Numerical instabilities of vector invariant momentum equations on rectangular C-grids
Bell, MJ; Peixoto, PS; Thuburn, J
Date: 21 October 2016
Article
Journal
Quarterly Journal of the Royal Meteorological Society
Publisher
Wiley
Publisher DOI
Abstract
The linear stability of two well known energy and enstrophy conserving schemes for
the vector invariant hydrostatic primitive equations is examined. The problem is analysed 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 ...
The linear stability of two well known energy and enstrophy conserving schemes for
the vector invariant hydrostatic primitive equations is examined. The problem is analysed 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 linearised
form of the shallow water equations (SWEs). As found by [Hollingsworth
~al.(1983)Hollingsworth, Kallberg, Renner and Burridge], (HKRB hereafter) the schemes are
linearly unstable in height coordinate models, due to a non-cancellation of terms in the
momentum equations. The schemes with the modified formulations of the kinetic energy
proposed by HKRB are shown to have Hermitain 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 the unstable schemes than the CharneyPhillips
grid. Test cases are proposed for assessing the stability of new numerical schemes using
the SWEs.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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