Nonlinear evolution of the horizontal shear instability in stratified rotating fluids under the complete Coriolis acceleration

This paper investigates the nonlinear dynamics of horizontal shear instability in an incompressible, stratified and rotating fluid in the non-traditional $f$ -plane, i.e. with the full Coriolis acceleration, using direct numerical simulations. The study is restricted to two-dimensional horizontal perturbations. It is therefore independent of the vertical (traditional) Coriolis parameter. However, the flow has three velocity components due to the horizontal (non-traditional) Coriolis parameter. Three different scenarios of nonlinear evolution of the shear instability are identified, depending on the non-dimensional Brunt–Väisälä frequency $N$ and the non-dimensional non-traditional Coriolis parameter $\tilde {f}$ (non-dimensionalised by the maximum shear), in the range $\tilde {f}\lt N$ for fixed Reynolds and Schmidt numbers $ \textit{Re}=2000$ , $ \textit{Sc}=1$ . When the stratification is strong $N\gg 1$ , the shear instability generates stable Kelvin–Helmholtz billows like in the traditional limit $\tilde {f}=0$ . Furthermore, when $N\gg 1$ , the governing equations for any $\tilde {f}$ can be transformed into those for $\tilde {f}=0$ . This enables us to directly predict the characteristics of the flow depending on $\tilde {f}$ and $N$ . When $N$ is around unity and $\tilde {f}$ is above a threshold, the primary Kelvin–Helmholtz vortex is destabilised by secondary instabilities but it remains coherent. For weaker stratification, $N\leqslant 0.5$ and $\tilde {f}$ large enough, secondary instabilities develop vigorously and destroy the primary vortex into small-scales turbulence. Concomitantly, the enstrophy rises to high values by stretching/tilting as in fully three-dimensional flows. A local analysis of the flow prior to the onset of secondary instabilities reveals that the Fjørtoft necessary condition for instability is satisfied, suggesting that they correspond to shear instabilities.