This paper describes Rossby waves propagating along a channel with a discontinuity in potential vorticity. A finite-amplitude, long-wave equation giving the displacement of the material interface separating two constant vorticity regions is derived and solved for both steady and unsteady finite-amplitude waves. Finite-amplitude Rossby waves that propagate in the opposite direction to infinitesimal waves are identified, and the mechanism for these waves is elucidated. The full parameter space of possible flows is described quantitatively. The weakly nonlinear limit of this finite-amplitude equation is also presented, including the novel case of kink solitons in an equation with only quadratic nonlinearity.
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