Abstract
We present examples of Padé approximations 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 the application of Padé approximants for the 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 to satisfactory accuracy for η , 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.
Highlights
Power series expansion of analytic functions is perhaps the most powerful tool of numerical analysis
We truncate the series (6) at orders up to 39, even terms missing. Roots of their Padé approximants quickly stabilize near ν = ν? ≈ 0.58, indicating a transition to negative eddy viscosity at lower molecular viscosities
The modes sk are functions of molecular eddy diffusivity η, meromorphic in this parameter. (By contrast, the slow-time growth rates of modes generated by the α-effect are not, because the square root present in (19) gives rise to branch points.) A power series expansion of sk in η −1 for large η can be constructed like in the hydrodynamic problem considered in the previous section
Summary
Power series expansion of analytic functions is perhaps the most powerful tool of numerical analysis. The analysis of equations makes it evident that eddy viscosity/diffusivity acquires unusual properties because it acts on mean fields only, i.e., essentially an open physical system is considered In this class of MHD systems, the inverse energy cascade is important, with energy proliferating from small scales toward large ones; the source of energy for the developing large-scale magnetic, hydrodynamic, or combined MHD perturbation is the forcing applied to maintain the perturbed ( sometimes called basic) fluid flow. When the parameter tends to the critical value for the onset of the small-scale instability (i.e., in the dynamo context, to the value for which the generation of small-scale magnetic fields starts), the tensors usually exhibit singular behavior [20,21,22,23] of a simple pole type, bounding from above the radius of convergence of the series This suggests trying Padé approximants for computing the tensors and the respective large-scale magnetic field/instability growth rates.
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