Ideal shocks in 2-layer flow - Part I: Under a rigid lid

Previous work on the classical problem of shocks in a 2-layer density-stratified fluid used either a parameterized momentum exchange or an assumed Bernoulli loss. We propose a new theory based on a set of viscous model equations. We define an idealized shock in two-layer density stratified flow under a rigid lid as a jump or drop of the interface in which (1) the force balance remains nearly hydrostatic in the shock, (2) there is no exchange of momentum between the two layers except by pressure forces on the sloping interface, and (3) dissipative processes can be treated with a constant viscosity. We proceed in two steps. First, we derive a necessary condition for shock existence based on a requirement for wave steepening. Second, we formulate and solve a set of viscous model equations. Some results are the following: Shocks require strong layer asymmetry; one layer must be much faster and/or shallower than the other layer. The linearized equations describing the shock tails provide boundary conditions and a proof of shock uniqueness. It is possible to derive an analytical solution for weak shocks if the steepening condition is met. The weak shock solutions provide closed form expressions for the Bernoulli loss in each layer. Bernoulli losses are strongly concentrated in the expanding layer as the relative layer depth change is much larger in that layer. Bernoulli losses are independent of layer viscosity. A sudden cessation of shock existence is found for strong shocks when the possible end state migrates into the supercritical regime. Surprisingly, the new ideal shock theory compares well with a 2-D, time-dependent shallow water model (SWM) with a flux formulation, but with no viscous formulation. Both the Bernoulli drop and shock cessation condition agree quantitatively.