Dispersive effects on equatorial shallow-water shock waves

The nonlinear dynamics of equatorial shallow-water waves is examined both theoretically and numerically. While unidirectional Kelvin waves are non-dispersive and can evolve into shocks, they also excite Poincaré waves which are dispersive. This dispersion significantly impacts the dynamics of the shock. A reduced model has been developed to describe the behaviours of the Kelvin wave and the first Poincaré wave, where the dispersive term appears in a form similar to that in the Ostrovsky equation. The findings indicate that a sharp spike forms in the profile of the first Poincaré wave, resulting in an overshoot of the shock. Theoretical predictions are validated through numerical simulations of the primitive equations.