Thumb - download.e-bookshelf.de · Christie,Charles Fisher, Sang-Mook Lee, and Sharon Givens (Eds.) 167 Recurrent Magnetic Storms: Corotating Solar Wind Streams Bruce Tsurutani, Robert

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Geophysical Monograph Series

IncludingIUGG Volumes

Maurice Ewing VolumesMineral Physics Volumes

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page ii mdash 2


164 Archean Geodynamics and Environments Keith BennJean-Claude Mareschal and Kent C Condie (Eds)

165 Solar Eruptions and Energetic ParticlesNatchimuthukonar Gopalswamy Richard Mewaldt andJarmo Torsti (Eds)

166 Back-Arc Spreading Systems Geological BiologicalChemical and Physical Interactions David MChristieCharles Fisher Sang-Mook Lee and SharonGivens (Eds)

167 Recurrent Magnetic Storms Corotating Solar WindStreams Bruce Tsurutani Robert McPherron WalterGonzalez Gang Lu Joseacute H A Sobral andNatchimuthukonar Gopalswamy (Eds)

168 Earthrsquos Deep Water Cycle Steven D Jacobsen andSuzan van der Lee (Eds)

169 Magnetospheric ULF Waves Synthesis and NewDirections Kazue Takahashi Peter J Chi Richard EDenton and Robert L Lysak (Eds)

170 Earthquakes Radiated Energy and the Physics ofFaulting Rachel Abercrombie Art McGarr HirooKanamori and Giulio Di Toro (Eds)

171 Subsurface Hydrology Data Integration for Propertiesand Processes David W Hyndman Frederick DDay-Lewis and Kamini Singha (Eds)

172 Volcanism and Subduction The Kamchatka Region JohnEichelberger Evgenii Gordeev Minoru Kasahara PavelIzbekov and Johnathan Lees (Eds)

173 Ocean Circulation Mechanisms and ImpactsmdashPast andFuture Changes of Meridional Overturning AndreasSchmittner John C H Chiang and Sidney R Hemming(Eds)

174 Post-Perovskite The Last Mantle Phase Transition KeiHirose John Brodholt Thorne Lay and David Yuen(Eds)

175 A Continental Plate Boundary Tectonics at SouthIsland New Zealand David Okaya Tim Stem and FredDavey (Eds)

176 Exploring Venus as a Terrestrial Planet Larry WEsposito Ellen R Stofan and Thomas E Cravens (Eds)

177 Ocean Modeling in an Eddying Regime Matthew Hechtand Hiroyasu Hasumi (Eds)

178 Magma to Microbe Modeling Hydrothermal Processesat Oceanic Spreading Centers Robert P Lowell Jeffrey SSeewald Anna Metaxas and Michael R Perfit (Eds)

179 Active Tectonics and Seismic Potential of Alaska JeffreyT Freymueller Peter J Haeussler Robert L Wesson andGoumlran Ekstroumlm (Eds)

180 Arctic Sea Ice Decline Observations ProjectionsMechanisms and Implications Eric T DeWeaver CeciliaM Bitz and L-Bruno Tremblay (Eds)

181 Midlatitude Ionospheric Dynamics and DisturbancesPaul M Kintner Jr Anthea J Coster Tim Fuller-RowellAnthony J Mannucci Michael Mendillo and RoderickHeelis (Eds)

182 The Stromboli Volcano An Integrated Study of the2002-2003 Eruption Sonia Calvari SalvatoreInguaggiato Giuseppe Puglisi Maurizio Ripepe andMauro Rosi (Eds)

183 Carbon Sequestration and Its Role in the GlobalCarbon Cycle Brian J McPherson and Eric T Sundquist(Eds)

184 Carbon Cycling in Northern Peatlands Andrew J BairdLisa R Belyea Xavier Comas A S Reeve and Lee DSlater (Eds)

185 Indian Ocean Biogeochemical Processesand Ecological Variability Jerry D Wiggert Raleigh RHood S Wajih A Naqvi Kenneth H Brink and SharonL Smith (Eds)

186 Amazonia and Global Change Michael KellerMercedes Bustamante John Gash and Pedro Silva Dias(Eds)

187 Surface Ocean-Lower Atmosphere Processes Corinne LeQuegraveregrave and Eric S Saltzman (Eds)

188 Diversity of Hydrothermal Systems on Slow SpreadingOcean Ridges Peter A Rona Colin W Devey JeacuterocircmeDyment and Bramley J Murton (Eds)

189 Climate Dynamics Why Does Climate Vary De-ZhengSun and Frank Bryan (Eds)

190 The Stratosphere Dynamics Transport and ChemistryL M Polvani A H Sobel and D W Waugh (Eds)

191 Rainfall State of the Science Firat Y Testik andMekonnen Gebremichael (Eds)

192 Antarctic Subglacial Aquatic Environments Martin JSiegert Mahlon C Kennicut II and Robert ABindschadler

193 Abrupt Climate Change Mechanisms Patterns andImpacts Harunur Rashid Leonid Polyak and EllenMosley-Thompson (Eds)

194 Stream Restoration in Dynamic Fluvial SystemsScientific Approaches Analyses and Tools AndrewSimon Sean J Bennett and Janine M Castro (Eds)

195 Monitoring and Modeling the Deepwater Horizon OilSpill A Record-Breaking Enterprise Yonggang Liu AmyMacFadyen Zhen-Gang Ji and Robert H Weisberg(Eds)

196 Extreme Events and Natural Hazards The ComplexityPerspective A Surjalal Sharma Armin Bunde Vijay PDimri and Daniel N Baker (Eds)

197 Auroral Phenomenology and MagnetosphericProcesses Earth and Other Planets Andreas Keiling EricDonovan Fran Bagenal and Tomas Karlsson (Eds)

198 Climates Landscapes and Civilizations Liviu GiosanDorian Q Fuller Kathleen Nicoll Rowan K Flad andPeter D Clift (Eds)

199 Dynamics of the Earthrsquos Radiation Belts and InnerMagnetosphere Danny Summers Ian R Mann DanielN Baker Michael Schulz (Eds)

200 Lagrangian Modeling of the Atmosphere John Lin (Ed)201 Modeling the Ionosphere-Thermosphere Jospeh D

Huba Robert W Schunk and George V Khazanov(Eds)

202 The Mediterranean Sea Temporal Variability andSpatial Patterns Gian Luca Eusebi Borzelli MiroslavGaCiC Piero Lionello and Paola Malanotte-Rizzoli(Eds)

203 Future Earth - Advancing Civic Understanding of theAnthropocene Diana Dalbotten Gillian Roehrig andPatrick Hamilton (Eds)

204 The Galapagos as a Natural Laboratory for the EarthSciences Karen S Harpp Eric Mittelstaedt David WGraham Noeacutemi drsquoOzouville (Eds)

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page iii mdash 3

Geophysical Monograph 205

Modeling Atmosphericand Oceanic Flows

Insights from Laboratory Experimentsand Numerical Simulations

Thomas von LarcherPaul D Williams

Editors

This Work is a co-publication between the American Geophysical Union and John Wiley amp Sons Inc

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page iv mdash 4


Published under the aegis of the AGU Publications Committee

Brooks Hanson Director of PublicationsRobert van der Hilst Chair Publications CommitteeRichard Blakely Vice Chair Publications Committee

copy 2015 by the American Geophysical Union 2000 Florida Avenue NW Washington DC 20009For details about the American Geophysical Union see wwwaguorg

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any meanselectronic mechanical photocopying recording scanning or otherwise except as permitted under Section 107 or 108 of the 1976United States Copyright Act without either the prior written permission of the Publisher or authorization through payment ofthe appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood Drive Danvers MA 01923 (978) 750-8400fax (978) 750-4470 or on the web at wwwcopyrightcom Requests to the Publisher for permission should be addressed to thePermissions Department John Wiley amp Sons Inc 111 River Street Hoboken NJ 07030 (201) 748-6011 fax (201) 748-6008 oronline at httpwwwwileycomgopermission

Limit of LiabilityDisclaimer of Warranty While the publisher and author have used their best efforts in preparing this book theymake no representations or warranties with respect to the accuracy or completeness of the contents of this book and specificallydisclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended bysales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situationYou should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit orany other commercial damages including but not limited to special incidental consequential or other damages

For general information on our other products and services or for technical support please contact our Customer CareDepartment within the United States at (800) 762-2974 outside the United States at (317) 572-3993 or fax (317) 572-4002

Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available inelectronic formats For more information about Wiley products visit our web site at wwwwileycom

Library of Congress Cataloging-in-Publication Data is available

ISBN 978-1-118-85593-5

Cover image The main image shows the sunset over St Brides Bay viewed from Broad Haven Pembrokeshire Wales on 4 August2012 as photographed by Paul D Williams The inset image shows a rotating fluid surface visualized by optical altimetryDifferent colors correspond to different values of the two components of the horizontal gradient of the surface elevation The flowis induced by a heated disk on a polar beta plane Baroclinic instability produces multiple meanders and eddies which are advectedby a zonal current that is flowing anti-clockwise and is driven by the sink in the top-right sector of the image This flow is alaboratory model of flows occurring in the atmospheres of rotating planets such as Earth and Saturn More details on theexperiment are given in Chapter 5 The image was produced at Memorial University of Newfoundland Canada by Y DAfanasyev and Y Sui

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page v mdash 5

CONTENTS

Contributors vii

Preface xi

Acknowledgments xiii

Introduction Simulations of Natural Flows in the Laboratory and on a ComputerPaul F Linden 1

Section I Baroclinic-Driven Flows

1 General Circulation of Planetary Atmospheres Insights from Rotating Annulus and Related ExperimentsPeter L Read Edgar P Peacuterez Irene M Moroz and Roland M B Young 9

2 Primary Flow Transitions in the Baroclinic Annulus Prandtl Number EffectsGregory M Lewis Nicolas Peacuterinet and Lennaert van Veen 45

3 Amplitude Vacillation in Baroclinic FlowsWolf-Gerrit Fruumlh 61

Section II Balanced and Unbalanced Flows

4 Rotation Effects on Wall-Bounded Flows Some Laboratory ExperimentsP Henrik Alfredsson and Rebecca J Lingwood 85

5 Altimetry in a GFD Laboratory and Flows on the Polar β-PlaneYakov D Afanasyev 101

6 Instabilities of Shallow-Water Flows with Vertical Shear in the Rotating AnnulusJonathan Gula and Vladimir Zeitlin 119

7 Laboratory Experiments on Flows Over Bottom TopographyLuis Zavala Sansoacuten and Gert-Jan van Heijst 139

8 Direct Numerical Simulations of Laboratory-Scale Stratified TurbulenceMichael L Waite 159

Section III Atmospheric Flows

9 Numerical Simulation (DNS LES) of Geophysical Laboratory Experiments Quasi-Biennial Oscillation(QBO) Analogue and Simulations Toward MaddenndashJulian Oscillation (MJO) AnalogueNils P Wedi 179

10 Internal Waves in Laboratory ExperimentsBruce Sutherland Thierry Dauxois and Thomas Peacock 193

11 Frontal Instabilities at DensityndashShear Interfaces in Rotating Two-Layer Stratified FluidsHeacutelegravene Scolan Roberto Verzicco and Jan-Bert Floacuter 213

v

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vi CONTENTS

Section IV Oceanic Flows

12 Large-Amplitude Coastal Shelf WavesAndrew L Stewart Paul J Dellar and Edward R Johnson 231

13 Laboratory Experiments With Abrupt Thermohaline Transitions and OscillationsJohn A Whitehead 255

14 Oceanic Island Wake Flows in the LaboratoryAlexandre Stegner 265

Section V Advances in Methodology

15 Lagrangian Methods in Experimental Fluid MechanicsMickael Bourgoin Jean-Franccedilois Pinton and Romain Volk 279

16 A High-Resolution Method for Direct Numerical Simulation of Instabilities and Transitionsin a Baroclinic CavityAnthony Randriamampianina and Emilia Crespo del Arco 297

17 Orthogonal Decomposition Methods to Analyze PIV LDV and Thermography Data of Thermally DrivenRotating Annulus Laboratory ExperimentsUwe Harlander Thomas von Larcher Grady B Wright Michael Hoff Kiril Alexandrovand Christoph Egbers 315

Index 337

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page vii mdash 7

CONTRIBUTORS

Yakov D AfanasyevProfessorDepartment of Physics and Physical OceanographyMemorial University of Newfoundland St JohnrsquosNewfoundland Canada

Kiril AlexandrovFormerly ResearcherDepartment of Aerodynamics and Fluid MechanicsBrandenburg University of Technology (BTU)Cottbus-Senftenberg Germany

P Henrik AlfredssonProfessor of Fluid PhysicsRoyal Institute of TechnologyLinneacute FLOW Centre KTH MechanicsStockholm Sweden

Mickael BourgoinResearcherLaboratoire des Eacutecoulements Geacuteophysiques etIndustrielsUniversiteacute de Grenoble amp CNRSGrenoble France

Emilia Crespo del ArcoProfessor of Applied PhysicsDepartamento de Fiacutesica FundamentalUniversidad Nacional de Educacioacuten a Distancia (UNED)Madrid Spain

Thierry DauxoisDirecteur de RechercheLaboratoire de PhysiqueEacutecole Normale SupeacuterieureLyon France

Paul J DellarUniversity LecturerOxford Centre for Industrial and Applied MathematicsUniversity of OxfordOxford United Kingdom

Christoph EgbersProfessorDepartment of Aerodynamics and Fluid MechanicsBrandenburg University of Technology (BTU)Cottbus-Senftenberg Germany

Jan-Bert FloacuterDirecteur de RechercheLaboratoire des Ecoulements Geacuteophysiques etIndustriels (LEGI)Grenoble France

Wolf-Gerrit FruumlhSenior LecturerSchool of Engineering amp Physical SciencesHeriot-Watt UniversityEdinburgh United Kingdom

Jonathan GulaAssistant ResearcherInstitute for Geophysics and Planetary PhysicsUniversity of California Los AngelesLos Angeles California United States of America

Uwe HarlanderProfessorDepartment of Aerodynamics and Fluid MechanicsBrandenburg University of Technology (BTU)Cottbus-Senftenberg Germany

Gert-Jan van HeijstProfessorDepartment of Applied PhysicsEindhoven University of TechnologyEindhoven The Netherlands

Michael HoffResearcherLeipzig Institute for MeteorologyUniversity of LeipzigLeipzig Germany

Edward R JohnsonProfessorDepartment of MathematicsUniversity College LondonLondon United Kingdom

Thomas von LarcherPostdoctoral ResearcherInstitute for MathematicsFreie Universitaumlt BerlinBerlin Germany

vii

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viii CONTRIBUTORS

Gregory M LewisAssociate ProfessorInstitute of TechnologyUniversity of OntarioOshawa Ontario Canada

Paul F LindenGI Taylor Professor of Fluid MechanicsDepartment of Applied Mathematics and TheoreticalPhysicsUniversity of CambridgeCambridge United Kingdom

Rebecca J LingwoodGuest Professor of Hydrodynamic StabilityRoyal Institute of TechnologyLinneacute FLOW Centre KTH MechanicsStockholm SwedenDirector of Continuing EducationInstitute of Continuing EducationUniversity of CambridgeCambridge United Kingdom

Irene M MorozLecturer in MathematicsMathematical InstituteUniversity of OxfordOxford United Kingdom

Thomas PeacockAssociate Professor of Mechanical EngineeringDepartment of Mechanical EngineeringMassachusetts Institute of TechnologyCambridge Massachusetts United States of America

Edgar P PeacuterezFormerly Graduate StudentMathematical InstituteUniversity of OxfordOxford United Kingdom

Nicolas PeacuterinetPostdoctoral FellowInstitute of TechnologyUniversity of OntarioOshawa Ontario Canada

Jean-Franccedilois PintonDirecteur de Recherche CNRSLaboratoire de PhysiqueEacutecole Normale Supeacuterieure de LyonLyon France

Anthony RandriamampianinaDirecteur de Recherche CNRSLaboratoire Meacutecanique Modeacutelisation amp ProceacutedeacutesPropres UMR 7340 CNRSAix Marseille UniversiteacuteMarseille France

Peter L ReadProfessor of PhysicsAtmospheric Oceanic amp Planetary PhysicsUniversity of OxfordOxford United Kingdom

Luis Zavala SansoacutenResearcherDepartamento de Oceanografiacutea FiacutesicaCICESE EnsenadaBaja California Meacutexico

Heacutelegravene ScolanPostdoctoral Research AssistantAtmospheric Oceanic amp Planetary PhysicsUniversity of OxfordOxford United Kingdom

Alexandre StegnerResearcherLaboratoire de Meacuteteacuteorologie Dynamique CNRSFranceAssociate ProfessorEacutecole PolytechniquePalaiseau France

Andrew L StewartPostdoctoral ScholarDepartment of Atmospheric and Oceanic SciencesUniversity of California Los AngelesLos Angeles California United States of America

Bruce SutherlandProfessor of Fluid MechanicsDepartments of Physics and of Earth amp AtmosphericSciencesUniversity of AlbertaEdmonton Alberta Canada

Lennaert van VeenAssociate ProfessorInstitute of TechnologyUniversity of OntarioOshawa Ontario Canada

ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page ix mdash 9

CONTRIBUTORS ix

Roberto VerziccoProfessor in Fluid DynamicsDepartment of Mechanical EngineeringUniversitagrave di Roma Tor VergataRome Italy

Romain VolkLecturerLaboratoire de PhysiqueEacutecole Normale Supeacuterieure de LyonLyon France

Michael L WaiteAssistant ProfessorDepartment of Applied MathematicsUniversity of WaterlooWaterloo Ontario Canada

Nils P WediPrincipal ScientistNumerical Aspects SectionEuropean Centre for Medium-Range Weather ForecastsReading United Kingdom

John A WhiteheadScientist EmeritusDepartment of Physical OceanographyWoods Hole Oceanographic InstitutionWoods Hole Massachusetts United States of America

Grady B WrightAssociate ProfessorDepartment of MathematicsBoise State UniversityBoise Idaho United States of America

Roland M B YoungPostdoctoral Research AssistantAtmospheric Oceanic amp Planetary PhysicsUniversity of OxfordOxford United Kingdom

Vladimir ZeitlinProfessorLaboratoire de Meacuteteacuteorologie DynamiqueEacutecole Normale SupeacuterieureParis FranceUniversiteacute Pierre et Marie CurieParis France


ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page xi mdash 11

PREFACE

The flow of fluid in Earthrsquos atmosphere and oceanaffects life on global and local scales The general circula-tion of the atmosphere transports energy mass momen-tum and chemical tracers across the entire planet and thegiant currents of the thermohaline circulation and wind-driven circulation perform the same function in the oceanThese established flows are reasonably steady on long timescales In contrast short-lived instabilities may developand result in transient features such as waves oscillationsturbulence and eddies For example in the atmospheresmall-scale instabilities are able to grow into heavy stormsif the conditions are right

It is therefore of interest to understand atmospheric andoceanic fluid motions on all scales and their interactionsacross different scales Unfortunately due to the complexphysical mechanisms at play and the wide range of scalesin space and time research in geophysical fluid dynam-ics remains a challenging and intriguing task Despite thegreat progress that has been made we are still far fromachieving a comprehensive understanding

As tools for making progress with the above chal-lenge laboratory experiments are well suited to studyingflows in the atmosphere and ocean The crucial ingredi-ents of rotation stratification and large-scale forcing canall be included in laboratory settings Such experimentsoffer the possibility of investigating under controlledand reproducible conditions many flow phenomena thatare observed in nature Furthermore immense computa-tional resources are becoming available at low economiccost enabling laboratory experiments to be simulatednumerically in more detail than ever before The inter-play between numerical simulations and laboratory exper-iments is of increasing importance within the scientificcommunity

The purpose of this book is to provide a comprehensivesurvey of some of the laboratory experiments and numer-ical simulations that are being performed to improve ourunderstanding of atmospheric and oceanic fluid motion

On the experimental side new designs of experimentson the laboratory scale are discussed together with devel-opments in instrumentation and data acquisition tech-niques and the computer-based analysis of experimentaldata On the numerical side we address recent devel-opments in simulation techniques from model formu-lation to initialization and forcing The presentation ofresults from laboratory experiments and the correspond-ing numerical models brings the two sides together formutual benefit

The book contains five sections Section I coversbaroclinic instability which plays a prominent role inatmosphere and ocean dynamics The thermally drivenrotating annulus is the corresponding laboratory setuphaving been used for experiments in geophysical fluiddynamics since the 1940s by Dave Fultz Raymond Hideand others Section II covers balanced and unbalancedflows Sections III and IV cover laboratory experimentsand numerical studies devoted to specific atmosphericand oceanic phenomena respectively Section V adressessome new achievements in the computer-based analysisof experimental data and some recent developments inexperimental methodology and numerical methods

We hope this book will give the reader a clear picture ofthe experiments that are being performed in todayrsquos labo-ratories to study atmospheric and oceanic flows togetherwith the corresponding numerical simulations We furtherhope that the lessons learnt from the comparisons betweenlaboratory and model will act as a source of inspiration forthe next generation of experiments and simulations

Thomas von LarcherFreie Universitaumlt Berlin

Germany

Paul D WilliamsUniversity of Reading

United Kingdom

xi

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ldquovonLarcher-Driverrdquo mdash 20141011 mdash 1019 mdash page xiii mdash 13

ACKNOWLEDGMENTS

We gratefully acknowledge support from the staff atthe American Geophysical Union (AGU) particularlyColleen Matan for guiding us through the book pro-posal process and Telicia Collick for assisting with the

external peer review of the chapters We also extend ourthanks to Rituparna Bose at John Wiley amp Sons Inc forsmoothly and professionally handling all aspects of thebook production process

xiii

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ldquovonLarcher-Driverrdquo mdash 20141011 mdash 957 mdash page 1 mdash 1

Introduction Simulations of Natural Flows in the Laboratoryand on a Computer

Paul F Linden

Humans have always been associated with natural flowsThe first civilizations began near rivers and humans devel-oped an early pragmatic view of water flow and the effectsof wind Experiments and calculations in fluid mechanicscan be traced back to Archimedes in his work ldquoOn floatingbodiesrdquo around 250 BC in which he calculates the posi-tion of equilibrium of a solid body floating in a fluid Heis of course attributed with the law of buoyancy knownas Archimedes principle The ancient Greeks also eluci-dated the principle of the syphon and the pump This workis essentially concerned with fluid statics and the firstattempts to investigate the motion of fluids is attributedto Sextus Julius Frontinus the inspector of public foun-tains in Rome who made extensive measurements of flowin aqueducts and using conservation-of-volume princi-ples was able to detect when water was being divertedfraudulently

Possibly the first laboratory experiment designed toexamine a natural flow was by Marsigli [1681] who deviseda demonstration of the buoyancy-driven flow associ-ated with horizontal density differences in an attemptto explain the undercurrent in the Bosphorus that flowstoward the Black Sea [Gill 1982] This is a remarkableexperiment in that it provides an unequivocal demonstra-tion that flow now known as baroclinic flow with no nettransport is possible even when the free surface is level sothat there is no barotropic (depth-averaged) flow Thesebuoyancy-driven flows occur almost ubiquitously in theoceans and atmosphere and are an active area of currentresearch

Another example of the early use of a laboratoryexperiment is the explanation of the ldquodead waterrdquo phe-nomenon observed by the Norwegian scientist Fridtjof

Department of Applied Mathematics and TheoreticalPhysics University of Cambridge Cambridge United Kingdom

Nansen [1897] who experienced an unexpected drag onhis boat during his expedition to reach the North Polein 1892 The responsible mechanism the drag associatedwith interfacial waves on the pycnocline was studied byEkman [1904] in his PhD thesis and a review of his workand some modern extensions using synthetic schlieren canbe found elsewhere [Mercier et al 2011]

This last reference nicely demonstrates one role of mod-ern laboratory experiments Although the basic mechanicshas been known since Ekmanrsquos study by careful observa-tion of the flow and making quantitative measurementsof the wave fields made possible with new image process-ing techniques it has been shown that the dead waterphenomenon is nonlinear The coupling of the large-amplitude interfacial and internal waves with significantaccelerations of the boat are an intrinsic feature of theenergy transmission from the boat to the waves Althoughthe essential features have been known for over a centurythese recent data provide new insights into the physics ofthe flows and show that the drag on the boat dependson the forms of the waves generated Experiments likethis provide insight and inspiration about the underlyingdynamical processes ideally motivating theory which cansubsequently refocus the experiments

Numerical methods were first devised to solve poten-tial flow problems in the 1930s and as far as I am awarethe first numerical solution of the Navier-Stokes equa-tions ie the first computational fluid dynamics (CFD)calculation applied to two-dimensional swirling flow waspublished by Fromm [1963] Since then there has beenenormous growth in computational power and this hasled to developments in both CFD and laboratory experi-mentation The reasons for the improvement in CFD areclear In order to calculate a flow accurately it is neces-sary that the discrete forms of the governing equationsare a faithful representation of the continuous partial

Modeling Atmospheric and Oceanic Flows Insights from Laboratory Experiments and Numerical SimulationsFirst Edition Edited by Thomas von Larcher and Paul D Williamscopy 2015 American Geophysical Union Published 2015 by John Wiley amp Sons Inc

1


2 INTRODUCTION

X Z

Figure 01 A sketch of Marsiglirsquos [1681] experiment illustratingthe counterflow driven by the density difference between thetwo fluids in either side of the barrier with flow along the surfacetoward the denser fluid and a countercurrent along the bottomin the opposite direction

differential equations For geophysical flows which aretypically turbulent this means that in order to avoidapproximations it is necessary to compute all the scales ofmotion which range from the energy input scales downto the smallest scales where viscous dissipation occursThis represents a huge range of scales Energy is input onglobal scales (106 m) and dissipated at the Kolmogorovscale (ν3ε)14 sim 10minus3 m This nine decade range of length(and associated time) scales remains well beyond the capa-bilities of current (and foreseeable) computing power andrepresents a huge challenge to the computation of geo-physical flows

Geophysical flows are stably stratified (buoyant fluidnaturally lies on top of denser fluid) and occur on arotating planet The stratification characterized by thebuoyancy frequency N is of the order of 10minus2 sminus1 and isroughly the same in the atmosphere and the oceans Therotation of Earth is characterized by the Coriolis param-eter f which with values of order 10minus4sminus1 introduceslonger time scales than those associated with the strati-fication Thus the atmosphere and the oceans viewed onglobal scales are strongly stratified weakly rotating fluids

Stratification provides a restoring force to verticalmotions through the buoyancy force associated with thedensity difference between the displaced fluid particle andthe background stratification Rotation provides a restor-ing force due to horizontal motions through the Coriolisforce (or by conservation of angular momentum viewedin an inertial frame) For motion with horizontal scale Land vertical scale H the balance between these forces isgiven by the Burger number B

B equiv NHfL

(01)

Stratification dominates when B gtgt 1 ie when hor-izontal scales are relatively small compared with theRossby deformation radius RD equiv NHf while rotationdominates when horizontal scales are large comparedwith RD and B ltlt 1 On global scales the oceans andthe atmosphere are thin layers of fluids with verticalto horizontal aspect ratios HL of order 10minus3 Conse-quently for motion on global scales in the atmosphereor basin scales in the oceans B sim 10minus1 and rotationaleffects dominate These flows can be modeled as essen-tially unstratified flows with Coriolis forces providing themain constraints Motion on smaller scales will generallylead to increasing values of B and increasing effects ofstratification Mesoscale motions in which buoyancy andCoriolis forces balance are typified by values of B sim 1 inwhich case the horizontal scale of the motion is compa-rable to the Rossby deformation radius RD which is onthe order of 1000 km in the atmosphere and 100 km inthe oceans

In order to examine the effects of rotation experimentsare conducted on rotating platforms These are generallyhigh-precision turntables capable of carrying significantweight and they present a significant engineering chal-lenge in their construction The requirements and per-formance of these turntables are discussed in Chapter 7which illustrates these by considering flows of thin fluidlayers in rotating containers of different diameters from01 to 10 m As the size of the turntable is increasedthe engineering requirements become more demandingand the cost increases Furthermore larger flow domainsrequire more fluid and if stratified this requires a morestratifying agent such as (the commonly used) sodiumchloride For these reasons most laboratory turntablesrange up to about 1 m in diameter and there needs to be acompelling reason to work on large-diameter turntables

One reason for increasing the experimental scale from1 to 10 m is to reduce frictional effects Reynolds numbersRe equiv ULν where U and L are typical velocity and lengthscales and ν is the kinematic viscosity are increased by afactor of 10 (equivalently Ekman numbers E equiv νfL2 arereduced by a factor of 100) and so the effects of bound-ary friction are reduced and damping times are increasedat large scale This can be an dominant factor when study-ing flows where separation or turbulence is dominant oras in the case discussed in Chapter 7 the motion of vor-tices driven by vortex interactions or interactions withtopography

On the other hand there is little to be gained in termsof the overall Rossby number or Burger number bothof which involve the product fL of rotation and lengthscale This product is the speed of the rim of the turntable


INTRODUCTION 3

and it is difficult for safety and operational reasons toincrease this very much above a few meters per secondwhich is easily obtainable with a 1 m diameter turntableHowever the size of the flow domain impacts the spatialand temporal resolution that is possible and needed Onlarger turntables structures such as vortices are increasedin size allowing better spatial resolution and lower Rossbynumbers

There has been a major resurgence in interest in lab-oratory experiments over the past decade due to newdevelopments in optics and in computing power mem-ory and the ability to read and write data rapidly usingsolid-state media Velocity measurements using particleimage velocimetry and particle tracking (see Chapter 15)now have the capabilities to resolve three-dimensionalflow fields with subpixel (01 pixel) spatial accuracy andmillisecond time resolution using high-speed camerasWith currently available megapixel cameras this impliesdata rates of gigabytes per second which requires ded-icated data storage and processing which are now cur-rently available and affordable There is every reason toexpect further technological improvements that will ren-der these techniques increasingly effective in the futureand inspire the development of new data analysis method-ologies (see Chapter 17) and also increasing study ofmultiscale phenomena

Other methods have been developed that also allownonintrusive measurements of the flow For example inChapter 5 Afanasyev describes optical altimetry whichprovides measurements of the free surface slope In arotating fluid it is then possible to infer the flow pro-vided certain assumptions are made about the dynamicsIf the fluid is homogeneous and the flow is in geostrophicand hydrostatic balance the free surface height is thestream function for the flow from which the velocityfield can be inferred This method can be extended totwo-layer flows if the depth of one layer can be measuredindependently This can be done by adding dye to onelayer and then measuring the absorption of light throughthe layer

Another method for nonintrusive measurements is syn-thetic schlieren described in Chapter 10 This methoduses the apparent movement of an image (say an array ofdots) placed behind an experiment as a result of refrac-tive index changes in the fluid between the camera and theimage This method has found many applications in thestudy of internal waves which produce refractive indexchanges by moving the stratified fluid as they propagateFrom the apparent movement of the dots say the changesto the density field can be measured and then assumingthese are caused by linear internal waves the associatedmotion can be determined This method has led to suc-cessful measurements of momentum and energy fluxesover a wave field and led to new interpretations of internal

wave dynamics Both synthetic schlieren and the opticalthickness method produce data integrated over the lightpath Thus in their simplest forms it is assumed thatthere are no variations in the flow along the light pathRecently attempts have been made to overcome this lim-itation so that fully three-dimensional flows can be mea-sured and this has had success in limited circumstances(see Chapter 10)

Computations in the form of numerical solution ofmodel equations or of approximate forms of the fullNavier-Stokes equations have also developed significantlyHowever perhaps the most interesting development is theincreasing capacity to carry our direct numerical simu-lations (DNSs) of the Navier-Stokes equations withoutmaking any approximations This capability has comefrom a combination of improved numerical schemes todeal with the discretized forms of the equations andof course from the rapid and continuing improvementsin computer power To that extent the development ofhigh-capacity computing has been the key to recent devel-opments in both computations and experiments whichleads to the interesting issue of how well either representreal geophysical flows

The challenge is to replicate the physical processes thatoccur on geophysical scales accurately in the laboratoryand in computations In the laboratory it is physicallyimpossible of course to work at full scale and numeri-cally the issue is to deal with the huge range of length andtime scales that are dynamically significant In the caseof stratified flows recent work on turbulence discussed inChapter 8 shows that in order to avoid viscous effects thebuoyancy Reynolds number

R equiv U3

νN2L (02)

where U is a typical velocity scale and L is a typicalhorizontal scale needs to be large Mesoscale motions inthe atmosphere have Rsim 104 and in the oceans Rsim 103and recent DNS calculations have achieved values up toRsim 102 This is very encouraging and these types of cal-culations have led to a significant reevaluation of theenergetics of stratified turbulence [Waite 2013]

Experiments on the other hand have so far been char-acterized by values of R sim 1 or less (see Chapter 8) whichis mainly a result of the fact that most of the relevantexperiments are concerned with decaying turbulence (seeBrethouwer et al [2007] Figure 18) It is however possibleby directly forcing the flow to achieve high values of RAn example is shown in Figure 02 In this experimenta water tank is partitioned into two sections both con-taining salt water with different densities A square ductthat may be inclined at an angle θ to the horizontal passesthroughout the central divide of the tank and connectsthe two sections of the tank which act as large reservoirs


4 INTRODUCTION

H

Higher density Lower density

L

θρρ + Δρ

Figure 02 Experimental setup

The flow is initiated by opening one end of the duct andmaintained until the fluid entering each reservoir beginsto affect the input at either end of the duct The flow ischaracterized by the inclination θ and the density differ-ence ρ between the two sections of the tank In theseexperiments we explored the flows in the range 0 le θ le 3(θ gt 0 means that the duct is inclined up toward the densereservoir) and 10 le ρ le 210 kgm3

The buoyancy forces establish a flow with the lightfluid flowing uphill in the upper part of the duct and thedense fluid flowing downhill in the lower part Betweenthese counterflowing layers there is an interfacial regionthat takes different forms depending on the density dif-ference and the angle of the duct At high angles andlarger density differences the interfacial region is stronglyturbulent with three-dimensional structures of Kelvin-Helmholtz type across its whole width Mass is transferredfrom the lower layer directly to the upper layer througheddies that span the entire thickness of the region Asthe density difference andor angle decrease the turbu-lence becomes less intense At a critical density differenceand angle combination a spatiotemporal intermittentregime develops where turbulent bursts and relaminariza-tion events occur Both Kelvin-Helmholtz and Holmboe-type modes appear in this regime and the interfacialregion has a complicated structure consisting of thinnerlayers of high-density gradients within it These layersand the instability modes display significant variability inspace and time At even smaller angles and density dif-ferences the flow is essentially laminar (or at least weaklydissipative) with a relatively sharp interface supportingHolmboe-type wave modes with the occasional break-ing event Shadowgraph images of these flow regimes areshown in Figure 03

Typical values of the flow in this experiment areU sim radic

gprimeH sim 01 ms L = 3 m N sim radicgprimeH sim 1 sminus1

ν asymp 10minus6m2s gives Rsim 102 which is comparable with thebest numerical computations Here we have taken a con-servative estimate of the flow speed and the length of theduct as the horizontal scale although Figure 03 suggeststhat a more likely value is L sim 01 m which will increasethe range of values of R by an order of magnitude and inthe range of geophysical flows

Turbulent

Intermittent

Laminar

Figure 03 Shadowgraphs of the turbulent intermittent andlaminar flow regimes in buoyancy-driven flow in an inclinedduct Buoyant fluid flows to the left above dense fluid flowingto the right Images taken by Colin Meyer [Meyer 2014]

As shown in Chapter 9 other examples where it ispossible to match the parameter ranges of DNS andlaboratory experiments are the numerical simulations ofexperiments that model the quasi-biennial oscillation andthe Madden-Julian oscillation These comparisons showthat the laboratory experiments by Plumb and McEwan[1978] represent the atmosphere rather better than hasbeen previously thought Once agreement between theexperiments and computations has been established thelatter can be used to investigate other effects such asthe dependence on the fluid properties for example thePrandtl number (see Chapter 16) which are difficult tovary in experiments

By revealing flow structures and allowing the physicsto be explored by varying parameters under controlledconditions both experiments and numerical simulationsallow fundamental insights to be revealed Nothing reallybeats looking at a flow either physically or using the won-derful graphics that are currently available to visualize theoutputs of numerical data to get a feel for and developintuition about the dynamics There is still a fascinationwith observing wakes (Chapter 14) the flows associatedwith boundary layers in rotating systems (Chapter 4)abrupt transitions in flow regimes caused by buoyancyforcing (Chapter 13) the forms of instabilities on fronts(Chapter 11) the form of shelf waves (Chapter 12) andthe amazing rich dynamics revealed by observations of theflow in a rotating heated annulus (Chapter 1)

Nevertheless as illuminating and as necessary as thesestudies are extracting the underlying physics and devel-oping an understanding of what these observations reveal


INTRODUCTION 5

requires a further component that is a model Withoutan underlying model which can be a sophisticated math-ematical model of say nonlinear processes in baroclinicinstability (see Chapters 6 and 3) or nonlinear waves cap-tured in a shallow water context (see Chapter 2) or indeeda more simplistic view based on dimensional analysisexperiments and numerical simulations only produce aseries of dots in some parameter space It is the modeland the understanding that are inherent in the simplifiedrepresentation of reality that joins the dots And when thedots are joined then one really has something

REFERENCES

Brethouwer G P Billant E Lindborg and J-M Chomaz(2007) Scaling analysis and simulation of strongly stratifiedturbulent flows J Fluid Mech 585 343ndash368

Ekman V W (1904) On dead water Norw N Polar Exped1893ndash1896 Sci results XV Christiana PhD thesis

Fromm J E (1963) A method for computing non-steadyincompressible fluid flows Tech Rep LA-2910 Los AlamosSci Lab Los Alamos N Mex

Gill A E (1982) Atmosphere-Ocean Dynamics AcademicPress New York

Marsigli L M (1681) Osservazioni intorno al bosforo tracioo vero canale di constantinopli rappresentate in lettera allasacra real maesta cristina regina di svezia roma Boll PescaPiscic Idrobiol 11 734 ndash 758

Mercier M J R Vasseur and T Dauxois (2011) Resurrect-ing dead-water phenomenon Nonlin Processes Geophys 18193ndash208

Meyer C R and P F Linden (2014) Stratified shear flowexperiments in an inclined duct J Fluid Mech 753 242ndash253doi101017jfm2014358

Nansen F (1897) Farthest North The Epic Adventure ofa Visionar Explorer Library of Congress Cataloging-in-Publication Data

Plumb R A and A D McEwan (1978) The instability of aforced standing wave in a viscous stratified fluid A laboratoryanalogue of the quasi-biennial oscillation J Atmos Sci 351827ndash1839

Waite M L (2013) The vortex instability pathway in stratifiedturbulence J Fluid Mech 716 1ndash4






1General Circulation of Planetary Atmospheres

Insights from Rotating Annulus and Related Experiments

Peter L Read1 Edgar P Peacuterez2 Irene M Moroz2 and Roland M B Young1

11 LABORATORY EXPERIMENTSAS ldquoMODELSrdquo OF PHYSICAL SYSTEMS

In engineering and the applied sciences the termldquomodelrdquo is typically used to denote a device or conceptthat imitates the behavior of a physical system as closelyas possible but on a different (usually smaller) scalepossibly with some simplifications The aim of such amodel is normally to evaluate the performance of sucha system for reasons connected with its exploitation foreconomic social military or other purposes In the con-text of the atmosphere or oceans numerical weather andclimate prediction models clearly fall into this categorySuch models are extremely complicated entities that seekto represent the topography composition radiative trans-fer and dynamics of the atmosphere oceans and surfacein great detail As a result it is generally impossible tocomprehend fully the complex interactions of physicalprocesses and scales of motion that occur within any givensimulation The success of such models can only be judgedby the accuracy of their predictions as directly verified(in the case of numerical weather prediction) against sub-sequent observations and measurements Similar modelsused for climate prediction however are often compara-ble in complexity to those used for weather prediction butare frequently used as tools in attempts to address ques-tions of economic social or political importance (egconcerning the impact of increasing anthropogenic green-house gas emissions) for which little or no verifying datamay be available

In formulating such models and interpreting theirresults it is necessary to make use of a different class of

1Atmospheric Oceanic amp Planetary Physics University ofOxford Oxford United Kingdom

2Mathematical Institute University of Oxford OxfordUnited Kingdom

model the ldquoconceptualrdquo or ldquotheoreticalrdquo model whichmay represent only a small subset of the geographicaldetail and physical processes active in the much largerapplications-oriented model but whose behavior may bemuch more completely understood from first principlesTo arrive at such a complete level of understanding how-ever it is usually necessary to make such models as simpleas possible (ldquobut no simplerrdquo) and in geometric domainsthat may be much less complicated than found in typi-cal geophysical contexts An important prototype of sucha model in fluid mechanics is that of dimensional (orldquoscalerdquo) analysis in which the entire problem reduces toone of determining the leading order balance of termsin the governing equations and the consequent depen-dence of one or more observable parameters in the formof power law exponents Following such a scale analysisit is often possible to arrive at a scheme of mathematicalapproximations that may even permit analytical solutionsto be obtained and analyzed The well-known quasi-geostrophic approximation is an important example ofthis approach [eg see Holton 1972 Vallis 2006] that hasenabled a vast number of essential dynamical processes inlarge-scale atmospheric and oceanic dynamics to be stud-ied in simplified (but nonetheless representative) forms

For the fundamental researcher such simplified ldquocon-ceptualrdquo models are an essential device to aid and advanceunderstanding The latter is achievable because simplifiedapproximated models enable theories and hypotheses tobe formulated in ways that can be tested (ie falsified inthe best traditions of the scientific method) against obser-vations andor experiments The ultimate aim of suchstudies in the context of atmospheric and oceanic sciencesis to develop an overarching framework that sets in per-spective all planetary atmospheres and oceans of whichEarth represents but one set of examples [Lorenz 1967Hide 1970 Hoskins 1983]


9


10 MODELING ATMOSPHERIC AND OCEANIC FLOWS

Warm water

(b)

Working fluidCold water

(a)

Figure 11 (a) Schematic diagram of a rotating annulus(b) schematic equivalent configuration in a spherical fluid shell(cf an atmosphere)

The role of laboratory experiments in fluid mechanicsin this scheme would seem at first sight to be as modelsfirmly in the second category Compared with a plane-tary atmosphere or ocean they are clearly much simplerin their geometry boundary conditions and forcing pro-cesses (diabatic and mechanical) eg see Figure 11 Theirbehavior is often governed by a system of equations thatcan be stated exactly (ie with no controversial param-eterizations being necessary) although even then exactmathematical solutions (eg to the Boussinesq Navier-Stokes equations) may still be impossible to obtain Unlikeatmospheres and oceans however it is possible to carryout controlled experiments to study dynamical processesin a real fluid without recourse to dubious approxima-tions (necessary to both analytical studies and numericalsimulation) Laboratory experiments can therefore com-plement other studies using complex numerical modelsespecially since fluids experiments (a) have effectively infi-nite resolution compared to their numerical counterparts(though can only be measured to finite precision and res-olution) (b) are often significantly less diffusive than theequivalent fluid eg in eddy-permitting ocean modelsand yet ( c) are relatively cheap to run

In discussing the role of laboratory experiments how-ever it is not correct to conclude that they have no directrole in the construction of more complex applications-oriented models and associated numerical tools (such asin data assimilation) Because the numerical techniquesused in such models (eg finite-difference schemes eddyor turbulence parameterizations) are also components ofmodels used to simulate flows in the laboratory undersimilar scaling assumptions laboratory experiments canalso serve as useful ldquotest bedsrdquo for directly evaluating andverifying the accuracy of such techniques in ways thatare far more rigorous than may be possible by compar-ing complex model simulations solely with atmosphericor oceanic observations Despite many advances in theformulation and development of sophisticated numeri-cal models there remain many phenomena (especially

those involving nonlinear interactions of widely differ-ing scales of motion) that continue to pose seriouschallenges to even state-of-the-art numerical models yetmay be readily realizable in the laboratory This is espe-cially true of large-scale flow in atmospheres and oceansfor which relatively close dynamical similarity betweengeophysical and laboratory systems is readily achiev-able Laboratory experiments in this vein therefore stillhave much to offer in the way of quantitative insightand inspiration to experienced researchers and freshstudents alike

12 ROTATING STRATIFIED EXPERIMENTS ANDGLOBAL CIRCULATION OF ATMOSPHERES AND

OCEANS

At its most fundamental level the general circulationof the atmosphere is but one example of thermal convec-tion in response to impressed differential heating by heatsources and sinks that are displaced in both the verticalandor the horizontal in a rotating fluid of low viscosityand thermal conductivity Laboratory experiments inves-tigating such a problem should therefore include at leastthese attributes and be capable of satisfying at least someof the key scaling requirements for dynamical similarityto the relevant phenomena in the atmospheric or oceanicsystem in question Such experimental systems may thenbe regarded [eg Hide 1970 Read 1988] as schemati-cally representing key features of the circulation in theabsence of various complexities associated for examplewith radiative transfer atmospheric chemistry boundarylayer turbulence water vapor and clouds in a way that isdirectly equivalent to many other simplified and approxi-mated mathematical models of dynamical phenomena inatmospheres and oceans

Experiments of this type are by no means a recent phe-nomenon with examples published as long ago as themid to late nineteenth century [eg Vettin 1857 1884Exner 1923] see Fultz [1951] for a comprehensive reviewof this early work Vettin [1857 1884] had the insightto appreciate that much of the essence of the large-scaleatmospheric circulation could be emulated at least inprinciple by the flow between a cold body (represent-ing the cold polar regions) placed at the center of arotating cylindrical container and a heated region (rep-resenting the warm tropics) toward the outside of thecontainer (see Figure 12) Vettinrsquos experiments used airas the convecting fluid contained within a bell jar on arotating platform As one might expect of a nineteenthcentury gentleman he then used cigar smoke to visual-ize the flow patterns demonstrating phenomena such asconvective vortices and larger scale overturning circula-tions However these experiments only really explored theregime we now know as the axisymmetric or ldquoHadleyrdquo


GENERAL CIRCULATION OF PLANETARY ATMOSPHERES 11

Figure 12 Selection of images adapted from Vettin [1884](see httpwwwschweizerbartde) Reproduced with permis-sion from the publishers showing the layout of his rotatingconvection experiment and some results

regime since the flows Vettin observed showed little evi-dence for the instabilities we now know as ldquobaroclinicinstabilityrdquo or ldquosloping convectionrdquo [Hide and Mason1975]

As an historical aside it is interesting to note that earlymeteorologists such as Abbe [1907] intended for labora-tory experiments of this type to serve also as modelsof the first kind ie as application-oriented predictivemodel atmospheres They realized that while it mightbe possible in principle to use the equations of atmo-spheric dynamics to determine future weather they werebeyond the capacity of mathematical analysis to solveThey hoped to use these so-called mechanical integra-tors [Rossby 1926] under complicated external forcingcorresponding to the observations of the day to repro-duce and predict very specific flow phenomena observed inthe atmosphere It was anticipated that many such exper-iments would be built representing different regions ofEarthrsquos surface or different times of year such as whenthe cross-equatorial airflow is perturbed by the monsoon[Abbe 1907] (although it is not clear whether such an

experiment was ever constructed) However following thedevelopment of the electronic computer during the firsthalf of the twentieth century and Richardsonrsquos [1922] pio-neering work on numerical weather prediction these morecomplex laboratory representations of the atmospherewere superseded

The later experiments of Exner [1923] explored a dif-ferent regime in which baroclinic instability seems to havebeen present The flows he demonstrated were evidentlyquite disordered and irregular likely due in part to theparameter regime he was working in but also perhapsbecause of inadequate control of the key parameters Itwas not until the late 1940s however that Fultz begana systematic series of experiments at the University ofChicago on rotating fluids subject to horizontal differen-tial heating in an open cylinder (hence resulting in theobsolete term ldquodishpan experimentrdquo) and set the subjectonto a firm footing Independently and around the sametime Hide [1958] began his first series of experiments atthe University of Cambridge on flows in a heated rotatingannulus initially in the context of fluid motions in Earthrsquosliquid core By carrying out an extensive and detailedexploration of their respective parameter spaces both ofthese pioneering studies effectively laid the foundationsfor a huge amount of subsequent work on elucidatingthe nature of the various circulation regimes identifiedby Fultz and Hide subsequently establishing their bifur-cations and routes to chaotic behavior developing newmethods of modeling the flows using numerical tech-niques and measuring them using ever more sophisticatedmethods especially via multiple arrays of in situ probesand optical techniques that exert minimal perturbationsto the flow itself

An important aspect of the studies by Fultz and Hidewas their overall agreement in terms of robustly identi-fying many of the key classes of circulation regimes andlocating them within a dimensionless parameter space Anotable exception to this at least in early work was thelack of a regular wave regime in Fultzrsquos open cylinderexperiments in sharp contrast to the clear demonstrationof such a regime in Hidersquos annulus As further discussedbelow this led to some initial suggestions [Davies 1959]that the existence of this regime was somehow depen-dent on having a rigid inner cylinder bounding the flownear the rotation axis This was subsequently shown notto be the case in open cylinder experiments by Fultzhimself [Spence and Fultz 1977] and by Hide and hisco-workers [Hide and Mason 1970 Bastin and Read 1998]and two-layer [Hart 1972 1985] experiments that clearlyshowed that persistent near-monochromatic baroclinicwave flows could be readily sustained in a system withouta substantial inner cylinder It is likely therefore that earlyefforts failed to observe such a regular regime in the ther-mally driven open cylinder geometry because of a lack of



close experimental control eg of the rotation rate or thestatic stability in the interior

Earlier studies in this vein were extensively reviewed byHide [1970] and Hide and Mason [1975] More recentlysignificant advances have been presented by variousgroups around the world including highly detailed exper-imental studies in the ldquoclassicalrdquo axisymmetric annulusof synoptic variability vacillations and the transitionsto geostrophically turbulent motions by groups at theFlorida State University [eg Pfeffer et al 1980 Buzynaet al 1984] the UK Met Office and Oxford Univer-sity [eg Read et al 1992 Fruumlh and Read 1997 Bastinand Read 1997 1998 Wordsworth et al 2008] severalJapanese universities [eg Ukaji and Tamaki 1989 Sug-ata and Yoden 1994 Tajima et al 1995 1999 Tamakiand Ukaji 2003] and most recently the BremenCottbusgroup in Germany [Sitte and Egbers 2000 von Larcherand Egbers 2005 Harlander et al 2011] and the Budapestgroup in Hungary [Jnosi et al 2010] These have beencomplemented by various numerical modeling studies[eg Hignett et al 1985 Sugata and Yoden 1992 Readet al 2000 Maubert and Randriamampianina 2002 Lewisand Nagata 2004 Randriamampianina et al 2006 Youngand Read 2008 Jacoby et al 2011] In addition the rangeof phenomena studied in the context of annulus exper-iments have been extended through modifications to theannulus configuration to emulate the effects of planetarycurvature (ie a β-effect) [eg Mason 1975 Bastin andRead 1997 1998 Tamaki and Ukaji 2003 Wordsworthet al 2008 von Larcher et al 2013] and zonally asymmet-ric topography [eg Leach 1981 Li et al 1986 Bernadetet al 1990 Read and Risch 2011] see also Chapters 2 37 16 and 17 in this volume

The existence of regular periodic quasi-periodic orchaotic regimes in an open cylinder was also a major fea-ture of another related class of rotating stratified flowexperiments using discrete two-layer stratification andmechanically-imposed shears Hart [1972] introduced thisexperimental configuration in the early 1970s inspiredby the theoretical work of Phillips [1954] and Pedlosky[1970 1971] on linear and weakly nonlinear instabilitiesof such a two-layer rotating flow system Because ofits simpler mode of forcing and absence of complicatedboundary layer circulations these kinds of two-layer sys-tesm were more straightforward to analyze theoreticallyallowing a more direct verification of theoretical pre-dictions in the laboratory than has typically proved thecase with the thermally driven systems Subsequent stud-ies by Hart [Hart 1979 1980 1985 1986 Ohlsen andHart 1989a1989b] and others [eg King 1979 Appleby1982 Lovegrove et al 2000 Williams et al 2005 2008]have extensively explored this system identifying variousforms of vacillation and low-dimensional chaotic behav-iors as well as the excitation of small-scale interfacial

inertia-gravity waves through interactions with the quasi-geostrophic baroclinic waves

In this chapter we focus on the ldquoclassicalrdquo thermallydriven rotating annulus system In Section 13 we reviewthe current state of understanding of the rich and diverserange of flow regimes that may be exhibited in thermalannulus experiments from the viewpoint of experimen-tal observation numerical simulation and fundamental(mainly quasi-geostrophic) theory This will include theinterpretation of various empirical experimental obser-vations in relation to both linear and weakly nonlinearbaroclinic instability theory One of the key attributesof baroclinic instability and ldquosloping convectionrdquo is itsrole in the transfer of heat within a baroclinic flow InSection 14 we examine in some detail how heat is trans-ported within the baroclinic annulus across the full rangeof control parameters associated with both the boundarylayer circulation and baroclinically unstable eddies Thisleads naturally to a consideration of how axisymmetricboundary layer transport and baroclinic eddy transportsscale with key parameters and hence how to parameterizethese transport processes both diagnostically and prog-nostically in a numerical model for direct comparisonwith recent practice in the ocean modeling communityFinally in Section 15 we consider the overall role ofannulus experiments in the laboratory in continuing toadvance understanding of the global circulation of plan-etary atmospheres and oceans reviewing the current stateof research on delineating circulation regimes obtainedin large-scale circulation models in direct comparisonwith the sequences of flow regimes and transitions in thelaboratory The results strongly support many parallelsbetween laboratory systems and planetary atmospheres atleast in simplified models suggesting a continuing impor-tant role for the former in providing insights for the latter

13 FLOW REGIMES AND TRANSITIONS

The typical construction of the annulus is illustratedschematically in Figure 11 and consists of a workingfluid (usually a viscous liquid such as water or siliconeoil though this can also include air [eg see Maubertand Randriamampianina 2002 Randriamampianina et al2006 Castrejoacuten-Pita and Read 2007] or other fluidsincluding liquid metals such as mercury [Fein and Pfeffer1976]) contained in the annular gap between two coax-ial circular thermally conducting cylinders that can berotated about their common (vertical) axis The cylin-drical sidewalls are maintained at constant but differenttemperatures with a (usually horizontal) thermally insu-lating lower boundary and an upper boundary that is alsothermally insulating and either rigid or free (ie withouta lid)



(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)

10ndash110ndash210ndash3

10ndash2

Lowersymmetrical

Uppersymmetrical

0382 times Taylor number

Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)

Figure 13 Schematic regime diagram for the thermally driven rotating annulus in relation to the thermal Rossby number

(or stability parameter prop minus2) and Taylor number T prop 2 showing some typical horizontal flow patterns at the top surfacevisualized as streak images at upper levels of the experiment

131 Principal Flow Regimes

Although a number of variations in these boundaryconditions have been investigated experimentally almostall such experiments are found to exhibit the same threeor four principal flow regimes as parameters such as therotation rate or temperature contrast T are variedThese consist of (I) axisymmetric flow (in some respectsanalogous to Hadley flow in Earthrsquos tropics and frequentlyreferred to as the ldquoupper-symmetric regimerdquo see below)at very low for a given T (that is not too small) (II)regular waves at moderate and (III) highly irregularaperiodic flow at the highest values of attainable Inaddition (IV) axisymmetric flows occur at all values of at a sufficiently low temperature difference T (a diffu-sively dominated regime termed ldquolower symmetricrdquo [Hideand Mason 1975 Ghil and Childress 1987] to distinguishit from the physically distinct ldquoupper-symmetricrdquo men-tioned above) The location of these regimes are usuallyplotted on a ldquoregime diagramrdquo with respect to the two (or

three) most significant dimensionless parameters Theseare typically

(a) a stability parameter or ldquothermal Rossby numberrdquo

=gαTd

[(b minus a)]2 (11)

providing a measure of the strength of buoyancy forcesrelative to Coriolis accelerations

(b) a Taylor number

T =2(b minus a)5

ν2d (12)

measuring the strength of Coriolis accelerations relativeto viscous dissipation and

(c) the Prandtl number

Pr =ν

κ (13)

Here g is the acceleration due to gravity α the thermalexpansion coefficient of the fluid ν the kinematic viscos-ity κ the thermal diffusivity and a b and d the radii of



the inner and outer cylinder and the depth of the annulusrespectively Figure 13 shows a schematic form of thisdiagram with the locations of the main regimes indicated

From a consideration of the conditions under whichwaves occur in the annulus (especially the location in theparameter space of the upper-symmetric transition) anda comparison with the results of linear instability the-ory it is clear that the waves in the annulus are fullydeveloped manifestations of baroclinic instability (oftenreferred to as ldquosloping convectionrdquo from the geometryof typical fluid trajectories for example see Hide andMason [1975]) Since these flows occur in the interiorof the annulus (ie outside ageostrophic boundary lay-ers) under conditions appropriate to quasi-geostrophicscaling a dynamical similarity to the large-scale midlat-itude cyclones in Earthrsquos atmosphere is readily apparentthough with rather different boundary conditions A moredetailed discussion of the properties of these flows isgiven below and by Hide and Mason [1975] and Ghil andChildress [1987] Associated with this conclusion is theimplication that the waves develop in order to assist inthe transfer of heat both upward (enhancing the staticstability) and horizontally down the impressed thermalgradient (ie tending to reduce the impressed horizontalgradient) The action of heat transport by the waves andaxisymmetric flows will be considered in the next section

132 AxisymmetricWave Transition and LinearInstability Theory

The previous section indicated the conditions underwhich baroclinic waves occur in the annulus and theirrole as a means of transferring heat upward and againstthe horizontal temperature gradient The Eady model ofbaroclinic instability has been commonly invoked as anidealized linearized conceptual model to account for theonset of waves from axisymmetric flow [Hide 1970 Hideand Mason 1975 Ghil and Childress 1987] Although theEady model is highly idealized it does seem to predict thelocation of the onset of large-amplitude waves remarkablyclose to the conditions actually observed at least at highTaylor number (note that the Eady problem in its ldquoclassi-cal formrdquo is inviscid) Apparent agreement can be madeeven closer if the Eady problem is modified to includeEkman boundary layers by replacing the w = 0 boundarycondition with the Ekman compatibility condition

w =E12

radic2Ro

nabla2ψ z = 0 1 (14)

where ψ is the stream function for the horizontal flowThis naturally brings in the Taylor number familiarto experimentalists (via Ekman number E) and leadsto a plausibly realistic envelope of instability at low

m = 1

2

3

4 103161003161000

= 10ndash4 10ndash3 10ndash2Δρρ

3160s 5

6789

101112

1314

15

10ndash8 10ndash7 10ndash6 10ndash5

108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1

Figure 14 Regime diagram based on the extension of Eadyrsquosbaroclinic instability theory to include Ekman layers and flathorizontal boundaries The wave number of maximum insta-bility is indicated by integer numbers and the transition curvesand contours of e-folding time are given on a Burger number(Bu sim see Hide and Mason [1975]) against T plot (Adaptedfrom Mason [1975] by permission of the Royal Society)

Taylor number (see Figure 14) supporting the hypothesis[Hide and Mason 1975] that the ldquolower symmetric transi-tionrdquo is frictionally dominated

The structure of the most rapidly growing instabilityhas certain characteristic features in terms of for examplephase tilts with height In the thermal annulus steadybaroclinic waves are also seen to exhibit many of thesefeatures as determined from experiment and numericalsimulation The extent to which Eady theory actually pro-vides a complete theoretical description of the instabilityproblem in annulus experiments however is a somewhatmore complicated question than it at first appears Thedominant instability in the Eady model relies on the exis-tence of horizontal temperature gradients on horizontalboundaries for the required change of sign in the lateralgradient of quasi-geostrophic potential vorticity partqpartyfor instability [eg Charney and Stern 1962] Elsewherethe flow is constructed such that partqparty = 0 In prac-tice however strong horizontal mass transports in theEkman layers result in almost no horizontal tempera-ture gradients at the boundaries in reality partqparty changessign smoothly in the interior (eg see Figure 117c later)Thus instability of an internal baroclinic jet is arguablya more appropriate starting point preferably including a






















































Editors













ISBN 978-1-118-85593-5



10 9 8 7 6 5 4 3 2 1


CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1
























































Editors













ISBN 978-1-118-85593-5



10 9 8 7 6 5 4 3 2 1


CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1




















































Editors













ISBN 978-1-118-85593-5



10 9 8 7 6 5 4 3 2 1


CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1









Editors













ISBN 978-1-118-85593-5



10 9 8 7 6 5 4 3 2 1


CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1















ISBN 978-1-118-85593-5



10 9 8 7 6 5 4 3 2 1


CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





CONTENTS

Contributors vii

Preface xi

















v


vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





vi CONTENTS









Index 337


CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





CONTRIBUTORS

















vii


viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





viii CONTRIBUTORS


















CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





CONTRIBUTORS ix











PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1






PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





PREFACE









Germany


United Kingdom

xi



ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1






ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





ACKNOWLEDGMENTS



xiii




Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1







Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1






Paul F Linden









1


2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





2 INTRODUCTION

X Z





B equiv NHfL

(01)






INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





INTRODUCTION 3








R equiv U3

νN2L (02)




4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





4 INTRODUCTION

H


L

θρρ + Δρ







Turbulent

Intermittent

Laminar






INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1





INTRODUCTION 5


REFERENCES



























9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1




















9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1



















9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1

















9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1
















9



Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1






Warm water

(b)


(a)






OCEANS






















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1























(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1















(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1






(a) (b)

(c) (d)

(e)

10210100

Ω2 (rad2 s2)


10ndash2

Lowersymmetrical

Uppersymmetrical


Regularwaves

105 106 107 108

Irregular

a

bc

d

e

fSta

bili

ty p

ara

mete

r

10ndash1

1

002 k01 k

02 k

ΔT = 6 k

1 k2 k

(f)







=gαTd

[(b minus a)]2 (11)


(b) a Taylor number

T =2(b minus a)5

ν2d (12)



Pr =ν

κ (13)








w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1










w =E12

radic2Ro

nabla2ψ z = 0 1 (14)


m = 1

2

3

4 103161003161000


3160s 5

6789

101112

1314

15


108107106

Ta

10510410ndash4

10ndash3

10ndash2

Bu

10ndash1

1




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