Nonlinear shear instabilities of alongshore currents on plane beaches

Abstract

Evidence for the existence in the nearshore surf zone of energetic alongshore propagating waves with periods O(100 s) and wavelengths O(100 m) was found from observations by Oltman-Shay et al. (1989). These oscillations have wavelengths that are much too short to be surface gravity waves at the observed frequencies. The existence and properties of the wave-like motions were found to be related to the presence, strength and direction of an alongshore current in the surf zone. Based on a linear stability analysis of a mean alongshore current with offshore scale O(100 m), Bowen & Holman (1989) described these fluctuations as unstable waves associated with a shear instability. Good agreement of wavelengths and wave speeds from observations and from predictions based on the most unstable linear mode was obtained by Dodd et al. (1992). The nonlinear dynamics of finite-amplitude shear instabilities of alongshore currents in the surf zone are studied here utilizing numerical experiments involving finite-difference solutions to the shallow water equations for idealized forced dissipative initial-value problems. Plane beach (i.e. constant slope) geometry is used with periodic boundary conditions in the alongshore direction. Forcing effects from obliquely incident breaking surface waves are approximated by an across-shore-varying steady force in the alongshore momentum equation. Dissipative effects are modelled by linear bottom friction. The solutions depend on the dimen-sionless parameter Q, which is the ratio of an advective to a frictional time scale. The steady frictionally balanced, forced, alongshore current is linearly unstable for Q less than a critical value Qc. The response of the fluid is studied for different values of ΔQ = Qc - Q. In a set of experiments with the alongshore scale of the domain equal to the wavelength 2π/k0 of the most unstable linear mode, disturbances that propagate alongshore in the direction of the forced current with propagation velocities similar to the linear instability values are found for positive ΔQ. The disturbances equilibrate with constant amplitude for small ΔQ and with time-varying amplitudes for larger ΔQ. For increasing values of ΔQ the behaviour of this fluid system, as represented in a phase plane with area-averaged perturbation kinetic energy and area-averaged energy conversion as coordinates, is similar to that found in low-dimensional nonlinear dynamical systems including the existence of non-trivial steady solutions, bifurcation to a limit cycle, period-doubling bifurcations, and irregular chaotic oscillations. In experiments with the alongshore scale of the domain substantially larger than the wavelength of the most unstable linear mode, different behaviour is found. For small positive ΔQ, propagating disturbances grow at wavelength 2π/k0. If ΔQ is small enough, these waves equilibrate with constant or spatially varying amplitudes. For larger ΔQ, unstable waves of length 2π/k0 grow initially, but subsequently evolve into longer-wavelength nonlinear propagating steady or unsteady wave-like disturbances with behaviour dependent on ΔQ. The eventual development of large-scale nonlinear propagating disturbances appears to be a robust feature of the flow response over plane beach geometry for moderate, positive values of ΔQ and indicates the possible existence in the nearshore surf zone of propagating finite-amplitude shear waves with properties not directly related to results of linear theory.