The Lagrangian form of the semigeostrophic equations has been shown to possess discontinuous solutions that have been exploited as a simple model of fronts and other mesoscale flows. In this paper, it is shown that these equations can be integrated forward in time for arbitrarily long periods without breaking down, to give a “slow manifold” of solutions. In the absence of moisture, orography and surface friction, these solutions conserve energy, despite the appearance of discontinuities.
In previous work these solutions have been derived by making finite parcel approximations to the data. This paper shows that there is a unique solution to the equations with general piecewise smooth data, to which the finite parcel approximation converges. It is also shown that the time integration procedure is well defined, and that the solutions remain bounded for all finite times.
Most previous results on the finite parcel solutions are restricted to the case of a Boussinesq atmosphere on an f-plane with rigid-wall boundary conditions. In this paper the results are extended to non-Boussinesq fluids, free-surface and periodic boundary conditions, and variable Coriolis parameter. Previous work on a version of the theory for axisymmetric flows is extended to approximately axisymmetric flows. The behavior of the equations on a sphere and the effects of external forcing are discussed.