## Variational Model Reduction for Non-hydrostatic Stratified Flows in the Mid-latitude and the Equator

- This thesis studies balance models for a rotating stratified three-dimensional fluid on a tangent plane with full Coriolis force. Derivations are done for two different regions, namely mid-latitude and equator, which we considered separately. Each model is studied via a variational approach which is based on Lagrangian dynamics assuming smallness of the Rossby number and allowing for anisotropy in the horizontal length scales. We assume semigeostrophic scaling, akin to the derivation of the L 1 model by Salmon (1985) for the rotating shallow water equations. Contrary to Salmon’s derivation, we start with an arbitrary change of coordinates and then choose the transformation to fix the degeneracy on the first order of the Lagrangian, L 1 , as suggested by Oliver (2006). In our setting, the full projection of the rotation vector of the Earth is considered, so that the horizontal component of the Coriolis vector is taken into account. For each model, conservation laws for the energy and the potential vorticity are valid because of the Hamiltonian structure. Our first model on f-plane is the most general model obtained so far in semi-geostrophic scaling. The other model concerns balance model on the equatorial β-plane. Under the additional assumption of construction of zero-meridional velocity as suggested by the leading order dynamics, an equatorial balance model is obtained.