Amplitude–Wavelength Relations of Nonlinear Frontal Waves on Coastal Currents

Abstract

Nonlinear, dispersive perturbations are applied to a single layer, geostrophic, density current of zero potential vorticity, which flows along a vertical wall (coast) and is bounded by a free streamline in the seaward direction. The temporal evolution of these perturbations is shown to be governed by the, well known, Korteweg-deVries equation. The amplitude of the solitary wave (standing wave) solutions of this equation depends on their width (wavelength) and on the undisturbed velocity of the free streamline. The correlation between the free streamline displacement and the velocity perturbation shows that the observable displacements will be in the seaward direction and will propagate upstream relative to the mean flow. The theoretical calculations of the amplitude-wavelength relations are shown to be consistent with winter observations of the Davidson Current (a northern extension of the California Current System) and with observations of the African current. Comparison with the finite amplitude, longwave, theory shows that dispersion eliminates the blocking waves but retains the breaking waves, bores and wedges. These solutions are encountered only for specific values of the undisturbed free streamline velocity. In the finite amplitude, long-wave theory these waves are permissible solutions for any value of that velocity.