UBC Theses and Dissertations
Baroclinic critical layers in rotating stratified shear flows Wang, Chen
In this thesis, we study baroclinic critical layers in rotating stratified shear flows. Baroclinic critical layers are characterized by strong amplitudes surrounding the singular points of the steady inviscid wave solution, and play crucial roles in the mixing and transition to turbulence in ocean, atmosphere and astrophysical disks. This thesis studies the baroclinic critical layers in strato-rotational instability and the forced baroclinic critical layers. The first problem we study is the baroclinic critical layer in strato-rotational instability. Strato-rotational instability (SRI) is normally interpreted as the resonant interactions between internal gravity waves or Kelvin waves. Using a combination of asymptotic analysis and numerical solution of the linear eigenvalue problem for plane Couette flow, it is shown that such resonant interactions can be destroyed by baroclinic critical layers. The critical-level coupling removes the requirement for resonance near specific wavenumbers, resulting in an extensive continuous band of unstable modes. The second problem we study is the forced baroclinic critical layers. Linear theory predicts the baroclinic critical layer dynamics is characterized by the secular growth of flow perturbations over a region of decreasing width. Once it enters the nonlinear regime, the nonlinear dynamics filters out harmonics and the modification to the mean flow controls the evolution. At late times, we show that the vorticity begins to focus into yet smaller regions whose width decreases exponentially with time, and that the addition of dissipative effects can arrest this focussing to create a drifting coherent structure. In the last problem, we show that the mean-flow defect generated in the forced baroclinic critical layer can make the flow unstable, and we study this 'secondary instability'. The instability is a horizontal shear instability with a distinct phase velocity compared to that of the forced baroclinic critical layers, and thus will excite new baroclinic critical layers. A WKB solution for the exponential growth is derived, which indicates the secondary instability grows faster than a common normal mode due to the unsteadiness of the mean-flow defect. At the later stage, the short-wave harmonics grow at extraordinarily high speeds and will finally make the linear problem ill-posed.
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