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.