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.