Computational Dispersion Properties of Vertically Staggered Grids for Atmospheric Models

The computational dispersion properties of vertically and time-vertically staggered grids, using corresponding centered-difference schemes for approximation of a linear baroclinic primitive equation system, are analyzed in terms of frequency and group velocity characteristics. The vertical scale ranges with group velocities of the wrong sign are pointed out. It is shown that among all possible vertical grids applicable to primitive equation atmospheric models the best vertical grids have computational dispersion properties corresponding to a regular (equidistant, unstaggered) grid with twice the vertical resolution. These best vertical grids are 1) two well-known vertically staggered grids, namely, the widely used Lorenz grid and the Charney-Phillips grid; 2) two other vertically staggered grids carrying both horizontal and vertical velocity components at the same levels; and 3) the new time-staggered versions of all the aforementioned grids, and the time-staggered regular vertical grid, if used with either the appropriate version of an economical explicit scheme or a semi-implicit scheme for approximations with these time-staggered grids. All these best vertical grids are computationally efficient due to their enhanced effective vertical resolution. Moreover, the time-vertically staggered grids considered here provide twice the effective vertical resolution of comparable vertically staggered grids for finite-difference approximations of the vertical derivatives in vertical advection and vertical diffusion terms. In other words, these time-vertically staggered grids provide uniformly twice the elective vertical resolution for the whole baroclinic model system. The application of higher- (fourth) order vertical-difference approximation results in some moderate improvement of vertical grid dispersion properties, primarily for the small vertical and large horizontal scale range, but it is definitely less significant than the effect of doubling the effective vertical resolution by staggering. Computational dispersion properties of vertical grids, along with other computational characteristics and requirements, may provide guidance for an optimal choice of an appropriate vertical grid for a primitive equation atmospheric model.