Self-Exciting Fluid Dynamos

540 Pages·2019·81.068 MB·English

by  Keith Moffatt, Emmanuel Dormy

Self-ExcitingFluidDynamos
Exploringtheoriginsandevolutionofmagneticfieldsinplanets,starsandgalaxies,
this book gives a basic introduction to magnetohydrodynamics, and surveys the
observational data with particular focus on geomagnetism and solar magnetism.
Pioneering laboratory experiments thatseek toreplicate particular aspects offluid
dynamo action are also described. The authors provide a complete treatment of
laminar dynamo theory and of the mean-field electrodynamics that incorporates
the effects of random waves and turbulence. Both dynamo theory and its coun-
terpart, the theory of magnetic relaxation, are covered. Topological constraints
associated with conservation of magnetic helicity are thoroughly explored, and
majorchallengesareaddressedinareassuchasfast-dynamotheory,accretion-disc
dynamo theory and the theory of magnetostrophic turbulence. The book is aimed
atgraduate-levelstudentsinmathematics,physics,earthsciencesandastrophysics,
andwillbeavaluableresourceforresearchersatalllevels.
KEITH MOFFATT FRS is Emeritus Professor of Mathematical Physics at the
University of Cambridge. He has served as Head of the Department of Applied
MathematicsandTheoreticalPhysicsandasDirectoroftheIsaacNewtonInstitute
for Mathematical Sciences in Cambridge. A former editor of the Journal of Fluid
Mechanics,hehaspublishedpapersinfluiddynamicsandmagnetohydrodynamics
andwasapioneerinthedevelopmentoftopologicalfluiddynamics.HeisaFellow
of the Royal Society, a member of Academia Europæa, and a Foreign Member of
theAcademiesofFrance,Italy,theNetherlandsandtheUSA.Hehasbeenawarded
numerous prizes, most recently the 2018 Fluid Dynamics Prize of the American
PhysicalSociety.
EMMANUEL DORMY is a CNRS Directeur de Recherche in the Department
of Mathematics and its Applications at the Ecole Normale Supe´rieure (ENS) in
Paris. He is also Professor at the ENS and at the Ecole Polytechnique, where he
teaches different aspects of fluid dynamics. Convinced of the need to embrace all
aspects of the dynamo problem, in 2006 he started a research group at the ENS
which promotes an interdisciplinary approach and jointly studies all geophysical
and astrophysical aspects of dynamo theory. He also founded and directed the
Dynamo-GDRE, which promotes exchanges among researchers working on all
aspectsofdynamotheorythroughoutEuropeandbeyond,andheorganiseswidely
attendedmeetings.
CambridgeTextsinAppliedMathematics
EDITORIALBOARD
ProfessorM.J.AblowitzUniversityofColoradoBoulder
ProfessorS.DavisNorthwesternUniversity,Illinois
ProfessorE.J.HinchUniversityofCambridge
ProfessorA.IserlesUniversityofCambridge
DrJ.OckendonUniversityofOxford
ProfessorP.J.OlverUniversityofMinnesota
Theaimofthisseriesistoprovideafocusforpublishingtextbooksinappliedmathematicsatthe
advancedundergraduateandbeginninggraduatelevel.Thebooksaredevotedtocoveringcertain
mathematicaltechniquesandtheoriesandexploringtheirapplications.
AlltitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslisting,visitwww.cambridge.org/mathematics.
GeometricandTopologicalInference
JEAN-DANIELBOISSONNAT,FRE´DE´RICCHAZAL&MARIETTEYVINEC
IntroductiontoMagnetohydrodynamics(2ndEdition)
P.A.DAVIDSON
AnIntroductiontoStochasticDynamics
JINQIAODUAN
Singularities:Formation,StructureandPropagation
J.EGGERS&M.A.FONTELOS
Microhydrodynamics,BrownianMotionandComplexFluids
MICHAELD.GRAHAM
DiscreteSystemsandIntegrability
J.HIETARINTA,N.JOSHI&F.W.NIJHOFF
AnIntroductiontoPolynomialandSemi-AlgebraicOptimization
JEANBERNARDLASSERRE
NumericalLinearAlgebra
HOLGERWENDLAND
Self-Exciting Fluid Dynamos
KEITH MOFFATT
UniversityofCambridge
EMMANUEL DORMY
EcoleNormaleSupe´rieure,Paris
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
79AnsonRoad,#06–04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107065871
DOI:10.1017/9781107588691
©KeithMoffattandEmmanuelDormy2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2019
PrintedintheUnitedKingdombyTJInternationalLtd.Padstow,Cornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Moffatt,H.K.(HenryKeith),author.|Dormy,Emmanuel,author.
Title:Self-excitingfluiddynamos/KeithMoffatt(UniversityofCambridge),
EmmanuelDormy(EcoleNormaleSupe´rieure,Paris).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Series:Cambridgetextsinappliedmathematics|Includesbibliographicalreferencesandindexes.
Identifiers:LCCN2018047454|ISBN9781107065871(hardback:alk.paper)
Subjects:LCSH:Fluiddynamics.|Magnetohydrodynamics.|
Dynamotheory(Cosmicphysics)|Geophysics.
Classification:LCCQA912.M652019|DDC523.01/886–dc23
LCrecordavailableathttps://lccn.loc.gov/2018047454
ISBN978-1-107-06587-1Hardback
ISBN978-1-108-71705-2Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Dedicatedtothememoryof
GeorgeKeithBatchelor
1920–2000

Contents
Preface pagexvii
PARTI BASICTHEORYANDOBSERVATIONS 1
1 Introduction 3
1.1 Whatisdynamotheory? 3
1.2 Historicalbackground 4
1.2.1 Thegeodynamo 4
1.2.2 Thesolardynamo 8
1.3 Thehomopolardiscdynamo 10
1.4 Axisymmetricandnon-axisymmetricsystems 12
2 MagnetokinematicPreliminaries 20
2.1 StructuralpropertiesoftheB-field 20
2.1.1 Solenoidality 20
2.1.2 TheBiot–Savartintegral 21
2.1.3 Linesofforce(‘B-lines’) 21
2.1.4 Helicityandfluxtubelinkage 22
2.2 Chirality 25
2.2.1 Therattleback:aprototypeofdynamicchirality 26
2.2.2 Meanresponseprovokedbychiralexcitation 28
2.3 Magneticfieldrepresentations 30
2.3.1 Sphericalpolarcoordinates 30
2.3.2 Toroidal/poloidaldecomposition 32
2.3.3 Axisymmetricfields 34
2.3.4 Two-dimensionalfields 34
2.4 Relationsbetweenelectriccurrentandmagneticfield 36
2.4.1 Ampe`re’slaw 36
viii Contents
2.4.2 Multipoleexpansionofthemagneticfield 37
2.4.3 Axisymmetricfields 38
2.5 Force-freefields 39
2.5.1 Force-freefieldsinsphericalgeometry 41
2.6 Lagrangianvariablesandmagneticfieldevolution 43
2.6.1 Changeoffluxthroughamovingcircuit 44
2.6.2 Faraday’slawofinduction 45
2.6.3 Galileaninvarianceofthepre-Maxwellequations 45
2.6.4 Ohm’slawinamovingconductor 46
2.7 Kinematicallypossiblevelocityfields 47
2.8 Freedecaymodes 48
2.8.1 Toroidaldecaymodes 49
2.8.2 Poloidaldecaymodes 50
2.8.3 Behaviourofthedipolemoment 51
2.9 FieldsexhibitingLagrangianchaos 53
2.10 Knottedfluxtubes 54
2.10.1 Twistsurgery 54
2.10.2 Helicityofaknottedfluxtube 56
3 Advection,DistortionandDiffusion 59
3.1 Alfve´n’stheoremandrelatedresults 59
3.1.1 Conservationofmagnetichelicity 60
3.2 Theanalogywithvorticity 62
3.3 Theanalogywithscalartransport 64
3.4 Maintenanceofafluxropebyuniformirrotationalstrain 64
3.5 Astretchedfluxtubewithhelicity 66
3.6 Anexampleofacceleratedohmicdiffusion 67
3.7 Equationforvectorpotentialandflux-functionunder
particularsymmetries 68
3.7.1 Two-dimensionalcase 69
3.7.2 Axisymmetriccase 69
3.8 Shearingofaspace-periodicmagneticfield 70
3.9 Oscillatingshearflow 73
3.9.1 Thecaseofsteadyrotationoftheshearingdirection 75
3.10 Fielddistortionbydifferentialrotation 76
3.11 Effect of plane differential rotation on an initially uniform
field:fluxexpulsion 77
3.11.1 Theinitialphase 78
3.11.2 Theultimatesteadystate 79
3.11.3 Flowdistortionbytheflowduetoalinevortex 81
Contents ix
3.11.4 Theintermediatephase 82
3.11.5 Fluxexpulsionwithdynamicback-reaction 84
3.11.6 FluxexpulsionbyGaussianangularvelocity
distribution 84
3.12 Fluxexpulsionforgeneralflowswithclosedstreamlines 86
3.13 Expulsionofpoloidalfieldbymeridionalcirculation 88
3.14 Generationoftoroidalfieldbydifferentialrotation 89
3.14.1 Theinitialphase 90
3.14.2 Theultimatesteadystate 90
3.15 Topologicalpumpingofmagneticflux 93
4 TheMagneticFieldoftheEarthandPlanets 99
4.1 Planetarymagneticfieldsingeneral 99
4.2 Satellitemagneticfields 104
4.3 SphericalharmonicanalysisoftheEarth’sfield 106
4.4 Variationofthedipolefieldoverlongtime-scales 113
4.5 Parametersandphysicalstateofthelowermantleandcore 116
4.6 TheneedforadynamotheoryfortheEarth 117
4.7 Thecore–mantleboundaryandinteractions 118
4.8 PrecessionoftheEarth’sangularvelocity 119
5 AstrophysicalMagneticFields 121
5.1 Thesolarmagneticfield 121
5.2 VelocityfieldintheSun 122
5.2.1 Surfaceobservations 122
5.2.2 Helioseismology 124
5.3 Sunspotsandthesolarcycle 126
5.4 ThegeneralpoloidalmagneticfieldoftheSun 131
5.5 Magneticstars 132
5.6 Magneticinteractionbetweenstarsandplanets 134
5.7 Galacticmagneticfields 136
5.8 Neutronstars 140
PARTII FOUNDATIONSOFDYNAMOTHEORY 143
6 LaminarDynamoTheory 145
6.1 Formalstatementofthekinematicdynamoproblem 145
6.2 Rate-of-straincriterion 146
6.3 Rateofchangeofdipolemoment 148
6.4 Theimpossibilityofaxisymmetricdynamoaction 149
6.4.1 Ultimatedecayofthetoroidalfield 150