The Riemann problem for the shallow water equations with horizontal temperature gradients

We consider the Riemann problem for the system of shallow water equations with horizontal temperature gradients (the Ripa system). The model under investigation has the form of a nonconservative system, and it is hyperbolic, but is not strictly hyperbolic. We construct all solutions of the Riemann problem. It turns out that there may be up to three distinct solutions. A resonant phenomenon which causes the colliding shock waves is observed, where multiple waves associated with different characteristic fields propagate with the same shock speed.