Modeling of turbulence using filtering, and the absence of ``bottleneck’’ in MHD Annick Pouquet Jonathan Pietarila-Graham &, Darryl Pablo Mininni^ and David Montgomery ! & MPI, Imperial College ! Dartmouth College ^ Universidad de Buenos Aires Cambridge, October 2008

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 Sun Scaling 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 Cadarache Numerical dynamo at a magnetic Prandtl number P M =/=1 (Nore et al., PoP, 4, 1997) and P M ~ 0.01 (Ponty et al., PRL, 2005). I n liquid sodium, P M ~ : does it matter?   R H=2R Bourgoin et al PoF 14 (‘02), 16 (‘04)… Experimental dynamo in 2007

ITER (Cadarache) Small-scale

The MHD equations Multi-scale interactions, high R runs P is the pressure, j = ∇ × B is the current, F is an external force, ν is the viscosity, η the resistivity, v the velocity and B the induction (in Alfvén velocity units); incompressibility is assumed, and .B = 0. ______ Lorentz force

Parameters in MHD R V = U rms L 0 / ν >> 1 Magnetic Reynolds number R M = U rms L 0 / η * Magnetic Prandtl number: P M = R M / R V = ν / η P M is high in the interstellar medium. P M is low in the solar convection zone, in the liquid core of the Earth, in liquid metals and in laboratory experiments And P M =1 in most numerical experiments until recently … Energy ratio E M /E V or time-scale ratio  NL /  A with  NL = l/u l and  A =l/B 0 (Quasi-) Uniform magnetic field B 0 Amount of correlations or of magnetic helicity Boundaries, geometry, cosmic rays, rotation, stratification, …

Small magnetic Prandtl number P M << 1: ~ in liquid metals Resolve two dissipative ranges, the inertial range and the energy containing range And Run at a magnetic Reynolds number R M larger than some critical value (R M governs the importance of stretching of magnetic field lines over Joule dissipation) Resort to modeling of small scales

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 modeling Slide from Comte, Cargese Summer school on turbulence, July 2007 Direct Numerical Simulations (DNS) versus Large Eddy Simulations (LES) Resolve all scales vs. Model (many) small scales 1D space & Spectral space

) Probability Higher 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 α V & α M, but not their sources (vorticity & current) Equations preserve invariants (in modified - filtered L 2 --> H 1 form) McIntyre (mid ‘70s), Holm (2002), Marsden, Titi, …, Montgomery & AP (2002)

Lagrangian-averaged model for Navier-Stokes & MHD Non-dissipative case ∂v/∂t + u s · ∇ v = −v j ∇ u j s − ∇ P * + j × B s, ∂B s /∂t + u s · ∇ B s = B s · ∇ u s The 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 E V = /2 for NS: E v α = /2 MHD: E T α, H c α and H M α are invariant (+ Alfven theorem)

Lagrangian-averaged NS & MHD dissipative equations ∂v/∂t + u s · ∇ v = −v j ∇ u j s − ∇ P * + j × B s + ∇ 2 v ∂B s /∂t + u s · ∇ B s = B s · ∇ u s +  ∇ 2 B B ~ k 2 B s --> hyperdiffusive term

Navier-Stokes: vortex filaments Alpha model DNS

MHD: magnetic energy structures at 50% threshold (nonlinear phase of a P M =1 dynamo regime) Alpha model, 64 3 DNS, grid

MHD decay NCAR on grid points Visualization freeware: VAPOR Zoom on individual current structures: folding and rolling-up Mininni et al., PRL 97, (2006) Magnetic field lines in brown At small scale, long correlation length along the local mean magnetic field (k // ~ 0)

3D Navier-Stokes: intermittency Chen et al., 1999; Kerr, 2002 Pietarila-Graham et al., PoF 20, DNS: X Largest filter length & smaller cost: more intermitency

Third-order scaling law for fluids (4/5th law) stemming from energy conservation v is the rough velocity and u s 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 ,  u s 3 ~ r 3, we have an advection by a smooth field, or E u ~ k -3, hence E  ~ E uv ~ k -1  r ~

Third-order scaling law stemming from energy conservation v is the rough velocity and u s is the smooth velocity,  is the filter length and  is the energy transfer rate Two ranges: –For small , Kolmogorov law –For large ,  u s 3 ~ r 3, we have an advection by a smooth field, or E u ~ k -3, hence E  ~ E uv ~ k -1 But we observe rather ~ k +1 Solid line:  model, for large  (k  =3)  r ~ Why? k +1 kk

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 DNS run ~ 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 DNS run ~ 10 -4

3D Run with large  (2  /10) Black:  u // 3 (r=2  /10) < 0.01 DNS run ~ 10 -4

3D Run with large  (2  /10) Black:  u // 3 (r=2  /10) < 0.01 DNS run ~ ``rigid bodies’’ (no stretching): u s (k) = v(k) / [ 1 +  2 k 2 ] and take limit of large  : the flow is advected by a uniform field U (no degrees of freedom)

u s =constant v ~ k 2 u s for large  u s v ~ k 2 ~ k E  (k) E  (k) ~ k +1 ``rigid bodies’’: u s (k) = v(k) / [ 1 +  2 k 2 ] and take limit of large  : the flow is advected by a uniform field U s (no degrees of freedom)

3D Run with large  (2  /10) Black:  u // 3 (l=2  /10) < 0.01 DNS run ~ Solid line:  model, for large  (k  =3) Dash line: same  model without regions of negligible transfer

Kinetic Energy Spectra in MHD Solid: DNS, Dash: LAMHD, Dot: Navier-Stokes, kk

Energy Fluxes Solid/dash: LAMHD (Elsässer variables) Dots: alpha-fluid Circulation conservation is broken by Lorentz force

Magnetic Energy Spectra Solid: DNS, grid Dash: LAMHD, kk

Energy transfer in MHD is more non local than for fluids Transfer of kinetic energy to magnetic energy from mode Q (x axis) to mode K =10 (top panel) K =20 K =30 Alexakis et al., PRE 72,

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

Comparison in 2D with LAMHD: cancellation exponent  (thick lines) & magnetic dissipation (thin lines) Graham et al., PRE 72, 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 resistivity associated with a lack of equipartition in the small scales  turb ~ E k V - E k M < 0 DNS Rädler; AP, mid ‘80s

Dynamo regime at P M =1: the growth of magnetic energy at the expense of kinetic energy : all three runs display similar temporal evolutions and energy spectra DNS at grid (solid line) and α runs ( or 64 3 grids, (dash or dot) Beltrami ABC flow at k 0 =3

Comparison of DNS and Lagrangian model R M = 41, R v =820, P M = 0.05 dynamo Solid line: DNS : LAMHD Linear scale in inset Comparable growth rate and saturation level of Direct Numerical Simulation and model

Beyond testing … Solid: DNS, , R ~ 1100 Dash: LAMHD, Dot: DNS, Temporal evolution of total energy (top), kinetic (bottom) and magnetic energies

Temporal evolution of total enstrophy j 2 +  2 Solid: DNS, Dash: LAMHD, Dot: DNS, 256 3

Magnetic energy spectra compensated by k 3/2 Solid: DNS, Dash: LAMHD, Dot: DNS, 256 3

Summary of results For 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 interactions The  -model in MHD allows a sizable savings over DNS (X6 in resolution for second-order correlations ) Applications: low-P M (experiments, Earth) and high P M (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, …

Conclusions Deal 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 flows –Models (many …) –Adaptive Mesh Refinement, keeping accuracy –Combine and contrast all approaches!

Conclusions Deal 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 flows –Models (many …) –Adaptive Mesh Refinement, keeping accuracy –Combine and contrast all approaches! Pietarila-Graham et al., PRE 76, (2007); PoF 20, (2008); and arxiv:

Thank you for your attention!

Scientific framework Understanding 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 challenges Pseudo-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 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 step A MHD grid, needed in order to resolve inertial interactions between scales, will require much more and represents a substantial computing challenge And add kinetic effects …

Some questions Are Alfvén 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 ν t (k,t) (à la Chollet-Lesieur) (no modification to the induction equation with K c 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 flow at a magnetic Prandtl number below P M ~ 0.25, down to 0.02 (Ponty et al., PRL 94, , 2005). Turbulent dynamo at P M ~ on the Roberts flow (Mininni, 2006). Turbulent dynamo at P M ~ 10 -6, using second-order EDQNM closure (Léorat et al., 1980) Critical magnetic Reynolds number R M c for dynamo action