Fourth-Order Horizontal Advection Schemes on the Semi-staggered Grid

Abstract

Horizontal advection schemes on the semi-staggered grid E are presented, which in their linearized versions have a fourth-order space accuracy. The first is a scheme for evaluation of the horizontal nonlinear terms in the momentum equation. It prevents false energy cascade in the manner of the Janji? scheme, i.e., by conserving rational C grid energy and enstrophy. However, it is derived by transformation of the fourth-order rather than of the second-order Arakawa Jacobian. In an extension of the case which includes the divergent part of the flow, care is taken to achieve the conservation of the physically important integral properties of the general flow. The other is a scheme for advection of a passive scalar variable. It is derived by a modification of the former scheme. The momentum equation scheme is tested in a long-term integration of the shallow water equations. As a consequence of the imposed conservation constraints, it is able to simulate general characteristics of the flow up to roughly 100 days of integration. The scalar advection scheme is tested in a traditional experiment with a rotating cone-shaped perturbation of the variable. The experiment demonstrates its advantages over a similar scheme, which is a modification of the original Janji? scheme.