  The purpose of this paper is to derive the anisotropic averaged Euler
equations and to study their geometric and analytic properties. These new
equations involve the evolution of a mean velocity field and an advected
symmetric tensor that captures the fluctuation effects. Besides the derivation
of these equations, the new results in the paper are smoothness properties of
the equations in material representation, which gives well-posedness of the
equations, and the derivation of a corrector to the macroscopic velocity field.
The numerical implementation and physical implications of this set of equations
will be explored in other publications.
