## Abstract

This paper is concerned with the absolute instability of the boundary-layer flow produced when an infinite disc rotates in otherwise still fluid. A greater understanding of the mechanisms and properties of the absolute instability is sought through the development of an analytic theory in the inviscid long-wave limit. It is shown that the fundamental basic flow characteristic of the absolute instability is a wall-jet in the radial direction superposed with an asymptotically small cross-flow, which generates a small reverse flow outside the boundary layer in the appropriately resolved basic velocity profile. The absolute instability is produced by a modal coalescence involving the interaction of *eight* saddle-points of the dispersion relation. An explicit expression for the growth rate has been obtained in terms of basic flow parameters. Most curiously, the pinch-point for absolute instability is shown to become asymptotically close to a branch-cut on the imaginary axis of the complex wavenumber plane, and unstable spatial branches emanating from the pinch-point cross this imaginary axis onto a Riemann sheet of the dispersion relation composed of solutions growing exponentially with distance from the disc. The existence of such modes contradicts the expectation of monotonic exponential decay of disturbances outside a boundary layer.

## 1. Introduction

When an infinite disc rotates about its axis of symmetry in an otherwise still fluid, viscous stresses at the disc surface drag fluid elements near the disc around in almost circular paths, and centrifugal forces then cause these elements to spiral outwards. The disc thus acts as a centrifugal fan with a radial flow component that has a wall-jet character directed away from the axis of rotation. The fluid thrown outwards in this way is replaced by an axial flow towards the disc surface. The azimuthal flow component has a typical boundary-layer profile, increasing monotonically from zero at the disc wall to a constant value proportional to the angular velocity of the disc and the distance to the axis of rotation (when considered, as here, in a frame of reference rotating with the disc). The importance of this cross-flow structure to the flow's stability was first recognized by Gregory *et al*. (1955) who observed a set of stationary vortices in an experimental study, and explained their appearance in terms of an inviscid inflexional ‘cross-flow’ instability. This cross-flow instability generates stationary vortices in many three-dimensional boundary layers of engineering interest, and they are believed to be involved in the laminar–turbulent transition process in many of these flows.

Those experiments showed that the stationary vortices are generated by points of surface roughness and grow in the direction of increasing radius, thus corresponding to a convective instability. However, Lingwood (1995) used Briggs' (1964) spatio-temporal analysis to show that this flow becomes absolutely unstable far enough from the axis of rotation. In an absolutely unstable flow, impulsive disturbances grow both upstream and downstream (for the rotating disc, radially inwards and outwards) of the disturbance source, and in time at the location of the source. Lingwood's location for convective–absolute transition coincides closely with the location of laminar–turbulent transition measured in many experimental studies; an observation that has generated much interest. Her study of the absolute instability was based on obtaining numerical solutions to the linearized disturbance equations. In this way she showed there is absolute instability for inviscid disturbances, and, by making the parallel-flow approximation, the viscous problem was also solved, allowing a quantitative estimate to be made for the location of the convective–absolute transition.

Numerical stability calculations are usually essential in order to make detailed comparisons with experiments, but do not always illuminate the fundamental physical mechanisms responsible for the observed phenomena. Lingwood's (1995) numerical solutions for the inviscid problem suggest that an asymptotic long-wave theory can be developed for the absolute instability. In addition to the insight into the nature and mechanisms of the absolute instability that a theory can give, there is a further motivation to investigate the long-wave regime provided by the results of Healey (2004). It was shown in that paper that the connection between the inviscid theory and the viscous theory is not as straightforward as originally supposed. The saddle-point giving the inviscid upper bound for absolutely unstable azimuthal wavenumbers turns out not to correspond to Lingwood's saddle-point for the upper branch of the neutral stability curve for absolute instability in the viscous version of the problem. Lingwood's viscous saddle-point becomes ‘non-pinching’ at a finite Reynolds number on the upper branch of the neutral curve, and a second family of saddle-points produces the pinch-points at higher Reynolds numbers. Nonetheless, Lingwood's saddle-point appears to remain the pinch-point along the lower branch of the neutral curve, which lies in the long-wave regime. Therefore, an analytic description of the long-wave inviscid absolute instability can form a starting point for a future theory for Lingwood's viscous absolute instability.

Turkyilmazoglou & Gajjar (2001) have already derived an asymptotic theory for the saddle-points that produce absolute instability in the rotating-disc boundary layer in the long-wave inviscid limit. In the present paper we extend this analysis, principally, by obtaining analytical solutions to the dispersion relation where they only obtained numerical solutions. In the process we shall bring forward a number of new features of the absolute instability, including an explicit formula for its growth rate, showing how it depends on flow parameters like the wall shear stress that may be useful for allowing one to anticipate how modifications to the basic flow would affect the strength of the absolute instability.

In addition, the characteristics of the basic flow that generate the absolute instability have been determined, as has the mechanism that produces the instability itself. However, probably the most interesting new finding is the way in which the pinch-point moves across the complex radial wavenumber plane as the azimuthal wavenumber approaches zero. It is shown that the pinch-point becomes asymptotically close to the imaginary axis of the radial wavenumber plane, where a branch-cut lies. The branch-cut is produced by a square-root term appearing in the solution far from the disc, and is placed along the imaginary axis to ensure that solutions decay exponentially with distance from the disc. As the azimuthal wavenumber is reduced, the pinch-point is predicted to become close enough to the branch-cut for the unstable spatial branches that emanate from the pinch-point to cross onto the Riemann sheet consisting of waves that grow exponentially with distance from the disc, thus seeming to violate the homogeneous boundary condition that requires disturbances to decay to zero far from the disc. The physical interpretation of this behaviour, and a detailed exploration of how the conventional behaviour found by Lingwood evolves into the new behaviour found here is presented in Healey (in press).

The paper is structured as follows. The governing equations are presented in §2, from which the basic flow solutions, and the linearized inviscid disturbance equations (the Rayleigh equation) are derived. The disturbance equations are solved in the long-wave limit in §3, and explicit expressions obtained for the location of saddle-points. It turns out there are *eight* families of saddle-points, and the one that corresponds to Lingwood's pinch-point is identified. The new insights into the nature of the absolute instability and the physical mechanisms that are operating are derived and discussed in §4. The conclusions are given in §5.

## 2. Problem formulation

The disc rotates at constant angular velocity *Ω*_{*} in an otherwise still viscous incompressible fluid of kinematic viscosity *ν*_{*} (in this paper all dimensional quantities have an asterisk subscript). The basic flow is obtained from the Kármán (1921) similarity solution. Batchelor (1951) showed that this flow is also a limiting case of a family of flows with similarity solutions, in which both the disc and the fluid far from the disc rotate with different angular velocities. This family also includes as limiting cases the Bödewadt (1940) layer, where the disc is stationary and the fluid rotates, and the Ekman (1905) layer, where fluid and disc co-rotate at almost the same angular velocity. The similarity structure persists when there is an axial flow towards the disc (see Hannah 1952), and through the disc wall, e.g. when there is wall suction (Stuart 1954) or blowing (Kuiken 1971). The inclusion of magnetic fields is also straightforward (e.g. Gorla 1992; Moresco & Alboussière 2004). The methods described in the present paper could, in principle, be applied directly to all of these flows.

### (a) Governing equations

We choose to work in cylindrical coordinates in a frame of reference rotating with the disc. The axial and radial coordinates are *z*_{*} and *r*_{*}, respectively, the azimuthal angle is *θ*, time is *t*_{*} and *ρ*_{*} is the density of the fluid. The velocities in the radial, azimuthal and axial directions are *u*_{*}, *v*_{*} and *w*_{*}, respectively, and the pressure is *p*_{*}. The governing equations are therefore(2.1)(2.2)(2.3)(2.4)where the differential operators are(2.5a)(2.5b)

Lengths are scaled by the characteristic viscous length-scale, and time by the angular velocity of the disc,(2.6a)(2.6b)(2.6c)Flow variables are separated into an axisymmetric steady basic flow, which respects von Kármán's similarity structure, and a more general unsteady part, whose amplitude is characterized by a small parameter *δ*≪1,(2.7)

(2.8)

(2.9)

(2.10)

### (b) Basic flow

Substituting (2.6*a*)–(2.10) into (2.1)–(2.4) and equating terms of *O*(*δ*^{0}) gives the basic flow similarity equations,(2.11)(2.12)(2.13)(2.14)to be solved subject to boundary conditions,(2.15a)(2.15b)The numerical solution of (2.11)–(2.14) subject to (2.15*a*) and (2.15*b*) is relatively straightforward, e.g. by a shooting method where (2.15*a*) provides three initial conditions, with two more initial conditions *U*′(0) and *V*′(0) chosen iteratively until (2.15*b*) has been satisfied at a suitable large finite value of *z* to within some prescribed accuracy.

However, the long-wave theory of §3 requires the asymptotic structure of the basic flow solution both as *z*→0 and *z*→∞. The behaviour in the limit *z*→0 is easily obtained by expanding all the dependent variables in (2.11) as power series in *z*. The first few terms are(2.16a)(2.16b)(2.16c)where the wall boundary conditions (2.15*a*) have been applied and the constants and are determined, in principle, by the remaining far-field boundary conditions (2.15*b*). The behaviour in the limit *z*→∞ is found by linearizing about the uniform flow state far from the disc. Substituting(2.17a)(2.17b)where *W*_{∞}<0 is the constant axial flow velocity towards the disc far from the disc, into (2.11)–(2.14), eliminating *U* and *P*, and linearizing in *δ*_{1} gives(2.18a)(2.18b)with boundary conditions following from (2.15*b*), and hence solution(2.19a)(2.19b)The three constants *U*_{∞}, *V*_{∞} and *W*_{∞} are determined, in principle, by the wall boundary condition (2.15*a*).

The exponential form of the solutions (2.19*a*) and (2.19*b*) and the quadratic nonlinearity in (2.11)–(2.14) suggests that the solution to (2.11)–(2.14) can be approximated by expanding all terms in powers of exp(*W*_{∞}*z*). The first few terms in these expansions for the *U*, *V* and *W* components are(2.20)(2.21)(2.22)In fact, the first numerical solutions to (2.11)–(2.14) were obtained by Cochran (1934) by matching the small and large *z* expansions (2.16*a*)–(2.16*c*) and (2.20)–(2.22) at a finite value of *z*. Fettis (1956) showed that the large *z* expansion converges even at *z*=0, and so was able to use this expansion alone to calculate the entire basic flow, thereby avoiding the need to match to the small *z* power series. More details of the method can be found in Benton (1966), who gives tabulated values of the solution, and Stuart (1966), who generalizes the method to solve the partial differential equations arising in oscillating boundary-layer flows. We have essentially followed Fettis; *U*_{∞}, *V*_{∞} and *W*_{∞} were evaluated by imposing (2.15*a*) on (2.20)–(2.22) for a certain number of terms, which was increased until the results became independent of the number of terms, to some prescribed level of precision. This led to using 32 terms, giving *U*_{∞}≈0.924864, *V*_{∞}≈1.20221 and *W*_{∞}≈−0.884474, and hence and , in agreement with Benton (1966). This large *z* expansion was also used for all numerical evaluation of the basic flow. The use of an analytic representation of the basic flow also greatly facilitates numerical solutions of the inviscid stability equations when the critical points (where the basic flow velocity coincides with a wave's phase velocity) require solution paths in the complex *z*-plane (see Healey in press). The basic flow components are shown in figure 1. Note the ‘wall-jet’ character of the radial velocity component, *U*, and the boundary-layer character of the other velocity profiles even though no boundary-layer approximation has been made.

### (c) Inviscid linearized equations

Substituting (2.6*a*)–(2.10) into (2.1)–(2.4) and equating terms of *O*(*δ*) gives the linearized disturbance equations,(2.23)(2.24)(2.25)(2.26)where(2.27a)(2.27b)The coefficients depend on both *r* and *z*, but not on *θ* nor *t*. Therefore, Fourier series can be taken in *θ* (since the flow field is periodic in *θ*) and Fourier transforms can be taken in *t*, but the disturbance equations remain partial differential equations, with variables depending on both *r* and *z*. Reduction to ordinary differential equations at leading order is only possible far from the axis of rotation, and this is the limit we shall work in.

Let *R*_{*} be the dimensional position of interest on the disc, then a Reynolds number, *Re*, can be introduced that is the ratio of *R*_{*} to the characteristic viscous length-scale,(2.28)(Some studies have defined *Re*^{2} to be the Reynolds number, corresponding to a Reynolds number based on length-scale *R*_{*} and local disc velocity *R*_{*}*Ω*_{*}.) We now introduce a new radial coordinate, *ρ*, given by(2.29)where *Re*≫1 and *ρ*=*O*(1) near the position of interest. If *Re* *α*≫1, where *α* is the radial wavenumber, then the radial wavelength is small compared with the distance to the axis of rotation, and the basic flow does not vary significantly on the length-scales associated with the disturbance. This separation between the length-scale of disturbance evolution and the length-scale of basic flow evolution allows a WKB formulation to be adopted for the disturbance structure, and so we let(2.30)(2.31)(2.32)(2.33)where *Re* *β* is an integer. We assume *Re* *β*≫1 and so will neglect the discretization of the scaled azimuthal wavenumber, *β*, which is discretized in units of *Re*^{−1}. Substituting (2.30)–(2.33) into (2.23)–(2.26) and neglecting terms of *O*(*Re*^{−1}), and smaller, gives the linearized inviscid disturbance equations,(2.34)(2.35)(2.36)(2.37)where primes denote partial differentiation with respect to *z*. These equations show that in the inviscid limit, not only are the viscous terms neglected, but also the Coriolis terms, streamline curvature terms, non-parallel terms (those which involve ∂/∂*ρ*) and the axial basic flow component, *W*. The absence of all these terms means that properties of the solutions of the inviscid equations can apply to a range of physical problems, and are not dependent on the rotation or axisymmetry of the disc problem. Equations (2.34)–(2.37) are the starting point for the long-wave theory developed in §3.

Note, however, that in dropping the non-parallel terms, we are only considering the local propagation properties of disturbances, which can differ from the global properties arising from the inhomogeneity of the basic flow. Davies & Carpenter (2003) have presented a direct numerical simulation of linearized disturbances in the full viscous non-parallel rotating-disc boundary layer, showing an ultimate decay of disturbances predicted to be absolutely unstable by the local theory. Nonetheless, as discussed in Healey (2004), local stability properties are still important because they are used as the starting point for constructing global mode theories.

Before proceeding to the long-wave theory, however, we note that eliminating *u*, *v* and *p* gives the Rayleigh equation,(2.38)where(2.39a)(2.39b)(2.39c)These equations are solved by choosing a real, scaled, azimuthal-wavenumber *β*/*ρ* and finding (possibly complex) *α* and *ω*/*ρ* that allow solutions to be found that satisfy the inviscid homogeneous boundary conditions,(2.40a)(2.40b)Due care is required with regard to any critical points, *z*=*z*_{c}, where *Q*(*z*_{c})=*c*. The integration path in the complex *z*-plane is chosen such that the resulting eigenvalues correspond to solutions of the associated viscous problem. The path is thus positioned with respect to critical points in accordance with the rule given by Lin (1955): when all the eigenvalues are real the path lies below a critical point if *Q*′(*z*_{c})>0 and above a critical point if *Q*′(*z*_{c})<0. Typically, this means taking paths along the real axis for unstable waves, and taking detours in the complex *z*-plane to avoid crossing critical points for stable waves. The correct path can also be chosen by investigating the behaviour of the critical points as the integration contours in the *ω*- and *α*-planes arising from taking inverse Fourier transforms are placed and deformed in accordance with an initial value calculation. It is found that the integration path can pass along the real *z*-axis for eigenvalues corresponding to pinch-points for the calculations presented here in the rest frame. However, in some frames of reference, the pinch-point (dominant saddle-point) requires a solution path in the complex *z*-plane, and for some spatial branches this path must taken a large distance from the real *z*-axis (see Healey 2005, in press for examples in this flow).

When *α* and *β* are real, *Q* can be interpreted as the basic flow resolved in the direction of the wavevector of the disturbance, and is shown in figure 2 for a range of wave-angles *ϕ*, where tan *ϕ*=*β*(*ρα*)^{−1}. Certain wave-angles are already known to be particularly important in the stability of this basic flow. The angle *ϕ*=39.64°, which corresponds to , gives zero shear at the wall and was shown by Hall (1986) to be the angle for long-wave viscous-Coriolis neutral stationary vortices; the angle *ϕ*=−37.57°, which corresponds to tan *ϕ*=−*U*_{∞}/*V*_{∞}, is the angle at which the inflexion point tends to infinity, and there is no inviscid instability for *ϕ*<−37.57°; the angle *ϕ*=13.22° has inflexion point at a height where *Q*=0 and gives the wave-angle for neutral inviscid stationary vortices found by Gregory *et al*. (1955) (see figure 2). One purpose of our theory will be to identify the characteristic profile *Q* corresponding to the absolute instability in the long-wave limit.

## 3. Asymptotic long-wave saddle-points

Lingwood (1995) obtained numerical solutions to the Rayleigh equation (2.38) and hence, using a shooting method to satisfy the boundary conditions (2.40*a*) and (2.40*b*), obtained sets of eigenvalues. A family of saddle-points satisfying d*ω*/d*α*=0 were found, and investigation of the associated spatial branches (zeros of the dispersion relation in the complex *α*-plane generated by horizontal contours in the complex *ω*-plane for various fixed Im(*ω*)) revealed them to be of pinching type according to the principles explained in Briggs (1964). Figure 4*a* of Lingwood (1995) indicates that these pinch-points produce absolute instability in the range 0<*β*/*ρ*<0.265.

The appearance of absolute instability for small values of *β*/*ρ* motivated Turkyilmazoglou & Gajjar (2001) to develop a long-wave theory for these saddle-points. However, while they only obtained numerical solutions to their dispersion relation, we have obtained analytical solutions, bringing out a number of new properties of the absolute instability. There are also logarithmic terms that were omitted in their calculation, but included here; another difference is that the contribution of the second critical point was neglected in their calculation, but included here (it turns out that its contribution to the stability is of larger order of magnitude than the contribution from the critical point near the wall). The logarithmic terms arise because of the logarithmic behaviour of the solution near critical points; the existence of a second critical point for long waves of small phase velocity is due to the flow reversal seen in figure 2 for small positive *ϕ*, and we shall show that this limit is important for the long-wave saddle-points. These differences make it useful to go through the calculation here, though it follows the same principles as Turkyilmazoglou & Gajjar's. These differences may also be what enable us to obtain better quantitative agreement with the numerical solutions than they obtained in their fig. 2*a*.

We shall let *ϵ*≪1 be a small parameter characterizing the smallness of the wavenumbers in the long-wave limit. We will consider *ϵ* scalings, such that there is a critical point close to the wall. In particular, let(3.1a)(3.1b)(3.1c)This results in a three-layered structure for the disturbance in the wall-normal direction, and we now derive the solutions in each layer separately.

### (a) Upper layer

Far from the wall the basic flow is uniform, *Q*″→0, and the Rayleigh equation (2.38) reduces to(3.2)with decaying solution(3.3)where the square-root sign indicates the root with positive real part, to satisfy (2.40*b*). Therefore, the solution decays over a length-scale of *O*(*ϵ*^{−1}) by (2.39*b*) and (3.1*a*)–(3.1*c*). This observation motivates the introduction of an upper-layer variable, *ζ*, where *z*=*ζ*/*ϵ* and *ζ*=*O*(1) in the upper layer. However, the dispersion relation is most quickly obtained not by working directly with the Rayleigh equation, but in terms of the continuity and momentum equations (2.34)–(2.37). The use of these equations instead brings out the importance of the coupling between the pressure and velocity fields, whereas working only with the velocity, i.e. solving the Rayleigh equation, requires working to higher order in the *ϵ* expansions to obtain the same dispersion relations.

We introduce the following upper-layer expansions:(3.4a)(3.4b)(3.4c)(3.4d)where the subscript *u* denotes an upper-layer variable and *u*_{u0}=*u*_{u0}(*ρ*,*ζ*), etc. The velocities and pressure are all of the same order of magnitude allowing them to interact in the upper layer. The absolute magnitude of these expansions is arbitrary, since we are solving linear homogeneous equations, but the *O*(*ϵ*) choice made here is convenient, because it turns out that the velocity components parallel to the disc in the upper layer are *O*(*ϵ*) smaller than in the rest of the boundary layer. Substituting (3.1*a*)–(3.1*c*), (3.4*a*)–(3.4*d*) and *U*=0, *V*=−1 into (2.34)–(2.37), then equating powers of *ϵ* leads to a series of differential equations for the upper-layer variables, and their solutions are(3.5a)(3.5b)(3.5c)(3.5d)(3.5e)(3.5f)where the square-root sign denotes the root with positive real part to satisfy (2.40*b*),(3.6)and *P*_{u0}=*P*_{u0}(*ρ*), etc. The solutions for the *u* and *v* variables can be expressed in terms of (3.5*a*)–(3.5*f*), but are not needed in the derivation of the dispersion relations.

### (b) Main layer

In the main layer, where *z*=*O*(1), we introduce the expansions(3.7a)(3.7b)(3.7c)(3.7d)where the subscript *m* denotes a main-layer variable and *u*_{m0}=*u*_{m0}(*ρ*,*z*), etc. The magnitudes of *w* and *p* have been chosen so that they match the upper-layer solutions; the magnitudes of *u* and *v* then follow from the requirement that they interact with *w* at leading order in the continuity equation (2.34). The pressure and velocity fields do not interact at leading order in the main layer. Substituting (3.1*a*)–(3.1*c*) and (3.7*a*)–(3.7*d*) into (2.34)–(2.37), then equating powers of *ϵ* leads to a series of differential equations for the main-layer variables, and their solutions can be written as(3.8a)(3.8b)(3.8c)(3.8d)(3.8e)(3.8f)where(3.9)the results and lim_{z→∞}*Q*_{0}(*z*)=−1/*α*_{0} from (2.16*a*)–(2.16*c*), (2.20)–(2.22) and (3.9) have been used, *z*_{c2} is a critical point lying away from the wall and *P*_{m0}=*P*_{m0}(*ρ*), *B*_{m0}=*B*_{m0}=(*ρ*), etc. Note that the singular part of *Q*_{0}^{−2} has been extracted from the integrand appearing in (3.8*f*), and ln(*z*−*z*_{c2})=ln(*z*_{c2}−*z*)+i*π* when *z*<*z*_{c2} since when the eigenvalues are real, so that this inviscid solution corresponds to the viscous solution as *Re*→∞. The form of the integral terms has been chosen to facilitate matching with the upper- and lower-layer solutions.

### (c) Lower layer

Assuming that , the scalings (3.1*a*)–(3.1*c*) then imply the existence of a critical point a distance of *O*(*ϵ*) from the wall. Accordingly, we introduce a lower-layer variable, *Z*, where *z*=*ϵZ* and *Z*=*O*(1) in the lower layer. In this layer we introduce the expansions(3.10a)(3.10b)(3.10c)(3.10d)where the subscript *l* denotes a lower-layer variable and *u*_{l0}=*u*_{l0}(*ρ*, *Z*), etc. The magnitudes of *w* and *p* have been chosen so that they match the main-layer solutions; the magnitudes of *u* and *v* then follow from the requirement that they interact with *w* at leading order in the continuity equation (2.34). The pressure and velocity fields interact at leading order in this layer. In the lower layer the series (2.16*a*)–(2.16*c*) apply and so we have(3.11a)(3.11b)(3.11c)(3.11d)Substituting (3.10*a*)–(3.10*d*) and (3.11*a*)–(3.11*d*) into (2.34)–(2.37), then equating powers of *ϵ* leads to a series of differential equations for the lower-layer variables, and their solutions are(3.12a)(3.12b)(3.12c)(3.12d)(3.12e)(3.12f)where is the position of the critical point near the wall at leading order, ln(−*Z*_{c})=ln *Z*_{c}−i*π* in accordance with Lin's rule and the wall boundary condition (2.40*a*) has been applied at each order in *ϵ*.

It is the logarithmic behaviour of (3.8*f*) and (3.12*f*) that forces the presence of logarithmic terms in (3.1*a*)–(3.1*c*), which are needed in order to complete the matching of the solutions between the layers.

### (d) Dispersion relations

Van Dyke's matching rule has been used to match *w* and *p* across all three layers. The results for the ‘0’ subscripted variables are(3.13a)(3.13b)(3.13c)(3.13d)and these relations admit non-zero solutions only if(3.14)Following the same steps with the results of the matchings at higher orders gives dispersion relations(3.15)(3.16)where the constants(3.17)

(3.18)

The expansion (3.1*a*)–(3.1*c*), and solutions (3.14)–(3.16), correspond to a temporal stability analysis when *α*_{0} is real, leading, in general, to complex frequencies. In principle, a spatial analysis could be carried out, in which (3.1*a*)–(3.1*c*) is replaced by a series for *α* with given *ω*, and dispersion relations obtained giving *α* explicitly in terms of *ω*. However, we shall investigate the spatio-temporal properties of the system by allowing *α*_{0} to become complex in (3.14)–(3.16).

### (e) Saddle-points

The condition for saddle-points, for given values of *β*/*ρ*, to the order of accuracy of our long-wave theory, is(3.19)which, in principle, can be evaluated numerically, as was done by Turkyilmazoglou & Gajjar (2001). However, we have identified an asymptotic scaling for the solutions of this equation as *ϵ*→0. It stems from the observation that if *α*_{0}≫1, then(3.20a)(3.20b)(3.20c)Therefore, ∂*ω*_{0}/∂*α*_{0} will be of the same order of magnitude as *ϵ*∂*ω*_{1}/∂*α*_{0} when *α*_{0}=*O*(*ϵ*^{−1/4}), i.e. *β*/*ρ*=*O*(*ϵ*) implies *α*=*O*(*ϵ*^{3/4}), and so this is the appropriate scaling for the saddle-points. Hence, let *σ*=*ϵ*^{1/4}, substitute(3.21)into (3.19) evaluated using (3.14)–(3.16), equate powers of *σ* to give *α*_{b0}, *α*_{b1}, etc. giving the wavenumber of the saddle-points, then substitute these results into (3.1*c*) using (3.14)–(3.16) to give the frequency at the saddle-points. The presence of the ‘ln ln’ term and inverse powers of the logarithm are explained below. By, at first, taking in (3.20*a*), we find four roots at leading order satisfying(3.22)which we shall denote by(3.23a)(3.23b)(3.23c)(3.23d)where *A*_{0} is the positive real root of (3.22). Each of these expressions generates a series for a family of saddle-points, but the second term of each series is the same and is given by(3.24)which is real and positive. Now recalling that we see that although the root whose series is generated by (3.23*b*) has negative real part, the other three have positive real parts. In fact, (3.23*b*) lies on the Riemann sheet corresponding to choosing square-roots with negative real parts in (3.3) and (3.5*a*)–(3.5*f*), and therefore have solutions that grow exponentially in the wall-normal direction and so do not satisfy the outer boundary condition (2.40*b*) (see Healey (in press) for the physical interpretation of this saddle-point). The expressions for subsequent terms in (3.21) are much more lengthy and will not be written out here.

The first terms in the expansion for the frequency at the saddle-points are(3.25)Therefore, the saddle-point with most positive Im(*ω*), i.e. the one first encountered as the frequency contour is lowered in the complex *ω*-plane, is produced by (3.23*d*). The full approximations to this family of saddle-points, with numerically evaluated coefficients, are found to be(3.26)(3.27)Note that in order to calculate *O*(*σ*^{6}) terms in (3.26) and *O*(*σ*^{12}) terms in (3.27) it would be necessary to include *O*(*ϵ*^{4}), *O*(*ϵ*^{4} ln *ϵ*) and *O*(*ϵ*^{4}(ln *ϵ*)^{2} terms in (3.1*c*), and corresponding higher-order terms in the expansions for the solutions in each layer.

This asymptotic long-wave theory for the family of saddle-points generated by root (3.23*d*) is compared in figure 3 with numerical solutions of the Rayleigh equation for the pinch-points found by Lingwood (1995). There is good quantitative agreement between the two as *β*/*ρ*→0, confirming that this asymptotic theory does describe the pinch-point in the inviscid long-wave regime. The corresponding series expansion for the saddle-point given by the root (3.23*a*) and a comparison with numerical solutions to the Rayleigh equation is given in appendix A. An investigation of the spatial branches in the complex *α*-plane associated with these saddle-points, and how they connect between the saddle-points, is presented in Healey (in press) in a numerical calculation that allows a global study of the complex *α*-plane to be made (the long-wave theory only applies near the origin of this plane). In that paper numerical solutions for the saddle-points corresponding to the roots (3.23*b*) and (3.23*c*) are also given, and parameter regimes identified where each of (3.23*a*), (3.23*b*) and (3.23*d*) represent the pinch-point. Even saddle-points with eigenfunctions that grow exponentially in the wall-normal direction can be pinch-points in some frames of reference.

Note that setting in (3.20*a*) leads to the calculation of another four saddle-points, giving a total of eight. At leading order in *β*/*ρ* these additional saddle-points can be obtained by multiplying each saddle-point in (3.23*a*)–(3.23*d*) by exp i*π*/4. However, as shown in Healey (submitted), the integration path cannot be deformed to pass through this second set of saddle-points. Furthermore, although the pinch-point calculation requires the composite expansion (3.19), there are saddle-points that can be calculated using only the first term in this expression, but these saddle-points are sub-dominant (see Healey submitted).

Now that we have derived the scalings for the saddle-points, it might seem more efficient, in retrospect, to have started the analysis with these scalings, e.g. to use *α*=*α*_{0}*σ*^{3}, *β*=*ρσ*^{4} and *ω*=*ω*_{0}*σ*^{8}+*ω*_{1L}*σ*^{12} ln *σ*+*ω*_{1}*σ*^{12} in place of (3.1*a*)–(3.1*c*). However, as discussed in appendix B, this seems not to be straightforward in this case.

The following physical insights into the nature of the absolute instability of the rotating-disc boundary layer can now be gained from the theory that were not readily apparent from the numerical solutions.

## 4. Properties of the pinch-points

The resolved basic velocity profiles, *Q*, are inflexional so it might be expected that the instability arises from the phase jumps at critical points, as is the case for the stationary vortices. However, the phase jumps at the critical points make contributions to Im(*ω*) at *O*(*σ*^{11}), the order at which the logarithmic terms first appear in (3.27), and so this is a secondary contribution to the growth rate compared with the *O*(*σ*^{9}) term, which is generated by substituting the purely imaginary (3.23*d*) into (3.25). The instability mechanism for these saddle-points is therefore a modal coalescence mechanism, produced by taking a complex fourth-root in (3.22), rather than a critical point mechanism.

It can also be noted that the existence of absolute instability for arbitrarily small *β*/*ρ* in the inviscid theory means that a wave of fixed azimuthal wavenumber, *β*, remains absolutely unstable for arbitrarily large radii *ρ*. Therefore, for inviscid waves, there is no outer radius at which the flow becomes convective again, and so the linear global mode theory of Monkewitz *et al*. (1993) does not apply in this context. Of course, this situation may be different in the more realistic viscous case.

The basic flow characteristics responsible for producing the absolute instability can also be easily extracted from the theory. It has been shown that the pinch-points have eigenvalues satisfying *β*(*ρα*)^{−1}=*O*(*σ*). Substituting (3.26) into (2.39*a*) gives(4.1)Therefore, the effective resolved profile *Q* is dominated by *U* in most of the boundary layer, and so has a predominantly wall-jet character. However, as *z*→∞, *U*→0 and *V*→−1, so that somewhere near the top of the main part of the boundary layer both the *U* and *V* components of *Q* will become of the same order, and far enough from the disc *V* eventually dominates. At large *z*, *Q* thus approaches a small constant value with negative real part, giving, overall, a weak reverse flow outside a wall-jet flow. This is the fundamental characteristic of the cross-flow structure that produces the absolute instability in the rotating-disc boundary layer in the inviscid long-wave limit.

This structure is also responsible for the appearance of the ‘ln ln’ terms and inverse powers of the logarithms in (3.21). For long waves with small phase velocity, as here, the critical points lie near where *Q*=0. Therefore, there is one critical point near the wall, and, because of (4.1), there is a second critical point, *z*_{c2}, relatively far from the disc, where *U* and *V* are of the same order of magnitude. It follows that *z*_{c2} can be estimated using the leading order terms in the large *z* expansions for the basic flow (2.20)–(2.22):(4.2)and hence(4.3)Substituting this expression into (3.16) then generates ln ln *σ*^{−1} terms and inverse powers of ln *σ*. Knowledge of the existence and location of multiple critical points is necessary in order to obtain numerical solutions to the Rayleigh equation.

We can also show that the strength of the critical point contribution from *z*_{c2} to the stability characteristics is larger than that due to the critical point at the wall. The phase-jump from the critical point at the wall is given by the −i*π* term in (3.16), with coefficient , while the phase-jump from *z*_{c2}, given by the +i*π* term in (3.16), has coefficient by (4.2) and (4.3). Nonlinearity will therefore first become important near the outer critical point.

There is a further important consequence of (3.26), which is that −Im(*α*)≫Re(*α*) as *σ*→0, and hence the pinch-point becomes asymptotically close to the negative imaginary axis of the complex *α*-plane as *β*/*ρ*→0. The significance of this observation lies in the presence of a branch-cut along this part of the imaginary axis of the complex *α*-plane. The branch-cut is due to the square-root appearing in (3.3) and was placed here to ensure that solutions decay to zero as *z*→∞ in accordance with the outer boundary condition (2.40*b*). We shall now show that this branch-cut must be moved in order for the Fourier inversion contour in the complex *α*-plane to pass between the pinching spatial branches (or, equivalently, for the inversion contour to lie within the valleys of the dominant saddle-point) for sufficiently small azimuthal wavenumbers.

Note that close to a pinch-point the dispersion relation takes the form(4.4)which are the first terms of a Taylor series at the saddle-point, where the subscript *s* denotes a quantity evaluated at the saddle-point (the pinch-point). The spatial branches emanating from the pinch-point are obtained by solving (4.4) for *α* for various *ω* with Im(*ω*)=Im(*ω*_{s}). Consider *ω*=*ω*_{s}+*ϵ*_{1}, where *ϵ*_{1} is small and real, and let *α*=*α*_{s}+*r*_{α} exp(i*θ*_{α}), where *r*_{α} and *θ*_{α} are real, so that *θ*_{α} is the angle of a spatial branch near the pinch-point. Substituting these expressions into (4.4) gives(4.5)and hence the angle *θ*_{α} is minus half the argument of (∂^{2}*ω*/∂*α*^{2})_{s}, with the other branch lying perpendicular to the first one (*ϵ*_{1}<0 produces a rotation by i*π*/2). Substituting (3.14)–(3.16) into(4.6)and then substituting the result for the pinch-point (3.23*d*) gives(4.7)which evaluates to(4.8)

Therefore, the spatial branches at the pinch-point lie at angles ±i*π*/4 to the horizontal, and it may be readily verified that (4.8) implies that for Im(*ω*)>Im(*ω*_{s}), the spatial branches lie above and below the pinch-point. This tells us that the steepest descent path for this saddle-point passes horizontally through the saddle and therefore crosses the imaginary axis onto the Riemann sheet with exponentially growing eigenfunctions. The situation is shown schematically in figure 4.

The branch-cuts produced by the square-root term in (3.3) terminate at branch-points determined by the condition that the argument of the square-root is zero. They lie, therefore, at *α*=±i*β*/*ρ*. Note that the dispersion relation is analytic in the strip −*β*/*ρ*<Im(*α*)<*β*/*ρ* if the branch-cuts radiate away from these points along the imaginary axes and away from the origin. Although, in principle, the inversion contour in the complex *α*-plane can then be placed along the real axis, thus avoiding the branch-cuts and other singularities in the *α*-plane, greater physical insight, and a large-time asymptotic estimate of the physical response, can be obtained by deforming the inversion contour to lie along the steepest descent path of the saddle-point. Figure 4 shows that this would require moving the branch-cut in the lower half-plane away from the imaginary axis to make the saddle-point's valleys accessible to the inversion contour. Moving the branch-cut in this way does not affect the behaviour along the real axis, and so does not affect the physical solution (since the integration could be obtained by integrating along the real axis instead). We therefore conclude that the long-time response has contributions from solutions that diverge exponentially in the wall-normal direction.

However, to demonstrate that the inversion contour does indeed pass through this saddle-point, it is necessary to conduct a global investigation of the whole complex *α*-plane. This is not possible with the present long-wave theory; a separate asymptotic long-wave theory is needed in order to describe the behaviour of the spatial branches in the upper half of the complex *α*-plane, and is presented in Healey (2005), and no long-wave theory can describe the whole complex *α*-plane. Instead, a numerical calculation has been carried out, and is presented in Healey (in press), that confirms that this saddle-point is indeed the pinch-point. The presence of unstable modes with exponentially growing eigenfunctions requires careful interpretation, since they do not satisfy homogeneous boundary conditions. These modes therefore resemble ‘leaky waves’ (see Crighton 1989) and can be understood through study of the associated initial value problem (see Healey in press).

## 5. Conclusions

An inviscid long-wave asymptotic theory has been derived for the saddle-point known to produce absolute instability in the rotating-disc boundary layer. It has been shown that as the scaled azimuthal wavenumber, *β*/*ρ*, tends to zero, the pinch-point in the complex radial wavenumber plane follows the scaling *α*=*O*(*β*/*ρ*)^{3/4}. In this limit the absolute instability is dominated by the radial wall-jet basic flow component, with a small, but essential, cross-flow component that generates a small reverse flow in the freestream. These are the fundamental basic flow characteristics that produce absolute instability in the rotating-disc boundary layer in the inviscid long-wave limit, and corresponds to a resolved basic flow profile with Re(*ϕ*)→0^{+} in figure 2.

Although the resolved basic velocity profiles are inflexional, the instability mechanism for the absolute instability is not due to phase jumps at critical points, but to a form of modal coalescence in which *eight* families of saddle-points interact. Solving for the pinch-point at leading order involves taking a fourth-root of a positive quantity, which gives two real, and two purely imaginary, saddle-points, one of the latter of which is the pinch-point. It has been shown that this imaginary term makes the dominant contribution to the temporal growth rate of the absolute instability, with the phase-jumps at the critical points making a higher-order contribution. An explicit expression for the growth rate of the absolute instability in terms of the basic flow can be obtained by substituting (3.23*d*) into (3.25) and taking the imaginary part of (3.25). Incidentally, the critical point far from the wall (at *z*=*O*(ln[*β*/*ρ*])) makes a larger contribution to the stability than the critical point near the wall (at *z*=*O*(*β*/*ρ*)). Nonlinearity will occur first near the outer critical point in the inviscid theory, and at large enough Reynolds numbers.

The theory also predicts that the pinch-points approach the imaginary axis of the complex *α*-plane as *β*/*ρ*→0. The position and orientation of the pinch-points are such that the inversion contour in the complex *α*-plane is constrained to pass across the imaginary axis, requiring the branch-cuts associated with a square-root term in the solution far from the disc to be moved away from this imaginary axis. The part of the dispersion relation revealed in this way is sometimes called the ‘unphysical’ Riemann sheet, since it comprises waves that grow exponentially with distance from the surface of the disc. However, these waves have been shown to give a useful description of the disturbance evolution in the wall-normal direction in an initial-value problem in the inviscid long-wave limit (see Healey in press) even though these modes themselves do not satisfy the homogeneous outer boundary condition. In that paper it is shown that modes crossing the imaginary axis of the complex *α*-plane in the manner predicted here generate physical behaviour corresponding to a convective spatial instability in the wall-normal direction. This extends the more familiar concept of convective spatial instability in the streamwise direction produced by modes crossing the real axis of the complex *α*-plane.

## Acknowledgments

This work has benefited from helpful discussions with M. Ruderman and V. I. Shrira.

## Footnotes

- Received July 14, 2005.
- Accepted November 30, 2005.

- © 2006 The Royal Society