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" one 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 1) Hamilton's principle for primitive equations with a complete Coriolis force, under
the hypothesis of columnar motion, 2) straightforward vertical averaging of the
"non-traditional" primitive equations, 3) 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 and other planets is discussed.
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