  We begin by placing the Generalized Lagrangian Mean (GLM) equations for a
compressible adiabatic fluid into the Euler-Poincar\'e (EP) variational
framework of fluid dynamics, for an averaged Lagrangian. We then derive a set
of approximate small amplitude GLM equations (glm) that 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 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).
The glm equations lead to generalizations of the Euler-alpha models to include
compressibility and magnetic fields.
