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

Kurzfassung/Abstract

We derive a semi-geostrophic variational balance model for the three-dimensional Euler–Boussinesq equations on the nontraditional f-plane under the rigid lid approximation. The model is obtained by a small Rossby number expansion in the Hamilton principle, with no other approximations made. We allow for a fully non-hydrostatic flow and do not neglect the horizontal components of the Coriolis parameter; that is, we do not make the so-called “traditional approximation.” The resulting balance models have the same structure as the “L1 balance model” for the primitive equations: a kinematic balance relation, the prognostic equation for the three-dimensional tracer field, and an additional prognostic equation for a scalar field over the two-dimensional horizontal domain, which is linked to the undetermined constant of integration in the thermal wind relation. The balance relation is elliptic under the assumption of stable stratification and sufficiently small fluctuations in all prognostic fields.
ACKNOWLEDGMENTS
This paper is a contribution to project M2 (Systematic Multi-Scale Analysis and Modeling for Geophysical Flow) of the Collaborative Research Center TRR 181 “Energy Transfers in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 274762653. Additional funding was received through the Ideen-und Risikofund 2020 at Universität Hamburg.