Interactions among stationary planetary waves in a low-order spherical stratospheric model are studied both analytically and numerically. A triad of interacting waves is examined using a steadystate hemispheric quasi-geostrophic model with a mean zonal wind in solid body rotation. Due to the structure of the model basic state, dissipation is required for the waves to interact. It is found that the nonlinear structure of the two gravest forced modes is independent of their relative position only if the third mode is unforced at the lower boundary. In this case the degree to which the nonlinearities act to amplify or phase shift the linear waves is shown to be dependent upon the vertical propagation characteristics of the waves. Large horizontal scale waves and weak zonal westerlies are found to be conditions which can result in significant amplitude changes due to the wave-wave interaction. As the scale of the wave is reduced and/or the speed of the mean zonal westerlies is increased, the predominant changes occur in the wave phases. Numerical solutions using realistic boundary forcing amplitudes and realistic dissipation are found to be only weakly nonlinear, despite the fact that the planetary waves distort considerably the polar vortex.