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The Lagrangian conservation law form of the semi-geostrophic equations used by Hoskins and others is studied further in two and three dimensions. A solution of the inviscid equations containing discontinuities corresponding to atmospheric fronts is shown to exist for all time under fairly general conditions, and to be unique if the potential vorticity is required to be nonnegative. Computational results show that this solution agrees with high resolution solutions of the viscous semi-geostrophic equations. The solution, however, disagrees with that obtained from the two-dimensional viscous primitive equations. An important aspect of the difference is that the semi-geostrophic solutions allow the front to propagate into the interior of the fluid while the primitive equation solutions do not. This is discussed. If correct, it may indicate a tendency for a separation effect in the atmosphere where frictional effects are present.
Cullen et al. (Tue,) studied this question.