Jump Conditions for Hyperbolic Systems of Forced Conservation Laws with an Application to Gravity Currents

Abstract

Weak solutions to systems of nonlinear hyperbolic conservation laws admit discontinuities that result from either an initial value or as part of the temporally developing solution itself. The propagation of such shocks or jumps is affected by forcing terms for the nonlinear system in a way that has not been investigated fully in standard references. Jump conditions for systems of conservation laws with discontinuous forcing terms are derived herein, following the method used to derive the Rankine–Hugoniot jump conditions, and the generalized results are illustrated for the one‐dimensional inviscid Burger's equation with discontinuous forcing. The main application of this type of jump condition, and the primary motivation for its study, is its application to a shallow‐water model of gravity currents previously described by the authors. Specifically, a new result relation between the front and height at a gravity current front is obtained by using the existing model. Front speeds for gravity currents resulting from instantaneous release are calculated numerically and used to determine the suitability of the jump conditions, which are then compared with existing theoretical expressions and experimental observations. New numerical results are portrayed for the gravity current model, suggesting that the standard method of modeling shallow‐water gravity currents with a simple Froude number front condition may tend to suppress some of the finer details of the flow resolved by the numerical scheme used by the authors.