Averaged Lagrangians and the mean dynamical effects of fluctuations in continuum mechanics

We begin by placing the Generalized Lagrangian Mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Lemma – that GLM averaged equations arise from GLM averaged Lagrangians in the EP framework. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuctions, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The new glm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the EP Averaging Lemma, we bridge between the GLM equations and the Euler-alpha closure equations, upon making the small-amplitude approximation resulting in the new glm equations in the EP variational framework. The glm equations also lead to generalizations of the Euler-alpha models to include compressibility and magnetic fields.