  In this paper we studied the weakly nonlinear stage of stationary convective
instability in a nonuniformly rotating layer of an electrically conductive
fluid in an axial uniform magnetic field under the influence of: a) temperature
modulation of the layer boundaries; b) gravitational modulation; c) modulation
of the magnetic field; d) modulation of the angular velocity of rotation. As a
result of applying the method of perturbation theory for the small parameter of
supercriticality of the stationary Rayleigh number nonlinear non-autonomous
Ginzburg-Landau equations for the above types of modulation were obtaned. By
utilizing the solution of the Ginzburg-Landau equation, we determined the
dynamics of unsteady heat transfer for various types of modulation of external
fields and for different profiles of the angular velocity of the rotation of
electrically conductive fluid.
