Stability of Leapfrog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations: Flat-Terrain Case

The stability of semi-implicit schemes for the hydrostatic primitive equations system has been studied extensively over the past 20 yr, since this temporal scheme and this system represented a standard for NWP. However, with the increase of computational power, the relaxation of the hydrostatic approximation through the use of nonhydrostatic fully elastic systems is now emerging for future NWP as an attractive solution valid at any scale. In this context, several models employing the so-called Euler equations together with a constant-coefficients semi-implicit time discretization have already been developed, but no solid justification for the suitability of this algorithmic combination has been presented so far, especially from the point of view of robustness. The aim of this paper is to investigate the response of this system/scheme in terms of stability in presence of explicitly treated residual terms, as it inevitably occurs in the reality of NWP. This study is restricted to the impact of thermal and baric residual terms (metric residual terms linked to the orography are not considered here). It is shown that, conversely to what occurs with hydrostatic primitive equations, the choice of the prognostic variables used to solve the system in time is of primary importance for the robustness with Euler equations. For an optimal choice of prognostic variables, unconditionally stable schemes can be obtained (with respect to the length of the time step), but only for a smaller range of reference states than in the case of hydrostatic primitive equations. This study also indicates that (i) vertical coordinates based on geometrical height and on mass behave similarly in terms of stability for the problems examined here, and (ii) hybrid coordinates induce an intrinsic instability, the practical importance of which is, however, not completely elucidated in the theoretical context of this paper.