Evolution of a stratified rotating shear layer with horizontal shear. Part I. Linear stability

Linear stability analysis is used to investigate instability mechanisms for a horizontally oriented hyperbolic tangent mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important parameters governing inviscid dynamics are maximum shear $S$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L$. Growth rates associated with the most unstable modes are explored as a function of stratification strength $N/ S$ and rotation strength $2\Omega / S$. In the case of strong stratification, growth rates exhibit self-similarity of the form $\sigma ({k}_{1} L, S{k}_{3} L/ N, 2\Omega / S)$. In the case of rapid rotation we also observe self-similar scaling of growth rates with respect to the vertical wavenumber and rotation rate. The unstratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ S)$ dependence while the strongly stratified cases show $\sigma ({k}_{1} L, 2\vert \tilde {\Omega } \vert {k}_{3} L/ N)$ dependence where $\tilde {\Omega } $ represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, $\Omega = S/ 4$. Stratification was found to stabilize the inertial instability for weak anticyclonic rotation rates. Near the zero absolute vorticity state, stratification and rotation couple in a destabilizing manner increasing the range of unstable vertical wavenumbers associated with barotropic instability. In the case of rapid rotation, stratification prevents the stabilization of low ${k}_{1} $, high ${k}_{3} $ modes that occurs in a homogeneous fluid. The structure of certain unstable eigenmodes and the coupling between horizontal vorticity and density fluctuations are explored to explain how buoyancy stabilizes or destabilizes inertial and barotropic modes.