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Helical MHD and a -effect

Helical MHD and a -effect. Axel Brandenburg ( Nordita, Copenhagen ) Kandaswamy Subramanian ( IUCAA, Pune ). arXiv:astro-ph/0405052 Phys. Rept. (244 pages, 62 figs). MHD equations. Magn. Vector potential. Induction Equation:. Momentum and Continuity eqns. Viscous force.

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Helical MHD and a -effect

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  1. Helical MHD and a-effect Axel Brandenburg (Nordita, Copenhagen) Kandaswamy Subramanian (IUCAA, Pune) arXiv:astro-ph/0405052 Phys. Rept. (244 pages, 62 figs)

  2. MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force Brandenburg: Helical MHD

  3. 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! Brandenburg: Helical MHD

  4. Pencil Code • 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 many people (CVS!) • Automatic validation (over night or any time) • Max resolution so far 10243 Brandenburg: Helical MHD

  5. Helical versus nonhelical Kida et al. (1991) helical forcing, but no inverse cascade Inverse cascade only when scale separation Brandenburg: Helical MHD

  6. Allowing for scale separation Position of the peak compatible with No inverse cascade in kinematic regime Decomposition in terms of Chandrasekhar-Kendall-Waleffe functions Brandenburg: Helical MHD

  7. Kazantsev spectrum (kinematic) Opposite limit, no scale separation, forcing at kf=1-2 Kazantsev spectrum confirmed (even for n/h=1) Spectrum remains highly time-dependent Brandenburg: Helical MHD

  8. 256 processor run at 10243 EM(k) not peaked at resistive scale, as previously claimed instead: kpeak~Rm,crit1/2kf ~ 6kf

  9. Structure function exponents agrees with She-Leveque third moment Brandenburg: Helical MHD

  10. Bottleneck effect: 1D vs 3D spectra Compensated spectra (1D vs 3D) Brandenburg: Helical MHD

  11. Relation to ‘laboratory’ 1D spectra Brandenburg: Helical MHD

  12. Bottleneck in the literature Porter, Pouquet, & Woodward (1998) using PPM, 10243 meshpoints Kaneda et al. (2003) on the Earth simulator, 40963 meshpoints Brandenburg: Helical MHD

  13. Helical MHD turbulence • Helically forced turbulence (cyclonic events) • Small & large scale field grows exponentially • Past saturation: slow evolution  Explained by magnetic helicity equation Brandenburg: Helical MHD

  14. Animations

  15. Effects of magnetic helicity conservation Early times: h=0 important Late times: steady state By the time a steady state is reached: net magnetic helicity is generated

  16. Slow-down explained by magnetic helicity conservation molecular value!! Brandenburg: Helical MHD

  17. Connection with a effect: writhe with internal twist as by-product clockwise tilt (right handed) W  left handed internal twist Yousef & Brandenburg A&A 407, 7 (2003) Brandenburg: Helical MHD

  18. Internal twist as feedback on a (Pouquet, Frisch, Leorat 1976) How can this be used in practice? Need a closure for <j.b> Brandenburg: Helical MHD

  19. Rm dependence of PFL formula St = turmskf not suppressed in Rm dependent fashion a is suppressed in Rm dependent fashion Brandenburg: Helical MHD

  20. Revised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982) Two-scale assumption Production of large scale helicity comes at the price of producing also small scale magnetic helicity Brandenburg: Helical MHD

  21. Express in terms of a  Dynamical a-quenching (Kleeorin & Ruzmaikin 1982) no additional free parameters Steady limit: consistent with VC92 (algebraic quenching) Brandenburg: Helical MHD

  22. Is ht quenched?can be in models with shear Larger mean field Slow growth but short cycles: Depends on assumption about ht-quenching! Brandenburg: Helical MHD

  23. Additional effect of shear Negative shear Positive shear Consistent with g=3 and Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999) Brandenburg: Helical MHD

  24. Effect of surface losses of current helicity • Large scale (LS) field: • Drainage on LS dynamo • Rm-dependent cutoff • Shortens saturation time • Small scale (SS) field • Enhancement of LS dynamo Brandenburg: Helical MHD

  25. The need for small scale losses initial slope Numerical experiment: remove field for k>4 every 1-3 turnover times • large scale losses: • lower saturation level 2) higher saturation level 3) still slow time scale

  26. How do magnetic helicity losses look like? N-shaped (north) S-shaped (south) (the whole loop corresponds to CME) Brandenburg: Helical MHD

  27. Sigmoidal filaments (from S. Gibson) Brandenburg: Helical MHD

  28. Examples ofhelical structures Brandenburg: Helical MHD

  29. Simulating solar-like differential rotation Brandenburg: Helical MHD

  30. Results for current helicity flux First order smoothing, and tau approximation Vishniac & Cho (2001 Expected to be finite on when there is shear Brandenburg: Helical MHD

  31. Conclusions • Homogeneous dynamos saturate resistively • Entirely magnetic helicity controlled • Inhomogeneous dynamo • Open surface, equator • Still many issues to be addressed • Current helicity flux important • Finite if there is shear • Avoid magnetic helicity, use current helicity Brandenburg: Helical MHD

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