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

## TL;DR

This paper develops Padé approximation methods to compute the $oldsymbol{ ext{α}}$-effect and eddy diffusivity tensors in fluid flows, providing accurate estimates of magnetic field growth rates and flow stability in dynamo theory.

## Contribution

It introduces Padé approximation techniques for tensors in kinematic dynamo problems, enabling better predictions of magnetic field growth and flow stability.

## Key findings

- Padé approximants accurately estimate growth rates for small magnetic diffusivity.
- Eddy diffusivity can tend to negative infinity near dynamo onset.
- First 49 coefficients are insufficient to capture certain behaviors.

## Abstract

We present examples of Pad\'e approximation 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 application of Pad\'e approximants for computation of tensors of magnetic $\alpha$-effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Pad\'e 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 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 is convergent. For one sample flow, we observe eddy diffusivity tending to negative infinity when $\eta$ tends from above to the point of the onset of small-scale dynamo action in a symmetry-invariant subspace where a neutral small-scale magnetic mode resides. However, 49 first coefficients in the power series in $1/\eta$ prove insufficient for Pad\'e approximants to reproduce this behaviour. We do computations in Fortran in the standard `double' (real*8) and extended `quadruple' (real*16) precision, as well as perform symbolic calculations in Mathematica.