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Pseudo-spectral methods for HPC of MHD turbulence
Luca BiferaleDept. of Physics and INFN University of Rome “Tor Vergata”
HPC Methods for Computational Fluid Dynamics and Astrophysics @Cineca
h"ps://www.fisica.uniroma2.it/~biferale/HPC-LEAP.html
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WHYPSEUDO-SPECTRAL?1) MULTI-SCALESACCURACY+EXPONENTIALACCURACYFORDERIVATIVES2) POTENTIALTOOLTOPERFORMEXPERIMENTSINSILICO:EXACTCONTROLOFTHEDEGREES-OF-FREEDOM
• Toomanyturbulences?• Canwedisentangleuniversalfromnon-universalproperXes?• CanweunderstanduniversalproperXes?• Does‘compuXng’mean‘understanding’?(Computoergosum?)
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smallparXcles:drag,addedmass,li]force,etc...
TurbulenceorTurbulences?
Flows with additives:Advection-diffusion-reaction of passive scalar/vectors (temperature, magnetic field, chemical reactions, etc...)Advection-diffusion of active scalars/vectors (convection, magnetic dinamo)Polymers (drag reduction) Bubbles/Droplets (two phase flows, rain formation, etc...)Swimmers (cooperative hydrodynamical interactions)
tracer bubble heavy
+boundarycondiXons
temperaturemagneXcfield
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+periodicboundarycondiXons
3DCASE:MAINLYUNSOLVED!
HOMOGENEOUS&ISOTROPICMHDTURBULENCE
-homogeneous-isotropic-Gaussian-white-noiseinXme-large-scale
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L.B.,G.Boffe"a,A.Celani,B.Devenish,A.Lano"eandF.ToschiPhys.Rev.Le+.93,064502,2004
K41 prediction
Multifractal prediction
Multifractal prediction
mean-field (k41) prediction
ACCELERATION
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Reynolds number ~ (Non-Linear)/(Linear terms)
Fully Developed Turbulence:1. Strongly non-linear & non-perturbative system
COMPLEXPHYSICSWITHSIMPLEFLOWS
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2. Out of Equilibrium (non perturbative)Dissipative anomaly 3. Small-scales PDFs strongly non-Gaussian
Anomalous scaling
COMPLEXPHYSICSWITHSIMPLEFLOWS
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1. inerXalrangeofscales:powerlaw(anomalous)2. extensionincreaseswithReynolds!
4. Many-body problem:
THE ENERGY CASCADE:
COMPLEXPHYSICSWITHSIMPLEFLOWS
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Number crunching approach: computo ergo sum.
state-of-the-art DNS (Kaneda’s group):
Isotropic, homogeneous Fully Periodic FlowsPseudo-Spectral Methods.
Resolution 12000^3
Reynolds : 10^7,Storage of 1 velocity configuration: 30 Tbyte
Moral: easy to saturate any computing power(present and/or future)
astrophys. flow atmosph. flow laboratory flow
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VISCOUSTERMEXACTLYINTEGRATEDADAMS-BASHFORTH2°ORDER
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WHATABOUTMHD????
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HTTP://WWW.P3DFFT.NET
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WHYPSEUDO-SPECTRAL?1) MULTI-SCALESACCURACY+EXPONENTIALACCURACYFORDERIVATIVES2) POTENTIALTOOLTOPERFORMEXPERIMENTSINSILICO:EXACTCONTROLOFTHEDEGREES-OF-FREEDOM
• Toomanyturbulences?• Canwedisentangleuniversalfromnon-universalproperXes?• CanweunderstanduniversalproperXes?• Does‘compuXng’mean‘understanding’?(Computoergosum?)
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LORENTZ
ADVECTION+STRETCHING
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LORENTZ
STRETCHING+ADVECTION
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+ boundarycondiXons
KinemaXcs+DissipaXonareinvariantunderRotaXon+TranslaXonNon-universalstaXsXcalbehaviour<->Anisotropy
Smallscalesvslargescales
Turbulentjet 3dConvecXveCell ShearFlow
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Thesimplestsetof0-ranktensor(SCALAR)observable:
LongitudinalStructureFuncXons
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Arad, V. L’Vov I. Procaccia PRE 59, 6753 (1999). Arad et al. PRL 82, 5040 (1999).
Arad et al. PRL 81, 5330 (1998).
3d rotation
Setof3n*(2j+1)EigenfuncXonsofgroupofrotaXonsin3d:
DecomposiXonintermsof(irreducible)invariantsubset-labelledbyq,j=0,1,2,…
n-ranktensorwhichdepends
ona3dvector
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rotaXonalinvariantoperator
FOLIATION!!!
+ so(3) ->
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J=0
J=2
J=4 Largescalephysics:
allsectorscoupledby
forcingterms
scaling?
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WorkingHypothesis
projecXononeachsectorhasauniversalscalingexponent,dependingonthatsectoronly.
Dependencyonlargescalephysicsshowsuponlyinprefactors
Purepowerlawsonlyineachseparetedsector:
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MatchingInfra-RedboundarycondiXons:
prefactor cannot be universal
Aboutuniversalityofscalingexponentsnothingcanbesaidrigorously,atleastfortheNSeqs.
RecoveryofIsotropySmall-ScalesUniversality
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WeperformedaDNSofaRandom-KolmogorovFlow
PeriodicboundarycondiXons
256x256x256
Hyperviscosity
HomogeneousbutAnisotropic
L.B. and F. Toschi, PRL 86, 4831 (2001)
L.B. I. Daumont, A. Lanotte and F. Toschi. PRE. 66, 056306 (2002)
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isotropicsector
beforeso(3)decomposiXon
Localslopes
ComparisonofscalingproperXes:isotropicsector(j=0,m=0)vsundecomposedstructurefuncXon
xyz
Arad et al. PRL 82, 5040 (1999)
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scalingism-independent
dimensional
j=6
j=4
j=0
Recoveryofisotropy
L.B. I. Daumont, A.S. Lanotte and F. Toschi PRE. 66, 056306 (2002)
ExtendingLumley’sanisotropictheory
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PARTIALREFERENCELISTOFINTERESTFORTHISMINI-COURSE
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