Instead of using complicated general circulation models (GCMs), a simple semi-analytical model based on ray theory has been used to study energy evolution and ray path of Rossby waves in slowly varying mean flows. Our model yields similar results to those calculated from barotropic models, and also provides a chance to study Rossby waves in the slowly varying flows with both vertical and meridional shears. The model results show that upward Rossby waves can only grow in westerlies, and decay when further ascend. The baroclinic Rossky waves are restrained by the β effect in lower latitude. In the westerly jet with meridional and vertical shears, the barotropic Rossby waves originated from south of the westerly jet, and these can grow while propagating upper-northward. The baroclinic Rossby waves originated from north of the westerly jet and can grow while propagating upward and southward. Such a semi-analytical model provides a simple forecasting tool to allow study of the local weather anomalies to the heating/topography forcing associated with the global warming. In fact, it is usually Rossby waves that transport energy in global teleconnection [7,20,21]. In sub-equatorial regions, Rossby waves can propagate in the westerly jet [13,20,22]. Such Rossby wave sources may come from tropical heating coupled with mean vertical shear [23], or higher-latitude topography [24].To understand global teleconnection, it is necessary to study the growth and ray path of Rossby waves under mean flow conditions. This allows prediction of the local weather or climate based on the pre-existent mean flows and wave sources. Earlier theories of Rossby waves identified the theoretical instability criteria [21,[25][26][27][28][29]: (a) consistent with the results of Hoskins and Karoly [7], "great circle" propagation was simulated in a uniform westerly; (b) the "guided" Rossby waves (l/k > 0) develop on the south of the westerly jet, where ∂U/∂y > 0, and the "trailing" Rossby waves (l/k > 0) develop on the north side of the westerly jet, where ∂U/∂y < 0 [21]; (c) in positive (∂U/∂y > 0) sheared mean flow, northward Rossby waves turn back at a turning point, while the southward Rossby waves approach the critical level directly [30,31]; (d) the divergent effect is necessary for the prop-