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

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 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.