Convective Wave Breaking in the KdV Equation

TL;DR

This paper investigates wave breaking phenomena in the KdV equation, identifying critical amplitudes where particle velocities match wave phase velocities, leading to wave breaking and model breakdown.

Contribution

It introduces a convective breaking criterion based on the local Froude number and applies it to both steady and time-dependent wave scenarios.

Findings

Wave breaking occurs when particle velocity exceeds wave phase velocity.

Critical amplitudes for solitary and cnoidal waves are identified.

Boundary forcing can induce wave breaking in undular bores.

Abstract

The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface boundary condition. The condition for breaking can be conveniently formulated as a convective breaking criterion based on the local Froude number at the wave crest. This breaking criterion can also be applied to time-dependent situations, and one case of interest is the development of an undular bore created by an influx at a lateral boundary. It is shown that this boundary forcing leads to wave breaking in the leading wave behind the bore if a certain threshold is surpassed.