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High-order codes for astrophysical turbulence

This document covers high-order numerical codes tailored for astrophysical turbulence, highlighting the Pencil Code's features and capabilities. Key topics include Kolmogorov spectrum behaviors, nonlinearity, and power law implications in cosmic structures. The study investigates direct simulations at resolutions like 10243 and discusses challenges such as hyperviscosity, normal diffusivity, and the bottleneck effect in turbulence. Additionally, it delves into the efficiency of high-order schemes, parallelization, and various modules for different astrophysical applications. This work is relevant for researchers focused on turbulent flow dynamics in astrophysical contexts.

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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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