The shallow water or hydraulic limit for a single layer of fluid is well understood. All waves eventually break, and, at shocks, conservation of mass and momentum completely define the hydraulic jump. The energy equation gives, a posteriori, the rate at which "internal" energy (mostly in the form of small scale turbulence) is generated. For the case of two-layer shallow water, even the evolution of smooth solutions was not completely understood since there is the possibility for the system to be elliptic, a remnant of the Kelvin-Helmholtz instability.
We first show that these flows are nonlinearly stable: smooth solutions never cross into the elliptic domain. Then, we consider shocks at the interface of miscible fluids - a problem of geophysical importance in the atmosphere and ocean. We require an additional postulate that would yield the mixing rate at the shock: one can imagine that the energy dissipated at the shock can now flow both into small scale turbulence and into mixing the fluid, but the partition between these sinks of macroscopic energy is unknown. We discuss two possibilities for deriving the additional constraint on the problem: kinematics and entropy maximization. We show that these are in fact equivalent and yield upper bounds on the mixing rates.
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