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

Fluids

SílvioM.A. Gama¹  Vladislav Zheligovsky²  Roman Chertovskih³ 

[1] Centro de Matemática (Faculdade de Ciências) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal;Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Ac. Sci., 84/32 Profsoyuznaya St, 117997 Moscow, Russian;Research Center for Systems and Technologies, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal;

关键词: incompressible fluid;    magnetic mode;    alpha-effect;    eddy diffusivity;    eddy viscosity;    Padé approximant;   

DOI  :  10.3390/fluids4020110

来源: DOAJ

【 摘 要 】

We present examples of Padé approximations 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 the application of Padé approximants for the 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 to satisfactory accuracy for $\eta$ , 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.

【 授权许可】

Unknown