Shear instability in a stratified fluid when shear and stratification are not aligned

The effect of an inclination angle of the shear with respect to the stratification on the linear properties of the shear instability is examined in the work. For this purpose, we consider a two-dimensional plane Bickley jet of width $L$ and maximum velocity ${U}_{0} $ in a stably stratified fluid of constant Brunt–Väisälä frequency $N$ in an inviscid and Boussinesq framework. The plane of the jet is assumed to be inclined with an angle $\theta $ with respect to the vertical direction of stratification. The stability analysis is performed using both numerical and theoretical methods for all the values of $\theta $ and Froude number $F= {U}_{0} / (LN)$. We first obtain that the most unstable mode is always a two-dimensional Kelvin–Helmholtz (KH) sinuous mode. The condition of stability based on the Richardson number $Ri\gt 1/ 4$, which reads here $F\lt 3 \sqrt{3} / 2$, is recovered for $\theta = 0$. But when $\theta \not = 0$, that is, when the directions of shear and stratification are not perfectly aligned, the Bickley jet is found to be unstable for all Froude numbers. We show that two modes are involved in the stability properties. We demonstrate that when $F$ is decreased below $3 \sqrt{3} / 2$, there is a ‘jump’ from one two-dimensional sinuous mode to another. For small Froude numbers, we show that the shear instability of the inclined jet is similar to that of a horizontal jet but with a ‘horizontal’ length scale ${L}_{h} = L/ \sin \theta $. In this regime, the characteristics (oscillation frequency, growth rate, wavenumber) of the most unstable mode are found to be proportional to $\sin \theta $. For large Froude numbers, the shear instability of the inclined jet is similar to that of a vertical jet with the same scales but with a different Froude number, ${F}_{v} = F/ \hspace *{-.1pc}\cos \theta $. It is argued that these results could be valid for any type of shear flow.