Observations of the stratorotational instability in rotating concentric cylinders

Abstract

We study the stability of density stratified flow between co-rotating vertical cylinders with rotation rates $Ω_o < Ω_i$ and radius ratio $r_i/r_o=0.877$, where subscripts $o$ and $i$ refer to the outer and inner cylinders. Just as in stellar and planetary accretion disks, the flow has rotation, anticyclonic shear, and a stabilizing density gradient parallel to the rotation axis. The primary instability of the laminar state leads not to axisymmetric Taylor vortex flow but to the non-axisymmetric {\it stratorotational instability} (SRI), so named by Shalybkov and Rüdiger (2005). The present work extends the range of Reynolds numbers and buoyancy frequencies ($N=\sqrt{(-g/ρ)(\partial ρ/\partial z)}$) examined in the previous experiments by Boubnov and Hopfinger (1997) and Le Bars and Le Gal (2007). Our observations reveal that the axial wavelength of the SRI instability increases nearly linearly with Froude number, $Fr= Ω_i/N$. For small outer cylinder Reynolds number, the SRI occurs for inner inner Reynolds number larger than for the axisymmetric Taylor vortex flow (i.e., the SRI is more stable). For somewhat larger outer Reynolds numbers the SRI occurs for smaller inner Reynolds numbers than Taylor vortex flow and even below the Rayleigh stability line for an inviscid fluid. Shalybkov and Rüdiger (2005) proposed that the laminar state of a stably stratified rotating shear flow should be stable for $Ω_o/ Ω_i > r_i/r_o$, but we find that this stability criterion is violated for $N$ sufficiently large; however, the destabilizing effect of the density stratification diminishes as the Reynolds number increases. At large Reynolds number the primary instability leads not to the SRI but to a previously unreported nonperiodic state that mixes the fluid.