Shallow water equations with a complete Coriolis force and topography

This paper derives a set of two-dimensional equations describing a thin inviscid fluid layer flowing over topography in a frame rotating about an arbitrary axis. These equations retain various terms involving the locally horizontal components of the angular velocity vector that are discarded in the usual shallow water equations. The obliquely rotating shallow water equations are derived both by averaging the three-dimensional equations and from an averaged Lagrangian describing columnar motion using Hamilton’s principle. They share the same conservation properties as the usual shallow water equations, for the same energy and modified forms of the momentum and potential vorticity. They may also be expressed in noncanonical Hamiltonian form using the usual shallow water Hamiltonian and Poisson bracket. The conserved potential vorticity takes the standard shallow water form, but with the vertical component of the rotation vector replaced by the component locally normal to the surface midway between the upper and lower boundaries.