Transport coefficients for the shear dynamo problem at small Reynolds numbers

Abstract

We build on the formulation developed in S. Sridhar and N. K. Singh [J. Fluid Mech. 664, 265 (2010)] and present a theory of the shear dynamo problem for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter. Specializing to the case of a mean magnetic field that is slowly varying in time, explicit expressions for the transport coefficients αil and ηiml are derived. We prove that when the velocity field is nonhelical, the transport coefficient αil vanishes. We then consider forced, stochastic dynamics for the incompressible velocity field at low Reynolds number. An exact, explicit solution for the velocity field is derived, and the velocity spectrum tensor is calculated in terms of the Galilean-invariant forcing statistics. We consider forcing statistics that are nonhelical, isotropic, and delta correlated in time, and specialize to the case when the mean field is a function only of the spatial coordinate X3 and time τ ; this reduction is necessary for comparison with the numerical experiments of A. Brandenburg, K. H. R¨adler, M. Rheinhardt, and P. J. K¨apyl¨a [Astrophys. J. 676, 740 (2008)]. Explicit expressions are derived for all four components of the magnetic diffusivity tensor ηij (τ ). These are used to prove that the shear-current effect cannot be responsible for dynamo action at small Re and Rm, but for all values of the shear parameter.