Invariant measures for the two-dimensional averaged-Euler equations

We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincar\'e recurrence theorem is used to show that the flow returns infinitely many times in a neighbourhood of the initial state.