  We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on
a rapidly rotating planet or star. The presence of a background toroidal
magnetic field is assumed, and the "shallow water" beta-plane approximation is
used. We derive a single equation for the slow large length scale dynamics. The
range of validity of this equation fits the MHD of the lighter fluid at the top
of Earth's outer core. The form of this equation is similar to the
quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the
parameters are essentially different. Our equation also implies the inverse
cascade; but contrary to the usual Q-G situation, the energy cascades to
smaller length scales, while the enstrophy cascades to the larger scales. We
find the Kolmogorov-type spectrum for the inverse cascade. The spectrum
indicates the energy accumulation in larger scales. In addition to the energy
and enstrophy, the obtained equation possesses an extra invariant. Its presence
is shown to imply energy accumulation in the $30^\circ$ sector around zonal
flow. With some special energy input, the extra invariant can lead to the
accumulation of energy in zonal flow; this happens if the input of the extra
invariant is small, while the energy input is considerable.
