Observations of the stratorotational instability in rotating concentric cylinders

We study the stability of density stratified flow between corotating vertical cylinders with rotation rates ${\mathrm{\Omega }}_o<{\mathrm{\Omega }}_i$ 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 a nonaxisymmetric stratorotational instability (SRI). The present work extends the range of Reynolds numbers and buoyancy frequencies $[N=\sqrt{(-g/\rho )(\partial \rho /\partial z)}]$ examined in previous experiments. We present the first experimental results for the axial wavelength $\lambda$ of the instability as a function of the internal Froude number, $\text{Fr}={\mathrm{\Omega }}_i/N$; $\lambda$ increases by nearly an order of magnitude over the range of $\text{Fr}$ examined. 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 [Astron. Astrophys. 438, 411 (2005)] proposed that the laminar state of a stably stratified rotating shear flow should be stable for ${\mathrm{\Omega }}_o/{\mathrm{\Omega }}_i>r_i/r_o$, but we find that this stability criterion is violated for $N$ sufficiently large. At large Reynolds number the primary instability is not the SRI but a previously unreported nonperiodic state that mixes the fluid.

Figure 1

(a) A schematic diagram of the experimental system. (b) Density as a function of height for $N=5.3{\mathrm{s}}^{-1}$. The black circles are measurements, and the solid red curve is a fit of the data to an exponential. (c) Viscosity of an aqueous solution of sodium polytungstate salt, as a function of density. Black triangles are the manufacturer's data for the kinematic viscosity $\nu$ at $25{}^{\circ }\mathrm{C}$ [21] normalized by the viscosity of water ${\nu }_{\mathrm{water}}$. The solid red curve is a cubic spline fit to the manufacturer's data.

Figure 2

Observations of the SRI onset for increasing anticyclonic shear as $\mu ={\mathrm{\Omega }}_o/{\mathrm{\Omega }}_i$ was decreased from solid body rotation ($\mu =1$) and ${\text{Re}}_i$ was held fixed (green circles, $N=1.57{\mathrm{s}}^{-1}$; red diamonds, $N=3.14{\mathrm{s}}^{-1}$; blue triangles, $N=4.71{\mathrm{s}}^{-1}$). For all three values of $N$ examined the SRI flow was observed in the region to the right of the vertical dot-dash $\mu ={\eta }^2$ line where stability is predicted by Rayleigh's criterion for an inviscid unstratified fluid [19]. The open symbols near the Rayleigh line identify conditions under which the base state became unstable to a nonperiodic state rather than to the SRI. For a viscous stratified fluid Shalybkov and Rüdiger [18] predict stability to the right of the dashed $\mu =\eta$ line, while our experiments with $N=4.71{\mathrm{s}}^{-1}$ exhibit the SRI in that region.

Figure 3

(a) At low Reynolds numbers ${\text{Re}}_i$ and ${\text{Re}}_o$ the vertical density gradient stabilizes the laminar flow compared with the curve for the unstratified Taylor vortex flow instability (dot-dash curve), but at ${\text{Re}}_i$ and ${\text{Re}}_o$ above about 350 a density gradient destabilizes the laminar flow before the onset of Taylor vortex flow. (b) For ${\text{Re}}_i$ and ${\text{Re}}_o$ an order of magnitude larger than in (a) the laminar flow is destabilized more for a larger density gradient, but the effect of stratification decreases at higher Reynolds numbers (green circles, $N=1.57{\mathrm{s}}^{-1}$; red diamonds, $N=3.14{\mathrm{s}}^{-1}$; blue triangles, $N=4.71{\mathrm{s}}^{-1}$). The open symbols near the Rayleigh line correspond to a nonperiodic state that mixes the fluid. The Taylor stability line [22] is the dot-dashed curve, which is indistinguishable from the Rayleigh line (labeled $q=2$ using the notation of Balbus and Hawley [1]) at high Reynolds numbers. The Keplerian velocity profile is the solid line ($q=3/2$), and the dashed line ($q=1$) is the Shalybkov and Rüdiger prediction for instability onset. The inset enlarges the region where the data cross the $\mu =\eta$ line. The perpendicular distance in Reynolds number between the $q=1$ and $q=2$ lines is at most 600.

Figure 4

(a) A power spectrum of a movie pixel time series for $\mu =0.80$ shows that the SRI flow is periodic. (b) The time evolution of a vertical strip of pixels reveals that the periodic state in (a) consists of interpenetrating spirals; the black horizontal bar shows one period (0.374 s) of an $m=6$ interpenetrating spiral, which corresponds to the dominant peak in the spectrum in (a), $2\pi f/{\mathrm{\Omega }}_{av}=5.941$; strong peaks from the $m=5$ and $m=7$ states are also present. The low intensity peaks are harmonics and sums and differences of the cylinder and flow frequencies. The interpenetrating spirals rotate at approximately the average rotation rate (${\mathrm{\Omega }}_{av}/2\pi =0.45$ Hz) of the inner and outer cylinders (${\mathrm{\Omega }}_i/2\pi =0.5$ Hz and ${\mathrm{\Omega }}_o/2\pi =0.4$ Hz). The axial wavelength in the image in (b) is $\lambda /d=2.38$ and is plotted as an open green circle in Fig. 6. [In (a) and (b) $N=1.9{\mathrm{s}}^{-1}$, ${\text{Re}}_i=571$, ${\text{Re}}_o=815$, and internal $\text{Fr}=1.32$.] (c) A power spectrum of a movie pixel time series for the $\mu =0$ (outer cylinder at rest) SRI state shows that it is also periodic. The six harmonics yield a fundamental frequency $2\pi f/{\mathrm{\Omega }}_{av}=0.9765$. (d) An image sequence showing the time evolution of a vertical strip from a movie of the flow state in (c); the white bar shows the period (17.23 s) corresponding to the fundamental frequency in (c). The inset in (d) shows a section of the pattern with 400% magnification. The axial wavelength in the image is $\lambda /d=1.18$ and is plotted as an open green circle in Fig. 6. The vertical distance between two white stripes, where one is directly above the other, is one half of the axial wavelength. [In (c) and (d) $N=1.57{\mathrm{s}}^{-1}$, ${\mathrm{\Omega }}_i/2\pi =0.119$ Hz, ${\text{Re}}_o=0$, ${\text{Re}}_i=169$, and $\text{Fr}=0.48$ (the lowest data point in Fig. 3).] Both sets of data were taken within 3% of the onset of the SRI states.

Figure 5

Images of the SRI pattern for four values of internal Froude number $\text{Fr}={\mathrm{\Omega }}_i/N$ with $N=1.57{\mathrm{s}}^{-1}$. The wavelengths determined from the images are plotted in Fig. 6. The length of the smallest scale bar is $\lambda =1.81d$, which is 1.07 cm. (${\text{Re}}_i=419,838,1257,1675$ and $\mu =0.768,0.823,0.830,0.835$, respectively.)

Figure 6

Axial wavelength $\lambda$ at the onset of the SRI, normalized by the gap width $d$, as a function of $\text{Fr}={\mathrm{\Omega }}_i/N$. The symbols are the same as in Figs. 2 and 3 (green circles $N=1.57{\mathrm{s}}^{-1}$; red diamonds, $N=3.14{\mathrm{s}}^{-1}$; blue triangles, $N=4.71{\mathrm{s}}^{-1}$). A linear fit to the data (the black dash-dot line) yields $\text{slope}=1.72$ and $\text{intercept}=0.32$; our results differ from the slope value 2.2 and intercept value zero predicted by linear inviscid theory (solid line) [Eq. (3)] [6]. The open green circles are the wavelengths from the images in Fig. 4, $\lambda /d=2.38$, and Fig. 4, $\lambda /d=1.18$.

Figure 7

Images showing the development of a spatially nonperiodic state from laminar flow at ${\text{Re}}_i=9000$ during a 9 min period after the ratio of the cylinder rotation rates had been reduced from $\mu =0.79$ to $\mu =0.78$ ($N=1.57{\mathrm{s}}^{-1}$).

Figure 8

(a) Density measurements (black circles) with an exponential fit (red curve) for a state with a 73% increase in density ($N=4.71{\mathrm{s}}^{-1}$) from the top to the bottom of the fluid. The green squares are the density measurements taken after the SRI was observed for more than 30 min over a 2 day period; thus the SRI does very little mixing even when the density variation is large. (b) Density measurements (black circles) with an exponential fit (red curve) before the fluid transitions from the laminar base state to a nonperiodic state for a fluid with a 6% change in density ($N=1.57{\mathrm{s}}^{-1}$). The fluid mixes much more rapidly for the noperiodic state (Fig. 7); green squares in the graph show the density 9 min after the transition to the nonperiodic state.