The Onset of Dynamo Action in Alpha-lambda Dynamos

Summary

Solutions of the Navier-Stokes equation are computed in a deep, incompressible, spherical shell, including a parametrization of the Reynolds stresses arising from anisotropic turbulence. Thus the purely dynamical problem has solutions with marked differential rotation. The critical dynamo number for the onset of dynamo action is determined for different hydrodynamic models for both axisymmetric and nonaxisymmetric magnetic fields.

INTRODUCTION

Although kinematic dynamo models reveal some of the basic features of Solar and stellar magnetic fields, a fully satisfactory model must allow the dynamics to emerge as part of the solution of the governing system of equations. This has been attempted in a number of mean-field studies starting with Proctor (1977). It has been shown that solutions exist in which axisymmetric fields become saturated at a finite energy by the action of the macroscopic Lorentz force acting on the fluid.

The mean-field formalism is used in turbulent convection zones to parametrise the effects of the small scale dynamics on the magnetic field (Steenbeck et al. 1966). The influence of the small-scale turbulence on the macroscopic motions can be similarly modelled by the ‘Λ-effect’, representing the Reynolds stresses of anisotropic turbulence induced in a rotating, stratified medium (Rudiger 1989). The resulting mean-field equations describe the evolution of quantities averaged over time or length scales greater than those of the turbulence.