A fourth-order ﬁnite difference method is proposed and studied for the primitive equations (PEs) of large-scale atmospheric and oceanic ﬂow based on mean vorticity formulation. Since the vertical average of the horizontal velocity ﬁeld is divergence-free,we can introduce mean vorticity and mean stream function which are connectedbya2-DPoissonequation. Asaresult,thePEscanbereformulatedsuchthat the 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 to the difﬁculty ofthe meanvorticity boundarycondition, while fourth-orderlong-stencil approximations are utilized to deal with transport type equations for computational convenience. The numerical values for the total velocity ﬁeld (both horizontal and vertical) are statically determined by a discrete realization of a differential equation at each ﬁxed horizontal point. The method is highly efﬁcient and is capable of producing highly resolved solutions at a reasonable computational cost. The full fourth-order accuracy is checked by an example of the reformulated PEs with force terms. Additionally, numerical results of a large-scale oceanic circulation are presented.

}, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7780.html} }