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Viscous perturbations of marginally stable Euler flow and finite-time Melnidov theory

Grenier, Emmanuel, Jones, Christopher K. R. T., Rousset, Frederic and Sandstede, Björn (2004) Viscous perturbations of marginally stable Euler flow and finite-time Melnidov theory Nonlinearity, 18 (2). pp. 465-483. ISSN 0951-7715


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The effect of small viscous dissipation on Lagrangian transport in two-dimensional vorticity conserving fluid flows motivates this work. If the inviscid equation admits a base flow in which different fluid regions are divided by separatrices, then transport between these regions is afforded by the splitting of separatrices caused by viscous dissipation. Finite-time Melnikov theory allows us to measure the splitting distance of separatrices provided the perturbed velocity field of the viscous fluid flow stays sufficiently close to vorticity-conserving base flow over sufficiently long time intervals. In this paper, we derive the necessary long-term estimates of solutions to Euler’s equation and to the barotropic vorticity equation upon adding viscous perturbations and forcing. We discover that a certain stability condition on the unperturbed flow is sufficient to guarantee these long time estimates.

Item Type: Article
Additional Information: This is a pre-copy-editing, author-prepared, peer-reviewed PDF of an article published in Nonlinearity, 18, 465-483. © 2005 Publishing Ltd and London Mathematical Society. Click here to access the publisher's version.
Uncontrolled Keywords: Viscous perturbations, Euler flow, Finite-time Melinkov theory
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:41
Last Modified: 23 Sep 2013 18:33

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