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

We develop a flux globalization-based well-balanced path-conservative central-upwind scheme on Cartesian meshes for the two-dimensional (2-D) two-layer thermal rotating shallow water equations. The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states. In the 2-D case, preserving general "moving-water" steady states is difficult, and to the best of our knowledge, none of existing schemes can achieve this ultimate goal. The proposed scheme can exactly preserve the x- and y-directional jets in the rotational frame as well as certain genuinely 2-D equilibria. Numerical experiments demonstrate the performance of the proposed scheme in computationally non-trivial situations: in the presence of shocks, dry areas, non-trivial topographies, including discontinuous ones, and in the case of hyperbolicity loss. The scheme works equally well in both the f-plane and beta-plane frameworks.