Consistent shallow-water equations on the rotating sphere with complete Coriolis force and topography

Consistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow-water wisdom to the case of the rotating sphere are exposed. Different derivations of the model are possible, being based, respectively, on (i) Hamilton’s principle for primitive equations with a complete Coriolis force, under the hypothesis of columnar motion, (ii) straightforward vertical averaging of the ‘non-traditional’ primitive equations, and (iii) a time-dependent change of independent variables in the primitive equations written in the curl (‘vector-invariant’) form, with subsequent application of the columnar motion hypothesis. An intrinsic, coordinate-independent form of the non-traditional equations on the sphere is then given, and used to derive hyperbolicity criteria and Rankine–Hugoniot conditions for weak solutions. The relevance of the model for the Earth’s atmosphere and oceans, as well as other planets, is discussed.