Geometric Lagrangian averaged Euler-Boussinesq and primitive equations

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler-Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler-Poincar\'{e} equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler-Boussinesq-$\alpha$ and primitive equations-$\alpha$ models, however feature a different regularizing second order operator.