Legendre-Transformable Semigeostrophic Theories

Abstract For semigeostrophic (SG) theories derived from the Hamiltonian principles suggested by Salmon it is known that a duality exists between the physical coordinates and geopotential, on the one hand, and isentropic geostrophic momentum coordinates and geostrophic Bernoulli function, on the other hand. The duality is characterized geometrically by a contact structure. This enables the idealized balanced dynamics to be represented by horizontal geostrophic motion in the dual coordinates while the mapping back to physical space is determined uniquely by requiring each instantaneous state to be the one of minimum energy with respect to volume-conserving rearrangements within the physical domain. It is found that the generic contact structure permits the emergence of topological anomalies during the evolution of discontinuous flows. For both theoretical and computational reasons it is desirable to seek special forms of SG dynamics in which the structure of the contact geometry prohibits such anomalies. It...