Stability of a two-time-level semi-implicit integration scheme for gravity wave motion

A study is made of the computational stability of semi-implicit treatments of gravity wave motion suitable for use with two-time-level advection schemes. The analysis is for horizontally uniform reference values of temperature and surface pressure, and for hybrid pressure-based vertical coordinates. Stability requires the use of reference temperatures that are warmer than those that can be used safely with the corresponding three-time-level scheme. The reference surface pressure should also be higher. When stable, the two-time-level scheme is damping, although the largest scales are damped less than by the three-time-level scheme if the latter uses a typical rime filtering. The first-order decentered averaging of gravity wave tendencies used in a number of semi-Lagrangian models reduces the need for a relatively warm reference temperature profile but causes a quite substantial damping of otherwise well-represented low-wavenumber modes. The low-wavenumber damping can be avoided by using an alternative, second-order averaging involving a third (past) time level. For this alternative averaging, an economical spatial discretization is proposed that requires no additional departure point. Phase speeds show little sensitivity to these changes in formulation. All variants of the semi-implicit method substantially reduce the phase speeds of the fastest high-wavenumber modes when use is made of the large time steps possible with semi-lagrangian advection.