Physica D 170 (2002) 253–286 Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics Darryl D. Holm Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, MS B284, Los Alamos, NM 87545, USA Received 20 February 2001; received in revised form 14 May 2002; accepted 31 May 2002 Communicated by I. Gabitov Dedicated to Rupert Ford (1968–2001) 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 Averag- ing 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 (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 fluctuations, 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 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 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. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Averaged Lagrangians; Ideal fluid dynamics; Taylor hypothesis; Turbulence closures 1. Brief review of generalized Lagrangian mean (GLM) theory for compressible fluids An exceptional accomplishment in formulating averaged motion equations for fluid dynamics is the GLM theory of nonlinear waves on a Lagrangian-mean flow, as explained in two consecutive papers of Andrews and McIntyre [2,3]. This section introduces the results that we shall need later from the rather complete description given in these papers. Even now, these fundamental papers still make worthwhile reading and are taught in many atmospheric science departments. Section 2 begins by placing the GLM equations for a rotating adiabatic compressible fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics in the Eulerian description. This is first done E-mail address: dholm@lanl.gov (D.D. Holm). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0167-2789(02)00552-3