  We develop a well-balanced central-upwind scheme for rotating shallow water
model with horizontal temperature and/or density gradients---the thermal
rotating shallow water (TRSW). The scheme is designed using the flux
globalization approach: first, the source terms are incorporated into the
fluxes, which results in a hyperbolic system with global fluxes; second, we
apply the Riemann-problem-solver-free central-upwind scheme to the rewritten
system. We ensure that the resulting method is well-balanced by switching off
the numerical diffusion when the computed solution is near (at)
thermo-geostrophic equilibria.
  The designed scheme is successfully tested on a series of numerical examples.
Motivated by future applications to large-scale motions in the ocean and
atmosphere, the model is considered on the tangent plane to a rotating planet
both in mid-latitudes and at the Equator. The numerical scheme is shown to be
capable of quite accurately maintaining the equilibrium states in the presence
of nontrivial topography and rotation. Prior to numerical simulations, an
analysis of the TRSW model based on the use of Lagrangian variables is
presented, allowing one to obtain criteria of existence and uniqueness of the
equilibrium state, of the wave-breaking and shock formation, and of instability
development out of given initial conditions. The established criteria are
confirmed in the conducted numerical experiments.
