A fourth order numerical method for the primitive equations formulated in mean vorticity

Jian-Guo Liu Duke University Cheng Wang the University of Tennessee

Numerical Analysis and Scientific Computing mathscidoc:1702.25035

COMMUNICATIONS IN COMPUTATIONAL PHYSICS, 4, (1), 26-55, 2008.7
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 connected by a 2-D Poisson equation. As a result, the PEs can be reformulated such that the prognostic equation for the horizontal velocity is replaced by evolutionary equa-tions 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 of the mean vorticity boundary condition, while fourth-order long-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 produc-ing 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. Addi-tionally, numerical results of a large-scale oceanic circulation are presented.
The primitive equations, mean vorticity, compact scheme, long-stencil approximation, one-sided extrapolation, large scale oceanic circulation
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@inproceedings{jian-guo2008a,
  title={A fourth order numerical method for the primitive equations formulated in mean vorticity},
  author={Jian-Guo Liu, and Cheng Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209004300981142349},
  booktitle={COMMUNICATIONS IN COMPUTATIONAL PHYSICS},
  volume={4},
  number={1},
  pages={26-55},
  year={2008},
}
Jian-Guo Liu, and Cheng Wang. A fourth order numerical method for the primitive equations formulated in mean vorticity. 2008. Vol. 4. In COMMUNICATIONS IN COMPUTATIONAL PHYSICS. pp.26-55. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209004300981142349.
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