Conservation law modelling of entrainment in layered hydrostatic flows

A methodology is developed for modelling entrainment in two-layer shallow water flows using non-standard conserved quantities, replacing layerwise 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 with other closures.