Numerical instabilities of vector-invariant momentum equations on rectangular C-grids

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• 作者：Michael J. Bell ; Pedro S. Peixoto and John Thuburn
• 刊名：Quarterly Journal of the Royal Meteorological Society
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：143
• 期：702
• 页码：563-581
• 全文大小：1239K
• ISSN：1477-870X

文摘

The linear stability of two well-known energy- and enstrophy-conserving schemes for the vector-invariant hydrostatic primitive equations is examined. The problem is analyzed for a stably stratified Boussinesq fluid on an f-plane with a constant velocity field, in height and isopycnal coordinates, by separation of variables into vertical normal modes and a linearized form of the shallow-water equations (SWEs). As found by Hollingsworth et al. the schemes are linearly unstable in height coordinate models, due to non-cancellation of terms in the momentum equations. The schemes with the modified formulations of kinetic energy proposed by Hollingsworth et al. are shown to have Hermitian stability matrices and hence to be stable to all perturbations. All perturbations in isopycnal models are also shown to be neutrally stable, even with the original formulations for kinetic energy. Analytical expressions are derived for the smallest equivalent depths obtained using Charney–Phillips and Lorenz vertical grids, which show that the Lorenz grid has larger growth rates for unstable schemes than the Charney–Phillips grid. Test cases are proposed for assessing the stability of new numerical schemes using the SWEs.