Toward a PV-based algorithm for the dynamical core of hydrostatic global models

The diabatic contour-advective semi-Lagrangian (DCASL) algorithms previously constructed for the shallow-water and multilayer Boussinesq primitive equations are extended to multilayer non-Boussinesq equations on the sphere using a hybrid terrain-following-isentropic (sigma-) vertical coordinate. It is shown that the DCASL algorithms face challenges beyond more conventional algorithms in that various types of damping, filtering, and regularization are required for computational stability, and the nonlinearity of the hydrostatic equation in the sigma- coordinate causes convergence problems with setting up a semi-implicit time-stepping scheme to reduce computational cost. The prognostic variables are an approximation to the Rossby-Ertel potential vorticity Q, a scaled pressure thickness, the horizontal divergence, and the surface potential temperature. Results from the DCASL algorithm in two formulations of the sigma- coordinate, differing only in the rate at which the vertical coordinate tends to with increasing height, are assessed using the baroclinic instability test case introduced by Jablonowski and Williamson in 2006. The assessment is based on comparisons with available reference solutions as well as results from two other algorithms derived from the DCASL algorithm: one with a semi-Lagrangian solution for Q and another with an Eulerian grid-based solution procedure with relative vorticity replacing Q as the prognostic variable. It is shown that at intermediate resolutions, results comparable to the reference solutions can be obtained.