Finite-amplitude magnetic Rossby waves in the magnetostrophic regime are studied. The slow mode, which travels in the opposite direction to the nonmagnetic/fast mode, may be relevant for understanding convection at high magnetic Prandtl and low Ekman numbers，and for applications to Earth，s core and explaining its magnetic secular variation. We consider a rapidly rotating spherical system in the presence of an imposed azimuthal magnetic field and flow by means of a quasi-geostrophic annulus model with sloping endwalls. The weakly-nonlinear， long-wavelength waves are analysed asymptotically using a reductive perturbation method. We find the evolution is described by the Korteweg-de Vries equation if either the background magnetic field, topography, or flow varies with cylindrical radius. This implies the nonlinear slow wave forms a coherent eddy， and that localised structures may interact in interesting ways.
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