Conservation Law Modeling of Entrainment in Layered Hydrostatic Flows

A methodology is developed for modeling entrainment in two-layer shallow water flows using non-standard conserved quantities, replacing layer-wise mass conservation by global energy conservation. Thus the energy that the standard model would regularly dissipate at internal shocks is instead available to exchange fluid between the layers. Two models are considered for the upper boundary of the flow: a rigid lid and a free surface. The latter provides a selection principle for choosing physically relevant conservation laws among the infinitely many that the former possesses, when the ratio between the baroclinic and barotropic speeds tends to zero. Solutions of the equations are studied analytically and numerically, applied to the lock-exchange problem, and compared to other closures.