Variational principles for Lagrangian-averaged fluid dynamics

Abstract The Lagrangian average (LA) of the ideal fluid equations preserves their fundamental transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its potential vorticity convection and helicity conservation. We show that Lagrangian averaging also preserves the Euler–Poincaré variational framework that implies the exact ideal fluid equations in the Eulerian representation. This is expressed in the Lagrangian-averaged Euler–Poincaré (LAEP) theorem proved here. We illustrate the LAEP theorem by applying it to incompressible ideal fluids to derive the Lagrangian-averaged Euler equations and thereby recover the generalized Lagrangian mean motion equation. Finally, we discuss recent progress in applications of these equations as the basis for new LA closure models of fluid turbulence.