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

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2 MTA - the minimal tau approximation 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?? (remains to be justified!)

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Brandenburg: non-Fickian diffusion3 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!)

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

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Brandenburg: non-Fickian diffusion5 System of mean field equations >>1 (!) mean concentration flux equation Damped wave equation, wave speed (causality!)

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Brandenburg: non-Fickian diffusion6 Wave equation: consequences >>1 (!) small tau i)late time behavior unaffected (ordinary diffusion) ii)early times: ballistic advection (superdiffusive) large tauintermediate tau Illustration of wave-like behavior:

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Brandenburg: non-Fickian diffusion7 Comparison with DNS Finite difference –MPI, scales linearly –good on big Linux clusters 6 th order in space, 3 rd order in time forcing on narrow wavenumber band Consider k f /k 1 =1.5 and 5

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Brandenburg: non-Fickian diffusion8 Test 1: initial top hat function Monitor width and kurtosis black: closure model red: turbulence sim. Fit results: k f /k 1 St= uk f 1.5 1.8 2.2 1.8 5.1 2.4

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Brandenburg: non-Fickian diffusion9 Comparison with Fickian diffusion No agreement whatsoever

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Brandenburg: non-Fickian diffusion10 Spreading of initial top-hat function

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Test 2: finite initial flux experiment Initial state: but with black: closure model red: turbulence sim. direct evidence for oscillatory behavior! Dispersion Relation: Oscillatory for k 1 /k f <3

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Brandenburg: non-Fickian diffusion12 Test 3: imposed mean C gradient >>1 (!) Convergence to St=3 for different Re

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Brandenburg: non-Fickian diffusion13 kf=5

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Brandenburg: non-Fickian diffusion14 kf=1.5

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Brandenburg: non-Fickian diffusion15 Comment on the bottleneck effect Comment on the bottleneck effect Dobler et al (2003) PRE 68, 026304

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Brandenburg: non-Fickian diffusion16 Bottleneck effect: 1D vs 3D spectra

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Brandenburg: non-Fickian diffusion17 Relation to laboratory 1D spectra Parseval used:

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Brandenburg: non-Fickian diffusion18 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, contribution arrives naturally Coriolis force & inhomogeneity straightforward

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