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Chandrasekhar-Kendall functions in astrophysical magnetism

Chandrasekhar-Kendall functions in astrophysical magnetism. Chandra’s lasting contributions to MHD. Accelerated growth!. Chandrasekhar number in MHD. Chandra’s interest in MHD. His most quoted MHD papers. His 1956 papers alone!. Toward magneto-rotational instability.

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Chandrasekhar-Kendall functions in astrophysical magnetism

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  1. Chandrasekhar-Kendall functions in astrophysical magnetism

  2. Chandra’s lasting contributions to MHD

  3. Accelerated growth!

  4. Chandrasekhar number in MHD

  5. Chandra’s interest in MHD

  6. His most quoted MHD papers

  7. His 1956 papers alone!

  8. Toward magneto-rotational instability Vertical field B0 Dispersion relation Alfven frequency:

  9. Alfven and slow magnetosonic waves Alfven slow magnetosonic Degeneracy lifted by q or W # 0

  10. Both were very much ahead of their time: No accretion discs were discovered yet!

  11. Emphesis on stability – not instability (1976)

  12. Chandrasekhar-Kendall functions Eigenfunctions of the curl operator: curl B = lB Fits to solar magnetograms Theory by B. C. Low

  13. CKF functions in plasma context Alladis et al. (2001) Prota-sphere experiment

  14. Magnetic helicity measures linkage of flux Therefore the unit is Maxwell squared

  15. Magnetic helicity conservation How J diverges as h0 Ideal limit and ideal case similar!

  16. CK functions in periodic space Fourier space Expand into longitudinal and polarized contributions so spectra

  17. Realizability condition Spectra Realizability condition Shell-integrated spectra Energies in positively and negatively polarized waves (Obtained just from the spectra)

  18. Cartesian box MHD equations Magn. Vector potential Induction Equation: Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

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

  20. Decaying fully helical turbulence helical vs nonhelical Initial slope M~k4 Christensson et al. (2001, PRE 64, 056405)

  21. Forced turbulence Brandenburg (2001, ApJ 550, 824) Expected from mean-field theory

  22. CK functions in linear/nonlinear regimes

  23. Slow-down explained by magnetic helicity conservation

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

  25. Effect of helicity fluxes Brandenburg (2005, ApJ) 1046Mx2/cycle

  26. 3-D simulations in spheres • The right dynamo regime? • Or a small scale dynamo? Brun, Miesch, & Toomre (2004, ApJ 614, 1073)

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

  28. a-effect dynamos (large scale) New loop Differential rotation (surface layers: faster inside) Cyclonic convection; Buoyant flux tubes Equatorward migration  a-effect

  29. How do magnetic helicity losses look like? N-shaped (north) S-shaped (south) (the whole loop corresponds to CME)

  30. Conclusions • He was close to getting a dynamo • Very much immersed into numerics • Always ahead of his time • Still gaining more citations every year!

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