The Lie-Poisson structure of the LAE-$\alpha$ equation
  This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed boundary conditions is smooth, a result already known for Dirichlet boundary conditions. 
Fran\c{c}ois Gay-Balmaz, Tudor S. Ratiu