Variational integrators for soundproof models on arbitrary triangular C-grids

We introduce variational integrators for soundproof approximations of the Euler equations on irregular triangular C-grids. In particular we provide explicit vector-matrix formulations and the solving algorithm for the previously derived discrete covariant equations of Boussinesq, anelastic, and pseudo-incompressible models of [4]. The derivation of these variational integrators is based on a discrete version of the Euler-Poincaré variational method, hence it provides structure-preserving discretizations that preserve Kelvin's circulation theorem and provide excellent long term energy behavior for arbitrary triangular C-grids. We verify the structure-preserving nature of these schemes for various scenarios of geophysical relevance: the hydrostatic adjustment process, the Kelvin-Helmholtz instability, and the cold and warm bubble test cases