Eur. Phys. J. B 60, 337-351 (2007)
https://doi.org/10.1140/epjb/e2007-00351-8

Eddy diffusivity in convective hydromagnetic systems

¹ Centro de Matemática da Universidade do Porto and Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
² International Institute of Earthquake Prediction Theory and Mathematical Geophysics, 84/32, Profsoyuznaya St, 117997 Moscow, Russian Federation
³ Institute of Mechanics, Lomonosov Moscow State University 1, Michurinsky ave., 119899 Moscow, Russian Federation
⁴ Observatoire de la Côte d'Azur, CNRS U.M.R. 6529, BP 4229, 06304 Nice Cedex 4, France

Corresponding author: ^a mbaptist@fc.up.pt

Received: 6 February 2007
Revised: 9 November 2007
Published online: 22 December 2007

Abstract

An eigenvalue equation, for linear instability modes involving large scales in a convective hydromagnetic system, is derived in the framework of multiscale analysis. We consider a horizontal layer with electrically conducting boundaries, kept at fixed temperatures and with free surface boundary conditions for the velocity field; periodicity in horizontal directions is assumed. The steady states must be stable to short (fast) scale perturbations and possess symmetry about the vertical axis, allowing instabilities involving large (slow) scales to develop. We expand the modes and their growth rates in power series in the scale separation parameter and obtain a hierarchy of equations, which are solved numerically. Second order solvability condition yields a closed equation for the leading terms of the asymptotic expansions and respective growth rate, whose origin is in the (combined) eddy diffusivity phenomenon. For about 10% of randomly generated steady convective hydromagnetic regimes, negative eddy diffusivity is found.