In this work, we neglect mean currents, but instead include the full effects of the earth's sphericity and the wave dispersion it induces.
Through a mix of perturbation theory and numerical computations using a Fourier/Newton iteration/continuation method, we show that for sufficiently small amplitude, there are nonlinear Kelvin traveling waves (cnoidal waves). As the amplitude increases, the branch of traveling waves terminates in a so-called ``corner wave'' with a discontinuous first derivative. All waves larger than the corner wave evolve to fronts and break.
The singularity is a point singularity in which only the longitudinal derivative is discontinuous.
As we solve the nonlinear shallow water equations on the sphere with increasing epsilon(``Lamb's parameter''), dispersion weakens, the amplitude of the corner wave decreases rapidly, and the longitudinal profile of the corner wave narrows dramatically.