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

1
Centro de Matemática (Faculdade de Ciências) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
2
Research Center for Systems and Technologies, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
3
Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Ac. Sci., 84/32 Profsoyuznaya St, 117997 Moscow, Russian
*
Author to whom correspondence should be addressed.
Fluids 2019, 4(2), 110; https://doi.org/10.3390/fluids4020110
Received: 13 May 2019 / Revised: 5 June 2019 / Accepted: 7 June 2019 / Published: 14 June 2019
(This article belongs to the Special Issue Multiscale Turbulent Transport)
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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. View Full-Text
Keywords: incompressible fluid; magnetic mode; alpha-effect; eddy diffusivity; eddy viscosity; Padé approximant incompressible fluid; magnetic mode; alpha-effect; eddy diffusivity; eddy viscosity; Padé approximant
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Gama, S.M.; Chertovskih, R.; Zheligovsky, V. Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation. Fluids 2019, 4, 110.

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