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Statistical properties and structure of turbulent convection [Elektronische Ressource] / Dan Škandera

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Technische Universita¨t Mu¨nchenMax-Planck-Institut fu¨r PlasmaphysikStatistical Properties and Structureof Turbulent ConvectionˇDan SkanderaVollsta¨ndiger Abdruck der von der Fakulta¨t fu¨r Physik der TechnischenUniversita¨t Mu¨nchen zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaftengenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. Rudolf GrossPru¨fer der Dissertation: 1. Hon.-Prof. Dr. Sibylle Gu¨nter2. Univ.-Prof. Dr. Katharina KrischerDie Dissertation wurde am 3.4.2007 bei der Technischen Universita¨t Mu¨ncheneingereicht und durch die Fakulta¨t fu¨r Physik am 18.7.2007 angenommen.Vˇenova´no m´ym nejbliˇzˇs´ım, bez jejichˇz l´asky, pochopen´ı, tolerancea bezmezn´e podpory by tato pra´ce nemohla vzniknout.ivAbstractThis thesis presents extensive numerical studies of turbulent convection. It fo-cuses on spectral properties of convection and its nonlinear dynamics. Withinthe frame of the thesis, several large-scaledirect numerical simulations are per-formed. Four particular systems are investigated and discussed in the work,namely: two- and three-dimensional hydrodynamic and magnetohydrodynamicturbulent convection. All systems are driven by a mean horizontal temperaturegradient. The numerical code uses a standard pseudospectral scheme. Spe-cial attention is paid to the comparison of various phenomenological theoriesof turbulence and their modifications for convective flows.

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Published 01 January 2007
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TechnischeUniversita¨tMu¨nchen
Max-Planck-Institutfu¨rPlasmaphysik

StatisticalPropertiesandStructure
ofTurbulentConvection

DanSkandera

Vollsta¨ndigerAbdruckdervonderFakulta¨tfu¨rPhysikderTechnischen
Universita¨tMu¨nchenzurErlangungdesakademischenGradeseines

DoktorsderNaturwissenschaften

genehmigtenDissertation.

Vorsitzender:Univ.-Prof.Dr.RudolfGross
Pru¨ferderDissertation:1.Hon.-Prof.Dr.SibylleGu¨nter
2.Univ.-Prof.Dr.KatharinaKrischer

DieDissertationwurdeam3.4.2007beiderTechnischenUniversita¨tMu¨nchen
eingereichtunddurchdieFakulta¨tfu¨rPhysikam18.7.2007angenommen.

Venova´nomy´mnejblizs´m,bezjejichzla´sky,pochopen´,tolerance
abezmezne´podporybytatopra´cenemohlavzniknout.

vi

Abstract

Thisthesispresentsextensivenumericalstudiesofturbulentconvection.Itfo-
cusesonspectralpropertiesofconvectionanditsnonlineardynamics.Within
theframeofthethesis,severallarge-scaledirectnumericalsimulationsareper-
formed.Fourparticularsystemsareinvestigatedanddiscussedinthework,
namely:two-andthree-dimensionalhydrodynamicandmagnetohydrodynamic
turbulentconvection.Allsystemsaredrivenbyameanhorizontaltemperature
gradient.Thenumericalcodeusesastandardpseudospectralscheme.Spe-
cialattentionispaidtothecomparisonofvariousphenomenologicaltheories
ofturbulenceandtheirmodicationsforconvectiveows.Structurefunctions,
probabilitydensityfunctionsandintermittencyeectsareexaminedinallindi-
vidualsystemsaswell.TheexistenceoftheBolgiano-Obukhovregimeoftur-
bulenceisconrmedinthetwo-dimensionalhydrodynamicsystem,whereasthe
three-dimensionalsystemsarefoundtooperateinKolmogorov-typeregimesof
turbulence.Thetwo-dimensionalmagnetohydrodynamicturbulentconvection
exhibitsquasi-oscillationsbetweentwodierentturbulentstates.Itisshown
thattheturbulentregimeoftwo-dimensionalmagnetoconvectiondependson
themutualalignmentbetweenvelocityandmagneticeld.

v

iv

Contents

Introduction1
1Fundamentalsofturbulentconvection5
1.1Generalconsiderations........................5
1.2Mathematicalformulation......................7
1.2.1BoussinesqMHDequations.................7
1.2.2Nondimensionalvariables..................8
1.3Turbulentregimes..........................11
1.3.1Kolmogorovphenomenology.................12
1.3.2Bolgiano-Obukhovphenomenology.............14
1.4Horizontaltemperaturegradient..................16
2Numericalcode19
2.1Pseudospectralscheme........................20
2.2Treatmentofaliasingerrors.....................21
2.3Temporaldiscretization.......................22
2.4Initialandboundaryconditions...................23
2.5Parallelizationandoptimization...................24
2.6Rayleigh-Be´nardtests........................26
32Dhydrodynamicconvection31
3.1Energybalance............................31
3.2Scalingofenergyspectra.......................33
3.3Nonlineartransport.........................36
3.3.1Transferfunctions......................36
3.3.2Shelltoshelltransfer.....................38
3.4Structurefunctions..........................39
3.5Intermittency.............................42
3.5.1Extendedself-similarity...................43
3.5.2Intermittencymodels....................43
3.5.3Probabilitydensityfunctions................46
3.6Realspacestructure.........................48
43Dhydrodynamicconvection53
4.1Idealinvariants............................54
4.2Energyspectra............................55
4.3Nonlineartransport.........................58
4.3.1Transferfunctions......................58

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CONTENTS

4.3.2Shelltoshelltransfer.....................60
4.4Structurefunctions..........................60
4.5Intermittency.............................63
4.5.1Intermittencymodels....................63
4.5.2Probabilitydensityfunctions................65
4.5.3Dissipativestructures....................67
52Dmagnetohydrodynamicconvection71
5.1Propertiesof2DMHD........................72
5.2Elsa¨sserelds.............................73
5.3Iroshnikov-Kraichnanphenomenology...............74
5.4Quasi-oscillationsbetweenturbulentregimes............75
5.4.1Integralcharacteristics....................76
5.4.2Iroshnikov-Kraichnanphase.................78
5.4.3Buoyancydominatedphase.................81
5.4.4Dynamicalmodelofquasi-oscillations...........83
5.5Structurefunctions..........................90
5.6Intermittency.............................93
5.7Spatialstructure...........................94
63Dmagnetohydrodynamicconvection99
6.1Roleofthemagneticeld......................100
6.1.1Idealinvariants........................100
6.1.2Conservativeformsofthenonlinearterms.........100
6.2Spectra................................102
6.2.1Energyspectra........................102
6.2.2Individualelds.......................104
6.3Inertial-rangedynamics.......................106
6.3.1Spectralenergytransfer...................106
6.3.2Detailedenergyexchange..................108
6.4Statisticalresults...........................110
6.4.1Structurefunctions......................111
6.4.2Probabilitydensityfunctions................111
6.4.3Intermittencymodels....................114
6.5Visualizationofelds.........................117
Conclusions121
Bibliography125

Introduction

Convectionrepresents,apartfromconductionandradiation,abasicmechanism
ofenergytransportinuids.Turbulenceisastateofowthatischaracterized
bychaoticandirregularvariationsofphysicalquantities,e.g.velocity,inspace
andtime.Bothphenomenaplayaveryimportantroleinanumberofphysical
systems.Prominentexamplesarethesolarconvectionzoneandtheliquidouter
coreoftheEarth.However,thefundamentalroleofturbulentconvectioncan
beidentiedinvariousothersystemstoo.TheEarth’satmosphereisheatedup
nearthesurface,andresultingconvectivemotionsinuenceitsglobaldynam-
ics.Dierentlevelsofsalinityinoceanicwatercauseconvectivecirculationsas
well.Observationsofallthesesystemsdemonstrateaninterestingfeature:the
presenceofconvectioninanaturalsystemoftenleadstoturbulentmotions.In
otherwords,wherevigorousconvectiontakesplace,turbulencecanbefound
aswell.Thisfundamentallinkagebetweenconvectionandturbulencecanbe
tracedtounderlyingphysicalreasons.Convection,beingtheconsequenceof
somekindofprimaryinstabilityinthesystem,providesasourceofenergyfor
turbulentuctuations.Theenergyinjectedtothesystemleadstoincreased
mixingratesandecienttransportofquantitiesadvectedbytheow.Con-
vectiveturbulenceisusuallydrivenbyameantemperaturegradient,butother
drivingmechanismsexistaswell,e.g.gradientsofchemicalsubstances.
MostofthevisiblematterintheUniverseisinaplasmastate.Plasmacan
bedescribed—undertheassumptionsgiveninsection1.1—asanelectrically
conductinguid.Inthiscase,convectionisaubiquitousagentinvolvedinthe
generationofmagneticelds.Thesearedynamicallyimportantforconvective
turbulence,andareabletochangeitspropertiesbynonlinearback-reactions
noticeably.Awell-knownexampleistheturbulentdynamo,i.e.thegeneration
ofalarge-scalemagnetic