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Variational principles for stochastic fluid dynamics.

Darryl D Holm1

  • 1Department of Mathematics , Imperial College , London, UK.

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|August 23, 2016
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Summary
This summary is machine-generated.

This study derives stochastic fluid dynamics equations using a stochastic variational principle. The findings show Stratonovich equations preserve fluid circulation and helicity, unlike their Itô counterparts.

Keywords:
cylindrical stochastic processesgeometric mechanicsmultiscale fluid dynamicsstochastic fluid modelssymmetry reduced variational principles

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Area of Science:

  • Fluid Dynamics
  • Stochastic Processes
  • Mathematical Physics

Background:

  • Stochastic partial differential equations (SPDEs) are crucial for modeling complex fluid dynamics.
  • Stochastic variational principles (SVPs) offer a rigorous framework for deriving such equations.
  • Understanding the mathematical properties of different stochastic calculus representations (Stratonovich vs. Itô) is essential.

Purpose of the Study:

  • To derive SPDEs for fluid dynamics from an SVP.
  • To analyze and compare the properties of stochastic fluid models, specifically their Stratonovich and Itô representations.
  • To investigate the behavior of stochastic geophysical fluid dynamics models.

Main Methods:

  • Derivation of stochastic Stratonovich fluid equations via variational calculus on an SVP.
  • Conversion of Stratonovich equations to their Itô representation.
  • Comparative analysis of stochastic models against each other and deterministic counterparts.
  • Derivation of motion equations for specific geophysical fluid dynamics approximations.

Main Results:

  • Stochastic Stratonovich fluid equations closely mimic deterministic ideal fluid models in preserving circulation.
  • Helicity of vortex field lines is preserved along stochastic Stratonovich paths in incompressible flows.
  • These properties are obscured in the Itô representation due to a quadratic covariation drift term.

Conclusions:

  • The Stratonovich formulation of stochastic fluid dynamics accurately captures key physical properties like circulation and helicity.
  • The Itô transformation can mask these physical characteristics due to drift terms.
  • The derived framework is applicable to geophysical fluid dynamics, including Euler-Boussinesq and quasi-geostrophic models.