THE IMPACT OF ROUGH FORCING ON SYSTEMS WITH MULTIPLE TIME SCALES

In a series of numerical experiments, Williamson and Temperton demonst rated that the interaction of the high-frequency gravity waves with th e low-frequency Rossby waves in a three-dimensional adiabatic model is very weak. However, they stated that this ''might not be the case whe n the model includes realistic physical processes, such as release of latent heat, which are strongly influenced by the vertical motion.'' T he bounded derivative theory is valid for inhomogeneous hyperbolic sys tems with multiple time scales, but the magnitude of any forcing term must be less than or equal to that of the horizontal advection terms i n the same equation. When diabatic effects are added to the basic dyna mical equations for the atmosphere, in the smaller scales of motion fo rcing terms can appear in both the entropy and pressure equations that do not satisfy this restriction. Assuming that the heating terms are only functions of the independent variables, the forcing term in the e ntropy equation can be eliminated so that only a large forcing term in the pressure equation remains. It is proved that a large forcing term in the pressure equation does not by itself preclude a smooth (in the bounded derivative sense) solution. However, the proof shows that the smoothness of the derivatives of the forcing determines the smoothnes s of the solution. If the spatial variation of the forcing in the pres sure equation is much larger than that of the advective component of t he solution of the homogeneous system, then no mathematical estimates of smoothness can be obtained and examples show a smooth solution does not exist. On the other hand, if the spatial derivatives of the forci ng are smooth, but the temporal derivatives are not, a smooth solution exists and the effect of the large variation of the forcing in time o n that smooth solution is small. When both spatial and temporal deriva tives of the forcing are smooth, a smooth solution also exists, and it is proved that it is extremely accurately described by the correspond ing reduced system; that is, the effect of the interaction of any grav ity waves generated by the prescribed forcing with the smooth solution is minimal. The implications of these results for atmospheric predict ion models are discussed.