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Numerical instabilities of vector‐invariant momentum equations on rectangular C‐grids
Author(s) -
Bell Michael J.,
Peixoto Pedro S.,
Thuburn John
Publication year - 2016
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.2950
Subject(s) - enstrophy , mathematics , shallow water equations , isopycnal , mathematical analysis , vector field , kinetic energy , hydrostatic equilibrium , primitive equations , momentum (technical analysis) , scalar (mathematics) , invariant (physics) , classical mechanics , physics , mathematical physics , vorticity , geometry , mechanics , partial differential equation , geology , vortex , quantum mechanics , economics , finance , climatology , numerical partial differential equations
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.