Abstract

The study of dam break flows in the inviscid shallow water equations dates back to the similarity solution of Ritter (1892). A natural modification to the problem, considered by Whitham and others, is to add a frictional drag force in the momentum equation. At early times this friction affects Ritter’s solution only in a small but expanding region near the tip of the dam break flow. The solution in this region can be described by a further similarity theory, developed by Hogg and Pritchard. Here we show that, for drag laws commonly used in applications, the Hogg-Pritchard equations – which are known to exhibit critical behaviour – do not admit continuous solutions and must be regularised by a rear-facing hydraulic jump or dispersive shock wave. The solutions therefore give – arguably – the simplest mathematical account of the origin of the gravity current `head’ seen in experiments. Interestingly, if a dispersive shock wave is used to regularise the Hogg-Pritchard equations it must be of a non-standard type, due to the dilation of the local coordinate system.