Variational principles for stochastic fluid dynamics
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Published version
Author(s)
Holm, Darryl D
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.
Date Issued
2015-04-08
Date Acceptance
2015-02-24
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015, 471 (2176)
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.
Identifier
https://www.ncbi.nlm.nih.gov/pubmed/27547083
Subjects
cylindrical stochastic processes
geometric mechanics
multiscale fluid dynamics
stochastic fluid models
symmetry reduced variational principles
Publication Status
Published
Coverage Spatial
England
Article Number
20140963
Date Publish Online
2015-04-08