Modeling the Effect of Friction on Quadratic Depth Profile

In the inviscid rip-current regime it was demonstrated that there is a longshore component of the rip current system, which does not need friction in order to exist. Hence here in the viscous regime we are not concerned with any frictionally determined currents, but only with how the frictional terms modify the already constructed inviscid solution. Note that the frictional terms are not invoked in the shoaling zone, and so the solution remains unchanged in that regime. The nearshore is characterized by the presence of breaking waves, and so we developed equation to be used outside surf zone, based on small-amplitude wave theory, and another set of equations to be used inside the surf zone, based on an empirical representation of breaking waves, Suitable matching conditions are applied at the boundary between the offshore shoaling zone and the nearshore surf zone. Both sets of equation are obtained by averaging the basic equations over the wave phase. Thus the qualitative solution constructed is a free vortex defined in both shoaling and surf zone where in the surf zone the free vortex is perturbed by a long shore component.