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.

The stratorotational instability of Taylor-Couette flows with moderate Reynolds numbers

“…The most prominent example for the instability is the potential flow with µ = 0.27. [15] argue that the instability of the stratified potential flow is the most unstable one in agreement with our numerical results. Figure 4 shows the lines of marginal instability derived from (13) for vanishing imaginary part of ω.…”

“…The inviscid equations of [34] for the potential flow combining a rigid inner boundary with an infinite gap width have been solved showing that the most unstable modes belong to high azimuthal wave numbers [15] . Weak stratification suppresses the instability.…”

Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor-Couette flow

“…Significantly, [13] explored a large range of rotation ratios and suggested a continuous connection between non-axisymmetric modes dominating on each side of the Rayleigh line. In contrast, [6] claimed later that stratorotational instabilities (SRI/RI) are much weaker than centrifugal instabilities (CI) when µ < η 2 , implying that a) the SRI/RI and CI instabilities are distinct, and b) CI are always stronger. A distinction between SRI/RI and CI instabilities certainly exists in the inviscid limit (the optimal axial wavenumber is bounded for the SRI/RI [6,7] whereas it is not for CI [4]) but this may not extend to the finite Reynolds numbers achieveable in experiments (consistent with [13]).…”

“…In contrast, [6] claimed later that stratorotational instabilities (SRI/RI) are much weaker than centrifugal instabilities (CI) when µ < η 2 , implying that a) the SRI/RI and CI instabilities are distinct, and b) CI are always stronger. A distinction between SRI/RI and CI instabilities certainly exists in the inviscid limit (the optimal axial wavenumber is bounded for the SRI/RI [6,7] whereas it is not for CI [4]) but this may not extend to the finite Reynolds numbers achieveable in experiments (consistent with [13]). Certainly having this distinction a) simplifies the identification of which instability mechanism dominates at a given point in parameter space but is not guaranteed.…”

“…The new instability -latter called the stratorotational instability or SRI in [5] -was interpreted as a resonance between boundarytrapped inertia-gravity waves. Using the same asymptotic framework as [4], [6] later showed that the SRI can become a radiative instability (RI) in the limit of an infinite gap (η → 0) so that the outer boundary 'goes to infinity'. The RI mechanism relies on a critical layer to extract energy from the base flow and radiate an evanescent wave radially outwards.…”