Stability of density-stratified viscous Taylor-Couette flows

Abstract

The stability of density-stratified viscous Taylor-Couette flows is considered using the Boussinesq approximation but without any use of the short-wave approximation. The flows which are unstable after the Rayleigh criterion (\hat μ<\hat η^2, with \hat μ=Ω_{out}/Ω_{in} and \hat η= R_{in}/R_{out}) now develop overstable axisymmetric Taylor vortices. For the considered wide-gap container we find the nonaxisymmetric modes as the most unstable ones. The nonaxisymmetric modes are unstable also beyond the Rayleigh line. For such modes the instability condition seems simply to be \hatμ<1 as stressed by Yavneh, McWilliams & Molemaker (2001). However, we never found unstable modes for too flat rotation laws fulfilling the condition \hat μ>\hat η. The Reynolds numbers rapidly grow to very high values if this limit is approached (see Figs. 3 and 4). Also striking is that the marginal stability lines for the higher $m$ do less and less enter the region beyond the Rayleigh line so that we might have to consider the stratorotational instability as a &#39;low-$m$ instability&#39;. The applicability of these results to the stability problem of accretion disks with their strong stratification and fast rotation is shortly discussed.