Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow

Abstract

The stability is investigated of the swirling flow between two concentric cylinders in the presence of stable axial linear density stratification, for flows not satisfying the well-known Rayleigh criterion for inviscid centrifugal instability, d(Vr)2/dr < 0. We show by a linear stability analysis that a sufficient condition for non-axisymmetric instability is, in fact, d(V/r)2/dr < 0, which implies a far wider range of instability than previously identified. The most unstable modes are radially smooth and occur for a narrow range of vertical wavenumbers. The growth rate is nearly independent of the stratification when the latter is strong, but it is proportional to it when it is weak, implying stability for an unstratified flow. The instability depends strongly on a non-dimensional parameter, S, which represents the ratio between the strain rate and twice the angular velocity of the flow. The instabilities occur for anti-cyclonic flow (S < 0). The optimal growth rate of the fastest-growing mode, which is non-oscillatory in time, decays exponentially fast as S (which can also be considered a Rossby number) tends to 0. The mechanism of the instability is an arrest and phase-locking of Kelvin waves along the boundaries by the mean shear flow. Additionally, we identify a family of (probably infinitely many) unstable modes with more oscillatory radial structure and slower growth rates than the primary instability. We determine numerically that the instabilities persist for finite viscosity, and the unstable modes remain similar to the inviscid modes outside boundary layers along the cylinder walls. Furthermore, the nonlinear dynamics of the anti-cyclonic flow are dominated by the linear instability for a substantial range of Reynolds numbers.

Experimental Analysis of the Stratorotational Instability in a Cylindrical Couette Flow

“…Further away from the threshold, the 2D-Fourier transforms are more complex and indicate other peaks, but in the following we only focus on the most energetic of them. As suggested by the analytical study of Shalybkov and Rüdiger [13], we did not observe any systematic variation of the wavenumber k = 2π q (R i (R o − R i )) 1/2 : over all our experiments, we measure a mean value of 10.6 with a standard variation of 1.5 in close agreement with analytical and numerical results [12,13]. This value corresponds to a wavelength equal to 1.17 time the gap, smaller than the standard value in the absence of stratification (i.e.…”

The strato-rotational instability of Taylor–Couette and Keplerian flows

“…This corresponds to the local approximation of the Keplerian flow for which both flows possess the same shearto-rotation ratio Ω /Om (Ω being the angular velocity) equal to −3/2. The stability properties of the flow in this small-gap limit have been obtained in Yavneh et al (2001) and Dubrulle et al (2005b). We shall see that they sensibly differ from those obtained for an unbounded Keplerian flow.…”

“…Small-gap configurations have been considered by Yavneh et al (2001) who have shown that the flow is unstable when the fluid is sufficiently stratified. In figure 1, results for a large-gap configuration (d = 5) are plotted.…”

Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data

“…This means that all anti-cyclonic flows are potentially linearly unstable. The role of dissipation on the instability has been studied numerically [41,42]) and experimentally [43,44]. It was found that stratification stabilizes the flow in the GSPC regime, while it destabilizes it in the GSBC anti-cyclonic regime.…”