Shallow water dynamics on linear shear flows and plane beaches

Abstract

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables based on the hodograph transform may be used to transform the system into a linear equation which may be solved exactly using the method of separation of variables. This method can be used to investigate the run-up of a long wave on a planar beach including the development of the shoreline.

Extreme Wave Runup on a Steep Coastal Profile

“…In the presence of bathymetry, it is somewhat more difficult to find the requisite change of variables than in the case of constant coefficients. Nevertheless, an appropriate hodograph transformation was found by Carrier and Greenspan 15 , and there have been a number of works seeking to extend and generalize that idea (see [16][17][18][19][20][21] and references therein).…”

A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model

“…In particular the case of a gravity current flowing upslope, as in a shallow water wave encountering a beach, is of importance and has seen new developments in recent years. [36][37][38] There is an exhaustive literature on the subject of hyperbolicity of the 1LSWE, [39][40][41][42][43][44] including the topic of well-balanced formulation, the more general E-balanced schemes, and the identification of resonant versus non-resonant regimes of flow. These points are mainly of interest for the numerical solution of the equations in specific regimes.…”