Joint Centre for Mesoscale Meteorology (JCMM)
JCMM INTERNAL REPORT NO.111
Holomorphic structures in hydrodynamical models of nearly geostrophic flow
by V N Roubtsov and I Roulstone
We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. In many of these models an elliptic Monge-Ampere equation defines the relationship between a "balanced" velocity field and the potential vorticity, and the Monge-Ampere operator defines an almost-complex structure on the model phase space. In this paper we show that a natural extension of the so-called geostrophic momentum transformation of semi-geostrophic theory, which has a special importance in both theoretical meteorology and numerical weather prediction, defines a special Kahler structure on phase space. Furthermore, analogues of the "geostrophic momentum coordinates" are shown to be special Lagrangian coordinates under certain conditions which depend upon the physical approximations under consideration.