Pseudo-spectral methods for HPC of MHD turbulence

Luca BiferaleDept. of Physics and INFN University of Rome “Tor Vergata”

biferale@roma2.infn.it

HPC Methods for Computational Fluid Dynamics and Astrophysics @Cineca

h"ps://www.fisica.uniroma2.it/~biferale/HPC-LEAP.html

WHYPSEUDO-SPECTRAL?1)  MULTI-SCALESACCURACY+EXPONENTIALACCURACYFORDERIVATIVES2)  POTENTIALTOOLTOPERFORMEXPERIMENTSINSILICO:EXACTCONTROLOFTHEDEGREES-OF-FREEDOM

• Toomanyturbulences?• Canwedisentangleuniversalfromnon-universalproperXes?• CanweunderstanduniversalproperXes?• Does‘compuXng’mean‘understanding’?(Computoergosum?)

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

+periodicboundarycondiXons

3DCASE:MAINLYUNSOLVED!

HOMOGENEOUS&ISOTROPICMHDTURBULENCE

-homogeneous-isotropic-Gaussian-white-noiseinXme-large-scale

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

Reynolds number ~ (Non-Linear)/(Linear terms)

Fully Developed Turbulence:1. Strongly non-linear & non-perturbative system

COMPLEXPHYSICSWITHSIMPLEFLOWS

2. Out of Equilibrium (non perturbative)Dissipative anomaly 3. Small-scales PDFs strongly non-Gaussian

Anomalous scaling

COMPLEXPHYSICSWITHSIMPLEFLOWS

1.  inerXalrangeofscales:powerlaw(anomalous)2.  extensionincreaseswithReynolds!

4. Many-body problem:

THE ENERGY CASCADE:

COMPLEXPHYSICSWITHSIMPLEFLOWS

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

VISCOUSTERMEXACTLYINTEGRATEDADAMS-BASHFORTH2°ORDER

WHATABOUTMHD????

HTTP://WWW.P3DFFT.NET

WHYPSEUDO-SPECTRAL?1)  MULTI-SCALESACCURACY+EXPONENTIALACCURACYFORDERIVATIVES2)  POTENTIALTOOLTOPERFORMEXPERIMENTSINSILICO:EXACTCONTROLOFTHEDEGREES-OF-FREEDOM

• Toomanyturbulences?• Canwedisentangleuniversalfromnon-universalproperXes?• CanweunderstanduniversalproperXes?• Does‘compuXng’mean‘understanding’?(Computoergosum?)

LORENTZ

ADVECTION+STRETCHING

LORENTZ

STRETCHING+ADVECTION

+ boundarycondiXons

KinemaXcs+DissipaXonareinvariantunderRotaXon+TranslaXonNon-universalstaXsXcalbehaviour<->Anisotropy

Smallscalesvslargescales

Turbulentjet 3dConvecXveCell ShearFlow

Thesimplestsetof0-ranktensor(SCALAR)observable:

LongitudinalStructureFuncXons

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

rotaXonalinvariantoperator

FOLIATION!!!

+ so(3) ->

J=0

J=2

J=4 Largescalephysics:

allsectorscoupledby

forcingterms

scaling?

WorkingHypothesis

projecXononeachsectorhasauniversalscalingexponent,dependingonthatsectoronly.

Dependencyonlargescalephysicsshowsuponlyinprefactors

Purepowerlawsonlyineachseparetedsector:

MatchingInfra-RedboundarycondiXons:

prefactor cannot be universal

Aboutuniversalityofscalingexponentsnothingcanbesaidrigorously,atleastfortheNSeqs.

RecoveryofIsotropySmall-ScalesUniversality

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)

isotropicsector

beforeso(3)decomposiXon

Localslopes

ComparisonofscalingproperXes:isotropicsector(j=0,m=0)vsundecomposedstructurefuncXon

xyz

Arad et al. PRL 82, 5040 (1999)

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

PARTIALREFERENCELISTOFINTERESTFORTHISMINI-COURSE

MulLfractalsEulerian&LagrangianTurbulence:A.Arneodo,R.Benzi,J.Berg,L.Biferaleetal.Phys.Rev.Le".100,254504(2008)R.Benzi,L.Biferale,G.Paladinetal.Phys.Rev.Le".67,2299(1991)L.Biferale,G.Boffe"a,A.Celanietal.Phys.Rev.Le".93,064502(2004)L.Biferale,G.Boffe"a,A.Celanietal.Phys.Fluids.17,021701(2005)L.Biferale,E.Bodenschatz,M.Cencinietal.PhysFluids20,065103(2008)Eulerian&LagrangianTurbulence:U.Frisch.‘Turbulence’(CambridgeUniversityPress,1996)R.Benzi,L.Biferale,R.Fisheretal.Journ.FluidMech.653,221(2010)L.Biferale,E.Bodenschatz,M.Cencinietal.PhysFluids20,065103(2008)G.Falkovich,K.Gawȩdzki,M.Vergassola.Rev.Mod.Phys.73,913(2001)T.Gotoh,D.FukayamaandT.Nakano.Phys.Fluids14,1065(2002)T.Ishihara,T.Gotoh,andY.Kaneda.Ann.Rev.FluidMech.41,165(2009)V.L’vovandI.Procaccia.Phys.Rev.E52,384(1995)V.L'vovandI.Procaccia.Phys.Rev.Le".76,2896(1996)J.P.L.C.SalazarandL.R.Collins.Ann.Rev.FluidMech.41,405(2009)B.SawfordandP.K.Yeung.Phys.Fluids23,091704(2011)K.R.Sreenivasan.Ann.Rev.FluidMech.25,435(1997)F.ToschiandE.Bodenschatz.Ann.Rev.FluidMech.41,375(2009)P.K.Yeung.Ann.Rev.FluidMech.34,115(2002)T.Baebler,L.Biferale.andA.S.Lano"ePhysRevERap.Comm85025301R2(012)InerLalParLcles:J.Bec,L.Biferaleetal.J.FluidMech.550,349(2006);J.BecetalPhys.Rev.Le".98,084502(2007)L.Biferale,J.Bec,M.Cencinietal.Phys.Fluids18,081702(2006)Anisotropies:L.BiferaleandI.Procaccia.Phys.Rep.414,43(2005)L.BiferaleandF.Toschi.Phys.Rev.Le".86,4831(2001)L.Biferale,E.Calzavarini,A.S.Lano"eetal.PhysicaA338,194(2004)K.Chang,G.P.BewleyandE.Bodenschatz.physics.flu-dynarXiv:1102.1197(2011)B.Jacob,L.Biferale,G.Iugoetal.Phys.Fluids16,4135(2004)S.Kurien,K.R.Sreenivasan.Phys.Rev.E62,2206(2000)X.Shen,Z.Warha].Phys.Fluids14,2432(2002)