Vertical discretization of hydrostatic models

Abstract

Although the spectral method can be used on the spherical surface of general circulation models, it is difficult to extend to the vertical direction since various boundary conditions are applied at the top and bottom of the atmosphere. Thus, the grid method is normally used for vertical discretization of general circulation models. We have seen that the spectral method of shallow-water equations on a sphere naturally assures conservation of various physical quantities. It will be shown that conservation in the primitive equation model is achieved by proper vertical discretization. Here we use the flux-form finite volume method for vertical discretization. We require the domain integral conservation of energy and that of potential temperature. We will also find a discretization that minimizes an error in pressure gradient force when the atmosphere is everywhere isentropic.