  Finite-amplitude hydromagnetic Rossby waves in the magnetostrophic regime are
studied. We consider the slow mode, which travels in the opposite direction to
the hydrodynamic or fast mode, in the presence of a toroidal magnetic field and
zonal flow by means of quasi-geostrophic models for thick spherical shells. The
weakly-nonlinear, long waves are derived asymptotically using a reductive
perturbation method. The problem at the first order is found to obey a
second-order ODE, leading to a hypergeometric equation for a Malkus field and a
confluent Heun equation for an electrical-wire field, and is nonsingular when
the wave speed approaches the mean flow. Investigating its neutral, nonsingular
eigensolutions for different basic states, we find the evolution is described
by the Korteweg-de Vries equation. This implies that the nonlinear slow wave
forms solitons and solitary waves. These may take the form of a coherent eddy,
such as a single anticyclone. We speculate on the relation of the anti-cyclone
to the asymmetric gyre seen in Earth's fluid core, and in state-of-the-art
dynamo DNS.
