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We present examples of Pad\'e approximation of the $\alpha$-effect and eddyviscosity/diffusivity tensors in various flows. Expressions for the tensorsderived in the framework of the standard multiscale formalism are employed.Algebraically the simplest case is that of a two-dimensional parity-invariantsix-fold rotation-symmetric flow, where eddy viscosity is negative, indicatingintervals of large-scale instability of the flow. Turning to the kinematicdynamo problem for three-dimensional flows of an incompressible fluid, weexplore application of Pad\'e approximants for computation of tensors ofmagnetic $\alpha$-effect and, for parity-invariant flows, of magnetic eddydiffusivity. We construct Pad\'e approximants of the tensors expanded in powerseries in the inverse molecular diffusivity $1/\eta$ around $1/\eta=0$. Thisyields the values of the dominant growth rate due to the action of the$\alpha$-effect or eddy diffusivity to satisfactory accuracy for $\eta$,several dozen times smaller than the threshold, above which the power series isconvergent. For one sample flow, we observe eddy diffusivity tending tonegative infinity when $\eta$ tends from above to the point of the onset ofsmall-scale dynamo action in a symmetry-invariant subspace where a neutralsmall-scale magnetic mode resides. However, 49 first coefficients in the powerseries in $1/\eta$ prove insufficient for Pad\'e approximants to reproduce thisbehaviour. We do computations in Fortran in the standard `double' (real*8) andextended `quadruple' (real*16) precision, as well as perform symboliccalculations in Mathematica.

Computation of kinematic and magnetic $\alpha$-effect and eddy diffusivity tensors by Pad\'e approximation