A new well-posed vorticity divergence formulation of the shallow water equations

•A new formulation of the two-dimensional shallow water equations is derived.•The new formulation includes four dependent variables instead of three.•It requires more equations and boundary conditions than the original formulation.•The new formulation uses gradients of height as dependent variables.•The new formulation leads to a well posed problem for limited area modelling.

A new vorticity–divergence formulation of the two-dimensional shallow water equations including boundary conditions is derived. The new formulation is necessary since the conventional one does not lead to a well-posed initial boundary value problem for limited-area modelling.The new vorticity–divergence formulation includes four dependent variables instead of three and requires more equations and boundary conditions than the conventional formulation. On the other hand, it forms a hyperbolic set of equations with well-defined boundary conditions that leads to a well-posed problem with bounded energy.