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

The stability is investigated of the swirling flow between two concentric c ylinders in the presence of stable axial linear density stratification, for flows not satisfying the well-known Rayleigh criterion for inviscid centri fugal 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 less than or equal to 0, which implies a far wider range of in stability than previously identified. The most unstable modes are radially smooth and occur for a narrow range of vertical wavenumbers. The growth rat e is nearly independent of the stratification when the latter is strong, bu t it is proportional to it when it is weak, implying stability for an unstr atified flow. The instability depends strongly on a nondimensional paramete r, S, which represents the ratio between the strain rate and twice the angu lar 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-os cillatory in time, decays exponentially fast as S (which can also be consid ered a Rossby number) tends to 0. The mechanism of the instability is an ar rest and phase-locking of Kelvin waves along the boundaries by the mean she ar flow. Additionally, we identify a family of (probably infinitely many) u nstable modes with more oscillatory radial structure and slower growth rate s than the primary instability. We determine numerically that the instabili ties persist for finite viscosity, and the unstable modes remain similar to the inviscid modes outside boundary layers along the cylinder walls. Furth ermore, the nonlinear dynamics of the anti-cyclonic flow are dominated by t he linear instability for a substantial range of Reynolds numbers.