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 is proven in this paper that this desideratum is equivalent to the existence of a mapping of geographical position to a Euclidean domain, combined with some position-dependent additive modification of the geopotential, which results in the SG theory being manifestly Legendre transformable from this alternative representation to its associated dual variables. Legendre-transformable representations for standard Boussinesq f-plane SG theory and for the axisymmetric gradient-balance version used to study the Eliassen vortex are already known and exploited in finite element algorithms. Here, two other potentially useful classes of SG theory discussed in a recent paper by the author are reexamined: (i) the nonaxisymmetric f-plane vortex and (ii) hemispheric (variable f) SG dynamics. The authors find that the imposition of the natural dynamical and geometrical symmetry requirements together with the requirement of Legendre transformability makes the choice of the f-plane vortex theory unique. Moreover, with modifications to accommodate sphericity, this special vortex theory supplies what appears to be the most symmetrical and consistent formulation of variable-f SG theory on the hemisphere. The Legendre-transformable representations of these theories appear superficially to violate the original symmetry of rotation about the vortex axis. But, remarkably, this symmetry is preserved provided one interprets the metric of the new representation to be a pseudo-Euclidean Minkowski metric. Rotation invariance of the dynamical formulation in physical space is then perceived as a formal Lorentz invariance in its Legendre-transformable representation.