1 / 54

Non-helical MHD at 1024 3

Non-helical MHD at 1024 3. Haugen, Brandenburg, & Dobler (2003, ApJ). Inverse cascade of magnetic helicity. argument due to Frisch et al. (197 5 ). and. Initial components fully helical:. and.  k is forced to the left. Magnetic helicity. Maxwell eqns. Vector potential.

abram
Download Presentation

Non-helical MHD at 1024 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Non-helical MHD at 10243 Haugen, Brandenburg, & Dobler (2003, ApJ)

  2. Inverse cascade of magnetic helicity argument due to Frisch et al. (1975) and Initial components fully helical: and  k is forced to the left

  3. Magnetic helicity Maxwell eqns Vector potential Uncurled induction eqn

  4. Magnetic helicity

  5. Slow saturation Brandenburg (2001, ApJ 550, 824)

  6. Periodic box, no shear: resistively limited saturation Brandenburg & Subramanian Phys. Rep. (2005, 417, 1-209) Significant field already after kinematic growth phase followed by slow resistive adjustment Blackman & Brandenburg (2002, ApJ 579, 397)

  7. Magnetic helicity conservation Steady state, closed box Early times

  8. Slow-down explained by magnetic helicity conservation molecular value!!

  9. Slow-down explained by magnetic helicity conservation

  10. With hyperdiffusivity Brandenburg & Sarson (2002, PRL) for ordinary hyperdiffusion

  11. Evidence from different simulations:strong fields only with helicity flux 3-D simulations, no mean-field modeling Forced turbulence in domain with solar-like shear Brandenburg (2005, ApJ 625, 539) Convective dynamo in a box with shear and rotation Käpylä, Korpi, Brandenburg (2008, A&A 491, 353) Only weak field if boxis closed

  12. Nonlinear stage: consistent with … Brandenburg (2005, ApJ)

  13. 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)

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

  15. Implications of tau approximation • MTA does not a priori break down at large Rm. (Strong fluctuations of b are possible!) • Extra time derivative of emf •  hyperbolic eqn, oscillatory behavior possible! • t is not correlation time, but relaxation time with

  16. Kinetic and magnetic contributions

  17. Connection with a effect: writhe with internal twist as by-product a effect produces helical field W clockwise tilt (right handed)  left handed internal twist both for thermal/magnetic buoyancy

  18. … the same thing mathematically Two-scale assumption Production of large scale helicity comes at the price of producing also small scale magnetic helicity

  19. Revised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982) Two-scale assumption Dynamical quenching Kleeorin & Ruzmaikin (1982) ( selective decay) Steady limit  algebraic quenching:

  20. General formula with magnetic helicity flux Rm also in the numerator

  21. 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

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

  23. Test fields Example:

  24. Validation: Roberts flow SOCA SOCA result Brandenburg, Rädler, Schrinner (2009, A&A) normalize

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

  26. Scale-dependence: nonlocality cf talk by Alexander Nepomnyashchy

  27. Time-dependent case Hubbard & Brandenburg (2009, ApJ)

  28. Importance of time-dependence

  29. From linear to nonlinear Brandenburg et al. (2008, ApJ) Use vector potential Mean and fluctuating U enter separately

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

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

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

  33. Is the field in the Sun fibril? Käpylä et al (2008) with rotation without rotation

  34. Takes many turnover times Rm=121, By, 512^3 LS dynamo not always excited

More Related