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

Abstract

We present examples of Padé approximation of the α-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 α-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/η around 1/η = 0. This yields the values of the dominant growth rate due to the action of the α-effect or eddy diffusivity to satisfactory accuracy for η, 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 η 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/η 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. (Sílvio M.A. Gama) unity. For finite-precision computations, however, the cancellation is not any more guaranteed, and the initial growth of individual terms can result in ultimate loss of accuracy.The other one stems from finiteness of the radius of convergence of most power series encountered in computational practice. A complementary technique is then needed to continue analytically a function defined by the power series outside its circle of convergence. This can be achieved by constructing the so-called Padé approximants [14,4,5], i.e., an implementation of the continuation in the form of the ratio of two polynomials. Let us cite the words of appraisal in [24]: "Padé approximation has the uncanny knack of picking the function you had in mind from among all the possibilities. Except when it doesn't! That is the downside of Padé approximation: it is uncontrolled. There is, in general, no way to tell how accurate it is, or how far out in x it can usefully be extended. It is a powerful, but in the end still mysterious, technique."A not less mysterious notion is that of eddy diffusivity [28], also known as eddy (or turbulent) viscosity when fluid viscosity, the source of diffusion in hydrodynamics, is considered. Like the magnetic α-effect and anisotropic kinetic alpha-(aka AKA) effect, eddy diffusivities are often encountered in magnetohydrodynamics when generation of large-scale magnetic fields by flows of electrically conductive fluids is considered. At first sight, i...

Some topological aspects of fluid dynamics

“…In general, and particularly in the first-order smoothing limit, turns out to be positive, and serves in a natural way as a ‘turbulent diffusivity’, augmenting the molecular diffusivity . It is possible, however, to construct space-periodic velocity fields which give rise to negative , and even to values of for which (Rasskazov, Chertovskih & Zheligovsky 2018; Gama, Chertovskih & Zheligovsky 2019). This ‘negative diffusivity’ situation is very curious, making the mean field (6.10) ill posed in a strict mathematical sense: catastrophic instability would result at the smallest scales (), and this is totally inconsistent with the two-scale approach postulated at the outset.…”

Negative magnetic eddy diffusivity due to oscillogenic α-effect

“…(Using Hölder inequality, it is easy to show that C ≤ S|v| 3 , where | · | p denotes the norm in the Lebesgue space L p (T 3 ) and S is a constant in the inequality |f | 6 ≤ S f following from the Sobolev embedding theorem for space-periodic zero-mean fields.) Therefore, series (55) is majorised by a geometric series with the ratio C/η and is guaranteed to converge for η > C. Numerical algorithms for computation of the α-effect and eddy diffusivity tensors based on the expansion (55) and employing Padé approximation will be considered in [5]. Relation (45.1) implies an expansion…”

“…whose solutions Z l are assumed to be zero-mean; the adjoint operator L * is defined by (5). Here and in what follows, the superscript "r" marks objects pertinent to the reverse flow −v(x):…”

“…Numerical algorithms for computation of the α-effect and eddy diffusivity tensors based on the expansion (55) and employing Padé approximation will be considered in [5]. Relation (45.1) implies an expansion…”

Editorial for Special Issue “Multiscale Turbulent Transport”

“…Gama et al (2019) [7] dealt with Padé approximations of the magnetohydrodynamic kinematic dynamo α-effect and of the eddy-viscosity or diffusivity tensors in different flows, employing expressions derived in the framework of the multiple-scale formalism. They performed computations in Fortran in the standard "double" (real*8) and extended "quadruple" (real*16) precision, as well as symbolic calculations in Mathematica.…”