Computation of kinematic and magnetic $\alpha$-effect and eddy diffusivity tensors by Padé approximation

Abstract

We present examples of Padé 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 Padé approximants for computation of tensors of magnetic $\alpha$-effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé 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 Padé 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.

References in corpus (6)

• Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of incompressible electrically conducting fluid
• Dynamo effect in parity-invariant flow with large and moderate separation of scales
• The Cauchy-Lagrangian method for numerical analysis of Euler flow
• Negative magnetic eddy diffusivities from test-field method and multiscale stability theory
• Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation
• Negative magnetic eddy diffusivity due to oscillogenic $\alpha$-effect

Cited by in corpus (1)

• Negative magnetic eddy diffusivity due to oscillogenic $\alpha$-effect