Shear instability of longshore currents and their interaction with gravity waves

When wind-generated gravity waves propagate towards the shoreline with an oblique incidence angle, they shoal and break in the surf zone, generating longshore currents. These currents are generally unstable and large amplitude shear waves are excited. Since wave-driven currents in the nearshore region greatly affect sediment transport, a good understanding of the wave-induced nearshore circulation system is important to accurately predict the transport of sediments. The dynamics of nearshore processes is also of scientific interest because various time and length scale motions coexist and they interact. Here the generation and evolution of low-frequency shear waves and their interaction with high-frequency incoming gravity waves are studied both analytically and numerically. Linear stability analyses for various longshore current profiles are carried out, and special attention is paid to the free surface effect. It is found that the free surface reduces both the linear growth rate and the unstable wave number. Dynamics of unstable longshore currents beyond the linear regime is studied by solving the shallow water equations numerically. Small disturbances grow into large amplitude shear waves and large-scale coherent vortices (eddies) are often formed. In order to shed light on the process of the formation of eddies, simple mathematical models are derived using systematic asymptotic expansion. Compared with full numerical simulations of the shallow water equations, numerical solutions of the new asymptotic models are shown to correctly capture the process of wave steepening, which eventually leads the waves to break. In order to better predict the nearshore circulation system, wave-current interaction is included and numerical simulations of the coupled shallow water and wave action equations are carried out. Our results show that this interaction process significantly changes the dynamics of wave-induced nearshore circulation.