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Modeling of turbulence using filtering, and the absence of ``bottleneck’’ in MHD

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

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Modeling of turbulence using filtering, and the absence of ``bottleneck’’ in MHD

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

  2. * The Sun, and other stars • * The Earth, and other planets - • including extra-solar planets • The solar-terrestrial interactions, • the magnetospheres, … Many parameters and dynamical regimes Many scales, eddies and waves interacting

  3. 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)

  4. Surface (1 bar) radial magnetic fields for Jupiter, Saturne & EarthversusUranus& Neptune(16-degree truncation, Sabine Stanley, 2006) Axially dipolar Quadrupole~dipole

  5. W R H=2R W Taylor-Green turbulent flow at Cadarache Bourgoin et al PoF 14 (‘02), 16 (‘04)… Numerical 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? Experimental dynamo in 2007

  6. Small-scale ITER (Cadarache)

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

  8. Parameters in MHD • RV = Urms L0 / ν >> 1 • Magnetic Reynolds number RM = Urms L0 / η * Magnetic Prandtl number: PM = RM / RV= ν / η PM ishigh 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=1in most numerical experiments until recently … • Energy ratio EM/EVor time-scale ratioNL/A with NL= l/uland A=l/B0 • (Quasi-) Uniform magnetic field B0 • Amount of correlations <v.B> or of magnetic helicity <A.B> • Boundaries, geometry, cosmic rays, rotation, stratification, …

  9. Small magnetic Prandtl number • PM << 1: ~ 10-6 in liquid metals Resolve two dissipative ranges, the inertial range and the energy containing range And Run at a magnetic Reynolds number RM larger than some critical value (RM governs the importance of stretching of magnetic field lines over Joule dissipation) Resort to modeling of small scales

  10. 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

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

  12. Higher grid resolutions, higher Reynolds numbers, more multi-scale interactions: study the 2Dcase (in MHD, energy cascades to small scales, and it models anisotropy …) ) • Probability

  13. Lagrangian-averaged (or alpha) Model for Navier-Stokes and MHD (LAMHD):the velocity & induction are smoothed on lengths αV & αM, 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)

  14. Lagrangian-averaged model for Navier-Stokes & MHD Non-dissipative case • ∂v/∂t + us · ∇v = −vj ∇u j s − ∇ P* + j × Bs, • ∂Bs/∂t + us · ∇Bs = Bs · ∇us • The above equations have invariantsthat differ in their formulation from those of the primitive equations: the filtering prevents the small scales from developing • For example, kinetic energy invariant EV = <v2>/2 for NS: Evα= < v2 + α2ω2 >/2 MHD: ETα, Hcα andHMα are invariant (+ Alfven theorem)

  15. Lagrangian-averaged NS & MHD dissipative equations • ∂v/∂t + us · ∇v = −vj ∇u j s − ∇ P* + j × Bs + ∇2 v • ∂Bs/∂t + us · ∇Bs = Bs · ∇us + ∇2 B B ~ k2 Bs --> hyperdiffusive term

  16. Navier-Stokes: vortex filaments DNS Alpha model

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

  18. MHD decay simulation @ NCAR on 15363gridpoints Visualization freeware: VAPORhttp://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 brown At small scale, long correlation length along the local mean magnetic field (k// ~ 0)

  19. 3D Navier-Stokes: intermittency DNS: X Largest filter length & smaller cost: more intermitency Chen et al., 1999;Kerr, 2002Pietarila-Graham et al., PoF 20, 035107 2008

  20. Third-order scaling law for fluids(4/5th law) stemming from energy conservation  r ~ < v us2 + [2 / r2 ] us3 > • v 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

  21. Third-order scaling law stemming from energy conservation  r ~ < v us2 + [2 / r2 ] us3 > • v 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+1 k • k+1 Solid line:  model, for large  (k =3) Why?

  22. 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  ~ 10-4 DNS run

  23. Regions with  /< > ~ 0 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 3D Run with large  (2 /10)  ~ 10-4 DNS run

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

  25. Black:u//3(r=2/10)< 0.01 3D Run with large  (2 /10)  ~ 10-4 DNS run ``rigid bodies’’ (no stretching): us(k) = v(k) / [ 1 + 2 k2] and take limit of large : the flow is advected by a uniform field U (no degrees of freedom)

  26. us=constant v ~ 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 of large : the flow is advected by a uniform field Us(no degrees of freedom)

  27. Black:u//3(l=2/10)< 0.01 3D Run with large  (2 /10)  ~ 10-4 DNS run Solid line:  model, for large  (k =3) Dash line: same  model without regions of negligible transfer

  28. Kinetic Energy Spectra in MHD k Solid: DNS, 15363 Dash: LAMHD, 5123 Dot: Navier-Stokes , 5123

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

  30. Magnetic Energy Spectra k Solid: DNS, 15363 grid Dash: LAMHD, 5123

  31. 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, 046301

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

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

  34. 2D - MHD, forcedKinetic (top) and magnetic (bottom) energiesand squared mag. potential growth: DNS vs. LAMHD

  35. Inverse cascade of <A2> associated with a negative eddy resistivityassociated with a lack of equipartition in the small scalesturb~ EkV - EkM< 0 Rädler; AP, mid ‘80s DNS

  36. Dynamo regime at PM=1: the growth ofmagnetic energyat the expense ofkinetic energy :all three runs display similar temporal evolutions and energy spectra DNS at 2563 grid (solid line) and α runs ( 1283 or 643 grids, (dash or dot) Beltrami ABC flow at k0=3

  37. Comparison of DNS and Lagrangian model • RM = 41, Rv=820, PM = 0.05 dynamo • Solid line: DNS • - - - : LAMHD • Linear scale in inset Comparable growth rate and saturation level of Direct Numerical Simulation and model

  38. Beyond testing … Temporal evolution of total energy (top), kinetic (bottom) and magnetic energies Solid: DNS, 15363, R ~ 1100 Dash: LAMHD, 2563 Dot: DNS, 2563

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

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

  41. 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-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, …

  42. 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!

  43. 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, 056310 (2007); PoF 20, 035107 (2008); and arxiv:0806.2054

  44. Thank you for your attention!

  45. 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 flowswith magnetic fields in three dimensions, taking into account long-range interactions between eddies and waves, and the geometrical shape of small-scale eddies

  46. 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 computationon a grid of 20483 points up to the peak of dissipationwill take ~ 22 days on 2000single core IBM POWER5 processors with a 1.9-GHz clock cycle, using ~230 s/ time step • A 40963 MHDgrid, needed in order to resolve inertial interactions between scales,will require much more and represents a substantial computing challenge • And add kinetic effects …

  47. 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? Some questions

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

  49. 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)

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