Run-up of long waves on background shear currents

Abstract

Long waves in shallow water propagating over a background shear current towards a sloping beach are investigated, and exact solutions are found using a hodograph transform and separation of variables. Inspired by the work of Carrier and Greenspan on steady waves over a uniform beach profile in the irrotational setting, we study waves which propagate over a background shear current. The shallow-water equations are obtained from the nonlinear Benney equations, and exact solutions are found with help of the hodograph transformation in conjunction with several further changes of variables. The hodograph transformation is effected by finding the Riemann invariants after the equations are written in the standard form of barotropic gas dynamics. In the current work, the background flow features zero mass flux, as would be required by a real flow at a beach. Moreover, in contrast with previous work, the present approach allows separate study of the influence of the strength of the shear current and the slope of the bottom profile. This enables us to provide an estimate of the run-up as a function of the shear flow while keeping the bottom slope constant.