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

Abstract

The linear inviscid stability of two families of centrifugally stable rotating flows in a stably stratified fluid of constant Brunt–Väisälä frequency N is analysed by using numerical and asymptotic methods. Both Taylor–Couette and Keplerian angular velocity profiles Ω TC = (1 − μ)/ r 2 + μ and Ω K = (1 − λ)/ r 2 + λ/ r 3/2 are considered between r = 1 (inner boundary) and r = d > 1 (outer boundary, or without boundary if d = ∞). The stability properties are obtained for flow parameters λ and μ ranging from 0 to +∞, and different values of d and N . The effect of the gap size is analysed first. By considering the potential flow (λ = μ = 0), we show how the instability associated with a mechanism of resonance for finite-gap changes into a radiative instability when d → ∞. Numerical results are compared with large axial wavenumber results and a very good agreement is obtained. For infinite gap ( d = ∞), we show that the most unstable modes are obtained for large values of the azimuthal wavenumber for all λ and μ. We demonstrate that their properties can be captured by performing a local analysis near the inner cylinder in the limit of both large azimuthal and axial wavenumbers. The effect of the stratification is also analysed. We show that decreasing N is stabilizing. An asymptotic analysis for small N is also performed and shown to capture the properties of the most unstable mode of the potential flow in this limit.