TY - JOUR
T1 - Holomorphic structures in hydrodynamical models of nearly geostrophic flow
JF - Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
SP - 1519
LP - 1531
M3 - 10.1098/rspa.2001.0779
VL - 457
IS - 2010
AU - Roubtsov, V. N.
AU - Roulstone, I.
Y1 - 2001/06/08
UR - http://rspa.royalsocietypublishing.org/content/457/2010/1519.abstract
N2 - We study complex structures arising in Hamiltonian models of nearly geostrophic flows in hydrodynamics. In many of these models an elliptic Monge–Ampère equation defines the relationship between a ‘balanced’ velocity field, defined by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge–Ampère operators define an almost–complex structure, and 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 theoretical meteorology, defines Kahler and special Kähler structures on phase space. Furthermore, analogues of the ‘geostrophic momentum coordinates’ 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 Kähler geometry.
ER -