@article {Roubtsov1519,
author = {Roubtsov, V. N. and Roulstone, I.},
title = {Holomorphic structures in hydrodynamical models of nearly geostrophic flow},
volume = {457},
number = {2010},
pages = {1519--1531},
year = {2001},
doi = {10.1098/rspa.2001.0779},
publisher = {The Royal Society},
abstract = {We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. In many of these models an elliptic Monge{\textendash}Amp{\`e}re equation defines the relationship between a {\textquoteleft}balanced{\textquoteright} velocity field, defined by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge{\textendash}Amp{\`e}re operators define an almost{\textendash}complex structure, and in this paper we show that a natural extension of the so{\textendash}called geostrophic momentum transformation of semi{\textendash}geostrophic theory, which has a special importance in theoretical meteorology, defines Kahler and special K{\"a}hler structures on phase space. Furthermore, analogues of the {\textquoteleft}geostrophic momentum coordinates{\textquoteright} are shown to be special Lagrangian coordinates under conditions which depend upon the physical approximations under consideration. Certain duality properties of the operators are studied within the framework of the K{\"a}hler geometry.},
issn = {1364-5021},
URL = {http://rspa.royalsocietypublishing.org/content/457/2010/1519},
eprint = {http://rspa.royalsocietypublishing.org/content/457/2010/1519.full.pdf},
journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences}
}