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High-order codes for astrophysical turbulence

High-order codes for astrophysical turbulence. Axel Brandenburg (Nordita, Stockholm) Boris Dintrans (Univ. Toulouse, CNRS). (...just google for Pencil Code ). Kolmogorov spectrum. nonlinearity. constant flux e [ cm 2 /s 3 ]. [ cm 3 /s 2 ]. E ( k ). a =2/3, b = - 5/3. e. e. k.

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High-order codes for astrophysical turbulence

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  1. High-order codes for astrophysical turbulence Axel Brandenburg (Nordita, Stockholm) Boris Dintrans (Univ. Toulouse, CNRS) (...just google for Pencil Code)

  2. Kolmogorov spectrum nonlinearity constant flux e [cm2/s3] [cm3/s2] E(k) a=2/3, b= -5/3 e e k

  3. Scintillations Big Power Law in Sky • Armstrong, Cordes, Rickett 1981, Nature • Armstrong, Rickett, Spangler 1995, ApJ

  4. Simulation of turbulence at 10243 (Porter, Pouquet,& Woodward 1998)

  5. Direct vs hyper at 5123 Biskamp & Müller (2000, Phys Fluids 7, 4889) Normal diffusivity With hyperdiffusivity

  6. Ideal hydro: should we be worried? • Why this k-1 tail in the power spectrum? • Compressibility? • PPM method • Or is real?? • Hyperviscosity destroys entire inertial range? • Can we trust any ideal method? • Needed to wait for 40963direct simulations

  7. Non-ideal equations

  8. Hyperviscous, Smagorinsky, normal height of bottleneck increased Haugen & Brandenburg (PRE, astro-ph/0402301) onset of bottleneck at same position Inertial range unaffected by artificial diffusion

  9. Bottleneck effect: 1D vs 3D spectra Why did wind tunnels not show this? Compensated spectra (1D vs 3D)

  10. Relation to ‘laboratory’ 1D spectra Dobler, et al (2003, PRE 68, 026304)

  11. MAGMA

  12. PencilCode • Started in Sept. 2001 with Wolfgang Dobler • High order (6th order in space, 3rd order in time) • Cache & memory efficient • MPI, can run PacxMPI (across countries!) • Maintained/developed by ~40 people (SVN) • Automatic validation (over night or any time) • Max resolution so far 10243 , 4096 procs • Isotropic turbulence • MHD, passive scl, CR • Stratified layers • Convection, radiation • Shearing box • MRI, dust, interstellar • Self-gravity • Sphere embedded in box • Fully convective stars • geodynamo • Other applications • Homochirality • Spherical coordinates

  13. Pencil formulation • In CRAY days: worked with full chunks f(nx,ny,nz,nvar) • Now, on SGI, nearly 100% cache misses • Instead work with f(nx,nvar), i.e. one nx-pencil • No cache misses, negligible work space, just 2N • Can keep all components of derivative tensors • Communication before sub-timestep • Then evaluate all derivatives, e.g. call curl(f,iA,B) • Vector potential A=f(:,:,:,iAx:iAz), B=B(nx,3)

  14. Switch modules • magnetic or nomagnetic (e.g. just hydro) • hydro or nohydro (e.g. kinematic dynamo) • density or nodensity (burgulence) • entropy or noentropy (e.g. isothermal) • radiation or noradiation (solar convection, discs) • dustvelocity or nodustvelocity (planetesimals) • Coagulation, reaction equations • Chemistry (reaction-diffusion-advection equations) Other physics modules: MHD, radiation, partial ionization, chemical reactions, selfgravity

  15. High-order schemes • Alternative to spectral or compact schemes • Efficiently parallelized, no transpose necessary • No restriction on boundary conditions • Curvilinear coordinates possible (except for singularities) • 6th order central differences in space • Non-conservative scheme • Allows use of logarithmic density and entropy • Copes well with strong stratification and temperature contrasts

  16. (i) High-order spatial schemes Main advantage: low phase errors Near boundaries: x x x x x x x x x ghost zones interior points

  17. Wavenumber characteristics

  18. Higher order – less viscosity

  19. In the exercises:

  20. Less viscosity – also in shocks

  21. (ii) High-order temporal schemes Main advantage: low amplitude errors 2N-RK3 scheme (Williamson 1980) 2nd order 3rd order 1st order

  22. Low order cheaper?

  23. Shock tube test

  24. Increase in # of auto tests

  25. Evolution of code size User meetings: 2005 Copenhagen 2006 Copenhagen 2007 Stockholm 2008 Leiden 2009 Heidelberg 2010 New York

  26. Pencil Code check-ins

  27. Faster and bigger machines

  28. Cartesian box MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

  29. Vector potential • B=curlA, advantage: divB=0 • J=curlB=curl(curlA) =curl2A • Not a disadvantage: consider Alfven waves B-formulation A-formulation 2nd der once is better than 1st der twice!

  30. Comparison of A and B methods

  31. 256 processor run at 10243

  32. Online data reduction and visualization non-helically forced turbulence

  33. Scalars on periphery of the box

  34. MRI turbulenceMRI = magnetorotational instability 2563 w/o hypervisc. t = 600 = 20 orbits 5123 w/o hypervisc. Dt = 60 = 2 orbits

  35. Vorticity and Density See poster by Tobi Heinemann on density wave excitation!

  36. Convection with shear and W Käpylä et al (2008) with rotation without rotation

  37. It can take a long time Rm=121, By, 512^3 LS dynamo not always excited

  38. Intrinsic Calculation Ray direction Transfer equation & parallelization Processors Analytic Solution:

  39. Analytic Solution: Communication Ray direction The Transfer Equation & Parallelization Processors

  40. Analytic Solution: Intrinsic Calculation Ray direction The Transfer Equation & Parallelization Processors

  41. Winter School 11-22 January

  42. Euler potentials • Has zero magnetic helicity, A=a grad b • Strictly correct only for h=0 • Or if a and b don’t depend on the same coords

  43. Gauge Choice & Dissipation

  44. Paper arXiv:0907.1906

  45. Roberts flow dynamo • Agreement for t<8 • For smooth fields • Not for delta-correlated • Initial fields • Exponential growth (A) • Algebraic decay (EP)

  46. Spectrum for MHD turbulencewith imposed field (no dynamo!)

  47. Reasons for disagreement • because dynamo field is helical? • because field is three-dimensional? • none of the two? EP too restrictive? • because h is finite?

  48. Visual comparison

  49. Spurious growth for h=0 • The two agree, but are underresolved

  50. Simple decay problem • Exponential decay for A, but not for EP!?

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