Weakly nonlinear shear waves

Alongshore propagating low-frequency O(0.01 Hz) waves related to the direction and intensity of the alongshore current were first observed in the surf zone by Oltman-Shay, Howd & Birkemeier (1989). Based on a linear stability analysis, Bowen & Holman (1989) demonstrated that a shear instability of the alongshore current gives rise to alongshore propagating shear (vorticity) waves. The fully nonlinear dynamics of finite-amplitude shear waves, investigated numerically by Alien, Newberger & Holman (1996), depend on a, the non-dimensional ratio of frictional to nonlinear terms, essentially an inverse Reynolds number. A wide range of shear wave environments are reported as a function of or, from equilibrated waves at larger a to fully turbulent flow at smaller a. When a is above the critical level a,, the system is stable. In this paper, a weakly nonlinear theory, applicable to a just below a,, is developed. The amplitude of the instability is governed by a complex Ginzburg-Landau equation. For the same beach slope and base-state alongshore current used in Alien et al. (1996), an equilibrated shear wave is found analytically. The finite-amplitude behaviour of the analytic shear wave, including a forced second-harmonic correction to the mean alongshore current, and amplitude dispersion, agree well with the numerical results of Alien et al. (1996). Limitations in their numerical model prevent the development of a side-band instability. The stability of the equilibrated shear wave is demonstrated analytically. The analytical results confirm that the Alien et al. (1996) model correctly reproduces many important features of weakly nonlinear shear waves.