Variational Principles for Lagrangian Averaged Fluid Dynamics

Abstract

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincaré (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincaré (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.

23 pages, 3 figures

References in corpus (5)

• The Navier-Stokes-alpha model of fluid turbulence
• Averaged Lagrangians and the mean dynamical effects of fluctuations in continuum mechanics
• The Three Dimensional Viscous Camassa-Holm Equations, and Their Relation to the Navier-Stokes Equations and Turbulence Theory
• Transient vortex events in the initial value problem for turbulence
• Navier-Stokes-alpha model: LES equations with nonlinear dispersion

Cited by in corpus (9)

• Analytical Study of Certain Magnetohydrodynamic-alpha Models
• Averaged Lagrangians and the mean dynamical effects of fluctuations in continuum mechanics
• On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Leray-alpha simulations of wall-bounded turbulent flows
• Energy-conserving Finite-beta Electromagnetic Drift-fluid Equations
• Variational nonlinear WKB in the Eulerian frame
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• A geometric look at MHD and the Braginsky dynamo