We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonally to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the stream-wise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number Re and the Froude number F r, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. F r ∼ 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, for instance the unstratified plane Couette flow in the shallow water approximation. The result is confirmed by fully non linear direct numerical simulations and to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. 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 values of F r and Re 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 slightly modified due to streamwise confinement.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list)

Jupiter’s dynamics shapes its cloud patterns but remains largely unknown below this natural observational barrier. Unraveling the underlying three-dimensional flows is thus a primary goal for NASA’s ongoing Juno mission that was launched in 2011. Here, we address the dynamics of large Jovian vortices using laboratory experiments complemented by theoretical and numerical analyses. We determine the generic force balance responsible for their three-dimensional pancake-like shape. From this, we define scaling laws for their horizontal and vertical aspect ratios as a function of the ambient rotation, stratification and zonal wind velocity. For the Great Red Spot in particular, our predicted horizontal dimensions agree well with measurements at the cloud level since the Voyager mission in 1979. We additionally predict the Great Red Spot’s thickness, inaccessible to direct observation: it has surprisingly remained constant despite the observed horizontal shrinking. Our results now await comparison with upcoming Juno observations.