Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Pade Approximation

We present examples of Pade approximations of the alpha-effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Pade approximants for the computation of tensors of magnetic alpha-effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Pade approximants of the tensors expanded in power series in the inverse molecular diffusivity 1/eta around 1/eta = 0. This yields the values of the dominant growth rate to satisfactory accuracy for eta, several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard "double" (real*8) and extended "quadruple" (real*16) precision, and perform symbolic calculations in Mathematica.