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Variational principles for stochastic fluid dynamics

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Title: Variational principles for stochastic fluid dynamics
Authors: Holm, DD
Item Type: Journal Article
Abstract: This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler-Boussinesq and quasi-geostropic approximations.
Issue Date: 8-Apr-2015
Date of Acceptance: 24-Feb-2015
URI: http://hdl.handle.net/10044/1/64380
DOI: 10.1098/rspa.2014.0963
ISSN: 1364-5021
Publisher: Royal Society, The
Journal / Book Title: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume: 471
Issue: 2176
Copyright Statement: © 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Keywords: cylindrical stochastic processes
geometric mechanics
multiscale fluid dynamics
stochastic fluid models
symmetry reduced variational principles
cylindrical stochastic processes
geometric mechanics
multiscale fluid dynamics
stochastic fluid models
symmetry reduced variational principles
math-ph
math-ph
math.MP
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status: Published
Conference Place: England
Article Number: 20140963
Online Publication Date: 2015-04-08
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences