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Numerical study on the incompressible Euler equations as a Hamiltonian system: Sectional curvature and Jacobi field

Ohkitani, K. (2010) Numerical study on the incompressible Euler equations as a Hamiltonian system: Sectional curvature and Jacobi field. Physics of Fluids, 22 (5). Art no.057101 . ISSN 1070-6631

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Abstract

We study some of the key quantities arising in the theory of [Arnold "Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits," Annales de l'institut Fourier 16, 319 (1966)] of the incompressible Euler equations both in two and three dimensions. The sectional curvatures for the Taylor-Green vortex and the ABC flow initial conditions are calculated exactly in three dimensions. We trace the time evolution of the Jacobi fields by direct numerical simulations and, in particular, see how the sectional curvatures get more and more negative in time. The spatial structure of the Jacobi fields is compared to the vorticity fields by visualizations. The Jacobi fields are found to grow exponentially in time for the flows with negative sectional curvatures. In two dimensions, a family of initial data proposed by Arnold (1966) is considered. The sectional curvature is observed to change its sign quickly even if it starts from a positive value. The Jacobi field is shown to be correlated with the passive scalar gradient in spatial structure. On the basis of Rouchon's physical-space based expression for the sectional curvature (1984), the origin of negative curvature is investigated. It is found that a "potential" alpha(xi) appearing in the definition of covariant time derivative plays an important role, in that a rapid growth in its gradient makes a major contribution to the negative curvature. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3407673]

Item Type: Article
Copyright, Publisher and Additional Information: © 2010 American Institute of Physics. This is an author produced version of a paper subsequently published in Physics of Fluids. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Preserving Diffeomorphism Group; Conservative-Systems; CLASSICAL MECHANICS; Volumorphism Group; Riemann Curvature; Ideal Fluids; Motion; Siingularities; Geodesics
Academic Units: ?? Sheffield.AMA ??
The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 25 Jun 2010 10:03
Last Modified: 08 Feb 2013 17:00
Published Version: http://dx.doi.org/10.1063/1.3407673
Status: Published
Publisher: American Institute of Physics
Identification Number: 10.1063/1.3407673
URI: http://eprints.whiterose.ac.uk/id/eprint/10965

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