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Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence

Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence. …and why First Order Smoothing seemed always better than it deserved to be. Brandenburg 1 , P. K äpylä 2,3 , A. Mohammed 4 1 Nordita, Copenhagen, Denmark 2 Kiepenheuer Institute, Freiburg, Germany

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Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence

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  1. Non-Fickian diffusion and Minimal Tau Approximation from numerical turbulence …and why First Order Smoothing seemed always better than it deserved to be Brandenburg1, P. Käpylä2,3, A. Mohammed4 1 Nordita, Copenhagen, Denmark 2 Kiepenheuer Institute, Freiburg, Germany 3 Dept Physical Sciences, Univ. Oulu, Finland 4 Physical Department, Oldenburg Univ., Germany

  2. MTA - the minimal tau approximation (remains to be justified!) 1) replace triple correlation by quadradatic 2) keep triple correlation 3) instead of now: 4) instead of diffusion eqn: damped wave equation i) any support for this proposal?? ii) what is tau??

  3. Purpose and background • Need for user-friendly closure model • Applications (passive scalar just benchmark) • Reynolds stress (for mean flow) • Maxwell stress (liquid metals, astrophysics) • Electromotive force (astrophysics) • Effects of stratification, Coriolis force, B-field • First order smoothing is still in use • not applicable for Re >> 1 (although it seems to work!) Brandenburg: non-Fickian diffusion

  4. Testing MTA: passive scalar “diffusion” primitive eqn fluctuations Flux equation  triple moment MTA closure >>1 (!)

  5. System of mean field equations mean concentration flux equation Damped wave equation, wave speed (causality!) Brandenburg: non-Fickian diffusion >>1 (!)

  6. Wave equation: consequences • late time behavior unaffected (ordinary diffusion) • early times: ballistic advection (superdiffusive) Illustration of wave-like behavior: small tau intermediate tau large tau Brandenburg: non-Fickian diffusion >>1 (!)

  7. Comparison with DNS • Finite difference • MPI, scales linearly • good on big Linux clusters • 6th order in space, 3rd order in time • forcing on narrow wavenumber band • Consider kf/k1=1.5 and 5 Brandenburg: non-Fickian diffusion

  8. Test 1: initial top hat function Monitor width and kurtosis black: closure model red: turbulence sim. Fit results: kf/k1St=tukf 1.5 1.8 2.2 1.8 5.1 2.4 Brandenburg: non-Fickian diffusion

  9. Comparison with Fickian diffusion No agreement whatsoever Brandenburg: non-Fickian diffusion

  10. Spreading of initial top-hat function Brandenburg: non-Fickian diffusion

  11. Test 2: finite initial flux experiment but with Initial state: black: closure model red: turbulence sim. Dispersion Relation: Oscillatory for k1/kf<3  direct evidence for oscillatory behavior!

  12. Test 3: imposed mean C gradient Convergence to St=3 for different Re Brandenburg: non-Fickian diffusion >>1 (!)

  13. kf=5 Brandenburg: non-Fickian diffusion

  14. kf=1.5 Brandenburg: non-Fickian diffusion

  15. Comment on the bottleneck effect Dobler et al (2003) PRE 68, 026304 Brandenburg: non-Fickian diffusion

  16. Bottleneck effect: 1D vs 3D spectra Brandenburg: non-Fickian diffusion

  17. Relation to ‘laboratory’ 1D spectra Parseval used: Brandenburg: non-Fickian diffusion

  18. Conclusions • MTA viable approach to mean field theory • Strouhal number around 3 • FOSA not ok (requires St  0) • Existence of extra time derivative confirmed • Passive scalar transport has wave-like properties • Causality • In MHD, <j.b> contribution arrives naturally • Coriolis force & inhomogeneity straightforward Brandenburg: non-Fickian diffusion

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