The linear instability of the stratified plane Couette flow*

We present the stability analysis of a plane Couette flow, stably stratified in the vertical direction orthogonally to the horizontal shear. We first perform the linear stability analysis of the flow in a confined domain in the cross-stream direction. The flow becomes unstable when shear and stratification are of the same order (i.e. Fr~1) and above a moderate value of the Reynolds number Re~700. The instability results from a resonance mechanism already known in the context of channel flows, e.g. in the non-stratified plane Couette flow in shallow water. The result is confirmed by fully nonlinear direct numerical simulations. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations, we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although modified due to streamwise confinement.