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We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal 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 streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number
and the Froude number
, 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.
) and above a moderate value of the Reynolds number
. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear 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
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.
We present an experimental study of the time evolution of an isolated anticyclonic pancake vortex in a laboratory rotating stratified flow. Motivations come from the variety of compact anticyclones observed to form and persist for a strikingly long lifetime in geophysical and astrophysical settings combining rotation and stratification. We generate anticyclones by injecting a small amount of isodense fluid at the centre of a rotating tank filled with salty water linearly stratified in density. The velocity field is measured by particle image velocimetry in the vortex equatorial plane. Our two control parameters are the Coriolis parameter
and the Brunt–Väisälä frequency
. We observe that anticyclones always slowly decay by viscous diffusion, spreading mainly in the horizontal direction irrespective of the initial aspect ratio. This behaviour is correctly explained by a linear analytical model in the limit of small Rossby and Ekman numbers, where density and velocity equations reduce to a single equation for the pressure. In particular for
, this equation ultimately simplifies to a radial diffusion equation, which admits an analytical self-similar solution. Direct numerical simulations further confirm the theoretical predictions that are not accessible to laboratory measurements. Notably, they show that the azimuthal shear stress generates secondary circulations, which advect the density anomaly: this mechanism is responsible for the slow time evolution, rather than the classical viscous dissipation of the azimuthal kinetic energy. The importance of density diffusivity is also analysed, showing that the product of the Schmidt and Burger numbers – rather than the bare Schmidt number – quantifies the importance of salt diffusion. Finally, a brief application to oceanic Meddies is considered.
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