  Generalized Lagrangian mean theories are used to analyze the interactions
between mean flows and fluctuations, where the decomposition is based on a
Lagrangian description of the flow. A systematic geometric framework was
recently developed by Gilbert and Vanneste (J. Fluid Mech., 2018) who cast the
decomposition in terms of intrinsic operations on the group of volume
preserving diffeomorphism or on the full diffeomorphism group. In this setting,
the mean of an ensemble of maps can be defined as the Riemannian center of mass
on either of these groups. We apply this decomposition in the context of
Lagrangian averaging where equations of motion for the mean flow arise via a
variational principle from a mean Lagrangian, obtained from the kinetic energy
Lagrangian of ideal fluid flow via a small amplitude expansion for the
fluctuations.
  We show that the Euler-$\alpha$ equations arise as Lagrangian averaged Euler
equations when using the $L^2$-geodesic mean on the volume preserving
diffeomorphism group of a manifold without boundaries, imposing a `Taylor
hypothesis', which states that first order fluctuations are transported as a
vector field by the mean flow, and assuming that fluctuations are statistically
isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged
Burgers' equations using the same argument on the full diffeomorphism group.
These results generalize an earlier observation by Oliver (Proc. R. Soc. A,
2017) to manifolds in geometrically fully intrinsic terms.
