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Planetary semi-geostrophic equations derived from Hamilton's principle

A new set of balanced equations (to be called planetary semi-geostrophic equations) for planetary-scale flow is derived from Hamilton's principle and constitutes a natural generalization of the semi-geostrophic equations of motion. Analogues of the global conservation of energy and the Lagrangian conservation of potential vorticity follow automatically by introducing approximations directly into the Hamiltonian in such a way that time and particle label symmetries are preserved. Two approximations are required: first, the kinetic energy associated with the component of velocity parallel to the axis of rotation is neglected; and secondly, the Lagrangian rate of change of the wind and pressure gradient directions (when projected onto the equatorial plane) must be small compared with twice the angular rotation rate of the system. Although the first of these approximations entails some loss of accuracy for application to the terrestrial atmosphere it is not nearly as severe as that for the Phillips type II geostrophic equations in which all of the kinetic energy is omitted from the Hamiltonian. The resulting equations take exactly the same form as the f-plane semi-geostrophic equations apart from a modification to the pseudo-density appearing in the continuity equation. They are also amenable to the geostrophic momentum coordinate transformation – a device which has had considerable impact on the theory of atmospheric fronts. In order to assess the accuracy of the equations, three different linearized eigenvalue problems on the sphere are solved and compared with the equivalent primitive equation problems. Eigenmodes are least accurate for high-zonal-wavenumber disturbances with grave meridional structure. Stationary, baroclinic planetary waves with zonal wavenumber less than ≈ 7 are shown to be accurately treated. The equations also support equatorially trapped Kelvin and Rossby modes which are accurate in the long-wave limit for meteorologically relevant equivalent depths.