On the Use of Implicit and Iterative Methods for the Time Integration of the Wave Equation

The very short-period oscillations inevit,ably undergone by any meteorological quantity predicted by a system of primit’ive equations are principally noise, if an atmospheric model is designed so as to forecast a large-scale and slowly moving meteorological wave. The noise appears as highfrequency gravitational waves. In this paper the words “gravitational wave” will be used in this sense. It is necessary to suppress the noise. Otherwise, an important meteorological wave can be masked by it. The problem of initia.lization of data has been st,udied to find a way to reduce the amplitude of noise (e.g., Hinkelmann [4], Phillips [7]). This can be att,ained by an appropriat,e adjustment between the fields of wind and pressure. However, the control of noise which may arise after t.he initial time has not yet been achieved. This problem is presumably serious when a model of the moist atmosphere is dealt with or when the influence of orography is taken into consideration. Namely, if a rapidly changeable process such as the release of latent heat due to condensation of water vapor is included without care in a prognostic equation, the maintained adjust,ment between the two fields will be destroyed and noise will be excited. Similarly, the motion which is forced by mountains is a source of noise too. In addition, noise will be amplified if the proceduye of numerical integration of the primitive equations does not satisfy the condition for computational stability. In the case cf tlhe “leapfrog” method, which is widely used and is also called the cent,ered-difference method, this condition places an upper limit on the t,ime interval of the marching process. The time interval thus specified is very small as compared with the characteristic time of the meteorological wave.