Vertical differencing of the primitive equations based on the Charney-Phillips grid in hybrid σ-p vertical coordinates

Abstract

Two types of vertical grids are used for atmospheric models: the Lorenz grid (L grid) and the Charney-Phillips grid (CP grid). Although the CP grid is the standard grid for quasigeostrophic models, it is not widely used in the primitive equation models because it is easier with the L grid to maintain some of the integral properties of the continuous system. In this paper, problems with the L grid are pointed out that are due to the existence of an extra degree of freedom in the vertical distribution of the temperature (and the potential temperature). Then a vertical differencing of the primitive equations based on the CP grid is presented, while most of the advantages of the L grid in a hybrid σ-p vertical coordinate are maintained. The discrete hydrostatic equation is constructed in such a way that it is free from the vertical computational mode in the thermal field. Also, the vertical advection of the potential temperature in the discrete thermodynamic equation is constructed in such a way that it reduces to the standard (and most straightforward) vertical differencing of the quasigeostrophic equations based on the CP grid. Simulations of standing oscillations superposed on a resting atmosphere are presented using two vertically discrete models, one based on the L grid and the other on the CP grid The comparison of the simulations shows that with the L grid a stationary vertically zigzag pattern dominates in the thermal field, while with the CP grid no such pattern is evident. Simulations of the growth of an extratropical cyclone in a cyclic channel on a β plane are also presented using two different σ-coordinate models, again one with the L grid and the other with the CP grid, starting from random disturbances. The L grid simulation is dominated by short waves, while there is no evidence of short-wave growth in the CP grid simulation.