  This paper extends the derivation of the Lagrangian averaged Euler
(LAE-$\alpha$) equations to the case of barotropic compressible flows. The aim
of Lagrangian averaging is to regularize the compressible Euler equations by
adding dispersion instead of artificial viscosity. Along the way, the
derivation of the isotropic and anisotropic LAE-$\alpha$ equations is
simplified and clarified.
  The derivation in this paper involves averaging over a tube of trajectories
$\eta^\epsilon$ centered around a given Lagrangian flow $\eta$. With this tube
framework, the Lagrangian averaged Euler (LAE-$\alpha$) equations are derived
by following a simple procedure: start with a given action, Taylor expand in
terms of small-scale fluid fluctuations $\xi$, truncate, average, and then
model those terms that are nonlinear functions of $\xi$. Closure of the
equations is provided through the use of \emph{flow rules}, which prescribe the
evolution of the fluctuations along the mean flow.
