A study on the vertical propagation of planetary waves and the effects of the upper boundary condition

Linear mid-latitudes models on a beta-plane were formulated to study the simulation of the vertical propagation of forced planetary waves in numerical models. Computations were made applying the rigid top boundary condition w=0 at various finite heights with different vertical resolutions and compared with exact solutions. The influence of the zonal wind distribution and of Newtonian cooling was examined using vertical profiles of winds and temperature representatives of winter, fall, and summer. The main conclusions are: 1) With constant values of the zonal wind and temperature, and without Newtonian cooling, no model is capable of representing the structure of a vertically propagating wave. This is due to the false reflection of planetary waves at the top. External modes, with no energy propagation, are well represented by the models. 2) With realistic distributions of temperature, zonal winds and Newtonian cooling, it is always possible to obtain a good simulation of planetary waves, provided there is enough vertical resolution. With good vertical resolution, the effect of the artificial top boundary condition is minimized during winter and summer because the mean zonal winds produce a natural reflective stratospheric layer. During fall, Newtonian cooling absorbs most of the upward energy before it reaches the top. 3) The absolute minimum number of levels necessary to obtain a relatively good wave structure simulation is ten. 4) The most important factor to obtain a good tropospheric representation is good resolution near the ground. For a fixed number of levels, the best results are obtained with high resolution at the extremes of the atmosphere. 5) For a fixed number of levels, the solutions are very sensitive to the position of the top. When the winds produce a reflective layer in the stratosphere, a relatively low top can improve the simulation. When Newtonian cooling is an important factor, a high top is better because more damping is allowed to occur. The optimum position of the top is therefore seasonally dependent. Two "sponge layer" mechanisms to reduce the reflection of wave energy at the upper boundary were also tested. In one of them, a complex index of refraction modified by a coefficient of viscosity, was introduced in a layer below the upper boundary, in such a way that the waves could be damped before they reached the top. A complex index of refraction that increases smoothly with height was found to produce better results that a constant one. It was found that there is an optimum value of the viscosity coefficient, which is a function of the frequency of the waves and the depth of the sponge layer. Lower values produce insufficient damping, and higher values produce internal reflections due to the strong vertical gradient of viscosity. The second approach uses a single layer at the top to eliminate the reflected wave by producing destructive interference between the wave reflected at the top and the wave reflected at the bottom of the sponge layer. A generalization of this method when several frequencies are present is proposed. Thesis Supervisor: Eugenia Kalnay-Rivas Title: Associate Professor