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Transport coefficients for the shear dynamo problem at small Reynolds numbers

We build on the formulation developed in Sridhar & Singh (JFM, 664, 265, 2010), and present a theory of the \emph{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, $α_{il}$ and $η_{iml}$, are derived. We prove that, when the velocity field is non helical, the transport coefficient $α_{il}$ 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 is non helical, isotropic and delta-correlated-in-time, and specialize to the case when the mean-field is a function only of the spatial coordinate $X_3$ and time $τ\,$; this reduction is necessary for comparison with the numerical experiments of Brandenburg, R{ä}dler, Rheinhardt & Käpylä (ApJ, 676, 740, 2008). Explicit expressions are derived for all four components of the magnetic diffusivity tensor, $η_{ij}(τ)\,$. These are used to prove that the shear-current effect cannot be responsible for dynamo action at small $\re$ and $\rem$, but for all values of the shear parameter.