The authors search the stationary solutions of the barotropic vorticity equation in spherical coordinates by numerically solving the equations with the Newton–Keller pseudoarclength continuation method. The solutions consist of planetary-scale Rossby waves superimposed on zonal wind profiles and forced by sinusoidal orography in near-resonance conditions. By varying the zonal wind strength across resonance, it is shown that multiple solutions with different wave amplitudes can be found: for small forcing and dissipation, the solution curve is the well-known bended resonance. The comparison between numerical results and theoretical predictions by a previously developed weakly nonlinear theory is successfully attempted.
The authors then extend the barotropic, weakly nonlinear theory to stationary Rossby waves forced by large-scale orography and dissipated by Ekman friction at the surface, in the framework of the quasigeostrophic model continuous in the vertical direction. The waves are superimposed on vertical profiles of zonal wind and stratification parameters taken from observations of the wintertime Northern Hemisphere circulation. In near-resonant conditions, the weakly nonlinear theory predicts multiple amplitude equilibration of the eddy field for a fixed vertical profile of the zonal wind. The authors discuss the energetics of the stationary waves and show that the form drag and Ekman dissipation can be made very small even if realistic values of the parameters are taken, at variance with the barotropic case.
This model is proposed as the theoretical base for such phenomena as atmospheric blocking, bimodality, and weather regimes.