A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance |
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Authors: | Roger Samelson Roger Temam Cheng Wang Shouhong Wang |
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Institution: | (1) College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331-5503, USA;(2) Institute for Scientific Computing and Applied Mathematics & Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA;(3) Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA |
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Abstract: | The planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form, and a fourth-order
finite difference numerical method of solution is proposed and analyzed in this article. In the reformulation, there is only
one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic
balance combined with the incompressibility condition. The key observation is that all the velocity profiles can be explicitly
determined by the temperature gradient, by utilizing the special form of the Coriolis parameter. This brings convenience and
efficiency in the numerical study. In the fourth-order scheme, the temperature is dynamically updated at the regular numerical
grid by long-stencil approximation, along with a one-sided extrapolation near the boundary. The velocity variables are recovered
by special solvers on the 3-D staggered grid. Furthermore, it is shown that the numerical velocity field is divergence-free
at the discrete level in a suitable sense. Fourth order convergence is proven under mild regularity requirements.
R. Samelson was supported by NSF grant OCE04-24516 and Navy ONR grant N00014-05-1-0891.
R. Temam was supported by NSF grant DMS-0604235 and the research fund of Indiana University.
S. Wang was supported by NSF grant DMS-0605067 and Navy ONR grant N00014-05-1-0218. |
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Keywords: | 35Q35 65M06 65M12 86A10 |
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