  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.
