A combination of system rotation and stable stratification is essential for geophysical applications, even if the former effect is significantly smaller than the latter in 3D flows, e.g., for scales much smaller than the synoptic ones in the atmosphere. As for pure rotating turbulence in Chapter 4 and purely stratified turbulence in Chapter 7, linear analysis, i.e., RDT, describes only neutral stability and will lead to the definition of both the wave-vortex eigenmode decomposition and dispersion frequencies of inertia gravity waves in the present chapter. Nonlinear dynamics is essential, and allows us to revisit a quasi-geostrophic (QG) cascade, which generalizes the toroidal cascade discussed in Chapter 7 with additional Coriolis effects.

Other coupled effects investigated in this chapter can create linear instabilities that can be analyzed within the RDT framework. These instabilities are associated with turbulence-production mechanisms, which are the main striking new physical phenomena when compared with other flows discussed in this book. Therefore only the linear approach will be emphasized in these cases. In the presence of mean shear, barotropic instabilities occur, with a strong analogy between the rotating-shear-flow case and the stratified shear flow. A special case combining the three ingredients, namely the mean shear, system rotation, and stable stratification, is shown to give new insight into the baroclinic instability. Finally, the very important elliptical-flow instability is investigated, which results from a coupled effect of mean vorticity with weak additional mean strain. This instability can trigger nonlinear cascade and turbulence in physical systems with...