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 Ω 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 2Ω/S. In the case of strong stratification, growth rates exhibit selfsimilarity of the form σ(k1L, Sk3L/N, 2Ω/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 σ(k1L, 2|Ω̃|k3L/S) dependence while the strongly stratified cases show σ(k1L, 2|Ω̃|k3L/N) dependence where Ω̃ represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, Ω = 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 k1, high k3 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.