Evolution of baroclinic critical layers

Recent work has suggested that three dimensional rotating stratified shear flows can be the setting of a self-replicating vortex instability (Marcus et al., Phys. Rev. Lett. 2013, 111(8): 084501). Such ``Zombie vortices'' replicate themselves by exciting internal waves that break at a novel type of critical layer. In this study, we examine the evolution of such ``baroclinic critical layers'' in the situation that the waves are directly forced by a steady disturbance. Linear theory predicts that the flow settles down to a steady wave everywhere but near the baroclinic critical levels, where vorticity and density grow linearly with time over a region whose width decreases with time. The critical layer must therefore become nonlinear. By developing a nonlinear baroclinic critical layer theory, we show that the secular linear growth becomes halted. However, at later times, the vorticity begins to grow exponentially over yet smaller regions that are shifted with respect to the original baroclinic critical levels.