Modeling of turbulence using filtering,and the absence of ``bottleneck in MHDAnnick Pouquet

Jonathan Pietarila-Graham& , Darryl Holm@, Pablo Mininni^

and David Montgomery! & MPI, Lindau @ Imperial College ! Dartmouth College ^ Universidad de Buenos AiresCambridge, October 2008 pouquet@ucar.edu

Many parameters and dynamical regimes

Many scales, eddies and waves interacting

* The Sun, and other stars* The Earth, and other planets -including extra-solar planets

The solar-terrestrial interactions, the magnetospheres,

Extreme events in active regions on the SunScaling exponents of structure functions for magnetic fields in solar active regions (differences versus distance r, and assuming self-similarity)

Abramenko, review (2007)

Surface (1 bar) radial magnetic fields for Jupiter, Saturne & Earth versus Uranus & Neptune

(16-degree truncation, Sabine Stanley, 2006) Axially dipolar

Quadrupole ~ dipole

Taylor-Green turbulent flow at CadaracheNumerical dynamo at a magnetic Prandtl number PM=/=1 (Nore et al., PoP, 4, 1997) and PM ~ 0.01 (Ponty et al., PRL, 2005).In liquid sodium, PM ~ 10-6 : does it matter?Bourgoin et al PoF 14 (02), 16 (04)Experimental dynamo in 2007

ITER (Cadarache)

Small-scale

The MHD equationsMulti-scale interactions, high R runsP is the pressure, j = B is the current, F is an external force, is the viscosity, the resistivity, v the velocity and B the induction (in Alfvn velocity units); incompressibility is assumed, and .B = 0.______ Lorentz force

Parameters in MHDRV = Urms L0 / >> 1Magnetic Reynolds number RM = Urms L0 /

* Magnetic Prandtl number: PM = RM / RV = / PM is high in the interstellar medium. PM is low in the solar convection zone, in the liquid core of the Earth, in liquid metals and in laboratory experiments And PM=1 in most numerical experiments until recently

Energy ratio EM/EV or time-scale ratio NL/A with NL= l/ul and A=l/B0(Quasi-) Uniform magnetic field B0 Amount of correlations or of magnetic helicity

Boundaries, geometry, cosmic rays, rotation, stratification,

Small magnetic Prandtl numberPM

Equations for the alpha model in fluids and MHD

* Some results comparing to DNS

The various small-scale spectra arising for fluids

The MHD case

Some other tests both in 2D and in 3D

* An example : The generation of magnetic fields at low magnetic Prandtl number and the contrast between two models

* Conclusion

Numerical modelingSlide from Comte, Cargese Summer school on turbulence, July 2007Direct Numerical Simulations(DNS)versusLarge Eddy Simulations(LES)

Resolve all scalesvs. Model (many) small scales1D space & Spectral space

)ProbabilityHigher grid resolutions, higher Reynolds numbers, more multi-scale interactions: study the 2D case

(in MHD, energy cascades to small scales, and it models anisotropy )

Lagrangian-averaged (or alpha) Model for Navier-Stokes and MHD (LAMHD):the velocity & induction are smoothed on lengths V & M, but not their sources (vorticity & current)Equations preserve invariants (in modified - filtered L2 --> H1 form)McIntyre (mid 70s), Holm (2002), Marsden, Titi, , Montgomery & AP (2002)

Lagrangian-averaged model for Navier-Stokes & MHD Non-dissipative case

v/t + us v = vj u j s P P* + j Bs,

Bs/t + us Bs = Bs usThe above equations have invariants that differ in their formulation from those of the primitive equations: the filtering prevents the small scales from developing

For example, kinetic energy invariant EV = /2 for NS: Ev = < v2 + 22 >/2

MHD: ET, Hc and HM are invariant (+ Alfven theorem)

Lagrangian-averaged NS & MHD dissipative equations

v/t + us v = vj u j s P P* + j Bs + 2 v

Bs/t + us Bs = Bs us + 2 B

B ~ k2 Bs --> hyperdiffusive term

Navier-Stokes: vortex filamentsAlpha modelDNS

MHD: magnetic energy structures at 50% threshold (nonlinear phase of a PM=1 dynamo regime)Alpha model, 643DNS, 2563 grid

MHD decay simulation @ NCAR on 15363 grid pointsVisualization freeware: VAPOR http://www.cisl.ucar.edu/hss/dasg/software/vapor Zoom on individual current structures: folding and rolling-up Mininni et al., PRL 97, 244503 (2006)Magnetic field lines in brownAt small scale, long correlation length along the local mean magnetic field (k// ~ 0)

3D Navier-Stokes: intermittencyChen et al., 1999; Kerr, 2002 Pietarila-Graham et al., PoF 20, 035107 2008DNS: XLargest filter length& smaller cost:more intermitency

Third-order scaling law for fluids(4/5th law) stemming from energy conservationv is the rough velocity and us is the smooth velocity, is the filter length and is the energy transfer rate

A priori, two scaling ranges:For small , Kolmogorov law (at high Reynolds number)

For large , us3 ~ r3, we have an advection by a smooth field, or Eu ~ k-3, hence E ~ Euv ~ k-1

r ~ < v us2 + [2 / r2 ] us3 >

Third-order scaling law stemming from energy conservationv is the rough velocity and us is the smooth velocity, is the filter length and is the energy transfer rate

Two ranges:For small , Kolmogorov law

For large , us3 ~ r3, we have an advection by a smooth field, or Eu ~ k-3, hence E ~ Euv ~ k-1

But we observe rather ~ k+1Solid line: model, for large (k =3) r ~ < v us2 + [2 / r2 ] us3 >Why?k+1k

Regions with /< > ~ 0 Black: u//3(r=2/10) < 0.01 Filling factor ff of regions with very low energy transfer ff~ 0.26 for DNS DNSrun ~ 10-4

Regions with /< > ~ 0 3D Run with large (2 /10) Black: u//3(r=2/10) < 0.01 Filling factor ff of regions with very low energy transfer (at scales smaller than ): ff~ 0.67 for LA-NS Versus ff~ 0.26 for DNS DNSrun ~ 10-4

3D Run with large (2 /10) Black: u//3(r=2/10) < 0.01

DNSrun ~ 10-4

3D Run with large (2 /10) Black: u//3(r=2/10) < 0.01 DNSrun ~ 10-4``rigid bodies (no stretching):

us(k) = v(k) / [ 1 + 2 k2]and take limit oflarge : the flow is advectedby a uniform field U (no degrees of freedom)

us=constantv ~ k2 us for large usv ~ k2 ~ k E(k) E(k) ~ k+1``rigid bodies:

us(k) = v(k) / [ 1 + 2 k2]and take limit oflarge : the flow is advectedby a uniform field Us (no degrees of freedom)

3D Run with large (2 /10) Black: u//3(l=2/10) < 0.01

DNSrun ~ 10-4Solid line: model, for large (k =3)Dash line: same model withoutregions of negligible transfer

Kinetic Energy Spectra in MHDSolid: DNS, 15363Dash: LAMHD, 5123

Dot: Navier-Stokes , 5123k

Energy FluxesSolid/dash: LAMHD (Elssser variables)

Dots: alpha-fluidCirculation conservation is broken by Lorentz force

Magnetic Energy SpectraSolid: DNS, 15363 gridDash: LAMHD, 5123k

Energy transfer in MHD is more non local than for fluidsTransfer of kinetic energy to magnetic energyfrom mode Q (x axis)

to mode K =10 (top panel)

K =20

K =30 Alexakis et al., PRE 72, 046301

Sorriso-Valvo et al., P. of Plas. 9 (2002)Currentsheets in 2D MHD

DNS

Comparison in 2D with LAMHD: cancellation exponent (thick lines)& magnetic dissipation (thin lines) Graham et al., PRE 72, 045301 r (2005)Solid: DNS

2D - MHD, forced

Kinetic (top) and

magnetic (bottom) energies

and

squared mag. potential growth: DNS vs. LAMHD

Inverse cascade of

associated with a negative eddy resistivityassociated with a lack of equipartition in the small scales

turb~ EkV - EkM < 0 DNSRdler;AP, mid 80s

Dynamo regime at PM=1: the growth of magnetic energy at the expense of kinetic energy : all three runs display similar temporal evolutions and energy spectraDNS at 2563 grid (solid line) and runs ( 1283 or 643 grids, (dash or dot)

Beltrami ABC flow at k0=3

Comparison of DNS and Lagrangian modelRM = 41, Rv=820, PM = 0.05 dynamo

Solid line: DNS- - - : LAMHDLinear scale in inset

Comparable growth rate and saturation level of Direct Numerical Simulation and model

Beyond testing

Solid: DNS, 15363, R ~ 1100Dash: LAMHD, 2563 Dot: DNS, 2563Temporal evolution of total energy (top),kinetic (bottom) andmagnetic energies

Temporal evolution of total enstrophy j2 +2Solid: DNS, 15363Dash: LAMHD, 2563Dot: DNS, 2563

Magnetic energy spectra compensated by k3/2Solid: DNS, 15363Dash: LAMHD, 2563Dot: DNS, 2563

Summary of resultsFor large , for fluids, the model has large portions of the flow with low energy transfer (67% vs. 26% for DNS) This results in an enhancement of spectra at small scales, akin to a bottleneck

This phenomenon is absent in MHD, perhaps because of nonlocal interactionsThe -model in MHD allows a sizable savings over DNS (X6 in resolution for second-order correlations)

Applications: low-PM (experiments, Earth) and high PM (interstellar medium) dynamos, MHD turbulence spectra, parametric studies (e.g., effect of resolution on high-order statistics, energy spectra, anisotropy, role of velocity-magnetic field correlations, role of magnetic helicity, )

There are other models in MHD,

ConclusionsDeal with peta and exa-scale computers: parallelism! But keep the absolute time of computation and usage of memory at their lowest, and watch for accuracy.

Collaborations on large projects (shared codes, shared data, )

Be creative:Tricks, as symmetric flowsModels (many )Adaptive Mesh Refinement, keeping accuracy

Combine and contrast all approaches!

ConclusionsDeal with peta and exa-scale computers: parallelism! But keep the absolute time of computation and usage of memory at their lowest, and watch for accuracy.

Collaborations on large projects (shared codes, shared data, ) GHOST: Geophysical High-Order Suite for Turbulence

Be creative:Tricks, as symmetric flowsModels (many )Adaptive Mesh Refinement, keeping accuracy

Combine and contrast all approaches!

Pietarila-Graham et al., PRE 76, 056310 (2007); PoF 20, 035107 (2008); and arxiv:0806.2054

Thank you for your attention!

Scientific frameworkUnderstanding the processes by which energy is distributed and dissipated down to kinetic scales, and the role of nonlinear interactions and MHD turbulence, e.g. in the Sun and for space weather

Understanding Cluster observations in preparation for a new remote sensing NASA mission (MMS: Magnetospheric Multi-Scale)

Modeling of turbulent flows with magnetic fields in three dimensions, taking into account long-range interactions between eddies and waves, and the geometrical shape of small-scale eddies

Computational challengesPseudo-spectral 3D-MHD code parallelized using MPI, periodic boundary conditions & 2/3 de-aliasing rule, Runge-Kutta temporal scheme of various orders, runs for ~ 10 turnover times at the highest Reynolds number possible in order to obtain multi-scale interactions. Parallel FFT with a 2D domain decomposition in real and Fourier space with linear scaling up to thousands of processors. Planned pencil distribution to scale to a larger number of processors. MHD computation on a grid of 20483 points up to the peak of dissipation will take ~ 22 days on 2000 single core IBM POWER5 processors with a 1.9-GHz clock cycle, using ~230 s/ time stepA 40963 MHD grid, needed in order to resolve inertial interactions between scales, will require much more and represents a substantial computing challengeAnd add kinetic effects

Some questions

Are Alfvn vortices, as observed e.g. in the magnetosphere, present in MHD at high Reynolds number, and what are their properties?

Is another scaling range possible at scales smaller than where the weak turbulence spectrum is observed (non-uniformity of theory)?

How to quantify anisotropy in MHD, including in the absence of a large-scale magnetic field? How much // vs. perp. transfer is there?

Universality, e.g. does a large-scale coherent forcing versus a random forcing influence the outcome?

And how can one travel through parameter space, at high Reynolds number, thus at high 3D resolution?

Large-Eddy Simulation (LES)Add to the momentum equation a turbulent viscosity t(k,t) ( la Chollet-Lesieur) (no modification to the induction equationwith Kc a cut-off wave-number

Taylor-Green flow

Energy spectrum difference for two different formulations of LES based on two-point closure EDQNM

Noticeable improvement in the small-scale spectrum

(Baerenzung et al., 2008)

The first numerical dynamo within a turbulent flowat a magnetic Prandtl number below PM ~ 0.25,down to 0.02 (Ponty et al., PRL 94, 164502, 2005).Turbulent dynamo at PM ~ 0.002 on the Roberts flow (Mininni, 2006).Turbulent dynamo at PM ~ 10-6 , using second-order EDQNM closure (Lorat et al., 1980)Critical magnetic Reynolds number RMc for dynamo action

Pablo, duane compare location and intensity of jmaxExponential filter As opposed to top hat or sinc=[d-dF]/2 (see Sorriso-Valvo et al. PoP, 2002)- - 128. 64