Flux globalization based well-balanced path-conservative central-upwind scheme for two-layer thermal rotating shallow water equations

Abstract

We develop a flux globalization based well-balanced path-conservative central-upwind scheme for the two-layer thermal rotating shallow water (TRSW) equations, which arise in both oceanography and atmospheric sciences. As it occurs in all of the two-layer shallow water models, the studied two-layer TRSW system loses hyperbolicity due to the well-known Kelvin-Helmholtz type instabilities arising when the vertical velocity shear between the layers becomes sufficiently large. Therefore, prior to developing the numerical method, we discuss the hyperbolicity criterion. Besides the hyperbolicity loss, there are additional challenges in the development of numerical methods for the studied system. They are related to the presence of nonconservative terms modeling layer interaction and also to a quite complicated structure of steady-states solutions a good well-balanced numerical method should be able to exactly preserve. In order to treat the nonconservative product terms, we use the path-conservative technique, which is implemented within the flux globalization framework: the source and nonconservative terms are incorporated into the fluxes, which results in a quasi-conservative system, which is then numerically solved using the Riemann-problem-solver-free central-upwind scheme. A well-balanced property of the resulting scheme is ensured by performing piecewise linear reconstruction for the equilibrium rather than the conservative variables, by development of special quadratures required for the flux globalization procedure, and by switching off a part of the numerical diffusion when the computed solution is near or at thermo-geostrophic equilibria. The advantages and excellent performance of the proposed scheme are demonstrated on a number of numerical examples.