Nonlinear Alpha Effect in Dynamo Theory

George B. Field; Eric G. Blackman; Hongsong Chou

doi:10.1086/306877

paper

Nonlinear Alpha Effect in Dynamo Theory

arXiv:astro-ph/9810345 · doi:10.1086/306877

Abstract

We extend the standard two-scale theory of the turbulent dynamo coefficient $α$ to include the nonlinear back reaction of the mean field $B ˉ$ on the turbulence. We calculate the turbulent emf as a power series in $B ˉ$ , assuming that the base state of the turbulence ( $B ˉ = 0$ ) is isotropic, and, for simplicity, that the magnetic diffusivity equals the kinematic viscosity. The power series converges for all $B ˉ$ , and for the special case that the spectrum of the turbulence is sharply peaked in $k$ , our result is proportional to a tabulated function of the magnetic Reynolds number $R_{M}$ and the ratio $β$ of $B ˉ$ (in velocity units) to the rms turbulent velocity $v_{0}$ . For $β \to 0$ (linear regime) we recover the results of Steenbeck et al. (1966) as modified by Pouquet et al. (1976). For $R_{M} ≫ 1$ , the usual astrophysical case, $α$ starts to decrease at $β \sim 1$ , dropping like $β^{- 2}$ as $β \to \infty$ . Hence for large $R_{M}$ , $α$ saturates at $B ˉ \sim v_{0}$ , as estimated by Kraichnan (1979), rather than at $B ˉ \sim R_{M}^{- 1/2} v_{0}$ , as inferred by Cattaneo and Hughes (1996) from their numerical simulations at $R_{M}$ =100. We plan to carry out simulations with various values of $R_{M}$ to investigate the discrepency.

41 pages, 1 Postscript figure, accepted for publication to ApJ

Nonlinear Alpha Effect in Dynamo Theory

Abstract

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