Holomorphic structures in hydrodynamical models of nearly geostrophic flow

Abstract

We study complex structures arising in Hamiltonian models of nearly geostrophic ®ows in hydrodynamics. In many of these models an elliptic Monge{Amp ere equation de nes the relationship between a`balanced' velocity eld, de ned by a constraint in the Hamiltonian formalism, and the materially conserved potential vorticity. Elliptic Monge{Amp ere operators de ne 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, de nes K ahler and special K ahler 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 K ahler geometry.

A geometric interpretation of coherent structures in Navier–Stokes flows

“…Roulstone & Sewell (1997) and McIntyre & Roulstone (2002), describe how contact and Kähler geometries provide a framework for understanding the basis of the various coordinate transformations that have proven so useful in this context. This present work has evolved from that of Roubtsov & Roulstone (2001) using the results of Banos (2002). 2 We note that the MongeAmpère structures developed by Lychagin et al (1993), Banos (2002) and Kushner et al (2007) are not present when (1.2) is studied in the context of the incompressible Navier-Stokes equations in three dimensions.…”

“…The work of Roubtsov & Roulstone (1997, 2001) has shown how quaternionic and hyper-Kähler structures emerge in models of nearly geostrophic flows in atmosphere and ocean dynamics. These results were based on earlier work by McIntyre and Roulstone, and were reviewed by them in McIntyre & Roulstone (2002).…”

Hyper-Kähler geometry and semi-geostrophic theory

“…SG theory is one of the many so-called 'balanced models' (balance in this context refers to the geostrophic balance between fluid velocity and pressure gradient in flows on a rotating domain), although SG theory retains a special significance because of its elegant geometrical properties. Roubtsov & Roulstone (1997, 2001 showed that a hierachy of balanced models possess symplectic, contact and Monge-Ampère structures akin to SG theory, and that these hitherto apparently disconnected features can be viewed as the component parts of a hyper-Kähler structure. However, it was believed that SG theory itself could not be formulated in terms of hyper-Kähler geometry, *Author for correspondence (s.b.delahaies@surrey.ac.uk). and several other questions concerning the relevance of hyper-Kähler structures to balanced models remained open.…”

“…Instead of focusing on the map between local symplectic coordinates and the Lagrangian fluid coordinates, we apply the methods of Kushner et al (2007) and re-derive the geometric properties using the theory of Monge-Ampère operators. The salient difference between the two approaches may be summarized as follows: Roubtsov & Roulstone (1997, 2001) studied the geometry associated with D(φ), where D denotes the determinant of the Hessian matrix of a dependent variable φ (this determinant is the Jacobian discussed above), whereas we study the Monge-Ampère equation D(φ) = q, where q is a given function of the independent variables. As a consequence, we are able to show that SG theory does indeed possess a hyper-Kähler structure, and the issues raised by McIntyre & Roulstone (2002) can be resolved.…”

“…The approach adopted by Roubtsov & Roulstone (1997, 2001) was based on the special significance attached to the Jacobian of the map between local symplectic coordinates and the Lagrangian configuration coordinates of the fluid.…”