Theory and Modeling of Rotating Fluids: Convection, Inertial Waves and Precession

541 Pages·2017·29.218 MB·English

by  Keke Zhang, Xinhao Liao

THEORY AND MODELING OF
ROTATING FLUIDS
A systematic account of the theory and modeling of rotating fluids that highlights the
remarkable advances in the area and brings researchers and postgraduate students in
atmospheres, oceanography, geophysics, astrophysics and engineering to the frontiers of
research.Sufficientmathematicalandnumericaldetailisprovidedinavarietyofgeome-
tries, such that the analysis and results can be readily reproduced, and many numerical
tablesareincludedtoenablereaderstocompareorbenchmarktheirowncalculations.Tra-
ditionally,therearetwodisjointedtopicsinrotatingfluids:convectivefluidmotiondriven
by buoyancy, discussed by Chandrasekhar (1961), and inertial waves and precession-
driven flow, described by Greenspan (1968). Now, for the first time in book form, the
authorspresentaunifiedtheoryforthreetopics–thermalconvection,inertialwaves,and
precession-drivenflow–todemonstratethattheseseeminglycomplicated,andpreviously
disconnected,problemsbecomemathematicallysimpleintheframeworkofanasymptotic
approachthatincorporatestheessentialcharacteristicsofrotatingfluids.
KEKE ZHANG,professorofGeophysicalandAstrophysicalFluidDynamicsattheUni-
versity of Exeter, UK, obtained his BSc in 1982 from the University of Nanjing, China,
andhisMScin1985andhisPhDin1987fromtheUniversityofCalifornia,LosAngeles,
USA.HeisaFellowofboththeAmericanGeophysicalUnionandtheRoyalAstronomi-
calSociety.ProfessorZhanghasauthoredover180publicationsinpeer-reviewedscientific
journals
XINHAO LIAO, professor of Celestial Dynamics at the Chinese Academy of Sciences,
obtainedhisBScin1983andhisPhDin1989fromtheUniversityofNanjing.
Established in 1952, the Cambridge Monographs on Mechanics series has
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areasaswavepropagation,fluiddynamics,theoreticalgeophysics,combus-
tion, and the mechanics of solids. Authors are encouraged to write for a
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pretation and experimental data, where appropriate. Whilst the research
literature is expected to be a major source for the content of the book,
authorsshouldaimtosynthesisenewresultsratherthanjustsurveythem.
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THEORY AND MODELING OF
ROTATING FLUIDS
Convection, Inertial Waves and Precession
KEKE ZHANG
UniversityofExeter
XINHAO LIAO
ChineseAcademyofSciences
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Informationonthistitle:www.cambridge.org/9780521850094
(cid:2)c KekeZhangandXinhaoLiao2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2017
PrintedintheUnitedStatesofAmericabySheridanBooks,Inc.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Names:Zhang,Keke.|Liao,Xinhao.
Title:Theoryandmodelingofrotatingfluids:convection,inertialwaves,and
precession/KekeZhang,UniversityofExeter,XinhaoLiao,ChineseAcademyofSciences.
Description:Cambridge:CambridgeUniversityPress,2017.|
Series:Cambridgemonographsonmechanicsseries|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017004135|ISBN9780521850094(hardback:alk.paper)
Subjects:LCSH:Rotatingmassesoffluid.|Fluidmechanics.|Fluiddynamics.
Classification:LCCQA913.Z432017|DDC532/.0595–dc23
LCrecordavailableathttps://lccn.loc.gov/2017004135
ISBN978-0-521-85009-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pagexi
Part1 FundamentalsofRotatingFluids 1
1 BasicConceptsandEquationsforRotatingFluids 3
1.1 Introduction 3
1.2 EquationsofMotioninRotatingSystems 4
1.3 TheHeatEquation 6
1.4 TheBoussinesqEquations 7
1.5 TheKineticEnergyEquation 10
1.6 Taylor–ProudmanTheoremandThermalWindEquation 11
1.7 AUnifiedApproach 13
Part2 InertialWavesinUniformlyRotatingSystems 15
2 Introduction 17
2.1 Formulation 17
2.2 FrequencyBound|σ|≤1 19
2.3 SpecialCases:σ =0andσ =±1 21
2.4 Orthogonality 23
2.5 ThePoincaréEquation 25
3 InertialModesinRotatingNarrow-gapAnnuli 27
3.1 Formulation 27
3.2 AxisymmetricInertialOscillations 29
3.3 GeostrophicMode 31
3.4 Non-axisymmetricInertialWaves 32
4 InertialModesinRotatingCylinders 35
4.1 Formulation 35
4.2 AxisymmetricInertialOscillations 36
4.3 GeostrophicMode 41
4.4 Non-axisymmetricInertialWaves 43
5 InertialModesinRotatingSpheres 50
5.1 Formulation 50
v
vi Contents
5.2 GeostrophicMode 52
5.3 EquatoriallySymmetricModes:m=0 54
5.4 EquatoriallySymmetricModes:m≥1 60
5.5 EquatoriallyAntisymmetricModes:m=0 71
5.6 EquatoriallyAntisymmetricModes:m≥1 75
5.7 AnExactNonlinearSolutioninRotatingSpheres 81
6 InertialModesinRotatingOblateSpheroids 83
6.1 Formulation 83
6.2 GeostrophicMode 91
6.3 EquatoriallySymmetricModes:m=0 92
6.4 EquatoriallySymmetricModes:m≥1 94
6.5 EquatoriallyAntisymmetricModes:m=0 96
6.6 EquatoriallyAntisymmetricModes:m≥1 99
6.7 AnExactNonlinearSolutioninRotatingSpheroids 102
7 AProofofCompletenessofInertialModesinRotatingChannels 105
7.1 SignificanceoftheCompletenessofInertialModes 105
7.2 Bessel’sInequalityandParseval’sEquality 107
7.3 AProofoftheCompletenessRelation 109
8 IndicationsofCompletenessofInertialModesinRotatingSpheres 118
8.1 SeekingSignsofCompleteness 118
8.2 AProofoftheVanishingDissipation-typeIntegral 119
Part3 PrecessionandLibrationinNon-uniformlyRotatingSystems 127
9 Introduction 129
9.1 Non-uniformRotation:PrecessionandLibration 129
9.2 Precession/LibrationinDifferentGeometries 130
9.3 KeyParametersandReferenceFrames√ 134
9.4 AsymptoticExpansionWithoutUsing Ek 135
10 FluidMotioninPrecessingNarrow-gapAnnuli 138
10.1 Formulation 138
10.2 ConditionsforResonance √ 141
10.3 AsymptoticSolutionatResonancewith(cid:3)= 3√ 142
10.4 AsymptoticSolutionatResonancewith(cid:3)=1/ 3 152
10.5 LinearNumericalAnalysis 155
10.6 NonlinearDirectNumericalSimulation 157
10.7 Comparison:Analyticalvs.Numerical 159
10.8 AByproduct:TheViscousDecayFactor 160
11 FluidMotioninPrecessingCircularCylinders 164
11.1 Formulation 164
11.2 ConditionsforResonance 166
11.3 DivergenceoftheInviscidPrecessingSolution 168
11.4 GeneralAsymptoticSolutionfor0<Ek(cid:5)1 172
Contents vii
11.5 AsymptoticSolutionatPrimaryResonances 180
11.6 LinearNumericalAnalysisUsingSpectralMethods 187
11.7 NonlinearPropertiesofWeaklyPrecessingFlow 190
11.8 NumericalSimulationUsingFiniteElementMethods 193
11.9 NonlinearPrecessingFlowatPrimaryResonances 195
11.9.1 DecompositionofNonlinearFlowintoInertialModes 195
11.9.2 TheStructureofNonlinearPrecessingFlow 199
11.9.3 SearchforTriadicResonance 205
11.10AByproduct:TheViscousDecayFactor 209
12 FluidMotioninPrecessingSpheres 213
12.1 Formulation 213
12.2 AsymptoticExpansionandResonance 215
12.3 AsymptoticSolution 217
12.4 NonlinearDirectNumericalSimulation 224
12.5 Comparison:Analyticalvs.Numerical 226
12.6 NonlinearEffects:MeanAzimuthalFlow 227
12.7 AByproduct:TheViscousDecayFactor 229
13 FluidMotioninLongitudinallyLibratingSpheres 231
13.1 Formulation 231
13.2 AsymptoticSolutions 232
13.2.1 WhyResonanceCannotOccur 232
13.2.2 AsymptoticAnalysis 233
13.2.3 ThreeFundamentalModesExcited 239
13.3 LinearNumericalSolution 244
13.4 NonlinearDirectNumericalSimulation 246
14 FluidMotioninPrecessingOblateSpheroids 250
14.1 Formulation 250
14.2 InviscidSolution 252
14.3 ExactNonlinearSolution 258
14.4 ViscousSolution 260
14.5 PropertiesofNonlinearPrecessingFlow 268
14.6 AByproduct:TheViscousDecayFactor 273
15 FluidMotioninLatitudinallyLibratingSpheroids 276
15.1 Formulation 276
15.2 AnalyticalSolution:Non-resonantLibratingFlow 279
15.3 AnalyticalSolution:ResonantLibratingFlow 283
15.4 NonlinearDirectNumericalSimulation 293
15.5 Comparison:Analyticalvs.Numerical 293
Part4 ConvectioninUniformlyRotatingSystems 297
16 Introduction 299
16.1 RotatingConvectionvs.Precession/Libration 299
viii Contents
16.2 KeyParametersforRotatingConvection 300
16.3 RotationalConstraintonConvection 302
16.4 TypesofRotatingConvection 303
16.4.1 ViscousConvectionMode 303
16.4.2 InertialConvectionMode 305
16.4.3 TransitionalConvectionMode 306
16.5 ConvectioninVariousRotatingGeometries 307
16.5.1 RotatingAnnularChannels 307
16.5.2 RotatingCircularCylinders 308
16.5.3 RotatingSpheresorSphericalShells 309
17 ConvectioninRotatingNarrow-gapAnnuli 313
17.1 Formulation 313
17.2 AFinite-differenceMethodforNonlinearConvection 316
17.3 StationaryViscousConvection 318
17.3.1 GoverningEquations 318
17.3.2 AsymptoticSolutionfor(cid:3)(Ta)1/6(cid:5)O(1) 320
17.3.3 AsymptoticSolutionfor(cid:3)(Ta)1/6=O(1) 325
17.3.4 NumericalSolutionUsingaGalerkin-tauMethod 327
17.3.5 Comparison:Analyticalvs.Numerical 329
17.3.6 NonlinearPropertiesofStationaryConvection 330
17.4 OscillatoryViscousConvection 332
17.4.1 GoverningEquations 332
17.4.2 SymmetrybetweenTwoDifferentOscillatory
Solutions 334
17.4.3 AsymptoticSolutionsSatisfyingtheBoundary
Condition 335
17.4.4 Comparison:Analyticalvs.Numerical 343
17.4.5 ComparisonwithanUnboundedRotatingLayer 348
17.4.6 NonlinearPropertieswith(cid:3)=O(Ta−1/6) 352
17.4.7 NonlinearPropertieswith(cid:3)(cid:6)O(Ta−1/6) 354
17.5 ViscousConvectionwithCurvatureEffects 356
17.5.1 OnsetofViscousConvection 356
17.5.2 NonlinearPropertiesofViscousConvection 359
17.6 InertialConvection:Non-axisymmetricSolutions 366
17.6.1 AsymptoticExpansion 366
17.6.2 Non-dissipativeThermalInertialWave 367
17.6.3 AsymptoticSolutionwithStress-freeCondition 369
17.6.4 AsymptoticSolutionwithNo-slipCondition 373
17.6.5 NumericalSolutionUsingaGalerkinSpectralMethod 383
17.6.6 Comparison:Analyticalvs.Numerical 385
17.6.7 NonlinearPropertiesofInertialConvection 386
17.7 InertialConvection:AxisymmetricTorsionalOscillation 393
Contents ix
18 ConvectioninRotatingCylinders 396
18.1 Formulation 396
18.2 ConvectionwithStress-freeCondition 399
18.2.1 AsymptoticSolutionforInertialConvection 399
18.2.2 AsymptoticSolutionforViscousConvection 405
18.2.3 NumericalSolutionUsingaChebyshev-tauMethod 408
18.2.4 Comparison:Analyticalvs.Numerical 410
18.3 ConvectionwithNo-slipCondition 412
18.3.1 AsymptoticSolutionforInertialConvection 412
18.3.2 AsymptoticSolutionforViscousConvection 418
18.3.3 NumericalSolutionUsingaGalerkin-typeMethod 419
18.3.4 Comparison:Analyticalvs.Numerical 421
18.3.5 EffectofThermalBoundaryCondition 424
18.3.6 AxisymmetricInertialConvection 426
18.4 TransitiontoWeaklyTurbulentConvection 430
18.4.1 AFiniteElementMethodforNonlinearConvection 430
18.4.2 InertialConvection:FromSingleInertialModetoWeak
Turbulence 431
18.4.3 ViscousConvection:FromSidewall-localizedModetoWeak
Turbulence 435
19 ConvectioninRotatingSpheresorSphericalShells 439
19.1 Formulation 439
19.2 NumericalSolutionusingToroidal/PoloidalDecomposition 442
19.2.1 GoverningEquationsunderToroidal/PoloidalDecomposition 442
19.2.2 NumericalAnalysisforStress-freeorNo-slipCondition 444
19.2.3 SeveralNumericalSolutionsfor0<Ek(cid:5)1 447
19.2.4 NonlinearEffects:DifferentialRotation 452
19.3 LocalAsymptoticSolution:ASmall-gapAnnularModel 459
19.3.1 TheLocalandQuasi-geostrophicApproximation 459
19.3.2 AsymptoticRelationfor0<Ek(cid:5)1 461
19.3.3 Comparison:Asymptoticvs.Numerical 463
19.4 GlobalAsymptoticSolutionwithStress-freeCondition 464
19.4.1 HypothesesforAsymptoticAnalysis 464
19.4.2 AsymptoticAnalysisforInertialConvection 465
19.4.3 SeveralAnalyticalSolutionsforInertialConvection 470
19.4.4 DifferentialRotationCannotbeSustainedbyInertialConvection 474
19.4.5 AsymptoticAnalysisforViscousConvection 476
19.4.6 TypicalAsymptoticSolutionsforViscousConvection 479
19.4.7 NonlinearEffects:DifferentialRotationinViscousConvection 481
19.5 GlobalAsymptoticSolutionwithNo-slipCondition 484
19.5.1 HypothesesforAsymptoticAnalysis 484
19.5.2 AsymptoticAnalysisforInertialConvection 485