Scalar transport and alpha-effect for a family of cat's-eye flows

Abstract

In this paper we study advection–diffusion of scalar and vector fields for the steady velocity field \[ (u, v, w) = \left(\frac{\partial \psi}{\partial y},-\frac{\partial\psi}{\partial x},K\psi\right),\quad \psi = \sin x \sin y + \delta \cos x \cos y. \] If δ > 0 the streamlines ψ = constant form a periodic array of oblique cat's-eyes separated by continuous channels carrying finite fluid flux. In the problems treated, advection dominates diffusion, and fields are transported both in thin boundary layers and within the channels. Effective transport of a passive scalar and the alphaeffect generated by interaction of the flow with a uniform magnetic field are examined. For the latter problem, we determine the alpha-matrix as a function of δ. Our results consist of (i) numerical solution of steady problems in the limit of large Reynolds number R with β = δR½ held fixed and O(1), and (ii) analytic asymptotic solutions for large R, obtained using the Wiener–Hopf technique, which are valid for large β. The asymptotic method gives reliable values of the effective diffusion and of alpha-matrices down to β ≈ 1.5.When β > 0 the transport and alpha-effect are greatly enhanced by flux down the channels. Consequently, the alpha-effect found here may have application to the construction of efficient fast dynamos, but this requires spatial dependence of the mean field and the inclusion of three-dimensional effects, as in the established fast-dynamo analysis with δ = 0.