Kelvin waves in the nonlinear shallow water equations on the sphere: nonlinear travelling waves and the corner wave bifurcation

Abstract

The Kelvin wave is the lowest eigenmode of Laplace's tidal equation and is widely observed in both the ocean and the atmosphere. In this work, we neglect mean currents and 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 Kelvin travelling waves (cnoidal waves). As the amplitude increases, the branch of travelling 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 ε (‘Lamb's parameter’), dispersion weakens, the amplitude of the corner wave decreases rapidly, and the longitudinal profile of the corner wave narrows dramatically.