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R. R. Kerswell
Abstract

We investigate the 2D instability recently discussed by Gallet et al. (2010) and Ilin & Morgulis (2013) which arises when a radial crossflow is imposed on a centrifugally-stable swirling flow. By finding a simpler rectilinear example of the instability - a sheared half plane, the minimal ingredients for the instability are identified and the destabilizing/stabilizing effect of inflow/outflow boundaries clarified. The instability - christened ‘boundary inflow instability’ here - is of critical layer type where this layer is either at the inflow wall and the growth rate is (as found by Ilin & Morgulis (2013)), or in the interior of the flow and the growth rate is where measures the (small) inflow-to-tangential-flow ratio. The instability is robust to changes in the rotation profile even to those which are very Rayleigh-stable and the addition of further physics such as viscosity, 3-dimensionality and compressibility but is sensitive to the boundary condition imposed on the tangential velocity field at the inflow boundary. Providing the vorticity is not fixed at the inflow boundary, the instability seems generic and operates by the inflow advecting vorticity present at the boundary across the interior shear. Both the primary bifurcation to 2D states and secondary bifurcations to 3D states are found to be supercritical. Assuming an accretion flow driven by molecular viscosity only so , the instability is not immediately relevant for accretion disks since the critical threshold is and the inflow boundary conditions are more likely to be stress-free than non-slip. However, the analysis presented here does highlight the potential for mass entering a disk to disrupt the orbiting flow if this mass flux possesses vorticity.

Instability Driven by boundary inflow]Instability driven by boundary inflow across shear: a way to circumvent Rayleigh’s stability criterion in accretion disks?

1 Introduction

Rotating flows are ubiquitous in nature and industrial applications so understanding their stability continues to be an important and active area of research. The extent of this activity is perhaps best epitomised by the huge body of work studying Taylor-Couette flow (Couette 1888; Mallock 1888; Taylor 1923) - the flow between two concentric cylinders rotating at different rates - which has become the laboratory paradigm of the subject (e.g. Tuckerman (2014); Fardin et al. (2014) and references herein). Despite all this work, however, only very recently has it been realised that imposed radial flow can destabilise an otherwise stable rotating flow (Gallet et al. 2010; Ilin & Morgulis 2013). The paper by Ilin & Morgulis (2013) is particularly revealing because it demonstrates that the (non-dimensionalised) flow

between two concentric cylinders at and is inviscidly unstable to infinitesimal 2D oscillatory disturbances for either inflow or outflow providing is not too large. This is surprising because, firstly, in the absence of radial flow, this rotating flow is (marginally) ‘Rayleigh-stable’ as the angular momentum of the flow nowhere decreases in magnitude radially (note this is strictly a condition on axisymmetric disturbances: Rayleigh (1917); Drazin & Reid (1981) §15.2). Secondly the flow also fails a requirement for 2D instability - the rotating flow version of Rayleigh’s inflexion point theorem (Rayleigh (1880): Drazin & Reid (1981) §15.3 and problem 3.2 p 121). Thirdly, it is also somewhat counterintuitive that the instability occurs for both converging and diverging radial flows since the former are stable and the latter generally unstable when the rotation is absent as in Jeffery-Hamel flow (e.g. Drazin (1999)).

The results of Ilin & Morgulis (2013) also possess many intriguing features of which four stand out. Firstly, the existence of an imposed normal flow through the cylinder walls increases the order of the linear operator describing the evolution of small inviscid disturbances from the normal 2 to 3. This means that an extra boundary condition has to be imposed beyond the usual no-normal-flow conditions at either cylinder wall. While it is straightforward to argue that this extra condition must be imposed at the inflow boundary (Ilin & Morgulis 2013), predicting the effect of a specific choice is less clear: if a no-slip condition is imposed there is instability whereas a vanishing vorticity condition gives stability. Secondly, in the limit of vanishing radial flow (), Ilin & Morgulis (2013) find growth rates which scale as rather than the generic one might expect. This unusual growth rate scaling arises because there are no discrete modes of the linear problem which can satisfy the usual 2 no-normal velocity conditions for . This gives rise to a non-standard singular perturbation analysis, the robustness of which is unclear to, say, changes in the rotation profile and/or to the addition of extra physics. Thirdly, the asymptotics presented seems incomplete. The instability described in Ilin & Morgulis (2013) has a critical layer character but only where this critical layer is actually at the inflow boundary. This suggests the existence of further instabilities with an interior critical layer separated from the boundary. Lastly, the mechanism for the instability is unclear. Crossflow and shear would seem obvious ingredients but rotation or curvature not necessarily so. It is also not apparent whether the energy to feed the instability comes wholly ’through the boundary’ or is extracted at least in part from the (interior) rotational energy of the underlying flow.

To keep this study manageable, the focus here is on the situation which represents small radial inflow on a predominantly rotating flow. The motivation for this (as in Gallet et al. (2010)) is the accretion disk problem where certainly for cold and hence weakly-ionised disks, the source of inferred disk turbulence remains a hotly contested issue (e.g. Dubrulle et al. (2005); Shariff (2009); Balbus (2011); Ji & Balbus (2013)). As a result there is considerable interest in uncovering robust linear instability mechanisms. Interestingly, the existence of the radial accretion flow is rarely included in theoretical models since it is so small - smaller than the azimuthal flow where is huge when based on a molecular viscosity (e.g. Dubrulle (1992) quote figures of ) - and presumably its presence only felt over timescales which are far too large to be relevant (e.g Shariff (2009) estimates that it would take longer than the age of the universe for molecular viscosity to diffuse momentum across a typical disk). However, the results of Ilin & Morgulis (2013) suggests that such a flow could actually drive linear instabilities over a much shorter timescale.

The plan of the paper is to start with perhaps the simplest example of the instability which is just a sheared half plane of fluid with imposed inflow. The effect of adding viscosity is discussed as well as the introduction of an outflow (suction) boundary so that the flow domain becomes a channel. Then the discussion turns to rotating flow with more general profiles, including the solidly Rayleigh-stable Keplerian profile , to examine the robustness of the instability. The effect of further physics in the form of viscosity, 3-dimensionality and compressibility are also broached. Finally, the nonlinear aspects of the instability are probed ranging from a weakly nonlinear analysis around the primary 2D bifurcation through to secondary bifurcations and the ensuing fully 3D finite amplitude solutions.

The findings of the paper, organised under the various questions posed above, are as follows.

  1. Is curvature or rotation important for the instability?. No, all the features of the instability are reproduced in a rectilinear flow with inflow described in §2.1. The instability operates by advecting a source of vorticity at the inflow boundary across shear.

  2. Are there other (non-boundary-layer) modes of instability caused by an inflow boundary? Yes, instabilities exist with interior critical layers distinct from the inflow boundary: see §2.1.2, §3.2 and expressions (2.0), (2.0) and (3.0). Their growth rates are , while much larger than , are smaller than the inviscid boundary-layer modes found by Ilin & Morgulis (2013) and the equivalent viscous boundary layer modes found by Gallet et al. (2010).

  3. Given the sensitivity to what is chosen for the extra boundary condition, how generic is the instability across possible boundary conditions? For finite the instability looks generic with only a no-vorticity boundary condition obviously ensuring stability: see §2.1. However for , any restriction on the normal derivative of the tangential velocity effectively kills the instability: see §2.1.4. The non-slip condition always seems to allow instability to occur (Gallet et al. 2010; Ilin & Morgulis 2013).

  4. How robust is the instability to different rotation profiles? Very robust. The form of the shear is unimportant for the boundary-layer instability and of secondary importance for the critical-layer instability: see §3.2.

  5. What is the effect of adding viscosity? The presence of viscosity introduces a threshold crossflow of in the rectilinear situation where long streamwise wavelengths are permissible (Nicoud & Angilella (1997) and §2.2) or for the rotational situation where the azimuthal wavenumber is an integer (Gallet et al. (2010) and §3.3).

  6. What is the effect of adding 3 dimensionality? Squire’s Theorem effectively holds for the boundary layer instabilities since only the shear at the boundary is important and curvature is secondary. As a result, adding 3 dimensionality leads to less unstable disturbances: see §5.

  7. How robust is the instability to adding further physics? The instability survives the addition of compressibility (see §4), 3-dimensionality (see §5) and viscosity (see §2.2 and §3.3). It is also insensitive to the exact shear present as long as it is non-vanishing (see §3.2).

  8. Has this instability been seen before Gallet et al. (2010) and Ilin & Morgulis (2013)? Yes, in a rectilinear form by Nicoud & Angilella (1997) who studied ‘generalised plane Couette flow’ where a streamwise pressure gradient is imposed to counterbalance the effects of crossflow in the streamwise momentum balance (see §2). Doering et al. (2000) also saw the instability without an imposed pressure gradient. In this case the introduction of crossflow not only adds a new flow component to the base state but also changes its cross-stream shear. From a stability perspective, how these two effects interact can be subtle and ‘suction’ (as it is typically called) can be found to stabilise or destabilise existing shear instabilities (e.g. Hains (1971); Fransson and Alfredsson (2003); Guha & Frigaard (2010); Deguchi et al. (2014)). The situation is similar in the Taylor-Couette problem where radial flow can either stabilise or destabilise the well known Taylor vortex instability (Chang & Sartory 1967; Bahl 1970; Min & Lueptow 1994; Kolyshkin & Vaillancourt 1997; Kolesov & Shapakidze 1999; Serre et al. 2008; Martinand et al. 2009). The importance of the work of Gallet et al. (2010) and Ilin & Morgulis (2013) is that they studied the effect of radial flow on centrifugally-stable Taylor-Couette flow.

  9. Is the instability supercritical or subcritical? It is a supercritical instability for bifurcations where the crossflow is fixed and non-slip boundary conditions are applied at the boundaries in the presence of viscosity: see §6.1 and Appendix B.

  10. Can secondary bifurcations off the 2D solutions reach crossflow values which are below the threshold for (primary) instability? There is no evidence for this. The six 3D bifurcations found are all supercritical leading to even higher crossflow values.

  11. What is the energy source for the inviscid instability? The instability draws its energy from the underlying shear. The energy of this is replenished by the pressure field, which drives the crossflow, doing work.

  12. Is this instability possibly relevant for accretion disks? In a quiescent disk, the (molecular) viscosity-driven accretion flow is whereas the critical threshold for linear instability is a radial flow of leaving aside any issues about the exact form of the outermost boundary conditions. This, together with the fact that no signs of subcriticality have been found in either the primary instability or of any secondary bifurcations, indicates that the instability is not operative in isolation . However, if some other process is able to generate a larger radial flow, then this instability may be triggered as a secondary consequence. The instability primarily derives its energy from the gravitational body force working on the accreting flow.

Ilin and Morgulis have themselves continued their investigation to consider the effect of viscosity (Ilin & Morgulis 2015) and 3-dimensionality (Ilin & Morgulis 2015b) albeit from a complementary perspective: given a crossflow what is the smallest critical swirling flow for instability? This is equivalent upon rescaling to the problem of fixing the swirling flow and finding when instability disappears as the crossflow is increased rather than decreased as studied here. That there is a finite range of crossflow for instability when there is both an inflow and outflow boundary appears a generic observation. There is also a tempting general interpretation - the inflow boundary is responsible for initially destabilising the flow as the crossflow is increased in magnitude from zero but ultimately the outflow (or more commonly labelled the ’suction’) boundary stabilises the flow again when the crossflow becomes large enough. The particularly simple half-plane problem treated below helps motivate this simple characterisation. We henceforth refer to the instability first seen by Nicoud & Angilella (1997), Doering et al. (2000), Gallet et al. (2010) and Ilin & Morgulis (2013) as the ‘boundary inflow instability’.

2 2D Instability in Simplest Form: The Half Plane

The boundary inflow instability operates by advecting vorticity across shear, occurs in 2D and does not need viscosity or underlying vorticity as shown in Ilin & Morgulis (2013). Here, we demonstrate that curvature or rotation are not necessary either by discussing a rectilinear example of the instability. The instability does need a boundary, however, so cannot be captured by a local analysis. To see this, consider the simplest possible set-up: a 2D shear flow

(2.0)

where there is a constant pressure gradient and conveniently solves both Euler equations and the Navier-Stokes equations (by designRemoving the pressure gradient means that the base state is no longer a constant shear in the viscous situation.). The linearised inviscid equation for the perturbation vorticity , where is the streamfunction (), is

(2.0)

which represents just advection of the vorticity and has solution

(2.0)

where is the initial vorticity distribution. In an unbounded domain, can be Fourier transformed so that it is sufficient to consider only . Then the stream function is

(2.0)

which exhibits transient growth but no linear instability just as in the case (Orr 1907; Farrell 1987).

2.1 Half Plane: Inviscid

A half plane, however, can permit a starting vorticity distribution which spatially grows towards the boundary. Consider a boundary at and so that this is an inflow boundary when the fluid domain is . Taking the Fourier transform in of (2.0) forces

(2.0)

where is an arbitrary function. If is to be a modal disturbance which depends exponentially on (and ), i.e. , then

(2.0)

is the only possibility ( is an arbitrary normalisation) and therefore

(2.0)

(no modal solution exists for ). We will show that this equation, where the initial vortical distribution decays exponentially as , has growing modal disturbances (i.e. ). No instability is possible if the boundary is an outflow boundary - i.e. - as is forced to be 0 by boundedness. This suggests that the instability exists providing the boundary is not a zero-vorticity boundary which would force . In what follows, we adopt a non-slip boundary condition as Ilin & Morgulis (2013) originally did but will revisit this issue in §2.1.4 ( equation (2.0) is a third order differential equation for integrated once to include an arbitrary constant - hence 3 boundary conditions are needed to specify a unique solution ).

To confirm there is instability, (2.0) must be solved to derive the dispersion relation. Using a Green’s function approach, setting to 1 (w.l.o.g.) and imposing the 2 (usual inviscid) boundary conditions that and , (2.0) has the solution

(2.0)

A third boundary condition needs to be applied to give the dispersion relation, and with no-slip at the influx boundary ( at ), this is the relation

(2.0)

There is no solution to this for indicating the absence of a 1D instability but there is instability in 2D (). The expression (2.0) is valid for finite but it is now useful to consider small where this instability can be understood as a critical layer instability. The (special) case where this critical layer is at the (influx) boundary (so it is in fact a boundary layer) was discussed by Ilin & Morgulis (2013) in their inviscid Taylor-Couette set-up. The growth rates for such modes are . The other (generic) case when the critical layer is distinct from the inflow boundary has not been discussed before. The growth rate here is smaller - (see below) - but is still larger than the default which would be expected.

The asymptotic form of the dispersion relation (2.0) can be derived as using standard steepest descent/saddle point ideas. Here we need two parts of the integrand to contribute at the same leading order and precisely cancel. This can happen in two ways since there is just one saddle point at : the contribution from an end point (clearly ) cancels the contribution from the saddle point (the interior critical layer case) or the saddle point and the end point are effectively one and the same asymptotically (the boundary layer case). In the former case, the leading contributions from the end point at (1st term on LHS in (2.0)) and from the saddle point (2nd term on LHS) must satisfy

(2.0)

Now, for the saddle point to be asymptotically separated from the endpoint , . This together with the fact that the magnitude of the saddle point contribution (where ) has to be requires either and then or and . The former case is inconsistent because the saddle point is in the wrong part of the complex plane leaving the contribution from the end point unbalanced. The latter situation, however, does yield solutions. Defining

(2.0)

with and both quantities and as , then (2.0) requires

(2.0)

and

(2.0)

so there is instability with asymptotic growth rate for a discrete set of frequencies

(2.0)

where is an integer of . The form of the corresponding eigenfunction is discussed in §2.1.2.

The other situation - the boundary layer case - arises when the saddle point is within of the endpoint at , i.e. . At this point, the endpoint and the saddle point contributions cannot be considered separately. Instead and must be rescaled so that (2.0) becomes

(2.0)

where and

(2.0)

This must be solved numerically but a good estimate for the eigenvalues can be found by treating the contributions from the end point and saddle point as if they are separated. This means taking the integer to be in the frequency expression (2.0) for the internal critical layer mode and calculating the corresponding growth using (2.0). This leads to the asymptotic form

(2.0)

which performs very well even for the first eigenvalue since is already then: see Table 1. Figure 1 shows a typical critical layer eigenfunction and 3 boundary layer eigenfunctions for and .

Figure 1: Unstable boundary layer eigenfunctions (upper plot: blue solid line is the most unstable, the red dashed line is the third most unstable and the black dash-dot line is the fifth most unstable mode) and an unstable critical layer eigenfunction (lower plot with the critical layer shown as a red dashed line) for and . In all, the real part of is shown.
asymptotic actual
  
100 -50.1012 0.17909 -50.10143 0.1790934
10 -15.7539 0.42268 -15.75720 0.4226325
9 -14.9387 0.43871 -14.93867 0.4386500
8 -14.0684 0.45724 -14.07266 0.4571595
7 -13.1449 0.47903 -13.14981 0.4789312
6 -12.1513 0.50526 -12.15722 0.5051295
5 -11.0690 0.53781 -11.07620 0.5376254
4 -9.8686 0.57997 -9.877886 0.5796846
3 -8.5004 0.63820 -8.513168 0.6377301
2 -6.8647 0.72800 -6.884666 0.7270530
1 -4.6894 0.90317 -4.732350 0.9003686
Table 1: The 10 most unstable eigenvalues from the dispersion relation (2.0) and asymptotic estimates from (2.0).

2.1.1 Inviscid Asymptotics for the Boundary Layer Instability

Here a modal streamfunction solution of the form is sought with a boundary layer of thickness as first uncovered by Ilin & Morgulis (2013) (see their §3.2). Defining the new boundary layer variable and splitting the streamfunction into an expansion of large-scale parts and a boundary layer corrections (hatted variables)

(2.0)

we look for an instability with vanishing frequency (= speed of the inflow boundary) to leading order,

(2.0)

The governing equation

(2.0)

is then simplified to

(2.0)

for the leading flow and

(2.0)

for its boundary layer correction. The interesting observation here is there is no large-scale solution which can handle both the boundary conditions that and . Crucially, this means that the boundary layer correction must be so as to contribute at leading order to fix up the boundary condition rather than the usual to satisfy the extra tangential boundary condition . As a consequence, there is an tangential velocity in the boundary layer which must vanish at . Since the large-scale flow cannot contribute at this order, this non-slip condition is solely on the boundary layer flow and is sufficient to determine the growth rate. Integrating (2.0) twice and incorporating the fact that , leads to

(2.0)

where is an arbitrary constant. Imposing at then forces which is precisely condition (2.0).

2.1.2 Inviscid Asymptotics for the Critical Layer Instability

An interior critical layer instability is constructed by looking for a mode of O(1) frequency. Adopting the expansion (2.0) where again both and are , the critical layer is centred on (so ) and as in the boundary layer case, has thickness . Defining the critical layer variable

(2.0)

the (leading order) streamfunction in the critical layer, , satisfies

(2.0)

which can be integrated once to give

(2.0)

(choosing the normalisation of the mode here for convenience later) and then two further times to give

(2.0)
(2.0)

where and are constants.

Outside the critical layer, the streamfunction consists of a WKB-type solution and simple exponentials which satisfy Laplace’s equation,

(2.0)
(2.0)

where , and are constants whose order of magnitude will be set by matching to the critical layer streamfunction. The fact that and must vanish as is imposed by including only the decaying exponential for : the WKB mode vanishes as when provided .

The governing equation (2.0) is third order and so and must be continuous everywhere. The oscillatory form of in the critical layer means it must match entirely to the outer WKB-type solution

(2.0)

either side of the critical layer so . This means that the outer WKB solution only contributes at to the tangential velocity near the critical layer whereas the critical layer streamfunction forces an tangential flow (see (2.0) ). As a result there must be a large-scale flow of in both and outside the critical layer: this explains the scaling of the integration constants in (2.0) and means that and are all . There are then six (complex) conditions to be satisfied at leading order by the five remaining constants and complex eigenvalue . The first four are matching conditions on () and () as ,

(2.0)

and

(2.0)

or eliminating and , simply two jump conditions across the critical layer

(2.0)

The two remaining (boundary) conditions, and , require

(2.0)

where, while the outer WKB-type solution contributes at to near the critical layer, it must contribute at to at the inflow boundary. The resulting dispersion relation is

(2.0)

which leads to the same leading expressions for and given in (2.0) and (2.0). The necessary change in the scaling of the WKB solution contribution gives the dominant contribution to and the exact numerical counterbalancing of the large scale tangential flow gives the subdominant contribution.

2.1.3 The Need for Shear and Inflow

To emphasize that shear is a key ingredient of the instability, the above problem can be solved for the shearless flow leading to the requirement that there must exist with

(2.0)

which is never satisfied. The presence of an inflow boundary is also crucial: converting the above inflow boundary to an outflow boundary () removes the instability. This is why the instability discussed here is not relevant to the considerable literature on suction boundary layers where suction is always a stabilizing effect (e.g. Joslin (1998)).

2.1.4 The Third Boundary Condition

Imposing the parametrised boundary condition as the second boundary condition at gives the modified dispersion relation

(2.0)

( recovers non-slip and a no-vorticity or equivalently stress-free condition as along ). For the boundary layer instabilities, the LHS is so this dispersion relation can only be assumed similar to the non-slip relation (2.0) if . In fact, numerical computations indicate that the boundary layer instability is suppressed by . For the critical layer instabilities, the condition (2.0) becomes a possible balance between 3 different terms

(2.0)

where now has an frequency as in (2.0). If , the dominant balance is between the 2nd and 3rd terms (as opposed to the 1st and 2nd for non-slip) and now leads to damped eigenvalues. It is then clear that for instability to occur, there should be no restriction on the normal derivative of the tangential velocity at the inflow boundary. In practice this means that the instability only really occurs for non-slip boundary conditions when which incidentally is the one choice which, in concert with the no-normal flow condition, means no disturbance kinetic energy is being advected into the domain through the inflow boundary.

2.2 Half Plane: Viscous

The base state (2.0) is unchanged (by design) when viscosity is introduced but the linearised disturbance equation (2.0) now includes diffusion of vorticity:

(2.0)

Looking for a modal solution leads to the equation

(2.0)

which has the bounded solution (as )

(2.0)

where is the Airy function bounded as with . The dispersion relation is then

(2.0)

and instability ( crosses through 0 to become positive ) occurs at a critical inflow . This integral is actually the same as that treated by Gallet et al. (2010) (see their expression (39)) after the transformations

(2.0)

( and from Gallet et al. (2010)). For the of interest (  ), and we can reuse their critical value of which has passing through the imaginary axis to give

(2.0)

where and (consistent with Gallet et al. (2010) who quote ‘4.58’ and ‘5.62i’ for and respectively). This threshold tends to zero as albeit with the unstable eigenfunction extending a distance in the direction. In terms of connecting this analysis to other problems, there are two notable cases: which is the interesting case in rotating flow where the wavenumber is forced to be an integer by periodicity, and which is gives the most unstable disturbance in a domain bounded in (i.e. a channel see Nicoud & Angilella (1997)). In the former case, and the implication from the scaling of the critical frequency is the growth rate away from criticality (i.e. ) will be or as before. For , which is consistent with the numerical findings in Nicoud & Angilella (1997) that the threshold ‘crossflow’ Reynolds number for inflow instability in their plane Couette flow is independent of the shear Reynolds number.

For any given , further boundary-layer-type instabilities exist as increases with the first 6 thresholds listed in Table 2 for . Within this ‘boundary-layer’ scaling of and for , asymptotic predictions for can be derived following the same route as in the inviscid case. This proves a little more involved leading to two coupled relations

(2.0)
(2.0)

which work reasonably well for given that higher order terms may only be smaller.

As in the inviscid case, there are also interior critical layer modes excited for even higher inflows: if the critical layer is at , then

(2.0)
asymptotic actual
6 35.2992 16.8776 35.183 16.501
5 29.5968 14.2990 29.482 13.885
4 23.8395 11.6711 23.730 11.201
3 18.0097 8.9777 17.9033 8.4197
2 12.0754 6.1886 11.9737 5.4762
1 5.9596 3.2301 5.8938 2.1875
Table 2: The 6 lowest inflow thresholds for instability in the dispersion relation (2.0) with as and asymptotic estimates from (2.0) & (2.0).

2.3 Inflow and Suction Together: Inviscid Plane Couette flow with Suction

The half plane system exhibits boundary inflow instability for all . This can be seen by a simple rescaling of space

(2.0)

where so that the equation (2.0) becomes

(2.0)

which is just the original equation with a new small number as . However, this is rather artificial since the applied pressure gradient also has to be increased with to maintain the constant shear in . Resorting to a constant pressure gradient instead now means the shear field decreases as increases again making it difficult to draw general conclusions for large . Introducing another boundary is then the only alternative and this must be an outflow boundary if the resulting base flow is to be steady and spatially non-developing. The simplest modification is to add an outflow boundary at which is moving at so that the constant-vorticity basic flow (2.0) is still a solution (Nicoud & Angilella 1997). The equivalent expression to (2.0) is then

(2.0)

with the dispersion relation

(2.0)

when non-slip is applied at the influx boundary for . This is essentially the same as the half plane dispersion relation and will have unstable eigenvalues as there is an inflow boundary. The simple rescaling (2.0), however, is disallowed and instability is lost if becomes too large (see figure 12 of Nicoud & Angilella (1997)). This could be interpreted as the stabilising influence of the newly-introduced suction boundary ultimately overpowering the destabilising inflow boundary. Further evidence for this comes from the pressure-gradient-free version of this flow which is also linearly unstable in the presence of viscosity (Doering et al. 2000). In this case, the base state varies exponentially in the cross-stream direction (see eqn 2.13 of Doering et al. (2000)) and possesses an area of linear instability in the plane (figure 3 of Doering et al. (2000) where is their proxy for ). The lower boundary of this instability region, , plausibly scales like which is the viscous threshold for the instability as described in §2.2 whereas the upper boundary has , which appears to be suction ultimately stabilizing the flow (Hocking 1974).

3 2D Swirling Flow with Radial Inflow

We now add curvature to the discussion and consider the basic, steady, axisymmetric solution

(3.0)

between two boundaries at and with (i.e. the inner radius and the angular velocity there are used as length and inverse timescales respectively) and so that there is a radial inflow. The 2D Navier-Stokes equations for the deviation of the total flow from the basic solution (3.0),

(3.0)

are

(3.0)
(3.0)

together with incompressibility

(3.0)

where

(3.0)

and is the kinematic viscosity. Introducing a streamfunction

(3.0)

reduces the system (3.0)-(3.0) to

(3.0)

where the Jacobian is defined as

(3.0)

and

(3.0)

is the vorticity of the basic flow (3.0). There are two special cases where the gradient of the vorticity vanishes: uniform rotation ( so ) and irrotational flow with originally considered by Ilin & Morgulis (2013).

3.0.1 Basic state

There are two usual scenarios: a) the swirl field is set by the boundary conditions as in Taylor-Couette flow or b) the swirl field is determined by an imposed body (gravitational) force as in the astrophysical context. In the former, the swirl is determined by the azimuthal momentum equation given a radial flow with the radial momentum equation determining the pressure field. For example Ilin & Morgulis (2013) discuss the inviscid, irrotational, axisymmetric Taylor-Couette-like flow

(3.0)

which is the only possibility with axisymmetric radial flow (recall by nondimensionalization). Gallet et al. (2010) consider the viscous Taylor-Couette situation where possible base flows form a 1-parameter family

(3.0)

with a constant set by the motion of the outer cylinder and is generally rotational (). In the (latter) astrophysical context, the acting gravitational (body) force sets the swirl field (via the radial momentum equation) which then sets the radial flow by the need to balance ensuing azimuthal viscous stresses. The focus here is on the latter situation and we consider the general set of profiles

(3.0)

in order to understand how robust the boundary inflow instability is. Profiles with have angular momentum increasing with radius and so are Rayleigh-stable (Rayleigh 1917). The gradient of the vorticity also does not change sign across the domain so that the flow is inviscidly stable (for disturbances which vanish at and ) by a rotating flow analogue of Rayleigh’s inflexion-point theorem (Rayleigh 1880). Particularly interesting choices for are which is the Keplerian profile for a thin accretion disk due to the radial force balance

(3.0)

(where is the gravitational constant and the mass of the central object) and which is used to model spiral galaxies. Since we work with deviations away from the basic state, the exact body force required to maintain the underlying rotation profile is not explicitly needed in what follows. In contrast to the Taylor-Couette situation, the azimuthal component of the Navier-Stokes equations forces the existence of a small radial flow

(3.0)

which is an inflow if (). Studying the consequences of this small accretion flow is the motivation for this work.

Figure 2: Inviscid 2D instabilities for , and with scaled growth rate plotted against frequency . The right (left) vertical dashed line is (). The eigenvalues from a full 2D eigenvalue calculation are shown as (the upper) black dots for () and (the lower) blue dots for (since here, the 15 most unstable eigenvalues are marked with a normal-sized dot and the rest by smaller blue dots). The red squares are the 20 most unstable modes from the boundary layer asymptotics (section 3.1) with the dash-dot red line tracing the path of the rest (eigenvalues calculated from the boundary layer equation (2.0) are indistinquishable from the asymptotics at this scale). The green dashed lines are the critical layer asymptotic expression (3.0) with and plugged in and appropriately rescaled: and with taken over . Notice that at even struggles to fully resolve the critical layer eigenvalues near the inner radius - see the hump in the numerical data which breaks the otherwise excellent correspondence with the asymptotic prediction. This hump is delayed to lower if is increased until it eventually disappears - e.g. the curve.
Figure 3: The most unstable boundary layer eigenfunction (upper plot) and an unstable critical layer eigenfunction with (lower plot) for , , and . In both, the green dashed line is and the blue solid line is . The critical layer position is shown as a red dashed line for the critical layer eigenfunction.

3.1 2D Inviscid Swirling Flow with

We study the simplest case of vanishing vorticity gradient in the base flow first (), so or , to show how the analysis mirrors that in the half plane case. The case of uniform rotation initially looks uninteresting because the base flow needs an azimuthal as well as radial body force to maintain it (since in (3.0) ). But it is worth considering as then only the enforced radial inflow creates shear in the base flow and the question is whether this is enough to generate instability. The inviscid, linearised governing equation (3.0) is just

(3.0)

which is the ‘curved’ analogue of (2.0) and again 3rd order rather than the usual 2nd order. As before, the solution can be written down using characteristics as

(3.0)

where is the initial vorticity. After a discrete Fourier transform, the component is

(3.0)

where is an arbitrary function. For modal growth, should only depend on through an factor for some complex constant which forces