Connections between centrifugal, stratorotational and radiative instabilities in viscous Taylor–Couette flow
The ‘Rayleigh line’ , where and are respectively the rotation and radius ratios between inner (subscript ‘’) and outer (subscript ‘’) cylinders, is regarded as marking the limit of centrifugal instability (CI) in unstratified inviscid Taylor–Couette flow, for both axisymmetric and non-axisymmetric modes. Non-axisymmetric stratorotational instability (SRI) is known to set in for anticyclonic rotation ratios beyond that line, i.e. for axially stably-stratified Taylor–Couette flow, but the competition between CI and SRI in the range has not yet been addressed. In this paper, we establish continuous connections between the two instabilities at finite Reynolds number , as previously suggested by M. Le Bars & P. Le Gal, Phys. Rev. Lett. 99, 064502 (2007), making them indistinguishable at onset. Both instabilities are also continuously connected to the radiative instability at finite . These results demonstrate the complex impact viscosity has on the linear stability properties of this flow. Several other qualitative differences with inviscid theory were found, among which the instability of a non-axisymmetric mode localized at the outer cylinder without stratification, and the instability of a mode propagating against the inner cylinder rotation with stratification. The combination of viscosity and stratification can also lead to a ‘collision’ between (axisymmetric) Taylor vortex branches, causing the axisymmetric oscillatory state already observed in past experiments. Perhaps more surprising is the instability of a centrifugal-like helical mode beyond the Rayleigh line, caused by the joint effects of stratification and viscosity. The threshold seems to remain, however, an impassable instability limit for axisymmetric modes, regardless of stratification, viscosity, and even disturbance amplitude.
There has been considerable recent interest in the effects of axial stratification on the Taylor–Couette problem following the discovery in 2001 Molemaker et al. (2001); Yavneh et al. (2001) that it leads to instabilities outside the centrifugally unstable region. This region is conventionally defined by Rayleigh’s criterion Rayleigh (1917) as
are respectively the rotation and radius ratios between inner and outer cylinders (denoted with indices ‘’ and ‘’ respectively). Rayleigh derived his criterion for axisymmetric perturbations in the inviscid limit and only comparatively recently has it been extended to non-axisymmetric, inviscid perturbations by Billant and Gallaire (2005), albeit only in the limit of large axial wavenumbers. Using an inviscid, small-gap analysis, Yavneh et al. (2001); Molemaker et al. (2001) uncovered non-axisymmetric stratified instabilities that could develop when the inner cylinder rotates faster than the outer one, despite the radial decrease in angular momentum; the so-called quasi-Keplerian regime . The new instability – latter called the stratorotational instability or SRI in Dubrulle et al. (2005) – was interpreted as a resonance between boundary-trapped inertia-gravity waves. Using the same asymptotic framework as Billant and Gallaire (2005), Le Dizès and Riedinger (2010) later showed that the SRI can become a radiative instability (RI) in the limit of an infinite gap () 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. More recently, Park and Billant (2013a) extended the instability range of stratified Taylor–Couette flow even further, reaching the striking conclusion that the flow is always unstable, except for the special case of solid-body rotation . Importantly, Park and Billant (2013a) relaxed the small-gap assumption initially made by Molemaker et al. (2001); Yavneh et al. (2001) (by using large axial wavenumber asymptotics) and uncovered the role played by a critical layer to achieve over-reflection between the two boundary-trapped waves causing SRI.
With the exception of Withjack and Chen (1974), pre-2001 laboratory experiments on stratified Taylor–Couette flow were always carried out with a fixed outer cylinder (Withjack and Chen, 1974; Boubnov et al., 1995, 1996; Caton et al., 1999, 2000) (so ) and the relevance of the Rayleigh line was not questioned. The first experimental evidence of the SRI came in 2007 Le Bars and Le Gal (2007) where non-axisymmetric instability was clearly observed in the centrifugally-stable regime. Significantly, Le Bars and Le Gal (2007) 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, Le Dizès and Riedinger (2010) claimed later that stratorotational instabilities (SRI/RI) are much weaker than centrifugal instabilities (CI) when , 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 (Park and Billant, 2013a; Le Dizès and Riedinger, 2010) whereas it is not for CI (Billant and Gallaire, 2005)) but this may not extend to the finite Reynolds numbers achieveable in experiments (consistent with Le Bars and Le Gal (2007)). Certainly having this distinction a) simplifies the identification of which instability mechanism dominates at a given point in parameter space but is not guaranteed. Also, plausibly, CI might exist beyond the Rayleigh line in the presence of stratification (consistent with Le Bars and Le Gal (2007)). To add to the uncertainty, statement b) seems inconsistent with the findings of Park and Billant (2013b) which showed that a RI could grow faster than a CI in the case of a stably-stratified Rankine vortex in a rotating frame.
The purpose of this paper is to shed some light on these seemingly contradictory statements by carrying out a stability analysis of stratified Taylor–Couette flow which bridges the gap between experimentally-relevant Reynolds numbers and large Reynolds numbers where inviscid analysis should hold in some fashion. The motivation for this study comes from an ongoing programme of experimental work Leclercq et al. (2016a, b, c) and the desire to be able to interpret the mechanistic origin of the instabilities observed there. The key questions to be addressed are as follows. 1) Are CI and SRI continuously ‘connected’ (defined at the end of section III) in parameter space or are they always distinct and thereby represent different instability mechanisms? 2) Can the SRI ever dominate CI in the centrifugally-unstable region ? And 3) can CI exist for and therefore beyond the Rayleigh line with stratification?
Viscous linear analyses already exist in the literature, starting with the contribution of Thorpe (1966) based upon numerous simplifying assumptions. Caton et al. (2000) considered axisymmetric perturbations only (small-gap limit and finite gap but no density diffusion) while Molemaker et al. (2001); Shalybkov and Rüdiger (2005); Rüdiger and Shalibkov (2009) considered the general case albeit only close to the marginal stability curve. Here the focus is to consider the dispersion relation for linear viscous disturbances over several decades of the Reynolds number up from the marginal stability curve to at least and sometimes . A large range of rotation ratios is also considered in order to assess the relevance of the Rayleigh line at finite and with stratification.
The plan of the paper is as follows. In §II, we briefly introduce the governing equations and numerical methods. In §III, we present the results of our parametric stability analysis and discuss the effect of the different control parameters on the dominant mode. In §IV, we assess the connection between CI and SRI at finite by exploring the discontinuities of the optimal axial wavenumber in the -plane. By doing so, we reveal several connections between the different instabilities, and show the limits of Rayleigh’s criterion in viscous stratified Taylor–Couette flow. We summarize our findings in §V.
Ii Problem formulation
ii.1 Governing equations
The geometry of an axially infinite container is characterised by the radius ratio , while rotation and shear are jointly characterised by the rotation ratio (see (1) ) and Reynolds number which is defined as
with the kinematic viscosity. The purely azimuthal basic velocity profile is of the form
with and . The basic density profile is linear in the axial direction and characterised by a constant buoyancy frequency , based on gravity and a reference density . Two nondimensional parameters characterise stratification: the Richardson and Schmidt numbers
where is the diffusivity of mass. For all the results presented in this paper, the Schmidt number was set to a value of 700 appropriate for salt-in-water experiments Leclercq et al. (2016a, b, c) with only a few exploratory calculations done with appropriate for heated-water experiments (see later in §V). In the following, we take as the unit of length, as the unit of speed and as the unit of density.
We consider the dynamics of infinitesimal perturbations of the velocity in cylindrical coordinates, pressure and density , around the linearly stratified basic flow (3). Perturbations can be written in the form of normal modes
with complex frequency , integer azimuthal wavenumber and real axial wavenumber . In the Boussinesq approximation, the linearized incompressible Navier–Stokes, advection-diffusion and continuity equations read:
where , is the Doppler-shifted frequency, is the axial vorticity of the basic flow and . The boundaries conditions are no slip () and no-flux at the walls. Symmetries are such that , where denotes the complex conjugate. Therefore, we consider only positive and without loss of generality.
ii.2 Numerical methods
The governing equations were discretized using Chebyshev collocation in the radial direction, leading to a generalized eigenvalue problem for and . This problem was solved using multi-threaded LAPACK routines with OpenBLAS (
http://www.openblas.net). The number of Chebyshev polynomials used for each dependent variable was set to 140 for calculations below and increased up to a maximum of 480 when required at higher (at and , doubling the resolution from 140 to 280 led to less than variation in the growth rate of the most unstable mode). The code was validated by reproducing figures 4 and 10 from (Park and Billant, 2013a). For a given , the most unstable mode was found by optimising the growth rate over , using a Newton–Raphson method. Standard continuation methods were used to follow local maxima of the growth rate in parameter space.
Iii Dominance diagrams
We start by presenting the azimuthal mode number of the fastest growing mode in the -plane, for three values of and two radius ratios: a ‘large’ gap case (used in Leclercq et al. (2016a, b, c)) and a ‘small’ gap case . The dominant was obtained after optimization of the growth rate over all possible sets of wavenumbers . Results are given in figure 1, for a large range of and . The vertical black line corresponds to the Rayleigh line .
The first observation is the rise of the marginal stability curve to higher indicating the stabilising effect of stratification for all and both considered, consistent with previous results in the literature. The second common feature of all the plots is that axisymmetric () steady vortices (hereafter referred to as Taylor vortices) only dominate in regions removed from the marginal stability curve. The dominance regions near to the marginal curve are distinct since they have non-zero frequency. These oscillatory instabilities have already be found numerically by Hua et al. (1997); Shalybkov and Rüdiger (2005), and experimentally by Caton et al. (1999, 2000) but were not apparently seen by Le Bars and Le Gal (2007) presumably because of their very restricted domain of dominance. Indeed, the critical instability is most often non-axisymmetric, with larger values becoming preferred as increases. In the small-gap case, the effect of the Rayleigh line is clearly visible: the dominant peaks to a maximum in its vicinity, and decreases on both sides. For the large-gap case though, only values dominate, and the Rayleigh line only seems to mark the limit of the dominant steady region.
The dominance diagrams show the existence of disconnected patches and kinks in their boundaries (e.g. for , and ). These features suggest that different instabilities corresponding to the same value of are competing for dominance. In order to assign an instability mechanism to each dominant mode, we track the loci of the discontinuities of – the axial wavenumber maximising the growth rate – over -space for every fixed in the next section. A discontinuity in indicates the coexistence of two global maxima in the growth rate curve (defined as the maximum growth rate at a given ): either side of this, the maxima switch dominance giving rise to the discontinuity in . If this discontinuity always separates the two competing instabilities in parameter space we refer to them as being distinct instabilities having different mechanisms. Conversely, if at some point the discontinuity terminates indicating that the local maxima have merged, we consider the two instabilities as being continuously connected in parameter space and therefore not distinct. (Formally, there is also the possibility that two distinct instabilities cross over momentarily having the same at but this would give rise to a discontinuity in which is never seen in this study.)
Iv Exploring the discontinuities of the optimal axial wavenumber in the -plane
Figures 2 and 3 break down each dominance diagram into contributions from in the large-gap case. We plot the contours of , in order to identify the loci of the discontinuities for each . The region where a given dominates overall is shaded. Figure 2 compares the weakly stratified case () to the unstratified one, while figure 3 compares the moderate () and strong () stratifications.
iv.1 Oscillatory axisymmetric mode as a collision between Taylor vortex branches
For , there is no discontinuity in but a dashed curve marks the limit between oscillatory (below the line) and steady axisymmetric vortices (above). For all values of , the critical instability is always oscillatory, as already found by Thorpe (1966); Hua et al. (1997) at large enough , but becomes subdominant to steady vortices at large enough . To understand this transition better, we plot in figure 4 the dependency of the frequencies and growth rates of the two dominant axisymmetric modes against for two values of : one below the dashed curve () and one above () (marked by red crosses in figure 3). This figure shows that the oscillatory vortices are created as decreases from by the collision between two steady vortex branches. This is clearly a joint effect of stratification and viscosity, as the ‘bubble’ in figure 4 only appears if is small enough and .
iv.2 Continuous connections between non-axisymmetric CI and SRI modes in the -plane
There are discontinuities in for all non-axisymmetric modes, including when , and these are indicated by thick red curves in figures 2 and 3. For , the discontinuity near the Rayleigh line disappears within the unstable regions, at low enough , indicating that the instabilities on either side are smoothly connected. For however, the discontinuity always seems to separate the instability region into two distinct zones. But tracking the discontinuity further up in Reynolds number in figure 5 shows that it terminates just above for , indicating that the competing instabilities are again smoothly connected.
At yet higher , a discontinuity reappears as it must do if the viscous analysis is to be consistent with inviscid predictions since there CI occurs at infinite (Billant and Gallaire, 2005) whereas remains finite for the SRI Park and Billant (2013a). The value of where the growth rates of both instabilities are equal may be computed by suppressing the viscous term in the momentum equations (5)–(8) for the SRI, and using the analytical expression given by (Billant and Gallaire, 2005) for the CI in the inviscid limit (denoted with superscript ): (independent of ), where is the Rayleigh discriminant. We find that the switchover from SRI to CI dominant occurs at a value of which is very close to but below . Indeed, exactly vanishes on the Rayleigh line, whereas does not. Therefore, there is a very narrow range of where non-axisymmetric SRI dominates CI in the inviscid limit. But this region is very small, which explains why Le Dizès and Riedinger (2010) claimed that CI dominates over SRI in the centrifugally unstable region. By a continuation argument, a discontinuity in must be found at large but finite , which asymptotes the inviscid value of where . The discontinuities for in figures 2 and 3 seem to directly approach this limit as . For and , the discontinuity forming at also asymptotes the inviscid limit at larger . This analysis of the discontinuities of in the -plane establishes the continuous connection between CI and SRI instability mechanisms at finite for and .
iv.3 A centrifugal instability mode localized at the outer cylinder in the unstratified case
A simple way to attempt to differentiate between CI and SRI is to suppress stratification: if the flow is stabilized, the instability was a SRI, otherwise it was a CI. This motivated us to compute the dispersion relation of modes in the unstratified case as well. As shown in figure 2, there is no longer an instability beyond the Rayleigh line when , but surprisingly, we still observe a discontinuity in asymptoting at large for and . This result was unexpected, as we previously associated such discontinuity to a CI and SRI instability competing in the inviscid limit. However, since SRI is caused by a resonance between boundary-trapped inertia-gravity waves (Molemaker et al., 2001; Yavneh et al., 2001), the mode dominating on the right of the discontinuity cannot be SRI when . This suggests that this previously-unreported CI branch on the right of the discontinuity for is connected to a SRI as increases from 0. This new CI mode dominating to the right of the discontinuity is localized exclusively at the outer cylinder, as can be seen in figure 6 at , whereas the one to the left of the discontinuity is localized at the inner cylinder (figure 6). These wall modes are reminiscent of the two families of neutral branches which create SRI in the inviscid limit (Park and Billant, 2013a). In that limit, the outer-wall mode can only become unstable by coupling with the inner one, under the effect of stratification, but here we find that it may become unstable alone with the help of viscosity. Since the discontinuity in asymptotes to the Rayleigh line as , this mode never dominates the inner-wall one in the inviscid limit: there is therefore no contradiction with the theoretical analysis of Billant and Gallaire (2005).
iv.4 Radiative instability mode connecting CI to SRI in finite gap
In figure 7, we compare the structure of the two modes on the discontinuity close to the Rayleigh line for , and . The instability dominating to the left (decreasing ) of the discontinuity must tend to a CI as , while the instability to the right must tend to a SRI. As way of confirmation, the right branch has a structure which is reminiscent of inviscid SRI: the mode is localized at the walls and has a critical layer (see figure 7), as described by Park and Billant (2013a). However, since remains small, the WKB framework of Park and Billant (2013a) does not obviously apply so there are not the oscillatory regions described by these authors.
The structure of the left branch, however, resembles the radiative instability mode described by Le Dizès and Riedinger (2010) in the limit where the gap and Reynolds numbers become infinite, while (and ). There is a critical layer and an oscillatory region of radially decaying amplitude in figure 7, similar to figure 3 in Le Dizès and Riedinger (2010). This region is bounded to the right by the critical point , as in these authors’ theory for weak stratification. This critical point effectively isolates the radiated wave from the outer cylinder, which may explain why we were able to find a RI mode in our finite gap geometry, whereas Le Dizès and Riedinger (2010) only refer to this instability in the limit of infinite gap. In the present case, the RI seems to mediate the continuous morphing from the SRI to a CI at finite .
iv.5 A helical mode propagating against the inner cylinder rotation
Finally, in the last two subsections we investigate the nature of the modes in regions of the -plane created by unexpected discontinuities of , i.e. discontinuities that do not separate CI from SRI in the inviscid limit. First we discuss the critical mode at large in the counterrotating regime (see the solid dot in figure 3). Surprisingly, the azimuthal velocity associated with that mode is negative, whereas it is always positive for other dominant modes. Such peculiar behaviour has not been reported before. Inspecting the structure of this mode in figure 8 indicates that it is not a SRI, as the amplitude of is negligible near the outer wall. The turning point such that does not seem to coincide with any particular feature of the mode structure, confirming that asymptotic theories of CI/RI/SRI are of little help at this low . Because of the apparent absence of a critical layer, we conclude that the mode is best classed as a CI.
iv.6 A centrifugal-type mode beyond the Rayleigh line destabilized by viscosity
Finally, we discuss the discontinuity located at the right of the Rayleigh line in figure 2 for . Since the dominant mode is ‘beyond’ the Rayleigh line and well-separated from the CI region by two discontinuities in , it is tempting to call this mode SRI. However, looking at the mode in figure 8 reveals a structure which is highly reminiscent of the CI found in the previous section. The only noticeable difference is the presence of a turning point (as defined in the previous section) exactly where , suggesting the mode may extract its energy from the base flow at the critical layer, as in the RI. In order to determine whether the mode is centrifugal or radiative, we plot isocontours of the growth rate in the -plane in figure 9. The local maxima of the growth rate in that plane are indicated with dashed red curves, becoming solid when the maximum is global. The plot shows that the dominant (only) instability at is created by the merging of two local maxima present in the ‘centrifugally-unstable region’ . This may explain why the mode is both reminiscent of CI and RI.
Pursuing this further, the three distinct branches, respectively dominating at are followed as in figure 10. Each branch is a global maximum of at and its respective value of , but only a local maximum at larger , hence the superscript to denote the associated , and . In figure 10, we observe very similar trends for the evolution of and between the unknown mode at and the CI at . Indeed, in both cases, and . Both trends were predicted analytically for () inviscid CI by Billant and Gallaire (2005), and that remains finite and for the RI Le Dizès and Riedinger (2010). This suggests that the mode dominating at connects with CI at large , despite being outside the so-called centrifugally unstable region. This observation does not contradict the theory in Billant and Gallaire (2005) as the growth rate of this mode tends to zero in the inviscid limit. The conclusion is therefore that this mode is a mixture between a CI and a RI mode, destabilised beyond the Rayleigh line by the joint effects of stratification and viscosity.
Our numerical results suggest that this intrusion of CI-type mode features beyond the Rayleigh line is only possible for non-axisymmetric modes. That inviscid axisymmetric modes still can’t exist beyond the Rayleigh line with stable stratification follows from a simple extension of Rayleigh’s criterion (see appendix A). A stronger result can be proved that no finite-amplitude axisymmetric state beyond the simple base flow can exist past the Rayleigh line for a viscous, stratified fluid in the thin-gap limit (see appendix B). But whether a general proof exists for finite curvature, viscosity and stable stratification is unclear.
In this paper, we have performed a thorough linear stability analysis of (axially) stably-stratified viscous Taylor–Couette flow motivated by ongoing experiments Leclercq et al. (2016a, b, c). We considered both a large-gap and a small-gap configuration and varied the rotation ratio , the Reynolds number and the Richardson number . This paper bridges the gap between the numerous inviscid analyses of the centrifugal, stratorotational and radiative instabilities (respectively CI, SRI and RI), and the computations of instability thresholds in the viscous case, by addressing the question of the dominant instability mechanism at finite beyond the marginal stability line.
As in previous studies, we found that the first bifurcation as increases above the marginal curve is always oscillatory, and usually non-axisymmetric. We showed that the oscillatory mode is created by a collision between steady Taylor vortex branches. This is an effect of viscosity and stratification (and large too, according to Thorpe (1966); Hua et al. (1997)), as this collision disappears at large or for .
By exploring the discontinuities in the optimal axial wavenumber , we were able to establish a continuous connection between CI and SRI instabilities in the -plane. Indeed the optimal wavenumber diverges for CI in the inviscid limit, whereas it remains finite for the SRI: as a consequence, there exists a value of where the growth rate of both instabilities are equal. This leads to a discontinuity in at which can be tracked down to finite , where it eventually disappears as the two local maxima of merge into one. The coalescence occurs within the instability region, so it is impossible to distinguish CI from SRI at onset. This explains why Le Bars and Le Gal (2007) observed a smooth evolution of non-axisymmetric patterns from the quasi-Keplerian to the centrifugally-unstable region in their experiment, whereas Le Dizès and Riedinger (2010) considered the two instabilities distinct in the inviscid limit. As argued by Le Dizès and Riedinger (2010), CI is indeed stronger than SRI when the two instabilities compete in the inviscid limit, but the discontinuity in bends back towards low values of as diminishes, making SRI dominant over a large portion of the Rayleigh-unstable domain, until the two instabilities can no longer be distinguished. We also established a connection between CI and RI induced by viscosity, without taking the limit of infinite gap as in Le Dizès and Riedinger (2010).
In the unstratified case, we did not expect to find similar discontinuities in for , as SRI is a resonance between boundary-trapped inertia-gravity waves. However, we did find two different branches competing in the vicinity of the Rayleigh line for a given when . Each of the competing branches corresponds to a wall-mode, localized either at the inner or the outer boundary. These are reminiscent of the two families of branches giving birth to SRI as they interact when Park and Billant (2013a). But here the outer-wall mode becomes unstable under the effect of viscosity, not because of a coupling with the inner-wall mode. This outer-wall CI mode, continuously connected to SRI as was previously unknown because it is always subdominant, yet it is destabilized at finite and large-gap widths.
We have also found a dominant mode propagating against the rotation of the inner cylinder in the counterrotating strongly stratified case. This behaviour has not been reported before, for either one of the three instabilities studied in this paper, since they all require to be able to extract energy from the base flow when . This is another surprising effect of viscosity on the instability mechanism, since this mode is only dominant at low enough .
Finally, we investigated the nature of a mode dominating beyond the Rayleigh-line in the weakly stratified case. This mode seems to have a critical layer, but since its maximum amplitude peaks at the inner cylinder only, it is reminiscent of a radiative mode. However, it shares features of CI, rather than RI modes as . In particular, the optimal wavenumber diverges while the azimuthal phase speed tends to the inner cylinder angular velocity: two properties of CI. This, however, does not violate the generalized Rayleigh’s criterion of Billant and Gallaire (2005) as the growth rate asymptotes zero in the inviscid limit. At finite , this mode seems connected to both CI and RI in the -plane, therefore, we conclude that it is a mixture of CI and RI, destabilized beyond the Rayleigh line by the coupled effects of viscosity and stratification. This reiterates that viscosity has a more complex impact on the stability properties beyond just expected stabilization.
For , we did not systematically investigate the discontinuities of in the -plane. None were found for both and at in the range but since a discontinuity must be present near the Rayleigh line in the inviscid limit, we conclude that these discontinuities form at larger as . This makes the distinction between SRI-type and CI-type instabilities even more problematic, and indicates that the effect of viscosity is heightened by reducing the gap size. We also produced dominance diagrams at a lower value of the Schmidt number , in the large-gap case, for which are qualitatively similar to and so not included. The number effects are expected to occur at even lower values of , according to Thorpe (1966); Hua et al. (1997).
We conclude by assessing the relevance of the Rayleigh line: even though non-axisymmetric centrifugal-type modes seem to be able to grow beyond with the help of viscosity and stratification, this limit appears to remain impassable to axisymmetric ones (whether steady or oscillatory). We were able to prove this result in two distinct limits: inviscid linear disturbances in finite gap and viscous finite-amplitude disturbances in thin gap. Whether a more general result suggested by our numerical results (and previous studies) can be proven remains an interesting question.
This work has been supported by the EPSRC (C. L.), under grant EP/K034529/1, and École Normale Supérieure de Paris (F. N.). We thank Stéphane Le Dizès for stimulating discussions.
Appendix A Rayleigh’s Criterion for Stably-Stratified Inviscid Flow
The Euler equations linearised around the basic flow (equations (5-9) with ) for an axisymmetric () incompressible disturbance can be reduced down to a 2nd order differential equation for , the radial perturbation velocity,
is the Rayleigh discriminant (Drazin and Reid (1981), p. 69). Multiplying by (where is the complex conjugate of ) and integrating from to gives
since . Rearranging
so providing for all , i.e. the basic flow is Rayleigh-stable, then all of the integrals are positive definite which implies and stability for all .
Appendix B Uniqueness of axisymmetric states beyond the Rayleigh line in the thin gap limit
Here we prove that the only streamwise-independent state that can exist in rotating, stably-stratified plane Couette flow beyond the Rayleigh line is one of simple shear implying that no other axisymmetric state beyond the base state can exist beyond the Rayleigh line in thin-gap stratified Taylor–Couette flow. The proof is a straightforward extension of the unstratified result presented by Hung et al. (1972) to include stratification. In a rotating frame where the shearing boundaries are at and gravity , there is the simple shear solution , and (stable stratification). Rayleigh’s criterion in this context is that centrifugal instability is only possible for (e.g. see Rincon et al. (2007)). The governing equations for disturbances away from this steady state, are
Defining and , then for streamwise-independent velocity and density fields taking , and leads to the ‘energy’ integrals,
where, periodicity across and either non-slip or stress-free velocity fields together with either Dirichlet () or Neumann conditions () for the density on kill all boundary terms which arise. Importantly, all the cubic nonlinear terms drop in these equations and so the kinetic energy in the field can be treated separately from that in and . As a result, generalised energy and dissipation functionals can be constructed as follows
and monotonic decay of the disturbance energy is ensured if
for any real . The maximum on the RHS can be minimised over to give the best energy stability result. Clearly, is a feature of the optimiser and we can rescale by defining to get an expression for the energy stability Reynolds number as
where the implication is that all streamwise disturbances decay for regardless of their amplitude. The latter maximisation corresponds to for an unstratified, non-rotating layer where Joseph and Munson (1970) under non-slip conditions. The minimisation problem has the minimum for and otherwise for real . As a result, we have
So, on and beyond the Rayleigh line , the (generalised) energy of all streamwise-independent disturbances, regardless of their amplitude, monotonically decays in time for any . To guarantee that (and hence the ultimate vanishing of all disturbance fields), we need a Poincaré inequality for some so that (21) becomes for some constant . Grönwall’s inequality then gives the required result. A Poincare inequality exists for non-slip conditions on the velocity field and either Dirichlet or Neumann conditions on the density field (in the latter case only if no mean flow is allowed in the direction of gravity). (Note that once for any , 2D spanwise-invariant disturbances are not assured to decay (Hung et al., 1972) so that there is no general global stability result for the basic state beyond the Rayleigh line.)
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