Variational balance models for the three-dimensional Euler–Boussinesq equations with full Coriolis force

We derive a semi-geostrophic variational balance model for the three-dimensional Euler–Boussinesq equations on nontraditional f-plane under rigid lid approximation. The is obtained by small Rossby number expansion in Hamilton principle, with no other approximations made. allow fully non-hydrostatic flow and do not neglect horizontal components of Coriolis parameter; that is, we make so-called “traditional approximation.” resulting models have same structure as “L1 model” primitive equations: kinematic relation, prognostic equation tracer field, an additional scalar field over two-dimensional domain, which linked to undetermined constant integration thermal wind relation. relation elliptic assumption stable stratification sufficiently fluctuations all fields.