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Lagrangian averaging of singular stochastic actions for fluid dynamics

Theo Diamantakis, Ruiao Hu

Abstract

We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation parts, and it is separated into the following steps. First, through Magnus expansion, the fluid velocity field is expressed in terms of fluctuation vector fields whose dynamics are assumed to be stochastic. Second, we use Malliavin calculus to give a regularised interpretation of the product of white noise when inserting the stochastic velocity field into the Lagrangian for Euler's fluid. Lastly, we consider closures of the mean velocity by making stochastic analogues of Talyor's frozen-in turbulence hypothesis to derive a version of the anisotropic Lagrangian averaged Euler equation.

Lagrangian averaging of singular stochastic actions for fluid dynamics

Abstract

We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation parts, and it is separated into the following steps. First, through Magnus expansion, the fluid velocity field is expressed in terms of fluctuation vector fields whose dynamics are assumed to be stochastic. Second, we use Malliavin calculus to give a regularised interpretation of the product of white noise when inserting the stochastic velocity field into the Lagrangian for Euler's fluid. Lastly, we consider closures of the mean velocity by making stochastic analogues of Talyor's frozen-in turbulence hypothesis to derive a version of the anisotropic Lagrangian averaged Euler equation.

Paper Structure

This paper contains 6 sections, 4 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Expansion of Kinetic Energy Lagrangian
  3. Closure via variational principle
  4. Conclusion
  5. Acknowledgments.
  6. Disclosure of Interests.

Key Result

lemma thmcounterlemma

In terms of the deformation vector fields $w_t$ and $\chi_t$, the fluctuation stochastic vector fields $\mathrm{d} v^0$, $\mathrm{d} v^\prime$ and $\mathrm{d} v^{\prime \prime}$ can be expressed as where $\nabla$ is a fixed torsion-free connection. For convenience we will use the Levi-Civita connection induced by the metric $\boldsymbol{g}$.

Theorems & Definitions (9)

  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark: Connection dependence and matrix Lie groups
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark: Expectation of advected quantities
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary