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Kaehler geometry and fluid mechanics

Le : 21/04/2006 15h00
Par : Ian Roulstone (Université de Surrey)
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Résumé : From slowly-evolving large-scale fluid flows, such as we observe in the atmosphere and oceans, to rapidly changing and turbulent flows, fluid mechanics is believed to be described accurately by the classical Navier--Stokes-based equations of motion. Detailed computations of the three-dimensional incompressible Navier--Stokes equations vividly illustrate the fact that vorticity has a tendency to accumulate on quasi one-dimensional tubes or filaments and on quasi two-dimensional sheets. On larger scales (such as in the atmosphere and oceans) and in the asymptotic regimes that are most relevant for weather and climate forecasting, it can be shown that the solutions of the fluid equations stay close over finite, but useful, time intervals to the solutions of much simpler dynamical systems. These approximate models seek to describe flows in which there is a dominant balance between the Coriolis, buoyancy and pressure-gradient forces on fluid particles, which can be described very succinctly using vortex dynamics. Recent research (Roubtsov and Roulstone (2001); McIntyre and Roulstone (2002); Roulstone et al. (2005)) suggests that ideas from kahler geometry may be important in understanding the principles that govern the vortex dynamics of both the incompressible Navier--Stokes equations and the equations that govern those regimes most important to weather and climate. In this lecture, I shall first explain how the complex manifold structures emerge from the underlying partial differential equations, and then describe some of the implications for the fluid mechanics. Ref. McIntyre M. E., Roulstone I. (2002) Are there higher-accuracy analogues of semi-geostrophic theory? In Large-scale atmosphere---ocean dynamics, Vol. II: Geometric methods and models}. J. Norbury and I. Roulstone (eds.); Cambridge: University Press. Roubtsov, V. N., Roulstone, I. (2001) Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. Lond., A 457, 1519-1531. Roulstone I., Banos B., Gibbon J. D. and Roubtsov V. N. 2005 Kahler geometry and the Navier-Stokes Equations. Arxiv: http//arxiv.org/abs/nlin.SI/0509023