WEAKLY NONLINEAR SHEAR-WAVES

Alongshore propagating low-frequency O(0.01 Hz) waves related to the d irection and intensity of the alongshore current were first observed i n the surf zone by Oltman-Shay, Howd & Birkemeier (1989). Based on a l inear stability analysis, Bowen & Holman (1989) demonstrated that a sh ear instability of the alongshore current gives rise to alongshore pro pagating shear (vorticity) waves. The fully nonlinear dynamics of fini te-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 ra nge of shear wave environments are reported as a function of or, from equilibrated waves at larger a to fully turbulent flow at smaller a. W hen a is above the critical level a,, the system is stable. In this pa per, a weakly nonlinear theory, applicable to a just below a,, is deve loped. The amplitude of the instability is governed by a complex Ginzb urg-Landau equation. For the same beach slope and base-state alongshor e current used in Alien et al. (1996), an equilibrated shear wave is f ound analytically. The finite-amplitude behaviour of the analytic shea r wave, including a forced second-harmonic correction to the mean alon gshore current, and amplitude dispersion, agree well with the numerica l 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 analytic al results confirm that the Alien et al. (1996) model correctly reprod uces many important features of weakly nonlinear shear waves.