  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.
