Asymptotic analysis of the eigenstructure of the two-layer model and a new family of criteria for evaluating the model hyperbolicity

Two-layer and multi-layer depth-averaged models have become popular for simulating exchange flows, seawater currents and geophysical flows. The partial differential equation systems associated with these models are similar to the single-layer shallow-water model. Yet, their eigenstructures are more complex owing to the pressure coupling between the layers. Such models occasionally lose their hyperbolic character, which may lead to numerical issues. A physical explanation is that Kelvin-Helmholtz type instabilities arise at the layers' interface, if the velocity difference between the layers becomes sufficiently large. A way to avoid the hyperbolicity loss is to locally introduce an extra momentum exchange between the layers, assessable from the system eigenstructure and aimed at roughly mimicking the dynamical effects of such instabilities. To better understand the hyperbolicity conditions, the eigenstructure of the two-layer model is methodically studied by an asymptotic analysis. The analysis for the limiting cases, where the layers' thicknesses are either comparable or very different from each other, reveals new stability criteria. These analytical criteria are, then, exploited to design a new family of approximate criteria, valid for any flow condition. Numerical investigations demonstrate the reliability of this approach, which can be easily implemented in numerical schemes for preserving the hyperbolicity.