Nonlinear α‐Effect in Dynamo Theory

We extend the standard two-scale theory of the turbulent dynamo coefficient$\alpha$ to include the nonlinear back reaction of the mean field $\bar B$ onthe turbulence. We calculate the turbulent emf as a power series in $\bar B$,assuming that the base state of the turbulence ($\bar B=0$) is isotropic, and,for simplicity, that the magnetic diffusivity equals the kinematic viscosity.The power series converges for all $\bar B$, and for the special case that thespectrum of the turbulence is sharply peaked in $k$, our result is proportionalto a tabulated function of the magnetic Reynolds number $R_M$ and the ratio$\beta$ of $\bar B$ (in velocity units) to the rms turbulent velocity $v_0$.For $\beta\to 0$ (linear regime) we recover the results of Steenbeck et al.(1966) as modified by Pouquet et al. (1976). For $R_M\gg 1$, the usualastrophysical case, $\alpha$ starts to decrease at $\beta \sim 1$, droppinglike $\beta^{-2}$ as $\beta \to \infty$. Hence for large $R_M$, $\alpha$saturates at $\bar B\sim v_0$, as estimated by Kraichnan (1979), rather than at$\bar B\sim R^{-1/2}_Mv_0$, as inferred by Cattaneo and Hughes (1996) fromtheir numerical simulations at $R_M$=100. We plan to carry out simulations withvarious values of $R_M$ to investigate the discrepency.Comment: 41 pages, 1 Postscript figure, accepted for publication to Ap