A Fourth Order Numerical Method for the Primitive Equations Formulated in Mean Vorticity (Communications in Computational Physics, 2008)
A fourth-order finite difference method is proposed and studied for theprimitive equations (PEs) of large-scale atmospheric and oceanic flow based on meanvorticity formulation. Since the vertical average of the horizontal velocity field isdivergence-free, we can introduce mean vorticity and mean stream function which areconnected by a 2-D Poisson equation. As a result, the PEs can be reformulated such thatthe prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity.The mean vorticity equation is approximated by a compact difference scheme due tothe difficulty of the mean vorticity boundary condition, while fourth-order long-stencilapproximations are utilized to deal with transport type equations for computationalconvenience. The numerical values for the total velocity field (both horizontal andvertical) are statically determined by a discrete realization of a differential equation ateach fixed horizontal point. The method is highly efficient and is capable of producing highly resolved solutions at a reasonable computational cost. The full fourth-orderaccuracy is checked by an example of the reformulated PEs with force terms. Additionally, numerical results of a large-scale oceanic circulation are presented.