Euler-Poincaré Models of Ideal Fluids with Nonlinear Dispersion

We propose a new class of models for the mean motion of ideal incompressible fluids in three dimensions, including stratification and rotation. In these models, the amplitude of the rapid fluctuations introduces a length scale, $\alpha$, below which wave activity is filtered by both linear and nonlinear dispersion. This filtering enhances the stability and regularity of the new fluid models without compromising either their large scale behavior, or their conservation laws. These models also describe geodesic motion on the volume-preserving diffeomorphism group for a metric containing the $H^1$ norm of the fluid velocity.