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.