Variational discretization framework for geophysical flow models

We introduce a geometric variational discretization framework for geophysical flow models. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D and produces schemes that preserve conservation laws such as mass and total energy. We illustrate the wide applicability of this discretization method that is based on a discrete version of the Euler-Poincaré variational method, in particular we present an overview of structure-preserving variational discretizations of various equations of geophysical fluid dynamics, such as the Boussinesq, anelastic, pseudo-incompressible, and shallow-water equations. We verify the structure-preserving nature of the resulting variational integrators for test cases of geophysical relevance. As descending from variational principles, the discussed variational schemes exhibit a discrete version of Kelvin circulation theorem and show excellent long term energy behavior.