Transport coefficients for the shear dynamo problem at small Reynolds numbers
Abstract
We build on the formulation developed in S. Sridhar and N. K. Singh [J. Fluid Mech.JFLSA70022112010.1017/S0022112010003745 664, 265 (2010)] and present a theory of the shear dynamo problem for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter. Specializing to the case of a mean magnetic field that is slowly varying in time, explicit expressions for the transport coefficients α_{il} and η_{iml} are derived. We prove that when the velocity field is nonhelical, the transport coefficient α_{il} vanishes. We then consider forced, stochastic dynamics for the incompressible velocity field at low Reynolds number. An exact, explicit solution for the velocity field is derived, and the velocity spectrum tensor is calculated in terms of the Galileaninvariant forcing statistics. We consider forcing statistics that are nonhelical, isotropic, and delta correlated in time, and specialize to the case when the mean field is a function only of the spatial coordinate X_{3} and time τ; this reduction is necessary for comparison with the numerical experiments of A. Brandenburg, K. H. Rädler, M. Rheinhardt, and P. J. Käpylä [Astrophys. J.AJLEEY0004637X10.1086/527373 676, 740 (2008)]. Explicit expressions are derived for all four components of the magnetic diffusivity tensor η_{ij}(τ). These are used to prove that the shearcurrent effect cannot be responsible for dynamo action at small Re and Rm, but for all values of the shear parameter.
 Publication:

Physical Review E
 Pub Date:
 May 2011
 DOI:
 10.1103/PhysRevE.83.056309
 arXiv:
 arXiv:1003.2787
 Bibcode:
 2011PhRvE..83e6309S
 Keywords:

 47.27.W;
 47.65.Md;
 52.30.Cv;
 95.30.Qd;
 Boundaryfree shear flow turbulence;
 Plasma dynamos;
 Magnetohydrodynamics;
 Magnetohydrodynamics and plasmas;
 Astrophysics  Astrophysics of Galaxies;
 Physics  Fluid Dynamics
 EPrint:
 27 pages, 5 figures, Published in Physical Review E