the deep part of the sea from the shallow one. Close to the Strait of Bosphorus the eddy energy also increases due to the convective processes there and the resulting disturbancies in potential vorticity to the East of the outcrop zone. 4.2 Experiment driven with seasonal variable boundary conditions The mean kinetic energy patterns in the surface layers resulting from the stages F2 and F3 are qualitatively very similar. However, the seasonal variability contributes to some very substantial changes of the eddy energy patterns (Fig. 4). In surface layers (not shown here) the patterns of eddy energy are qualitatively similar to those in the model driven by annual mean forcing. However, some fundamental differences can be observed. In the model driven by seasonally variable forcing, the eddy energy is substantially higher and its patterns are much more regular than in the model driven by annual mean forcing. This result is pronounced in the western part of the sea. Thus, it appears that the sea response to the permanent change in atmospheric forcing results in "better organization" o f the eddy field. There i s a substantial increase of eddy energy along the coast (also in the eastern Black Sea) and the area with maximal values o f eddy energy coincides with the core of the main current.

431

DEG

N

us. 1

a

Y4.U

93.4

U2.4

41.11

YD.9

,26.2

,a.2

,30.2

.32.2

,34.2

,36.2

.38.2

,110.2 DEG

E

DEG N

b U5.Y

10. U ,26.2

,2e.2

,30.2

.32.2

,34.2

,36.2

.38.2

,UO.2

DEG __ E

Fig. 4. As in Fig. 3 . but for the experiment driven by seasonal variable forcing. (a) 210 m, (b) 800 m. Similar results are more easily observed i n deep layers. Unlike the experiment driven by annual mean forcing, in the experiment driven by seasonally variable forcing the maximum of eddy variability in 210 m and 800 m follows the core of the current (compare Fig. 4a with Fig. 3b). Also, although very small, the transient processes along the current's core can be very well distinguished in deep layers (Fig. 4. b).

432

These results can be explained by taking into account the fact that seasonal forcing contributes to some changes in halocline depth. I n winter, when the circulation intensifies, the dome like form of the halocline is more pronounced. Its slope can be observed along the main current (over the continental slope). During the year this slope changes, which contributes to increasing the variability of the velocity field, thus to an increase in the energy of the transient processes. The regularity in the seasonal cycle has as a response regular variations in the halocline slope (approximately over the continental slope), thus increasing the regularity in the eddy energy patterns in the model. The model results can be compared with the experimental results of Blatov et al. (19841, see Fig. 1. As can be seen from this figure the r.m.s. deviations of temperature in the upper 75 m layer correspond qualitatively to the eddy energy distribution (Fig. 3 and Fig 4). Also, there i s a qualitatively good agreement between the r.m.s. deviations in salinity in the layer 100200 m and the model computed eddy energy in 200 m. ACKNOWLEDGEMENTS This study was carried out during the stay of the author as a visiting scientist at the Institute o f Oceanography, Hamburg, ans was supported by the Alexander-von-Humboldt-Foundation. I express my gratitude to H. Friedrich for his support and discussions on the results. Thanks are due to S. Lisse for preparation of manuscript. 5 REFERENCES

Blatov, A.S., N.P. Bulgakov, V. A. Ivanov, A. N. Kosarev and V. 5. Tujilkin, 1984. Variability of hydrophysical fields in the Black Sea, Hydrometeoizdat, Leningrad, 240 pp. (Russian) Bryan, K., 1969. A numerical method for the study of the circulation o f the World Ocean. Journal of Computational Physics, 3, 3, 347-378. Deuser, W.G., 1972. Late Pleistocene and Holocene history o f the Black Sea as indicated by stable isostope studies, J. Geophys. Res., 77 (61, 1071-1077. Friedrich, H. J. and E. V . Stanev, 1987. Parameterization of vertical diffusion in a numerical model of the Black Sea. Proceedings of the 19-th International Liege Colloquium on Ocean Hydrodynamics. (submitted) Hellermann, S. and M. Rosenstein, 1983. Normal monthly wind stress over the World Ocean with error estimates. Journal of Physical Oceanography, 13, 1093-1 104. Makerov, Yu. V., 1961. Heat balance of the Black Sea. Trudy GOIN, MOSCOW, 61, 169-183 (Russian). Semtner, A.J., 1974. An oceanic general circulation model with bottom topography. Numerical simulation of weather and climate. Technical Report, N9. UCLA, Los Angeles, 99 pp. Semtner, A.J. and Y. Mintz, 1977. Numerical simulation o f the Gulf Stream and midocean eddies. J. Phys. Oceanogr. , 7, 208-230.

433

Sorkina, A.I. ( E d i t o r ) , 1974. Reference book on the Black Sea climate, 1974, MOSCOW, Hydrorneteoirdat, 406 pp. (Russian). Stanev, E.V., Trukchev, D.I. and Roussenov, V.M., 1988. The Black Sea c i r c u l a t i o n and numerical modelling o f the Black Sea currents. Sofia U n i v e r s i t y Press, Sofia, 240pp, (Russian).


435

THE DYNAMICAL BALANCE OF THE ANTARCTIC CIRCUMPOLAR CURRENT STUDIED WITH AN EDDY RESOLVING QUASIGEOSTROPHIC MODEL

J.-0. WOLFF, Max-Planck-Institut fur Meteorologie, Hamburg, FRG and

D. J. OLBERS, Alfred-Wegener-Institut fur Polar- und Meeresforschung, Bremerhaven, FRG

ABSTRACT The dynamics of a wind-driven current in a zonal channel is reviewed and investigated with a quasigeostrophic P-plane model with two layers and eddy resolution. The model structure is similar to the one used by McWilliams, Holland and Chow (1978) and the channel has dimensions of 4000 km x 1500 km. The experiments with this model address the problem of the relative role of bottom friction and bottom form drag in the balance of a current driven by a steady eastward surface windstress. The response of the system is investigated for different values of the friction parameter and various locations of topographic barriers in the bottom layer of the channnel. The principal momentum balance emerging from these experiments support the concept of Munk and Palm& (1951) for the dynamics of the Antarctic Circumpolar Current proposing that the momentum input by the windstress is transferred to the deep ocean - in the present model by vigorous eddy activity - where it may leave the system by bottom form drag. Frictional effects in the balance of the circumpolar flow may thus be of minor importance.

1. INTRODUCTION AND A SHORT REVIEW OF CIRCUMPOLAR OCEAN DYNAMICS

Our theoretical understanding of ocean circulation and the dynamical balance of current systems is largely oriented at the large scale, wind-driven flow in ocean basins which is rather uneffected by topography or detailed structures of small-scale dissipation mechanisms and generally governed by Sverdrup type dynamics with some sort of boundary layer. Of course, each current has its pecularities but the one that drops out of the general picture in most respects is the Antarctic Circumpolar Current (ACC) found in the Southern Ocean. The ACC flows almost unconstrained by significant continental barriers in the only zonally unbounded region of the world ocean so that Sverdrupian dynamics do not apply. Though mainly winddriven the current extends with a strong barotropic component down to the bottom and the path of the current clearly shows the influences of the mid-ocean ridges, plateaus and sea mounts in the Southern Ocean (see e.g. the dynamic topographies of Gordon et al., 1978 or the @-spiralmodel of Olbers and Wenzel, 1988). A further important feature of the Southern Ocean - and difference to most other oceanic regions - is the presence of a vigorous eddy field (see e.g. Patterson, 1985) which of course

436

must bear some relations to the dynamics of the ACC and the heat balance of the ocean and the atmosphere. Indeed, contrary to the conditions in the ocean basins to the north, there is convincing observational evidence for an overwhelming contribution of the eddy field to the meridional heat transport in the ocean (deSzoeke and Levine, 1981, see also the review by Bryden, 1983). The suggestion that eddies also may play a significant role in the dynamics of the ACC is however mainly based on theoretical grounds. The basic investigations in this direction have been performed by J. McWilliams and W. Holland with quasigeostrophic models of the flow in a zonal channel on a /3-plane (McWilliams, Holland and Chow, 1978, McWilliams and Chow, 1981). The work on the role of topography and mesoscale eddies in the dynamical balance of the ACC which we present here is an extension of the results of MHC. The dynamical balance of the ACC has long been considered as a mystery. Unlike other current systems the problem with the ACC is not to identify the basic driving mechanism but rather to find the sink for the momentum which is generously imparted to the ocean by the observed surface windstress. Continental barriers are lacking in the Southern Ocean to support a net zonal pressure gradient and thereby oppose the acceleration of the current by the wind. Large-scale models of flow in a zonal channel with the usual diffusive parametrisation of momentum transport by eddies or reasonable magnitudes of bottom friction gave rather unsatifactory results (Hidaka and Tsuchiya, 1953, Gill, 1968 and others). The current amplitudes in these models are proportional to the applied windstress and inversely proportional to the values of the frictional parameters, e.g. for lateral friction the transport scales as .r.D3/Awhere T is the windstress amplitude, D the current width and A the lateral viscosity. Hidaka's dilemma then was to explain the observed transport values with reasonable values for the model friction in view of the observed windstress, i.e. the traditionally accepted values for horizontal and vertical eddy viscosity results in transport values for the ACC which are an order of magnitude above the observed values in the Drake Passage, and conversely, the fit of the models to the observed transport requires unrealistically large eddy coefficients (for T = m2/s2 and D = 1000 km a transport of 10' m3/s yields an A of lo6 m2/s). One of the very early papers on this problem (Munk and Palm&, 1951) already suggested a solution which today still appears as the only acceptable way out of a highly frictionally controlled balance of the current. Though continental barriers are absent there are significant submarine ridges to build up net zonal pressure gradients in the deeper blocked part of the water column. This could enable horizontal momentum to flux out off the system to the solid earth. Of course, a balance with the windstress can occur only if the flow establishes pressure differences across the ridges with the appropriate sign (higher pressure value on the western side for westerly winds) and magnitude: to balance a stress of m2/s2 which is by and large observed in the belt of polar westerlies a pressure difference of only a few dynamic centimeters is required across a ridge of a few thousand kilometers. Furthermore, the momentum given to the ocean at the surface must be transferred somehow down to the blocked depths where fluby pressure differences across topography - the so called bottom form drag - can be effective. This transport of momentum must be extreme large compared

437

to other oceanic areas: to transmit the above value for the windstress in the observed vertical current shear of about 0.1 m/s over 1000 m by vertical diffusion - if this concept is meaningful at all under these conditions - would need a value of a vertical eddy viscosity of 1 m2/s. This is two or even three orders of magnitude larger than elsewhere (large scale oceanic models generally use m2/s). Both requirements in this momentum balance can hardly be tested by observations. The determination of the bottom form drag needs measurements of bottom pressure differences on a geopotential surface correct to a dynamic centimeter which is far beyond todays capabilities. The accurate measurement of the vertical eddy flux of horizontal momentum - i.e. the correlation between the vertical and horizontal velocity fluctuations - is a traditional problem which is still on the margins of the present technology. Models thus appear the only way to a solution. Unfortunately, different models gave different answers. Ocean circulation models with coarse resolution (grid size much larger than the eddy scale) generally denied the balance proposed by Munk and Palm& whereas eddy resolving models support this concept quite well. An early prototype of the coarse model research is the set of experiments reported by Gill and Bryan (1971). In these experiments the combined action of topography (or the idealized submarine ridge in form of the submerged sluice gate in the Drake Passage of Gill and Bryan’s box ocean) and baroclinicity increased the flow, i.e. the bottom form drag worked with the wind to accelerate the current even more. The series of experiments of Bryan and Cox (1972) and Cox (1975) with a coarse global m2/s and ocean circulation model (2” x 2” horizontal resolution with eddy viscosities 4.104 mZ/sfor vertical and horizontal transport) are harder to interpret. The first experiment with a flat bottom homogeneous ocean clearly demonstrates Hidaka’s dilemma: the transport through the Drake Passage has increased to 600 Sv (1 Sv is lo6 m3/s) after 90 days of integration and is still increasing. The introduction of the topography reduces the transport drastically to only 32 Sv. Hidaka’s dilemma is circumvened here by the peculiar pattern of the contours of planetary vorticity f/H in the region of the Drake Passage: the sill in this passage expells almost all contours. The few which pass must stick to the continental margins and thus run far into the adjacent ocean basins to the north. Since the barotropic flow must follow f/H contours (in case of weak forcing) the circumpolar current in the homogeneous topographic ocean resides predominantly in the basins north of the circumpolar belt, where the wind can be balanced by net zonal pressure forces and Sverdrup dynamics apply. In the experiments which include the baroclinicity the ACC gets a transport of 180 Sv. Though the solution presented in Cox’s (1975) paper is definitely not in equilibrium we certainly can draw the conclusion that the baroclinicity enables the system to increase the transport of the ACC. This is confirmed by the many investigations with numerical global models following Bryan and Cox’s work. There main reason why coarse resolution models do not support the balance concept of

438

Munk and Palm& (1951) must be searched in the parametrisation of the turbulent momentum transport. All of these models use diffusive parametrisations with vertical viscosities in the traditional range m2/s and very large horizontal viscosities dictated by to consideration of boundary layer resolution (a 5” resolution like the one by Han (1984) uses 8 lo6 m2/s which is quite in the uncomfortable range of Hidaka’s dilemma). Since the models approximately reproduce the observed vertical current shear the momentum does not flux to the deep ocean with a stress comparable to the windstress because of the low vertical viscosity but the wind input must diffuse away laterally and leaves the Southern Ocean to the northern basins or at the continental boundaries of Antarctica. The bottom form drag is thus not required to act as an effective sink of momentum so that the baroclinic field may evolve quite indepently from the constraint of the overall momentum balance of the current. 1

The work of McWilliams, Holland and Chow (1978) desribes the only eddy resolving experiments in channel geometry with topography we are aware of. They use a two layer quasigeostrophic model of a wind-driven flow in a zonal channel with partial meridional barriers in some experiments to represent a Drake Passage. Lateral boundary conditions allow the flow to freely slip along the walls while the bottom is frictional. There are two experiments related to our question of the role of the bottom form drag: one has a flat bottom and one has idealized topography in the center of the passage - a small Gaussian shaped mound entirely contained in the deep layer. The flat bottom case reflects Hidaka’s dilemma: the friction at the model bottom is too low to keep the transport through the gap to grow to values exceeding 600 Sv. The balance of the zonal momentum resulting for the topographic case is a perfect replica of Munk and Palmkn’s concept: in the passage the upper layer momentum is in balance between the wind input and the interfacial form drag which transfers the momentum to the deep layer where it leaves the system via the bottom form drag. The transport stays below 100 Sv. The interfacial form drag is entirely due to the eddy field which is fed vigorously by the baroclinic instability of the mean current. It is worth to note that the lateral Reynolds stresses exerted by the eddies on the mean flow tend to transfer eastward momentum into the center of the eastward current and concentrate the jet, quite in contrast to the outward momentum diffusion which is inherent in coarse models. The present paper is a report on similar eddy resolving experiments in channel geometry. We compare various experiments with different values of the bottom frictional parameter and various simple topographies with a basic case which is similar to the above described topographic experiment of McWilliams, Holland and Chow (1978). Particular emphasis is put on a detailed analysis of the balance of the time and zonally averaged zonal momentum in all experiments. The work is currently continued with a three layer model with realistic geometry and topography of the Southern Ocean.

2. THE QUASIGEOSTROPHIC CHANNEL MODEL We consider a rotating, hydrostatic, adiabatic fluid with two immiscible layers of different densities in a zonal channel of length L and width W. The upper layer is driven by a prescribed

439 windstress and in the lower layer there is dissipation by Rayleigh bottom friction. The numerical model is based on quasigeostrophic dynamics and in most aspects identical to McWilliams et al. (1978). The basic equations are potential vorticity balances in each layer

Di

[nil

= F ; , i = 1,3

(1)

where index 1 denotes the upper layer and index 3 the lower layer. The potential vorticities, multiplied by the square of the average layer thicknesses Hi, are given by

f:

- V53) + H i f

qi = HrV*$i - ,($I 9 q3

= H3Va$3

+ fo29

- $3)

;($I

(2)

+ H3f + foB

(3)

where the +i are horizontal geostrophic velocity streamfunctions (ui = -& ,vi = &), f = fo @(y - yo) is the Coriolis frequency on a @-planewith the midlatitude of the channel at y = yo, g’ is the reduced gravity and B ( z ,y ) the bottom topography. The operator Di in eqs. (1) denotes the geostrophic substantial time derivative

+

[ a ]

where J is the Jacobian operator. The friction terms in eqs. (1) are taken as T

F, = curl- - A4HlV6$I Pl

F3 = -eH3V2$3

-

A4H3V6$3

where r / p I is the windstress normalized by the upper layer mean fluid density, A4 and coeffients for biharmonic and bottom friction, respectively.

e

are

The boundary conditions for eqs. (1) are C i N ( t ) on the northern solid wall 6 a ~ cis(t) on the southern solid wall 6!&

V z $ , = V4$i = 0

on both solid walls

(i = 1,3).

These conditions (8) assure that there is no flow through the solid walls and no sources or sinks of momentum or volume integrated energy at these boundaries. All quantities are assumed to be periodic on the basin length L. The model domain of a periodic annulus is a multiply connected domain with an inner island (Antarctica) so that four auxiliary conditions are required to determine each of the constants c i in eqs. ( 8 ) . The following conditions were chosen for the numerical calculations (McWilliams, 1977, McWilliams et al., 1978).

+

H I C I N H ~ C ~= N0 on 6 0 ~

(9)

440

Here y* is the meridional coordinate of a zonal line in the middle of the channel, s and n (with unit vectors s^ and A ) define coordinates parallel and perpendicular to the southern boundary (li is antiparallel to the direction of y). The condition (9) results in an elimination of the dynamically irrelevant pressure signal on the northern solid boundary SRN. Condition (10) is an integral of the time rate of change of the interfacial deviation 9 = f0(+3 - 31)/g' over the model domain R. Under the assumption of vanishing of the ageostrophic normal velocity on the boundaries this condition prevents a netto mass exchange between the two fluid layers, i.e. the total volumes of the two different watermasses remain constant during the time of integration. The partial time derivative can be omitted if the integration starts from a state of rest. The two conditions (11) and (12) garanty the continuity of the ageostrophic pressure fields in the two layers around the southern island. The auxiliary conditions can be derived from the continuity equation and the ageostrophic momentum balance. Eqs. (1) - (12) fully determine the evolution of the model variables in both layers. The numerical integration of the model equations uses standard, second order finitedifference discretizations on an Arakawa-C-type grid for velocities and streamfunction. For the calculation of the Jacobian terms an energy and enstrophy conserving formulation is used (Arakawa and Lamb, 1977). The time discretization was done with the leap-frog-method where three timesteps are mutually connected. To avoid time splitting of the developing solutions a projection of the timesteps n 1 and n - 1 on the timestep n was done every 100 timesteps. The bottom friction and viscosity terms are calculated at the timelevel n - 1 to avoid (linear) numerical instability (Richtmyer, 1967).

+

The experiments considered here are carried out in a periodic circumpolar two-layered channel with 1500 km width and 4000 km length centered at 60" S. The southern boundary of the channel is viewed as an idealized coastline of Antarctica. The mean depths of the two layers are 1000 m for the upper layer and 4000 m for the lower layer. The eastward windstress T is assumed to be zonally constant and meridionally varying in a sinusoidal shape with a maximum amplitude TO = 0.1 N/m2 at the central latitude of the model area T

XY = To sin (-)

L,

The reduced gravity g' = gAp/p is 0.02 and the biharmonic friction parameter is A4 = 10'0m4s-'. For the values of the bottom friction parameter in the following experiments (e = lO-'s-' and E = 6 . 10-7s-1) bottom friction is the dominating effect. The spin down times are e-' M 116 days and 19 days, respectively, whereas the lateral friction has a spin down time of AT'L: M 3 . lo6 years (channel scale flow) or A;'(Ax)~ M 170 days (gridsize A x = 20 km). The timestep was chosen to be 2 hours.

441

3. FLOW CHARACTERISTICS OF THE BASIC EXPERIMENT The flow in the quasigeostrophic model has been investigated for various cases of different topography and numerical values of the bottom friction parameter. We started here to describe the basic case (CASE A) where the bottom friction is e = lO-'s-' and the bottom topography is a Gaussian mound with a maximum amplitude of 500 m (Fig. 1). Notice the little shift to the south from the center latitude of the channel. The spinup of the model is characterized by a barotropic response of the water column to the onset of the wind, i.e. both layers immediately build up a meridional pressure gradient through Ekman transport in the upper layer, which leads to a geostrophic transport along the channel. The transport of the lower layer is 4 times larger than the transport of the upper layer because of the layer thicknesses ratio of 1 to 4 (Fig. 2a). The barotropic response timescale is roughly 3 months. The streamfunctions and interfaces during this phase are exemplified in Fig. 3a. The baroclinic timescale, i.e. the time which is needed to build up sufficiently strong velocity gradients to allow for the mechanisms of barotropic and baroclinic instabilities, is much greater. Baroclinic instability transforms available potential energy into kinetic energy on the scale of the internal Rossby radius of deformation which is 32 km for the parameters under consideration. This instability process can be seen to start acting in the time series of the potential and kinetic energies (Fig. 2b) after the phase of relatively smooth build-up of the baroclinic flow over a few years. After about 8 years the development of the potential energy changes from a smooth increasing behaviour into a strong irregular oscillation and the kinetic energies start to vary with higher amplitudes and on a higher level in both layers. The potential energy is roughly 10 times larger than the kinetic energies in this state of quasistationary equilibrium. Fig. 3b shows the model state at the end of the experiment after 22 years. The flow field is characterized by a vigorous eddy field in both layers and greatly deformed interfaces. Typical instantaneous streamfunction patterns during the quasistationary equilibrium are shown in Fig. 4. A remarkable permanent feature of the flow is a strong jet stream in the northern part of the channel in both layers, which can be seen clearly in the time mean streamfunctions (the average over the lwt 11 years). The topography tends to concentrate the more or less broad upstream motion into a strong jet and deflects the direction to roughly 45" to the south. Downstream of the topography barrier the time mean jet broadens again with a superimposed standing lee wave pattern. This time mean solution shows remarkable similarities to the analytical solution for the flow of a zonal current over an isolated seamount of McCartney (1976)and Spillane (1978). The time mean transports of the basic case are 161 Sv for the upper layer and 180 Sv for the lower layer ( I Sv = lo6 m3/s) with maximum zonal velocities of 26.4 cm/s and 11.6 cm/s, respectively. The typical horizontal scale of the transient eddies in the upper layer is 150 - 350 km and in the lower layer 200 - 500 km.

442

,'.I

ZONAL SECTION OF BOTTOM-TOPOGRAPHY

A

0

L

Fig. 1. Bottom topography (CI=50 m), a sltctched zonal scction of thc channel and thc shape of the applied wind stress for the basic experiment (CASE A). sv 400 1

a) .LAYER 3

320

240 160 80

YEARS

630

-

560

-

490

-

420

-

350 -

/

70

YEARS

Fig. 2. a) Time series of the transports in the uppcr and lower laycr b) Timc scrics of the potential and layer lcinetic encrgies of CASE A.

443

Fig. 3. Velocity streamfunctions and intcrfaces aftcr a) 80 days and b) 22 years with arbitrary amplitudes and contour intervals.

Fig. 4. Instantaneous and time mean streamfunctions of CASE A.

445

4. T H E BALANCE OF ZONAL MOMENTUM

The complex flow described above is ultimately forced by input of zonal momentum into . understand the the upper layer by the simply shaped, purely zonal wind stress T = ~ ( y ) To intriguing mechanisms of redistribution and transfer of momentum we consider the balance of the zonal momentum h;ui (i = 1,3) given by

+

where hi are the layer thicknesses. The pressure fields are pl = gC and p3 = gC 9/71 with J T and TL the surface and interface deviations C and 7, respectively. For the upper layer T ~ = is an interfacial frictional stress T I , whereas for the lower layer TU = TI and TL = ~h3u3,the bottom frictional stress. Restricting our interest to the time and zonal average momentum in the quasistationary state

( h;u; )

1 =

A

AL L

1 =-

L

dx

T

dx

1 .J To

dt hiu,

we notice (since (hivi) = 0) that only the advective, pressure and friction terms remain in the balance. With no explicit vertical momentum diffusion (i.e. TI = 0) as considered here in the eddy resolving experiments the only net transport of mean momentum from the surface layer to the deeper layer is by the interfacial pressure drag ~

~

aPi apt -(hi-) = -(q1-)-

ax

ax

aix a(-) ax

where 71 = ( - 71 M -71 and 713 = q - B. The second term vanishes because of the auxiliary conditions (11) and (12). The balance of the mean zonal momentum is then expressed as

The actual layer thicknesses h; may be replaced here by the constants Hi. It may be worth to mention that these balances of the mean zonally averaged momentum are effectively statements about the vorticity dynamics of the system. It can be shown, using the auxiliary conditions (9 - 12) of McWilliams (1977) in the original momentum form, that eqs. (16) and (17) are the balance of the time averaged circulation around a loop consisting of the respective latitude line, the northern or southern boundary and closing meridional sections at the periodic eastern and western boundaries. Equivalently, eqs. (16) and (17) represent the balance of the time averaged vorticity, integrated over the area enclosed by this loop. In the quasigeostrophic approximation we have u; = --qbiv, vi = and 71; 0, since v < 0 and h < 0. Thus hlt< 0 and and the DWBC, v1 1 2g 1Y 2g 1Y the upper layer thickness decreases locally. As the layer thins, potential vorticity is conserved via the flow gaining anticyclonic relative vorticity and/or leaving the coast at a more southerly latitude, in effect shifting the Gulf Stream southward along the continental slope, as indicated in the 20 Sv (dashed-dot) curve of Fig. 4.

.

3.4 Deep flow Fig. 5a shows the mean sea surface for Exp. 5 (DWBC =20 Sv) after statistical equilibrium is reached while the corresponding deep mean pressure and the deep mean velocity vectors are shown in Fig. 5b. The mean velocities readily indicate the location of the DWBC along the continental slope and tend to follow the contours of f/h. The deep pressure pattern is somewhat similar to that without the DWBC but is considerably strengthened. With the exception of eddy energy estimates, the most extensive data set which can be compared in the western North Atlantic over long time periods is the deep mean flow. Hogg

473

Fig. 5. (a) Mean SSH for Exp. 5a.

Contour interval

is 5

cm.

(b)

Density-

normalized lower-laye5 pressure anomaly and deep mean currents. Contour interval is .1 m sec-

.

474

(1983) compiled all of the long-term deep current meter measurements in this region. He has recently updated that map (Hogg et al., 1986). While there are numerous detailed comparisons to be made we will focus on four specific areas: (1) the recirculation zone near the region of the HEBBLE experiment at 41 N, 63 W , ( 2 ) the foot of the continental rise near 68 W and 37 N, (3) the region of the "northern recirculating gyre" north of 40 N along 55 W, and (4) the flow around isolated seamounts such as Corner Rise. Hogg (1983) identified two separate deep flow regimes, the classical DWBC flowing along the continental rise and a second system of recirculating classical slope water gyres further offshore. However these regimes are not always distinct. For example, Hogg (1983) and Hogg et a1.(1986) found that the Wunsch and Grant (1982) circulation scheme, deduced from an inversion of hydrographic data, could not easily account for the HEBBLE data. It suggested as much as 40 Sv flowing along the 4000 m isobath near 63 W was not evident at 70 W. Hogg surmised that there was a decoupling between the two currents such that a recirculation existed in the deep flow in the HEBBLE area. Such a recirculation is very much evident in the deep mean flow from the model with the magnitude of the recirculation being about 35 Sv. This circulation is indeed decoupled from that near 70 W. In fact, a separate "near-mesoscale" recirculation zone exists to the southwest along the foot of the continental rise centered near 37 N and 68 W and may explain the Rise Array data. In particular, the shoreward flow at the 4800 m isobath near 37 N and 68 W is found in the observations. The model results clearly point toward the need for additional measurements to determine if a closed gyre exists. In Hogg's 1983 schematic the current meter array near 40 N, 59 W was not available and in fact Hogg suggested the deep flow in this region was westward. However, new measurements suggest that the currents in this area are flowing toward the northeast. The model results show that the eastern side of the "Hebble recirculation" is also the western side of a small anticyclonic gyre which splits what Hogg termed the "northern recirculating gyre" into two separate closed cyclonic gyres. The anticyclonic circulation is reminiscent of that suggested by Schmitz (1980) and by Richardson (1981) from one degree box averaged temperature data at 450 m., although the model gyre is further north and west of that indicated in the observations. The eastern cyclonic gyre centered near 41 N and 55 W is dynamically similar to that outlined by Hogg and Stommel (1985) in that this deep gyre is nearly potential vorticity conserving. The significant influence of the DWBC on the Gulf Stream path clearly extends into the model interior. This change in the interior has radical consequences for the deep eddy field as well. In Fig. 6 we have plotted the deep eddy kinetic energy for the 0/20/40 Sverdrup DWBC experiments (4, 5a, 6 respectively). In the case with no DWBC (Fig. 6a) the deep EKE is largely

475

Fig. 6. (a) Deep eddy kinetic energy for

5a sec

(20 Sv DWBC), -2

.

(a) Exp.

Exp. 6 (40 Sv DWBC).

4

(no DWBC),

(b)

Exp.

Contour interval is 10 cm 2 Crosshatched areas indicate deep EKE values greater than 40 cm sec-’. and

(c)

2

476

confined to a region west of the NESC with maximum values less than 80 cm2sec-2 The NESC is also evident in the EKE contour pattern. In Fig. 6b we have plotted the EKE for the 20 Sverdrup experiment. The distribution of EKE is more equally distributed east and west of the NESC with maximum values near 50 cm2sec-2 to the east and 90 cm2sec-2 to the west. For the 40 Sverdrup DWBC case (Fig. 6c) the EKE field is almost entirely shifted to east of the NESC 2 with maximum values near 100 cm sec-’. These three experiments demonstrate that increasing the DWBC shifts the EKE distribution further eastward and toward higher amplitude as the mean path of the Gulf Stream shifts southward. The intercomparison of deep EKE suggests that the model amplitudes are too small by roughly a factor of two in comparison with the observations. All else being the same? time-dependent fluctuations in the inflow transport (or angle) and the wind driving should both act t o increase deep EKE. While fluctuations in direct wind driving are unlikely to contribute significantly to abyssal EKE (Niiler and Koblinsky, 1985), fluctuations in volume transport on seasonal to interannual time scales near Cape Hatteras may be much more critical. We do not suggest that the vertical and horizontal resolution, friction parameters, and stratification are less important in altering the amplitude and distribution of EKE in the model. In fact model sensitivity studies suggest that they may be more important than time dependent forcing in this regard. However, for the experiments presented in this paper, time-dependent forcing would yield larger, and thus more realistic EKE amplitudes. In Fig. 7a we show the rms SSH variability of Exp. 5a (20 Sv DWBC). The maximum value is about 39 cm to the west of the NESC chain. This corresponds well in amplitude with values determined by Marsh et al. (1984) based on SEASAT and GEOS-3 crossover data although location of the maximum is several degrees too far west in the model results. With the advent of GEOSAT we now have additional data on variability of the SSH in the Gulf Stream region to compare with these model results (Hallock et al., 1988). Fig. 7b compares the maximum rms variability from Exp. 5b at each longitude with Dantzler’s (1977) calculation of the rms variability of the 15 C isotherm at 37 N. and 39 N. The magnitude of the rms thermocline variability is 10-20% higher in the experiment with bottom friction (5b) than without (5a). We are encouraged by two aspects of this comparison. First, the amplitudes of rms thermocline variability of the model are only slightly lower over the range of longitudes than observed. Since our model thermocline has a mean depth of 1000 m the variability of the 15 C isotherm is probably underestimated. This is offset by the fact that model variability was chosen not along constant latitudes lines but the latitude of maximum variability. Second, the model rms variability did not decrease signficantly downstream to 45 N. at the outflow boundary. This suggests the boundary condition did not artificially constrain stream variability near the outflow.

.

477

RMS SEA SURFACE HEIGHT

YIN

65 I

N ATUNTIC 585332 B B

DF =

=

500

CM

UAX = 4 1 233

1.268

W. Long. (deg.) 55 45 I

1

35 I

250

r"

150

150

50

50 I

I

I

75

65

I

I

I

55

45

35

W. Long. (deg.) Fig. 7. (a) Root Mean Square of SSH for Experiment 5a. Contour interval is 5 cm. (b) RMS of 15 C isotherm from Dantzler (1977) at 37 N and 39 N and maximum variability from the model thermocline from Exp. 5b.

478

Fig. 8 shows a histogram for the mass transport at 55 W for the (a) upper layer, (b) for the sum of the upper and lower layer, averaged over one degree latitude bands. The recirculation zones are dramatically illustrated in these figures. Note that the total eastward transport, which includes a significant eddy-flux induced circulation, is 194 Sv between 39 and 41 N (recall the specified inflow at Cape Fear is 50 Sv). This is much larger than the total mean transport of 93 Sv at 55 W by Richardson (1985). However, the appropriately weighted difference between model upper and lower layer transports associated with the model Gulf Stream analogue yields a value (relative to the bottom) of 93 Sv, close to the 90 Sv estimate by McCartney et al. (1980) for the same calculation from observations at 55 W. (See also Schmitz, 1980). 4 SUMMARY AND CONCLUSIONS A primitive-equation numerical model has been applied as a two-layer analogue of the Gulf Stream System to a limited area from Cape Hatteras to east of the Grand Banks (78-45 W, 30-48 N). Experiment 1 consisted of a flat bottom regime driven by wind forcing only. Realistic inflow transport above the main thermocline was then prescribed for two different outflow specifications at the eastern boundary in Experiments 2 and 3 . Three other model runs included (4) bottom topography, ( 5 ) a deep western boundary current with 20 Sv total transport added to Exp. 4 , and (6) increasing the Deep Western Boundary Current (DWBC) in Exp. 5 to 40 Sv. Without a prescribed inflow (Florida Current Transport) the resulting velocity fields are much too weak, as expected. For a realistic inflow transport exiting at a fixed narrow outflow location both the mean and eddy fields are quite unrealistic. For an entirely open eastern boundary the modelled Gulf Stream System with realistic topography is more energetic at abyssal depths but is located too far north relative to observations. The addition of the DWBC moves the mean path of the midlatitude jet southward toward a more realistic configuration, shifting the region of increased abyssal EKE eastward thus improving the comparison with deep observations at 55 W. The 20 Sv DWBC experiment (Exp 5 ) also yields a deep mean flow which corresponds to the observational data in several areas, including recirculations north and south of the Gulf Stream and a deep cyclonic gyre just to the east of the northern part of the NESC. Another deep cyclonic circulation at the foot of the continental rise centered near 37 N and 68 W and anticyclonic deep flows around several isolated topographic features such as the Corner Rise Seamounts and Bermuda are also indicated in the model results but as yet have little or no observational data to confirm them. The amplitudes of upper level eddy kinetic energies and the mean flow are within

479

SECTION POSITIONS

TRANSPORT 58533 2

8.8

NA TFUNSPORT 58533.2

8.8

NA

LAYER 1

SECTION POSITIONS LAYERS I + 2

45N

40N

35N

30N

Fig. 8. (a) Mean zonal transport histogram averaged over 1 degree latitude bands at 55 W. from Exp. 5a for the (a) upper layer and (b) sum of upper and lower layer. Each degree longitude represents 10 Sv of transport.

480

at least a factor of two of the data base and the model equivalent Gulf Stream transport at 55 W is potentially relevant. ACKNOWLEDGEMENTS This paper is a contribution to the Regional Energetics Experiment (REX) program sponsored by the Office of Naval Research to the Naval Ocean Research and Development Activity under the Accelerated Research Initiatives "Ocean Dynamics from Altimetry" (ONR 32-05-3F) and the "GEOSAT Exact Repeat Mission" (ONR 32-05-36) (supporting JDT). WJS was supported in this investigation by the ONR and the Institute for Naval Oceanography. The IN0 is sponsored by the Navy and administered by the Office of the Chief of Naval Research. Thanks are due to Harley Hurlburt, Jim Mitchell, George Heburn, John Kindle, and Alan Wallcraft for assistance in model development and analysis and many aspects of manuscript preparation. Appreciation is extended to Lina Lo and Woody Woodyard of JAYCOR for programming assistance and figure preparation. Calculations were performed on the Cray XMP-24 at the Naval Research Laboratory in Washington, D.C. and on the Institute for Naval Oceanography's VAX-8800 at the Stennis Space Center, Mississippi. Graphics software was in part provided by the National Center for Atmospheric Research (sponsored by the National Science Foundation). Appendix A List of Symbols

A b' f g Hi hi

Pi t -f

i'

horizontal eddy viscosity coefficient of bottom friction Coriolis parameter acceleration due to gravity reduced gravity, equal to g(p2 - p l ) / p thickness of layer i at rest instantaneous layer thickness of the ith layer density normalized pressure for layer i time current velocity in layer i with x-directed component u.and y-directed component v i volume transport, hivi geostrophic current velocity

481

tangent plane Cartesian coordinates: x positive eastward, y positive northward, and z positive upward time step in the numerical integration horizontal grid increments for each independent variable

At Ax, AY

free surface height anomaly (FSA), height of the free surface above its initial elevation densities of seawater x- and y-directed tangential stresses at the top (i) and bottom (itl) of layer i respectively REFERENCES Adamec, D., 1988: Numerical simulations of the effects of seamounts and vertical resolution on strong ocean flows. J. Phys. Oceanogr., 18, 258269.

Auer, S. J., 1987: Five-year climatological survey of the Gulf Stream System and its associated rings. J. Geophys. Res., 92, 11709-11726. Bogue, N.M., R.X. Huang, and K. Bryan, 1986: Verification experiments with an isopycnal coordinate model. Phys. Res:, 16,985-990. Boris, J.P. and D.L. Book, 1973: F ux correcte transport. I. SHASTA, A fluid transport algorithm that works. J. Com ut. Ph s . , 11, 38-69. Busbee, B.L., Dorr, F.W., George, J.A., Golu;, G.H.): 1 9 7 E The direct solution of the discrete Poisson equation on irregular regions. SIAM J.N.A., 722-

i.

736.

Cox, M.D., 1985: An eddy resolving numerical model of the ventilated thermocline. J. Phys. Oceanogr., 10, 1312-1324. Dantzler, H.L., 1977: Potential energy?;iaxima in the tropical and subtropical North Atlantic. J. Ph s . Oceano-r., 7, 512-519. Fisher, A., 1977: Histori:al limitsgof the northern edge of the Gulf Stream. Gulfstream. 3 , 265-273. Halliwell, G.R., Jr. and C.N.K. Mooers, 1983: Meanders of the Gulf Stream downstream from Cape Hatteras. J. Ph s. Oceano ., 13, 1275-1292. Hallock, Z.R., J.L. Mitchell, and J.0: Thompson: 1988: Sea surface height variability near the New England Seamounts: an intercomparison among in situ observations, GEOSAT altimetry, and numerical simulations. (to be submitted to J. Geophys. Res.). Harrison, D.E., 1982: On deep mean flow generation mechanisms and the abyssal circulation of numerical model gyres. Dynamics of Oceans and Atmospheres, 6, 135-152. Hellerman, S. and M. Rosenstein, 1983: Normal monthly wind stress over the world ocean with error estimates. J. Phys. Oceanog., 13, 1093-1104 Hockney, R.W., 1965: A fast direct solution of Poisson’s equation using Fourier Analysis. J. Assoc. Comput. Mach., 12, 95-113. Hogg, N.G., 1983: A note on the deep circulation of the western North Atlantic: Its nature and causes. Dee -Sea Res 30, 945-961 Hogg, N.G. and H. Stommel, 1985: On t h h ; e G the deep circulation and the Gulf Stream. 32, 1181-1193. Hogg, H.G., R.S. Pickart, R X . Hendry, and W.J. Smethie, Jr., 1986: The northern recirculation gyre of the Gulf Stream. Deep Sea Res., 22, 11391165.

Holland, W.R., and W.J. Schmitz, 1985: Zonal penetration scale of midlatitude jets, J. Phys. Oceanog., Is, 1859-1875. Hurlburt, H.E., 1984: The potential for ocean prediction and the role of altimeter data. Mar. Geod., 8, 17-66. Hurlburt, H.E., and J.D. Thompson,-1973: Coastal upwelling on a beta plane. JPhys. Oceanog., 2, 16-32.

482

Hur burt, H.E., and J.D. Thompson, 1980: A numerical study of Loop Current intrusions and eddy shedding. J. Phys. Oceano ., 10, 1611-1651. Hur burt, H.E., and J.D. Thompson, 1984: Pre1imina;y Results from a Numerical Study of the New England Seamount Chain Influence on the Gulf Stream. Proceedings of the Workshop on Predictability of Fluid Motions, American Institute of Physics. Ed. G. Hollaway. Lai, D.Y., 1984: Mean flow and variabilities in the Deep Western Boundary Current. J. Phys. Oceanog., 14, 1488-1498. Marsh, J.G., R.E. Cheney, J.J. McCarthy, and T.V. Martin, 1984: Regional mean sea surface based upon GEOS-3 and Seasat altimeter data. Mar. Geod.,

s,

385-402.

McCartney, M.S., L.V. Worthington, and M.E. Raymer, 1980: Anomalous water mass distribution at 55 W in the North Atlantic in 1977. J. Mar. Res., 147172.

Mellor, G.L., C.R. Mechoso, and E. Keto, 1982: A diagnostic calculation of the general circulation of the Atlantic Ocean. Deep-sea Research, 29, 11711192.

Niiler, P.P. and C.J. Koblinsky, 1985: A local time dependent Sverdrup balance in the Noortheast Pacific, Science, 229, 754-756. Olbers, D.J., Wenzel, M., and J. -ant 1985: The inference of North Atlantic circulation from climatological hydrographic data. Rev. Geo h s 23, 313-356. Orlans&* n 7 6 : A simple boundary condition for unbounded hyperbolic flows. J. Compi9;2s., 21, 251-269. Richardson, P.L., GulfStream trajectories measured with free-drifting buoys. J. Ph s . Oceano r., 11, 990-1010. Richardson, P.L., ;983: Eddygkinesc energy in the North Atlantic from surface drifters. J. Geophys. Res., 88, 4355-4367. Richardson, P.L., 1985: Average velocity and transport of the Gulf Stream near 55 W. J. Mar. Res., 43, 83-111. Richardson, W.S., W.J. Schitz, and P.P. Niiler, 1969: The velocity structure of the Florida Current from the Straits of Florida to Cape Fear. DeepSea Res., Suppl., 225-231. Riser,=. Freeland, and H.T. Rossby, 1978: Mesoscale motions near the deep western boundary current of the North Atlantic. Deep-sea Res., 11,

s,

1179-1191.

Schmitz, W.J., 1980: Weakly depth-dependent segments of the North Atlantic circulation. J. Mar. Res., 38, 111-133. Schmitz, W.J., Jr. and W.R. Holland, 1982: A preliminary comparison of selected numerical eddy-resolving general circulation experiments with observations. J. Mar. Res., 40, 75-117. Schmitz, W.J., and W.R. Holland, 1986: Observed and modelled mesoscale variability near the Gulf Stream and Kuroshio Extension. J. Geophys. Res., 91, 9624-9638. Stornmem.M, 1958: The abyssal circulation. Deep-sea Res., 5 , 80-82. Swallow, J.C., and L.V. Worthington, 1961: An observation of a deep countercurrent in the western North Atlantic. Deep-sea Res., 8, 1-19. Thompson, J.D., 1986: Altimeter data and geoid error in mesoscale ocean prediction: some results from a primitive equation model. J. Geophys. Res., 91, 2401-2417. ThompscJ.D., and H.E. Hurlburt, 1982: A numerical study of the influence of the New England Seamount Chain on the Gulf Stream: Preliminary results. Proceedings: Workshop on Gulf Stream Structure and Variability, Office of Naval Research, Ed. J.M. Bane, Jr., Univ. of N.C., Chapel Hill. 346362.

Thompson, J.D., and W.J. Schmitz, 1989: A limited-area model of the Gulf Stream: design, initial experiments, and model/data intercomparisons. JPh s Oceano (to appear). Van *;., 1979: SYNBAPS, Volume I . Data sources and data preparation. Naval Ocean Research and Development Activity, NSTL, Ms., NORDA Tech. Note 35. Wallcraft, A.J., 1980: Capacity Matrix Techniques. Ph. D. Thesis. Dept. of Mathematics, University of London. 267 pp.

483 Wallcraft, A . J . , and J.D. Thompson, 1984: Ocean modelling and drifters. 1984 Drifting Buoy Workshop Procedings. Mar. Tech. SOC., Gulf Coast Sec., Nat. Space Tech. Lab., Miss. 81-98. Wunsch, C. and B. Grant, 1982: Towards the general circulation of the North Atlantic Ocean. Progress in Oceanography, 11, 1-59.


485

A SYNOPSIS OF MESOSCALE EDDIES IN THE GULF OF MEXICO A.W. INDEST, A.D. KIRWAN, JR. Old Dominion University, Department of Oceanography, Norfolk, Virginia 23529-0276 (USA)

J.K. LEWIS Science Applications International Corporation, College Station, Texas 77480 (USA) P. REINERSMAN University of South Florida, Department of Marine Sciences, St. Petersburg, Florida 33701 (USA)

ABSTRACT Here we review hydrographic and drifter observations of Gulf of Mexico rings. Typical sizes of these structures are 200 km in diameter as inferred from the maximum temperature gradient with swirl speeds often reaching one d s . They have been observed to transport 30 sverdrups of Caribbean water to the western boundary of the Gulf. As many as three of these structures have been observed in the Gulf of Mexico at one time. They can exhibit elliptical shapes particularly as they approach the western slope or other eddies. Eddies have been observed to follow a west southwest path over the abyssal plain, or to take a more nonhernly route roughly paralleling the northern continental slope. We also compare kinematic characteristics as inferred by drifter observations with Lagrangian data generated by the Hurlbuflhompson eddy resolving general circulation model. This model produces eddy paths that are remarkably similar to the observed transits across the abyssal Gulf. The model also shows some elliptical structure in the eddies as well as their northward migration along the western slope. During the latter period, the agreement between the model and the observations is remarkably good. This results in the unusual situation of the prediction of mesoscale features by an eddy resolving general circulation model actually improving in time. However, not all of the simulated kinematics agree with the observations. Some of the model swirl velocities for outer orbits significantly exceed observed swirl velocities despite the fact that the rotational period of the model (22-23 days) is larger than the observed rotational period (11-15 days). INTRODUCTION Prior to 1980, the concensus picture of the circulation in the Gulf of Mexico (GOM) was a vigorous Loop Current on the eastern side of the basin and a poorly defined anticyclonic circulation in the interior. This was based on qualitative descriptions of hydrographic and geomagnetic electro-kinetograph data. A reevaluation of this data, along with the climatological surface winds by Elliot (1979, 1982) showed that the anticyclone structures were strong and likely represented rings spawned by the Loop Current. This gave credence to earlier but generally ignored work by Ichiye (1962) who had postulated that rings could pinch off the

486

Loop Current and migrate into the Western GOM. As will be seen shortly, these structures are quite large (diameters - 200 km as inferred from the maximum temperature gradient) and intense (swirls - 1 m/s). In renospect, it is surprising that such features were consistently misinterpreted by most workers in the 1960’s and 1970’s. Since 1980, an impressive amount of data has been amassed on the pinch off of Loop Current rings and their subsequent migration across the GOM. Kirwan et al. (1984a,b, 1988) provided the first direct evidence of the westward migration of Loop Current rings. Merrell and Morrison (1981) and Merrell and Vazquez (1983) showed that dong the western slope the rings often were paired with cyclones and that the offshore transport between pairs could be 30 sverdrups, the same as that entering the Gulf through the straits of Yucatan. Lewis and Kirwan (1985, 1987) have used drifter and hydrographic data to study both ring topography and ring-ring interaction and the separation of a ring from the Loop Current. Concurrent with the observations has been a substantial amount of theoretical and modeling studies. This started with the Hurlburt and Thompson (1980) two-layer, nonlinear, primitive equation model of eddy shedding by the Loop Current and the general circulation of the GOM. That study stimulated further modeling efforts by Hurlburt (1986), Kindle (1986), and Thompson (1986), all of whom looked at various aspects of the problem of incorporating altimeter data into GOM eddy-resolving general circulation models. The interaction of the Loop Current rings with the continental slope on the western side of the basin has also stimululated theoretical studies. Smith (1986) examined this interaction in some detail with a twolayer nonlinear primitive equation model and Nakamoto (1986) has applied solitary wave theory to explain the northward migration of the rings along the western slope. The purpose here is to summarize the recent observational and modeling results on the synoptic motions in the GOM. Perhaps the best overview of this circulation can be obtained not from the historical hydrography but from the Hurlburt-Thompson eddy-resolving general circulation model of the GOM. Section 2 provides a synopsis of events as determined by that model. Then in Section 3, hydrographic and drifter observations given by Lewis and Kinvan (1987) on the spawning of a ring by the Loop Current and subsequent migration of the ring along the northern part of the GOM are presented. Section 4 summarizes the results of Kirwan et al. (1988) on the southwest movement of rings across the central part of the GOM and on the comparison of this movement with predictions of the Hurlburt-Thompson model. Finally, in the last section, we discuss some of the implications of these results. 2

SYNOPSIS OF MESOSCALE CIRCULATION HURLBURT-THOMPSON MODEL

AS

DETERMINED BY THE

Perhaps the picture that best represents current thinking on the mesoscale motions in the GOM comes from the Hurlburt-Thompson model. This is a two-layer (upper layer 200 m) primitive equation model with realistic bottom topography below 500 m. The general sequence of events along with numerous other details is depicted in Figure 1, taken from Wallcraft (1986).

487

Fig. 1. Upper layer velocities for model days 063 and 153 for panels a and b. The top layer is 200 m deep. Maximum plotted speeds for panels a and b are .95 and .98 d s .

488

Fig. 1 (cont.). Upper layer velocities for model days 243 and 333 for panels c and d. The top layer is 200 m deep. Maximum plotted speeds for panels c and d are .94 and 1.10 m/s.

489

Panel a shows the situation with the Loop Current fully extended into the GOM. There are several mesoscale features that should be noted. Just to the northwest of the extension there is a small cyclonic flow structure. Farther to the west there is another cold core eddy. West and south of this latter feature are two anticyclones. These may be the remnants of a previously shed Loop Current ring. Finally, note the anticyclonic circulation within the loop just to the northwest of Cuba. Lewis and b a n (1987) have speculated that this "Cuba Eddy" can be a critical factor in the ring spawning process. In panel b, the Loop Current has pinched off with the result that there is a large anticyclonic ring in the central part of the Gulf. The north-south extension of the maximum velocities of this ring spans the region between 23"N and 27"N. The ring itself is distorted, probably due to interactions with the bottom topography to the north and south as well as the cyclones and anticyclones farther to the west and south. To the north and west of the ring are two smaller anticyclones and a cyclone. These are remnants of the features noted in panel a. The fourth cyclonic feature in the first panel is now in the extreme northwest comer of the Gulf. Note the Loop Current has reformed with a direct connection between inflow through the Yucatan channel and out flow through the Straits of Florida and, south of the flow, there is an indication of the Cuban Eddy. Panel c shows the Loop Current beginning to extend into the Gulf. North of the Loop Current, a small anticyclone has already formed. Farther to the west the ring has acquired an elongated "peanut" shape. This characteristic is often seen in drifter paths. As suggested by Lewis and K h a n (1985), this is probably due to strong interactions with the continental slope to the west and other cyclones and anticyclones. Just to the north of the ring, there is the persistent anticyclone noted in the previous panels. One cyclone is northeast of the latter anticyclone. A second cyclone is directly south of the ring and a third is pushed up on the shelf. The final "snapshot" of this sequence is shown in panel d. Here the ring has fissioned into two anticyclones with the possibility that the western fission has merged with the older anticyclone observed previously. Northeast of these two anticyclones is an anticyclonecyclone pair and farther to the south is a well developed cyclone. Also, in the Bay of Campeche, there is a well-developed cyclone. In the east, the Loop Current is continuing its penetration into the interior of the GOM. 3

GOM RINGS - OBSERVATIONAL EVIDENCE OF THEIR FORMATION AND INITIAL MOVEMENTS

Do the general characteristics of the mesoscale circulation as depicted in the model results agree with observations? In the following synopsis of Lewis and Kirwan, 1987, this question is addressed using hydrographic and drifter observations of ring fornation and early movements across the gulf.

490

Lewis and Kirwan (1987) reported on hydrographic sections taken across the Loop Current from late May to early July 1985. During this period, a ring separated from the Loop Current and migrated westward across the GOM. The temperature field along three sections across the Loop Current is shown in Figure 2. Section A (panel b) shows quite clearly the thermal characteristics of an extended Loop Current. In the core of the current, the 20'

Fig. 2. (a) Cruise tracks for XBT data collected during May 26-31, 1985. The arrows denote the flow at the edges of the Loop Current (maximum horizontal temperature gradient) based on the vertical temperature structure shown in (b) temperature data from the E.M. Queeny cruise, May 26-17, 1985, (c) temperature data from the Stena Hispania cruise, May 27-28, 1985, and (d) temperature data from the Stena Hispania cruise, May 30-31, 1985 (from Lewis and Kinvan, 1987).

491

isotherm is depressed to about 300 m. There are sharp thermal gradients at both the northwest (stations 3 and 4) and southeast (stations 13 and 14) edges of the Loop Current. There is just the barest hint of a small cyclone at the northwestern edge of this section. Section B (panel c) is a north-south cut along the western limb of the Loop Current. As in the previous section, the 20° isotherm is depressed to 300 meters. The northern edge of this section shows more clearly than before the cyclonic structure. Note that the Loop Current

Fig. 3. (a) Cruise tracks for XBT data collected during June 26 to July 2, 1985. The arrows denote the flow at the edges of the Loop Current (maximum horizontal temperature gradient) based on the vertical temperature structure shown in (b), temperature data from the Stena Hispania cruise, June 26-28, 1985, (c) temperature data from the Stena Hispania cruise, June 29-30, 1985, and (d) temperature data from the E. M. Queeny cruise, July 1-2, 1985 (from Lewis and Kirwan, 1987).

492

extends from 24ON to 27'N. Stations 53-55 show a small cold dome persisting down to 300 meters. Section C (panel d) is a section down the axis of the current. There is no evidence of ring separation in any of these panels. Temperature profiles collected from 26 June to 2 July 1985 are shown in Figure 3. Perhaps the most striking feature of these sections is seen in Section C (panel d). Here the eastwatd extent of the southerly flow has shifted from 96.5OW (Figure 2) to 88OW (see stations 9-12, panel d). On the other hand, Section B (panel c) shows easterly flow south of this (stations 14-19). Section A (panel b) depicts the north-south extent of the flow field which is

Fig. 4. (a) Cruise tracks for XBT data collected during July 16-19, 1985. The arrows denote the flow at the edges of the Loop Current and ring (maximum horizontal temperature gradient) based on the vertical temperature structure shown in (b) and (c) both are temperature data from the M/V Nat Co 6 cruise, July 16-19, 1985, and (d) temperature data from the Stena Hispania cruise, July 16-17, 1985.

493

the same as in Figure 2. Note the cyclonic feature at the northern edge of the flow field (stations 23-27). The cold doming extends down to 400 m. In Figure 4, the temperature sections for the period 16-19 July, 1985 clearly show a ring has detached from the Loop Current. From Section A, shown in panel b, the north-south extent of the ring is evident. The east-west extent of the ring is clearly shown in panel c (stations 33-49). These are very nearly the same as that shown in the model results, Figure 1. Panel d shows a northwest-southeast section just to the east of the ring. Comparing this with panel c shows a region of very strong horizontal shear. Also, panel d shows a very intense cyclonic structure just to the north of the northern edge of the reformed Loop Current (stations 8-13). In mid-June 1985, a satellite-tracked drifter was deployed in the Loop Current. After the ring was detached, a second drifter was deployed. Representative paths for these two drifters are shown in Figure 5, taken from Lewis and Kinvan (1987). This figure also shows the

30"

25"

20"

95"

90"

85"

80"

Fig. 5. Trajectories of drifter 3354 from mid-June through mid-September 1985 and of drifter 3378 from mid-July through September 1985. Also shown is the location of Ghost Eddy during August and September 1985. Squares denote the beginning positions of the drifter trajectories and triangles denote the end positions.

494

location of "Ghost Eddy" which was an older ring being tracked by another drifter. This ring detached approximately two months before. Frank Kelly (personal communication) also has current meter data in the north-westem part of the Gulf which shows the presence of yet another presumably even older ring. Thus, there are three documented rings in the western GOM with ring shedding of approximately three to four months. The velocity record (panel c), the displacement from flow center (panel b), and the frequency of rotation of drifter 3354 are shown in Figure 6. This was the drifter originally seeded in the Loop Current. This figure indicates velocity components in the Loop Current approaching 75 cm/s with a radius of about 100 km. The period of rotation is surprisingly constant at about 8 to 9 days. LOOP CURRENT (33541

LOOP CURRENT I33541 75

; i

5 0

w

B

50 25

o

2

'

B8

-25

oL

-50

180

-75

a

JULIAN DAYS (1985)

b

C

JULIAN DAYS (1985)

200

220

240

260

JULIAN DAYS (1985)

Fig. 6. Time histories of velocity (panel a), distance from flow center (panel b), and rotation frequency (panel c) of drifter 3354. In the first two panels, the dark curves refer to northsouth components and the light curves refer to the east-west components.

100 7

RING 3378

FQ

0.15 r

Y 0 >

m $ e 3

RING 3378

2

0

t

$ a

JULIAN DAYS (1985)

b

JULIAN DAYS (1985)

C

O 200 . 0.05

220 l

240O 260 L

JULIAN DAYS (1985)

Fig. 7. Time histories of velocity (panel a), distance from flowcenter (panel b), and rotation frequency (panel c) of drifter 3378 (from Lewis and Kirwan, 1987). In the first two panels, the dark curves refer to north-south components and the light curves refer to the east-west components.

495

Depicted in Figure 7 are the same quantities determined from the path of drifter 3378. The velocity record is more regular than that in Figure 6, but the essential flow characteristics are the same; component velocities of 75 cm/s and a radius of about 80 km. Note, however, that the period of rotation has increased to about 12 to 13 days. This suggests that drifter 3354 may have been affected more by the "Cuban" eddy than by the Loop Current proper. KINEMATICS OF RING PATHS - COMPARISON OF OBSERVATIONS A N D SIMU-

4

LATIONS Not all rings follow the northerly path traced by drifter 3378. Other rings have followed west-southwest paths as well. Kirwan, et al. (1984a,b, 1988) have reported rings in 1980/1981 and 1982/1983. Figure 8, taken from the above papers, shows the path inferred ring center and velocity record for one of the drifters in the 1980/1981 ring in the mid-Gulf regions. Note the peanut shaped orbit (encircled with dashed lines) in the middle of the path. Velocities are in the neighborhood of 80 cm/s.

100

26'

25.

50

-

24.

p

5

E , >

3 23'

ca 3

22.

50

- Path ......

21'

.....

20'

100

96"

a

94'

LonglIude

92'

90'

350

b

450

400

"

I 500

Time

Fig. 8. Path and inferred center (panel a) and velocity (panel b) of drifter 1599 in the 1980/1981 ring (from Kinvan et al., 1988). In (a), the dashed line encircles the peanut shaped orbit. In (b), u, v refer to north-south, east-west velocity components, respectively. The same kinematic characteristics for the drifter in the 1982/1983 ring are shown in Figure 9. As in the previous figure, there is an extended orbit near 95'W. This characteristic is not inconsistent with the simulated results shown in Figure 1. The velocities for this drifter are somewhat less than those shown in Figure 8. This is attributed to the fact that the drifter was in an orbit that was somewhat closer to the center in the latter ring.

496

The simulated drifter path and velocity record obtained from the Hurlburt-Thompson model is shown in Figure 10. Depicted in panel a is the path, the center inferred from the path, and the paths of the surface and interface maximum pressure anomalies. This path shows a hint of the characteristic peanut or\ seen in ,e previous two figures (encircled with dashed lines) and in Figure 1. 1DO

I

.....

50

2 $ 0

> 3.c

3

-

50

Palh

......

C.3"lM

1982

a

I

I

96'

94. Longtude

1

92.

100

I

I

1983

,

t

L

90.

b

Fig. 9. Path and inferred center (panel a) and velocity (panel b) of drifter 3374 in the 1982/1983 ring (from Kirwan et al., 1988). In (a), the dashed line encircles the peanut shaped orbit. In (b), u, v refer to north-south, east-west velocity components, respectively.

25' -

50

24' -

B

-

4

23'

-

22.

-

l> o * a

-50

-100

100

b

200

300

TIRE

Fig. 10. Path and centers as inferred from path data, the surface and interface pressure anornolies for the simulated drifter from the Hurlburt-Thompson model (panel a), and velocity record (panel b) (from K h a n et al., 1988). In (a), the dashed line encircles the peanut shaped orbit. In (b), u, v refer to north-south, east-west velocity components, respectively.

497

In the mid-Gulf region (94'W - 9OoW), the three centers deviate by an amount that cannot be accounted for by expected numerical errors. At present, there is no explanation for this, although it is noted that Smith and Reid (1982) showed centers of various kinematic properties such as maximum pressure anomaly, energy and enstrophy do not coincide. Also, note that the velocities for this simulated drifter are considerably less than the velocities from the two observed rings. A comparison of the centers for three drifters in the 1980/1981 ring, the 1982/1983 ring, and the three centers from the model simulations is given in Figure 11. Considering the

26 '

25

24" a,

-0 3 L ._

5

23"

1980J81 Ring 1982~83Ring

7

22"

'.d

Model Drifter Surface Pressure Interface Pressure

----

21

20

96'

94"

92"

90"

Longitude

Fig. 11. Center paths as inferred from the path data and the surface and interface pressure anomalies from the Hurlburt-Thompson model.

498

diversity of sources the agreement of the centers is quite good. It seems then that at least two rings in two different years have confirmed the movement predicted by the HurlburtThompson model. After intersecting the continental slope, the two observed rings migrated slowly to the north along the slope. The same character was observed in the simulations. Along the slope, the agreement is even better than the mid-Gulf region, the spread between the paths being essentially attributed to numerical noise. This is an example of a situation in which the model predictions actually improve with time.

5

DISCUSSION

The data shown above demonstrate that mesoscale motions in the GOM are among the most vigorous in the world ocean. Horizontal diameters of the Loop Current rings are of the order of 200 km as inferred from the maximum temperature gradient and swirl speeds approach 1 m/s. They tend to propagate westward across the Gulf. Even after intersecting the continental slope on the western side of the basin, these rings retain a strong hydrographic signature. For example, the ring studied by Merrell and Morrison (1981) showed a transport of 30 sverdrups, essentially that of the Loop Current as it enters the Gulf. Evidence from the drifter tracks shows that the rings can be identified and tracked by Lagrangian techniques for periods of approximately one year. Moreover, quantitative comparisons of the kinematics of these tracks with simulated tracks by the Hurlburt-Thompson eddy resolving general circulation model show excellent agreement. This suggests that Lagrangian data can be used to rigorously assess the predictive capability of these models. It appears from the data presented above that the Loop Current can shed rings as often as once every three to four months. Given the lifetime of at least a year, it follows that at any one time there can be three to five rings in the western Gulf. Because of the size, it is likely that the dynamics in the western Gulf and the western continental slope is governed by ringtopography and ring-ring interactions. Three to five rings in the western Gulf in one year implies that approximately 90 to 150 sverdrups of Caribbean water is being transported into the western Gulf. This is almost twice the transport of the Gulf Stream. Yet the GOM central water, being slightly cooler and fresher, is distinct from Caribbean water. The question then arises as to why the central GOM water isn’t essentially the same as the Caribbean water. In the case of temperature, it is possible that exchange of heat with the atmosphere, especially through hurricanes, could reduce the temperature of the Caribbean water to that of the central water. Hurricanes could also produce a negative E-P which potentially could lower the salinity as well, although this is not at all clear. Another source of freshwater is run off. However, in both cases, the freshening processes are stable. Thus, it is not clear how the Caribbean water can be modified to form Gulf Central Water. This, perhaps, is the most important scientific question on the circulation in the Gulf of Mexico.

499

6

ACKNOWLEDGMENTS

Most of this work was supported by the Mineral Management Service, Gulf of Mexico Physical Oceanography Study through a contract with SAIC and by contract N00014-88-K0203 between the Office of Naval Research and Old Dominion University. A.D. Kirwan, Jr. acknowledges the support of the Samuel L. and Fay M. Slover endowment of Oceanography to Old Dominion University. 7 REFERENCES Elliot, B.A., 1979. Anticyclonic rings and the energetics of the circulation of the Gulf of Mexico. Ph.D. dissertation, Texas A&M University, College Station, Texas. Elliot, B.A., 1982. Anticyclonic rings in the Gulf of Mexico. J. Phys. Oceanogr., 12: 1292-1309. Hurlburt, H.E., 1986. Dynamic transfer of simulated altimeter data into subsurface information by a numerical ocean model. J. Geophys. Res., 91(C2): 2372-2400. Hurlburt, H.E. and Thompson, J.D., 1980. A numerical study of Ioop current intrusions and eddy shedding. J. Phys. Oceanogr., 10: 1611-1651. Ichiye, T., 1962. Circulation and water mass distribution in the Gulf of Mexico. Geofis. Int., 2: 47-76. Kindle, S.C., 1986. Sampling strategies and model assimulation of data for area modeling and prediction. J. Geophys. Res., 91(C2): 2418-2432. Kirwan, A.D., Jr., Lewis, J.K., Indest, A.W., Reinersman, P. and Quintero, I., 1988. Observed and simulated kinematic properties of Loop Current rings. J. Geophys. Res., 93(C2): 1189-1198. Kirwan, A.D., Jr., Merrell, W.J., Jr., Lewis, J.K. and Whitaker, R.E., 1984a. Lagrangian observations of an anticyclonic ring in the western Gulf of Mexico. J. Geophys. Res., 89(C3): 3417-3424. Kirwan, A.D., Jr., Merrell, W.J., Jr., Lewis, J.K., Whitaker, R.E. and Legeckis, R., 198413. A model for the analysis of drifter data with an application to a warm core ring in the Gulf of Mexico. J. Geophys. Res., 89(C3): 3425-3438. Lewis, J.K. and Kirwan, A.D., Jr., 1987. Genesis of a Gulf of Mexico ring as determined from kinematic analyses. J. Geophys. Res., 92(C11): 11,727-11,740. Merrell, W.J. and Morrison, J.M., 1981. On the circulation of the western Gulf of Mexico with observations from April 1978. J. Geophys. Res., 86(C5): 4181-4185. Merrell, W.J. and Vazquez, A.M., 1983. Observations of changing mesoscale circulation patterns in the western Gulf of Mexico. J. Geophys. Res., 88(C12): 7721-7723. Nakamoto, S., 1986. Application of solitary wave theory to mesoscale eddies in the Gulf of Mexico. Ph.D. dissertation, 50 pp., Department of Oceanography, Texas A&M University, College Station. Smith, D.C., IV, 1986. A numerical study of Loop Current eddy interaction with bottom topography in the western Gulf of Mexico. J. Phys. Oceanogr., 16(7): 1260-1272. Smith, D.C., IV and Reid, P.O., 1982. A numerical study of non-friction decay of mesoscale eddies. J. Phys. Oceanogra., 12(3): 244-255. Thompson, J.D., 1986. Altimeter data and geoid error in mesoscale ocean prediction: Some results from a primitive equation model. J. Geophys. Res., 91(C2): 2401-2417.

500

Wallcraft, A.J., 1986. Gulf of Mexico circulation modeling study, annual progress report: Year 2. Report to Mineral Manage. Sew., 94 pp., JAYCOR, Vienna, VA.

501

MESOSCALE EDDIES AND SUBMESOSCALE,COHERENT VORTICES: THEIR EXISTENCE NEAR AND INTERACTIONS WITH THE GULF STREAM J.M. BANE and L.M. O'KEEFE Marine Sciences Program, University of North Carolina Chapel Hill, North Carolina 27599-3300 (USA)

D.R.WATTS Graduate School of Oceanography, University of Rhode island Narragansett, Rhode island 02882 ( U S A )

ABSTRACT An array of tall current meter moorings, bottom-mounted inverted echo sounders and pressure gauges was deployed for one year in an area about 150-500 km northeast of Cape Hatteras, North Carolina. The Gulf Stream jet was observed by the upper current meters in the array, and the deep western boundary current was observed by several of the deep instruments. Numerous mesoscale eddies and submesoscale, coherent vortices (SCVs) were observed progressing through the region. Two types of mesoscale eddies were seen. Three cold-core Gulf Stream rings were observed, two of which interacted with the Stream, ultimately coalescing with the jet within the array area. Associated with the first ring-Stream interaction was a lateral shift of the position of the Gulf Stream's axis. This shift repositioned the jet about 100 km seaward, and this new path lasted for several months. The second ring-Stream interaction was followed by another shift in the Stream's path. The bottom pressure signal observed during each ring passage through the area gave clear evidence of a vertical "tilt" to each ring's low pressure center. The second type of mesoscale eddy observed was an intriguing, subsurface, anticyclonically swirling warm eddy which moved through the array area along the seaward side of the Gulf Stream in the Sargasso Sea. This 160 km diameter eddy had maximum observed swirl velocities over 50 cm/sec and a positive temperature anomaly of about 4 O C, both at about 900 m below the sea sufrace. No clear surface temperature expression was observed by satellite, and the in s i t u velocity observations suggest that there was little surface velocity, implying that this energetic feature was an interior eddy. Its movement was complex, with a northeastward progression through the array area followed by a seaward turn and another transit through the array area towards the southwest. Progression speed in either direction was about 12-15 km/day. The origin of this eddy and its level of interaction with the Gulf Stream are not clear at this point in the analysis. The SCVs were seen mostly below and beside the Gulf Stream jet, with both cyclonically and anticyclonically swirling SCVs observed. That none were observed in the Gulf Stream jet implies that either SCV transit speeds there were too fast to allow their detection, or that SCVs become sheared apart by the Stream and lose their identity. Of the nineteen SCVs identified, anticyclones outnumbered cyclones almost two to one. The anticyclones typically had larger diameters and higher swirl velocities than the cyclones.

502

1. INTRODUCTION

An array of tall current meter moorings, bottom-mounted inverted echo sounders and pressure gauges was deployed for one year in an area about

150-500 km

northeast of Cape Hatteras, North Carolina. The Gulf Stream jet was observed by the upper current meters in the array, and the deep western boundary current was

observed by several of the deep instruments (Fig.

1). During the study period,

several isolated eddies were observed to progress through the array. These eddies may be roughly divided into three classes: Gulf Stream cold-core rings; a subsurface, anticyclonically swirling warm eddy; and submesoscale, coherent vortices (SCVs). This paper presents a description of each eddy type, and discusses the apparent relationship which each type had to the Gulf Stream in this area. The first section gives an overview of the sctting of the field experiment, and the next three sections discuss the eddies. 2. T H E OBSERVATIONAL SETTING

2.1 The Gulf Stream Dynamics Experiment

The moored array shown in Fig.

1 comprised the central component of the

oservational study, which we refer to as the Gulf Stream Dynamics Experiment (Bane and Watts, 1986). The array was composed of five current meter/bottom pressure gauge moorings and twenty inverted echo sounders (IES). The five current meter mooring sites each had four levels instrumented from 500 m below the surface to near the bottom. An IES with a bottom pressure gauge was located at the base of each mooring. A high performance design was used for the current meter moorings, which allowed them t o extend high into the strong current. This was the first 3-dimensional array of current meters to span through the main thermocline and strong vertical shear in a region where the Gulf Stream flows in deep water. To withstand the strong currents, each mooring was constructed with small diameter (3/16") jacketed wire to reduce drag, and high floatation ( c a . 2000 Ibs. positive buoyancy).

These

high

performance,

"stiff"

moorings

survived

a

one-year

deployment with little indication of adverse effects. The tilting of the moorings due to currents was well within design specifications for the Aanderaa current meters

used, and the amount of vertical excursion was somewhat lower than the design target. All of the moorings and current meters were safely recovered, and data returns of 85% for velocity and 90% for temperature were achieved. The IESs were recovered in both January and May 1985, with only one instrument loss and one data tape failure (both from the second deployment period) for a data success rate of 95%

on the 19-month-long combined records. The failure of the electrical circuitry

503

Fig. 1. Gulf Stream Dynamics Experiment study region showing year-long mean velocities at (a) the 400 m level, and (b) D-500 m level. The locations of the five current meter moorings are shown with solid circles labelled B2 through C3 in both panels, and the IES locations are shown on the D-500 m panel. Note the rotated coordinate system. (No data available at C3 in panel (a).)

504

controlling one of the five bottom pressure gauges resulted in a data return of only 80%. However all the bottom temperature sensors functioned properly. 2.2 The current meter records

The array was located in a region where Gulf Stream meanders are known to propagate and grow in the downstream direction. In Fig. 1 solid circles on lines B and C denote current meter moorings. Aanderaa current meters were placed on each mooring at nominal depths of 400 m, 900 m and 1900 m from the surface and 500 m from the bottom (D-500 m). They recorded current speed, current direction, and temperature at one-hour intervals. Each instrument is identified by its mooring and position on the mooring. The shallowest meter, at the 400 m level, is designated by a "

1". followed sequentially by

the deeper meters. Thus, on the B2 mooring, the

current meter at 400 m is identified as B2-1, the current meter at 900 m as B2-2, and

so on. Figs. l a and l b also show the mean flow vectors at the 400 m and D-500 m levels. Note that a rotated coordinate system has been defined according to the 400 m level mean currents, with the x , or downstream direction being positive towards 040° T r u e and the y , or cross-stream direction being positive towards 310° True. During much of 1984 the Stream flowed along a course which was north of its usual path. This condition resulted in our array being positioned within the anticyclonic side of the Stream, as may be seen in the 400 m mean currents. The northernmost mooring (CI) was near the Stream center during much of the deployment period. In contrast, the D-500 m currents at the two southern moorings show the presence

of a deep

southwestward mean flow, counter to that of the surface Gulf Stream. This is likely the Deep Western Boundary Current described by many authors (Richardson,

1977;

Joyce et al., 1986). Times series of forty-hour low-pass filtered (40 HRLP) downstream speed (u), cross-stream speed (v). and temperature (T) measured by the four instruments on mooring B2 are shown in Figs. 2a through 2d. Mooring B2 was the westernmost in the array, located near 35.6 N and 73.5 W. The depth of the top instrument is also shown in the top panel in Fig. Za, to provide an indication of the mooring's performance (r.m.s. vertical excursions of about 40 m at the mean depth of 395 m, excursion range from 350 to 585 m). Several aspects of the Gulf Stream environment within the study area during 1984 and early 1985 may be seen from visual inspection of the Fig. 2 time series. The uppermost instrument on B2 was on the southern fringes of the Gulf Stream jet during the first and last portions of the period, while from about year-day 120 to day 350 the Stream had moved far enough south that its high velocity core flowed through the array near mooring B2. A general decrease in temperature was seen at the lowest three levels of B2 during this period, associated with the southward shift throughout the water column of the baroclinic temperature field along with the

Fig. 2. Forty-hour low-pass filtered time series of downstream (u) and cross-stream ( v ) velocity components, temperature (T) and instrument depth (top meter only) for the B2 mooring. Nominal instrument depths are 400 m (B2-1). 900 m (B2-2). 1900 m (B2-3) and 500 m above the bottom (B2-4).

~n

ul

m 0

62-3

B2-4

Fig. 2. (continued)

Day.

507

current. The uppermost instrument was within eighteen degree water for most of the year. Two strong events occurred near day 115 and day 265 in the B2-1 record. Using velocity and temperature signatures at this instrument, plus delay times between instruments on the other moorings, it was determined

that

these events were

cold-core, cyclonic eddies moving to the northeast. Satellite data confirm that the events were cold-core Gulf Stream rings coalescing with the main current and travelling "downstream." The earlier of these two events is discussed in Section 3 of this paper. A third cold-core ring was observed farther offshore and progressing southwestward through the array. This ring was sufficiently far from the Gulf Stream that it had no apparent interaction with it. With the exception of the cold-core rings, the largest and most energetic isolated eddy to be observed was an anticyclonically swirling, subsurface warm eddy which progressed first northeastward through the array on the Sargasso Sea side of the Gulf Stream, then returned travelling southwestward on a trajectory which was somewhat farther offshore. The eddy extended from at least the 400 m level to about the 1900 m level, and was about 160 km in diameter. It left its signature in the B2, B3.

C2, and C3 mooring instruments during December 1984 (first transit through the array) and January 1985 (second transit). This eddy is discussed in Section 4. Two velocity fluctuations may be seen at the 1900 m level (Instrument B2-3) near days 150 and 220. These are believed to have been caused by cyclonically swirling, submesoscale coherent vortices transiting through the array. Each eddy was about

40-50 km in diameter, and had swirl velocities near 15 cm/sec. At least seventeen other SCVs were observed by this array. The SCVs are discussed in Section 5 .

3. COLD-CORE RINGS As

shown above, velocity

and temperature

signatures of two cold-core rings

(CCRs) were left in the B2 mooring time series near days 115 and 265 as these events progressed through the array. Another CCR was observed by the seaward current meter moorings (B3 and C3) near day 67. The rings were designated CCR I, CCR I1 and

CCR 111 in order of their passage through the instrument array, beginning with the day 67 event. CCR I was well seaward of the Gulf Stream and not interacting with it, and it was seen to move southwestward. Each of CCR I1 and CCR 111 was interacting with the Gulf Stream, and as a result was being advected in the northeastward direction. Each of these two rings met its ultimate fate by being absorbed into the Gulf Stream as it exited the current meter array area. Although the movements of these three rings (and the fates of two) were similar to those of rings observed earlier in this region (see Richardson, 1983, pp. 31-39), the detailed observations made during the Gulf Stream Dynamics Experiment have revealed the following two interesting aspects of ring/Gulf Stream behavior.

508

3.1 Relationship between Gulf Stream path and ringiStream interactions Satellite and IES observations of CCR I1 show clearly that it coalesced with the Gulf Stream jet as it exited the current meter array area. Redrawn versions of two satellite sea surface temperature (SST) images showing CCR I1 before and during this interaction

with

the

Stream

are

given

in

Fig.

3.

Prior

to

this

ring/Stream

interaction the path of the Gulf Stream jet was well to the northwest (shoreward) of its longer term mean position. A four-year-long time series of monthly averaged Gulf Stream location ( i . e . distance of the Stream's shoreward SST front from the shelfbreak) in the array area was determined from satellite imagery by Brown and Evans (1987). and it is presented in Fig. 4. It shows this northwestward Gulf Stream location prior to the CCR I1 passage in April 1984 and that the Stream moved a considerable distance southeastward (seaward) following the coalescence of CCR I1 with the Gulf Stream. Following the similar coalescence of CCR 111 into the Stream in September 1984, Fig. 4 shows that the Gulf Stream moved northwestward. These events

suggest

that

such

energetic

eddy-current

interactions

may

play

an

important role in adjusting the path of the Gulf Stream on time scales of several days, after which the general course of the Stream may remain relatively constant for up to a few months. 3.2 Cold-core ring vertical structure

Data from mooring B2 (solid square in Fig. 3) reveal an interesting aspect of the vertical structure of two of these three rings. As each ring passed mooring B2, the velocity and temperature time series at the upper levels indicated the time of closest passage of ring center to that mooring. On the ocean floor, the B2 bottom-pressure gauge showed the passage of a pressure minimum associated with each ring. Fig. 5 shows a subset of the mooring

B2 data, which

indicates

that the

upper and

near-bottom centers in each of CCR I and CCR I1 were not positioned over one another. That is, the axis of the ring is not vertical, but rather is somewhat "tilted" with respect vertical. This implies a vertical phase shift of the ring currents, which in turn has implications in terms of ring energetics. The lag time between the passage of the upper ring center and the bottom pressure minimum, coupled with the ring's direction of propagation shows that the lower portion of the ring was located to the southwest of the upper ring center, in a direction roughly along the bottom topography. This was true for both CCR I and CCR 11; however, the bottom pressure signal associated with CCR 111 was not as clear as the ones shown in Fig. 5.

so its vertical structure was not as well determined. 4. SUBSURFACE, MESOSCALE WARM EDDY During the last two months of

the current meter measurement

program,

an

unusual eddy of unknown origin was observed to transit through the current meter array twice, both times seaward of the Gulf Stream jet. Best measured during its first

509

Fig. 3. Redrawn versions of two satellite SST images showing CCR I1 approaching the array area from the southwest (24 April) and beginning its coalescence with the Stream in the array area (26 April). The solid square shows the position of mooring B2.

Gulf Stream Position 0

Y 0

a,

-100-

-E L

(u

S

v)

-200-

E

E

c

E

X

-300-

0

10

20

30

40

50

60

Month (Beginning January 1982) Fig. 4. Four-year-long time series of monthly averaged Gulf Stream position within the array area. The times at which two of the observed CCRs were absorbed into the Stream are denoted with arrows. Note the long term trend of northwestward movement of the Stream, (northwestward is towards the top of the figure) punctuated with rapid shifts in position, usually associated with strong ring/Stream interactions. (Figure from Brown and Evans, 1987.)

510

transit, it was seen to be a subsurface anticyclone with swirl speeds of greater than

50 cm/sec, diameter of about 160 km and a positive temperature anomaly of about 4O C. This feature was large enough that it was detected by several of the current meters

and the

IES

array.

Combining data from

several

sources allowed the

determination of the eddy's trajectory and speed through the array, and in turn this allowed the Eulerian measurements to be combined to give a more complete picture

of the feature. The eddy's movement through the study area was complex. It first progressed towards the northeast at about 12-15 km/day and was close enough to the Gulf Stream that interaction between the two was likely. Such an interaction may be the reason for the eddy's direction of travel at that time. After the eddy exited the current meter array area, it made an anticyclonic, seaward turn and returned to

Fig. 5 . 400 m level velocity and temperature data and bottom-pressure gauge data from mooring B2. [Stick vectors are oriented in the top panel such that up is in the downstream ( x ) direction.] The signature of CCR I1 was left in these time series during its northeastward transit through the array. CCR I was also measured, but its effects were smaller at this mooring due to the more seaward location of its southeastward line of transit. Note the bottom-pressure minimum associated with each ring's passage, and that it did not occur precisely in phase with the passage of the upper portion of the ring. The time lag and ring propagation direction for each case indicate an offset of the bottom ring center to the southwest of the upper ring center.

511

progress once again through the array. During this second transit it progressed on a southwestward heading, again at about 12-15 km/day, but this time was about 50 km

farther seaward from its earlier path. Approximately sixteen days elapsed

between the time the eddy exited the area and was first seen again on its return passage. Unlike two of the CCRs described above, this eddy was not absorbed by the Gulf Stream during the time of its northeastward movement even though it was on the same side of the Gulf Stream as the CCRs. It is interesting to speculate that the anticyclonic swirl direction of this eddy reduced its chance of absorption, while the cyclonic swirl of the CCRs enhanced theirs (cf. Stern and Flierl, 1987). Note that anticyclonically swirling, warm-core Gulf Stream rings on the opposite side of the Gulf Stream have been observed to have a range of interactions with the Gulf Stream extending from "glancing" encounters to complete absorption (Richardson, 1983; Evans er al., 1985). Velocity and temperature data were combined to give views of the eddy's internal structure in eddy coordinates ( i x . , a s functions of depth and radial distance from eddy center). Fig. 6 shows the resulting swirl speed and temperature sections, presented as vertical slices through the eddy's center. Note that data used for contouring were available at the locations of the dots. The typical isolated eddy structure is apparent in these presentations, with the warm eddy core and the subsurface swirl velocity maximum clearly evident. The level of maximum velocity is around the 900 m level, the radius to maximum velocity is about 30 km, and maximum eddy radius is about 80 km. It is of interest to determine if this eddy possessed any surface expression. Examination of satellite SST imagery available for this time period did not reveal a clear surface temperature signature of the eddy, a property consistent with the subsurface

temperature

data.

The

subsurface

data

showed

little

fluctuation

in

temperature at 400 m during the eddy's passage, while at 900 m the maximum fluctuation of about 4 O C was seen. The surface velocity was not determined from the field measurements. The values shown in Fig. 6 are a result of the objective analysis routine used for the plotting. If a surface velocity value of zero is imposed, the general subsurface structure of the eddy swirl velocity field remains unchanged. It appears that a maximum surface swirl speed of about 20 cm/sec is reasonable, which implies a maximum high in the dynamic topography in the seasurface of around 5 cm due to this eddy. The origin of the eddy is unknown at this stage of the analysis. Unfortunately, no hydrographic

measurements

were

made

within

the

eddy, so

directly measured

temperature-salinity information is lacking. Assuming the eddy is in gradient wind balance, the existence of a warm core at the level of maximum swirl velocity ( i . e . at the level of no vertical shear and, thus, no lateral density gradient) implies a positive salinity anomaly within the core. A temperature anomaly of +4O C must be balanced by a salinity anomaly of about + 1 ppt. suggesting a core salinity of around

512

Fig. 6. Subsurface structure of the mesoscale, warm eddy as observed during December 1984. Using the transit speed of the eddy, data have been transformed into eddy coordinates. (a) The typical isolated eddy swirl velocity structure is apparent. (Positive swirl velocities are into the page.) (b) The warm eddy core contains a positive temperature anomaly of about 4 O C. In each panel, an elliptical outline indicates the approximate “boundary” of the eddy, as suggested by the data, and the dots indicate actual data point locations.

513

37 ppt. Further study is required before more progress may be made on determining

the eddy's origin.

5. SUBMESOSCALE, COHERENT VORTICES 5.1 SCV signatures

Numerous signatures in the current meter velocity and temperature time series have been interpreted to be due to the passage of submesoscale, coherent vortices (SCVs) through the array. Fig. 7 shows an expanded portion of the B2-3 time series, in which two such signatures may be seen (arrows near days 153 and 224). These velocity fluctuations, more noticeable in the v time series, have a pattern similar to that which would be left by an isolated vortex progressing horizontally past the current meter. If that is the case, and it is also assumed that each eddy was simply embedded within, and moving with the broader scale "background" flow, then the eddy swirl direction may be determined and the eddy diameter and maximum swirl speed estimated. Accordingly, the two eddies in Fig. 7 were cyclonically swirling, had diameters of 54 and 38 km, and had maximum recorded swirl velocities of about

CYCLONICSCVs (82 @f 1900 m)

Days

I

135

185

160

235

Din-38km

Dia- 54 km

3

I

210

2 I4

24

MAY

3

13

23

3

JUNE

13

JULY

23

2

I2

22

AUGUST

1984

Fig. 7. A subset of the B2-3 time series which contains the signatures of two cyclonic SCVs. Eddy diameters, maximum swirl speeds and temperature anomalies are indicated.

514

15 cm/sec. None of the events like these was seen to have a vertical extent sufficient to be observed in more than one current meter on a given mooring, implying that

vertical scales were at most a few hundred meters, and none was immediately preceeded or followed by another such event, supporting the notion that each was in fact isolated. Taken together, the evidence suggests that these isolated eddies were in the submesoscale range; thus, each such eddy may be categorized as an SCV ( c f . McWilliams, 1985). 5.2 SCV census for the Cutf Stream Dynamics Experiment Using the assumptions discussed

above,

all

current meter time

series were

inspected for SCV signatures. Nineteen SCVs could be identified. It is possible that more

passed

through the

array

but

were

undetected

because

their

recorded

signatures did not "stand out" from the background flow as did those of the two SCVs shown in Fig. 7. This could be due to either the eddy being relativly weak, or to it having passed through the array such that the strong flow segment of the eddy was not observed by any current meter. Since the Gulf Stream Dynamics Experiment array was designed to sample the Gulf Stream variability field, its geometry was not optimum for sampling the SCV population. Consequently, the total of nineteen SCVs identified should be viewed as a lower bound. Several SCV properties were tabulated as a result of this census, and they are displayed graphically in Figs. 8 through 10. Since not every SCV was sampled through its center, the true magnitude of certain properties

( e . g . maximum swirl speed and temperature anomaly) could not be

determined. Thus, this census should be regarded as suggestive, not conclusive. The depth dependence of SCV number, swirl direction and maximum observed swirl speed may be seen in Fig. 8. Of the nineteen SCVs included in the census, anticyclones outnumbered cyclones almost two to one. McWilliams (1985) argues that no cyclonic SCVs should exist in the world ocean as a result of the mixing process which he proposes as the SCV generation mechanism. D'Asaro (1988a,b) has presented evidence for the existence of cyclonic SCVs in the Beaufort Sea, and has proposed that generation may occur through the action of viscous torques applied by a side wall in the flow domain. Such a mechanism may produce submesoscale cyclones. Although the data presented here give little clue as to the generation mechanism(s) of the observed SCVs, an anticyclonic bias is apparent. It is notable that no SCVs were identified at the 400 m level. That none were observed in the Gulf Stream jet implies that either SCV transit speeds there were too fast to allow their detection, or that SCVs become sheared apart by the Stream and lose their identity. It was at 900 m. the level of the approximate base of the jet, that the strongest swirl speeds were observed (-30 decrease in swirl speed with depth from there.

cm/sec), and there was a general

515

n

E v I F n w P

1000

2000

3000

4000

0

2

4

6

8

10

Number of SCVs Observed

2 2 D B

1000

4000

BB

f

I

I

I

0

10

20

30

40

Swirl Speed (cmlsec)

Fig. 8. Results from the SCV census for the Gulf Stream Dynamics Experiment. (a) SCV number and swirl direction are shown as functions of depth. Note that none were identified at the 400 m level, within the Gulf Stream jet. (b) An indication of general decrease in swirl speed with depth was found for both cyclones and anticyclones.

Fig. 9 indicates a general trend of increasing swirl speed with increasing eddy diameter; however, there is considerable scatter about the trend. There was no clear distinction between anticyclones and cyclones in this general relationship. For many of the SCVs, a temperature fluctuation was recorded simultaneously with its velocity fluctuation (see the second eddy in Fig. 7, for example). The magnitudes of these temperature fluctuations were determined, and they

are shown plotted

against swirl speed in Fig. 10. This figure suggests that there is a tendency for a faster swirling anticyclone to have a larger temperature anomaly.

516

30

-

20

-

10 -

R = 0.7 0

1

0

.

I

20

'

40

I

-

I

60

80

.

100

DIAMETER (km) Fig. 9. The SCV census suggests a direct relationship between eddy diameter and swirl speed. The correlation is not very strong, however. No difference between cyclones and anticyclones is apparent.

1.o

0.6 0.4 0.8

0.2

-

Anticyclonic

Cyclonic

0

EJ

2

Fig. 10. Some anticyclones were found to have a positive temperature anomaly. An increase in the magnitude of this anomaly with increased swirl speed is suggested.

6 . SUMMARY Data from the Gulf Stream Dynamics Experiment have been used to describe three types of isolated eddies that exist near, and have some interaction with the Gulf Stream. The study region was approximately 150-500 km downstream from Cape Hatteras. North Carolina. Two types of mesoscale eddies were observed: cold-core Gulf Stream

rings

and

a

subsurface,

mesoscale

warm

anticyclone.

The

data

show

northeastward movements for the mesoscale eddies when they were in close enough

517

proximity to the Gulf Stream to be interacting with it. Two of the cyclonic, cold-core rings were absorbed by the Stream during their northeastward transit through the study area, whereas the anticyclonic, warm eddy was not. It is possible that eddy swirl direction plays a role in determining absorbtion likelihood. Each of the two cold-core ring absorbtions was followed by a lateral shift in the Gulf Stream's path

of about 100 km. The mesoscale eddies that were sufficiently far offshore to not interact with the Stream progressed in a southwestward direction. Cold-core ring structure was observed to have a central axis "tilted" somewhat from the vertical. The warm eddy was shown to have a typical subsurface, isolated eddy structure, with maximum swirl speed of over 50 cm/sec, a core with a positive temperature anomaly of 4O C and an implied positive salinity anomaly of about 1 ppt. Nineteen submesoscale, coherent vortices were also observed, and form the third class of isolated eddy. These eddies were seen mostly below and beside the Gulf Stream jet, with both cyclonically and anticyclonically swirling vortices observed. That none were observed in the Gulf Stream jet implies that either their transit speeds there were too fast to allow their detection, or that an eddy of this type will become sheared apart by the Stream and lose its identity. A census of these vortices indicates that the anticyclones have larger diameters (up to about 60 km) and swirl speeds (up to about 20 cm/sec) than do the cyclones, are more numerous than the cyclones by about a factor of two. and are associated with positive temperature anomalies of up to several tenths of a degree Celsius.

ACKNOWLEDGEMENTS We wish to thank Russell Auk, John Schultz and Karen Tracey for their assistance with the data processing and presentation. Helpful discussions with Bill Dewar and Tom Rossby are acknowledged. We are grateful to the crews of RIV Oceanus and R I V

Endeavor for their efforts at sea during the instrument deployment and recovery phases of this project. We also thank the Technical Services group at the University of Rhode Island and the Buoy Lab at Woods Hole Oceanographic Institution for much effort in the preparation of instruments. Special thanks go to Gerry Chaplin and Mike Mulroney

for their engineering and preparation

of the IES and pressure

gauge instrumentation. Support for this research has been provided by the Office of Naval

Research

(N00014-77-C-0354

through

contracts

to

and N00014-87-K-0233)

the and

University

of

the University

North

Carolina

of Rhode

Island

(N00014-81-C-0062), and by the National Science Foundation through grants to the University of Rhode Island (OCE82-01222 and 0CE85-37746).

REFERENCES Bane, J.M., and Watts, D.R., 1986. The Gulf Stream downstream from Cape Hatteras: The current and its events during 1984. Trans. Amer. Geophys. Un., 66: 1277.

518

Brown, O.B., and Evans, R.H., 1987. Satellite infrared remote sensing. In: Casagrande (Editor), Study of physical processes on the U . S . mid-Atlantic slope and rise. Science Applications International, Raleigh, NC, pp. IV-67 - IV-97. D'Asaro, E., 1988a. Observations of small eddies in the Beaufort Sea. J . Geophys. Res., 93: 6669-6684. D'Asaro, E., 1988b. Generation of submesoscale vortices: A new mechanism. J . Geophys. Res., 93: 6685-6693. Evans, R.H., Baker, K.S., Brown, O.B., Smith, R.C., 1985. Chronolgy of warm-core ring 82B. J. Geophys. Res., 90: 8803-8811. Joyce, T.M., Wunch, C., and Pierce, S.D., 1986. Synoptic Gulf Stream velocity profiles through simultaneous inversion of hydrographic and acoustic Doppler data. J . Geophys. Res., 91: 7573-7585. McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Rev. Geophys., 23: 165-182. Richardson, P.L., 1977. On the crossover between the Gulf Stream and the Western Boundary Undercurrent. Deep-sea Res., 2 4 : 139-159. Richardson, P.L., 1983. Gulf Stream rings. In: A.R. Robinson (Editor), Eddies in Marine Science. Springer-Verlag, New York, pp. 19-45. Stern, M.E., and Flierl. G.R., 1987. On the interaction of a vortex with a shear flow. J . Geophy. Res., 92: 10733-10744.

5 19

A SUMMARY OF THE OPTOMA PROGRAM'S MESOSCALE OCEAN PREDICTION STUDIES IN THE CALIFORNIA CURRENT SYSTEM.

MICHELE M. RIENECKER and CHRISTOPHER N.K. MOOERS Institute for Naval Oceanography, Stennis Space Center, MS 39529 (U.S.A.) ABSTRACT The OPTOMA program sampled the California Current System mainly on the continental rise off Northern California from March 1982 to November 1986. Maps of surface dynamic height from OPTOMA surveys during summer show intense offshore jets and associated eddies in the coastal transition zone. The current meanders during summer tend to be sharper than those during winter so that eddies often have smaller horizontal scale. The mean, the standard deviation and the range of the dynamic height, and maximum surface current speeds are larger in summer than in winter. In both seasons, the horizontal structure at 200 db, relative to 450 db, is weak; current filaments are less prominent at this depth than at the surface and the regions of strongest flow are associated with eddies. An array of three moorings, separated by about 100 km, is influenced by small-scale eddies which may account for the lack of large-scale coherence between velocity or temperature variations at the three moorings. Some features appear to propagate offshore, and the temperature at the offshore mooring clearly shows the influence of onshore/offshore advection. From dynamical model hindcast experiments, as well as from dynamic height maps, both local instability processes and wind stress curl forcing are important to the evolution of mesoscale features off Northern California.

1 INTRODUCTION. The large-scale variability of the California Current System (CCS) has been investigated

by the California Cooperative Oceanic Fisheries Investigations (CalCOFI) program since 1949. Although the coarseness of the CalCOFI grid (station spacing of 74 km or more offshore) has not resolved the mesoscale variability (especially north of San Francisco where the longshore station spacing is usually about 2")' it has shown the existence of mesoscale meanders and possible eddies superposed on the larger-scale equatorward flow of the California Current (Hickey, 1979; Wyllie, 1966). The OPTOMA (Ocean Prediction Through Observations, Modeling and Analysis) Program aimed to provide an improved description of the mesoscale variability of the CCS in order to gain a better understanding of its governing dynamics as well as to investigate the feasibility of and limitations to practical mesoscale ocean prediction. OPTOMA made 45 oceanographic surveys in various areas of the CCS, from Cape Mendocino to P t Sur, between March 1982 and November 1986. Most of this effort concentrated offshore P t Arena/Pt Reyes, although surveys from Monterey to Cape Mendocino were conducted as well as surveys off Pt Sur (see Rienecker et al., 1987b). Observations were acquired primarily by ship survey; aircraft surveys were used to obtain synoptic descriptions and thus test whether the asynoptic surveys were aliasing the signal to an inappropriate description of the variability.

520

The oceanographic surveys included three experiments designed for ocean prediction studies (summer, 1983 and 1984 and spring, 1986). In these experiments, a domain was sampled several times to acquire initial and boundary data for dynamical model hindcast and forecast experiments. Data were also acquired for verification purposes (Robinson et al., 1986; Rienecker et al., 1987a, henceforth referred to as RMR87). Other surveys were undertaken to investigate the seasonal variability of the mesoscale eddies and current filaments and to determine seasonal and interannual variations in the regional statistics used to initialize and update the dynamical model by objective mapping methods. The observational strategy for the ocean prediction experiments was such as to provide data which were as synoptic as feasible, necessitating small domains (initially about 150 km square covered in 540-9 days). These experiments usually sampled only a portion of any (inferred) eddy. Larger, almost regional, domains were found to be useful to place these features and their evolution in the context of the larger-scale circulation and to relate that circulation to available satellite imagery. When possible, surveys not directly related to ocean prediction experiments covered, and repeated, large domains so that a more complete picture of the variability in this region of the CCS could be gained. The ocean prediction experiments used only mass field data for initial and boundary data for the quasi-geostrophic model and so flow relative to some pressure surface was used rather than absolute flow. This use of a reference level presumes an association between baroclinic and barotropic components of flow which can be justified somewhat by the realism of the dynamical model predictions, but which can only be fully tested by direct current measurements. The OPTOMA Program (in association with Robert L. Smith of Oregon State University) acquired current meter data from a triad of moorings on the continental rise offshore Pt ReyeslPt Arena (Fig. 1). These moorings were set, from October 1984 to July 1985, to investigate current variability and its relation to the variability inferred from maps of dynamic height, the temporal and spatial relationship between barotropic and baroclinic flow and the relation between ocean variability and wind forcing. The mooring locations were chosen to conduct three statistically independent studies of the vertical modal structure. This paper summarizes mesoscale variability from the hydrographic surveys by describing the seasonal and interannual differences in the mesoscale circulation of the CCS off Northern California. The upper ocean variability (to 350 m) and barotropic and baroclinic flow components estimated from current meter data off Pt Arena are also discussed. The OPTOMA Program’s ocean prediction experiments are summarized.

2 BACKGROUND. The larger scale variability from CalCOFI data shows the California Current (CC) as a near-surface (0-300m), permanent, equatorward flow whose core is generally located 300to-400 km offshore California and whose western boundary occurs about 900 km offshore (Hickey, 1979; Lynn and Simpson, 1987). Typical speeds are 4-to-12 cm/s in the upper 150 m of the water column. Although there are exceptions (e.g., Simpson et al., 1984),

521

Fig. 1. Bathymetry in meters and location of current meter moorings. M1, M2 and M3 comprise the OPTOMA array, from 3 October 1984 to 1 July 1985; W10 is the LLWOD mooring, from 23 September 1982 to 1 September 1983. most of the vertical shear is contained in the upper 200 m and the horizontal variability at 500 db, relative to 1000 db, is insignificant (Wyllie, 1966). At depth, the poleward flow of the California Undercurrent (CU) occurs along the coast, to about 150 km offshore. The highest speeds in the core of this flow (about 30 cm/s over 20 km horizontally and 300 m vertically) are found along the continental slope. During periods of minimum monthly mean alongshore wind stress forcing (October to February), the CU surfaces and becomes the Inshore Countercurrent (IC), confined primarily over the continental shelf and slope but sometimes observed as far as 100 km offshore. During periods of maximum equatorward wind stress (March to August), surface equatorward flow overlies the CU. Along the coast, the strongest equatorward flow occurs in spring. The southward flowing CC moves inshore during April/May, overlying the poleward flow at the surface and sometimes displacing it at 200 m. From CalCOFI data, the main southward current at 200 m is usually farther offshore than at the surface (Wyllie, 1966). Interannual variations in the IC (inferred from dynamic height and coastal sea level) appear primarily due to remote forcing by poleward-propagating coastally-trapped waves originating in the tropics and to local forcing by anomalous onshore Ekman transport in winter. The variations do not appear to be related to interannual variability of the local wind stress curl; however the curl may contribute to the long-term tendency for poleward alongshore currents (McClain and Thomas, 1983; Chelton, 1982). The North Pacific subtropical high, the Aleutian low, and the thermal low over southwest North America govern the seasonal variations in the wind field over the CCS. During fall and winter, the Aleutian low is relatively strong and the subtropical high and the thermal

522

low are both relatively weak, as is the resulting wind field off Northern California which is dominated by the passage of storm systems at intervals of 3-to-5 days. During spring the high strengthens and moves northward; the thermal low also strengthens and the enhanced pressure gradient between the low and the high results in strong, persistent southeastward winds off Northern California. The passage of cyclones and associated fronts is often followed by a northeastward intensification of the high, producing strong upwelling events along the California coast (Halliwell and Allen, 1987). The climatological longshore winds over the CCS reach their maximum speed between 200 and 400 krn offshore resulting in a change in sign of the wind stress curl, with positive curl inshore and negative curl offshore (Nelson, 1977). Anomalous positive wind stress curl over the CC generates anomalous nearshore counterflow and upwelling of the thermocline in a region roughly parallel to the coast, approximately 200-to-300 km offshore (Chelton, 1982). This region of open ocean upwelling contrasts with coastal upwelling in a narrow (20-to-50 km) zone close to the coast. Based on CalCOFI data south of San Francisco, three subdomains of the CCS can be identified (Lynn and Simpson, 1987): the oceanic, the coastal and the intervening transition zone centered about 200 km offshore (at least south of San Francisco). The transition zone is defined by a band of maximum standard deviation of dynamic height. Eddy activity significantly contributes to this standard deviation as do spatial variations in the position of the core of the CC which may themselves be associated with eddy activity. The eddies of this transition zone seem to be primarily anticyclonic (Lynn and Simpson, 1987). Off Northern California, this is supported by the few maps from the CalCOFI program, although, as noted above, the horizontal station spacing is 74 km (or greater) in the offshore direction and about 2" alongshore. The mesoscale variability off Northern California is also known to include near-surface, intense offshore jets (e.g., Davis, 1985; Rienecker et al., 1985; Kosro and Huyer, 1986), especially in the summer when the winds are usually favorable for upwelling. The variability of both ocean currents and the surface wind field over the shelf and slope between P t Arena and Pt Reyes during the coastal upwelling season of 1981 and 1982 has been investigated during CODE (Coastal Ocean Dynamics Experiment, e.g., Winant et al., 1987; Huyer and Kosro, 1987). Mesoscale variability south of San Francisco has been described by Bernstein et al., (1977), Chelton (1984), Lynn and Simpson (1987), Breaker and Mooers (1987), Simpson et al. (1984), inter alia, and will not be described here. 3 DATA PREPARATION AND OBJECTIVE MAPPING METHODS. The seasonal mesoscale variability in the CCS off Northern California is investigated through maps of surface dynamic height relative to 450 db. These maps are produced from CTD, XBT and AXBT data from surveys with along-track station spacing of about 15 km (compared with a Rossby radius of deformation of 25 km). To calculate dynamic height from the (A)XBT temperature (T) profiles, a corresponding salinity (S) profile is estimated using either an average S(T) relation over the whole observational domain or, if the density of CTD stations is sufficient, from average T and S profiles calculated from CTDs within

523

50 km of the XBT. Some AXBTs provided data only to 300 m; to estimate the dynamic height relative to 450 db for these profiles, 19 dyn cm was added to the dynamic height relative to 300 m. This number represents the average shear between 300 and 450 db from deeper stations; the standard deviation is only 1 dyn cm. The vertical shear tends to be slightly higher in anticyclonic regions and slightly lower in cyclonic regions but the maximum difference (3 dyn cm) only slightly exceeds uncertainty in the dynamic height (about 2 dyn cm) and so the use of a constant value will not change the qualitative information in the maps below. A deeper reference level would show slightly more variability: the standard deviation of dynamic height shear between 450 and 750 db is 2 dyn cm with, again, higher values in the anticyclonic regions and lower values in the cyclonic regions. The generally low variability in dynamic height at depth is consistent with the observations from the CalCOFI program; however, the variability at depth can be nontrivial, as vertical excursions (of about 150 m) of isotherms, isohalines or isopycnals are evident to at least 700 m (from figures not shown). An anticyclone, offshore Pt Conception, with a pronounced subsurface maximum speed (up to 30 cm/s) at 250 m, has been reported (Simpson et al., 1984). For that case, horizontal variability below 750 m was markedly less than that above that depth. The repeated stations of OPTOMA11, 15 and 17 allowed estimation of anisotropic spacetime covariance functions. From OPTOMAll and 15 the fields at zero time-lag are nearly isotropic for distance-lags up to about 50 km (e.g., RMR87) and the predominant time variation indicated southwestward propagation at about 5 km/day for the first survey and southward propagation at about 2 km/day for the second. From OPTOMA17, the covariance was slightly anisotropic for lags greater than 20 km with high correlation for spatial separation oriented southeast-northwest. No distinct time variation was observable. For most surveys, the observations are dense enough that any anisotropic nature of the covariance function should have little influence on the Bnalyzed fields and so isotropy is assumed here. Since feature propagation varied from one survey to the next (and since the predominant time variation during any survey usually does not apply to all features encountered during the survey), no attempt is made here to form synoptic maps from the quasi-synoptic surveys. (In fact, the duration of these surveys is short enough that features would only be shifted by about 15 km, a difference that is not readily observable in these maps). The surface dynamic height data, relative to 450 db, from the larger (regional) observational domains (OPTOMA5 in June 1983, OPTOMAll in June/July 1984, OPTOMA15 in January/February 1985 and OPTOMA17 in August 1985) were used to estimate an isotropic spatial covariance function, Fig. 2. The general shape of the covariance function is consistent from all six surveys, although there is a spread of about 0.15 between extreme values at low spatial lags. The difference from one survey to the next is probably mainly due to differences in sampling patterns and number of observations. For example, OPTOMAll had many repeated stations with a denser coverage of the observational domain than OPTOMA5 (although along-track station spacing was comparable, there were many more stations for OPTOMA11). The other four surveys were intermediate between these two and have more consistent covariances. The ensemble covariance function was used t o fit the form

524

C

0 F

-0

0

Fig. 2. Isotropic covariance function for surface dynamic height calculated from OPTOMA5D2 (B);OPTOMA17-Dl D, A1 (+); OPTOMA11-D1, D2, A2 ( x ) ; OPTOMA16D1 (0); ( 0 ) ; and D2 (0). The solid line shows the ensemble covariance function used to fit the functional form of Section 3 and the dashed line shows the fitted function derived from this ensemble. The number of observations used for the calculations is 161, 364, 182, 197, 195 and 215, respectively, so that the lower bound for the 95% significance level is 0.1.

(where r is spatial lag). The calculated covariance at zero-lag is the sum of the mesoscale signal variance, Co, and the noise variance, E , due to sub-mesoscale processes as well as instrument noise. No estimate can be made of the covariance function associated with the sub-mesoscale processes because of the relatively coarse station spacing; here, it is assumed to be nonzero only at zero-lag. This assumption is acceptable if the station spacing is comparable to or larger than the analysis grid (Clancy, 1983), as is the case for this study. An estimate of E , and also of C, is made by extrapolating the function C ( r ) ,fit to the covariance at nonzero lags, to r = 0. From the ensemble covariance, E = 0.1, a = 90 km and b = 110 km. This covariance function was used in a statistical objective analysis (OA) model (e.g., Bretherton et al., 1976) to produce maps of the surface dynamic height. 4 MESOSCALE VARIABILITY FROM OCEANOGRAPHIC SURVEYS. Here, the alongshore wind stress and its curl are investigated in association with maps of mesoscale ocean variability from the OPTOMA program. The wind stress was cplculated, using the Large and Pond (1981) formulation, from Fleet Numerical Oceanography Center (FNOC) wind analyses on a Northern Hemisphere polar stereographic grid. These analyses, which are a blend of wind data and a six-hour forecast from NOGAPS (the Navy Opera-

525

tional Global Atmospheric Prediction System), are intended to represent wind at a height of 19.5 m above the surface. For subsequent analysis, the FNOC values were reduced to a height of 10 m, using a neutral flux profile with no rotation of the wind direction (Summers, 1986). Wind stress curl estimates based on these FNOC analyses (at a grid spacing of about 330 km) will provide lower bounds to the actual values. In the data presented below, the alongshore wind stress is taken to be the component 30” W of N. 4.1 The circulation in summer. During summer, the alongshore wind stress is predominantly equatorward and is favorable to coastal upwelling (Fig. 3a); strong wind stress events tend to be associated with positive wind stress curl (Fig. 3b). Intense offshore jets (about 50 km wide, with maximum speeds in excess of 50 cm/s relative to 450 db) advect cool coastal water several hundred kilometers offshore to produce the cool filaments so prominent in AVHRR images off the west coast of North America during summer (e.g., Fig. 4). The intense jet is often associated with a pair of counter-rotating eddies (Mooers and Robinson, 1984). The maps of surface dynamic height for each year clearly show strong current filaments inextricably associated with an eddy field both offshore and in the coastal transition zone. The large scale maps from OPTOMA5 and OPTOMAll (Figs 5a, b) show very similar structures except that the meandering current filament near 38N is less intense during OPTOMAS. (Some of the differences could be associated with the coarser sampling pattern in 1983 and two-week difference in sampling period.) The strongest gradients (speeds up to 60 cm/s) were found in June 1984 (Fig. 5d) when equatorward wind stress larger than 2 dyne cm-’ persisted from 13 to 20 June. During this time the wind stress curl was consistently positive and averaged about 3 x lo-* dyne cm-’. The repeated observations during summer 1984 allowed a glimpse of the evolution of the jet from early June (Fig. 5c), when the seaward extent of the jet was about 125.5W, to early August (Fig. 5f) when, in the same observational domain, the jet was seaward of 125.5W. The reorientation of the jet from late June to early August is associated with offshore propagation of the anticyclone to the north of the jet (Fig. 5b) and a broadening of the cyclonic region south of the jet (Figs 5e, f ) under the influence of the wind stress curl (RMR87) which is consistently positive for most of July. In comparison, in 1985 the anticyclone does not appear to propagate offshore between 2 1 July and mid-August (Figs 5g, h) although there is a reorientation of the jet/anticyclone and an apparent diminishing of the cyclonic zone. The differences between 1984 and 1985 are likely associated with differences both in the wind stress and its curl. During July 1985, the curl is much weaker and events of positive curl are less sustained. The equatorward wind stress events are also weaker and less prolonged; for most of August both quantities are near zero. From 21 July to mid-August, the intensity of the anticyclone decreases. From mid to late August both the anticyclone and the cyclone to the south of the offshore jet propagate to the northwest. The inshore cyclone south of Cape Mendocino propagates about 50 km to the southwest (Fig. 5i). The increased meander and appearance of a small anticyclone in the southeast “corner” of the large anticyclone in late August is possible evidence of a local

526

1983 4 hl

0 -

x TJ

4 -

)

1984 0 JUNE

3

JULY

AUGUST

iga5

0

A

-4 1

8

IS

22

29

6

JUNE

13

20

27

3

JULY

10

17

24

31

AUGUST

1986

-10

3 1

E t -

>U 03

8

15

22

28

6

JUNE

10 -

13

20

27

3

JULY

10

17

24

31

24

31

AUGUST

1983

-10 1

10 -

B

15

JUNE

22

29

5

13

20

JULY

27

3

10

17

AUGUST

1984

0

Fig. 3. Time series of (a) alongshore wind stress and (b) wind stress curl during summer.

521

Fig. 4. AVHRR image (Channel 4, with a 1" grid superposed) from the NOAA'I satellite on 20 June 1984.

528

OPTOMAll - D1, 02 - 10 July, 1984

OPTOMA5 - D, A1 12 - 22 June, 1983

23 June

.... .. 9

.

401

381

.. .. .

3?!

kv

126W

I

124W

124w

3q!

Y

OPTOMAll - A1 5 - 15 June, 1984

126W

124W

122w

OPTOMA11 - A2, D1 21 - 30 June, 1984

40N

40N

38N

38N

f..

-.

., .

I

i/

126W

124W

122w

3Yi!8w

126W

124W

122w

OPTOMAll - P 18 July, 1984

I

e

401

Fig. 5. Surface dynamic height relative to 450 db during summer. The contour interval is 2 dyn cm. The dots show station positions.

38f

I

126W

124W

122w

529

OPTOMA17 - P 21 July, 1985

OPTOMAl1 - D3 27 July - 5 August, 1984

40N

38N

38y8W

126W 124W 12 w OPTOMA17 - D2 23 August - 5 September, 1985 N

126W 124W OPTOMA17 - D1 10 - 22 August, 1985

4or

40P

381

38F

v

3T

N

126W 27 July

40N

3%8w

40N.

126w

124W Fig. 5 ctd.

124W

OPTOMA22 - 5 August, 1986

I 122w

530

instability process. This small anticyclone has a larger amplitude a t depth than the large anticyclone (see Fig. 8d). The evolution during summer 1983 cannot be determined unambiguously from the repeated sampling in a small domain (120 km square); however (from maps not shown, see Robinson et al., 1986), there is a marked increase in horizontal shear associated with an intensification of the anticyclonic region t o the north of the jet from mid-June to mid-July. Dynamical interpolation between observation periods using a baroclinic quasi-geostrophic model showed the evolution to be subject to barotropic instability processes (Robinson et al., 1986). The variability in both wind stress and its curl in 1983 was similar to that in 1985. There are some similarities between the variability and evolution in 1984 and that in 1986 (Fig. 5e c.f. 5j and 5f c.f. 5k). The anticyclone to the north of the jet has a higher amplitude in 1984 and the corresponding horizontal shear is slightly stronger; however, in 1986, there is also a strong anticyclone west of the seaward extent of the offshore jet. The subsequent evolution shows a reorientation of the jet as the northern anticyclone propagates offshore and apparently intensifies. The cyclonic zone broadens and poleward flow is evident inshore. The wind forcing in 1986 contains several sustained equatorward wind stress events especially during July. On 2 1 July there is an abrupt cessation of a strong event, after which there would be a relaxation of coastal upwelling. The stress throughout June and July is noticeably weaker and less sustained than for corresponding months in 1984. The shorter burst-like events may account for the multiple fronts and filaments evident in AVHRR images in July 1986 (see Rienecker and Mooers, 1988). The wind stress curl during 1986 has a prominent cycle of events of strong positive curl followed by weak, negative curl during June and early July. On 21 July, there is a transition to consistently positive curl until 14 August (Fig. 3b). Thus, there are consistent features in the summertime surface dynamic height fields off Northern California in several different years. Interannual differences could be related to differences in wind stress curl forcing and in the strength and persistence of coastal upwelling events. Anomalous onshore advection sometimes associated with El Nino events (Simpson, 1984; Rienecker and Mooers, 1986) can also perturb the seasonal signal. During 1984 and 1986 the evolution of the offshore jet and associated eddy field appeared similar and the jet was probably unstable in that the intensity of the anticyclone increased even though the cyclonic regions broadened under the influence of the wind stress curl. In 1985, the evolution was different from the surrounding years and the eddy field appeared to “spin down” slightly in that the amplitude of the anticyclone diminished from mid-July to early August and the cyclonic region did not broaden to dominate the observational domain. However, there was also evidence of local instability in the anticyclonic region. The intense offshore jets are associated with onshore flow, sometimes as part of a meandering current filament (Fig. 5a) or in the form of ‘return flow’ south of the jet (e.g., Fig. 5b). Most of these summertime fields display poleward flow over the continental slope between Pt Reyes and P t Arena.

531

4.2 The circulation in winter. The wintertime forcing is much more variable in direction than that in summertime. For the years shown (Fig. 6a) the stress is weaker in 1983/4 and 1984/5 than for summer; however, the poleward alongshore stress in January/February 1986 is far stronger than in the other years or in summer and is associated with strong, negative wind stress curl forcing (Fig. 6b). The horizontal gradients of surface dynamic height are noticeably weaker in winter than in summer. There is equatorward flow (expected of the CCS) in December 1983 (Fig. 7a) and (in the southwest corner) in January 1985 (Fig. 7b). Changes in the eddy field are apparent from the OPTOMA15 surveys. The synoptic map of 15P (Fig. 7b) was acquired during the initial sampling of 15D1 (Fig. 7c). The two maps are in agreement in the offshore region where 15D1 sampled first, although the weakening of the equatorward current filament in the southwest is evident; this weakening continues through mid-February (Fig. 7d). From the inshore portion of 15D1, the anticyclone centered at about 126W evolves so that the major axis is re-oriented east-west and a cyclone develops south of Cape Mendocino. From late January to mid-February, the anticyclone appears to propagate southeastward and intensify slightly. The inshore cyclonic center propagates equatorward. The field from OPTOMAIS (Fig. 7e) is similar to that of 15D2 inasmuch as there is a very narrow region of flow reversal parallel to the coast with a cyclone inshore, northwest of P t Arena. However, the positions of the offshore high and low are reversed. 4.3 Mesoscale Structure at 200 db. The horizontal patterns of dynamic height have some coherence over the upper water column. The structure at 200 db (Fig. 8), mid-to-lower thermocline in this region, is fairly weak relative to 450 db. Current filaments are less prominent than at the surface and the regions of strongest flow are associated with eddies. Weak poleward flow is evident inshore, especially during summer. The poleward flow at this depth is strongest during OPTOMA22 (Fig. 8e) when the flow is about 10 cm/s (compared with 15 cm/s at the surface). In comparison, the maximum equatorward flow farther offshore is about 5 cm/s at this depth (compared with about 30 cm/s at the surface). There is no evidence in these maps to support Wyllie’s (1966) observation that the main equatorward current is farther offshore at 200 m than at the surface, primarily because of the predominance of eddies (which would not be well-resolved in the CalCOFI observations) rather than equatorward flow. In summary, eddies are ubiquitous features of the CCS off Northern California. The main difference between summer and winter regimes is the presence of the intense, near-surface (0-150m) current filaments associated with the summer coastal upwelling regime. The current meanders during summer tend to be sharper than those during winter so that the eddies often have a smaller horizontal scale. The anticyclones of summer also have larger amplitude, with maximum dynamic height reaching 100 dyn cm (compared with about 88 dyn cm in winter). The dynamic height lows do not vary much from summer to winter. Six summer and five winter surveys had sufficient station density in a common region (37-39N,

532

8 r 4 0 -4 1

B

15

22

29

5

DECEMBER 1983

hl

1

8

15

22

12

19

26

2

JANUARY

9

16

23

FEBRUARY 1984

29

5

DECEMBER 1984

12

19

26

2

JANUARY

I6

9

23

FEBRUARY

h

1985

8 r

A 1

8

15

22

28

5

DECEMBER 1985

m

x

U 00

19

26

2

9

16

23

FEBRUARY 1986

o

10

0

1

g

I2

JANUARY

-10

-

1

B

15

22

29

5

I2

I9

DECEMBER 1984

JANUARY

DECEMBER 1985

JANUARY

26

2

9

16

25

FEBRUARY 1985

0

B

-10 FEB~UARY 1986

Fig. 6. Time series of (a) alongshore wind stress and (b) wind stress curl during winter.

533

OPTOMA8 10 December, 1983

3%w:!

OPTOMA15 - P 27 January, 1985

I 126w

124w

122w

OPTOMA15 - D1 24 January - 6 February, 1985

OPTOMA15 - 02 8 - 23 February, 1985 r

40N

38N

122w

12iw 124w 122w

3q2gw

OPTOMA19

8

- 13 February, 1985

40N

Fig. 7. Surface dynamic height relative t o 450 db during winter. The contour interval is 2 dyn cm. The dots show station positions.

-

3%!8W

126W

124W

12

w

534

OPTOMA11 - DI, 02 2 3 June - 10 July, 1984

OPTOMA8 10 December, 1983

40N

40N

38N

38N

..

I

a .

~

'Wi

JV

126W

124W

12 I

% ' i'

OPTOMA15 - D1 24 January - 6 February, 1985

3% 8w !

v

126W 124W 122w OPTOMA17 - 02 23 August - 5 September, 1985

126W

124W

122w

126W

124W

I

122w

OPTOMA22 27 July - 5 August, 1986

Fig. 8. Dynamic height at 200 db relative to 450 db. The contour interval is 2 dyn cm. The dots show station positions.

38Ysw

126W

124W

12 W

535

124-126W) to allow a direct comparison of variability. Because of differences in station density, the comparison is done using OA maps, all using 10 km horizontal grid spacing. From the summer surveys, the average dynamic height was 83 dyn cm and the standard deviation 5 dyn cm; the dynamic range varied from 12 to 22 dyn cm (from OPTOMA17-D2 and 11D1, D2, respectively) and the maximum speed from 25 to 55 cm/s (from OPTOMA22 and 11-Dl, D2, respectively). From the winter surveys, the average dynamic height was 80 dyn cm and the standard deviation 3 dyn cm; the dynamic range varied from 6 to 12 dyn cm (from OPTOMA15-P and 8, respectively) and the maximum speed was consistently about 20 cm/s in all winter surveys. 5 MESOSCALE VARIABILITY FROM CURRENT METER DATA. The current meter moorings (see Fig. 1 for locations) acquired data, from October 1984 to July 1985, from about 150 m below the surface to 200 m above the ocean bottom in water depths of 3500 to 4300 m (Rienecker et al., 1988, henceforth referred to as RMS88). The moorings were separated by about 100 km and those closest to shore were about 50 km from the base of the continental slope. The currents at each of the moorings are surfaceintensified (at M1 the ratio of root mean square variability at 175 to that at 375 m is 4.3; at M2 the corresponding ratio is 2.6) and show high vertical coherence in the upper 1200 m of the water column. The signals at depth are apparently constrained by local topography near the continental slope; farther offshore the variability 200 m off the bottom has a larger amplitude (Stabeno and Smith, 1987). The variability near 150 m is mainly in the northsouth direction at M1 and M2. (Measurements were not available at this depth at M3). There is no obvious seasonal cycle in the current or temperature time series (Fig. 9), except possibly for the presence of fairly persistent poleward flow at M2 from October to February and at M1 from January to April. Although the core of the CU (and the associated surface IC) is over the continental slope, poleward flow has been observed as far as 150 km offshore (Lynn and Simpson, 1987). This tendency for poleward flow at M1 and M2 leads to a gradual increase in temperature at these two moorings; but fluctuations associated with mesoscale flow variability are superposed. Neither the velocity nor the temperature variations at MI, M2 and M3 seem to have much correlation. At M2 and M3, the temperature fluctuations from mid-March to the end of June appear to be influenced by fluctuations in northward transport. Some events at MZ(340) appear to have a similar signature at M3 about a month later. For example, the sharp rise (by about 0.5C) in temperature a t M2 towards the end of November is mimicked at M3 at the beginning of January, consistent with the offshore propagation of mesoscale features at about 1-to-2 km/day. This feature has a more distinct signal at 350 m than at 150 m where there is greater variability in current strength and direction. There is also evidence for offshore advection in the frequency domain: the offshore flow at M3(350) is coherent with T(350) at M3 at periods of 8.2-to-9 days with temperature lagging by 90". The rotary nature of the current at M1 and the sudden increase in T(175) (by about 1.5C) is evidence for an anticyclone impinging on the array in late October/early November. The anticyclone barely encounters M2 and appears to propagate off to the

536

Fig. 9a. Time series of the upper ocean current from the OPTOMA array.

537

M1 T(175) 7.5

7.0

-

0

6.0 5.0

M1 T(375)

1

1

M2 T(145) 7.0 r 0 0,

a,

6.0

U

5.0

'

M2 T(340) J

0 0-l

a

U

7.5

7.0 0 0,

a

U

6.0 5.0

M3 T(145)

'

J

1'

1

M3 T(350)

1

"

'

'

'

NOV 1984

l

'

'

DEC

'

'

"

.

JAN

.

'

l

'

FEE

'

'

'

MAR

~

'

.

'

APR 1985

.

~

'

'

MAY

'

.

I

~

~

~

~

JUN

Fig. 9b. Time series of the temperature measurements from the OPTOMA array.

~

~

~

r

.

l

538

northwest. At M1 and M3 the temperature decreases, towards the end of June, apparently due to offshore advection of coastal, upwelled water, consistent with the offshore jets evident in summertime surface dynamic height patterns. At M1, the offshore jet is surface-intensified with speeds (from mid-to-late June) up to 16 cm/s at 175 m and up to 8 cm/s a t 375 m. At M3, the speeds at this time reach 13 cm/s at 350 m; T(175) at M1 drops to as low as 8.1C and T(145) at M3 drops rapidly to as low as 7.7C while T(145) at M2 rises due to poleward flow. At least for these observations, the offshore jet is north of M2 and the jet axis must be slightly south of M1 or the jet accelerates offshore to M3. Prior to the appearance of the offshorejet, there is strong, persistent equatorward flow at both M1 and M2. At M2 this flow appears to advect cold water southward, although the rapid drop in temperature to as low as 7.5C indicates the likelihood of some other mechanism. The equatorward flow at M1 is not accompanied by a marked temperature decrease. Comparison with available oceanographic data (Fig. 1Oc) shows that M2 was under the influence of an inshore cyclone (which may have advected cool coastal water from P t Arena to M2) and that both M1 and M3 were influenced by a larger-scale offshore anticyclone. Oceanographic data from other OPTOMA surveys concurrent with the moorings also show the influence of eddy activity or current meanders on the array (Fig. IOa, b). The first two surveys sampled the same anticyclone which appeared to intensify slightly from the first survey to the second. The cyclone to the south of the array during 15D1 had decreased in diameter during 15D2 and an elongated weak anticyclone developed in the vicinity of M2. The slightly rotary nature of the currents at M1 in late February/early March suggests that, subsequent to 15D2, the intensification of either the large or the elongated anticyclone may have continued and its presence may have been detected at MI. The larger anticyclone of OPTOMA16 appears, from the consistently southward flow at M1 and offshore flow at M3 (Fig. 9a), to have persisted (with some evolution) during May and early June. The presence of the anticyclone was apparently felt almost simultaneously at M1 and M3 on about 1 May and continued until about 16 June when there was an abrupt change in flow direction at these two moorings. The southward flow at M2, associated with a seemingly smaller scale cyclone nearer the coast was evident for a shorter time (viz., about 11 May to 2 June). The superposition of current meter flow vectors and the geostrophic flow at 150 m, relative to 450 db, shows greatest discrepancies in direction (as may be expected) at low speeds. Speed estimates agree best in the larger anticyclone of OPTOMAl6. Elsewhere, when the shear is greater between 150 and 350 m, the geostrophic flow estimate at 150 m is about 50% of the measured current. For the stronger flows, the direction given by the geostrophic estimate is fairly good. At M2, the agreement between the vertical geostrophic shear and the current meter shear is better than for the currents per se (RMS88), indicating that the geostrophic reference level should be deeper than 450 db. For M1, the discrepancy in shear is comparable to that in the velocities, so differences cannot be attributed solely to an inappropriate reference level: for OPTOMA16, the geostrophic speed at Ml(150) is

539

!W

37N'

37"

-12kw

12kW

-

124W

1: IW

124W

1 !W

Fig. 10. Dynamic height at 150 m, relative to 450 m, from (a) OPTOMA15-D1, 24 January to 6 February, 1985; (b) OPTOMA15-D2, 8 to 23 February, 1985 and (c) OPTOMAIG, 20 May to 11June, 1985. The currents at 150 m (solid arrowhead) and 350 m (open arrowhead), from the mooring array, are shown with the geostrophically estimated currents at 150 m, relative to 450 m (dashed).

540

equivalent to the current meter speed. In summary, the variability determined from the current meter data is consistent with that from concurrent quasi-synoptic surveys. The array is seen to be influenced by eddies, which are of small enough scale that they may account for the lack of large-scale coherence between velocity or temperature variations at the three moorings. Although there is some evidence for the offshore propagation of eddy variability from M2 to M3, there is no firm evidence that eddies which influence M1 propagate southward to dominate the variability at M2. 6 THE OCEAN PREDICTION EXPERIMENTS. 6.1 The model. Three of OPTOMA's series of surveys were designed as ocean prediction experiments. From these surveys several forecast experiments were conducted by initializing a dynamical model with observed fields and allowing the model to evolve the fields according to assumed dynamics and various boundary data. The model used for the forecast experiments (Robinson et al., 1986; RMR87) is the open-ocean, quasi-geostrophic (QG) model presented in Haidvogel et al. (1980) and Miller et al. (1983). A model integration requires initial specification of streamfunction and vorticity throughout the domain and, at each subsequent time step, specification of streamfunction along the boundary and vorticity at inflow points along the boundary (Charney et al., 1950). In addition, the vertical velocity, w , at the top and bottom boundaries must be specified at each time step. At the surface, w is approximated by w = ir

. {V x . / p f } ,

where r is the surface wind stress and 1; is the unit vector in the positive z direction. Bottom topography, B(x,y), is included merely as the kinematic effect of the bottom slope (which should be no larger than O ( E ) where , E is the Rossby number, V / f o L = 0.02): w = u V B , where u = -1; x VG, and 1c, is the QG streamfunction. Nonzero vertical velocity forces density changes through the relation

) N i / N * ( z ) , with L a horizontal length scale of where '?I = f o L 2 / N i H 2 and ~ ( z = the motion (50 km), f o the Coriolis parameter (0.91 x s-'), Nz the stratification scale, (NO= 0.011 s-'), NZ(z)the average (in space and time) stratification, and H the scale of thermocline depth variations (150 m). Thus, the surface and bottom values of a&, needed for the solution of $, are provided by the integration in time of the above equation, i.e., =

-

f, [I'-2w + u . V (a&)] dt

at z = 0, and -B(x,y).

In the model integration experiments finite differences are used in the vertical with the local bottom topography (slope is 0(10-3) in the domain shown in Fig. 5c). For the OP-

541

TOMAll experiment the model evolves the streamfunction at 50, 150, 400, 1070, 2150 and 3380 m, for an average ocean depth of 4000 m. The horizontal grid spacing is 8.3 km in a 130 km square model domain (the square region evident in Fig. 5c) and the time step is 2 hours. 6.2 The initialization and boundary updating strategy. The initial and boundary data are obtained from OA fields. The initial field is always the best estimate at the central day of a particular cruise, i.e., it is formed from data throughout the cruise by statistical interpolation in space and, if possible, in time. Dynamical experiments are hindcasts if the boundary data are obtained by statistical interpolation between surveys and forecasts if the boundary data are obtained by statistical extrapolation using data only from the initializing survey. The horizontal fields of streamfunction and vorticity are required at each level used by the model. The mass field data acquired during OPTOMA surveys do not permit estimation of the total (barotropic plus baroclinic) streamfunction. For the prediction experiments this deficiency is compensated for by assuming a constant reference level throughout the domain over the entire forecast period. The reference level is chosen, by trial-and-error, to give the “best” forecast at all levels. Since the maximum depth of most data is only 450 m, the streamfunction at the deeper levels of the dynamical model (or at a deeper reference level) is estimated by a vertical extension process. This process uses the eofs (empirical orthogonal functions) of density, determined from deep (3000 m) casts acquired during the experiment, to extrapolate shallow profiles to the bottom (RMR87). For the OPTOMAS experiment, a 1550 m reference level was used; for the OPTOMAll experiment, 750 m. These reference levels led to low speeds a t the deep levels of the model (10 cm/s or less), consistent with the current meter data. The difference in reference level in these two experiments is due to differences in the vertical shear at depth inferred from the first eof. For OPTOMA11, the signal was very small below 200 m. Differences in the eofs are, at least partially, due to interannual effects such as El Nino which had a large subsurface (maximum at 100 m) signal in the CCS off Northern California during summer 1983 (Rienecker and Mooers, 1986). The use of a constant reference level presumes that the baroclinic and barotropic components of flow are phase-locked. The current meter data were used to look at the barotropic and baroclinic signals off Northern California and any association between them. The first three modal profiles (Fig. 11) determined from OPTOMA CTD data were fit, in a linear least-squares manner, to the current meter data a t all available depths at M1 and M2. The fit at M3 proved untenable because data were available only in the mid-water column (RMS88). At M2, the long-term ratio of barotropic (mode 0) to first baroclinic (mode 1) energy is 0.62. This is equivalent to a root mean square modal amplitude ratio of about 0.8 and, since the amplitudes are positively correlated with almost no temporal or directional offset, indicates deep flow nearly at rest. At M1, the estimated ratio of mode 0 energy to mode 1 energy is 1.1; the barotropic energy is equivalent to that at M2, but the baroclinic energy is slightly

,542

-2

-4

0

a

I

a

I

m

0 '

*

'

*'

/

1-

I

2-

3-

-

4

Fig. 11. The barotropic and first two baroclinic dynamical modes calculated from the N2 profile determined from CTD data in the vicinity of the OPTOMA array. less. Again, the modal amplitudes are positively correlated at zero lag (although the maximum correlation is mode 0 leading mode 1at 5 days) with no significant directional offset, so that the deep flow is nearly at rest. Farther offshore (at a Low-Level-Waste-Ocean-Disposal Program Mooring, see Stabeno and Smith, 1987), about 250 km northwest of the OPTOMA array (see Fig. 1 for location), the long-term ratio of mode 0 energy to mode 1 energy is 1.6. This dramatic change from M2 is due to an increase in mode 0 energy which is three times that at M2. The mode 1, and even mode 2, energy is comparabIe at the two sites. Modes 0 and 1 a t this offshore site are generally not well correlated; however, there is coherence in broad period bands (between 5 and 32 days) corresponding to mesoscale activity and in these bands the modes are out of phase. Hence, at this offshore site, the modal relationship is such as to enhance the flow at depth and indicates a shallow reference level (near 600 m, based on the energy at various depths). This spatially-varying modal decomposition complicates the strategy for ocean prediction. 6.3 Forecast experiments.

For the OPTOMA5 experiment (Robinson et al., 1986), the observations were not adequate to define temporal variation in the covariance function. Hindcasts used boundary data estimated by linear interpolation between OA fields at the central day of surveys (separated by 14 days) to evolve the QG streamfunction and potential vorticity. Forecasts used boundary data which persisted from the initialization field. The hindcasts, with their improved boundary data, provided accurate evolution of the observed features and, as noted above,

543

TABLE 1 Error measurest for streamfunction at 50 m Experiment No curl FNOC curl constant curl vorticity source

C NRMSD NRMSS NRMSU 0.73 123% 89% 85% 0.82 68% 68% 96% 0.96 30% 3% 28% 28% 0.94 38% 26%

t The error measures are those defined by Willmott et al. (1985 . C is the correlation; NRMSD is the root mean square difference normalized by the stan ard deviation of the OA field; NRMSS is the normalized systematic root mean square difference which highlights linear biases in the fields; NRMSU is the normalized unsystematic root mean square difference which highlights differences in patterns.

d

indicated that this evolution was partly governed by barotropic instability processes. For the OPTOMAll experiment (RMR87), the predominant temporal dependence of the covariance function was southwestward propagation at about 5 km/day. This information was used for statistical interpolation between surveys and for synoptic estimates of fields at the central dates of surveys, so,that better estimates of initial and boundary data could be obtained. Each survey was of limited duration (usually less than 7 days in the forecast domain) so that asynopticity of the initial field was not detrimental to the model evolution; however, the evolution based on synoptic fields was slightly better than that based on asynoptic fields. The OPTOMAll hindcasts demonstrate the importance of wind stress curl forcing when the curl was strong and positive for a sustained period (about 20 days), viz., from the field of Fig. 5d to that of 5e. Although the hindcast from Fig. 5e to 5f using FNOC wind stress curl has less error than that with no surface forcing (Table l), the error is still high and there is far too much anticyclonic curvature in the jet (Fig. 12b, c c.f. 12a), possibly due to inadequacy of the surface and/or boundary forcing. As noted above, the FNOC curl is likely an underestimate of the true curl. In addition, small-scale variability will not be resolved by the coarse FNOC grid, so that in this region near the zero curl line, the sign of the FNOC curl could be incorrect. On experiment used an ad hoc constant curl of 6 x10-* dyn cm-’ during the integration between the streamfunction fields of Fig. 12a. This value of the curl gives a fairly good agreement between model hindcast and observations (Table 1);no attempt was made to refine the curl estimate since this forcing is artificial. The main discrepancy between the model-generated field and the observed field, the intensity of the cyclonic region to the south of the jet, is possibly related to errors in the initialization field which was determined from the shallow AXBT data. Another possible source of error lies in the boundary data. Miller and Bennett (1988), in similar open ocean simulations, show how the vorticity information on the boundary cannot propagate into the interior of the model domain when the streamlines are tangent to the open boundary, so that the

544

Fig. 12. Nondimensional streamfunction at 50 m. The spatial and temporal mean has been removed. Contour interval is 0.5. (a) OA; (b) dynamical model hindcast with no wind stress curl; (c) model hindcast using FNOC wind stress curl; (d) model hindcast using a constant ~ cm-'; (e) model hindcast using a vorticity source along the wind stress curl of 6 X ~ O -dyn northern boundary.

545

model may evolve the vorticity near the boundary independently of the ‘observed’ vorticity. The use of a horizontal filter, which eliminates possible instabilities associated with large vorticity gradients due to the incompatibility of the fields near the boundary, can produce streamfunction fields which differ from those observed. Clearly, the streamfunction field approximates this singular case along the northern boundary of Fig. 12a and Fig. 5e. In addition, the vorticity along the northern boundary may not be accurately defined by the coarse station spacing of Fig. 5e. A hindcast experiment was run in which a constant vorticity source was applied along the northern boundary. This source was equivalent to about 20% of the planetary vorticity. The model-generated field (Fig. 12e) is comparable to that of Figs 12a and d, except that the cyclonic region has lower amplitude. In fact, the small closed circulation in the lower half of the domain is anticyclonic. Either the required surface forcing is strong (about 6 X ~ O -dyn ~ cm-’) throughout the model integration, unlike the FNOC forcing, or some combination of increased surface forcing and additional boundary forcing is required to evolve the streamfunction fields consistent with observations. Given accurate initial, boundary and surface forcing data, the QG open ocean model developed at Harvard University and documented in Miller et al. (1983) evolves the mesoscale fields of the CCS consistent with observations. Dynamical instability processes have been shown to operate in the CCS and the importance of wind stress curl has been identified.

7 SUMMARY. During summer, alongshore wind stress off the coast of Northern California is predominantly equatorward and favorable for coastal upwelling. Maps of surface dynamic height from OPTOMA surveys during summer show intense offshore jets and associated eddies in the coastal transition zone. There is nearshore poleward flow (usually weak) in most of the summertime fields between P t Reyes and P t Arena, consistent with fields produced in theoretical modeling studies (e.g., Ikeda et al., 1984a, b). Such studies show, in the surface pressure field, the development of a series of highs and lows (meanders) alongshore as a result of linear baroclinic instability of the equatorward coastal upwelling jet and poleward undercurrent. Mutually induced offshore velocity causes the offshore propagation of dipole eddies which separate from the equatorward flow due to nonlinear instability. In contrast to the observations, the numerical results show cyclones which are much more intense than the anticyclones and the remnant of equatorward surface flow at the coast. Observations show the coastal current to be spatially inhomogeneous, occasionally with strong equatorward flow, but usually with weak, variable flow (Kosro, 1987). During summer, the current meanders tend to be sharper than those during winter so that eddies often appear to have smaller horizontal scale; however, the covariance functions estimated from the larger regional surveys have consistent structure in summer and winter and in different years. The mean dynamic height, the standard deviation, the dynamic range and maximum current speeds are larger in summer than in winter. In both seasons, the horizontal structure at 200 db, relative to 450 db, is weak; current filaments are less prominent than at the surface and the regions of strongest flow are associated with eddies.

546

This pattern contrasts with numerical simulations of the instability of the upwelling jet which show the strongest flow at the surface to be associated with eddies and at depth with the meandering poleward current filament. The variability determined from the current meter data is consistent with that from concurrent quasi-synoptic surveys. The array of three moorings, separated by about 100 km, is influenced by small-scale eddies and larger eddies which do not propagate across the entire array, which may account for the lack of coherence between velocity or temperature variations at the three moorings. Some features appear to propagate offshore from M2 to M3; the temperature at M3 clearly shows the influence of onshore/offshore advection. Ocean prediction experiments were undertaken to help identify the feasibility of and limitations to open ocean forecasting. Hindcast experiments in which the mesoscale features observed during a survey were dynamically interpolated, using a QG model, to the time of the next survey showed that given accurate initial, boundary and surface forcing data, the mesoscale fields of the CCS could be predicted with QG dynamics. Dynamical instability processes were shown to operate and the importance of wind stress curl identified. Thus, dynamical interpolation helped, not only to fill data gaps, but also to identify the significance of physical processes operating in the CCS. The strategy of ocean prediction from mass field data alone is complicated by the spatial and temporal variations in the relation between the barotropic and baroclinic flow fields. These variations, plus the lack of information on boundary and surface forcing, highlight the importance of four-dimensional data assimilation to studies of mesoscale ocean variability. The surveys of the OPTOMA program have helped identify the ubiquitous nature of eddies in the coastal transition zone of the California Current System off Northern California. The existence of offshore jets and associated eddies has been inferred from satellite imagery and from drifter tracks in summer (e.g., Davis, 1985). From the maps presented here, eddy variability is also evident in winter and, consistent with Lynn and Simpson’s (1987) observations, in both seasons the anticyclones tend to be more energetic and have larger scale than the cyclones. ACKNOWLEDGEMENTS The OPTOMA Program was a joint program between the Naval Postgraduate School (PI: Chris Mooers) and Harvard University (PI: Allan Robinson), sponsored by the Office of Naval Research whose support is gratefully acknowledged. The surveys could not have been completed without the cooperation of the captains and crews of each of the research vessels (the R/V Acania, the R/V P t Sur, the NAVOCEANO’s USNS De Steiguer and the NOAA Ship McArthur) and the flight crews from US Navy Patrol Squadron Ninety-one and the Naval Air Reserve Center at NAS Moffett Field, CA. Of course, the contributions of the scientific crews and the Chief Scientists (Arlene Bird, Skip Carter, Marie Colton, Gordon Groves, Pat Kelley Jr, Chris Mooers, John Rendine, Allan Robinson, Jerome Smith, Leonard Walstad and Paul Wittmann) were essential to the data acquisition and are gratefully acknowledged. Computer time was provided by the Institute for Naval Oceanography

541

Computer Center. The Institute for Naval Oceanography is sponsored by the Navy and administered by the Office of the Chief of Naval Research. Contribution No. 8 from the Institute for Naval Oceanography. REFERENCES Breaker, L.C. and C.N.K. Mooers, 1986. Oceanic variability off the central California coast. Prog. Oceanogr., 17: 61-135. Bretherton, F.P., R.E. Davis and C.B. Fandry, 1976. A technique for objective analysis and design of oceanographic experiments applied to MODE-73. Deep-sea Res., 23: 559-582. Charney, J.G., R. Fjortoft and J. von Neumann, 1950. Numerical integration of the barotropic vorticity equation. Tellus, 2: 237-254. Chelton, D.B., 1982. Large-scale response of the California Current to forcing by the wind stress curl. In: CalCOFI Rept. 23, Calif. Coop. Ocean. Fish. Invest., La Jolla, CA, pp. 130-148. Chelton, D.B., 1984. Seasonal variability of alongshore geostrophic velocity off central California. J. Geophys. Res., 89: 3473-3486. Clancy, R.M., 1983. The effect of observational error correlations on objective analysis of ocean thermal structure. Deep-sea Res., 30: 985-1002. Davis, R.E., 1985. Drifter observations of coastal surface currents during CODE: The method and descriptive view. J. Geophys. Res., 90: 4741-4755. Haidvogel, D.B., A.R. Robinson and E.E. Schulman, 1980. The accuracy, efficiency and stabilitv of three numerical models with application t o open ocean problems. J. -Compu-t. Phys., 34: 1-53. Halliwell, G.R. and J.S. Allen, 1987. The large-scale coastal wind field along the west coast of North America. 1981-1982. J. GeoDhvs. Res.. 92: 1861-1884. Hickey, B.M., 1979. The California Curreh-System’- hypotheses and facts. Prog. Oceanogr., 8: 191-279. Huyer, A. and P.M. Kosro, 1987. Mesoscale surveys over the shelf and slope in the upwelling region near P t Arena. J. Geophys. Res., 92: 1655-1682. Ikeda, M., W.J. Emery and L.A. Mysak, 1984a. Seasonal variability in meanders of the California Current system off Vancouver Island. J. Geophys. Res., 89: 3487-3505. Ikeda, M., L.A. Mysak and W.J. Emery, 1984b. Observation and modeling of satellite-sensed meanders and eddies off Vancouver Island. J. Phys. Oceanogr., 14: 3-21. Kosro, P.M., 1987. Structure of the coastal current field off Northern California during the coastal ocean dynamics experiment. J. Geophys. Res., 92: 1637-1654. Kosro, P.M. and A. Huyer, 1986. CTD and velocity surveys of seaward jets off Northern California, July 1981 and 1982. J. Geophys. Res., 91: 7680-7690. Large, W.G. and S. Pond, 1981. Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr., 11: 324-336. Lynn, R.J. and J.J. Simpson, 1987. The California Current System: The seasonal variability of its physical characteristics. J. Geophys. Res., 92: 12947-12966. McClain, D.R. and D. H. Thomas, 1983. Year-to-year fluctuations of the California Countercurrent and effects on marine organisms. In: CalCOFI Rept. 24, Calif. Coop. Ocean. Fish. Invest., La Jolla, CA, pp. 165-181. Miller, R.N., A.R. Robinson and D.B. Haidvogel, 1983. A baroclinic quasi-geostrophic open ocean model. J. Comput. Phys., 50: 38-70. Miller. R.N. and A.F. Bennett, 1988. On the need for interior data assimilation in open ocean forecasting. Tellus (submitted). Mooers. C.N.K. and A.R. Robinson. 1984. Turbulent iets and eddies in the California Current and inferred cross-shore transports. Science, 223: 51-53. Nelson, C.S., 1977. Wind stress and wind stress curl over the California Current. NOAA Tech. Rept. NMFS SSRF-714, U.S.Dept of Commerce, 87pp. Rienecker, M.M., C.N.K. Mooers, D.E. Hagan and A.R. Robinson, 1985. A cool anomaly off Northern California: an investigation using IR imagery and in situ data. J. Geophys. Res., 90: 4807-4814. Rienecker, M.M. and C.N.K. Mooers, 1986. The 1982-1983 El Nino signal off Northern California. J. Geophys. Res., 91: 6597-6608.

648

Rienecker, M.M. and C.N.K. Mooers, 1988. Mesoscale eddies, jets and fronts off P t Arena, July 1986. J. Geophys. Res. (submitted). Rienecker, M.M., C.N.K. Mooers and A.R. Robinson, 1987a. Dynamical interpolation and forecast of the evolution of mesoscale features off Northern California. J. Phys. Oceanogr., 17: 1189-1213. Rienecker, M.M., C.N.K. Mooers and R.L. Smith, 1988. Mesoscale variability in current meter measurements in the California Current System off Northern California. J. Geophys. Res., 93: 6711-6734. Rienecker, M.M., C.H. Reed and C.N.K. Mooers, 1987b. Mesoscale variability in the California Current System during 1982 to 1986: Maps of surface dynamic height, sea surface temperture, temperature at 50 m, mixed layer depth and depth of the 8C isotherm from observations during the OPTOMA Program. Institute for Naval Oceanography Tech. Rept. 87-01, 138pp. Robinson, A.R., J.A. Carton, N. Pinardi and C.N.K. Mooers, 1986. Dynamical forecasting and dynamical interpolation: an experiment in the California Current. J. Phys. Oceanogr., 16: 1561-1579. Robinson, A.R. and D.B. Haidvogel, 1980. Dynamical forecast experiments with a barotropic open ocean model. J. Phys. Oceanogr., 10: 1909-1928. Simpson, J.J., 1984. El Nino-induced onshore transport in the California Current during 1982-1983. Geophys. Res. Lett., 11: 223-236. Simpson, J.J., T.D. Dickey and C.J. Koblinsky, 1984. An offshore eddy in the California Current system, I, Interior dynamics. Prog. Oceanogr., 13: 5-49. Stabeno, P.J. and R.L. Smith, 1987. Deep-sea currents off Northern California. J. Geophys. Res., 92: 755-771. Summers, S.J., 1986. Wind-current relationships in the OPTOMA domain off the Northern California coast. M.S. Thesis, Naval Postgraduate School, Monterey, CA, NTIS ADA-176019, 80 pp. Willmott, C.J., S.G. Ackleson, R.E. Davis, J.J. Feddema, K.M. Klink, D.R. Legates, J. O’Donnell and C.M. Rowe, 1985. Statistics for the evaluation and comparison of models. J. Geophys. Res., 90: 8995-9005. Winant, C.D., R.C. Beardsley and R.E. Davis, 1987. Moored wind, temperature, and current observations made during Coastal Ocean Dynamics Experiments 1 and 2 over the Northern California continental shelf and upper slope. J. Geophys. Res., 92: 1569-1604.

549

GEOMETRY-FORCED COHERENT STRUCTURES AS A MODEL OF THE KUROSHIO LARGE MEANDER T. YAMAGATA and S. UMATANI Research Institute for Applied Mechanics, Kyushu University, Kasuga 8 16, Japan

ABSTRACT We discuss the bimodality of the Kuroshio path south of Japan from a new viewpoint of direct interaction of current with local coastal geometry. By solving the barotropic quasi-geostrophic equation in a channel with step-like coastal geometry, we demonstrate that the model Kuroshio can actually show the localized, bimodal behavior for a reasonable range of inlet current speed. The amplitude of the large meander is approximately given by 2 6 . In contrast to all “non-local” model results, our local coherent structures have nothing to do with the basin-size geometry such as Kyushu and the Izu-Ogasawara Ridge. In general, the present study suggests that even a small feature of coastline geometry may trigger a big change in a near-shore current. 1 INTRODUCTION Not a few oceanographers have been fascinated with the remarkable bimodal behavior of the Kuroshio path south of Japan for more than 50 years. In particular, the problem has been received considerable attention in recent years from the viewpoint of interaction of the current with coastal geometry (Robinson and Niiler, 1967; White and McCreary, 1976; Charneyand Flierl, 1981; Chao and McCreary, 1982; Masuda, 1982; Chao , 1984; Yasuda et al., 1985; Yamagata and Umatani, 1987; Yoon and Yasuda, 1987). As in the study of the atmospheric blocking, it will be possible to classify the theories proposed for the Kuroshio large meander into the two categories of “local” and “non-local” (cf. Pierrehumbert and Malguzzi, 1984; Haines and Marshall, 1987; Malanotte-Rizzoli and Malguzzi, 1987). The “non-local” approach considers the Kuroshio large meander in terms of planetary Rossby waves satisfying suitable boundary conditions such as Kyushu and the Im-Ogasawara Ridge (see for example Yoon and Yasuda, 1987). Although almost all theories can be categorized as this “nonlocal” approach, one typical example of the “local” approach has been recently presented by Yamagata and Umatani( 1987). They discussed the Korteweg-de Vries equation forced by coastal step-like geometry as a simple conceptual model of the Kuroshio meander. The present paper adopts the “local” approach and demonstrates, by use of the Q-G equation, how a localized, coherent structure can be generated by such a modest steplike coastal geometry as Cape Shionomisaki*. It is outside the scope of the present short article to discuss relative advantage of “local”versus “non-local” theories. *Shionomisaki means “Capeof the Kuroshio” literally in Japanese.

550

2 BRIEF DESCRTPTIONOF THE MODEL We have numerically integrated the barotropic quasi-geostrophic equation on the beta-plane:

using Arakawa’s (1966) formulation for the Jacobian term with a leap-frog scheme. The Coriolis parameter takes the form f = fo + p(x sine + ycose), since the axes (x,y) are rotated anticlockwise with respect to the conventional set by the angle e . This is because we take the coastline inclination into account. Therefore the potential vorticity q is defined as q=VZyr+p( xsin0 +ycos0),

(2)

The model ocean is a channel (2000 km x 1000 km) with coastal geometry. As put forward in Yamagata and Umatani( 1987), we adopt step-like coastal geometry as our model shoreline. This step, of which amplitude &is60 km, corresponds to that of Cape Shionomisaki. It should be noted here that the step-like geometry can give rise to a localized source of potential vorticity by interacting with a nearshore current. The grid spacings are Ax = Ay = 20 km in all experiments. The free slip boundary condition is adopted at all lateral walls. The profile of the inflow is fured at the inlet; it is always given by

where H is the Heaviside step function and L is the typical width of the flow. The above jet takes the maximum value U-at y = yo - L. The outflow condition is given byy, = Vzqr, 0 as in Matsuura and Yamagata (1986). In order to check the influence of this outflow condition, we ran the case in which the channel length was doubled. The results for the same parameters were almost identical to the standard m e reported here. All parameters used in this study are given in Table 1. TABLE 1 Model parameters Parameters j3 (beta parameter) 0 (inclination angle of the coastline) v (lateral viscosity coefficient) L (jet width) yo (position of the northern end of the jet) 6 (amplitude of coastal step)

values 1 . 9 2 ~1013cm-~s-’ 20” 4~106cm2s-1 50 km -20 km 60 km

551

k 113

Meander Amplitude

GOO[

0

I-i-

11

111

0.5

1

I

1.5ms-’

Umax

Fig. 1. Amplitude of the meander as a function of maximum inlet speed Urn=, The solid line shows the state realized by the spin-up experiment of Section 3. I. The dashed line shows the state realized by the experiment of Section 3.2. The system vacillates between the two states in Regime II. 3 NUMERICAL RESULTS

3. I Three different regimes of evolution In this section, numerical solutions are presented when the model is initialized with the jet along the northern boundary. In each experiment, the model is run for 620 days. The results are summarized in Fig. 1, where the maximum amplitude of meander is shown as a function of maximum inlet speed Urn. This curve aids us in classifying the following three different regimes of evolution. Figure 2 shows the time evolution of the streamfunction pattern for the three typical regimes. The amplitude of meander increases as U, increases in the Regime I. The curve of Fig. 1 is approximated mther well by 2 G i n this regime. This suggests that both advection of relative vorticity and production of planetary vorticity are important ingredients of the meander. With increasing U, (Regime II), the jet vacillates between a large meander state and a small meander state and detaches a cyclonic eddy just after reachig a maximum meander. This shedding process is quite similar to that observed (Fig. 3). The vacillation period increases with increasing U, For example, it takes 100 days at Urn 0.52 ms-1, whereas it takes 500 days at Urn, = 0.87 ms-1. Although we failed to quantify the period by use of a simple formula, we expect that the period is given by the time needed to spin up the cyclonic eddy of which total potential vorticity is related to the maximum amplitude of the meander (cf. Section 4.3 of Yamagata and Umatani, 1987). The

552

amplitude of this maximum meander lies almost in the curve given by 2 G as in Regime I. As Urn increases further above the critical value of 0.94 ms-1 (Regime 111), the jet always stays in a small meander state. This tendency does not change even if Umax is increased further. This means that the advection of relative vorticity and the lateral diffusion of vorticity are important to the small meander of this regime.

Fig. 2a. Streamfunction at days 20, 60, 100, 140, 200, 240, and 300. Urn===0.3 ms-1. Contours are plotted in units of 104 m%1.

-"

553

T = 20.00

I.

:pTl :mp ;$ .#Jg :mT T = 220. 00

0

+

0

0

-

0 D

0

I

'-1000

0

X

I000

$-lOOO

/=-

-

X

0

-----

T-

1000

260.00

D 0

~l000

0

X

1000

'-1000

0

X

I000

T = 3 0 0 . 00

0

F

-

0

'-1000

-

0 m 0

0 0

0

X

1000

'-11100

0

Fig. 2b. Streamfunction at days 20, 60, 100, 140, 200, 240, and 300. U,, are plotted in units of l(Yrnk-*.

X

I000

= 0.6 ms-l.

Contours

554

Fig. 2c. Streamfunction at days 20, 60, 100, 140, 200, 240, and 300. Urn=- 1 ms-*.Contours are plotted in units of 104mzS-1.

555

J

I

,

I

135'E Fig. 3 Evolution of the path of the Kuroshio observed in 1977. The large tongue-like meander sheds a cyclonic eddy eventually.

3.2 Multide states In order to check a possibility of multiple equilibrium states, two sets of experiments have been performed. The first set is as follows. The steady solution for U,, = Ims-'(Regime 111) is first realized and then Urn, is decreased up to a value in Regime I or 11. When the final value of U, is within Regime I, a small meander state, which is different from the large meander state described in Section 3a, is realized. However, when the final value of U,, is within Regime 11, the result is the same as those in Section 3a: the vacillation between the large meander state and the small meander state is observed. The other set of experiments is as follows. First, the time-dependent solution for U, = 0.8 ms-1 is realized, and then Urn, is increased up to values within Regime 111when the meander is large. In the present case, the meander grows and intensifies the cyclonic eddy. This eddy finally detaches when the amplitude of meander reaches its maximum. Thus the small meander state described in Section 3a is again realized. The maximum amplitude of the temporary large meander is again predicted well by 2 6 ~ .

556

3.3 Evolution of the a -w scatter diagram The large meander state is always associated with the cyclonic eddy. Since the radius of the eddy is related with the quantity,-2 it is of interest to see how well the inviscid, steady state balance is satisfied for the eddy. To see this, it is best to check the relation between the potential vorticity q and the streamfunction ly. Figure 4 shows the time evolution of the q-ly scatter diagram 0.6111s- 1 in Section 3.1. Figure 5 shows the evolution of the potential for the case of U,, vorticity pattern for the same,U At t = 20 days the cyclonic eddy is not generated yet as we see in Fig. 1. Therefore the q-ly scatter diagram has no dots in the second and third quadrants. At t = 60 days the cyclonic eddy is already spun up and has a linear functional relationship between q and q~ in the second quadrant of Fig. 4. Thus the eddy is in the“inertial” or “almost-free” limit (see for example Greatbatch,l987). At t = 100 days the maximum q of the same quadrant becomes nearly constant. This means that the potential vorticity is homogenized in some area inside the eddy (see Yamagata and Matsuura, 1981; Rhines and Young, 1982). This area of homogenized q increases as time elapses(Fig.5). Although the typical diffusion time calculated with U ,, 0-1 y-1 gives a rather long time such as 904 days, the homogenization may be accelerated by the shear dispersion in the present case. Up to t = 260 days the dots in the second quadrant disappear totally. The corresponding streamfunction pattern shows almost no flow inside the tongue-like area surrounded by the winding jet(Fig. 2b). Just after this event the jet detaches the cyclonic ring into the ocean interior and then returns to the small meander state. This process is quite similar to that observed (Fig. 3). At t = 300 days the cyclonic eddy associated with the large meander is again well developed. Thus the q ly scatter diagram at this stage resembles the one at t = 100 days.

4 SUMMARY AND DISCUSSION We have demonstrated that the model Kuroshio can take a localized, large meander path as a result of direct interaction between the current and the steplike coastal geometry. In addition, for a wide range of the maximum inlet speed, our model shows vacillation between the small meander and the large meander. It is also found that the cyclonic eddy associated with the large meander is in the “almost-free” limit. Since no analytic form is known for such a localized eddy embedded in a westerly jet on the barotropic beta-plane, it is tather dificult to develop an analytic “local”theory for the Kuroshio meander. The present numerical work, however, seems to be sufficient to throw doubt on the several %on-1ocal”models which stress the importance of the basin-size geometry such as Kyushu and the Izu-Ogasawara Ridge. Rather, we suggest that even a small feature of coastline geometry may trigger a big change in a nearshore current. It is noteworthy that the above issue is almost parallel to that of the atmospheric blocking. It is of interest to comment briefly on a question whether the large, cyclonic meander of the kind discussed here is unique to the Kuroshio. It is known that the Somali Current develops intense, anticyclonic gyres and cyclonic wedges of cold sea-surface temperature during the Southwest Monsoon(Schott, F, 1983). Several model results suggest the importance of the zero slip condition

557

Fig. 4. Evolution of the q y scatter diagram for Urn,= 0.6 ms-I. q,y are in units of 10-5s-1 and 104 m2s-1, respectively.

558

Fig. 5. Potential vorticity at days 20, 60, 100, 140, 180, 220, 260, and 300 for Urn, = 0.6 ms-1.

559

as the source of positive vorticity for the cold, cyclonic wedges on the northern flanks of main anticyclonic gyres(Cox, 1979; Luther et al., 1985; McCreary and Kundu, 1988). Interestingly enough, the above is also the case of Yoon and Yasuda (1987)’s model for the Kuroshio cyclonic meander wrapping around the main anticyclonic gyre (private communication). In contmst to the Somali Current, however, the Kuroshio flows along the more irregular coastline and often hits the Cape Shionomisaki as summarized in Yamagata and Umatani (1987). The present study, therefore, stress the importance of the direct interaction between the current and the coastline geometry as a direct source of positive vorticity for the cold cyclonic meander. The effect of the different time variability between the two western boundary currents is another important topic somehow related to the present issue but is out of the scope of the present short article.

560

5 ACKNOWLEDGEMENT

We would like to thank Drs. R. Greatbatch, M. Yamanaka and S. Yoden for positive comments of the manuscript. This research is supported in part by the Grant-in-Aid for General Scientific Research from the Ministry of Education, Science and Culture. It is also carried out in conjunction with the Ocean Research Project of RIAM, Kyushu University.

6 REFERENCES Amkawa,A , 1966. Computational design for long-term numerical integmtion of the equations of fluid motion :Two dimensional incompressible flow. Part I. I. J. Comput. Phys., 1: 119-143. Chao, S.-Y., 1984. Bimodality of the Kuroshio. J. Phys. Oceanogr.,l4: 92-103. Chao, S-Y. and McCreary J.P., 1982. A numerical study of the Kuroshio south of Japan. J. Phys. Oceanogr., 12: 680-693. Charney, J.G. and Flierl, G.R., 1981. Ocean analogues of large-scale atmospheric motions. Evolution of Physical 0ceanography.h: B.Warren and C.Wunsch, Eds., The MIT Press,pp504-548. Cox, M.D., 1979. A numerical study of Somali Current eddies. I. Phys. Oceanogr., 9: 31 1-326. Greatbatch, R.J., 1987. A model for the inertial recirculation of a gyre. J. Mar. Res.,45: 601-634 Haines, K. and Marshall, J., 1987. Eddy-forced coherent structures as a prototype of atmospheric blocking. Q. J. R. Meteorol. SOC.,113: 68 1-704. Luther, M.E., OBrien, J.J. and Meng, A.H., 1985. Morphology of the Somali Current system during the Southwest Monsoon. In: J.C.J. Nihoul (Editor), Coupled Ocean-Atmosphere Models. Elsevier, Amsterdam, pp.405-437. Malanotte-Rizzoli, P. and Malguzzi,P., 1987. Coherent structures in a baroclinic atmosphere. Part 111: Block formation and eddy forcing. J.Atmos. Sci., 44: 2493-2505. Masuda, A., 1982. An interpretation of the bimodal character of the stable Kuroshio path. DeepSea Res., 29: 471-484. Matsuura, T. and Yamagata, T., 1986. A numerical study of a viscus flow past a right circular cylinder on a P-plane. Geophys. Astrophys Fluid Dynam., 37: 129-164. McCreary, J.P.and Kundu, P.K., 1988. A numerical investigation of the Somali Current during the Southwest Monsoon. J. Mar. Res., 46: 25-58. Pierrehumbert, R.T. and Malguzzi, P., 1984. Forced coherent structures and local multiple equilibria in a barotropic atmosphere. J. Atmos. Sci., 41: 246-257. Rhines, P.B.and Young, W.R., 1982. Homogenization of potential vorticity in planetary gyres. J. Fluid Mech., 122: 347-368. Robinson, A.R. and Niiler, P.P., 1967. The theory of free inertial currents. I. Path an structure. Tellus, 19: 269-291. Schott, F., 1983. Monsoon response of the Somali Current and associated upwelling. Progr. Oceanogr. 12: 357-381. White,W.B. and McCreary, J.P.,1976. The Kuroshio meander and its relationship to the largescale ocean circulation. Deep-sea Res., 23: 33-47. Yamagata, T. and Matsuura, T., 1981. A generalization of Prandtl-Batchelor theorem for planetary fluid flows in a closed geostrophic contour. J. Met. SOC.Japan, 59: 615-619. Yamagata T. and Umatani, S., 1987. The capture of current meander by coastal geometry with possible application to the Kuroshio Current. Tellus, 39A: 16 1- 169. Yasuda, T., Yoon, J.-H. and Suginohara, N., 1985. Dynamics of the Kuroshio large meanderbarotropic model I. J. Oceanogr. Soc.Japan, 41: 259-273. Yoon, J.-H. and Yasuda, I., 1987. Dynamics of the Kuroshio large meander: two-layer model. J. Phys. Oceanogr., 17: 66-81.

561

THE BEHAVIOR OF KUROSHIO WARM CORE RINGS NEAR THE EASTERN COAST O F JAPAN T. MATSUURA and M. KAMACHI Faculty of Engineering, Ibaraki University, Hitachi 316, (Japan) Research Institute for Applied Mechanics, Kyushu University 87, Kasuga 816, (Japan)

ABSTRACT The behavior of Kuroshio warm core rings in the area east of Japan is investigated using a numerical model(rigid-lid, two layer primitive equation,on a p -plane). Specifically, the evolution, migration, decay and barotropic-baroclinic energy exchange of such vortices are clarified. Moreover, w e compare the results of numerical experiments w i t h observations of a Kuroshio w a r m core ring obtained f r o m hydrographic data and from W satellite infrared images. 1

INTRODUCTION

The Kuroshio-Oyashio confluence zone is a region with great eddy activity similar to the region of the Gulf stream. Observations of Kuroshio warm core rings (KwQls) in this region have been reported since the 1950's (Kawai, 1955; Sugiura, 1955; Ichiye, 1955). The features and the behavior of KWCRs have been recently investigated using NOAA satellite infrared (IR) images. I t is well-known that KWa3s are formed about twice a year (Kimura, 1 9 7 0 ) , have a long lifetime and change their shape from circular to elliptical (Kitano, 1975). The migration of warm eddies in the area east of Japan is complex. Warm eddies off Sanriku and East Hokkaido, which may occur from the Oyashio front, move to the north o r the northeast (Hata, 1 9 7 4 ; Kitano, 1 9 7 5 ; Saito et al., 1 9 8 6 ; Tomosada, 1986). Kw(Rs in the region east of 150'E move to the west, and KWCRs off Jyoban move to the northwest or the southwest. KWCRs moving southwestward coalesce with the Kuroshio (Tomosada, 1986). The prediction of the location of fish schools by using IR images has been developed because the fish schools migrations are influenced by KWCRs and Oyashio w a r m eddies (Saito et al., 1 9 8 6 ) . There is also an attempt to estimate the surface velocity distribution by using several IR images (Emery et al., 1 9 8 6 ; Kamachi, 1 9 8 9 ) , but the vertical structure and the detailed properties of WCRs cannot be obtained through that approach. Hence, the numerical analysis and forecasting of KWCRs are interesting and needed.

562

T h e e v o l u t i o n a n d m i g r a t i o n of i s o l a t e d v o r t i c e s h a v e b e e n investigated numerically by Bretherton and Karweit (1975), M c W i l l i a m s and Flier1 (1979), M i e d and Lindeman (1979), Matsuura and Yamagata (1982) from the viewpoint of geophysical fluid dynamics. The interaction between isolated vortices and bottom topography has been examined numerically by Smith and O'Brien (1983) and S m i t h (1986) for the w a r m vortices in the Gulf of Mexico. Moreover, the passive tracer anomaly of vortices has been investigated numerically by Davey and Killworth (1984) and Holloway et al. (1986). However, comparisons between the numerical experiments and field observations have not yet been conducted in order to understand the actual ocean vortex phenomena. I n order to investigate the evolution, migration, passive tracer anomaly, and energetics of anticyclonic isolated vortices, a study of such vortices over a strong bottom slope is carried out using a n u m e r i c a l h y d r o d y n a m i c a l m o d e l (rigid-lid, two-layer primitive equations, beta plane). T h e focus of o u r interest is to understand the relation between the upper and lower motion of vortices over the western bounding bottom slope. Moreover w e discuss a comparison between the present numerical results and real observations by IR images and hydrographic data. This research is important because the phenomenon has the large variability of coastal seas and i t also has a great influence on the inshore fishery and the local climate. 2 O B S E R V A T I O N A L DATA

W e shall discuss the evolution and migration of a KWCR formed in the IR images and area east of Japan in April 1985 because various hydrographic data are available. T h e KWCR moved to the northwest at a mean speed of 6 c m s-l w h i l e changing its shape from circular to elliptical and reducing its size. After the middle of J u n e , 1985, i t moved t o the south and coalesced with the Kuroshio by August, 1985. The Kw(R may be influenced by the western bounding bottom slope because its center w a s located over the slope at a depth of 2500111 and its western edge was located over the slope at a depth of 500m. Figures l(a) and (b) are the IR images of the KWCR at 0411(JST) on J u n e 5, 1985 and at 0400 (JST)J u n e 6 , 1985, respectively. W a r m e r temperatures correspond to darker shades i n the figures. The land area is relatively w a r m and appears nearly black o r dark gray. T h e KWCR is 38'10") located off Kinkazan (the center of the KWCR is at 143'20'E, and its shape is that of an ellipse. Figure 2 shows the surface velocity pattern of the

Kw(R

on June 6, 1985 a s

analysed by the pattern matching

563

Fig. 1. infrared images of the Ku r o s h i o w a r m core ring off Jyoban. (a): 0 4 1 1 ( J S T ) J u n e 5 , 1 9 8 5 . ( b ) : 0 4 0 0 ( J S T ) J u n e 6 , 1 9 8 5 .

1 4 0 OE

1 4 2 OE I

1 4 4 OE

146'E

I

I

-. .

5 0 [CM/SI

4 1 ON

41°N

39 ON

39 O N

37 O N 140°E

-

37 O N 142"E

1 4 4 OE

Fig.2. A d v e c t ive s u r f a c e v e l o c i t y field.

1 4 6 OE

564

method applied to Figs. l(a) and (b) (Kamachi, 1989). Because the northern part of the vortex is covered with cloud on June 5, the surface velocity of the Kw(R can only be obtained for the southern part. Fig.2 shows that the velocity on the east-side of the Kw(R is faster than that on the west-side. The maximum velocity of the KwcR reaches 60 c m s - ' . Figure 3 shows some of the results of hydrographic observations carried out by the Tohoku Regional Fisheries Research Laboratory from June 1 to June 1 1 , 1985. In order to reveal the structure of the KWCR, three sections were taken and the distance between observation points was taken as 10' ( 14.6 km). DBT, XBT (from the surface to the depth of 1000m), GIX (surface), and STD were used to observe the structure of the KWCR. The temperature distributions show that the vortex shape is elliptical (see Fig. 3), that the vertical axis of the vortex center leans toward the east, and that its vertical structure reaches more than 1000 m. The horizontal temperature distribution at the depth of lOOm is in good agreement with the IR image on June 6, 1985 for the location and the shape of the vortex (cf. Fig. l(b) and Fig. 3). The arrows in Fig. 3 show the surface velocity field observed by GEK. Figure 3 also shows that the azimuthal velocity on the east-side is faster than that on the west-side. The maximum velocity of the vortex reaches llOcm s - ' . The horizontal velocity distribution agrees well with that obtained by the pattern matching method. 3 NUMERICAL MODEL An important requirement for the numerical model is that the steep

continental slope may be taken into account. The Japan trench runs along the north-south direction about 200 k m away from the shoreline of northern Honshu (Tohoku). The northwest pacific basin is rather flat, with the depths of 5000-6000 m, east of the Japan trench. Figure 4 shows the depth profile at 38'N and the depth profile of the model. Because the thermocline of KWCRs is at a depth of nearly 500 m (Cheney, 1977), the boundary between upper and lower layers i s taken at 500 m below the surface in o u r model. The values of the model parameters are as follows: h l = 500 m , h2 = 5000 m, H (mean water depth) = 5500 m, dmax (maximum height of topography)

=

-

4900 m, a (horizontal scale of topography)

= 160

km,

PT (

dmaxfo/(aH) : topographic beta ) = 5 . 0 ~ 1 0 -m-ls-', ~~ L (pressure efolding radius) = 60 km, Rd (internal Rossby's radius of deformation) = 34 k m , f = fo+ py = ~ . O X ~ O - ~ ( +S -l.8x10-11(m-1s-1) ~) y (3aDN), g' = =

2.0x10-2 m s - ' ,

and Vo

(particle speed)

= 0.86

m

s-'.

From these

565

41

40

39

38

37

36

Fig.3. Kuroshio warm core ring off Jyoban. Isotherms at 100 m with surface velocity vectors (GEK) observed from June 1 to 11, 1985.

Fig.4. A side view of the two-layer model with the western bounding bottom slope.

566

dimensional values w e get the dimensionless parameters: Q ( = VO / PL2) = 13.3, dZ(=L/Rd) = 1.76, RO (=VO/fOL) = 0.159, Ro/(dmax/E) = 0.178, and tT/ p = 27.8. W e use the depth-averaged Boussinesq equations governing the mot ion o f a two-layer ocean and the continuity equations as model e q u a t i o n s (cf. M i e d and L i n d e m a n , 1979). T h e lateral d i f f u s i o n = 2 . 5 ~ 1 0 cm2s-l ~ and the bottom Ekman layer coefficient coefficient Ev = 1/(100 days) are used i n our model. The primitive equations are solved numerically as a mixed initial boundary value problem. W e impose a free-slip boundary condition at the side walls and use the Rayleigh damping in the lower layer as the Ekman dissipation. We use the finite-difference scheme formulated by Holland and Lin (1975). The integration is carried out in 640 k m and 640 k m square boxes. A time step o f 10 m i n and a grid spacing o f 10 k m are required to give a stable numerical solution in o u r model and to adequately resolve the details of vortices. The familiar Gaussian vortex is used as an initial state. Three different initial vortices are considued, that is, pure barotropic vortex ( u 1 = 1.0 m s-l and u2 = 1.0 m s-l 1, nearly barotropic vortex ( u 1 = 1.0 m s-' and u2 = 0.8 m s - ' 1, and nearly baroclinic vortex ( u 1 = 1.0 m s-l and u2 = 0.2 m s - l ). Since the numerical results for a nearly barotropic vortex are the most similar to the field observations (June 6, 19851, that case is discussed in detail in the following sections. 4 NUMERICAL RESULTS During the initial adjustment,the anticyclonic Gaussian vortex reduces its scale and generates a Rossby wave wake by the planetary beta effect ( t i l l T = 2 0 day in Figs. 5 and 6 w h i c h s h o w time evolution o f upper and lower vortices, respectively). During these 2 0 days, the kinetic energy of the lower layer (TKE2) decreases rapidly due to the Ekman and lateral dampings and the available potential energy (APE) increases gradually (see Fig. 7(a) which s h o w s time history of f o u r kinds o f energy i n total area. I n the figure TE is the total energy, TKE1 is the kinetic energy of the upper layer, TKE2 is the kinetic energy of the lower layer, and APE is the available potential energy.). By T=40 day, topographic Rossby waves occur due to the western bounding bottom slope and a cyclonic eddy builds up northeast of the initial a n t i c y c l o n i c v o r t e x (Figs. 5 and 6). T h e v o r t e x m o v e s to the southsouthwest with a speed o f 6.4 c m s-l (the west component Cw=2.7 c m s-' and the south component Cs=5.8 c m s-') t i l l T=40 day. W e can understand this vortex propagation using vortex trajectories which are

567

568

0 0

I d

N

m

-W!

m

*

d m

a !

0

a

569

shown in Fig. 8. The figure shows four trajectories with the elapsed time (day). I n the figure, TI0and T20 are trajectories of the initial anticyclonic vortices in the upper and the lower layers, respectively. Trajectories of two vortices in the lower layer (Tzl, Tz2), which are induced in the south and the north of the initial vortex at T = 40 day, are also shown from 40 o r 50 day. McWilliams and Flierl (1979) showed that for a mixed mode vortex, Cs/Cw increases as the barotropic component increases. They also showed that when both components are equal, Cs/Cw is nearly equal to 2. In the present case, the barotropic component amounts to 80% and Cs/Cw is 2.1. This result is in agreement with that of McWiIIiams and Flierl (1979) although they used the quasigeostrophic equation. Since the Itp2 vortex'' (i.e., the lower layer part of the initial anticyclonic vortex) is directly influenced by the bottom slope, its shape changes from circular to elliptical by T=40 day and i t forms a dipole vortex by T=50 day (Fig. 6 ) . Because this phenomenon takes place rapidly, the instability of the p2 vortex may be forced by the bottom topography. During this period (T=40 day to T = 5 0 day), APE increases while TKE2 decreases rapidly (Fig. 7(a)). The Itpl vortex", which means the upper layer part of the initial anticyclonic vortex, develops an elliptical shape by T=50 day, associated with the development of the p2 vortex changing into a dipole vortex. The p 1 vortex remains of elliptical shape as the horizontal axis rotates clockwise, decouples from the lower layer, and becomes the upper ocean vortex by T=80 day. The interface amp1 i tude of the vortex center lqlmax increases rapidly from T = 3 0 day to T=80 day. The time dependence of the amplitude is similar to the dependence of APE. A large anticyclonic eddy forms in the northwest corner from T = 5 0 day to T=70 day (Figs. 5 and 6 ) . From T=50 day to T=80 day, the initial anticyclonic p2 vortex moves to the southeast because i t changes into a dipole vortex due to the bottom topography(Fig. 6 ) . The cyclonic part of the dipole vortex rotates clockwise around the center from T=50 day to T=60 day (see Fig. 8, T21). On the other hand, the anticyclonic part of the dipole vortex rotates counterclockwise around the center from T=60 day to T=80 day (see Fig. 8, T20). The p1 vortex moves to the northwest from T=40 day to T=60 day (Fig. 8, T10) because its shape becomes elliptical. Flierl (1977) showed that an elliptical linear vortex, of which the long axis points to the northwest, has a northward velocity component. From T=70 day to T=110 day, the p 1 vortex moves to the southwest at a mean speed o f 3.8 cm ,-I (the westward velocity component is 2.0 c m s - l

570

h

-

wo

-I c9-

3(3

7

-

I p o

0

c‘1-

‘ I -

X

-

Y

s o

& ywo

Z

w

0 0

,O.OO

I

I

I

20.00

I

1

40.00

1

1

1

80.00

1

I

100.00

,

I

120.00

I

,

140.00

T(DAY 1

c90 h

1

60.00

-

wo

‘4-

-I 3 0

D 7

-

9 0 0

4-

- 0

X

-

Y

$ 0

cc c‘1W 0

Z

w

0

I

0

0.00

I

20.00

I

I

40.00

I

I

I

60.00

I

80.00

I

I

I

100.00

I

1

120.00

1

1

140.00

T(DAY 1 Fig.7. Time history of seven kinds of energy. Descriptions are given in the text.

Fig.8. Vortex trajectory based on maximum values of p1 and pa.

and the southward velocity component 3 . 2 cm s - ' ) . T h e southward component of the translation velocity is smaller than that obeserved before T=40 day because the initial vortex has become an upper ocean vortex after T=80 day. After T=110 day, the p l vortex m o v e s along the side boundary by the effect of the mirror image. The p1 vortex goes over the bottom slope by T=130 day and forms the anticyclonic part of a modon like vortex under the influence of the western lateral boundary. Because this vortex is exchanging some of the baroclinic kinetic energy over the slope (SBCK) into barotropic kinetic energy over i t (SBTK), the vortex component of the lower layer is generated (see Fig. 6 and Fig. 7(b) which shows time history of three kinds of energy in the region with the bottom slope. In this figure, SBK is the total kinetic energy in the region.). F r o m T=140 day to T = 1 5 0 day, the p 1 vortex moves to the north at a mean speed of 13.9 c m s - I . After T=90 day, the anticyclonic part of the p 2 d i p o l e v o r t e x is s t r e n g t h e n e d by t h e c o r r e s p o n d i n g p1 anticyclonic vortex and i t becomes a single anticyclonic vortex. T h e monopole p2 vortex moves to the westnorthwest t i l l T = 1 2 0 day as the westward translating speed is increasing(Fig. 8). 5 DISCUSSION In this section, we compare the numerical results with the IR images obtained from late April to late June, 1985 (figures are omitted except for Figs. l(a) and (b)) and with hydrographic data collected from June 1 to J u n e 1 1 , 1985. I f the pannel at T=20 day in Fig. 5 is the s a m e condition as the IR image o n April 29, the pannel at T=60 day in Fig. 5 (and Fig. 9(a) corresponds to the IR image on J u n e 6 (Fig. i(b)). T h e evolution of the vortex obtained from the numerical model is in good agreement with that of the IR images. In Fig. 9, velocity arrows in the strongest vortices s h o w anticyclonic vortices. I n the figure, a small cyclonic vortex is also present northeast of the elliptical anticyclonic vortex; this feature appears also in Fig. l(b). As discussed in Section 4 , w e can conclude that the shape of the p 1 circular vortex changes into the elliptical one because the p2 vortex is influenced by the bottom slope, forms a dipole vortex, and moves to the southeast. With a nearly baroclinic vortex as initial condition, the p1 vortex changes into an upper ocean vortex because the p2 vortex disperses rapidly as barotroPic Rossby waves. The vortex remains of circular shape even i f i t goes over the bottom slope. When the initial vortex is pure barotropic, both the p 1 and the p2 vortices change into dipole eddies and m o v e to the southeast (figures are omitted).

512

(b)

64t

T=EO. 00 ./. ......... .......... a

Y

32t

.. ..,.. . .. 0 F i g . 9. C o n t o u r s of p 1 a n d u 1 ( a ) , a n d p 2 a n d u 2 (b).

0.5

VEL. 0.0

-0.5

-1.0

Fig.10.

T a n g e n t i a l v e l o c i t y d i s t r i b u t i o n on J u n e 6,1985.

513

Fig. 10 shows several tangential velocity profiles along the E-B section of Fig. 3. The profiles are obtained from ( 1 ) GEK data ( G E K ) , ( 2 ) the analysis of XBT data by using the centrifugal balance (XBT), ( 3 ) the analysis of the IR images by using the pattern matching method (PM), and (4) the numerical analysis (NA). Profiles ( 1 ) and ( 2 ) are i n good agreement with each other: both shows the asymmetry of the velocity profile along the east-west direction and have similar absolute values o f the maximum velocity o n both side of the center. The analysis by using the pattern matching method can resolve only the southern half of the vortex (Fig. 2 ) because the northern part in Fig. l(a) is covered with clouds. When w e compare the tangential velocity between profiles ( I ) , ( 2 ) and ( 3 ) , they all show an asymmetric profile along the eastwest direction. Because the section corresponding to ( 3 ) is to the south of the E-B section, the velocity of ( 3 ) is smaller than that of ( 1 ) and ( 2 ) . The position of the maximum velocity in profile ( 4 ) closer to the point of reversal is than in ( 1 ) and ( 2 1 , and the asymmetry of the velocity profile is less pronounced than in ( 1 1 , (2), and ( 3 ) . This may be because the two layer ocean is a poor aptroximation of the real vertical structure of the oean (Flierl, 1978). Judging from the comparison between ( l ) , ( 2 ) , ( 3 ) , and (41, the Oyashio front flowing southward along the northern Honshu coast may be a most important factor for the asymmetric velocity prof i le. I t is well known from field observations that KwQis have a northward migrating component. The Kw(R formed in late April, 1985 also moved to the northwest at a mean speed of 2.1 c m s - ' from April 29 to June 6. The mirror image has been suggested as the northward migration model of KWCRs (Yasuda et al., 1986). However since the vortex center on June 6 is 110 km away from the 500 m isobath, the effect of the mirror image cannot affect the migration of the vortex (cf. Fig.5). I n the present numerical result, an elliptical anticyclonic vortex, whose long axis turns to the northwest, moves to the northwest (Fig. 8). The migration velocity toward the northwest is 2 . 4 5 c m s-' from T=40day to T=60 day in the numerical model and the migration velocity obtained from the IR image is 2.1 c m s - ' . These results are in good agreement. After T=60, the vortex in the numerical simulation moves to the southwest, but the observed vortex moved further to the northwest as the horizontal axis kept rotating clockwise until June 1 7 , 1985. The observed vortex then changed its shape into a circle, moved to the south on July 2 8 and finally coalesced with the Kuroshio.

574

6

CONCLUSIONS I t has been shown that a vortex which has a significant barotropic

component initially is strongly influenced by the western bounding bottom slope w h e n i t propagates westward. T h e p1 vortex changes its shape from circular to elliptical because i t separates from the p2 vortex. The latter forms a dipole vortex by interacting with the bottom slope and moves to the southeast. KWCRs propagating near the western coast m a y change their shape from circular to elliptical due to the above cause (Fig, I(b)). The elliptical p l vortex, whose long axis turns to the northwest, moves to the northwest. This is the same tendency as that obtained by Flier1 ( 1 9 7 7 ) from the linear analysis of a slightly el 1 iptical vortex. The analysis of IR images, of GEK and XBT data, and of the numerical model are compared to each other. T h e velocity profiles obtained from the pattern matching method applied to the IR images, from the centrifugal balance by using a vertical temperature profile, and from the numerical simulation agree qualitatively well with one another (Fig. 1 0 ) . Thus, i t m a y be possible to use the velocity profile of the upper layer, which is obtained by interpolating the above results, as an initial velocity profile for the numerical model : such an element of data assimilation may be useful for the forecast of m s .

ACKNOWLEDGMENTS Both authors would I ike to thank Dr. Yamagata for providing useful suggestions. We also thank Tohoku Regional Fisheries Reserch Laboratory for providing hydrographic data. The present work is partly supported by a Grant-in-Aid for Encouragement of Young Scientist from the Ministry of Education, Science and Culture. O n e of the authors (MK) acknowledges support from the Ocean Project of Research Institute for Applied Mechanics, Kyushu University. Computations were made with HITAC S810 in Ibaraki University.

REFERENCES Bretherton, F.P. and Karweit, M.J., 1975. Mid-ocean mesoscale modeling. I n : Numerical models of ocean circulation. Ocean Affairs Board, Nat. Res. Counc. U.S. Nat. Acad. Sci., Washington, M=: 237-249. Cheney, R.E., 1977. Synoptic observations of the oceanic frontal system east of Japan. J . Geophys. Res., 8 2 : 5 4 5 9 - 5 4 6 8 . Davey, M.K. and Ki I Iworth, P.D., 1984. Isolated waves and eddies in a shallow water model. J. Phys. Oceanogr., 14: 1047-1064. Emery, W.J., Thomas, A.C., Collins, M.J., Crawford, W.R. and Mackas, D.L., 1 9 8 6 . An objective method for computing advective surface velocities from sequential infrared satel I i te images. J . Geophys. Res., 91: 12865-12878.

515

Flierl, G.R., 1977. The application of linear quasi-geostrophic dynamics to Gulf Stream rings. J.Phys. Oceanogr., 7: 3 6 5 - 3 7 9 . Flierl, G.R, 1978. Models of vertical structure and the calibration of two-layer models. Dyn. Atmos. Oceans., 2: 341-381. Hata, K., 1974. Behavior of a w a r m eddy detached from the Kuroshio. J. Meteor. Res., 2 6 : 2 9 5 - 3 2 1 . Holland, W.R. and Lin, L.B., 1975. On the generation of mesoscale eddies and their contributions to the oceanic general circulation. I . A preliminary numerical experiment. J. Phys. Oceanogr., 5: 642-651. Holloway, G., Riser, S.C. and R a m s d e n , D., 1 9 8 6 . Tracer anomaly evolution in the flow field of an isolated eddy. Dyn. Atmos. Oceans., 5 : 1-41. Ichiye,T., 1955. On the behavior of the vortex in the polar front region. Oceanogr. Mag., 7 : 115-132. Kamachi, M., 1989. Advective surface velocities derived from sequential Nihoul and B.M. Jamart (Editors), images o f eddy fields. I n : J.C.J. Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Elsevier Oceanography Series, Elsevier, Amsterdam (this volume). Kawai, H., 1955. On the polar frontal zone and its fluctuation in the waters to the northeast of Japan. Bull. Tohoku Reg. Fish. Res. Lab., 4 : 1-46 (in Japanese). Kimura, K., 1970. A mechanism of fishing ground format ion due to w a r m eddies cut off from the Kuroshio Front. Marine Science Monthly., 2 : 30-35 (in Japanese). Kitano, K., 1975. Some properties of the w a r m eddies generated in the confluence zone of the Kuroshio and the Oyashio currents. J. Phys. Oceanogr., 5 : 2 4 5 - 2 5 2 . McWilliams, J.C.and Flierl, G.R., 1979. On the evolution of isolated nonlinear vortices. J. Phys. Oceanogr., 9 : 1155-1182. Matsuura, T. and Yamagata, T., 1982. On the evolution o f nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr., 12: 440-456. Mied, R.P. and Lindemann, G.J., 1979. The propagation and evolution of cyclonic Gulf Stream rings. J. Phys. Oceanogr. 9: 1183-1206. Saito, S . , Kosaka, S. and Iisaka, J., 1 9 8 6 . Sattelite infrared observations of Kuroshio warm-core rings and their application to study of Pacific saury migration. Deep-sea Res., 33: I601 -1615. Smith, D.C.,IV. and O'Brien, J.J., 1983. The interaction of a two layer isolated mesoscale eddy with topography. J. Phys. Oceanogr., 1 3 : 1681-1697.

Smith, D.C.,IV, 1986. A numerical study of Loop Current eddy interaction with topography in the western Gulf of Mexico. J . Phys. Oceanogr., 16: 1260-1272.

Sugiura, J., 1955. On the transport in the eastern sea of Honshu (Part 1 ) . Oceanogr. Mag., 6 : 1 5 3 - 1 6 3 . Tomosada, A., 1986. Generat ion and decay of Kuroshio warm-core rings. Deep-sea Res., 3 3 : 1475-1486. Yasuda,l., Okuda, K. and Mizuno, K., 1986. Numerical study o n the vortices near boundaries - considerations on warm core rings in the Bulletin of Tohoku Regional vicinity of east coast of Japan Fisheries Reserch Laboratory., 48: 6 7 - 8 6 (in Japanese).

-.


577

ADVECTIVE SURFACE VELOCITIES D E R I V E D FROM SEQUENTIAL IMAGES

OF EDDY FIELDS M.KAMACH I Research Institute for Applied Mechanics, Kyushu University 87, Kasuga 8 1 6 (Japan).

ABSTRACT An objective technique for the calculation of advective surface velocities from sequential images is developed. The procedure is pattern matching with the identification of m a x i m u m cross correlation coefficient (MCC) between a template window in the first image and search areas in the second image. T h e advective velocities are calculated from the displacement and the elapsed time. W e examined the limitations and the resolution of M C C m e t h o d in two cases : ( 1 ) eddy size (LE) much smaller than the Rossby's radius of deformation (LD); and ( 2 ) LE LD. In the first case, an eddy is regarded as a particle. Maximum detectable time period is about one day. For the second case we developed the MCC method to detect rotational motion ( M I X ) . W e discuss a comparison between the velocity derived from the M I X method applied to a tracer field and a real velocity field calculated by numerical analysis of quasigeostrophic flow. W e also apply the MCXR method to an eddy formation i n the Kuroshio-Oyashio confluence zone.

I.INTRODUCTION There has been increasing interest in observing surface patterns from satellite images. Almost all studies have used the imagery as a flow visualization picture to interpret the ocean phenomena. In order to elucidate the ocean phenomena and use the data of the satellite imagery for the ocean predict ion, suitable methods of obtaining physical quantities (e.g., velocity) from satellite images have been needed. Recently two methods have been developed. The f i r s t one is called feature tracking. I t is a subjective method to track submesoscale features in sequential satel I i te images (Tanaka et al., 1 9 8 2 ; Vastano and Borders, 1 9 8 4 ; Vastano and Reid, 1 9 8 5 ) . W e can estimate surface advective velocity, but cannot get all velocity values at all grid points. The second method is a maximum cross correlation (MCC)method in which a cross correlation between sequential images is used. This method is an objective one (Ninnis et al., 1986 ; Emery et al., 1 9 8 6 ) . W e can get velocity values at all grid points. This method has been widely used in satel I i te image registrat ion (McGi I lem and Svedlow, 1 9 7 7 , 1 9 7 8 ;

578

Svedlow et al., 1 9 7 8 ) . I t has also been applied to detect cloud motion and get 'satellite wind' field (Leese et al., 1971 ; Smith and Phillips, 1 9 7 2 ; Hamada, 1 9 7 9 ) . I t is known that the M E method cannot be applied to detect two kinds of motion, rotation and deformation. Therefore Emery et a1.(1986) used the method with very short time intervals between two images (4-5hours), because 'smal I segments of rotational motion wi 1 1 appear as translation' as they mentioned. They also point out that ' i t is likely that for longer image separations ( 1 2 - 2 4 hours) the MCC velocity vectors will less faithfully represent rotational motions' (Emery et al., 1 9 8 6 , p. 1 2 8 7 0 ) . Therefore, a development of the MCC method to detect rotational motion with a longer time period between images would be u s e f u l . I n this paper, we examine and make clear the limitations of the Mcc method and we develop the method to detect a rotational motion (e.g., an eddy) using a long time separation between images (more than one day). W e review the MCC m e t h o d briefly in section 2. W e e x a m i n e the limitations of the method and develop i t for rotational motion i n section 3. W e describe applications of the method to the tracer field of a numerical experiment and t o an eddy field of the Kuroshio-Oyashio confluence zone observed with satellite images in section 4. Section 5 is devoted t o the conclusion. 2.

MCC METHOD

W e review the M E method briefly in this section. Successive pairs of images are treated according to the following processes: 1. The first image is divided into LTxLT pixel template windows. The second image is also divided into corresponding LsxLs pixel search area. 2 . Two-dimensional cross correlation coefficients between one template and each search areas are computed. 3. The maximum value of the coefficients is detected. The surface velocity is then calculated from the position of the maximum value and the time interval between the two images. Before the f i r s t process, a running mean f i 1 ter i s used in order to reduce small scale image noise. As another filter temperature gradient is used because the gradient operation improves the registration

performance (Svedlow et al., 1 9 7 8 ; Van Woert, 1 9 8 2 ; Emery et al., 1 9 8 6 ) . When w e use this matched f i I ter for imagery of NOAA/AVHRR, w e derive a standard deviation of registration error C = 0.60/(km), where SNR is the signal-to-noise ratio, using a method of McGillem and

579

Svedlow ( 1 9 7 6 ) for LANDASAT images. T h e standard deviation of the registration error is much smaller than the resolution (1.lkm). I n the first process, the values of LT and Ls have been set at 21 and 3 1 , respectively, for mesoscale features (cf. Emery et al., 1 9 8 6 ) . W e examine the value for eddy fields in subsequent sections. C o m p l e t e estimations about the selection of the size of the template window are described by Takagi ( 1 9 8 5 ) for general image analyses. The use of the fast Fourier transform to perform the calculation of the cross correlation coefficient (process 2 ) reduces computation time. I t should be noted that this Mx process cannot respond to two kinds of m o t ion, rotat ion and deformat ion, because the coordinates of the template window and search area are fixed. The distribution of the values of the coefficient (matching surface) often has the second largest value nearly equal to the m a x i m u m value (process 3). For this case the resultant velocity vectors are assessed by examining the following five items of the matching surface on threshold values : ( I ) m a x i m u m value, ( 2 ) the difference of the first m a x i m u m and the second m a x i m u m values, (3) sharpness of the matching surface, (4) the distance of the positions of two m a x i m u m values, (5) maximum lag value between the template and the search area (Hamada, 1979 ; and references therein). We calculate the surface velocity in process number 3 only when the maximum value of the cross correlation coefficient is above 0.4. This is in agreement w i t h the error analysis of Emery et al., 1 9 8 6 (see also Ninnis et al., 1 9 8 6 ) . 3. LIMITATIONS AND DEVELOPMENT OF THE MCC METHOD In this section, w e examine the limitations of the MCC method and develop the Mx method for rotational motion. W e examine two different cases, depending o n the relation between the eddy size (LEI and the Rosbby's radius of deformat ion (LD). 3.1 Eddy-particle analogy (L, 10. As a result of that instability, an S - l i e form was formed or a decay to two isolated vortices occurred. However, when 6, was about 0.1, the instability was not observed. Therefore a particular attention was paid to the 6, parameter. The ratio &,RV as a function of 6,. where & , is the lens radius at the beginning of the instability, is shown in Fig. 4. In logarithmic coordinates all points are alined, which indicates a very strong dependence: R,,IR,. ' ' : 6 From this formula, one can see that as $ gmws, R, decreases

-

697

(since R, - const. in these experiments) down to the value Rv for 6, = 1. On the other hand, we have not found a significant influence of S on the lens instability.

Fig. 4: Ratio stratified fluid.

/ R, versus fractional depth 6, for variable volume lenses in a linearly

3.2. Constant volume vortices Vortical lenses of constant volume were produced at the interface of a two-layer fluid by two methods. In the fist method, the lens results from the collapse of a constant volume of homogeneous dyed fluid, mixed in a cylinder which is placed in the centre of the tank. The second method is similar to that used for the production of variable-volume lenses. In this case, after supplying the required volume of lens water, the constant flux of fluid is stopped. The same method was used in the experiments performed by Griffiths and Linden (1981) and Zatsepin (1983), but those lenses were all unstable in contrast to the experiments by Kitamura and Nagata (1983) and Kostianoy and Shapiro (1986). This discrepancy may be explained by the fact that the experiments of the first two studies were performed with values of B and S = Q f / (g’)3 (the parameter which characterizes the inclination of the lens front during the inviscid stage of evolution) greater than in the second set of studies.

698

In our investigation, we concentrated on the study of stable lens evolution. By the "collapse" method, we performed 35 experiments for wide ranges of the main parameters: 0 5 f I 4 s-', 29 I V I 131 cm3, 3 I & 17.7 cm. 1.96 I g' 57.6 cm s - ~ , v = 0.01 cm2 s-', where Ho is the total fluid depth. In addition, 9 experiments were performed using the "intrusion" method, with the parameters: 25 I V 5 100 cm3, 1 5 f I 2.1 C', g' = 7.6 cm s - ~ , & = 6 cm and variable time of lens formation (Kostianoy and Shapiro, 1986). The dimensionless parameters varied in the ranges: 0.1 < Buo < 4, 0.8 5 I 3.5, 80 = 1, and < E c lo-'.

-

In each experiment, the radius of the growing lens is a power function of time, R ta, where a varies from 0.07 to 0.25 depending on the Burger number Buo = g'& / (f &)2 according to CL = 0.25 B u $ ~ ~ . In the experiments by Kitamura and Nagata (1983), when the constant supply of fluid was stopped, the lens radius continued to increase as t1'2, since they considered only the initial stage of the lens relaxation. It was also found that the lens radius is a function of volume, R - V0.3, and of reduced gravity, R - (g')'.'. These results are in good agreement with formula (4), obtained for very thin lenses. The experimental data on the lens radius, plotted in dimensionless universal coordinates R" = R I (V I (2 h,)") and t" = g' h z t / (v f ) , are shown in figure 5. One can see the common functional dependence of R" on t", which changes by three orders of magnitude over the ranges of the main parameters. It can also be noted that the functional dependence is the same for both methods of lens generation.

I

0.1 0.01

1

.

I

.

I

0.I

1

t"

I

1

10

Fig. 5: Vortex lens dimensionless radius R" vs dimensionless time t". Dots: lenses generated by the collapse method, plus signs: lenses generated by the intrusion method.

699

Several experiments were performed to study the instability process of constant-volume lenses. Under some conditions (for particular values of the Burger number), the resultant anticyclonic lens broke up into two, three or four well-defined smaller secondary vortices. The number of these vortices, n, was found to follow n 1.8 B U ? ~ ~which , is in good agreement with the results of Saunders (1973) and Griffiths and Linden (1981).

To understand how these experimental results may be used for ocean lenses, let us compare some dimensionless parameters that determine the lens dynamics. For example, in the ITE observed in the Canary upwelling system (Kostianoy and Rodionov, 19863, the Ekman and Burger numbers are 4.10-5 and 0.2, respectively, and the ratio of the lens thickness to its diameter is In our experiments with constant-volume lenses, the Ekman number varied from to 1, the Burger number from 50 to 3.10-3, and the aspect ratio was - 10-’. The dimensionless criteria of similarity are sufficiently satisfied to suppose that the experimental dependences may be true for the Ocean conditions as well. On the basis of the experimental dimensionless relationship R”(t”), the lifetime of the Canary ITE (i.e., the time it takes for the orbital velocity to decrease to the background value, say, 1 cm s -’), is about 1.6 years. The same estimates for the largest Mediterranean lenses, discovered by Armi and Zenk (1984), give more than 5 years, which is in agreement with indirect estimates of travel time across the North Atlantic, based on the general circulation velocity field. 4. CONCLUSIONS

Laboratory simulations of ocean Ill3 have been used to determine the general features of vortex viscous relaxation. In the ocean, this idealized situation is superimposed on different other processes: heat and mass exchanges with the ambient water, intrusion layering which may change the lens dynamics, etc. However, the lifetime of different ITE’s, estimated on the basis of our experimental results, is in good agreement with observations in the ocean and with estimates based on the mean general circulation velocity. This confirms the usefulness of laboratory experiments to model the general features of ITE dynamics.

In spite of several investigations of baroclinic instability of vortices, the results obtained in this study cannot identify clearly the physical mechanism of instability and determine an instability criterion. Such a criterion is a multiparameter function of the Burger and Ekman numbers, and of the fractional depth parameter. The latter parameter may be one of the most important in regions where the depth is comparable to the eddy thickness. For lack of instability observations in the ocean, further laboratory investigations may give an answer to the question of how ITE disappear in the ocean.

700

5. REFERENCES Army, L. and Zenk, W., 1984. Large lenses of highly saline Mediterranean water. J. Phys. Oceanogr., 14: 1560-1576. Belkin, I.M., Emelyanov, M.V., Kostianoy, A.G. and Fedorov, K.N., 1986. Thennohaline structure of intermediate waters of the ocean and intrathermocline eddies. In: K.N. Fedorov (Editor), Intrathermocline Eddies in the Ocean, P.P. Shirshov Institute of Oceanology, Moscow, pp. 8-34 (in Russian). Dugan, J.P., Mied, R.R., Mignerey, P.C. and Schuetz, A.F., 1982. Compact, intrathermocline eddies in the Sargasso Sea. J. Geophys. Res., 87: 385-393. Griffiths, R.W. and Linden, P.F., 1981. The stability of vortices in a rotating stratified fluid. J. Fluid Mech., 105: 283-316. Hebert, D., 1985. Oceans seem to mix in lens-shaped chunks. Canadian research, 11: 21-23. Kitamura, Y. and Nagata, Y., 1983. The behaviour of a fresh water lens injected at the surface of a uniformly rotating ocean. J. Oceanogr. SOC.of Jap., 39: 89-100. Kostianoy, A.G., 1984. On the stability of vortices in a rotating stratified fluid. In: Problemi gidromekhaniki v osvoenii okeana, p.1, Kiev, pp. 190-191 (in Russian). Kostianoy, A.G., 1987. Laboratory modelling of ocean intrathermocline eddies and slow density currents. Ph.D. Thesis, P.P. Shirshov Institute of Oceanology, Moscow, 117 pp. (in Russian). Kostianoy, A.G. and Rodionov, V.B., 1986. On the generation of intrathermocline eddies in the Canary upwelling region. Okeanologiya, 26: 892-895 (in Russian). Kostianoy, A.G. and Shapiro, G.I., 1984. Theoretical and laboratory study of the dynamics of an axisymmemc mesoscale eddy. In: Tonkaya struktura i sinopticheskaya izmenchivost morei i okeanov, p. 2, Tallinn, pp. 37-39 (in Russian). Kostianoy, A.G. and Shapiro, G.I., 1985. Theoretical and laboratory modelling of anticyclonic Ocean eddies. Morskoi gidrofisicheskyjurnal, 5: 14-21 (in Russian). Kostianoy, A.G. and Shapiro, G.I., 1986. The evolution and structure of an intrathermocline eddy. Izv. Acad. Sci. USSR, ser. Physics of Atmos. and Ocean, 22: 1098-1105 (in Russian). McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Reviews of Geophysics, 23: 165-182. Saunders, P.M., 1973. The instability of a baroclinic vortex. J. Phys. Oceanom., - 3: 61-65. Zatsepin, A., 1983. Some experiments on rotating baroclinic vortices. Tech. Report, WHOI83-41, pp. 272-285.

701

LONG-LIVED SOLITARY ANTICYCLONES IN THE PLANETARY ATMOSPHERES AND OCEANS, IN LABORATORY EXPERIMENTS AND IN THEORY M.V. NEZLIN I.V. Kurchatov Institute of Atomic Energy, Moscow, USSR G.G. SUTYRIN P.P. Shirshov Institute of Oceanology, Academy of Sciences USSR, Krasikova Street 23, 117218 Moscow, USSR

ABSTRACT A new class of stationarily translating monopolar Rossby solitons (anticyclones) in a rotating shallow fluid is studied both experimentally and theoretically. Unlike the KdV solitons, the two-dimensional monopolar solitons are found to have trapped fluid which is transported westward at the phase speed. These dualistic structures appear to be vortices on the one hand and solitons on the other. Owing to the trapped fluid, vortical solitons collide inelastically and they have a "memory" of the initial disturbance (which was responsible for the formation of the soliton). As a consequence, there is no definite relationship between the amplitude of the soliton and its characteristic size. These vortical properties are connected with the geostrophic advection of local vorticity. The solitary properties (non-spreading and stationary translation) are due to a balance between the Rossby wave dispersion and the nonlinear steepening of the elevation in anticyclones. Monopolar cyclones, due to depressions, are dispersive and non-stationary features. This difference in the dispersive properties of cyclones and anticyclones is thought to be one of the reasons for the observed predominance of anticyclones among long-lived vortices in the atmospheres of large planets and among intrathermocline oceanic eddies. 1. INTRODUCTION Geophysical fluid flows display many forms of ordered and long-lived structures over a wide range of scales. The most persistent ones are strong monopolar vortices in which a fluid rotates around the vertical axis. The largest of these vortices is the Great Red Spot (GRS) in the Jovian atmosphere. This atmospheric vortex, which is significantly larger than the Earth, has been coherent for at least 300 years. Other Jovian long-lived vortices, the Large Ovals, have been known for several tens of years (Smith et al., 1979). Similar large monopolar vortices such as Big Bertha, the Brown Spots, Anne's Spot, etc., have been recently detected in the atmosphere of Saturn (Smith et al., 1982). In the Earth's oceans, we also find long-lived monopolar vortices such as the Gulf Stream rings or the recently discovered compact intrathermocline eddies which can exist coherently for several years in a highly variable eddy field (Kamenkovich et al., 1986). Observations show that these long-lived vortices have trapped fluid which they transport over thousands of kilometers. Owing to their high intensity, long lifetimes and transport properties, the role of coherent vortices in geophysical fluids is very important. This has stimulated the development of theoretical and experimental modelling.

702

As planetary waves, the long-lived isolated structures are non-spreading nonlinear Rossby wave packets, in which the dispersion spreading (characteristic of a linear wave packet) is balanced by nonlinear effects. Some theoretical models of long-lived vortices such as Rossby solitons (localized and stationarily translating) have been developed over the last few years (Flierl, 1986). Since the horizontal spatial scales of these long-lived structures are much larger than the characteristic depths of the atmosphere and ocean, the shallow water approximation is generally used in their modelling. The validity of that approximation has been discussed in detail by Williams (1985). The previous theories of monopolar Rossby solitons ignore the trapping of fluid and are based on the wave concept without a transport of fluid. A particular solution with an analytic relationship between the potential vorticity and the stream function has been found only when the scale is between the deformation radius and the planet’s radius, i.e., at the so-called intermediate scale (Flierl, 1979; Mikhailova and Shapiro, 1980; Petviashvili, 1980; Charney and Flierl, 1981). This analytic solution describes a geostrophic anticyclonic vortex with increased thickness of fluid; there is a definite relationship between the vortex radius and its amplitude; the soliton drifts westward at a speed somewhat higher than the maximum linear wave speed. In this case, the basic balance is between a weak wave dispersion and a weak nonlinear steepening as for KdV soliton. At this intermediate scale, the predominance of anticyclones has been later demonstrated numerically (Matsuura and Yamagata, 1982; Williams and Yamagata, 1984; Williams and Wilson, 1988). Laboratory experiments in shallow water, stimulated by the above mentioned theory, were initiated at the beginning of 1981 (Antipov et al., 1981) and they have gone through several stages (Nezlin, 1986). In the first stage, the Rossby soliton was regarded as a non-stationary vortex produced by the short-time action of a generator, existing without spreading significantly more than a linear wave packet and decaying gradually due to viscous losses of momentum. Experiments have been carried out with vortices larger than the deformation radius and they displayed the cyclonic-anticyclonic asymmetry in qualitative agreement with the then-existing theory. These experiments have also revealed some obvious weaknesses of the theory, and, most importantly, they have revealed a number of new fundamental properties of Rossby solitons. This broadened the concept of monopolar Rossby solitons and stimulated the development of a new theory of vortical solitons with trapped fluid (Sutyrin, 1985). In the second stage, stationary structures - chains of monopolar vortices generated by unstable, oppositely directed, zonal flows - were obtained (Antipov et al., 1983). In the third stage, a single self-organized, stationary anticyclonic vortex - a Rossby autosoliton - was generated in a system of flows (Antipov et al., 1986). This structure can be regarded as a physical, experimental model of the GRS, constructed by the method of analogy simulation in shallow water. Other laboratory and numerical models of geophysical vortices like the GRS have been recently developed by Sommeria et al. (1988) and Marcus (1988), but without a deformation of

703

the rotating fluid surface. This corresponds to an infinite radius of deformation, unlike the conditions in the Jovian atmosphere. An important point made in those papers, as pointed out by Ingersoll (1988), is that turbulent mixing tends to produce a state of almost uniform potential vorticity. In this case initial non-uniformities are shown to be swept up gradually into isolated patches which cannot propagate without a background potential vorticity gradient. The experiments of Sommeria et al. (1988) and the simulation of Marcus (1988) lead to the conclusion that isolated vortices should form in planetary zones of strong shear and uniform potential vorticity. However, they do not consider such important questions as the observed cyclone-anticyclone asymmetry, the westward drift of the GRS showing the influence of the peffect, the generation of the GRS by shear flows and its uniqueness on the perimeter of Jupiter.

In the following sections, we consider these questions on the basis of our laboratory experiments and of the corresponding theory. 2. EXPERIMENTAL ARRANGEMENT: THE PARABOLIC MODEL OF THE ATMOSPHERE (OR THE OCEAN) OF PLANETS Experiments to observe and study Rossby solitons in a thin layer of rotating fluid were carried out with near parabolically shaped vessels (Fig. 1). The free surface of a fluid rotating in the gravitational field with a constant angular speed 0 around a vertical axis assumes a parabolic shape:

where z and R are the distances of a point on the surface from its lowest point, measured along the rotational axis and in a perpendicular direction, respectively; g is the acceleration of gravity. In order for the thickness of the layer % to be constant along the normal to the surface of the fluid, vessels of somewhat gentler slope than (1) were used in all of the experiments to be described; we shall refer to those vessels briefly as paraboloids.

In our experiments, vortices were excited by two methods. In the first method, a "pumping disk, placed at some "latitude" of the paraboloid in the plane of the bottom, was switched on for several seconds. The diameter of the disk could be varied from one experiment to another. In the second method, a jet of water was injected in a pulsed manner into the working fluid (water) from a short tube near the bottom of the paraboloid; the action of the Coriolis force on this jet formed an anticyclonic vortex. Both methods gave the same results. It is important to note that, for methodological reasons associated with the sensitivity of the experimental procedures used, the amplitude of the vortices, i.e., the maximum relative rise in the , was not small: h, = max(h) > 0.15. fluid, h = ( H - % ) /

704

Fig. 1: Layout of the experimental arrangements for exciting and studying either solitary vortices (a), or chains of vortices in zonal flows and Rossby autosolitons (b). 1) Vessel with a parabolic bottom (the radius of the small vessel is 14 cm, that of the large vessel 37 cm, the height of the vessel is about equal to its radius); 2) surface of the fluid; 3) camera, rotating with the vessel or together with a vortex; 4) rotating pumping disk; 5)-6) rotating rings creating a zonally sheared flow. In the view from above, the solid arrows indicate the direction of rotation of the vessel, the anticyclonic direction of rotation of the pumping disk and the directions of the zonal flows; the broken arrow shows the direction of the drift of the solitary vortex in layout (a).

3. REQUIREMENTS ON THE MODEL PARABOLOID The theoretical possibility of observing a monopolar Rossby soliton of intermediate scale in a layer of shallow fluid rotating together with a parabolically shaped vessel was noted by Petviashvili (1980). However, it became clear in preparing the laboratory experiments that it was necessary to use quite steep paraboloids for the identification of Rossby solitons. Indeed, in order to identify an isolated vortex as a soliton in an experiment, it is necessary to show that it exists without dispersion decaying for a period of time T which exceeds the time of dispersion spreading of a linear two-dimensional packet of Rossby waves of the same size, Td. The characteristic lifetime Td, in the P-plane approximation, was found by Flier1 (1977) for a Gaussian circular packet, i.e., for a perturbation defined by: h = h, exp

[3]

705

The minimum spreading time corresponds to the diameter of the packet 2L = ~ L and R is given by: min( Td ) =

where LR =

LR VR

(3)

v, is the radius of deformation, V, the speed of long gravity waves, f the Coriolis f

parameter, and V, the Rossby speed (the maximum linear Rossby wave phase speed). In a paraboloid with & = const., we have

[s] If2

v,= VR =

;f=2SLcosa

(4)

-

SL sina SL2 R ; tga = 2 g

In the experiments with water, the thickness of the Ekman friction layer was much less than the depth of the fluid (H, = 0.3 - 5 cm). Thus, the viscous time of a spindown TE, taking into account the free surface, can be estimated by the formula (Pedlosky, 1979):

where v is the viscosity coefficient. The quantity T is bounded from above by the viscous time, TE, which does not exceed several tens of seconds in the experiments using water as a working fluid. Therefore in order to satisfy T > Td, it follows immediately from (3)-(5) that the working region (characterized by a)must be located quite far away from the pole: sin(2a) >

16 g TE V,

Q2

(7)

The two vessels used in the experiments are quite steep and this condition holds. 4. OBSERVATION OF FREE VORTICES WITHOUT ZONAL FLOWS

In the experiments, the lifetime of a vortex is defined as the e-folding time of decrease of the maximum azimuthal velocity, V,. As shown in Fig. 2, the lifetime of an anticyclone increases with the depth of the fluid and it corresponds to the viscous time TE (given by equation 6), which exceeds the dispersion time Td(given by equation 3 ) by about a factor 2. During its lifetime, the anticyclone drifts westward (opposite to the direction of rotation of the vessel) at a speed u > VR until its amplitude h, is sufficiently large. The radius of the vortex (L) exceeds slightly the deformation radius LR and the observed range over its lifetime equals 6 times the diameter of the vortex, 2L, i.e., it is approximately an order of magnitude

706

0 a

0

0

0 0 0

0

/

0

lo

x

x % *

Fig. 2: Observed lifetime of (a) anticyclones (circles) and (b) cyclones (crosses) as a function of the depth of the fluid for the large paraboloid (Antipov et al., 1988); the broken line corresponds to formula (6)for viscous decay. longer than the range of a linear wave packet. It is in this feature that the solitary nature of the vortex under study is manifested. Concerning cyclones, the experiments have shown that their lifetime and drift speed (they drift in the same direction as the anticyclones) are significantly smaller than those of anticyclones of the same size (2L = 2.5L~)(Fig. 2b). If the rates of dispersion decay and viscous decay could be added, an estimation of the dispersion spreading time of a cyclone, T,, can be obtained from the experimental data: T, = min( Td), see equation (3). Correspondingly, the observed range of the cyclone under study does not exceed one and a half diameter of the vortex over its lifetime, i.e., it is on the order of the range of a linear wave packet. Thus, under the described experimental conditions, a cyclone does essentially not differ from a linear wave packet, unlike an anticyclone. The observed cyclone-anticyclone asymmetry (further manifestation of which are described below) is a very fundamental dispersion - nonlinear property of monopolar vortices not only at the intermediate scale L > LR, as pointed out by the previous theory and numerical modelling results (Matsuura and Yamagata, 1982), but at all the scales beyond the deformation radius LR. It is a direct consequence of the strong steepening under a significant elevation which can balance the strong dispersion in an anticyclone of size L > LR.

707

These properties of the monopolar vortices under study can be predicted and interpreted on the basis of the concept of solitary wave, l i e the KdV soliton, propagating in a medium like a wave, without trapped fluid. However, the experiments show that the monopolar Rossby soliton under study contains a region of trapped fluid. Consequently, the soliton transports very effectively trapped particles (either contained in the vortex at the time of generation or injected from above) and it does not allow particles which it encounters during its drift to enter it. An example of the efficient transport of particles by a solitary anticyclone is shown in Fig. 3.

Fig. 3: Drift of an anticyclone in the direction opposite to the direction of rotation of the vessel (Antipov et al., 1982). The vortex is formed in pure water (near the pumping disk I), it is colored with particles of potassium permanganate (introduced from above) in position 2 and it drifts clockwise in the small paraboloid.

A region of trapped fluid inside the monopolar vortex exists only if u > V,. This condition holds in all experiments under the above-indicated condition (h, > 0.15). The drift speed, u, of a solitary anticyclone is about one third of the typical maximum azimuthal velocity, V,. The fact that monopolar Rossby solitons contain trapped fluid, first discovered experimentally, will be taken into account in the theory described below, The question of the nature of the collisions of solitary waves is fundamental in the theory. The experiments under discussion showed that the solitary anticyclones collide inelastically: they either coalesce in one vortex (if they approach each other with a sufficiently high speed) or they mutually destroy each other, transforming into a "mean" flow (when the approach speed is low). An example of the coalescence of solitary anticyclones (a large-amplitude vortex, approaching from behind, catches up with the vortex in front of it) is shown in Fig. 4. An

708

analogous behavior of long-lived vortices is also observed in the Jovian atmosphere (Smith et al., 1979) and it has been simulated numerically in the intermediate geostrophic approximation (Williams and Yamagata, 1984). It follows from the numerical simulations that an inelastic collision of monopolar Rossby solitons is assumed to be a consequence of their vortical properties in the region of trapped fluid (Sutyrin and Yushina, 1986). The most important property that anticyclonic Rossby solitons manifest is the property of being an attractor. The experiments under study have shown that an arbitrary initial perturbation of the fluid (sufficiently extended and having large amplitude) rapidly evolves into wellformed anticyclonic vortices (solitons). (A monopolar cyclone, as shown above, does not have this property). The experimental data indicate also that these solitons are relatively stable.

5. EXPERIMENTS WITH SHEARED ZONAL FLOWS In the experiments without zonal flows described in the preceding section, the solitary vortex has a limited lifetime which is determined by the viscosity of the medium. In this section, we consider a system with oppositely directed zonal flows, as observed in planetary atmospheres. Such flows could compensate the viscous and, possibly, other losses of localized vortices and prevent their decay. The following method, illustrated on the right-hand side of Fig. 1, was used in our experiments to generate oppositely directed geostrophic flows in rotating "shallow water". Two wide ring-shaped slots, oriented along the parallels and separated from each other by some distance (d) along the meridian, are made in the thick bottom of a paraboloid. Zonal rings, which can freely rotate relative to the paraboloid in the plane of its bottom, are inserted into these slots. As they rotate, these rings entrain the layers of fluid lying above them, thereby creating oppositely directed zonal flows. By changing the distance d between the rings from one experiment to another, it is possible to modify the characteristic size of the transverse gradient of the flow velocity. Experiments with a smooth velocity profile (d > LR)show that large-scale stable smctures do not arise if the shear is cyclonic. Only in an anticyclonic shear is a chain of large-scale (L > LR) vortices (anticyclones) generated. This is another effect of the cyclone-anticyclone asymmetry beyond the deformation radius. To support the anticyclones in a stable state, a "pumping" of the structures by the weak unsteady flow with a smooth velocity profile is sufficient. But such a weak pumping is not able to support a stable state of quickly spreading cyclones. A stationary chain of large-scale cyclones is generated only if d a LR,when the strong unsteady zonal flow regenerates the cyclones and overcomes their tendency to dispersion spreading. This reason could be responsible for the existence of the prolonged cyclonic "barges" observed in the very strong shear flow of the Jovian atmosphere, which is almost the only example of long-lived cyclones in the atmospheres of large planets. The number of vortices (m) in the chain on the perimeter of the system is determined by the velocity of the flows: when the velocity is relatively low, 8 to 10 vortices are observed, for

709

Fig. 4: Different stages of the approach and coalescence of two solitary anticyclones created by the pumping disk method (the small paraboloid, Antipov et al., 1983).

710

a relatively high velocity, 2 or 3 anticyclones are observed. The decrease of the mode number, m, as the velocity of the flows increases, is a fundamental property which makes it possible, under other experimental conditions, to form one Rossby autosoliton on the perimeter of the system: m = 1. Figure 5 shows a photograph of an autosoliton generated with a large distance between the flows: d = 11 cm (this photograph was taken using a camera rotating together with the vortex, unlike the photographs presented previously which were taken with a camera rotating with the paraboloid). The parameters of the vortex are characteristic of the vortical Rossby soliton described above. In particular, the diameter of the vortex is 2L = (3 to 4) LR, the streamlines in the core are closed, and it effectively transports trapped fluid westward at a drift speed u > VR. Its vorticity is several times larger than that of the surrounding flow, in analogy to what is observed in large vortices in the atmospheres of large planets. The lifetime of the observed autosoliton has no limits and this structure can be regarded as a stationary one.

Fig. 5: The Rossby autosoliton, a stationary model of the GRS (the small paraboloid, Antipov et al., 1985).

This single vortex on the perimeter of the laboratory vessel is a result of self-organisation of the soliton structure within a system of oppositely directed unstable flows. Like the uniqueness of the GRS on the perimeter of Jupiter, this phenomenon is not a simple consequence of merging of small vortices into a large one as in the laboratory experiments of Sommeria et al. (1988) and in the numerical simulations of Marcus (1988) or Williams and Wilson (1988). The GRS should be considered as the first (strongly nonlinear and localized) instability mode of a

711

sheared zonal flow; this mode is realized if the flow width is large enough (Nezlin, 1986). 6. THE SHALLOW WATER EQUATIONS

With Ekman friction, the shallow water equations are: ah + V [(l + h)ifl = ErotV', at

E=Ho

W + F [ ~ X J+] VP = 0, at

V i (- a

F=f+rotV',

1iZ

1

ax

P=hVz+K,

[%] a

(9)

3 % )

K = 2W

(10)

where x is the coordinate along a parallel (the eastward direction being positive) and y is the coordinate along a meridian (the poleward direction being positive), f is the Coriolis parameter, F is the absolute vorticity, K is the dynamical pressure, ??is the unit vertical vector, V, is the long gravity wave speed, and E is the Ekman number. In the P-plane approximation, f = fo + Py and (3 = const. From equations @)-(lo), we obtain:

-

a EFrotV' (x+?.V) - = l l f h l

(l+h)*

Without friction (E = 0), Ertel's theorem on the conservation ("frozen-in'' nature) of the potential vorticity,

l+h'

in fluid particles follows immediately from (11).

For two-

dimensional motion at low frequencies, the first term of (9) is much smaller than the others and we can obtain an approximate expression for the velocity: F V ' = F x V P ] - at d

[TI;

F=f+V[F];

K

= 2 F2W

In this manner, the inertia-gravity waves with frequencies of the order o f f are filtered out of equations (11)-(12), without any limitations on the elevation h or on the local vorticity rot? (Sutyrin, 1986). Substituting (12) into (8), we obtain

v, ap + L,2J ax V,(1

+ h)"

-

.]

J v;J[y,

- K, - v :,I[ v [$] =

a(pat

[P,

v

I]$[

+

=

F

is the local deformation radius. In an anticyclone, L, is greater than F the ambient deformation radius LR, and in a cyclone L, < LR. As a result, the local Rossby speed, V, = PL:, is larger than VR in an anticyclone and smaller than VR in a cyclone.

where L, =

712

Due to the nonlinearity of the first term on the right-hand side of (13), connected with a variation of the local Rossby speed V, , there is a cyclone-anticyclone asymmetry relative to the Rossby wave dispersion. The second term on the right-hand side of (13) describes the horizontal advection of local vorticity, taking into account trapped fluid and the vortical properties of strong vortices. The terms connected with the dynamical pressure K are important only when the elevation h is significant. 7. ANALYTIC ROSSBY SOLITON For a Rossby soliton without friction (E = 0), a translating reference frame moving with speed u can be introduced. Thus, equations (11)-(12) become: J[B,

-&I

[L + y(fo+ F)]

= 0; B = P + u F ap aY

It follows from (14) that B is a functional of the potential vorticity,

For isolated l+h'

features that are decaying in x and y. this functional is defined as follows outside the closed isolines of the potential vorticity, i.e., outside the region of trapped fluid

The special case of a single-valued well-behaved Bernoulli functional (15) inside the region of closed isolines of

yields a particular analytic solution for the anticyclonic solil+h

ton, which can be obtained without a limitation on the amplitude h, (Sutyrin, 1985). In that solution, the elevation profile is defined by a function of the hyperbolic secant type; it is shown in Fig. 6 by the broken lines. In this case, there is a definite relationship between the radius, L, of the vortex and its amplitude, h,: LR L = 1.7 -

hA'2

It is interesting that the amplitude of the particular anticyclone with the Bernoulli functional (15) is limited by h, = 1.03, for which F = 0, Le., rot V = - f, at the center of vortex. This case has been observed experimentally when the thickness of the working fluid was very = 0.3 cm (Fig. 6a). For such an analytic soliton, the radial profile of the angular shallow, vo f rotation speed, w = -, is monotonic with an extremum wo= - - in the center. This feature r 2 has been observed in the ocean for intrathexmocline anticyclones. For example, the velocity measurements made in an Arctic eddy (Newton et al., 1974) are in good agreement with the analytic soliton of highest amplitude in the reduced-gravity one-layer model, with LR = 4 km defined independently by the density field in the eddy (Fig. 7).

713

I

1.0 h

0

-4

I

1

-1

0

I

I

1

3

x/L,

Fig. 6: Comparison of experimental profiles of the elevation in solitary anticyclones (circles) with the theoretical profiles of the vortical soliton. The solid curve corresponds to the numerical calculation, demonsmting the existence of a ’memory’ of the initial conditions in the region of trapped fluid. The broken lines show the particular analytic solution discussed in section 7. This solution differs from the classical one by the fact that the restriction h LR and h 4: 1. As pointed out by Charney and Flier1 (1981), the condition for the existence of a stable monopolar Rossby soliton without a meridional twisting requires:

where R, is the radius of curvature of the system. These authors. however, did not remark on an important consequence of (16) and (17). namely that the maximum azimuthal velocity. V , exceeds the drift speed which is on the order of the Rossby speed. Indeed,

114

Fig. 7: Comparison of the observed profile of angular speed in an intrathermocline anticyclone (circles, data of Newton et al., 1974) with the theoretical profile calculated for the particular analytic solution of highest amplitude (Sutyrin, 1986). Thus, the analytic Rossby soliton must have trapped fluid and vortical properties as a particular case of a wider class of vortical solitons.

8. NUMERICAL SIMULATION OF VORTICAL SOLITONS The possibility of the existence of a non-analytic class of monopolar Rossby solitons has been confirmed by numerical simulations using the filtered shallow water model described by equations (1 1)-(12) and simplified for for the case where the angular speed of particles rotation, a,is small with respect to f (Sutyrin and Yushina, 1988). The numerical simulations presented here were conducted with initially Gaussian anticyclones (see equation 2) of the same amplitude hm = 0.6 and different sizes, given VR = 0.02Vg, which is an order of magnitude less than Vm. The calculations show that the anticyclones begin to move west-southward at a speed u < VR and accelerate gradually (Fig. 8). After taking the form of a soliton, the anticyclones translate stationady westward at a speed u > VR without changing their shape. In the numerical simulations, the range is about 10 vortex diameters, as for the laboratory analogue. Figure 9 shows the evolution of the potential vorticity field. The region of trapped particles is characterized by closed isolines, and could be compared with Fig. 3, demonstrating the transport of fluid. Inside the region of trapped particles, we see only little change in the profile of

in comparison with the initial profile (Fig. 9c). l+h

Outside the closed isolines of poten-

tial vorticity, the profile becomes gentler in agreement with the Bernoulli functional (15). The

715

1-

t lr I

0

5

I

10

Fig. 8: The westward speed of an initially Gaussian anticyclone as a function of time. The 2xV,t latter is measured by the period of particles revolution around the center, z = L .

-

final vortical solitons can have quite different sizes for the same amplitude, depending on the initial structure. This feature can be regarded as the manifestation of a kind of memory of the system. The Ekman friction in equations @)-(lo) leads to the viscous decay of the monopole vortices in agreement with the estimation (6). The difference in viscous decay between a cyclone and an anticyclone, due to the difference in their local radii of deformation, does not exceed 30% under our experimental conditions. Thus, the cyclone-anticyclone asymmetry observed in laboratory experiments is really connected with the dispersion properties, as mentioned above.

9. DISCUSSION Our experimental and theoretical investigations, which are in agreement with each other, reveal a variant of the anticyclonic monopolar Rossby soliton of size L > LR with trapped fluid in the central region, substantially different than that of the previous theory. The trapped particles, rotating around the axis of the vortex, give the soliton qualitatively new properties. Among these properties, we note first the existence of a "memory" of the initial disturbance inside the region of trapped fluid, or, in other words, the absence of a definite relationship between the amplitude of the soliton and its characteristic size. It is important to note also that such solitons, whose sizes and amplitudes are quite arbitrary (in the sense just indicated), are attractors. A particular case of this wide class of vortical Rossby solitons exhibiting "memory" in the region of trapped fluid is the analytic soliton for which the relationship between the Bernoulli

716

a

b

x/L, a

-10

0

io

Fig. 9: (a) Contour plots of the potential vorticity for an initially Gaussian vortex of size LlLR = 1.8; (b) same as (a), but for the final vortical vortex soliton (the region of trapped fluid is shaded); (c) zonal profiles of potential vorticity for the initial vortex (right-hand side of the graph, x > 0) and for the final soliton (x < 0); the solid lines denoted 1 and 2 correspond to values of L/L, equal to 2.5 and 1.8, respectively. The dashed line denoted 3 corresponds to the analytic soliton of equation (15).

717

function B and the potential vorticity is described by equation (15). This case has a very small specific "statistical weight" and, apparently, a correspondingly low probability of realization. The Rossby soliton studied in this paper is a "real" vortex, which efficiently transports particles trapped in it and which collides inelastically. On the other hand, the dispersive properties of this vortex (cyclone-anticyclone asymmetry, direction of the drift and drift speed) are well predicted on the basis of the wave concept, according to which this structure is a result of the balance between dispersion and nonlinearity. Thus, the Rossby soliton is an explicitely dualistic object, and for this reason the following question often arises: "Is it a vortex or is it a wave?". This question is obviously not completely correctly posed. Such an object can be equally well called either a solitary (Le., non-spreading) vortex or a vortical soliton - depending on which of its properties are being studied. In this connection, the English term "solitary vortex" (Williams and Yamagata, 1984) is appropriate, since it combines the concepts of a vortex and of a solitary wave (soliton). Our results lead to the conclusion that the dispersive properties of cyclones and anticycIones are quite different not only at the intermediate scale, but at all scales beyond the deformation radius. In the shallow water approximation, isolated anticyclones evolve into stationary vortical solitons translating westward; monopolar cyclones, on the other hand, translate not only westward but also towards the pole and their decay is not only frictional but also due to Rossby wave radiation. This difference is assumed to be one of the reasons for the observed cycloneanticyclone asymmetry in the planetary oceans and atmospheres. With a few exceptions (the Brown Ovals, or "barges", of Jupiter at 14 N.L. or the "UV-spot" of Saturn at 24 N.L.) all large long-lived vortices on large planets are anticyclones. The cyclone-anticyclone asymmetry is also observed among the intrathemocline oceanic eddies on the Earth; as a rule, these eddies are anticyclones (McWilliams, 1985; Belkin et al., 1986). It should be noted that the rate of nonfrictional decay of an isolated cyclone decreases as its intensity increases, as shown by the numerical simulations of McWilliams and Flier1 (1979) and of Smith and Reid (1982). This effect has recently been shown to be connected with a quick fluid rotation in the region of trapped particles, which prevents any significant deviation from axisymmetry (Sutyrin, 1987). This is a main reason why very intense cyclones, such as Gulf Stream rings or tropical cyclones, can exist without significant dispersion spreading much longer than a linear Rossby wave packet.

10. REFERENCES Antipov, S.V., 1982. Rossby soliton in the laboratory. Soviet Phys. JEW, 55 : 85-95. Antipov, S.V., 1985. Rossby autosoliton and laboratory model of the Jovian Great Red Spot. Soviet Phys. JEW, 59: 201-215. Antipov, S.V., 1986. Rossby autosoliton and stationary model of the Jovian Great Red Spot. Nature, 323: 238-240.

718

Antipov, S.V., Nezlin, M.V., Snezhkin, E.N. and Trubnikov, AS., 1981. Rossby soliton. JETP Lett., 33: 351-355. Antipov, S.V. and Rodionov, V.K., 1983. Rossby solitons: stability, collisions, asymmetry and generation by sheared flows. Soviet Phys. JETP, 57: 786-797. Antipov, S.V., Rcdionov, V.K., Rylov, A.Y. and Hutoretskii, A.V., 1988. Properties of drift solitons in plasma, following from laboratory experiments with rapidly rotating shallow water. Fiz. Plazmy, 14: 1104-1 121. Belkin, I.M., Emelyanov, M.V., Kostianoy, A.G. and Fedorov, K.N., 1986. Thermohaline smcture of intermediate waters of the ocean and intrathermocline eddies. In: K.N. Fedorov (Ed.), Imathermocline Eddies in the Ocean. Inst. Oceanol. Acad. Sci. USSR, MOSCOW, pp. 8-34. Chamey, J.C. and Flierl, G.R., 1981. Oceanic analogues of large scale atmospheric motions. In: B.A. Warren and C. Wunsch (Eds.), Evolution of Physical Oceanography. The MIT Press, pp. 504-548. Flierl, G.R., 1977. The application of linear quasigeostrophic dynamics to Gulf Stream rings. J. Phys. Oceanogr., 7: 365-379. Flierl, G.R., 1979. Planetary solitary waves. Polymode News, 62: 1, 7-14. Flierl, G.R., 1986. Isolated eddy models in geophysics. Ann. Rev. fluid Mech., 19: 493-530. Ingersoll, A.P., 1988. Models of Jovian vortices. Nature, 331: 654-655. Kamenkovich, V.M., Koshlykov, M.N. and Monin, AS., 1986. Synoptic Eddies in the Ocean. Reidel Publ. Comp., the Netherlands. Marcus, P.S., 1988. Numerical simulation of Jupiter’s Great Red Spot. Nature, 331: 693-696. Matsuura, T. and Yamagata, T., 1982. On the evolution of nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr., 1 2 440-456. McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Rev. Geophys., 23: 165-182. McWilliams, J.C. and Flierl, G.R., 1979. On the evolution of isolated nonlinear vortices. J. Phys. Oceanogr., 9: 1157-1182. Mikhailova, E.I. and Shapiro, N.B., 1980. Two-dimensional model of synoptic disturbances evolution in the ocean. Izv. Acad. Sci. USSR, Phys. Atmos. Okeana, 16: 823-833. Newton, J.L., Aagaard, K. and Coachman, L.K., 1974. Baroclinic eddies in the Arctic ocean. Deep-sea Res., 21: 707-720. Nezlin, M.V., 1986. Rossby solitons (Experimental investigations and laboratory model of natural vortices of the Jovian Great Red Spot type). Soviet Phys. USPEKHI, 29(9): 807842. Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer-Verlag, Berlin Petviashvili, V.I., 1980. The Jovian Red Spot and drift soliton in plasma. JETP Lett., 32: 632-635. Smith, B.A., Soderblom, L.A., Johnson, T.V. et al., 1979. The Jupiter system through the eyes of Voyager-1. Science, 204: 951-972. Smith, B.A., Soderblom, L.A., Batson, R. et al., 1982. A new look at the Saturn system: The Voyager-2 images. Science, 215: 504-537. Smith, D.C. IV and Reid, R.O., 1982. A numerical study of nonfrictional decay of mesoscale eddies. J. Phys. Oceanogr., 12: 244-255. Sommeria, J., Meyers, S.D. and Swinney, H.L., 1988. Laboratory simulation of Jupiter’s Great Red Spot. Nature, 331: 689-693. Sutyrin, G.G., 1985. On the theory of solitary anticyclones in a rotating fluid. Dokl. Acad. Sci. USSR, 280: 1101-1105. Sutyrin, G.G., 1986. Intrathermocline eddies as solitary Rossby waves. In : K.N. Fedorov (Ed.), Intrathermocline Eddies in the Ocean. Inst. Oceanol. Acad. Sci. USSR, Moscow, pp. 93-100.

719

Sutyrin, G.G.,1987. On the influence of the P-effect on the evolution of a localized vortex. DOH.Acad. Sci. USSR, 296: 1076-1080. Sutyrin, G.G.and Yushina, I.G., 1986. Interaction of synoptic eddies of finite amplitude. Dokl. Acad. Sci. USSR, 288: 585-589. Sutyrin, G.G. and Yushina, LG.,1988. Formation of the vortical soliton. Dokl. Acad. Sci. USSR, 299: 580-584. Williams, G.P.,1985. Jovian and comparative atmospheric modeling. Adv. Geophys., 28A: 381-429.

Williams, G.P.and Yamagata, T., 1984. Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Amos. Sci., 41: 453-478. Williams, G.P. and Wilson, R.J., 1988. The stability and genesis of Rossby vortices. J. Atmos. Sci., 45: 207-241.


721

NUMERICAL MODELLING OF THE FORMATION, EVOLUTION, INTERACTION AND DECAY OF ISOLATED VORTICES

G.G.SUTYRIN and I.G. YUSHINA P.P. Shirshov Institute of Oceanology, Acad. Sci. USSR, 23 Krasikova Street, 117218 Moscow, USSR

ABSTRACT The filtered shallow water model which allows order-one variations of the potential vorticity is used for modelling such physical processes as the formation of a vortex from a meander of a frontal current, the vortex motion due to the p-effect, the non-frictional decay of a vortex, and the merging of two vortices. The numerical simulations display asymmetries between cyclones and anticyclones at scales beyond the deformation radius: for the pinching off of a vortex, the amplitude of a cyclonic meander must be larger than for the detachment of an anticyclonic meander; a cyclone moves to the north-west with Rossby wave radiation and nonfrictional decay unlike an anticyclone which evolves into a vortical soliton translating stationarily westward without change of shape; two cyclones merge slower than two anticyclones if the vortex pairs have the same potential vorticity anomalies. These results are interpreted as a consequence of a significant variation of the local deformation radius.

1. INTRODUCTION There is an increasing number of observations of isolated long-lived vortices in the ocean. Among the most intense are frontal rings which are formed by the pinching off of large amplitude meanders of the Gulf Stream and other stream currents in frontal mnes (Fedorov, 1986). These vortices are highly baroclinic, strongly nonlinear, and they have a long lifetime and slow decay rates. By virtue of their formation process, eddies represent an efficient mechanism by which heat, salt and momentum are transferred across frontal mnes which otherwise act as barriers to mixing between different water masses. These reasons have encouraged theoreticians to develop models to describe the formation, the motion, the interaction and the decay of these coherent features. Nearly geostrophic balance is a traditional approximation for low-frequency geophysical motions which are known to be controlled mainly by a redismbution of the potential vorticity (PV). In the widely used quasi-geostrophic equations, the average dismbution of the PV is prescribed, and the equations apply only to slight departures from the prescribed state (Kamenkovich et al., 1986). However, a characteristic feature of frontal zones and rings is the order-one variability of the PV mainly due to strong spatial variations in the density stratification. In this case the primitive equations are appropriate, but they include the relatively fast inertia-gravity waves

122

that can make numerical integration very costly. Thus, intermediate models have been developed, which filter out the inertia-gravity waves and still retain some degree of physical simplicity compared to the primitive equations (McWilliams and Gent, 1980). In this paper, we consider a filtered reduced-gravity shallow water model (Sutyrin, 1986), which is simpler than the balanced model of Gent and McWilliams (1984) or the nearly geostrophic model of Salmon (1983, 1985). The present model also allows the conservation of both the energy and the PV with its order-one variations. For scales beyond the deformation radius, we use the simplified filtered model which has only one variable, as the quasi-geostrophic model. The accuracy of that model is higher in respect to the Rossby number or inverse Froude number than the accuracy of the general geostrophic model (Williams, 1985) or of the frontal geostrophic model (Cushman-Roisin, 1986). The examples of numerical simulation display mainly the quantitative differences between cyclones and anticyclones due to strong variation of the PV. These differences are explained using the concept of local deformation radius.

2. THE FILTERED SHALLOW WATER MODEL Consider a homogeneous layer of fluid of undisturbed depth H, and density pl, overlying a much deeper layer of slightly larger density p2. The flow is assumed to be confined to the upper layer, and it is governed by the shallow water equations with reduced gravity g‘ =

(” - ”)

.

Including a turbulent viscous horizontal exchange of momentum, the nondi-

P2

mensional equations are:

ah -+V.

[(l

w +w -

[?xq

at

at

+ h)3] = 0;

h = -.H - H o , V = ( -

H,

f w=+ rot?

+ Vp = a A?

a -)a ay

ax’

p =h

f0

131 +2

where H is the thickness of the upper layer, the velocity ?= (vx. vy) is scaled by the gravity wave speed V, = (g’Q)’’2, the time t is scaled by fi’, and the horizontal distances by the V deformation radius LR = A,where f0

f0 is the rotational rate of the whole system about the 2

vertical axis. The x and y horizontal axes are oriented to the east and north, respectively, w is the absolute vorticity, 2 is the unit vertical vector, and a is the nondimensional viscosity f coefficient. In the P-plane approximation, - = 1 + py; f0

p = -V,R

v,

where VR is the maximum

speed of linear Rossby waves. The above equations have the well-known and important property that without viscous losses of momentum (a= 0), the PV = because

is conserved for each vertical column of fluid, l+h

723

a + (x

3.V)

0 =0 l+h

(3)

In that case, there are other "globally" conserved quantities such as the total energy E and the total potential enstrophy Q, defined as

considering a fixed area A which either encloses entirely any disturbance or has periodic boundaries.

a

For any slow motions (- = E a: I), an approximate expression for the velocity can be at obtained from (2) (Sutyrin, 1986):

a?'=E x Vp] + [aA - 2 12 at

w

Equations (I), (5) and (6) represent the common filtered model which allows order-one variations of the PV connected with either the elevation h, or the planetary vorticity py, or the local c

vorticity rot ?= V.

-

I.J

.

When a = 0, it is easy to check that this system conserves exactly

the PV in fluid columns, the total energy E and the enstrophy Q. In this system, all variables are expressed by p and W, instead of three variables h, vx, vy in the shallow water equations (1)-(2). For finite amplitude motions (h

= 1) of scale beyond the deformation radius (L > LR), the

Rossby number is proportional to the inverse Froude number

LR2 = d'' L2

(Cushman-Roisin,

1986). Substituting (5) into (l), we obtain the simplified filtered equation for the tendency of ah the layer thickness at

ah - V. [PT V at where PT =

l+h = (PV)-' w

$1

= J(PT, p) - V. (a PT VAh)

(7)

is the potential thickness which has been considered by Stommel

(1987) for the large scale oceanic circulation when PT = -. We include the local vorticity l+PY in the definition of PT, which is conserved for a column of fluid if a = 0 as the inverse potential vorticity. Comparing (7) with the quasi-geostrophic model we see that: L, = L, (PT)ln

(8)

724

can be interpreted as the local deformation radius, which differs from LR due to variations of PT. Considering the expressions (6) with the accuracy of

LRZ =

we obtain the general

L2

geostrophic model (Williams, 1985), slightly modified for the exact conservation of the PV (Sutyrin and Yushina, 1986a).

The frontal geostrophic model of Cushman-Roisin (1986) does not account for the dispersion .term on the left side of (7) because, with respect to the first term, it is also of the order of dR.To obtain the accuracy of E, the right part of (7) has to be calculated more accurately, i.e.,

While the relations L4 > L$ and L2 > L i are mathematically equivalent, in practice the former is much less severe, and, with a 10% relative accuracy as is usually acceptable in oceanography, the required assumption is L4 2 10 L$ or L 2 1.8 LR as is characteristic for oceanic meanders and rings. The proposed filtered shallow water model describes the cycloneanticyclone asymmetry as in the full shallow water model but it has only one unknown function, h, as the quasi-geostrophic model. Other kinds of filtered shallow water models, the balanced model (Gent and McWilliams, 1984) or the nearly geostrophic model (Salmon, 1983), are more complicated because they involve more than one variable. The numerical calculations presented in this paper are based on a finite-difference representation of (7) in an east-west periodic canal, with free-slip northern and southern boundaries. The finite-difference formulas are centered and second-order in both space and time; in particular, the nonlinear Jacobian operations are calculated by the spatially conservative formula of Arakawa (1966). The spatial resolution scales are dx = dy = 1/4; the nondimensional viscosity coefficient is 01 = 5 . 10"'.

3. PINCHING OFF OF LARGE AMPLITUDE MEANDERS The first problem we shall consider is that of a single disturbance of a stream current connected with a potential vorticity front, without p-effect. The development of baroclinic instability of a zonal stream current and the pinching off of an eddy have been studied numerically by Ikeda and Appel (1981) with a two-layer quasigeostrophic model. For isolated disturbances, these authors find that the physical mechanism of eddy detachment involves the generation of broad recirculations to the north and south of the stream. Once established, these recirculations are able to cut off the tips of intruding meanders through simple advection. Some cases in which large and steep meanders of the Gulf Stream resist detachment for long periods of time have been noted by Pratt and Stem (1986). In trying to isolate the detachment mechanism and to simplify the overall problem, they used a reduced-

125

gravity quasi-geostrophic model of a potential vorticity front which is stable relative to small disturbances. Their attention was concentrated on inertial effects which come into play when the amplitude of the disturbance becomes finite, and these may be independent of the instability processes. Numerical simulations by the method of contour dynamics suggest that fairly extreme initial conditions (i.e., small width and large amplitude compared with the deformation radius) are required to allow single lobe disturbances to pinch off. We consider a frontal zone with a monotonic distribution of the PV corresponding to a stable stream current directed eastward

I

PV = 1 + qo tanh [(y-y)s:y-y'

where 2q0 is the amplitude of the relative variation of potential vorticity across the frontal zone of finite width 2s, and Y(x) describes the position of the maximum gradient of the PV (for an undisturbed rectilinear front, Y = 0). In this case, the local deformation radius L, defined LR far from the front to the south according to (8) decreases monotonically from Ls = (1 - 40)'" LR (y -+- o), to LN = far from the front to the north (y 4 -). The case of s e 1 (1 + qo)''2 and qo a 1 corresponds to the quasi-geostrophic front of zero width considered by Pratt and Stem (1986). In our simulations, we use qo = 0.4 to model the difference between L, and LN for the Gulf Stream. A single lobe meander is prescribed by the initial condition

~ ( x =) yo e-xz'w2

(12)

consisting of a ridge of amplitude Yo and half-width w. The results for an anticyclonic meander with Yo =

2, w = s = 1 are shown in Fig. LR

1. In this case, the anticyclonic

meander does not pinch off due to the large width of the frontal zone. For a narrower frontal zone (s = OS), these amplitude and width of the meander are sufficient to lead to the detachment of an anticyclonic core in the PV field (Fig. 2, left). However, closed isolines of h do not appear because of the small size of the detached core which remains near the frontal zone and is translated slowly downstream (Fig. 2, right). For the same narrow frontal zone (s = 0.5), a cyclonic meander of the same amplitude 5LN - 3.5LS (yo = -) fails to pinch off in agreement with the results of Pratt and Stem (1986).

-

=R

LR

For the detachment of a single lobe cyclonic meander, a larger amplitude relative to the local -5Ls deformation radius is needed. Such a case, with Y o = -, is presented in Fig. 3. Owing to LR the larger size of the detached cyclonic core, closed isolines of h do appear at t = 100 (see Fig.

726

Fig. 1: The PV of an anticyclonic meander for a wide frontal zone at t = 0 (left) and t = 150 (right).

3, right). However, because the south side of the stream is wider in this case, in connection with the larger deformation radius, the cyclonic eddy is absorbed by the stream at t = 200. In order to investigate whether an eddy will be =-absorbed or be removed from the near vicinity of the stream, one should take into account the P-effect and the lower layer motion. Both of these effects appear to play a role for the eddies observed in the Gulf Stream. Our simulations indicate that the differences between cyclonic and anticyclonic meanders in the detachment process are connected with the strong variations of the local deformation radius and that the frontal zone has to be narrow for the detachment of an isolated eddy to occur. 4. VORTEX EVOLUTION DUE TO THE P-EFFECT

When a ring is removed from the stream, the ambient vorticity is much smaller than the angular speed in the central region of such an isolated vortex. The interaction of a vortex with its surroundings is influenced by the ambient PV gradient connected in the simplest case with the p-effect. Much analysis has been performed on the role of the p-effect in geophysical coherence. Numerical simulations with a reduced-gravity, one-layer model on the P-plane show that a monopolar vortex of a scale on the order of the deformation radius moves predominantly westward, with a meridional component of motion induced by nonlinear advective effects (McWilliams and Flierl, 1979; Smith and Reid, 1982). These experiments reveal that nonlinearity stabilizes the vortex against Rossby wave dispersion and allows an intense vortex

cc 727

I

Fig. 2: Evolution of an anticyclonic meander in the PV (left) and h (right) fields. The time interval between snapshots is 100.

728

Fig. 3: Evolution of a cyclonic meander of large amplitude in the PV (left) and h (right) fields. The time interval between snapshots is 100.

729

to propagate as a stable unit for longer periods of time than their linear counterparts considered by Flierl (1977). Analytic center-of-mass estimations predict that an anticyclone travels westward faster than the linear Rossby waves (Nof, 1983). This prediction is confirmed by numerical simulations using the shallow water equations (1)-(3) (Davey and Killworth, 1984). Our simulations using a modified general geostrophic model confirm these results and lead to the conclusion that the nonfrictional decay of an intense vortex can be estimated by its meridional displacement which does not increase for an anticyclone after its transformation into a vortical soliton stationarily translating westward. The evolution of an isolated eddy is simulated far from the stream current in order to display the 0-effect only. The initial conditions for the numerical calculations are the following: PV = 1 + qo [ch (hr)]43,

r = [(x-xo)~+

(13)

where r is the radial coordinate and (XO,yo) is the initial location of the vortex center. The parameters qo and h characterize the amplitude of the potential vorticity disturbance and the vortex size, respectively. If

h=0.6 Iqol" then (13) describes approximately the structure of the anticyclonic Rossby soliton considered by Petviashvili (1980) for h a 1. Our calculations show that an anticyclonic vortex satisfying the condition (14) translates westward faster than the linear Rossby waves (u > p) without a change of shape as a soliton, if p < h3 in agreement with the prediction of Charney and Flierl (1981). In this case, there is a region of closed isolines of the PV, with trapped fluid which is effectively transported at the phase speed u a V, (V, is the maximum azimuthal velocity of the vortex). When the value of p increases, the vortex does not maintain its circular shape. Therefore, in the initial stage, an anticyclone moves not only westward but also southward and it takes on the form of a vortical soliton. The trajectory of an anticyclone for qo = - 0.5 and p = 0.02 is shown in Fig. 4a. A wider class of vortical solitons with a memory is considered by Nezlin and Sutyrin (1989). Unlike the anticyclone, a cyclone (qo = 0.5) translates nonstationarily north-westward at a speed -u, < p and uy < 4 (Fig. 4b) in agreement with previous numerical results obtained 4

under the shallow water approximation (Smith and Reid, 1982). As in the quasi-geostrophic approximation, the westward drift speed of an intense vortex approaches the Rossby speed p (McWilliams and Flierl, 1979), the differences between cyclones and anticyclones are connected with the increase of the local Rossby speed V, = p PT in an anticyclone (PT > 1) and the decrease of V, in a cyclone (PT < 1).

730

Fig. 4 Trajectories of (a) an anticyclone; and (b) a cyclone. Successive positions are marked by a circle every 100 time units.

The behavior of a cyclone in the shallow water model does not differ greatly from its behavior in the quasi-geostrophic model. Thus, for the simulation of the long-range evolution of a cyclone, we can use the quasi-geostmphic model with a constant deformation radius. The evolution of an initial disturbance with PV = 1 + qo e-'

and

qo

= 0.03 is shown in Fig. 5.

In this case, about ten particles revolutions around the vortex center occur during one 2n: synoptic period T = - and the region of closed PV isolines containing trapped fluid remains

P

near circular for 6 synoptic periods (Fig. 5, left). Here, the condition vy = 0 is applied at all boundaries to allow the propagation Rossby waves. It is very important to note that in an intense cyclone (qo w P), the quick fluid rotation in the region of trapped fluid prevents a significant deviation from the circular form. As a consequence of Lagrangian invariance of the PV, the potential vorticity anomaly q = PV - 1 - Py inside the trapping region changes only due to the meridional displacement Y: q = qi(r) - PY (see Fig. 6). Thus, the nonfrictional decay of an intense vortex could be estimated as:

This estimate of the nonfrictional decay is much smaller than the rate of linear dispersive decay calculated by Flier1 (1977):

1.9.4

" = - qo dt

8

For an intense vortex yl w y and nonfrictional decay is unimportant for such intense vortices as Gulf Stream cyclones, for which qo = 10 and y = y1 (Sutyrin, 1988). In the shallow water approximation both anticyclones, as vortical solitons, and cyclones, as intense vortices, are long-lived and the relaxation of oceanic rings should be regarded as mainly connected with internal mixing.

731

a

a

Fig. 5: Evolution of a cyclone due to the 0-effect in the PV (left) and h (right) fields, at t = 0.5 T (upper panel) and t = 6 T (lower panel).

732

Fig. 6: Zonal profile of the potential vorticity anomaly at (a) t = 0 and (b) t = 5T.

5. MERGING OF TWO VORTICES The merging of two like-signed vorticity monopoles plays an essential role in the longtime evolution of geophysical flows such as quasi-two-dimensional macroturbulent-like states (McWilliams, 1984). The contour dynamics computations show that the inviscid merger process may be conveniently viewed as the long-time behavior of the evolution of an instability of perturbed steady-state corotating "V-states" with constant vorticity cores (Overman and Zabusky, 1982). According to these numerical experiments, the end result of symmetric merging is a near elliptical constant vorticity core surrounded by a pair of filaments. Although narrow, the filaments play a crucial role in the mass and angular momentum balances (Cushman-Roisin, pers. comm.). The axisymmetrization also plays an important role in the merging process, especially for spatially smooth vorticity cores (Melander et al., 1987a). The asymmetric merger involving vortices of different size and vorticity amplitude causes one of the vortices to dominate (Melander et al., 1987b). In all cases, the merger will occur if and only if the distance between the vorticity cores is less than a critical value. In most of these studies, the non-divergent barotropic model with an infinite radius of deformation was used. But an increase of the vortex core size relative to the radius of deformation may result in the weakening of the vortices interaction due to shielding effects. Consider the interaction of two vortices on the f-plane with the following initial distribution of the PV (Sutyrin and Yushina, 1986b):

PV = 1 + q1 [ch

+ 92 [ch

[;]r

733

a

b

Fig. 7: Merging of two cyclones in the PV (left) and h (right) fields at (a) t = 0 and (b) t =

800.

734

where the qj's are the amplitudes of the potential vorticity anomalies, the rj's the distances from the centers of vortices, and r, is the radius of maximum azimuthal velocity in the vortices. Results for ql = 0.5,q2 = 0.7 and the distance between the vortices centers Y = 3.5 r, (r, = 2.5) are shown in Fig. 7. In this case, the maximum velocity of the first cyclone is 0.045 and that of the second is 0.06. While rotating around the common center, the cores approach each other. After a 90' rotation, the stronger vortex absorbs the weaker one (Fig. 7b). Anticyclones of the same amplitudes (ql = - 0.5, q2 = - 0.7) and with the same distance between their centers have maximum velocities of 0.13 and 0.2 respectively. Such a difference in the intensities of cyclones and anticyclones is caused by the strong variation of the local deformation radius. In the center of a strong anticyclone, the value of L, is more than twice the value of L, in the center of a strong cyclone. In this case, the anticyclones merge after a rotation of about 40' and the merging occurs about ten times faster than the merging of the cyclones. This asymmetry appears to be caused by the difference in both the intensities and the local deformation radii of the vortices. For a larger distance between the vortices centers (Y = 7rm), both cyclones and anticyclones of the kind described by (17) do practically not interact. The P-effect is regarded to be one of the causes for the relative motion of isolated eddies and for their cores to converge or diverge. The approaching of vortices due to the P-effect and their merging is discussed by Nezlin and Sutyrin (1989). 6. DISCUSSION Our numerical simulations with the filtered shallow water model reveal some common features of isolated disturbances and also differences between cyclonic and anticyclonic vortices. For the detachment of an isolated eddy from a stream current, the frontal zone has to be narrow. The problem of the eddy removal from the vicinity of the stream needs to take into account additional physical factors, e.g., the p-effect and the lower layer motion. Due to the p-effect, both cyclones and anticyclones of scale larger than the deformation radius drift predominantly westward. The nonfrictional decay inside the vortex core is much smaller than for a linear Rossby wave packet. If vortices of the same sign approach each other, the stronger vortex absorbs the weaker one. These processes play an essential role for long-lived oceanic eddies. Our main purpose is to understand the physical significance of order-one variations of the potential vorticity or the potential thickness for the dynamics of isolated eddies. If the relative potential vorticity anomaly is small, then variations of the deformation radius are not important and the quasi-geostrophic model can be used without essential dependence on the sign of the eddy. In this case, it is only at the intermediate scale, which is much larger than the deformation radius, that the p-effect influence on an isolated eddy depends on its sign (Chamey and Flierl, 1981).

735

We have considered situations where the variation of the local deformation radius is important and we have observed asymmetries between cyclones and anticyclones with the same potential vorticity anomalies: for the pinching off of a cyclonic meander, the amplitude must be larger than for the detachment of an anticyclonic meander; the westward motion of a cyclone is slower than that of an anticyclone; a cyclone moves northward with Rossby wave radiation and nonfrictional decay, whereas an anticyclone evolves into a vortical soliton translating stationarily westward without changing its shape; two cyclones merge much slower than two anticyclones. These quantitative differences are explained qualitatively using the concept of local deformation radius in the shallow water model. 7. REFERENCES Arakawa, A,, 1966. Computational design for long term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part 1: J. Comput. Phys., 1: 119143. Chamey, J.C. and Flierl, G.R., 1981. Oceanic analogues of large scale atmospheric motions. In: B.A. Warren and C. Wunsch (Eds), Evolution of Physical Oceanography. The MIT Press, pp. 504-548. Cushman-Roisin, B., 1986. Frontal geostrophic dynamics. J. Phys. Oceanogr., 16: 132-143, Davey, M.K. and Killworth, P.D., 1984. Isolated waves and eddies in shallow water model. J. Phys. Oceanogr., 14: 1047-1064. Dewar, W.K. and Flierl, G.R., 1985. Particle trajectories and simple models of transport in coherent vortices. Dyn. Atmos. Oceans, 9: 215-252. Fedorov, K.N., 1986. The Physical Nature and Structure of Oceanic Fronts. Springer-Verlag, Berlin, 333 pp. Flierl, G.R., 1977. The application of linear quasigeostrophic dynamics to Gulf Stream rings. J. Phys. Oceanogr., 7: 365-379. Gent, P.R. and McWilliams, J.C., 1984. Balanced models in isentropic coordinates and the shallow water equations. Tellus, A36: 166-171. Ikeda, M. and Appel, J.R., 1981. Mesoscale eddies detached from spatially growing meanders in an eastward-flowing oceanic jet using a two-layer quasi-geostrophic model. J. Phys. Oceanogr., 11: 1638-1661. Kamenkovich, V.M., Koshlykov, M.N. and Monin, A.S., 1986. Synoptic Eddies in the Ocean. Reidel Publ. Comp., The Netherlands. Matsuura, T. and Yamagata, T., 1982. On the evolution of nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr., 12: 440-456. McWilliams, J.C., 1984. The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech., 146: 21-43. McWilliams, J.C. and Flierl, G.R., 1979. On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr., 9: 1 155-11 82. McWilliams, J.C. and Gent, P.R., 1980. Intermediate models of planetary circulations in the atmosphere and ocean. J. Atm. Sci., 37: 1657-1678. Melander, M.V., McWilliams, J.C. and Zabusky, N.J., 1987a. Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech., 178: 137-159. Melander, M.V., Zabusky, N.J. and McWilliams, J.C., 1987b. Asymmetric vortex merger in two-dimensions: Which vortex is "victorious" ? Phys. Fluids, 30: 2610-2612. Nezlin, M.V. and G.G.Sutyrin, 1989. Long-lived anticyclones in the planetary atmospheres and oceans, in laboratory experiments and in theoy. In: J.C.J. Nihoul and B.M. Jamart

736

(Editors), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Elsevier Oceanography Series, Elsevier, Amsterdam (this volume). Nof, D., 1983. On the migration of isolated eddies with application to Gulf Stream rings. J. Mar. Res., 41: 399-425. Overman, E.A. and Zabusky, N.J., 1982. Evolution and merger of isolated vortex structures. Phys. Fluids, 25: 1297-1305. Petviashvili, V.I., 1980. The Great Red Spot of Jupiter and the drift soliton in plasma. JEW Lett., 32: 632-635. Pratt, L. and Stem, M.E., 1986. Dynamics of potential vorticity front and eddy detachment. J. Phys. Oceanogr., 16: 1101-1120. Salmon, R., 1983. Practical use of Hamilton’s principle. J. Fluid Mech., 132: 431-444. Salmon, R., 1985. New equations for nearly geostrophic flow. J. Fluid Mech., 153,461-477. Smith, D.C. IV and Reid, R.O., 1982. A numerical study of nonfnctional decay of mesoscale eddies. J. Phys. Oceanogr., 12: 244-255. Stommel, H., 1987. A View of the Sea. Princeton Univ. Press, Princeton, New Jersey, 165 pp. Sutyrin, G.G., 1986. Synoptic motions of finite amplitude. Dokl. Acad. Sci. USSR, 280: 1101-1105. Sutyrin, G.G., 1988. On the motion of an intense vortex on a rotating sphere. Izv. Acad. Sci. USSR, Mekh. Zhidk. Gaza, 2: 24-34. Sutyrin, G.G. and Yushina, I.G., 1986a. On the evolution of isolated eddies in a rotating fluid. Izv. Acad. Sci. USSR, Mekh. Zhidk. Gaza, 4: 52-59. Sutyrin, G.G. and Yushina, I.G., 1986b. Numerical modelling of the merging process of intrathermocline eddies. In: K.N. Fedorov (Editor), Intrathermocline Eddies in the Ocean. Inst. Oceanol. Acad. Sci. USSR, Moscow, pp. 101-104. Williams, G.P., 1985. Geostrophic regimes on a sphere and a beta-plane. J. Atm. Sci., 42: 1237-1243.

737

EDDY-RESOLVING MODEL OF IDEALIZED AND REAL OCEAN CIRCULATION D.G. SEIDOV, A.D. MARUSHKEVICH and D.A. NECHAEV Institute of Oceanology, Acad. Sci. USSR, 23 Krasikova Street, 117218, Moscow, USSR

ABSTRACT The model discussed in this paper is a new version of the eddy-resolving model of the large scale ("general") Ocean circulation, ERGCM, described by Seidov (1985). The model was designed and used for the study of the North Atlantic circulation (Seidov et al., 1985, 1986). Some additional calculations for an idealized geometry and some of the results of the real circulation modelling (Seidov et al., 1985, 1986) are presented in this paper for the purpose of comparison.

1. DESCRIPTION OF THE MODEL The eddy-resolving model discussed in this paper is derived from the basic system of the momentum equations together with the continuity and state equations, with the Boussinesq, incompressibility and hydrostatic equilibrium approximations (e.g., Kamenkovich, 1977). We reduced this initial system to a simpler one by splitting the horizontal velocity vector into a vertically averaged (barompic) component and a shear (deviations from barotropic) component, and, also, by assuming geostrophic equilibrium for the shear component field. It can be shown that the reduction leads to three prognostic equations for the relative vorticity of the total water fluxes Q, the temperature T, and the salinity S, together with several diagnostic equations:

where 1 case

Fx = a

a [$1u"u"dz + -cos$ I u"v"dz - sine ae H

[

n

H

H

0

0

u"v"dz

1 a case & [ u"v"dz + -cos$ I v"v"dz + sine] u"u"dz a@

FQ= a

aT + --aT at

acosQ) ah

+

--aT

a &$I

H

H

0

0

+ W -aT =A~AT+KTaz

a2T az2

1

738

as +--++at

as

aces$ ah

a

as + W- as

a$

= As AS

a2s + Ks -

(3)

az2

aZ

k$ = div v V + div 9$ + w - 2u"o tg$ av*g

+

aZ

a

u = cj + u"; v = 9 + v"; 9 = (u,v)

H

In these equations, u, v, w denote the velocity vector's components in the s~..xical layer x x (h, 4, z), where h is the longitude, $ the latitude (-- I$ I-) and where the z-axis is 2 2 directed downward; p, T, S denote the density, the temperature and the salinity of sea water; P is the pressure; is the total stream function; H denotes the depth; f = 2 0 sin$;

w

P= M a , R being the angular speed of the Earth's rotation ; w =

x + w" is the relative Q

vorticity of the horizontal velocity vector; A, and KM are the coefficients of turbulent momentum exchange in the horizontal and vertical directions; AT, As, KT, Ks are the coefficients for turbulent exchange of heat and salt; ? is the wind stress vector, UE, vE are the Ekman's drift

I&[

velocities at the surface (6, = 1 for z = 0, and

a=

= 0 for z > 0);

112

E=

2aH '

; A denotes the Laplacian operator, which is slightly simplified:

A=1 a2 cos$

[$+ [$ cos$

cos$

$11

where

739

Along the side boundary (C), the conditions of no-flux and free-slip reduce, in the (Q, W)-system, to:

wlz= Qlc=O

(12)

The heat and salt balances at the surface are:

where QT and Qs are the heat and salt fluxes through the sea surface. At the side boundaries and at the bottom, we have no-flux conditions:

Finally, there is a specified momentum flux into the ocean:

In the experiments discussed below, we used either fresh water, so that p = p(T), or a constant salt approximation wherein p = p (T, So), with SO = const. Thus at the surface, we only need the heat balance condition which is written as:

where y is an an empirical coefficient that roughly parameterises all forms of air-sea heat exchange (Haney, 1971). It should also be mentioned that equations (1) and (6) are obtained using the "rigid-lid' approximation: w I = = 0. Equation (4) is just a continuity equation for the total fluxes. Equations (1)-(3) are to be integrated in time, and all other equations are simply diagnostic relations which need only trivial direct calculations, except for equation (4) which should be solved as the Poisson equation. As mentioned earlier there are only three prognostic variables: Q, T, and S, and ten diagnostic variables: v, P", Pk, p, li,0, u", v", a",and w. Since there are geostrophic relations for u", v" instead of primitive equations, fast internal gravity waves and inertial oscillations are filtered out as soon as the rigid-lid is used. All these filterings allow us to use a rather long time step, up to 2 lo4 s with 6h = 6t$ = 0.5'.

The numerical schemes are in general the same as in Seidov (1980, 1985), and for timedependent problems are based on the procedure described by Marchuk (1973), with the LeithRichtmyer approximation for the advection step. The Poisson's equation (4) is solved using an iterative procedure with an over-relaxation accelerative technique (Roache, 1976). The tendency equation for the vertical velocity, i.e., equation (6), is diagnostic with backward differences.

a m calculated by at

740

2. EXPERIMENTS WITH AN IDEALIZED BASIN

Our knowledge of the role played by synoptic-scale eddies in the large-scale ocean thermohydrodynamics is largely based on numerical experiments with geometrically idealized oceanic basins (e.g., Holland and Lin, 1975; Holland, 1978; Robinson et al., 1977; Seidov, 1980, 1985). The ERGCM presented above was designed and used for the simulation of a real ocean basin. However, it is necessary to know how the same model represents the circulation and the eddies in both an idealized basin and in a real one. In addition, it is easier to interpret the most general features, such as the role of the eddies in a ventilated thermocline (Cox, 1985), for the simplified geometry case. In our idealized experiments, we use a basin with a flat bottom located in the subtropical latitudes, with dimensions of 18' in latitude and 12' in longitude. The depth is 2 km, as in Seidov (1985). The zonal temperature of the air above the sea surface decreases linearly from 19OC at the southern boundary to 6OC at the northern boundary. A zonal wind stress is specified in such a way that in the barornopic case a symmetric two-gyres wind-driven circulation is formed, with the total flux of the western boundary currents being about 30 lo6 m3/s. The initial distribution of the temperature is taken as homogeneous in the horizontal, with an initial profile described by T = 19OC, 17OC. 14OC, 7OC and 4OC at the model's five vertical levels (0, 200, 600,1200 and 2000 m). The parameters of the idealized experiment discussed in this paper are: AM = 5 lo6 cm2 s-', AT = 6 lo6 cm2 s-', E = 6 lo-* s-', y = 5 lo4 cm-', KT = 2.5 cm2 s-', 6 1 = &I = 0.5".6t = 2.16 lo4 s. We ran the model for 40 years of model time. Fig. 1 shows the evolution of the basinaveraged kinetic energy (KE). One can see that after 3 years the system approaches a statistically-stable regime of dynamical equilibrium between the eddies and the large-scale currents. The dashed line in figure 1 represents the available potential energy (APE). The fact that there are sharp peaks in the KE time series is in good qualitative agreement with the observations made during POLYMODE (Kamenkovich, 1982), and with earlier ERGCM's results (Seidov, 1985). This kind of behavior led some of us to use the well-known nonlinear dynamical model known as the brusselator to explain some special features of the energetics of the ocean circulation (Seidov, 1986, 1989; Seidov and Marushkevich, 1988). Fig. 2 shows the instantaneous total flux function and the temperature at 200 m for t = 4800 days. In the field of the bamtropic currents (or yt. which is the same for the flat bottom case), it is easy to see a rather strong and meandering jet at the basin's mid latitudes. This jet produces rings of both signs, and it radiates Rossby-wave-type eddies. The rings are not as stable as in reality, since the resolution is still rather coarse. The smallest eddies have diameters of about 200 km (which is at the edge of the resolution limit, about five grid-points for one eddy). There are also bigger eddies with diameters of 300-400 km. The strongest eddy activity is observed in a small area located south-west of the mid-ocean jet, in the so-called recirculation gyre. It has been shown, by the way, that this gyre is itself produced by the eddies (Seidov et al., 1986).

J 0 m

r

00

I

I 0

2

0

r

2

0 d

m r

'

(D

cu

r

\i t (days 1

e

4

741

Fig. 1: Time evolution of the basin-averaged KE (solid line) and APE (dashed line) obtained in an idealized ERGCM experiment.

742

c.

3

z

m m

0

Fig, 2 : (a) Instantaneous total stream function \v (in lo6 m3 s-l); (b) instantaneous temperature at the 200 m level Tzo0(in "C).

743

s,

The time-averaged (mean) fields of ?;200 and iC2m are shown in Fig. 3, while the deviations from the mean are presented in Fig. 4. It can be easily inferred from those figures that a rather high correlation exists between T&, and w ; ~ . This is characteristic for the whole thermocline, with less correlation in the deep layers. The areas of upward and downward motion correspond to the frontal or the tail zones of the eddies. The southern and the northern parts of the basin are almost completely isolated from each other in the case without eddies. The latters transport heat and vorticity through the jet, i.e., the eddies accomplish something like tunnel effect (Seidov, 1985). As already mentioned, the ventilation of the thermocline depends significantly on the eddy mixing along isopycnals (Cox, 1985). This model shows them to play the same role. This conclusion is based on a comparison of the eddy-resolving and the non-eddy-resolving experiments. It is easy to see that the eddy dynamics of the northern gyre differs very much from the dynamics of the southern gyre. Furthermore, the gyres are more asymmetric when the eddies are resolved. There is an area to the north of the jet where isopycnals outcrop to the surface and produce ventilation. The selfsharpening of the jet after it leaves the western boundary follows strictly the scenario proposed by Monin and Seidov (1982). According to this scenario, the negative viscosity process dominates in the area to the south of the axis of the jet and the positive viscosity due to eddy mixing dominates north of the jet. This leads to the sharpening of the jet and to the instability growing further to the east. We conclude from those results that the new ERGCM is a reliable model, at least for the idealized studies. Since there were significant modifications both in the equations and in the coding of the computer program compared to the earlier version (Seidov, 1985), one can say that the model is a system with a rather strong structural stability.

3. EXPERIMENTS WITH A REALISTIC DOMAIN CONFIGURATION To compare the case of the North Atlantic with the idealized study, we want to look at the results of the real circulation simulation stressing some interpretations of Seidov et al. (1985, 1986). In thar study of the North Atlantic, the basin extends from 81' to 3' W and from 13' to 61' N. The bottom topography and the geometry are approximated on a 2Ox2' grid, with 7 levels in the vertical (0, 200, 500, 800, 1200, 2000, 3000 m). Among other calculations, we have carried out several experiments which we group here as follows: 1. Prognostic experiments, with the simulation either of both temperature and velocity fields on a coarse (2Ox2') grid or of the barotropic velocity on a fine grid (2/3'~2/3') with temperature (and shear currents) calculated on the coarse grid. In the latter case, which one could call "barotropically-eddy-resolving"or "semi-eddy-resolving", we simulated the evolution of the barotropic component of the vorticity on the fine grid using temperature and shear current velocities (which are calculated geostrophically from the temperature field)

7

I

a

b

Fig. 3 : (a) Time-averaged stream function (in lo6 m3 s-l); (b) time-averaged temperature at 200 m Tzoo(in "C); (c) time-averaged vertical velocity at 200 m w~~ (in lo4 cm s-1).

C

Fig. 4 : Eddy-fields y ~ ’(a), T& (b), and w& (c); units as in Fig. 3.

746

interpolated from the coarse grid onto the fine one for the calculation of the baroclinic and nonlinear terms of the vorticity equation (1).

2. Diagnostic experiments (density field specified and fixed), which we have run with both the coarse and the fine grid. In this case, the vorticity of the total fluxes is a time-dependent variable which is calculated using the two-dimensional balance equation. The prognostic experiments differ from the diagnostic ones in a very important way. The diagnostic calculations are performed using the real density field, i.e., p = p (T,S), while in the prognostic experiments only the temperature field is simulated, p = p (T,So) with SO = 35°/0, specified everywhere. As mentioned, we operated with 7 levels on a 2"x2" grid, although the (T,S)-fields we used (Levitus, 1982) exist on a laxlo grid and at 31 levels. We could use a resolution of laxlo and 7 levels with the same computer facilities, but we thought that it would be interesting to show that the nonlinear model has the ability to sharpen barotropic jets, to reconstruct the total fluxes field, and to lead to a "better" general circulation pattern (i.e., a pattern which is much closer to that obtained without eddies on a laxlo grid than to that obtained without eddies for the case where the density is known only on a 2Ox2" grid). The parameters used in the experiments with the coarse grid are: AM = 5 lo7, E = lo4, AT = 6 lo7, KT = 1 (all in CGS units; AT and KT are used only in the prognostic experiments). The same parameters for the "semi-eddy-resolving" calculations are: AM = 3 1 6 and (AT and KT being naturally the same as in the non-eddy-resolving experiments). E =5 Fig. 5 shows the velocity field at the 500 and 1200 levels for the diagnostic experiment without eddies (a,b), and for the diagnostic experiment with barotropic eddies resolved on the 2/3"x2/3" grid (c,d). Naturally, the barotropic component is the same at all levels, but the ratio of barotropic to barwlinic components is different. The role of the barotropic mode is much more significant at the deep levels than at the upper ones. In Fig. 6, this mode is presented separately by the v-function maps without eddies (a), and for the "semi-eddy-resolving'' case (b). Let us stress again that the field in Fig. 6a is much smoother than that obtained by Holland and Hirshman (1972), because our density field from Levitus (1982) is specified on the 2Ox2O grid with poor vertical resolution, whereas Holland and Hirshman use a loxlo grid and fine spacing in the vertical. We also have an oversmoothed bottom topography (on the 2"x2O

grid). The most important difference, however, comes from the highly idealized wind stress distribution. It is interesting, keeping in mind all these "deficiencies", that for this poorly resolved density field on a 2Ox2" grid with only 7 levels, we obtain barotropic currents which are qualitatively closer to the Holland and Hirshman study when barotropic eddies are resolved than in the case without eddies. Some important details of the fine structure of the vorticity field (and, therefore, of v) appear to be strongly dependent on the nonlinear interactions between eddies (here only their barotropic mode) and the ocean general circulation.

0

9

0

I

r7 n

I

4

R

Fig. 5: Velocity field of the North Atlantic in the diagnostic experiments at 500 and 1200 m: 747

(a,c) experiment without eddies (a - 500m; c - 12OOm). (b,d) the barompic eddy-resolving experiment 0, - 5OOm; d - 1200 m).

748

0

m

(D

r -

m

0

c

K

i

749

750

0

m

c

0

aD

0

c

W

W

c

0

0

0

2

2

m

0

m

1;

0 c

Q)

751

Comparing figures 6a and b, the diagnostic total transport is significantly intensified by the eddies in the vicinity of the Gulf Stream, the Labrador Current, and in the eastern part of the North Atlantic. There is a recirculation zone to the south of the Gulf Stream which becomes much more pronounced in the case with eddies. Comparing this result with the idealized studies, one can a f h that this zone is really produced by the eddies. The total flux of the Gulf Stream is 1.5 times greater in the presence of eddies than in the case without them. The Labrador Current's total transport is almost 3 times greater in the case with eddies. It is interesting to note that in general the y-field in this barotropic-eddy-resolvingcase correlates closely to the f isolines of -. H Another zone with highly variable currents is situated in the eastern part of the basin. There is a core of saline Mediterranean water which is represented in the specified density field. This core seems to be a source of eddies (although we resolve only the barotropic mode). A chain of eddies generated in this region propagates to the south-west. The eddies penetrate the trade-wind current area and, presumably, they dissipate there. To the south of the Azores, there is a deep westward jet which is rather intensive in the eddy-resolving case. The velocities are as strong as 4-5 cm s-' at the 1200 m levels (compare Figs. 5 b, d). It so happened that, for this very same level, the SOFAR floats experiment of the WHO1 (Price et al., 1986) has also shown this strong westward jet, with similar current speed. It should be stressed that the jet is formed mainly by the barotropic component since the surface currents have an eastward direction. We can add, also, that we believe that this jet is produced due to the eddies interactions, because the jet is almost completely absent in the case without eddies or in traditional diagnostic calculations of the steady currents. Hence, the jet is not present in the density field and it appears as a barotropic jet due to some cause(s) other than a baroclinic drive. The wind has no significant pecularities in this area. Our modelling results point towards eddy dynamics as the cause for the jet formation in this area. We can also suppose that the eddies work in a similar way in other regions. These observations on the role of the eddies also apply to the "prognostic" eddy-resolving experiments. The synergetic behavior demonstrated above for the barotropic mode of the motion appears to be more pronounced in the prognostic experiments, although they are not as realistic as in the diagnostic case since the salinity field is homogeneous. On the other hand, the dynamics itself is more realistic because the temperature field is not fixed and can produce an additional instability into the barotropic mode. We also want to mention that it was found that the role of salinity extends to the production of the countercurrents under the Gulf Stream, the eastern North Atlantic current system, and in some other important regions. In the case of the Gulf Stream, the salinity plays a decisive role in the formation of a deep countercurrent. The total transport in the case of "real" density (taking into account contributions by both T and S) is about 70-80 lo6 m3 s-'. In the case of constant salinity, So = 34O/,, the transport increases to 150 lo6 m3 s-'. This interesting fact does not seem to have been widely discussed in the literature.

152

3 D

0

c

3 co

753

Fig. 6 : Mean total stream function in the diagnostic calculations. (a): Case without eddies (contour interval = 5 lo6 m3 s-'); (b): Eddy-resolving experiment (contour intental = 8 lo6 rn3 s-').

754

81

Step= 2.000

47

Step= 2.000

81

Fig. 7: Instantaneous eddy total stream function fields vicinity of the Gulf Stream.

47

(w'= w - v) taken 6 days apart in the

Coming back to the role of the eddies, we note that the sharpening and the intensifying of the jet is not the only effect of the eddy-mean flow interactions. These interactions modify significantly the whole flow system, and they lead to changes in the interior as well as in the jet area. For example, the jet of the Gulf Stream is pushed southward and offshore by the cyclonic gyre in the northern part of the basin, and this "pushing gyre" is stronger in the presence of the eddies. In Fig. 7, we present several maps of the eddy part of the total fluxes for the eddyresolving experiment in the Gulf Stream area: I$ = - @ (the mean 7 is obtained by averaging y~ over a 200-day interval). The eddies exhibit the characteristic features of nonlinear interacting Rossby waves. There is an intense energy partitioning between eddies as they propagate to the south-west as if they were moving along a line located south of the jet axis. The propagation speed varies significantly along the path of the eddies. The eddies speed up or slow down as if they were under the influence of some external force (we think it is mainly a bottom topography effect on the Rossby waves, which can be transformed into topographic Rossby waves over the slope). On the average, the phase speed is about 10 to 20 cm s-',

w

755

although there are times when this speed can go up to 75 cm s-I. From our point of view, these trains of eddies are responsible for the formation of the tight recirculation zone.

4. CONCLUSIONS The main conclusion from our experiments with this ERGCM is that the eddies should be taken into account in models of the real ocean general circulation, even in diagnostic calculations. It should be kept in mind, also, that these eddies play their role in two ways. First, they are definitely important for the ventilation of the thermocline, as they provide the effective mixing along isopycnals (Cox, 1985). Second, and more importantly, the eddies are responsible for various synergetic processes in the barotropic mode of the motion: intensification or "selfsharpening" (see Monin and Seidov, 1982) of the jet currents, alignement of the deep flows along f the - isolines, formation of the recirculation zones of the jets, and so on. H Our concluding remark, therefore, can be formulated as follows. The synergetic attributes of the local and global thermodynamical processes with scales from several days to several years in the ocean result, most probably, from the eddy-mean flow interactions as a mechanism of the reaction of the ocean structures to instabilities. These eddies can rapidly restructure some hydrophysical fields in order to compensate and to suppress such instabilities and to keep some integrals conserved. 5. REFERENCES Cox, M., 1985. An eddy-resolving numerical model of the ventilated thermocline. J. Phys. Oceanogr., 15: 1312-1324. Cox, M. and Bryan, K., 1984. A numerical model of the ventilated thermocline. J. Phys. Oceanogr., 14: 674-687. Haney, R.L., 1971. Surface thermal boundary condition for ocean circulation models. J. Phys. Oceanogr., 1: 241-248. Holland, W.R., 1978. The role of mesoscale eddies in the general circulation of the ocean: numerical experiments using a wind-driven quasi-geostrophic model. J. Phys. Oceanogr., 8: 363-392. Holland, W.R. and Hirshman, A.D., 1972. A numerical calculation of the circulation of the North Atlantic ocean. J. Phys. Oceanogr., 2: 187-210. Holland, W.R. and Lin, L.B., 1975. On the generation of mesoscale eddies and their conmbution to the oceanic general circulation. J. Phys. Oceanogr., 5: 642-669. Kamenkovich, V.M., 1977. Fundamentals of Ocean Dynamics. Elsevier, Amsterdam. Kamenkovich, V.M., Koshlyakov, M.N. and Monin, AS., 1986. Synoptic Eddies in the Ocean. Reidel, New York. Luyten, J.R., Pedlosky, J. and Stommel, H., 1983. The ventilated thermocline. J. Phys. Oceanogr., 13: 293-309. Marchuk, GI., 1973. Methods of Computational Mathematics, Novosibirsk, Nauka Publ., 352 pp (in Russian). Monin, AS. and Seidov, D.G., 1982. On the generation of jet currents by negative viscosity. Proc. of the USSR Acad. Sci., 268(2): 454-457. Price, J., McKee, T.K., Valdes, J.R., Richardson, P.L. and Armi, L., 1986. SOFAR Float Mediterranean Outflow Experiment. Woods Hole Oceanogr. Inst. Tech. Rept., WHOI-

756

86-31, Woods Hole. Roache, P.J., 1976. Computational Fluid Dynamics. Hermosa, Albuquerque. Robinson, A.R., Harrison, D.E., Mintz, Y. and Semtner, A.J., 1977. Eddies and general circulation of an idealized oceanic gyre. J. Phys. Ocemogr., 7: 182-207. Seidov, D.G., 1980. Synoptic eddies in the ocean: numerical experiments. Izvestiya, Atmospheric and Oceanic Physics, 16(1): 46-55 (in Russian). Seidov, D.G., 1986. Auto-oscillations in the system "large-scale circulation and synoptic Ocean eddies". Izvestiya, Atmospheric and Oceanic Physics, 2 2 679-685 (English edition). Seidov, D.G., 1989. Synergetics of the Ocean circulation. In: J.C.J. Nihoul and B.M. Jamart (Editors), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, Elsevier Oceanography Series, Elsevier, Amsterdam (this volume). Seidov, D.G. and Marushkevich, A.D., 1988. A model of energetics of the Ocean cumnts (brusselator with excitation), Izvestiya of the USSR Acad. Sci., Atmospheric and Oceanic Physics, 24(2): 159-169 (in Russian). Seidov, D.G., Marushkevich, A.D. and Nechaev, D.A., 1985. Modelling of synoptic variability of the large-scale Ocean circulation using the North Atlantic as an example. Oceanology, 26(6): 669-673 (English edition). Seidov, D.G., Marushkevich, A.D. and Nechaev, D.A., 1986. Synoptic eddies and the circulation of the Atlantic Ocean. Ocean Modelling, Nr. 71.

757

ON THE EVOLUTION OF INTENSIVE CYCLONIC-ANTICYCLONIC VORTEX PAIRS G.I. SHAPIRO P.P. Shirshov Institute of Oceanology, USSR Academy of Sciences, 23 Krasikova Street, 117218 Moscow, USSR V.N. KONSMN Institute of Computer Aided Design, USSR Academy of Sciences, 40 Vavilova Street, 117333 Moscow, USSR

ABSTRACT Intensive baroclinic eddies in the ocean are shown to have a new type of nonlinearity which is not taken into account by the traditional quasi-geostrophic equation. The evolution and decay of dipole eddies are investigated using a generalized near-geostrophic model. The time interval after which the traditional approach is no longer valid is estimated analytically. Two sets of numerical calculations are presented. In the first one, the evolution of "modons" is simulated and shown to depend on their intensity and size. It is found that additional nonlinear effects change the trajectory of relatively small eddies rather than their structure. In the second set of experiments, we consider the process of coupling of initially monopole eddies in a shear flow.

1. INTRODUCTION Mesoscale dipole eddies have been observed in many parts of the World Ocean. They are often referred to as "mushroom-like currents", a term introduced by Professor Konstantin Fedorov (Ginsburg and Fedorov, 1984). Highly organized cyclonic-anticyclonic pairs can also be realized in laboratory experiments (Ginsburg et al., 1987; Flierl et al., 1983). It has been shown theoretically in the work of Flierl et al. (1983) that any slowly varying and isolated disturbance in a stratified fluid on a P-plane must have zero net relative angular momentum, so that the dipole is one of the simplest dynamically consistent representation of such a disturbance. Analytical models of dipole eddies based on the solution of the Quasi-Geostrophic Equation (QGE)have been presented by Stem (1975), Larichev and Reznik (1976), and others. In this paper some aspects of the evolution of cyclonic-anticyclonic vortex pairs are investigated mainly by numerical simulation. The present level of theoretical knowledge and the available observational data do not allow to give a proper numerical forecast of the behaviour of real ocean mesoscale eddies for a durable period of time of the order of several years. Thus, theoreticians concentrate their atten tion on the detailed study of certain idealized situations which are useful to understand the physical picture of the dynamical processes and to obtain reasonable estimates of the lifetime and trajectories of eddies.

758

The observational data (Kamenkovich et al., 1982; Belkin et al., 1986; Ivanov et al., 1986) show that, for intense baroclinic eddies, the thickness of the fluid layer confined between two fixed isopycnal surfaces deviates significantly from the equilibrium state. This results in finite amplitude disturbances of the density field and Brunt-Valkala frequency. The question then arises as to whether there are any qualitatively new features in the behaviour of eddies of finite amplitude in comparison with less intensive eddies. Eddies associated with small fluctuations of the density field are often described analytically by the QGE (Pedlosky, 1979; Kamenkovich et al., 1986) which is suitable to investigate the evolution of barotropic and baroclinic synoptic scale eddies. However, as follows from its derivation, this equation is not adequate for the description of disturbances with large amplitude fluctuations of the density field. In this case, one can use the so-called Generalized Near-geostrophic Equation (GNE) derived by Williams (1985) and Shapiro (1986, 1989). The GNE has no restrictions on the amplitude of the isopycnals displacement. This equation filters out high-frequency gravity waves, as does the traditional QGE, and it also includes some additional nonlinear terms. In this paper, we consider intense baroclinic eddies on a P-plane using the reduced gravity approximation. It is assumed that the motion is sufficiently slow that f T* 1 (where f = fo + Py is the Coriolis parameter and T, is the time scale for the eddy evolution) and that the geostrophic balance dominates, so that Ki =

ur f, L*

0.04OC (in the lens centre AT = + 0.28OC)) was 130 km3. This eddy-like "megaplume" lens was apparently created by a brief but massive release of high-temperature hydrothermal fluids. The megaplume was formed in a few days, yet its volume equaled the annual output of 200-2000 high-temperature chimneys. As a result of geostrophic adjustment, the megaplume gained a lens-like shape and anticyclonic vorticity, i.e. a hydrothermal ITE was formed. This very exotic mechanism of ITE generation was predicted by us in 1985 (Belkin et al., 1986). Hundreds to thousands of such hydrothermal ITE may be generated each year along the global ridge system, playing an important role in the heat- and mass-transfer in the deep ocean (Baker et al., 1987).

836

7. CONCLUSIONS To conclude, we wish to outline the main ways to conduct a systematic study of the generation, propagation, evolution and geographic distribution of ITE in the Ocean: 1) Expeditions. Future surveys must cover the probable regions of ITE formation. Such

a study in the Eastern North Atlantic (the area of "Meddies" formation) resulted in the discovery of a large ITE with AT = 4OC and AS = lo/oo(Egorikhin et al., 1987). A few interesting regions are: the Arabian Sea, the Gulf of Aden, the Tasman Sea and the Black Sea. In the Arctic Ocean, we need to investigate why ITE are concentrated in the Canadian Basin. Is this the result of the closed circulation of the Canadian Basin or of a short lifetime of the ITE ? For the study of ITE generation, the Beaufort and Chukchi Sea may be chosen.

2) Detailed analysis of MODE and POLYMODE data. The results of the statistical analysis of isopycnal S,02-anomalies, based on the LDE data, have shown that 31 ITE were found during two months of observations in a 200 km circle (Lindstrom and Taft, 1986). 3) Analysis of historical hydrological data. It is necessary to study the distribution of isopycnal tracers (salinity, etc) anomalies, first of all in the "preferred' density ranges for ITE locations (e.g., in the Sargasso Sea, such a range is a8 = 26.7-27.8).

8. REFERENCES Armi, L. and Stommel, H., 1983. Four views of a portion of the Nonh Atlantic Subtropical Gyre. J. Phys. Oceanogr., 13: 828-857. Armi, L. and Zenk, W., 1984. Large lenses of highly saline Mediterranean water. J. Phys. Oceanogr., 14: 1560-1576. Armi,L., Hebert, D., Oakey, N., Price, J.F., Richardson, P.L., Rossby, H.T. and Ruddick, B., 1988. Two years in the life of a Mediterranean salt lens. J. Phys. Oceanogr., 18, in press. Bagriantsev, N.V., 1985. Thermohaline structure and water circulation in the Weddell Gyre. Ph.D Thesis, Arctic and Antarctic Resemh Institute, Leningrad, 154 pp. (in Russian). Bagriantsev, N.V. and Sarukhanyan, E.I., 1984. Weddell Polynya as a consequence of hydrophysical processes in the Weddell circulation. Dokl. Acad. Sci. USSR, 276: 1238-1242 (in Russian). Baker, E.T., Massoth, G.J. and Feely, R.S., 1987. Cataclysmic hydrothermal venting on the Juan de Fuca Ridge. Nature, 329: 149-151. Bannov-Baykov, Yu.L. and Bulatov, L.V., 1985. On disturbances of currents in deep layers of the Arctic Basin. Trudy AANII, 389: 52-54 (in Russian). Barton, E.D. and Hughes, P., 1982. Isolated homogeneous layers over the continental slope off northwest Africa. Deep-sea Res., 29: 135-143. Beliakov, L.N., 1972. Triggering mechanism of deep episodic currents in the Arctic Basin. Problemy Arktiki i Antarktiki, 39: 25-32 (in Russian). Beliakov, L.N. and Volkov, V.A., 1980. On the spatial distribution of mesoscale subsurface currents in the Amerasian Subbasin of the Arctic Ocean. Dokl. Acad. Sci. USSR,

837

254: 752-754 (in Russian). Beliakov, L.N. and Volkov, V.A., 1985. Mesoscale subsurface currents in the Arctic Basin. Trudy AANII, 389: 46-51 (in Russian). Belkin, I.M., 1986. Typical profiles. In: The POLYMODE Atlas, WHOI, Woods Hole, Ma., pp. 175, 183-184. Belkin, LM. and Kostianoy, A.G., 1988. Lenses of Mediterranean Water in the North Atlantic. In: V.G. Kort (Editor), Hydrophysical Investigations in the "Mesopolygon" Experiment, Nauka, Moscow, pp. 110-123 (in Russian). Belkin, I.M. and Mikhailitchenko, Yu.G., 1986. Thermohaline structure of the frontal zone of the Northwest Pacific Ocean at 160'E. Okeanologiya, 26: 70-72 (in Russian). Belkin, I.M., Emelianov, M.V., Kostianoy, A.G. and Fedorov, K.N., 1986. Thermohaline structure of intermediate waters of the Ocean and intrathemocline eddies. In: K.N. Fedorov (Editor), Intrathermocline Eddies in the Ocean, P.P. Shirshov Inst. of Oceanology, Moscow, pp. 8-34 (in Russian). Berestov, A.L., Egorikhin, V.D., Ivanov, Yu.A., Kort, V.G., Koshlyakov, M.N., Lukashev, Yu.F., Monin, AS., Morozov, E.G., Moroshkin, K.V., Ovchinnikov, LM., Paka, V.T., Tsybaneva, T.B., Shadrin, I.F., Shapovalov, S.M., Tscherbinin, A.D. and Yaremchuk, M.I., 1986. Thermohaline, hydrochemical, and dynamical characteristics of an intrusive lens of Mediterranean water as based on the data collected during "Mesopolygon-85" expedition in the Tropical Atlantic. In: K.N. Fedorov (Editor), Intrathermocline Eddies in the Ocean, P.P. Shirshov Inst. of Oceanology, Moscow, pp. 35-49 (in Russian). Bogdanov, K.T., Iliychev, V.I., Lobanov, V.B. and Medjitov, R.D., 1985. Investigation of anticyclonic eddy in the northwest part of the Pacific Ocean. DOH. Acad. Sci. USSR, 281: 1210-1213 (in Russian). Bogorodskii, V.V., Baranov, G.I. and Gusev, A.V., 1980. Space-time variability of hydrophysical fields of the Arctic Basin. Dokl. Acad. Sci. USSR, 253: 967-970 (in Russian). Boland, F.M., 1971. Temperature-salinity anomalies at depths between 200 m and 800 m in the Tasman Sea. AJMFR, 22: 55-62. Bourke, R.H., 1983. Currents, fronts and fine structure in the marginal ice zone of the Chukchi Sea. Polar Record, 21: 569-576. Brundage, W.L. and Dugan, J.P., 1986. Observations of an anticyclonic eddy of 18°C water in the Sargasso Sea. J. Phys. Oceanogr., 16: 717-727. Bryden, H.L. and Stommel, H.M., 1984. Limiting processes that determine basic features of the circulation in the Mediterranean Sea. Oceanol. Acta, 7: 289-296. Bubnov, V.A., 1971. Structure and dynamics of the Mediterranean waters in the Atlantic Ocean. Okeanol. issled., 22: 220-278 (in Russian). Clarke, R.A., 1984. Transport through the Cape Farewell-Flemish Cap section. Rapp. et Roc.-Verb. Cons. Int. Expl. Mer, 185: 120-130. Coachman, L.K. and Newton, J.L., 1972. Water and ice motion in the Beaufort Sea, spring 1970. AIDJEX Bull. 12, Univ. of Washington, Seattle, pp. 61-91. Cresswell, G.R., 1982. The coalescence of two East Australian Current warm-core eddies. Science, 215: 161-164. D'Asaro, E.A., 1988. Observations of small eddies in the Beaufort Sea. J. Geophys. Res., 93,C6: 6669-6684. Dugan, J.P., Mied, R.R., Mignerey, P.C. and Schuetz, A.F., 1982. Compact, intrathermocline eddies in the Sargasso Sea. J. Geophys. Res., 87: 385-393. Ebbesmeyer, C.C., Taft, B.A., McWilliams, J.C.,Shen, C.Y., Riser, S.C., Rossby, H.T., Biscaye, P.E. and Ostlund, H.G., 1986. Detection, structure and origin of extreme anomalies in a Western Atlantic Oceanographic section. J. Phys. Oceanogr., 16: 591612. Egorikhin, V.D., Ivanov, Yu.A., Kort, V.G., Koshlyakov, M.N., Lukashov, Yu.F., Morozov, E.G., Ovchinnikov, I.M., Paka, V.T., Tsybanjova, T.B., Shadrin, I.F. and

838

Shapovalov, S.M., 1987. The Mediterranean Water intrathermocline lens in the tropical part of the North Atlantic. Okeanologiya, 27: 165-175 (in Russian). Elken, Yu., 1987. Lens-like eddies in the Baltic Sea and Atlantic Ocean. Proceedings of 3rd Meeting of Soviet Oceanographers, Gidrometeoizdat, Leningrad, pp. 191-192 (in Russian). Elliot, B.A. and Sanford, T.B., 1986a. The subthermocline lens D1. Part I: Description of water properties and velocity profiles. J. Phys. Oceanogr., 16 : 532-548. Elliot, B.A. and Sanford, T.B., 1986b. The subthermocline lens D1. Part II: Kinematics and Dynamics. J. Phys. Oceanogr., 16: 549-561. Emelianov, M.V. and Kostianoy, A.G., 1989. Intrathermocline eddies in the Atlantic region of the Antarctic, Antarktika, 28 (in Press) (in Russian). Emelianov, M.V. and Fedorov, K.N., 1985. Structure and transformation of intermediate waters of the Mediterranean Sea and Atlantic Ocean. Okeanologiya, 25: 206-214 (in Russian). Emelianov, M.V. and Shapiro, G.I., 1987. Mesoscale structure of the frontal zone and the subsurface vortical lens in the Atlantic sector of the Antarctic. Proceedings of 3rd Meeting of Soviet Oceanographers, Gidrometeoizdat, Leningrad, pp. 76-77 (in Russian). Fedorov, K.N., 1976. The Thermohaline Fine Structure of the Ocean. Gidrometeoizdat, Leningrad, 184 pp. (in Russian), translated by Pergamon Press, Oxford, 1978, 170 pp. Fedorov, K.N. and Belkin, I.M., 1984. Some peculiarities of the frontal eddies of the East Australian Current. Okeanologiya, 24: 204-212 (in Russian). Fedorov, K.N., Ginsburg, A.I. and Zatsepin, A.G., 1978. On the thermohaline disturbances which accompany eddies in the Sargasso Sea. In: K.N. Fedorov (Editor), Studies of the Variability of Physical Processes in the Ocean, P.P. Shirshov Inst. of Oceanology, Moscow, pp. 8-34 (in Russian). Fedorov, K.N., Ginsburg, A.I., Zatsepin, A.G. and Shapovalov, S.M., 1979. The peculiarities of spatial-temporal variability of T,S characteristics in the central part of the POLYMODE area. Okeanol. issled., 30: 51-62 (in Russian). Galt, J.A., 1967. Current measurements in the Canada Basin of the Arctic Ocean, summer 1965. Tech. Rep. 184, Univ. of Washington, Seattle, 17 pp. Gilmour, A.E., and Cole, A.G., 1979. The Subtropical Convergence east of New Zealand. N.Z.J. Mar. Freshwat. Res., 13: 553-557. Godfrey, J.S., Jones, I.S.F., Maxwell, J.G.H. and Scott, D.D., 1980. On the winter cascade from Bass Strait into the Tasman Sea. Aust. J. Mar. Freshwat. Res., 31: 275-286. Golmen, L.G. and Hackett, B., 1986. Sea Sour cruises in the Norwegian Coastal Current, winter and spring 1985. Rep. Geoph. Inst. Univ. Bergen, 64,38 pp. Gordon, A.L., 1983. The Weddell Gyre. Ant. J. US, 18: 135-136. Gordon, A.L. and Greengrove, C.L., 1986. Abyssal eddy in the southwest Atlantic. Deep-sea Res., 33: 839-847. Gordon, A.L. and Huber, B.A., 1984. Thermohaline stratification below the Southern Ocean sea ice. J. Geophys. Res., 89: 641-648. GrUndlingh, M.L., 1981. On the observation of a solitary event in the Mediterranean outflow west of Gibraltar. "Meteor" Forsch.-Ergebnisse, 23: 15-46. GrUndlingh, M.L., 1985. Occurrence of Red Sea water in the south-westem Indian Ocean, 1981. J. Phys. Oceanogr., 15: 207-212. Hallock, Z.R., Teague, W.J., and Broome, R.D., 1981. A deep, thick, isopycnal layer within an anticyclonic eddy. J. Phys. Oceanogr., 11: 1674-1677. Hamon, B.V., 1967. Medium-scale temperature and salinity structure in the upper 1500 m in the Indian Ocean. Deep-sea Res., 14: 169-182. Hart, J.E. and Killworth, P.D., 1976. On open ocean baroclinic instability in the Arctic. Deep-sea Res., 23: 637-645.

839

Hayes, S.P., 1975. The temperature and salinity fine suucture of the Mediterranean water in the western Atlantic. Deep-sea Res., 22: 1, 11. Heath, R.A., 1976. Models of the diffusive-advective balance at the Subtropical Convergence. Deep-sea Res., 23: 1153-1164. Hebert, D., 1988. A Mediterranean Salt Lens. Ph.D Thesis, Dalhousie University, Halifax, NS, 187 pp. Howe, M.R., Abdullah, M.I. and Deetae, S., 1974. An interpretation of the double T,S maxima in the Mediterranean outflow using chemical tracers. J. Mar. Res., 32: 377386. Hunkins, K., 1974. Subsurface eddies in the Arctic Ocean. Deep-sea Res., 21: 10171033. Hunkins, K., 1980. Review of the AIDJEX Oceanographic Program. In: R.S. Pritchard (Editor), Sea Ice Processes and Models, Univ. of Washington, Seattle, pp. 34-45. Johannessen, J.A., Johannessen, O.M., Svendsen, E., Shuchman, R., Manley, T., Campbell, W.J., Josberger, E.G., Sandven, S., Gascard, J.C., OIaussen, T., Davidson, K. and Van Leer, J., 1987. Mesascale eddies in the Fram Strait marginal ice zone during the 1983 and 1984 Marginal Ice Zone Experiment. J. Geophys. Res., 92: 6754-6772. Karlin, L.N., Klyuikov, E.Yu. and Kut’ko, V.P, 1988. Small-scale structure of hydrophysical fields of the upper layer of the ocean. Gidrometeoizdat, Moscow, 164 pp. (in Russian). KBse, R.H. and Zenk, W., 1987. Reconstructed Mediterranean salt lens trajectories. J. Phys. Oceanogr., 17: 158-163. Katz, E.J., 1970. Diffusion of the core of Mediterranean water above the Mid-Atlantic Ridge crest. Deep-sea Res., 17: 611-625. Khrapchenkov, F.F., 1987. The study of Eastern Kamchatka coast eddies. Proceedings of 3rd Meeting of Soviet Oceanographers, Gidrometeoizdat, Leningrad, pp. 181-182 (in Russian). Kinder, T.H. and Bryden, H.L., 1987. The 1985-1986 Gibraltar Experiment: Data collection and preliminary results. EOS, 68: 786, 787, 793-795. Klyausov, A.V. and Lanin, V.I., 1987. Position and some peculiarities of the near-shelf frontal zone structure in the Sodrujestvo Sea. Okeanologiya, 27: 384-390 (in Russian). Kostianoy, A.G. and Rodionov, V.B., 1986a. Coastal upwelling zones - a source of intrathermocline eddy formation. In: K.N. Fedorov (Editor), Intrathermocline Eddies in the Ocean, P.P. Shirshov Inst. of Oceanology, Moscow, pp. 50-52 (in Russian). Kostianoy, A.G. and Rodionov, V.B., 1986b. On the formation of intrathermocline eddies in the Canary upwelling region. Okeanologiya, 26: 892-895 (in Russian). Latun, V.S., 1980. Thermohaline and dynamic inhomogeneities in the western region of the Lomonosov current. In: Structure, Kinematics and Dynamics of Synoptic Eddies, Marine Hydrophysical Institute, Sevastopol, pp. 20-30 (in Russian). Leetmaa, A., 1976. Recent observations of eddies south-west of Bermuda. POLYMODE News, 10: 1. Levitus, S., 1982. Climatological Atlas of the World Ocean. NOAA Prof. Pap. No. 13, U.S. Govt. Printing Office, Rockville, Md., 173 pp. Lindstrom, E.J., Ebbesmeyer, C.C. and Owens, W.B., 1986. Structure and origin of a small cyclonic eddy observed during the POLYMODE Local Dynamics Experiment. J. Phys. Oceanogr., 16: 562-570. Lindstrom, E.J. and Taft, B.A., 1986. Small water property transporting eddies: statistical outliers in the hydrographic data of the POLYMODE Local Dynamics Experiment. J. Phys. Oceanogr., 16: 613-631. Manley, T. and Hunkins, K., 1980. Oceanographic measurements at the FRAM I ice station. EOS, 61: 278. Manley, T. and Hunkins, K., 1985. Mesoscale eddies of the Arctic Ocean. J. Geophys. Res.. 90: 491 1-4930.

840

Manley, T., Codispoti, L.A., Hunkins, K., Jackson, H.R., Jones, E.P., Lee, V., Moore, S., Morison, J., Packard, T.T. and Wadhams, P., 1982. The FRAM rI1 Expedition. EOS, 63: 627-636. McDowell, S.E., 1977. Discovery of a "Meddy" in the Hatteras Abyssal Plain. POLYMODE News, 36: 5-6. McDowell, S.E., 1986. On the origin of eddies discovered during the POLYMODE Local Dynamics Experiment. J. Phys. Oceanogr., 16: 632-652. McDowell, S.E. and Rossby, H.T.. 1978. Meditteranean water: An intense mesoscale eddy off the Bahamas. Science, 202: 1085-1087. McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Reviews of Geophysics, 23: 165-182. Newton, J.L., Aagaard, K. and Coachman, L.K., 1974. Barwlinic eddies in the Arctic Ocean. Deep-sea Res., 21: 707-719. Nilsson, C.S., Andrews, J.C. and Scully-Power, P., 1977. Observations of eddy formation off east Australia. J. Phys. Oceanogr., 7: 659-669. Nilsson, C.S. and Cresswell, G.R., 1981. The formation and evolution of East Australian Current warm-core eddies. Prog. Oceanogr., 9 133-183. Paquette, R.G., Bourke, R.H., Newton, J.F. and Perdue, W.F., 1985. The East Greenland Polar Front in autumn. J. Geophys. Res., 90: 4866-4882. Peters, H., 1977. Meanders and eddies off Northwest Africa. POLYMODE News, 25: 1, 4. Piip, A.T., 1969. Large cells of Mediterranean Water in the Madeira-Canaries region. EOS, 50: 193. Plakhin, E.A. and Smirnov, V.G., 1982. The Mediterranean Water in the Central Atlantic Ocean. Okeanologiya, 22: 569-572 (in Russian). Pounder, E.R., 1980. Physical oceanography in the central Arctic. EOS, 61: 278. Riser, S.C., Owens, W.B., Rossby, H.T. and Ebbesmeyer, C.C., 1986. The structure, dynamics, and origin of a small-scale lens of water in the western North Atlantic thermocline. J. Phys. Oceanogr., 16: 572-590. Rossby, H.T., Riser, S.C. and McDowell, S.E., 1981. On the origin and structure of a small-scale lens of water observed in the North Atlantic thermocline. Univ. of Rhode Island, Graduate School of Oceanography, Kingston, R.I. (preprint). Shirshov, P.P., 1938. Oceanological observations. Dokl. Acad. Sci. USSR, 8: 569-580. Siedler, G., 1968. Die Haufligkeits verteilung von wasserarten im ausstrombereich von meeresstrassen. Kieler Meeresforschungen, 24: 59-65. Simpson, J.J., Koblinsky, C.J., Haury, L.R. and Dickey, T.D., 1984. An offshore eddy in the California Current system. Prog. Oceanogr., 13: 1-111. Somov, M.M., 1955. Observational data of the scientific research drifting station of 1950-1951, v.1, Morskoi Transport, Leningrad (Trans. by Amer. Met. Soc. for A.F. Cambridge Research Centre, ASTIA, No. AD 117133). Stanton, B.R., 1983. Low frequency variability in the Mediterranean Outflow west of Gibraltar. Deep-sea Res., 30: 743-761. Swallow, J.C., 1969. A deep eddy off Cape St. Vincent. Deep-sea Res., 16: 285-296. Thorpe, S.A., 1976. Variability of the Mediterranean undercurrent in the Gulf of Cadiz. Deep-sea Res., 23: 71 1-727. Tomczak, M., 1981. Bass Strait Water intrusions in the Tasman Sea and mean temperature-salinity curves. Aust. J. Mar. Freshwat. Res., 32: 699-708. Vinogradov, A.S. and Pavelson, Yu. H., 1980. Fine water stratification of the Sargasso Sea under the main thermocline. Okeanol. issled., 31: 56-63 (in Russian). Wadhams, P., Gill, A.E. and Linden, P.F., 1979. Transects by submarine of the East Greenland Polar Front. Deep-sea Res., 26: 1311-1327.

841

Wadhams, P. and Squire, V.A., 1983. An ice-water vortex at the edge of the East Greenland Current. J. Geophys. Res., 88: 2770-2780. Zantopp, R. and Leaman, K., 1982. Gulf of Cadiz Water observed in the thermocline eddy in the Western North Atlantic. J. Geophys. Res., 87: 1927-1934. Zenk, W., 1975. On the Mediterranean outfiow west of Gibraltar. Meteor Forsch.Ergebnisse, 16: 23-24. Zubin, A.B. and Ozmidov, R.V., 1987. A lens of Mediterranean Water near the Amper and Josephine underwater mountains. Dokl. Acad. Sci. USSR, 292: 716-719 (in Russian).


synoptic coherent structures in geophysical turbulence: proceedings of the 20th International Liege Colloquium on Ocean Hydrodynamics

Bottom turbulence: proceedings of the 8th International Liege Colloquium on Ocean Hydrodynamics

Small-scale turbulence and mixing in the ocean: Proceedings of the 19th International Liege colloquium on ocean hydrodynamics

Ocean Hydrodynamics: Ecohydrodynamics 12th: International Colloquium Proceedings

Hydrodynamics of the Equatorial Ocean: Proceedings of the 14th International Liege Colloquium on Ocean Hydrodynamics (Elsevier Oceanography Series)

Marine Forecasting: Predictability and Modelling in Ocean Hydrodynamics : Proceedings of the 10th International Liège Colloquium on Ocean Hydrodynamics

Ocean Hydrodynamics: Hydrodynamics of Semi-enclosed Seas 13th: International Colloquium Proceedings (Elsevier oceanography series)

Ocean Hydrodynamics: Remote Sensing of Shelf Sea Hydrodynamics 15th: International Colloquium Proceedings

Ocean Hydrodynamics: Hydrodynamics of Estuaries and Fjords 9th: International Colloquium Proceedings (Elsevier oceanography series ; 23)

Frontiers in Turbulence and Coherent Structures: Proceedings of the Cosnet Csiro Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and ... Lecture Notes in Complex Systems)

Frontiers in Turbulence and Coherent Structures: Proceedings of the Cosnet Csiro Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and ... Lecture Notes in Complex Systems)

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Three-dimensional models of marine and estuarine dynamics: [proceedings of the 18th International LieРњР‚ge Colloquium on ocean hydrodynamics, 1986]

Hydrodynamics Around Cylindrical Structures (Advanced Series on Ocean Engineering)

Combinatorial Mathematics: International Colloquium Proceedings

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Coherent Structures in Complex Systems

Progress in Turbulence III: Proceedings of the iTi Conference in Turbulence 2008 (Springer Proceedings in Physics)

Differential Geometry: Proceedings of the VIII International Colloquium

Logic Colloquium 1984: Proceedings

IUTAM Symposium on Elementary Vortices and Coherent Structures: Significance in Turbulence Dynamics: Proceedings of the IUTAM Symposium held at Kyoto International ... 2004 (Fluid Mechanics and Its Applications)

An Introduction to Ocean Turbulence

An Introduction to Ocean Turbulence

Proceedings of the Thirteenth Symposium on Naval Hydrodynamics

Group Theoretical Methods in Physics: Proceedings of the XVIII International Colloquium Held at Moscow, Ussr, 4-9 June 1990 (International Colloquium on ... Theoretical Methods in Physics: Proceedings)

Time and matter: Proceedings of the international colloquium on the science of time

Logic Colloquium '88: Proceedings

Logic Colloquium 1976: Proceedings

Logic: Colloquium Proceedings, 1977

Logic Colloquium 1982: Proceedings

synoptic coherent structures in geophysical turbulence: proceedings of the 20th International Liege Colloquium on Ocean Hydrodynamics

Bottom turbulence: proceedings of the 8th International Liege Colloquium on Ocean Hydrodynamics

Small-scale turbulence and mixing in the ocean: Proceedings of the 19th International Liege colloquium on ocean hydrodynamics

Ocean Hydrodynamics: Ecohydrodynamics 12th: International Colloquium Proceedings

Hydrodynamics of the Equatorial Ocean: Proceedings of the 14th International Liege Colloquium on Ocean Hydrodynamics (Elsevier Oceanography Series)

Marine Forecasting: Predictability and Modelling in Ocean Hydrodynamics : Proceedings of the 10th International Liège Colloquium on Ocean Hydrodynamics

Ocean Hydrodynamics: Hydrodynamics of Semi-enclosed Seas 13th: International Colloquium Proceedings (Elsevier oceanography series)

Ocean Hydrodynamics: Remote Sensing of Shelf Sea Hydrodynamics 15th: International Colloquium Proceedings

Ocean Hydrodynamics: Hydrodynamics of Estuaries and Fjords 9th: International Colloquium Proceedings (Elsevier oceanography series ; 23)

Frontiers in Turbulence and Coherent Structures: Proceedings of the Cosnet Csiro Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and ... Lecture Notes in Complex Systems)

Frontiers in Turbulence and Coherent Structures: Proceedings of the Cosnet Csiro Workshop on Turbulence and Coherent Structures in Fluids, Plasmas and ... Lecture Notes in Complex Systems)

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Three-dimensional models of marine and estuarine dynamics: [proceedings of the 18th International LieРњР‚ge Colloquium on ocean hydrodynamics, 1986]

Hydrodynamics Around Cylindrical Structures (Advanced Series on Ocean Engineering)

Combinatorial Mathematics: International Colloquium Proceedings

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Proceedings of the 20th International Conference on Fluidized Bed Combustion

Coherent Structures in Complex Systems

Progress in Turbulence III: Proceedings of the iTi Conference in Turbulence 2008 (Springer Proceedings in Physics)

Differential Geometry: Proceedings of the VIII International Colloquium

Logic Colloquium 1984: Proceedings

IUTAM Symposium on Elementary Vortices and Coherent Structures: Significance in Turbulence Dynamics: Proceedings of the IUTAM Symposium held at Kyoto International ... 2004 (Fluid Mechanics and Its Applications)

An Introduction to Ocean Turbulence

An Introduction to Ocean Turbulence

Proceedings of the Thirteenth Symposium on Naval Hydrodynamics

Group Theoretical Methods in Physics: Proceedings of the XVIII International Colloquium Held at Moscow, Ussr, 4-9 June 1990 (International Colloquium on ... Theoretical Methods in Physics: Proceedings)

Time and matter: Proceedings of the international colloquium on the science of time

Logic Colloquium '88: Proceedings

Logic Colloquium 1976: Proceedings

Logic: Colloquium Proceedings, 1977

Logic Colloquium 1982: Proceedings

Recommend Documents