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A new era for mean fields: test fields and tests

Discover the latest advancements in mean field theory for predictive modeling in astrophysical fluid dynamics, examining effects of boundary conditions, helicity, and angular velocity. Explore open domains, magnetic buoyancy, and turbulent electromagnetic effects.

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A new era for mean fields: test fields and tests

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  1. A new era for mean fields:test fields and tests Axel Brandenburg (Nordita, Stockholm)

  2. Output so far 25%

  3. Mean field theory is predictive • Open domain with shear • Helicity is driven out of domain (Vishniac & Cho) • Mean flow contours perpendicular to surface! • Excitation conditions • Dependence on angular velocity • Dependence on b.c.: symmetric vs antisymmetric

  4. Best if W contours ^ to surface Example: convection with shear  need small-scale helical exhaust out of the domain, not back in on the other side Magnetic Buoyancy? Tobias et al. (2008, ApJ) Käpylä et al. (2008, A&A)

  5. To prove the point: convection with vertical shear and open b.c.s Magnetic helicity flux Käpylä, Korpi, & myself (2008, A&A 491, 353) Effects of b.c.s only in nonlinear regime

  6. Calculate full aij and hij tensors turbulent emf • Imposed-field method • Convection (Brandenburg et al. 1990) • Correlation method • MRI accretion discs (Brandenburg & Sokoloff 2002) • Galactic turbulence (Kowal et al. 2005, 2006) • Test field method • Stationary geodynamo (Schrinner et al. 2005, 2007) a effect and turbulent magnetic diffusivity

  7. Calculate full aij and hij tensors Original equation (uncurled) Mean-field equation fluctuations Response to arbitrary mean fields

  8. Test fields Example:

  9. Validation: Roberts flow SOCA SOCA result normalize

  10. Kinematic a and ht independent of Rm (2…200) Sur et al. (2008, MNRAS)

  11. The hxx and hyy are now the same Brandenburg & Sokoloff (2002, GAFD) Brandenburg (2005, AN)

  12. Scale-dependence

  13. Time-dependent case

  14. From linear to nonlinear Use vector potential Mean and fluctuating U enter separately

  15. Nonlinear aij and hij tensors Consistency check: consider steady state to avoid da/dt terms Expect: l=0 (within error bars)  consistency check!

  16. Application to passive vector eqn cf. Cattaneo & Tobias (2009) Verified by test-field method Tilgner & Brandenburg (2008)

  17. ht(Rm) dependence for B~Beq • l is small  consistency • a1 and a2 tend to cancel • to decrease a • h2 is small

  18. Test field tests • l = 0 • l = l(s) • l = l(B)

  19. The Future • Test fields will continue to provide guidence • Test flows: eddy viscosity • vorticity dynamo? • Maxwell stresses • Turbulent flux collapse • Negative turbulent mag presure • Global dynamo • Shell sectors 1046 Mx2/cycle

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