In the past twenty years, semigeostrophic equations have become a prominent model for describing certain atmospheric motions on a synoptic scale, including the presence of fronts. Theoretical studies of them have revealed Hamiltonian features, and novel numerical methods, motivated by the need to improve weather forecasts, have been explored. A shallow–water theory analogue has been used as a paradigm for some aspects.
This paper sets out to uncover the mathematical structure of the semigeostrophic equations that has been essential to finding solutions and developing numerical techniques. We study the shallow–water and atmospheric theories side–by–side, and we introduce a generalized form which encapsulates the differences between them. When the Coriolis parameter, f, is a constant, it is found that a lift transformation is at the heart of the theory, and the consequences of this are developed. When f is not a constant, the role of the lift transformation is, in some respects, looser; we explore the extent to which it still offers a worthwhile guide. In particular, it can be viewed as motivating a generalization of the geostrophic momentum transformation for planetary semigeostrophic equations.
The paper is broadly self–contained, and it takes account of several different strands in the existing literature.