From instabilities to turbulence in rotating stratified flows with horizontal and vertical shear

Two idealized rotating and stratified flows are explored using linear stability analysis and three-dimensional direct numerical simulations. The first problem explores barotropic mean flow containing horizontally-oriented shear flow in the form of a mixing layer. The second problem explores a uniform baroclinic mean flow in the form of a homogeneous density front in thermal wind balance with uniform vertical shear. Both flows are explored using Richardson and Rossby numbers appropriate for submesoscale shear flows, with lateral length scales of roughly 1 to 20 kilometers. The horizontal shear flow results in simultaneous inertial and barotropic instabilities provided the mixing layer vorticity is opposite in sign and substantially greater in magnitude than the Coriolis parameter. When the vorticity of the mixing layer is opposite in sign and similar in magnitude to the Coriolis parameter the barotropic instability is fundamentally altered. The vertical wavenumber associated with this new instability increases by an order of magnitude destabilizing barotropic vortices leading to increased turbulence intensity well outside of the inertially unstable regime.;Symmetric instability (fluctuations have no along-front variation and are aligned with isopycnals) has been identified in the literature as a potential route to turbulence at fronts as an alternative to wind-driven boundary layer mixing. Linear analysis and simulations of a uniform baroclinic flow in initial geostrophic balance performed here suggest that the instability responsible for initiating transition to turbulence should be near­symmetric and not exactly symmetric as predicted for asymptotically large time scales. Owing to near-symmetry, the instability fundamentally differs from the purely symmetric instability due to currents crossing surfaces of constant density and tapping the reservoir of potential energy available in the front. The presence of strong vertical shear only intensifies this effect as time increases. A highly-resolved turbulent simulation demonstrates a pathway to turbulence from quiescent flow via near-symmetric currents which succumb to shear-convective instabilities which in turn act to destabilize vorticity fluctuations aligned with the mean vorticity in the base flow. Once these fluctuations are sufficiently strong enough, the flow three-dimensionalizes and rapidly breaks down into turbulence throughout the domain.