Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation
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Centro de Matemática (Faculdade de Ciências) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
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Research Center for Systems and Technologies, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
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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
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Revised: 5 June 2019
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Accepted: 7 June 2019
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Published: 14 June 2019
(This article belongs to the Special Issue Multiscale Turbulent Transport)
We present examples of Padé approximations of the -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 -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 around . This yields the values of the dominant growth rate to satisfactory accuracy for , 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.
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Keywords:
incompressible fluid; magnetic mode; alpha-effect; eddy diffusivity; eddy viscosity; Padé approximant
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MDPI and ACS Style
Gama, S.M.A.; Chertovskih, R.; Zheligovsky, V. Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation. Fluids 2019, 4, 110. https://doi.org/10.3390/fluids4020110
AMA Style
Gama SMA, Chertovskih R, Zheligovsky V. Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation. Fluids. 2019; 4(2):110. https://doi.org/10.3390/fluids4020110
Chicago/Turabian StyleGama, Sílvio M.A., Roman Chertovskih, and Vladislav Zheligovsky. 2019. "Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation" Fluids 4, no. 2: 110. https://doi.org/10.3390/fluids4020110
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