Nonlinear Evolution of Shear Instabilities of the Longshore Current

Surface gravity waves breaking in the nearshore region force a longshore surf zone current. This current can be unstable to longshore periodic perturbations. The continuity and momentum conservation equations averaged over the short wave time scales and over depth present a suitable basis for the modeling of these motions. The governing equations are in the form of the well-known shallow water equations with additional terms accounting for short wave forcing and dissipation effects. The objective of this study is to analyze the finite amplitude behavior of instabilities of the surf zone longshore current utilizing numerical experiments. For this purpose a solution method for the shallow water equations governing wave motions in the nearshore environment is developed. Spatial derivatives contained in these equations are computed using spectral collocation methods. A high-order time integration scheme is used to compute the time evolution of the velocities and water surface elevation given initial conditions. The model domain extends from the shoreline to a desired distance offshore and is periodic in the longshore direction. Properly posed boundary conditions for the governing equations are discussed. A curvilinear moving boundary condition is incorporated at the shoreline to account for wave run-up. An absorbing-generating boundary is incorporated offshore. The boundary treatments are tested using analytical and numerical results. The model is applied to the prediction of neutral stability boundaries and equilibrium amplitudes of subharmonic edge waves. Numerical results are compared to weakly nonlinear theory and are found to reproduce the theory well. The solution method is utilized to simulate instabilities of an analytic longshore current profile over a plane beach. The instabilities are observed to grow and equilibrate at amplitudes up to 50% of the original peak mean longshore current. For long domains in the longshore direction the long time behavior is observed to be dominated by subharmonic transitions that result in a reduction of the number of waves in the domain. The resulting longshore periodic fiow structures exhibit strong offshore directed velocities and propagate in the longshore direction at a fraction of the peak current speed. Details of the subharmonic transitions as well as the effect of non-linearity on the flow structures are analyzed. Next, the shear instability climate during the SUPERDUCK field experiment is simulated. Observations of undulations in the longshore current were first made during this field experiment by Oltman-Shay et al. (1989), who stated that the frequency range less than 0.01 Hz is dominated by these motions. Due to uncertainties in the friction and lateral mixing coefficients, numerical simulations are carried out for a realistic range of values for these coefficients. The resulting flow structures can be characterized as unsteady vortices propagating in the longshore direction. These vortices interact, occasionally merge and are shed offshore. During the shedding process, locally strong offshore directed currents are generated. Lateral mixing induced by the finite amplitude shear instabilities is analyzed and found to be of comparable magnitude to other mixing processes in the surf zone. Results from simulations of shear instabilities on plane and barred beaches show the existence of localized, migrating, offshore directed currents. Since the short wave field can be affected by these flow features, the modeling effort is extended to include the effects of time-varying short wave forcing and interactions between the short wave and current fields. The extension involves the solution of the time-dependent energy equation for the short wave motions and refraction equation due to variations in the bathymetry as well as current fields. The inclusion of a more realistic bottom friction treatment is also discussed.