Baroclinic instability of flows along sloping boundaries

In the two-layer quasi-geostrophic model with boundaries sloping perpendicular to the basic flow, the ratios of the slopes of the bottom and the top to that of the interface between the fluid layers in the basic state are important parameters in the expression of the growth rate of unstable waves. When Eady's (1949) model is extended to include sloping bottom and top boundaries, the growth rates of unstable waves depend on the ratios of the slopes of the bottom and the top to that of the isentropes of the basic state. For the Eady model with sloping bottom, an important parameter characterizing the instability is the ratio between the vertical and horizontal heat transports by the wave divided by the slope of the isentropes of the basic state. An interpretation of these ratios and their relations clarifies the stabilization of the system for large slopes, the variation of the wavelength of the most unstable wave with the bottom slope, and the destabilization of some short waves for negative bottom slopes. It is found that the most unstable wave of the system has zero vertical energy flux convergence at the sloping bottom.