STABILITY OF A 2-TIME-LEVEL SEMIIMPLICIT INTEGRATION SCHEME FOR GRAVITY-WAVE MOTION

A study is made of the computational stability of semi-implicit treatm ents of gravity wave motion suitable for use with two-time-level advec tion schemes. The analysis is for horizontally uniform reference value s of temperature and surface pressure, and for hybrid pressure-based v ertical coordinates. Stability requires the use of reference temperatu res that are warmer than those that can be used safely with the corres ponding three-time-level scheme. The reference surface pressure should also be higher. When stable, the two-time-level scheme is damping, al though the largest scales are damped less than by the three-time-level scheme if the latter uses a typical rime filtering. The first-order d ecentered averaging of gravity wave tendencies used in a number of sem i-Lagrangian models reduces the need for a relatively warm reference t emperature 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 economic al spatial discretization is proposed that requires no additional depa rture point. Phase speeds show little sensitivity to these changes in formulation. All variants of the semi-implicit method substantially re duce the phase speeds of the fastest high-wavenumber modes when use is made of the large time steps possible with semi-lagrangian advection.