Mixing and shocks in geophysical shallow water models

In the first section, a reduced two-layer shallow water model for fluid mixing is described. The model is a nonlinear hyperbolic quasilinear system of partial differential equations, derived by taking the limit as the upper layer becomes infinitely deep. It resembles the shallow water equations, but with an active buoyancy. Fluid entrainment is supposed to occur from the upper layer to the lower. Several physically motivated closures are proposed, including a robust closure based on maximizing a mixing entropy (also defined and derived) at shocks. The structure of shock solutions is examined. The Riemann problem is solved by setting the shock speed to maximize the production of mixing entropy. Shock-resolving finite-volume numerical models are presented with and without topographic forcing. Explicit shock tracking is required for strong shocks. The constraint that turbulent energy production be positive is considered. The model has geophysical applications in studying the dynamics of dense sill overflows in the ocean.; The second section discusses stationary shocks of the shallow water equations in a reentrant rotating channel with wind stress and topography. Asymptotic predictions for the shock location, strength, and associated energy dissipation are developed by taking the topographic perturbation to be small. The scaling arguments for the asymptotics are developed by demanding integrated energy and momentum balance, with the result that the free surface perturbation is of the order of the square root of the topographic perturbation. Shock formation requires that linear waves be nondispersive, which sets a solvability condition on the mean flow and which leads to a class of generalized Kelvin waves. Two-dimensional shock-resolving numerical simulations validate the asymptotic expressions and demonstrate the presence of stationary separated flow shocks in some cases. Geophysical applications are considered.; Overview sections on shock-resolving numerical methods and the shallow water equations are also included.