Nonlinear Advection Schemes on the Octagonal Grid

Abstract

Abstract Successful treatment of nonlinear momentum advection is one of the outstanding challenges for the application of rectangular quasi-uniform spherical grids in global circulation models. Quasi-uniform grids (e.g., cubic and octagonal), which are virtually assembled by connecting a set of regional domains along their boundaries, appear to be an excellent choice for the expansion of regional atmospheric models to global coverage. However, because of an unavoidable lack of orthogonality of these grids in the proximity of the singular points (i.e., the corner points connecting three neighboring rectangular tiles), a common-sense approach is to first generalize underlying numerical schemes to the general curvilinear coordinates, and then to apply globalization. In this procedure, assuming that a “weak conservative formulation” for the generalization is applied, the advective formalism of the Arakawa-type momentum schemes and some of their properties, especially those important for the long-term “climate type” simulations, may be lost. This paper discusses challenges faced in the application of Arakawa-type nonlinear advection schemes on the quasi-uniform semistaggered grids and suggests a solution that is based on discretization of the momentum equation in the vector form. Both the second- and the fourth-order energy-conserving nonlinear advection schemes are considered. The potential merits of this approach are demonstrated in a series of benchmark test integrations of a shallow-water model on the octagonal quasi-uniform grid.