Numerical solution of the two-layer shallow water equations with bottom topography

We present a simple, robust numerical method for solving the two-layer shallow water equations with arbitrarybottom topography.Using the techniqueof operator splitting,we write the equationsas a pair of hyperbolic systems with readily computed characteristics, and apply third-order-upwind differences to the resulting wave equations. To prevent the thickness of either layer from vanishing, we modify the dynamics, inserting an artiŽ cial form of potential energy that becomes very large as the layer becomes very thin. Compared to high-order Riemann schemes with  ux or slope limiters, our method is formally more accurate, probably less dissipative, and certainly more efŽ cient. However, because we do not exactly conservemomentum and mass, bores move at the wrong speed unless we add explicit, momentum-conserving viscosity. Numerical solutions demonstrate the accuracy and stability of the method. Solutions corresponding to two-layer, wind-driven ocean  ow require no explicit viscosity or hyperviscosity of any kind; the implicit hyperdiffusion associated with third-order-upwinddifferencing effectively absorbs the enstrophycascade to small scales.