Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

Abstract

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 Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations ( gℓ m 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 gℓ m equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓ m derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓ m 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 gℓ m) 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 closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new gℓm turbulence closure equations for compressible fluids in the EP variational framework.