Global bifurcation to travelling waves with application to narrow gap spherical Couette flow

Physica D 177 (2003) 122–174

Global bifurcation to travelling waves with applicationto narrow gap spherical Couette flow

Derek Harrisa, Andrew P. Bassoma,b, Andrew M. Sowarda,∗a School of Mathematical Sciences, University of Exeter, Exeter EX4 4QE, UK

b School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia

Received 26 November 2001; received in revised form 22 July 2002; accepted 28 September 2002Communicated by F.H. Busse

Abstract

In a previous paper [Physica D 137 (2000) 260], an inhomogeneous complex Landau equation was derived in the context ofthe amplitude modulation of Taylor vortices between two rapidly rotating concentric spheres, which bound a narrow gap andalmost co-rotate about a common axis of symmetry. In this weakly nonlinear regime the latitudinal vortex width is comparableto the gap between the shells. The vortices are located close to the equator and are modulated on a latitudinal length scale largecompared to the gap width but small compared to the shell radius. The system is characterised by two parameters:λ, which isproportional to the Taylor number, andκ, which provides a measure of phase mixing. Only when the inner and outer spheresalmost co-rotate isκ of order unity; otherwiseκ is large. In [Physica D 137 (2000) 260], it was shown that there is a finiteamplitude steady solution branch at fixedκ that connects the first two bifurcation pointsλ0 andλ1. For sufficiently largeκ, thebranch lies onλ0 ≤ λ ≤ λ1, but for smallerκ it extends beyondλ1; there onλ1 < λ < λN a large and small amplitude solutionco-exist and coalesce at the nose of the branch,λN. In this paper we investigate both analytically and numerically the stability ofthe steady solutions and their subsequent evolution. Two types of modes exist—one (SP) preserves the reflectional symmetryof the steady solutions with respect to the equatorial plane, while the other (SB) breaks it. Three SP-global bifurcation scenariosare identified. Each lead to limit cycles, which correspond to vortices drifting towards the equator from both sides. For smallκ,a heteroclinic connection is made between the steady nose solution and its reversed flow state (opposite sign). The same occursfor moderateκ except that two oppositely signed small amplitude steady solutions are connected. For largeκ a homocliniccycle forms joining the undisturbed state to itself and this leads to a gluing bifurcation. This homoclinic cycle evolves fromthe vacillating wave limit cycle shed by an SP-Hopf bifurcation of the large amplitude solution. An SB-pitchfork bifurcationof the steady solutions leads to asymmetric drifting-phase solutions (travelling waves). Time-stepping reveals that they aregenerally unstable and evolve into larger amplitude periodic solutions, which for small and moderateκ are SP-states. For largeκ, the SB-drifting-phase solutions are strongly subcritical. The realised larger amplitude periodic state, to which they evolve,may be either of SB- or SP-type depending on the value ofλ. These complicated solutions consist of trains of stationary pulseseach modulating travelling waves with distinct frequencies. The asymmetric SB-waves correspond to vortices drifting acrossthe equator; yet far from it, where these vortices are very weak, they drift towards the equator as in the case of the SP-waves.© 2002 Elsevier Science B.V. All rights reserved.

Keywords: Couette flow; Taylor vortices; Global bifurcations; Pulse-trains

∗ Corresponding author. Tel.:+44-1392-263976; fax:+44-1392-263997.E-mail addresses: [email protected] (A.P. Bassom), [email protected] (A.M. Soward).

0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0167-2789(02)00709-1

D. Harris et al. / Physica D 177 (2003) 122–174 123

1. Introduction

The motion of incompressible viscous fluid confined between concentric spheres of radiiR1 andR2 (>R1),caused by rotating them both about a common axis with distinct angular velocitiesΩ1 andΩ2, respectively, iscalled spherical Couette flow. Unlike classical Couette flow between concentric cylinders which consists of pureazimuthal flow, a secondary axisymmetric meridional circulation is induced as a consequence of the sphericalgeometry. In most experimental configurations the outer sphere is held at rest (Ω2 = 0) and for that reason many ofthe numerical investigations have also focussed on that case. An important parameter in determining the character ofbifurcations from the basic state is the gap aspect ratioε := (R2−R1)/R1. For the case of the stationary outer sphere,the experiments highlight three parameter regimes, namely the narrow(ε < 0.12), medium(0.12< ε < 0.24) andwide (0.24< ε) gap geometries. For narrow to medium-sized gaps the undisturbed flow first becomes susceptibleto axisymmetric vortices; they are akin to the ubiquitous Taylor vortices which occur in the classical cylindricalgeometry. The vortices form in the vicinity of the equator of the spherical system—in some cases the cells have beenobserved to be symmetric with respect to the equatorial plane while in other circumstances they are asymmetric[2].The numerical results of Marcus and Tuckerman[3,4], who largely focus their attention on the medium gap case,find that even the transitions between the basic state and alternative steady states may be very complicated. Furtherbifurcations may lead to time-dependent states, which may be axisymmetric[5,6] or non-axisymmetric[7]. Theexperimental studies[7–9] for the medium gap case(ε = 0.14) reveal that the non-axisymmetric flow consists ofspiral vortices. These have also been identified in the numerical results at the same gap width[10] and for a narrowgap withε = 0.06 [11].

Wimmer [12,13] reported experimental results for a variety of gap widths over the range 0.0063 ≤ ε ≤ 0.6.His results for the narrow gap limit show that, at the onset of instability, axisymmetric Taylor vortices with theirroughly square cross-section are localised in the vicinity of the equator. Such a configuration may be analysed bymultiple scale asymptotic methods, based on the idea that the structure of each vortex on its short O(εR1) lengthscale is determined by conditions locally whereas its amplitude varies over a longer latitudinal length scale fixedby a higher order theory. Early studies[14–16]assumed that the critical Taylor number is simply a perturbation ofthe value obtained by approximating the equatorial shellular region by infinite cylinders. This gives a condition for‘local instability’ but does not give the correct criterion for ‘global instability’, which determines the critical Taylornumber when the effects of the spatial modulation are correctly accounted for. This distinction between ‘local’ and‘global’ instability is well known in the context of spatially evolving shear flows[17].

There are two important physical ingredients involved in the latitudinal modulation of the vortices. One is that ofboundary curvature, while the other is the secondary meridional circulation present in the basic state (the primaryflow being azimuthal). Both lead to the physical mechanism of phase mixing (a name coined in astrophysicalcontexts[18]), whose nature we explain in more detail below. For the limitε ↓ 0, Soward and Jones[19] obtainedthe true critical (or global) Taylor number and showed that it exceeded the local value. Another example involvingphase mixing, where local theory gives a fallacious answer, concerns the critical Rayleigh number for the onsetof thermal convection in a rapidly rotating sphere. This problem has a long history and only recently[20] has thecorrect asymptotic solution been obtained.

In our previous paper[1] we extended the Soward and Jones[19] linear analysis for axisymmetric Taylor vorticesinto the weakly nonlinear regime by including the Stuart–Landau term as derived by Davey[21]. To appreciatethe nature of the asymptotic method we recap briefly the notation employed in[1]. Natural parameters which arepertinent in the small-ε limit are the Ekman and effective Reynolds numbers

E := ν

R21Ω1 + R2

2Ω2, Ra := ε

ν(R2

1Ω1 − R22Ω2), (1.1a)

124 D. Harris et al. / Physica D 177 (2003) 122–174

whereν denotes the kinematic viscosity of the fluid. These can be combined to construct the alternative independentdimensionless parameters

T := (1 + 12ε)

−3ε2E−1 Ra, δ := ε−1E Ra, (1.1b)

where the latter is related to the ratio of the angular momenta of the two shellsµ := R22Ω2/R

21Ω1 by

δ ≡ (1 − µ)/(1 + µ). The Soward and Jones theory[19] places no restriction on the angular momentum ra-tio µ. However, to construct our amplitude equation we are obliged to restrict attention to the case ofδ 1which means that the angular momentum ratio is close to unity and for instability the spheres need to rotate ex-tremely fast. As we explained in[1] this is not a parameter range easily accessed by experiment. In principle thetheory could be extended to encompass non-axisymmetric modes, which would be relevant, for example, to spi-ral modes. Nevertheless, for O(1) azimuthal wavenumber this complication is only likely to introduce terms ofhigher order than we have retained and so these small asymmetries would not modify our conclusions at leadingorder.

The essential idea is that the realised solution for the departure from the basic flow is localised near the equatorialplane of the shell. In terms of the latitude−θ , its leading order approximation is of separable form with harmonicdependence

a(x, t)F(r)exp(iε−1kcylθ)+ c.c., (1.2a)

whereF(r) is the radial structure of some measurable quantity, the wavenumberkcyl is the local critical value

kcyl ≈ 3.116 (1.2b)

based on the cylinder approximation and c.c. denotes complex conjugate. Here the complex amplitudea is dependenton a suitably scaled timet and stretched length

x := ∆−1/2θ, ∆ := O(ε). (1.2c)

In the small-δ limit, it was shown in[19] thata(x, t) is governed by the amplitude equation

∂a

∂t= (λ+ 2iκx − x2 − |a|2)a + ∂2a

∂x2, (1.3)

where we have included the nonlinear Stuart–Landau term−|a|2a as argued in[1]. Localisation of the solutionin the neighbourhood of the equatorial plane requires that|a| → 0 as|x| → ∞. The scaling which leads to thebalance of the terms−x2a and∂2a/∂x2 in (1.3) fixes the size of the parameter∆, while the further balance withthe term∂a/∂t ties down the time scale: accordingly the results of Soward and Jones[19] determine

∆ ≈ 0.262ε, κ ≈ 0.270δ

ε1/2. (1.4a)

In turn the constantλ provides the parametric representation of the Taylor number

T ≈ Tcyl + 967λε, where Tcyl = 1707.76− 13.0δ2 (1.4b)

is the critical Taylor number based on the cylinder approximation.Whereas the term−x2a causes strong damping as|x| increases, which leads to localisation of the disturbance

in the vicinity of the equator, the role of the term 2iκxa is more subtle. To appreciate its nature, we note that takenin isolation the balance∂a/∂t = 2iκxa has solutions proportional to exp(2iκxt). This means that the temporalfrequency 2κx varies with latitude; a process known as phase mixing[18]. From a different perspective, the spatialstructure can be regarded as evolving with time and characterised by a wavenumber 2κt . The upshot is that the


vortices have a tendency to shift their wavelength away from the local critical cylinder valuekcyl and so inducestabilisation. To reinforce this point, we note that the first bifurcation of the trivial zero amplitude state obtainedfrom solving the linear version of(1.3)gives the steady solution

as(x) ∝ exp(iκx − 12x

2) when λ = κ2 + 1, (1.5)

which by(1.4a)leads to the surprising implication thatλε = O(δ2) independent ofε. It means that the magnitudeof the Taylor vortices is modulated by a Gaussian due to the damping term−x2a, while their wavenumberk isincreased to

k = kcyl + εκ

∆1/2= kcyl + 0.529δ (1.6a)

with

T = Tcyl + 70.8δ2 + 967ε (1.6b)

due to phase mixing. The point stressed in[19] is that neither the critical wavenumber nor the critical Taylor numberreduce to the cylinder values in the limitε → 0 at fixedδ. Indeed phase mixing has decreased the wavelength andincreased the stabilisation as anticipated. In[1] we obtained the steady finite amplitude solutionsas(x) that occurfollowing the bifurcation.

Many experiments are carried out with the outer sphere at rest, for whichδ = 1. To make contact with them atheory applicable toδ = O(1) is required such as the linear theory developed in[19]. Unfortunately, that is notreadily extended into the nonlinear regime, which is why we make our small-δ ansatz. Indeed strictly our expansionprocedure assumes that

δ = O(ε1/2) (1.7)

and so to make any contact withδ = O(1), we require solutions of(1.3)for largeκ. Fortunately, we are able to makeanalytic progress in this regime using techniques developed in[1] and these aspects are described in due course.

In this paper we study the nature of the complex solutionsa(κ; x, t) of (1.3) following the bifurcations from thesteady solutionsas(x) obtained in[1]. (Further results are reported in[22].) Before outlining the direction that ourstudy takes we note four basic symmetries:

(I) Solutions for negativeκ are generated from those for positiveκ through the identity

a(−κ; x, t) = a(κ; −x, t). (1.8)

Throughout this paper we will restrict attention toκ ≥ 0, since non-negativeκ (see(1.4a)) is the signappropriate to our problem. Nevertheless, the symmetry(1.8)permits us to generate solutions for negativeκ

possibly relevant to other physical situations.(II) Solutionsa(x, t) of (1.3) are rotationally invariant in the sense thata(x, t)exp(iϕ) also solves(1.3) for any

real constantϕ, i.e.

a(x, t) ↔ a(x, t)exp(iϕ). (1.9)

Though this appears to be a trivial remark, it has important physical repercussions. Significantly the vortexpattern that(1.5)modulates may be shifted latitudinally at will through appropriate choices ofϕ. Furthermore,since we have not identified what features of the vortices the radial structure functionF(r) in (1.2a)defines,we have no way of linking arga to (say) the vortex boundaries! To emphasise the point, if we make theinterchanges(1.9)andF(r) ↔ F(r)exp(−iϕ) simultaneously, the realised solution is unchanged. Only whenargF(r) directly influences the asymptotics can the vortex boundary location be fixed. This happens at some


higher order of approximation, which involves a more careful specification of the fluid dynamic problem andlies outside the scope of our present study. This lack of determinacy means, for example, that we cannotdistinguish between cases in which the equator is a vortex boundary, a vortex centre or something asymmetricintermediate between the two.

(III) For every solutiona(x, t), the complex reflectiona∗(−x, t) is a solution too, where the asterisk is used todenote the complex conjugate. We say that any solution with the property

a(x, t) = a∗(−x, t) (1.10)

is a symmetry preserving (or SP) solution.After rotation through a suitable angleϕ, as mentioned in (II), the steady solutionsas(x) of (1.3)obtained

in [1] can be cast in a form exhibiting the property thatas(x) = a∗s(−x). Though with (1.2) this implies that

Taylor vortex patterns defined by SP-solutions possess a certain symmetry with reflection in the equatorialplane, the failure of the theory to identify the phase angleϕ means that we cannot locate the cell boundariesas we explained above. Thus the realised solution might be phase shifted and not be symmetric in the usualsense.

(IV) With the above provisos, we say that solutions not linked to(1.10)under phase rotation, i.e. those solutionswith the property

a(x, t)exp(iϕ) = a∗(−x, t)exp(−iϕ) (1.11)

for any choice of constantϕ, are symmetry breaking (or SB) solutions and lack reflectional symmetry in theequatorial plane.

There is important class of SB-solutions which are characterised by a drifting-phaseΩt with constant rotationalfrequencyΩ(= 0). In view of the complex reflection identified in (III), they come in pairs and may be expressedin the form

a+(x, t) := A(Ω; x, t)exp[iΩ(t − t0)], a−(x, t) := A∗(Ω; −x, t)exp[−iΩ(t − t0)], (1.12a)

wheret0 is an arbitrary constant andA(Ω; x, t) satisfies

∂A

∂t= (λ− iΩ + 2iκx − x2 − |A|2)A+ ∂2A

∂x2. (1.12b)

Hocking and Skiepko[15] proposed a simple time-dependent SB-solution of the above type for whichA = Ad(Ω; x)is steady and satisfies

d2Ad

dx2+ (λ− iΩ + 2iκx − x2 − |Ad|2)Ad = 0 (1.13)

with Ad → 0 as|x| → ∞. We find it convenient to choose the time origint0 such thatAd(Ω; 0) = A∗d(Ω; 0) (i.e.

real). However, the presence of the real non-zero constantΩ in (1.13)guarantees thatAd(Ω; x) = A∗d(Ω; −x)

so breaking the symmetry(1.10). The limiting caseΩ → 0 recovers its bifurcation from the SP-steady solu-tion as(x) = Ad(0; x) = A∗

d(0; −x). Since the modesa+(x, t) = Ad(Ω; x)exp[iΩ(t − t0)] and a−(x, t) =A∗

d(Ω; −x)exp[−iΩ(t − t0)] are distinct whenΩ = 0, the bifurcation to them atΩ = 0 from the steady solutionas(x) is a pitchfork.

A clear understanding of the nature of the Taylor vortices determined by the Hocking–Skiepko solutions is vitalfor our physical interpretation of the more complicated time-dependent solutions which we find. To that end weexpressAd in its polar form

Ad(Ω; x) = Ad(Ω; 0)R(Ω; x)exp

[i∫ x

0K(Ω; x′)dx′

], (1.14)


whereR(Ω; x) = R(−Ω; −x) andK(Ω; x) = K(−Ω; −x) are both real but lack the symmetry with respect tothe interchangex ↔ −x at fixedΩ. Whena+ is combined with(1.2a), we see that the modulusAd(Ω; 0)R(Ω; x)determines the magnitude (or strength) of the Taylor vortices, while the argument fixes the local wavenumberε−1kcyl + ∆−1/2K(Ω; x) (in units ofθ ). Locally the magnitude of|K| has little significance for it simply deter-mines small O(ε3/2) variations in the O(ε) width of the vortices. The phase integral

∫ x0 K dx′, however, has an

accumulative effect on the latitudinal length scale∆1/2 = O(ε1/2). This may lead to an O(1) variation of theO(ε−1/2) number of vortices present. Now the propagation velocity of the vortices is−εΩ/kcyl, whereas that ofthe complex amplitudea+ is −∆1/2Ω/K and hence the vortices propagate at a speed O(ε1/2) smaller than thecomplex amplitudea+. Here the relative signs ofK andkcyl is important: for the positiveκ case relevant to theTaylor vortices, we find thatK andkcyl always have the same sign and so the direction of propagation of the complexamplitudea+ is that of the vortices. This is not the situation for negativeκ, for whichK andkcyl then have theopposite sign.

Unlike the steady SP-solutions|as(x)|, which are maximised atx = 0, the amplitude|Ad(x)| of the SB-travellingwaves are maximised elsewhere at, say,xMAX (= 0). Such modes were first identified and found numerically byHocking and Skiepko[15] for the caseκ = 1. Our numerical solutions of the nonlinear eigenvalue problem(1.13)for Ω andAd show that they bifurcate from the steady solution atλ = λHS ≈ 4.3780, as illustrated by the plots ofthe maximum amplitudes|as(0)| andAMAX ≡ |Ad(xMAX )| in Fig. 1; the frequencyΩ increases in concert withλandAMAX . The structure ofAd(x) whenλ = 10 (andΩ ≈ 2.0007) is depicted inFig. 2where the eigenfunctionhas been normalised such that ImAd(0) = 0. This modulated wavea+(x, t) is similar to the steady solutions(1.5)but, unlike them, it lacks any reflectional symmetries aboutx = 0, is maximised atxMAX ≈ 0.8 and propagatesas a travelling wave in the negativex direction; remember thata−(x, t) is its complex reflection inx = 0 and so

Fig. 1. Hocking–Skiepko SB-solution for the caseκ = 1. The continuous curve designates the amplitudeAMAX as a function ofλ while, forreference, the amplitude of the underlying steady solution|as(0)| is indicated by the long-dashed curve. The frequencyΩ is plotted vs.λ by theshort dashed line. Hocking and Skiepko’s numerical results[15] are identified by the solid dots.


Fig. 2. The eigenfunctionAd vs.x for the caseκ = 1, λ = 10 withΩ ≈ 2.0007. The magnitude|Ad| and ReAd (ImAd) are identified bythe continuous and long (short) dashed curves, respectively.

is maximised atxMAX ≈ −0.8 and propagates in the positivex direction. Though all our analytic and numericalresults suggest that the Hocking–Skiepko travelling waves are only stable in limited parameter ranges, their structureis important for our study of the nature of the time-dependent solutions of(1.3).

We outline the organisation of this paper. In order to appreciate the character of the Hocking–Skiepko travellingwaves, we consider their nature forκ 1 in Section 2. Since the steady SP-solutions are of small amplitudein this large-κ limit, the Hocking–Skiepko modes sufficiently close to their bifurcation from them are of smallamplitude too. They have a WKB representation dominated bya± ∝ exp[i(κx ± Ωt) − x2/2] similar to (1.5)above, for whichR ≈ exp(−x2/2) andK ≈ κ in the polar representation(1.14)—for our spherical Couette flowapplication remember that the sign ofκ is that ofkcyl (see(1.2b) and (1.4a)). At a higher order of accuracy the WKBrepresentation captures small reflectional asymmetriesR(Ω; x) = R(Ω; −x) andK(Ω; x) = K(Ω; −x) (similarto those portrayed inFig. 2 for κ = 1). The asymptotic theory outlined inAppendix A is used to show that thelarge-κ Hocking–Skiepko mode is subcritical and without doubt unstable. Nevertheless, since these time-dependentmodes provide the building blocks for the more complicated finite amplitude solutions that are realised numericallyfrom our time-stepping integration, the asymmetries noted turn out to have considerable significance.

A comprehensive stability analysis of the steady solutionsas(x) is performed inSection 3(see alsoAppendix B).This is complicated by the fact that, forκ < κN

1 ≈ 1.55, there exist a pair of finite amplitude solutions over a rangeof λ which terminates when the two solutions coalesce at someλ = λN (say). We refer to our pair of steady formsas the large (small) amplitude upper (lower) branch solutionsaU

s (aLs ) for λ < λN and the nose solutionaN

s as thecoalesced form whenλ = λN. The main conclusions are that with respect to SP-perturbations the large amplitudesolutions are stable forκ ≤ κN

H (≈ 1.05). They undergo a Takens–Bogdanov bifurcation at the nose, whenκ = κNH

and lose stability via a Hopf bifurcation forκ > κNH , whose locationλH retreats from the noseλN along the upper

branch on increasingκ. The SP-Hopf bifurcation is followed by a limit cycle, which corresponds to vacillating flow;


small amplitude, large-κ theory is outlined inAppendix C. In addition, an SB-pitchfork bifurcation occurs via aHocking–Skiepko mode of zero frequency for all values ofκ. The key qualitative features of the small-κ stabilityresults are faithfully illustrated by a low-order system investigated inAppendix D.

In Section 4we explore the time-dependent SP-states that ensue. Forκ ≤ κNhe(≈ 0.85), a heteroclinic connection

is made, whenλ = λN, between two steady nose statesN± : ±aNs of equal magnitude but opposite sign via their

respective saddle node bifurcations. (For some values ofκ this description is an over-simplification as the lowerbranch solution, from which it emerges, may be a saddle-focus rather than a saddle.) The combination forms what wewill call anN-heteroclinic cycle and remark that similar behaviour has been identified by Barkley and Tuckerman[23,24]. The low-order model ofAppendix Dis also used to capture the small-κ nonlinear dynamics includingtheN-heteroclinic cycle; the model and the nature of its predictions have some points in common with[25]. Onincreasingκ the pair of heteroclinic connections move off the nose and link lower branch unstable steady statesL± : ±aL

s instead; we will call the resulting combination, which occurs onκNhe < κ < κG

1 (≈ 1.07)anL-heterocliniccycle. Whenκ reachesκG

1 , the amplitude|aLs | of the steady state making the heteroclinic connection collapses to

zero. Thereafter, whenκ > κG1 , two oppositely signedZ-homoclinic cycles are glued at the zero stateZ : a = 0 and,

for sufficiently largeκ, this gluing bifurcation is subcritical. All these transitions to limit cycles correspond to globalbifurcations. They are accompanied by a change of character of the Taylor vortices from stationary flow to travellingwaves associated with the drifting of the vortices towards the equator symmetrically from both sides. At first sight,we might have expected an equal distribution of left and right travelling waves appropriate to an SP-solution to leadto standing waves, which would have been the outcome in a spatially homogeneous situation. That does not happenbecause of the spatial asymmetry of travelling waves. Instead the incoming wave dominates leading to the travellingwave character of the solution. For sufficiently largeκ there is an important preliminary Hopf bifurcation of thestationary state to vacillating solutions. They describe Taylor vortices, whose cell width pulsates or ‘breathes’. Onvaryingλ the pulsation period lengthens and when infinite leads to the aforementioned glued homoclinic cycles.We remark that we have not constructed low-order models, like that ofAppendix D, to capture the nonlineardynamics which occurs whenκ is not small. Nevertheless, in the context of a thermal convection problem Siggers[26] has derived a low-order system from a weakly nonlinear truncation and found that the solutions exhibit similarbehaviour to all the various SP-scenarios which we have identified from the solutions of our partial differentialequation.

Higher order bifurcations are examined inSection 5. In Section 5.1we show that the SP-solutions acquire asecond resonant frequency which appears to be adequately explained through the notion of Arnold tongues. Guidedby these results, we discuss inSection 5.2the possible mathematical structure of SB-solutions which evolve fromunstable Hocking–Skiepko states. Numerical simulations of this evolution were reported in the first author’s thesis[22]. At small to moderateκ the final forms were generally SP-solutions while for largerκ SB-solutions were oftenreached that exhibited complicated but generally periodic temporal behaviour. These may occur subcritically andat values ofλ below which any SP-solutions can exist. InSection 5.3we consider in detail the caseκ = 4, whichappears to be just sufficient to capture the large-κ behaviour. Interestingly the numerical results show evidence ofpulse-trains—by this we mean that pulses akin to the Hocking–Skiepko modes are spatially localised by phasemixing about points determined by their frequencies and together roughly fill the region in which such modes areunstable by local theory. The location and frequency of each pulse increases monotonically in concert. In view ofthe importance of the large-κ limit, we believe that this strong subcriticality and the related complexity of the flowmay have far-reaching implications.

One natural question that arises from our studies is how do the results relate to physical phenomena that may berealisable? In view of the complicated picture that we have summarised above, we leave this interpretation until theconcluding section after we have presented our detailed findings. There we also comment further on the significanceof the results in both experimental and numerical contexts.


2. Asymptotic nature of the Hocking–Skiepko modes for large κ

Harris et al.[1] showed that, for fixedκ, the trivial zero amplitude solution bifurcates to the steady finite amplitudesolutionsas(x) at λ = λn := 1 + 2n + κ2 for n = 0,1,2, . . . , and these solutions exist on the lobes connectingconsecutive pairs ofλn. When|κ| 1, the finite amplitude solution on the first lobeλ0 < λ < λ1 is very smalland can be approximated uniformly by its linear WKB-solution

as(x)

as(0)≈ w0(ζ ) :=

(iζ

κ

)αexp

[−1

2(κ2 + ζ 2)

], (2.1a)

where

ζ := x − iκ, (2.1b)

α := 12(λ− λ0). (2.1c)

The nonlinearity is a higher order effect which does not influence the above first asymptotic approximation but itdoes ultimately determine the small amplitude|as(0)|, which was isolated by the asymptotic theory of Harris et al.[1]. That theory is encompassed by the development here inAppendix A. There,(A.4b) and (A.5)shows that themaximum value of|as(0)| on the lobeλ0 < λ < λ1 is attained atλ = λmax := λ0 + 2αmax, where

αmax ≈ 1

2 ln(√

3|κ|/2)− (1 − γ ), γ = 0.57721. . . (2.2)

small or, more precisely, O(1/ ln |κ|).Provided that the two Hocking–Skiepko modes, which bifurcate from the steady solution(2.1a), remain small,

they too have the similar linear WKB-solution

Ad(Ω; x)Ad(Ω; 0)

≈ W0(ζ ) :=(

iζ

κ

)α−iΩ/2

exp

[−1

2(κ2 + ζ 2)

]. (2.3)

On substitution into(1.12a), this determines two distinct solutions

a± ≈ Ad(0)R±(x)exp[iΦ±(x, t)], Ad(0) ≡ Ad(±Ω; 0) (2.4a)

related by the symmetries of(1.12a), where in the notation of(1.14)

R± ≡ R(±Ω; x) ≈(

1 +(xκ

)2)α/2

exp

[−1

2x2 ± 1

2Ω tan−1

(xκ

)], (2.4b)

Φ± ≈[κx + α tan−1

(xκ

)]±Ω

[(t − t0)− 1

4ln

(1 +

(xκ

)2)]

(2.4c)

with the local wavenumber given by

K± ≡ K(±Ω; x) = ∂Φ±∂x

≈ κ + κα ∓ 12Ωx

κ2 + x2. (2.4d)

The steady solutionas(x) = as(0)R exp(iΦ) is recovered whenΩ = 0. Its wave-like character is exhibited by asymmetric wavenumberK(0; −x) = K(0; x) := K±|Ω=0, while its symmetric envelopeR(0; −x) = R(0; x) :=R±|Ω=0 is strongly damped at a great distance by the exponential.

The time-dependent Hocking–Skiepko solutionsa± defined by (2.4) have some interesting features. Unlike thesteady solution, the drifting-phaseΩt leads to waves which travel with phase speed given correct to lowest order by


∓Ω/κ. (Though, of course, the drifting velocity of the vortices measured in the same units is±εΩ/(∆1/2kcyl+εκ).)Significantly, the wave envelope is stationary and asymmetric:

R±(x)R±(−x) ≈ exp

[±Ω tan−1

(xκ

)](2.5)

with its amplitudeR± maximised atxMAX ≈ ±Ω/2κ. This means that the amplitude of a wave approaching theoriginx = 0, at a given distance|x| from it, is larger (and has longer wavelength, see(2.4d)) than the correspondingoutgoing wave at the same distance. The larger amplitude is clearly linked to the smaller dissipation resulting fromthe longer wavelength and, despite being properties derived under the assumption of largeκ, these asymmetricfeatures are evident even for the caseκ = 1, λ = 10 illustrated inFig. 2. Exactly the same amplitude asymmetry(see(D.6a)) is predicted by the low-order model relevant to the small-κ case outlined inAppendix D.

Returning to the large-κ limit we determine inAppendix A the amplitude|Ad(0)| (close toAMAX , since|xMAX | 1) and frequencyΩ as the solution of(A.3). According to (A.4) the pitchfork bifurcation (Ω = 0)to the Hocking–Skiepko modesa± occurs at

αHS ≈ 1

2 ln(|κ|/√3)− (1 − γ ), (2.6)

just after the maximum (αHS > αmax, cf. (2.2)) of the steady solutionsas. Close to the bifurcation, whereα andΩare O(1/ ln |κ|), the solution of(A.3) is

|Ad(0)|2 ≈√α2 + 1

4Ω2

(4√3|κ|

)2α √8 exp

(−1

2|κ|2

)(2.7a)

provided that

α ≈ 1

2Ω cot

[Ω ln

( |κ|2√

3

)]. (2.7b)

(The corresponding steady solution|as(0)| for small-α is recovered by settingΩ = 0 in (2.7a).) Interestingly thepitchfork bifurcation is subcritical, because as|Ω| increases soα decreases—indeed on the local scaling, it decreasesindefinitely to−∞. The subcritical bifurcation is illustrated inFig. 3 for κ = 4 and here|Ad(0)| is plotted ratherthanAMAX (cf. Fig. 1) since it is thex = 0 values that we use to measure the amplitude of our other time-dependentsolutions in later sections. The subcriticality should be contrasted with the supercritical bifurcation illustrated inFig. 1for the caseκ = 1 and predicted (see(D.7)) by the small-κ low-order model ofAppendix D.

Before continuing we point out that for givenα the solutionΩ of (2.7b) is multi-valued with correspondingmultiple solution branches. We have only described the first branch but believe that the others also provide genuinesolutions. We speculate that they might originate from bifurcations from the steady state solutions on lobes con-nectingλ2m to λ2m+1, wherem = 1,2,3, . . . . If so, each lobe generates an unstable Hocking–Skiepko solutionbranch, which (2.7) identifies in the neighbourhood ofλ = λ0.

3. Stability analysis for κ = 0

We consider small perturbations to the steady solutionas(x) of the form

a(x, t)− as(x) = a(x)exp[p(t − t0)] + a∗(x)exp[p∗(t − t0)], (3.1a)

where

p := σ + iω (3.1b)


Fig. 3. As inFig. 1except that hereκ = 4 and that the amplitude of the Hocking–Skiepko solution is taken as|Ad(0)| rather thanAMAX . Theminimum value ofλ for a solution isλHS

min ≈ 16.5486.

is the constant complex growth rate andt0 is an arbitrary real constant. Whenω = 0, the complex functionsa anda∗ satisfy

L(p; a, a∗) = 0 (3.2a)

and

L(p∗; a∗, a) = 0, (3.2b)

where the differential operatorL is defined by

L(p; a, b) ≡ ∂2a

∂x2+ (λ− p + 2iκx − x2 − 2|as|2)a − a2

s b∗. (3.2c)

On taking the complex conjugate of(3.2a) and (3.2b), it is readily seen that symmetry preserving and symmetrybreaking solutions are possible. Recalling thatas has the SP-symmetryas(x) = a∗

s(−x), we may identify SP- andSB-solutions with the properties

a∗(x) = a∗(−x) (3.3a)

and

a∗(x) = −a∗(−x) (3.3b)

respectively. Such SP-modes satisfy

a∗(0) = a∗(0), (3.4a)

da∗dx

(0) = −da∗

dx(0), (3.4b)


while the SB-modes satisfy

a∗(0) = −a∗(0), (3.5a)

da∗dx

(0) = da∗

dx(0). (3.5b)

We remark that for the special caseω = 0 the second terma∗(x)exp[p∗(t − t0)] in (3.1a)may be ignored and thesingle equationL(p; a, a) = 0 solved rather than the pair(3.2a) and (3.2b). Further, the subscript asterisks in thesymmetry properties (3.3)–(3.5) can then be dropped.

Given a finite amplitude solutionas(x), there is a set of eigenvaluesp with associated eigenfunctionsa(x). Onlythe mode with the largest growth rate is relevant for determining instabilities. Nevertheless, some of the higher ordermodes are portrayed inFigs. 5–8to help clarify the nature of the spectrum.

Some analytic progress is possible on two fronts. First, inAppendix D, we consider a low-order model whichcaptures the key features of the small-κ behaviour. Secondly, inSection 3.1a perturbation theory is developed that isapplicable for allκ when the steady solution has small amplitude (|as| 1) close to the bifurcations atλ = λn. Forlargeκ, however,|as| is small for allλ. Accordingly, in this limit we are able to use our perturbation technique to con-siderable effect inSection 3.2to determine where the steady state loses stability. Indeed we go further inSection 3.5and analyse the weakly nonlinear vacillating solution which follows the SP-Hopf bifurcation isolated inSection 3.2.

Numerical results for SP- and SB-modes, obtained using the AUTO package[27], are described inSections 3.3and 3.4, respectively. In view of the symmetry conditions, system (3.2) is solved on 0< x < ∞ subject to theboundary conditions atx = 0 of either (3.4) for SP-modes or (3.5) for SB-modes. To account for the amplitudedegeneracy of our linear system, we normalise our solutions by the additional requirementsa(0) = 1 for SP-modesanda(0) = i for SB-modes. Boundedness at infinity is met by the requirement that

a → 0 and a∗ → 0 as x ↑ ∞. (3.6)

Despite the apparent simplicity of the linear equations (3.2), when decomposed into their real and imaginary parts,they constitute four coupled second order differential equations for the four unknowns Rea, Ima, Rea∗ andIma∗. The system was solved on the finite range 0< x < x∞ with the four boundary conditionsa(x∞) =a∗(x∞) = 0 and the six stated conditions atx = 0. That is sufficient to determine the eigenfunctions and thecomplex eigenvaluep = σ + iω, whenω = 0. Some simplifications occur whenω = 0, as the order of oursystem for the modified eigenvalue problem described above is halved. Of course, throughoutx∞ was chosen to besufficiently large to ensure that proper convergence had been achieved.

Attention is largely restricted to the first finite amplitude lobe which bifurcates from the trivial zero amplitudestate atλ = λ0 = 1+ κ2 and returns to it atλ = λ1 = 3+ κ2. The lobe lies entirely within the rangeλ0 ≤ λ ≤ λ1

when|κ| ≥ κN1 (see, e.g.,Fig. 5(c)), where

κN1 :=

√1 +

√2 = 1.5538. . . (3.7)

(see Eq. (3.4a) of[1] with λ1 = 0), but extends beyondλ1 (see, e.g.,Figs. 7(c) and 8(c)), when|κ| < κN1 , up toλN

(say) which we call the nose of the lobe. In the (λ, κ)-plane, the curveλ = λN(κ) terminates at(κN1 , λ

N1 ), where

λN1 := λN(κ

N1 ) = λ0 + 2 ≈ 5.4142. InFig. 4we plotλN − λ0 vs.κ; it is the curve labelled N that exists forκ < κN

1and cuts out at the point denoted N1.

3.1. Small amplitude theory: |as| 1

Whenever the magnitude of the steady finite amplitude solutionas is small as it is, for allκ, sufficiently close tothe bifurcation pointsλ = λ0, λ = λ1, we may employ its linear approximation(2.1a). In the same spirit, our small


Fig. 4. A partition of theκ, (λ − λ0)-plane for SP-disturbances. The nose and Hopf curvesλ = λN(κ) andλ = λH(κ) (see below(3.7) andtowards the end ofSection 3.3) are identified by the continuous lines, which are labelled N and H respectively. The gluing bifurcation curveλ = λG(κ) (seeSection 4.1) is the broken line G. The homoclinic cycle curveλ = λho(κ) (seeSection 4.1) is the short schematic broken lineconnecting the solid dots NH and G1. The heteroclinic cycle curveλ = λhe(κ) (seeSection 4.2) is the broken line he. The various end points atκN

he(≈ 0.85), κNH (≈ 1.05), κG

1 (≈ 1.07) andκN1 (≈ 1.55) are identified by the solid dots labelled Nhe, NH, G1 and N1 respectively.

perturbations (3.1) to it may be cast in the form

a

a(0)≈ wp(ζ ) :=

(iζ

κ

)α−p/2exp

[−1

2(κ2 + ζ 2)

](3.8)

similar to(2.3)and inAppendix Bwe outline the theory which determines the complex growth ratep.In the case of SP-modes, on linearising the dispersion relation(B.2) nearα = 0 and using (B.3), we obtain

p ≈

−2(λ− λ0)

for 0 ≤ λ− λ0 1.−2 + 1

2(5κ2 + 1)(λ− λ0)

(3.9)

These are the growth rates of the first two modes at the beginning of the lobe. A similar calculation nearα = 1yields the corresponding growth rate at the end of the lobes

p ≈

−2(λ− λ1)

for |λ− λ1| 1.

2 + 3κ4 + 8κ2 − 1

3(κ4 − 2κ2 − 1)(λ− λ1)

(3.10)

Note that strictlyλ− λ1 ≤ 0, when|κ| > κN1 andλ− λ1 ≥ 0, when 0≤ |κ| < κN

1 , appropriate to theλ-range overwhich the finite amplitude solution exists (see(3.7)).

In the case of SB-modes, there is always a trivialp = 0 solution, which corresponds to a rotational phase shiftϕ of the steady solution (as(x) → as(x)exp(iϕ), see(1.9)). In addition to that we have as above another mode


with properties

p ≈ −2 − 12(κ

2 + 1)(λ− λ0) for 0 ≤ λ− λ0 1, (3.11)

and

p ≈ 2 + 3κ4 + 4κ2 + 3

3(κ4 − 2κ2 − 1)(λ− λ1) for |λ− λ1| 1. (3.12)

3.2. Large-κ asymptotics: linear theory

Whenκ is large the finite amplitude steady solutionas remains small independent of its proximity to the bifurcationpointsλ0 andλ1. This permits further reduction of the dispersion relation(B.2) to the form(B.4). Indeed most ofthe interesting behaviour occurs close toλ0, where the solution of(B.4) is given by(B.5) and, accordingly, we findit convenient to rescaleα andp so that

α := 2α ln

( |κ|2

), (3.13a)

p := p ln

( |κ|2

). (3.13b)

Solutions for SP-modes correspond to the upper plus sign in(B.5), which reduces under the small-p approximation(p = O(1)) to

α = p

1 − 3 e−p . (3.14)

The eigenvaluesp double up when dα/dp = 0 which happens atp = pc− (< 0) andp = pc+ (> 0), defined asthe negative and positive solutions of

epc± = 3(1 + pc±). (3.15)

The corresponding values ofα are thenαc± = 1 + pc± and the resulting topological structure of the solutions isexactly the same as that for our numerical results for theκ = 2 case portrayed inFig. 5(a). One real branch withp ≤ 0begins at(α, p) = (0,0), ends at(α, p) = (0,−2) and maximisesα atαc− := αc−/[2 ln(|κ|/2)]. The other realbranch withp ≥ 0 begins at(α, p) = (1,2), ends at(α, p) = (1,0) and minimisesα atαc+ := αc+/[2 ln(|κ|/2)].Forα = O(1) the solution is described by the asymptotesα = p andp = ln 3 of formula(3.14)for largeα, whichdeterminep ≈ 2α and 0, respectively. Over the rangeαc− < α < αc+, the growth rate is complex,p = σ + iωwith

α = ω cotω + σ and eσ = 3

[cosω +

(sinω

ω

)σ

]. (3.16)

From this we deduce that there is a SP-Hopf bifurcation withσ = 0 and

ω = ωH := cos−1 13 ≈ 1.2310 (3.17a)

at

α = αH := 1

2√

2cos−1 1

3 ≈ 0.4352. (3.17b)

The SB-modes (corresponding to the lower minus-sign in(B.4)) have a similar topological structure but do exhibitsome noteworthy differences. There are two modes—one is the trivialp = 0 solution which simply corresponds to


Fig. 5. The stability characteristics of the steady finite amplitude solutionas(x) for the caseκ = 2. The real (σ ) and imaginary (ω) parts ofp are plotted vs.λ − λ0 betweenλ = λ0 = 5 andλ1 = 7; they are identified by continuous and broken lines, respectively. The SP-Hopfand SB-Hocking–Skiepko bifurcations are represented by the solid dots and labelled H or HS as appropriate: (a) symmetry preserving modes:p ≡ pSP; (b) symmetry breaking modes:p ≡ pSB; (c) the steady solution amplitudes|as(0)|.

the rotational phase shift mentioned above. The other non-trivial solution is determined for smallα andp by (B.5),which gives

α = p

1 − e−p . (3.18)

This shows thatp increases monotonically and vanishes at

α = αHS := 1, (3.19)

which is precisely the SB-pitchfork bifurcation to finite amplitude Hocking–Skiepko modes discussed inSection 2.To establish the equivalence of(A.3) with Ω = ω and(B.4) with p = iω we need to appreciate in(B.4) that2− (

√3)−p ∼ (

√3)p asp → 0. We remark that a better approximation toαHS follows quickly from(2.6)and that,

for p 1, (3.18)has the asymptotic behaviourp = α. This determines the linear increase ofp (= 2α) from zeroto 2 onαHS α ≤ 1. In the opposite limit−p 1, relation(3.18)admits the asymptotic behaviourα = −p ep

which determines the rapid decrease ofp towards−2 atα = 0.


That said, however, it should be appreciated that the approximation(3.18)is slightly misleading in the limitα ↓ 0.From(B.5) we see that the appropriate refinement of(3.11)is

κ2α ≈ −(p + 2)(12|κ|)p+2 for 0 < −(p + 2) 1. (3.20)

The value ofα determined by(3.20) is maximised for realp at pc− ≈ −2 − 1/ ln(|κ|/2). This fixes the valueαc− := α(pc−) at which the two roots forp double up and become complex forp > pc−, as illustrated inFig. 5(b). The roots remain complex until the local minimumαc+ := α(pc+) (< αHS) of (B.5) is reached atpc+ ≈ −2( ln 2/ ln 3)+ 1/ ln(|κ|/2), just above the value of the zero of 2− (

√3)−p.

Given these analyses of the large-κ structures for both SP- and SB-modes, we are able to make some helpfulcomparisons. Significantly, the SP-Hopf bifurcation occurs atαH ≈ 0.4352 (from(3.17b)) prior to the SB-pitchforkbifurcation atαHS = 1 (see(3.19)). Such behaviour is seen inFig. 5(c) relating to a comparatively modest value ofκ = 2. Notice also that the valuesαc± at which the real branches of the SP-disturbances cut out (see below(3.15))sandwichαH—once more this prediction is supported by the numerical evidence inFig. 5(a).

3.3. Symmetry preserving modes including the Hopf bifurcation

A comprehensive stability study involves the exploration of the entire (λ, κ)-parameter space. We tackle the taskby restricting attention to various fixed values ofκ and studying the stability characteristics that arise on varyingλ.An extremely rich structure is revealed and plausible transitions between the selectedκ, which illustrate the mainfeatures, are outlined.

We begin by noting that we can categorise the eigenfunctions for the perturbations to the finite amplitude steadysolutionsas(x) by pairing them with the eigenfunctions of the perturbations to the trivial zero amplitude solutionat the bifurcation pointsλ = λn := 1 + 2n + κ2 (n = 0,1,2, . . . ). The complex growth rates, whenas(x) issmall in the neighbourhood of these points, can be determined by perturbation theory as explained inSection 3.1.In Figs. 5(a)–8(a), we plot the real and imaginary partsσ andω of the principal complex eigenvaluesp computednumerically. The slopes of the tangents to the real growth ratesp = σ atλ = λ0 andλ1 are correctly predicted by(3.9) and (3.10).

The topological features of the large-κ behaviour of the first two eigenvalues, predicted inSection 3.2, is ex-emplified by the caseκ = 2 in Fig. 5(a) over the rangeλ0 (= 5) < λ < λ1 (= 7). The eigenvalues form acomplex conjugate pair throughoutλc− (≈ 5.1374) ≤ λ ≤ λc+ (≈ 6.7854) and a Hopf bifurcation occurs atλ = λH ≈ 5.7146, whereσ = 0 andω ≈ 1.15. This relatively straightforward situation persists while the steadystate solutionsas exist only onλ0 < λ < λ1 (so in particular do not overshoot pastλ1)—by our earlier result(3.7)this is guaranteed for|κ| > κN

1 ≈ 1.5538.When|κ| < κN

1 the finite amplitude solutions extend beyondλ1 up to the nose locationλN(κ). Onλ1 < λ < λN

there are two steady solutions: one large (small) amplitude solutionaUs (aL

s ) on the upper (lower) solution branch.We denote the complex growth rates for the modes associated with the upper (lower) branch bypU (pL). ThesolutionsaU

s andaLs double up and take the valueaN

s (say) at the noseλ = λN of the steady solution branches, asillustrated inFigs. 7(c) and 8(c); there, by necessity, one of the complex conjugate growth ratespN vanishes.

Figs. 7(a) and 8(a)illustrate the nature of the|κ| < κN1 growth rate structure. The transitions on the figures between

the upper and lower branch growth rates occur asp passes its various nose valuespN. When the nose has just formedκN

1 − |κ| 1, the first two nose growth ratespN1 andpN

2 are real with 0< 2 − pN1 1 andpN

2 = 0; while onλ1 < λ < λN, we have 0< pN

1 −pU1 1, 0< pU

2 1 andpN1 < pL

1 < 2, 0≤ −pL2 1. This configuration near

the nose is partially illustrated inFig. 6(a). (We say ‘partially’ because withκ = 1.35 the double root (pU1 = pU

2 ) atλc+ is visible having advanced beyondλ1.) Elsewhere, onλ0 < λ < λ1 the picture is much the same as it was beforethe nose formed (seeFig. 5(a)). As|κ| is decreased further,pN

1 decreases towardspN2 = 0, while simultaneously the


Fig. 6. The complex growth ratepSP for SP-modes at values ofκ < κN1 so that the steady solution lobe possesses a nose which extends beyond

λ = λ1. Real and imaginary parts of the growth rates are shown by continuous and broken lines, respectively, are plotted vs.λ − λ1 and arelabelled as U or L depending upon whether they are associated with the upper or lower branch of the steady solution. The point where they meetat the noseλ = λN is labelled either N (forp = 0) or SN (forp = 0): (a) the caseκ = 1.35 withλ1 = 4.8225 andλN ≈ 4.8693; (b) the caseκ = 1.05 ≈ κN

H with λ1 = 4.1025 andλN ≈ 4.4299≈ λNH.


Fig. 7. As inFig. 5together with the notation ofFig. 6, but hereκ = 1 with λ0 = 2, λ1 = 4 andλN ≈ 4.4089.

λc+ advances towardsλN. When it is reached at|κ| = κNH ≈ 1.05, we havepN

1 = pN2 = 0 andλc+ = λH = λN =

λNH ≈ 4.43. The coincidence here of the upper branch Hopf and the lower branch saddle, as illustrated inFig. 6(b),

marks a crucial transition linked to the occurrence of a Takens–Bogdanov bifurcation (see[28]). Importantly for|κ| > κN

H the upper branch is unstable between the Hopf bifurcation atλH and the nose atλN. The curveλ = λH(κ)

terminates at(κNH , λ

NH), whereλN

H = λH(κNH ) ≈ 4.43. InFig. 4we plotλH − λ0 vs.κ; it is the curve labelled H.

For |κ| ≤ κNH the upper branch is completely stable, while the lower branch is always unstable. These features

are captured by the low-order model considered inAppendix D(see(D.3a)). As |κ| decreases belowκNH , we keep

pN1 = 0, while pN

2 decreases to negative values. The transition is illustrated inFig. 6 and the new topology isillustrated forκ = 1 in Fig. 7(a). The behaviour on decreasing|κ| to smaller values is illustrated inFig. 8(a) forκ = 0.5. Curiously at this value ofκ, though the leading nose eigenvaluepN

1 is real and positive atλN ≈ 6.1274,the following two come as a complex conjugate pairpN

2 ≈ −2.106±1.005i with negative real part. This means thatthe lower branch solutionL± in the neighbourhood of the nose has the character of a saddle-focus with implicationsfor the heteroclinic cycles which we discuss inSection 4.2.

3.4. Symmetry breaking modes including the pitchfork bifurcation

As we explained inSection 3.1, the rotational degeneracy of the finite amplitude stateas(x)means that there is al-ways a trivialp = 0 SB-mode of disturbance but we do not portray it in our figures. Its existence is important because


Fig. 8. As inFig. 7, but hereκ = 0.5 with λ0 = 1.25,λ1 = 3.25 andλN ≈ 6.1274.

the SB-bifurcation is by necessity associated with a double zero of the growth ratep. It corresponds to a pitchforkbifurcation into two distinct Hocking–Skiepko modes with frequencies of equal magnitude but of opposite sign.

The topological features of the large-κ behaviour predicted inSection 3.2is exemplified by the caseκ = 2illustrated inFig. 5(b). Here the eigenvalues form a complex conjugate pair over the rangeλc− (≈ 5.0766) ≤ λ ≤λc+ (≈ 6.2373) and the Hocking–Skiepko pitchfork bifurcation occurs atλHS ≈ 6.3468 beyondλc+ in thep realrange. For|κ| 1, the complex conjugate pair are compressed into a small region nearλ = λ0, where Rep isnegative and increases from−2 to−2 ln 2/ ln 3.

As κ is decreased from large values, the Hocking–Skiepko pitchfork bifurcationλHS advances along the upperbranch (see parts (b) and (c) ofFigs. 5 and 7) and reaches the nose atκ = κN

HS ≈ 0.82. Forκ < κNHS the upper branch

is stable to SB-disturbances and asκ is decreased belowκNHS so the bifurcationλHS retreats along the lower branch

(seeFig. 8(b) and (c)) leaving it stable to SB-disturbances onλHS ≤ λ ≤ λN, a property which is also captured bythe low-order model considered inAppendix D(see(D.3b)).

In the large-κ limit the asymptotic results(3.17b) and (3.19)show thatλHS > λH—a feature still evident forκas small as 2 (seeFig. 5(c)). Asκ is decreased andλHS advances along the upper branch we find thatλH catchesit up atκ = κH

HS ≈ 1.10, whereλHS = λH. As κ is decreased furtherλH overtakesλHS and reaches the nose first,whenκ = κN

H ≈ 1.05. The SB-Hocking–Skiepko bifurcation continues along the upper branch (see, e.g.,Fig. 7(c))reaching the nose whenκ = κN

HS ≈ 0.82.


3.5. Large-κ asymptotics: weakly nonlinear theory

The stability theory developed inSection 3.2valid for largeκ may be adapted to derive a nonlinear equationgoverning the evolution of small amplitude solutions following the SP-Hopf and the SB-pitchfork bifurcations. Tobegin, we note that though(B.4) has a complicated dependence on the growth ratep, the treatment of the nonlinearterms is insensitive to the small value ofp in the large-κ asymptotic reduction leading to(3.14) and (3.18). Bycarefully incorporating the nonlinear terms (see also(A.1)–(A.3)), it is possible to construct from(B.5)the amplitudeequation (C.1), which is equivalent to the delay equation(

d

dτ− α

)a(τ + 1) = −|a(τ )|2a(τ ), (3.21a)

where

a(τ ) := 2−1/4 eα/2√

ln

( |κ|2

)exp

(1

4κ2)a(0, t), τ :=

[ln

( |κ|2

)]−1

(t − t0), (3.21b)

andt0 is an arbitrary time origin chosen at our convenience. The amplitude ofa(0, t) has been normalised so thatthe steady solutionas(0) becomesas = √

α. Likewise the Hocking–Skiepko solutionAd(0)exp(iΩt) becomesAd exp(iΩτ ), where

|Ad |2 =√α2 + Ω2, α = Ω cotΩ (3.22)

is equivalent at lowest order to (2.7).Our main application of (3.21) is to determine the nature of the SP-vacillating solution that emerges from the

SP-Hopf bifurcation atα = αH (see (3.17)). The results so obtained exhibit key features that continue to apply atmore moderateκ in the fully nonlinear regime discussed inSection 4. In the immediate vicinity ofαH, we may seekSP-periodic solutions with harmonic expansion

a(τ ) = a0 + 12[∆a0 eiωτ +∆2a2 e2iωτ + c.c.] + O(∆3), 0 ≤ ∆ 1, (3.23)

wherea0 =√αH + O(∆2) is a real constant and∆ is a small real non-negative expansion parameter. The weakly

nonlinear theory outlined inAppendix Cshows that

α = αH +∆2α2, (3.24a)

ω = ωH +∆2ω2, (3.24b)

where

α2 = 119(27αH − 5), (3.24c)

ω2 = −1019

√2. (3.24d)

With αH ≈ 0.4352 (see(3.17b)), it is clear from(3.24c) and (3.24d)that α2 > 0 andω2 < 0. This means that ourSP-Hopf bifurcation, which is the first instability to occur asλ is increased, is supercritical in the large-κ limit.

The finite amplitude solution constructed from the first harmonic proportional to exp(iωτ ) and its complexconjugate corresponds to the superposition of two Hocking–Skiepko travelling wavesa±(x, t) defined by(1.12a).The substitution of (2.3) and (2.4) withΩ = ω into (3.1) yields the vacillating solution

a ≈ as(x)

1 +∆ cosh

[1

2ω tan−1

(xκ

)+ iω

(t − 1

4ln

((xκ

)2 + 1

))]. (3.25)


Consider the second term proportional to∆. It defines two travelling waves whose superposition might be expectedto lead to a standing wave; this view is over-simplistic due to the asymmetry noted inSections 1 and 2. Indeed theinward propagating wave at large|x/κ| is greater than the outward propagating wave by a factor exp(|ω|π) (see(2.5)). This idea is generic to the entire problem and is not limited to large-κ or small amplitude theory—it has farwider implications as we will see. Note also that the dominance of the inward propagating wave is independentof the sign ofκ although a word of caution is appropriate. The direction of propagation of the vortex structuresthemselves would reverse ifκ andkcyl happened to take opposite signs, which according to the result (1.6a) of[19]is not the case. Now, since the second term proportional to∆ is small,(3.25)is dominated by the leading steadycontributionas(x). As a consequence, the travelling wave effect only causes the solution to pulsate and that is whywe refer to(3.25)as the vacillating solution.

The temporal behaviour atx = 0 of this vacillating mode(3.25), namely

a(0, t)

as(0)= 1 +∆ cosωt, (3.26a)

∂a/∂x(0, t)

ias(0)=(ακ

+ κ)(1 +∆ cosωt)+∆

ω

2κsinωt, (3.26b)

is also of some interest in relation to the phase portraits of Rea(0, t) vs. Im∂a/∂x(0, t) plotted inFigs. 11(c), 13and 14(b). Our large-κ weakly nonlinear result (3.26) describes a small elliptical limit cycle with its origin at eitherof the steady state solutions, marked on the figures, which is followed in an anti-clockwise sense. The major axis ofthe ellipse is almost aligned to the vector from the origin to the stationary solution, while its ellipticity linked to theterm proportional to sinωt in (3.26b)is very small for|κ| 1. Evidently, those figures show more fully developednonlinear behaviour. Nevertheless, all trajectories investigated, irrespective of the value ofκ, were found to exhibitan anti-clockwise sense of rotation, which significantly appears to be a very robust feature.

4. Symmetry preserved periodic solutions

In this section we restrict attention to SP-solutions satisfyinga(x, t) = a∗(−x, t). On the one hand, we solvedinitial value problems on the semi-infinite interval 0< x < ∞ subject to the symmetry conditions

Ima(0, t) = 0, Re

∂a

∂x(0, t)

= 0, (4.1)

and determined the periodic solutions to which they approached after a long time. To that end, a finite-differencescheme was adopted to solve the system of nonlinear parabolic partial differential equations, which was implementedusing the numerical algorithm group (NAG) routine D03PCF. On the other hand, the harmonic representation

a(x, t) =∞∑

n=−∞an(x)exp(inωt), (4.2)

where the symmetry implies thata∗n(x) = a−n(−x), was employed to track periodic solutions in parameter space.

As for the initial value problem, we solved the resulting nonlinear eigenvalue problem on the semi-infinite intervalsubject to the boundary conditions

a∗n(0) = a−n(0),

da∗n

dx(0) = −da−n

dx(0). (4.3)

Two types of solutions exist. One is the vacillating solution with non-zero-mean part(a0(x) = 0); the other isthe travelling wave, which has zero-mean and the propertyan(x) = 0 for all evenn. As we will see, the latter


travelling waves have an important status being the generic form of the solution following a global bifurcation,which occurs at some value ofλ(κ) for all κ. Since the zero-mean cycles are often unstable, it was necessary toapply the harmonic method to locate them. Specifically, we considered the equations for the non-zero coefficientsa2m+1(x) with (−M ≤ m ≤ M − 1); outside that range we setan(x) = 0. This truncation leads to a finite set ofcoupled ordinary differential equations, which we solved using finite difference methods. For practical reasons welimitedM to values at most 15 which worked well except at small values of|ω|.

It is important to appreciate that though our boundary condition(4.1) removes the arbitrariness in the phaseangle, we may still identify two distinct steady solutions±as(x), which we need to account for when classifyingthe natures of our periodic solutions.

4.1. Homoclinic cycles and the gluing bifurcation |κ| > κNH

When|κ| (> κNH ≈ 1.05) is fixed, the upper branch steady solutionsU± : a = ±aU

s (x)undergo a Hopf bifurcationatλH to limit cyclesC± with non-zero-mean.

The situation is relatively straightforward whenκ is large. Indeed the results ofSection 3.5(see (3.24)–(3.25))indicate that the Hopf bifurcation is supercritical and that these cycles expand with increasingλ and lengthen theirtemporal period. Numerical results relating to this large-κ scenario were obtained using the initial value approachwhenκ = 2 (> κN

1 ≈ 1.55). Asλ increases fromλ0 = 5, the steady upper branch solutionsU± are stable until theylose stability at the Hopf bifurcation atλH ≈ 5.7146. The time-series for Rea(0, t) appropriate to the ensuingnon-zero-mean limit cyclesC±, whenλ = 5.77, is illustrated inFig. 9(a). This shows that on increasingλ theoscillations aboutU± (with as(0) ≈ ±0.3071) expand and reduce their frequencyω lingering longer close to theunstable trivial solutionZ : a = 0. Asλ ↑ λG (say), the frequency tends to zero (ω → 0) andZ-homoclinic cyclesform with their vertices at the saddle pointZ. In fact, the merger of the two distinct limit cyclesC± whenλ = λG

corresponds to a gluing bifurcation, after which their combination leads to a limit cycleC with zero temporal mean,as illustrated inFig. 10for the caseλ = 5.7735(> λG). The gluing bifurcation signifies a change in character ofthe solution from vacillations (seeFig. 9(b)) to travelling waves, whose nature becomes clearer at larger values ofλ. Indeed, onceλ = 6,Fig. 11(b) illustrates clearly the waves propagating towardsx = 0 from both plus and minusinfinity. This feature is consistent with our earlier large-κ result(2.5), which shows that inward travelling wavesare of larger amplitude than outgoing waves. So even though SP-solutions possess left and right travelling waves inequal proportions, locally the inward propagating wave dominates because of their spatial asymmetry. The temporalevolution towards the limit cycle is illustrated inFig. 11(c) by a phase plane plot of Rea(0, t) vs. Im∂a/∂x(0, t),which shows the results from two initial value problems. The unstable finite amplitude solutionsU± are perturbedand their subsequent spiralling out to the limit cycleC is followed. Note the anti-clockwise sense of the spirallingwhich conforms with the prediction of (3.26).

The gluing bifurcation just described can only occur atZ when it is a saddle point, which requiresλG < λ1. Todetermine the dependence ofλG onκ, we used the harmonic representation for the zero-mean limit cycles and trackedλ againstκ for various given small values ofω on the basis thatλ → λG asω → 0. This method loses accuracy forsmallω but its implementation was necessitated by the subcriticality of the zero-mean limit cycles at largeκ. In thisway, we were able to isolate the curveλ = λG(κ)which terminates at(κG

1 , λ1), whereκG1 ≈ 1.07 has been defined by

the propertyλG(κG1 ) = λ1. The form ofλG−λ0 as a function ofκ is shown inFig. 4; it is distinguished as the dashed

curve labelled G. The termination ofλG happens after the nose has formed but before the upper branch stabilises andsoκN

1 > κG1 > κN

H ; in fact,κG1 ≈ 1.07 andκN

H ≈ 1.05 are very close to each other. Significantly, the curves G andH cross at aboutκ ≈ 1.6. This means that though the supercritical scenario described for the vacillating solutions atκ = 2 may exist for largerκ, the situation for smallerκ remains unclear. It is certain, however, that theZ-homocliniccycles are subcritical, relative to the Hopf bifurcation, asλ decreases at values ofκ less than roughly 1.6.


Fig. 9. The vacillating solutionsC± which oscillate aboutU± for the caseκ = 2, λ = 5.77(> λH ≈ 5.71 but < λG): (a) the time-series ofRea(0, t); (b) contours of constant Rea(x, t) and Ima(x, t) in the (x, t)-plane.


Fig. 10. The travelling wave solutionC for the caseκ = 2, λ = 5.7735(> λG) illustrated by the time-series of Rea(0, t).

Let us denote the maximum value of|a(0, t)| over timet for our periodic solutions byamax: this quantity andthe frequencyω of zero-mean limit cyclesC are plotted vs.λ for various fixed values ofκ in Fig. 12. Eachω-curvefor given κ starts atω = 0. That end point corresponds to the gluedZ-homoclinic cycle for the caseκ = 4 ofFig. 12(b) but only the caseκ = 2 ofFig. 12(a); the end points of theκ = 0.5 and 1 curves correspond to heterocliniccycles as we discuss in the following subsection. As remarked above, the solution in the neighbourhood of the gluedhomoclinic cycles is difficult to obtain because large numbers of terms need to be retained in the harmonic expansionto guarantee a reliable solution—hence the data inFig. 12does not extend right down toω = 0. For that reason,whether the zero-mean limit cycles following the gluing bifurcation for cases withκ > 2 are super- or sub-criticalremains unclear. Still, the caseκ = 4 is certainly subcritical and probably remains so thereafter for largerκ.

For the caseκ = 4, the gluing bifurcation occurs atλ ≈ 17.4 (recall thatλ0 = 17). A lower limit cycle branch isthen tracked on decreasingλ down toλ ≈ 14.74, after which an upper branch is followed on increasingλ. It seemslikely that the lower branch behaviour can be determined by the amplitude equations (3.21) but, since this lowerbranch is evidently unstable, we have not taken the trouble to attempt solutions. Nevertheless, a crude truncation ofthe harmonic expansion by its first harmonic leads to solutions of the form (3.25)–(3.26) under the limit∆ → ∞, i.e.with the temporally mean part dropped. The point here is that such a severe truncation leads to the amplitude equationsimilar to (2.7) for the Hocking–Skiepko modes, in whichΩ = ω and|Ad(0)| = √

3/2amax. The subcritical lowerbranch behaviour of our SP-solutions inFig. 12(b) has similar features to that for the Hocking–Skiepko solutionsillustrated inFig. 3. There is loose quantitative agreement, which is all that can be expected.

Over the small rangeκG1 > |κ| > κN

H the expanding non-zero-mean limit cyclesC± become two disconnectedL-homoclinic cycles, whenλ = λho (say), with their vertices at the saddle points of the lower branch steadysolutionsL±, rather thanZ. The gluing bifurcation no longer exists andZ is now an unstable node. Following thedestruction of the limit cyclesC± via theL-homoclinic cycles, it is plausible that a distinct zero-mean limit cycleC remains originating from heteroclinic limit cycles as we will explain in the following subsection. Asκ decreasestowardsκN

H , the upper and lower branch steady solutionsU± andL± approach each other and so the homoclinic


Fig. 11. The travelling wave solutionC for the caseκ = 2, λ = 6: (a) the time-series of Rea(0, t); (b) contours of constant Rea(x, t) andIma(x, t); (c) a transient phase portrait of Im∂a/∂x(0, t) vs. Rea(0, t). The unstable upper branch equilibriaU± and the trivial solutionZare identified by the labelled solid dots. The outward and anti-clockwise spiral (emphasised by the direction of the arrows) fromU+ (U−) drawncontinuous (broken) illustrates evolution towardsC.


Fig. 11. (Continued ).

orbits shrink at the Takens–Bogdanov bifurcation point(κNH , λ

NH). The homoclinic bifurcation curveλ = λho(κ)

links the Takens–Bogdanov bifurcation point(κNH , λ

NH) and the point(κG

1 , λ1) marking the saddle node bifurcationof the glued homoclinic orbits. InFig. 4we provide a schematic sketch ofλho−λ0 vs.κ; it is the unlabelled dashedcurve (see the figure caption). It must be stressed that noL-homoclinic cycles were found numerically. Since theyonly exist over a very smallκ range, and in view of the difficulty in calculatingκG

1 , we cannot even be entirelycertain thatκG

1 > κNH .

It must be emphasised that the vacillating solutions are of limited interest because they exist over a relativelyshortλ-range. There they describe the nature of the transition between the Hopf bifurcation of the steady state andthe global bifurcation to travelling waves.

4.2. The heteroclinic cycles |κ| < κG1

When the nose exists for|κ| < κN1 , the trivial solutionZ undergoes a supercritical pitchfork bifurcation shedding

two non-trivial solutionsL± asλ increases beyondλ1. More precisely, asλ − λ1 becomes positive,Z makes thetransition from an unstable saddle to an unstable node, whereasL± emerge as unstable saddles. So on following thetrack of the gluing bifurcationλ = λG(κ) back through decreasing values ofκ, these gluedZ-homoclinic cyclesare torn apart by the saddle node bifurcation at(κG

1 , λ1). After the bifurcation, the saddlesL± can still connect butin one of the two ways determined by distinct values ofλ. Firstly, they may join toL± for λ = λho forming theL-homoclinic cycle discussed in the previous subsection, or, secondly, they may connect toL∓ for λ = λhe therebycreating two heteroclinic connections whose combination form anL-heteroclinic cycle. This latter case sheds azero-mean limit cycleC, which forms the basis for our discussion below.

The situation over the rangeκG1 > |κ| > κN

H is unclear, but plausibly both zero-mean and non-zero-mean limitcycles may co-exist at certain values ofλ.


Fig. 12. Characteristics of the SP-limit cyclesC, which emerge from the global bifurcations, and plotted vs.λ. The maximum amplitudeamax

over the cycle and frequencyω are drawn continuous and broken, respectively: (a) the casesκ = 2 (gluedZ-homoclinic), 1 (L-heteroclinic)and 0.5 (N-heteroclinic), as labelled, forλ ≤ 8; (b) the caseκ = 4 (gluedZ-homoclinic). The small region of stability on the upper branch islabelled S. The realised valueamax andω of another periodic solution atλ = 15.9 is identified by the dotamax ≈ 2.8622 and starω ≈ 0.98175,respectively; both are labelled LA.

When|κ| < κNH the upper branchU± is stabilised and so plausibly no non-zero-mean limit cycles form about

it. Certainly, that is the case forκ = 1(< κNH ≈ 1.05) solutions illustrated inFigs. 13 and 14. To understand their

character, we note that the results portrayed inFig. 7(a) indicate that the upper branch solutionsU± are stable,whereas the lower branch solutionsL± are unstable. Forλ > λ0 = 2, the trivial solutionZ is unstable and the


Fig. 13. A transient phase portrait of Im∂a/∂x(0, t) vs. Rea(0, t) for the caseκ = 1, λ = 4.1(< λhe). The outward anti-clockwise spiralsfrom the unstable lower branch equilibriaL± and the trivial solutionZ to the stable upper branch equilibriaU+(U−) are drawn continuous(broken). The exterior curves spiralling intoU± are shown long-dashed.

solution evolves either toU+ orU− ast → ∞. Whenλ > λ1 = 4 the small amplitude solutionsL+ orL− are alsounstable and all solutions again evolve either toU+ or U−. Fig. 13illustrates these features by phase plots of thetemporal evolution obtained from numerical integrations for the caseλ = 4.1(< λhe). The corresponding phaseportrait following theL-heteroclinic bifurcation is shown inFig. 14(b) for the caseλ = 4.21(> λhe). It shows howsome solutions, particularly those fromZ andL±, spiral intoU±, while others spiral onto the limit cycleC, all inan anti-clockwise sense. The limit cycle time-series inFig. 14(a) illustrates the fact that the solution lingers in thevicinity of the saddleL±, which receives a close pass.

On decreasingκ the picture just described continues to hold. The main point of interest is that the lower branchsolutionsL± : ±aL

s providing the vertices to theL-heteroclinic cycles shift towards the nose solutionsN± : ±aNs .

When reached atκ = κNhe ≈ 0.84, anN-heteroclinic cycle forms atλ = λN(κ

Nhe) = λN

he ≈ 4.47 with the nosesolutionsN± as vertices. Thereafter, for|κ| < κN

he, the picture with increasingλat fixedκ is relatively straightforwardand is illustrated inFigs. 15 and 16for the caseκ = 0.5. Whenλ = 5(< λN), almost all solutions approach one ofthe upper branch stable solutionsU± ast → ∞; Fig. 15(a) shows the tracks to them, which start from the unstableequilibriaZ andL±. Asλ increases towardsλN, the short tracksL± toU± contract and evaporate atλN so formingourN-heteroclinic cycle whenλ = λN. The limit cycle solution which emerges atλ = 6.2 > λN is illustrated inFig. 15(b) and the corresponding temporal plots for the slightly larger valueλ = 7 are illustrated inFig. 16. Froma more general point of view, the saddle node scenario for the formation of theN-heteroclinic cycle at the nosefor smallκ is nicely illustrated by our simple model described inAppendix D(see(D.5)). We remark also that thesaddle node scenario was previously identified in a related convection problem by Barkley and Tuckerman[23,24](Fig. 1 of both references).

On a technical matter, though the largest growth ratepL1 for SP-disturbances to the lower branch solutionsL±,

which exist for|κ| < κNH , is real and positive, the second growth ratepL

2 , which has negative real part, may be


Fig. 14. The caseκ = 1, λ = 4.21(> λhebut < λN): (a) the periodic time-series of Rea(0, t); (b) the transient phase portrait as inFig. 13.The exterior curves spiralling into the limit cycleC are indicated long-dashed.

complex. When it is real,L± is a saddle point, but when it is complex, there is a saddle-focus atL±. For thoseL-heteroclinic cycles involving the saddle-focus, the incoming tracks spiral in and out of the neighbourhood ofL± exhibiting Shil’nikov’s mechanism (see, e.g.[28]). For the caseκ = 1 the value ofλhe evidently lies in therange 4.1 < λhe < 4.21 asFigs. 13 and 14indicate. It is clear, however, fromFig. 7 thatpL

2 is real and negativeimplying thatL± is a saddle. Nevertheless, on decreasingκ, the unstable equilibriumL± atλ = λhe(κ) advancestowards the nose, where it almost certainly enters a region withpL

2 complex and so becomes a saddle-focus. As


Fig. 15. As inFig. 13, transient phase portraits for the caseκ = 0.5: (a)λ = 5(< λN ≈ 6.13). The curves from the unstableL± andZ to thestableU+ (U−) are drawn continuous (broken); (b)λ = 6.2 (> λN). The curves from unstableZ, as in (a), spiralling onto the limit cycleC.


Fig. 16. The caseκ = 0.5, λ = 7(> λN ≈ 6.13): (a) the time-series of Rea(0, t); (b) contours of constant Rea(x, t) and Ima(x, t).

for theN-heteroclinic cycles, the one atκ = 0.5 illustrated inFigs. 15 and 16definitely involves a saddle-focus(seeFig. 8). The stability results for the steady solutions atκ = 0.2 reported in[22] suggest that on decreasingκ the lower branch recovers its saddle characteristics near the nose and that this will persist as|κ| ↓ 0. Plausibly,Shil’nikov’s mechanism operates over a continuous range ofκ containing bothL- andN-heteroclinic cycles.


Finally, recall that the maximum amplitudeamax and frequencyω of our zero-mean limit cycles are plotted vs.λ in Fig. 12. For the casesκ = 2, 1 and 0.5 illustrated inFig. 12(a), theω = 0 end point corresponds to a gluedZ-homoclinic cycle, anL-heteroclinic cycle and anN-heteroclinic cycle respectively. Note also the kink on theupper branch of the curves inFig. 12(b) for the caseκ = 4. The results of running time-stepping numerical codesindicated that the solutions are stable on the upper branch as far as the kink. After the kink the mode becomesunstable to new modes which are discussed in the following section. This scenario also occurs in the casesκ = 1andκ = 2 but at values ofλ beyond the right-hand border ofFig. 12(a).

5. Higher order bifurcations

5.1. Symmetry preserving resonant interactions

The SP-periodic zero-mean limit cyclesC : a = aC(x, t) (say), which are shed at the global bifurcations discussedin the previous section, have the half-period reflectional symmetry

aC

(x, t + π

ω

)= −aC(x, t). (5.1)

We may consider small perturbations to them of the form

a(x, t)− aC(x, t) = a(x, t)exp[p(t − t0)] + a∗(x, t)exp[p∗(t − t0)], (5.2a)

where the growth rate

p := σ + i(ω − ω) (5.2b)

(Floquet multiplier) is a complex constant andt0 is an arbitrary real constant. The complex functionsa anda∗ havethe half-period reflectional symmetry(5.1)of aC and satisfy

∂a

∂t= L(p; a, a∗) (5.3a)

and

∂a∗∂t

= L(p∗; a∗, a), (5.3b)

where the differential operatorL is defined by(3.2c). On taking the complex conjugate of(5.3a) and (5.3b), it isreadily seen that solutions with one of two symmetries are possible. SinceaC has the symmetrya∗

C(x, t) = aC(−x, t),SP- and SB-solutions with

a∗(x, t) = a∗(−x, t) (5.4a)

and

a∗(x, t) = −a∗(−x, t), (5.4b)

respectively, are possible. Within this generalised framework, we may essentially use the language ofSection 3,which we used to describe the stability characteristics of the steady solutionas.

In this section we restrict attention to SP-bifurcations. The limit cycle is unstable to perturbations(5.2a)whenσ := Rep = 0. Without loss of generality we may limitω to the range|ω− ω| < |ω|. In general, for fixedκ andincreasingλ, the ratioω/2ω will be irrational at the value ofλ where stability is lost. Nevertheless, it is helpful to


regardκ as an additional parameter and consider the critical value ofω/2ω to be a function ofκ. Asλ is increasedbeyond critical, we expect to locate Arnold tongues of stability corresponding to finite amplitude solutions withrational ratios ofω/2ω (see p. 322 of[29]).

Some evidence for the above scenario was found whenκ = 1. On time-stepping our numerical code at variousvalues ofλ, the period appeared to triple at aboutλ roughly 9 maintaining the half-period reflectional symmetry(5.1). To test the hypothesis we tracked our solutions using the zero-mean harmonic representation. On the onehand, we followed the period 2π/ω of the originalω-mode asλ was increased. On the other hand, we set

ω − ip = ω ≈ 13ω (5.5)

and followed the new period-tripledω-mode. For this special choice, (5.2) can provide an exact representation ofthe solution; no further harmonics are generated by nonlinear interactions—this is often denoted a resonant triad.Referring toFig. 17, the larger of the two period-tripled frequencies (the upper branch of the dashed curve) wasisolated by the initial value problem. Representing that solution in terms of its harmonic expansion enabled usto track its evolution for both increasing and decreasingλ. On decreasing, we eventually rounded the ‘nose’ atλ ≈ 9.11 and thereafter on increasingλ were able to determine the lower branch. Time-series and phase portraitsof the two bifurcated solutions atλ = 9.5 are illustrated inFigs. 18 and 19, respectively. Note that, due to thelimited number of harmonics that we were able to retain in the expansions, the phase portraits are not quite properlyresolved.

The maximum amplitudeamax of the originalω-mode is illustrated inFig. 17by the lower continuous curve. Itsdepression (or kink) signals instability of the mode, while its occurrence, prior to the emergence of the hangingbranch of the period-tripledω-mode, is reminiscent of an imperfect bifurcation. This feature is consistent with our

Fig. 17. Continuation ofFig. 12(a) toλ ≥ 8 for the caseκ = 1. The upper branch of bothamax andω of the period-tripled solution correspondto the same stable solution.


Fig. 18. The stable period-tripled large amplitude SP-periodic solution for the caseκ = 1,λ = 9.5 withω ≈ 0.7076 (amax ≈ 3.0351) identifiedby the continuous curves. The originalω ≈ 1.9041 (amax ≈ 2.4729) solution is illustrated by the broken curves: (a) the time-series of Rea(0, t);(b) the phase portrait of Im∂a/∂x(0, t) vs. Rea(0, t).


Fig. 19. As inFig. 18but for the unstable period-tripled small amplitude SP-periodic solution withω ≈ 0.6243 (amax ≈ 2.9343).

application of the Arnold tongues. If instability had set in withω = ω/3 exactly, we would expect theamax-curvesto respond at the same location as the emergence of the period-tripled solution as in a pitchfork bifurcation. Also,weakly nonlinear theory would then only determinea to within a factor±1 so generating the solutions linked tothe upper and lower curves of the hanging branch. Put another way, on increasingλ the originalω-mode was found


by application of our time-stepping code to be unstable atλ ≈ 8.75 near the start of the kink. The time-dependentsolutions realised in the small range 8.75< λ < 9.11 between that instability and the emergence of the period-tripledsolution showed a complicated time-dependent structure consistent with more complicated frequency ratiosω/2ω.We also remark that the instability for other values ofκ is also signalled by kinks. In the caseκ = 2 there is a kinkof the amplitude and frequency curves atλ = 8.3 just off the right-hand side ofFig. 12(a).

For the caseκ = 4 the situation for the subcritical zero-mean modes illustrated inFig. 12(b) is even moredramatic. There the kink is located on the upper branch at roughlyλ ≈ 15.35 and is associated with amplificationrather than suppression ofamax. In consequence, stable finite amplitude SP-periodic solutions are only located onthe upper branch on the small range between the nose atλ ≈ 14.74 and the kink at roughlyλ ≈ 15.35. This intervalis completely subcritical relative to the onset of the steady solutionU± : as(x) = 0, which occurs atλ0 = 17.It was generally difficult to identify any periodic structure of the solutions once instability sets in. Nevertheless,we illustrate inFig. 20 one relatively simple periodic solution which occurs atλ = 15.9. Evidently the period(frequencyω ≈ 0.98175 marked by the star inFig. 12(b)), has increased fivefold and, interestingly, the solutionlacks the half-period reflectional symmetry(5.1). This suggests that the instability in the (λ, κ)-plane to which itis linked is an even (rather than odd) multiple ofω/5, plausiblyω = 2ω/5. By nonlinearity this generates allthe harmonicsnω/51 with n taking all integer values including zero. Thus the resulting solution is likely to havenon-zero-mean albeit apparently very small.

5.2. Symmetry breaking solutions

The upper branch steady solutionsaUs (x) generally lose stability first through SP bifurcations except for a

small range ofκ close to unityκNHS (≈ 0.82) < |κ| < κH

HS (≈ 1.10) (see, e.g.,Fig. 7). More precisely, theSP-Hopf bifurcation occurs for smallerλ than the SB-pitchfork when|κ| > κH

HS, while the upper branch is stable toSB-modes when|κ| < κN

HS; then stability is lost at the noseaNs (x) to an SP-bifurcation (saddle node or saddle-focus).

The SB-pitchfork is supercritical for|κ| less than roughly 3 as illustrated inFig. 1 and so sufficiently close tothe bifurcation the Hocking–Skiepko mode is stable over the rangeκN

HS < |κ| < κHHS. Elsewhere, however, the

SB-Hocking–Skiepko solutionsa±(x, t) (see(1.12a)) appear to be unstable. This is hardly surprising when thebifurcation to them occurs from lower branch steady solutionsaL

s (x), as it does for|κ| < κNHS (see, e.g.,Fig. 8). This

viewpoint is supported by our low-order model ofAppendix D, for which(D.9) demonstrates the instability of allHocking–Skiepko modes. For|κ| greater than roughly 3 the Hocking–Skiepko mode bifurcates subcritically. In thecase ofκ = 4 it exists down toλ = λHS

min ≈ 16.55 where the upper and lower solution branches ofFig. 3meet. Sinceboth branches are unstable, it is of interest to ascertain where such solutions evolve to. Harris[22] time-stepped thesolutions and found that forκ less than about 3 they generally evolved into the SP-periodic solutions whereas forlarger values ofκ that is no longer the case.

On writing a(x, t) ≡ eiΩtA(x, t) we may, in principle, consider small perturbations to the Hocking–SkiepkosolutionAd(x) of the form

A(x, t)− Ad(x) = A(x)exp(pt)+ A∗(x)exp(p∗t), (5.6a)

where

p := σ + iω (5.6b)

is the constant complex growth rate. The complex functionsA andA∗ satisfy

H(p; A, A∗) = 0 (5.7a)

1 Thisω/5 corresponds to theω ≈ 0.98175 marked by the star inFig. 12(b).


Fig. 20. As inFig. 18but for the stable SP-periodic solution corresponding to the points LA inFig. 12(b) atκ = 4, λ = 15.9.


and

H(p∗; A∗, A) = 0, (5.7b)

where the differential operatorH is defined by

H(p; A, B) ≡ ∂2A

∂x2+ (λ− iΩ − p + 2iκx − x2 − 2|Ad|2)A− A2

dB∗. (5.7c)

Bifurcations through marginal disturbances withp = iω lead to SB-periodic solutions with the harmonic structure

a(x, t) ≡ eiΩtA(x, t) = eiΩt∞∑

n=−∞An(x)exp(inωt), (5.8)

similar to (4.2). We found no numerical evidence of continuous bifurcation to stable structures of this type. Nev-ertheless, unstable structures possibly exist which bifurcate onto tori characterised by the frequencyω and a newfrequencyω much as the SP-solutions studied in the previous subsection. We may also suppose that rational ratiosω/ω = M/N (say;M andN (> M > 0) integers) will lead to robust structures similar to the Arnold tonguesdiscussed above. Under that scenario, we would expect to find periodic solutions of the form

A(x, t) =N−1∑m=0

Am exp(im

Nωt), (5.9a)

where

Am(x, t) =∞∑

n=−∞Am,n(x)exp(inωt). (5.9b)

Indeed, when they first emerge, the tongue idea would suggest that the contributionA0(x, t) dominates, while thedisturbance to it originates primarily from the two modesAM(x, t) andAN−M(x, t), if distinct. Of course, thefrequencyω/N in (5.9) byω, the solution structure (5.9) is not different to(5.8). Accordingly, the classification(5.9) is not unique but appears to be helpful as the numerical results of the following section suggest.

5.3. Pulse-trains: κ = 4

Throughout the remainder of this section we report numerical results forκ = 4, which appears to be just largeenough to capture the large-κ behaviour. Our initial value time-stepping integrations were performed by perturbingthe upper branch (see, e.g.,Fig. 3) Hocking–Skiepko modes. Our results showed evolution into yet much largeramplitude solutions with apparently very complicated behaviour. Nevertheless, under closer inspection the solutionswere generally seen to be periodic with the structure (5.9).

The simplest solution that we managed to locate is the subcritical caseλ = 15.5(< λHSmin ≈ 16.55) shown in

Fig. 21which illustrates properties of both the functionA(x, t) (see (5.9)) anda(x, t) = eiΩtA(x, t). From theplot of |a(0, t)| given inFig. 21(a) the beating of the oscillation identifiesN = 6 (see alsoFig. 21(b)), while theabsence of any apparent sub-beating suggests thatM = 1; the beat periodT ≈ 3.89 is clearly visible, which in turndetermines the frequencyω := 2πN/T ≈ 9.6913, also evident inFig. 21(f). We then estimatedΩ as a solution of

[exp(iΩT )] ≡ exp

(i2π

NΩ

ω

)= a(0, t0 + T )

a(0, t0), (5.10)

where t0 is any fixed time chosen at our convenience. Of course,(5.10) only determinesΩ up to an integermultiple ofω/N but that multiple was chosen so that the phase plots of the real and imaginary parts ofA(0, t) =


Fig. 21. The periodic SB-solution for the caseκ = 4,λ = 15.5 withΩ ≈ 2.31,ω ≈ 9.69 andN = 6 (see (5.9)): (a) the time-series of|a(0, t)|;(b) the phase portrait of|∂a/∂x(0, t)| vs. |a(0, t)|; (c) likewise for ImA(0, t) vs. ReA(0, t); (d) the magnitudes|a| and Rea (Ima) vs.x at timet = 5 identified by the continuous and long (short) dashed curves, respectively; (e) contours of constant Rea(x, t); (f) likewise for|a(x, t)| with band boundaries atx(n)MAX (n = −2,−1,0,1,2; seeTable 1).



a(0, t)exp(−iΩt) remained close to its mean value as illustrated inFig. 21(c). Under this criterion we determinedΩ ≈ 2.3131. (Other choicesΩ +Lω/N for integerL would lead to phase diagrams such that the path would orbitthe originL times.)

Remember that, given a solutiona(x, t) of (1.3), we may generate other solutions eiϕa(x, t) with any givenconstant rotationϕ. In that sense,Fig. 21(c) is not unique and may be rotated arbitrarily about the originA = 0.



The fact thatA(0, t) on the figure stays in the vicinity of 2i suggests that the displayed solution is dominated atx = 0 by the single Hocking–Skiepko drifting mode

A0,0(x)exp(iΩt) (5.11a)

of the double expansion (5.9) withA0,0(0) roughly 2i. Furthermore,Ω is the value of the frequency identified onthex = 0 axis of the space–time contour plotFig. 21(e) and for that matter throughout the central region of lateralextent roughly−1 < x < 1. The wave fronts there are tilted and thus indicate the existence of a leftwards travellingwave consistent with our large-κ asymptotic (essentially linear) interpretation of such a Hocking–Skiepko mode inSection 2. That theory also shows that the mode takes its maximum amplitude (see below(2.5)) atx(0)MAX ≈ Ω/2κ ≈0.29 (seeTable 1). Note that we only plot contours of constant Rea(x, t) in Fig. 21(e); in view of the dominanceof the mode(5.11a)the corresponding plot of Ima(x, t) is essentially the same in the central region−1 < x < 1but shifted backwards in time by the quarter periodπ/2Ω.

Table 1The locationx(n)MAX (see(5.12)) determined by the frequenciesΩ ≈ 2.31 andω ≈ 9.69 of theκ = 4, λ = 15.5 periodic SB-solution

n = −2 n = −1 n = 0 n = 1 n = 2

Ω + nω −17.07 −7.38 2.31 12.00 21.69x(n)MAX −2.13 −0.92 0.29 1.50 2.71


The fact that the curves inFig. 21(c) only loopA = 2i once over the period 2π/ω suggest that the next mostimportant modes in the vicinity ofx = 0 are

A0,1(x)exp[i(Ω + ω)t ] + A0,−1(x)exp[i(Ω − ω)t ]. (5.11b)

Interestingly, the positive frequencyΩ + ω ≈ 12.00 can be loosely identified on the positive range 1< x < 2 ofFig. 21(e). The structure on this interval is naturally associated with a leftward travelling Hocking–Skiepko wave.Indeed, asx is increased further the next harmonic with frequencyΩ + 2ω ≈ 21.69 can be identified towards theright edge ofFig. 21(e). Likewise we can associate the rightward travelling wave on the negative range−2 < x < −1with the negative frequencyΩ−ω ≈ −7.38. Again more structure is visible at the left edge ofFig. 21(e) associatedwith the next harmonicΩ − 2ω ≈ −17.07. Note that the large-κ theory ofSection 2predicts that the maximumamplitude of a Hocking–Skiepko mode frequencyΩ + nω occurs at

x(n)MAX ≡ Ω + nω

2κ, (5.12)

whose values we list inTable 1.Our interpretation of the leading termA0(x, t) of the series(5.9a)leads to the notion of a solution composed of

a train of stationary Hocking–Skiepko pulses

A0,n(x)exp

i

[∫Kdx + (Ω + nω)t

](5.13a)

with

K = κ, (5.13b)

A0,n(x) ∝ exp[−12(x − x

(n)MAX )

2] (5.13c)

when|x| |κ| (see (2.3) and (2.4)), localised by phase mixing in the vicinity ofx(n)MAX . From a quantitative point

of view, this essentially linear notion is over-simplistic. Though the locationx(n)MAX of the pulse maximum(5.12)is

uniquely defined (independent of the nonlinearity) as the point of vanishing phase mixing, nonlinearity requires thelocal wavenumbersK(x) to be smaller than the valueκ given by(5.13b). In fact, a crude nonlinear WKB calculationgives the rough estimates

λ− x2 −K2 ∼ |a|2 ≈∞∑

n=−∞|A0,n(x)|2, (5.13d)

A0,n(x) ∝ exp

[−|κ|(x − x

(n)MAX )

2

2|K(x(n)MAX )|

], (5.13e)

which is consistent with(5.13b) and (5.13c)in the linear limit |a| = 0, when|x| |κ| andλ/κ2 = 1 + o(1).Significantly, it places an estimated bound|a|2 < λ−x2 on the amplitude; that also determines the range|x| < √

λ

over which the finite amplitude solution may be expected to exist. All the reported results on our time-steppingnumerical calculations were done on the interval−5 ≤ x ≤ 5, which was sufficiently long to satisfy the criterion√λ < 5. Nevertheless, calculations were undertaken on broader ranges the check their validity.Further evidence for the validity of our nonlinear pulse picture is provided byFig. 21(d) and (f). In our snapshot

(i.e. fixed time plot)Fig. 21(d), we may identify a smoothly varying wavenumberK(x) common to each pulse.The spatial variation ofa(x, t) illustrated captures the details of a cross-section (fixed time) of the contour plotsin Fig. 21(e) and (f). The contour plot of|a(x, t)| in Fig. 21(f) shows a checkerboard-like pattern with band-edges


Table 2The characteristics of the SB-periodic solutions, whenκ = 4, for various values ofλa

λ amax N ω Ω

13.5 2.7793 46 9.2503 –15.5 3.2094 6 9.6913 2.313115.9 2.8622 10 9.8175 4.908716.0 3.2485 9 9.7498 1.634618.0 3.5819 67 9.9733 0.8194

a Data for the SP case ofFig. 20is given in italics.

located roughly atx(n)MAX (n = −2,−1,0,1,2). The pattern reflects the interaction of two neighbouring pulses withdistinct frequencies. Though any individual pulse has a roughly temporally constant amplitude|A0,n| at fixedx,the interaction of two neighbouring pulses beats with frequencyω, as illustrated inFig. 21(f). More specifically,on the rangex(n)MAX < x < x

(n+1)MAX the primary interaction is between theA0,n-pulse of frequencyΩ + nω, which

decreases in magnitude to the right of its maximum atx(n)MAX , and theA0,n+1-pulse of frequencyΩ + (n + 1)ω,

which decreases in magnitude to the left of its maximum atx(n+1)MAX .

The beating behaviour with frequencyω/6 = 2π/T ≈ 1.6152 evident inFig. 21(a)–(c) must be generated bythe next termA1(x, t) together withA5(x, t) of the series(5.9a)and associated specifically with the harmonicsA1,0(x) andA5,0(x, t) of (5.9b).

Asλwas increased beyond the valueλ = 15.5, we obtained similar results with interesting trends listed inTable 2.The frequencyω tended to grow very slightly although both the beat periodT andN increased dramatically, whileM probably retained the value unity. An intriguing feature of the solutions for the casesλ = 15.5 and 18 is the dip inthe curves at the top ofFigs. 21(b) and 22(b)giving them heart-like appearances. This dip deepens with increasingλ leading eventually to small loops similar to those portrayed inFig. 18(b).

It is worth comparing the results illustrated inFig. 21(a) and (b) for our SB-solution atλ = 15.5 with thecorresponding SP-solution that occurs atλ = 15.9 and illustrated inFig. 20. Perversely we can recast that harmonicstructure into our symmetry broken format (5.9). According to the data listed in the caption toFig. 12(b), we canfirst identify the frequencyω/N ≈ 0.98175, while fromFig. 20(b) we determineN = 10. (That is because thelack of half-period reflectional symmetry implies that the plot of moduli of the data inFig. 20(b) would double thenumber of loops from 5 to 10.) That said, the frequency which dominates the oscillations in the plot inFig. 20(a) is5ω/N ≈ 4.9087 and arguably that is the value we should attribute toΩ. These characteristics are listed inTable 2;the frequencyω particularly compares favourably with that found for the SB-solutions. Note also the dip at thetop (and bottom) of the curves inFig. 20(b), which has its counterpart inFig. 21(b). Incidentally, the maximumamplitudes of the SB-solutions are comparable but slightly larger than those for the SP-solutions.

On decreasingλ below the valueλ = 15.5 of Fig. 21no further reduction inN was found. Instead there was evi-dence of periodic solutions withN increasing as illustrated inFig. 23for the caseλ = 13.5. The amplitude|a(0, t)|oscillates on a local frequencyω ≈ 9.2503 while its envelope beats over 23 oscillations. Nevertheless, there is evi-dence of period doubling, in the sense that the pattern only repeats itself over twice that period givingN = 46. Thedifferences are subtle and identified primarily by some barely discernible curve-splitting inFig. 23(b) but robust in thesense that when the initial value problem was run over much longer time intervals the differences persisted. It provedhard to determineΩ uniquely according to our criterion below(5.10)and so we have refrained from specifying a valuein Table 2. Similar SB-solutions appeared to exist as far down asλ roughly 13 but whether or not they are periodicwas unclear. Certainly, if they are periodic, we may reasonably suppose that their structure is very complicated. In-terestingly, our SB-solutions exist at values ofλwell below the minimum valueλ ≈ 14.74 for SP-periodic solutions.


Fig. 22. As inFig. 21(a) and (b) but hereκ = 4, λ = 18 withΩ ≈ 0.82,ω ≈ 9.97 andN = 67.


Fig. 23. As inFig. 21(a) and (b) but hereκ = 4, λ = 13.5 withω ≈ 9.25 andN = 46.

Over the range 14.74< λ 15.35, on which the stable SP-modes exist on the upper branch curve inFig. 12(b), la-belled S, we found no stable SB-solutions, as the solutions were attracted to the SP-states. On the other hand, forλ 15.35, after the bifurcation of the SP-modes, either the SP- or SB-modes could be reached depending on the initialconditions.


6. Discussion

There are two aspects of our study. Firstly, in the previous sections we have unravelled the complicated bifurcationstructure of the solutions to the complex Ginzburg–Landauequation (1.3). Secondly, the physical interpretation ofthe results as applied to the narrow gap spherical Couette flow is of interest and we will expand on that aspect here.From that point of view, we stress that whenκ = O(1) the angular momentum ratio of the outer to inner spheresis close to unity (see (1.1)-(1.4)), which is an unusual limit and not a parameter range adopted in experiments.Those usually have finite angular momentum ratio for whichκ 1. Indeed often the outer sphere is at rest; thisconfiguration is well outside the parameter range for which our analysis is valid.

With regard to our mathematical results, we found a wide variety of nonlinear finite amplitude states. Indeed atvarious values ofκ andλ, both stable symmetry preserved and symmetry broken states were located and sometimesthese could co-exist. Nevertheless, there appeared to be a tendency for the symmetry preserved states to be preferredfor small-κ, while symmetry broken states appear to be preferred for largeκ.

On restricting attention to symmetry preserved solutions, the highlight of our analysis is the identification of theglobal bifurcations to a zero-mean limit cycle, either viaN- orL-heteroclinic cycles connecting reversed sign steadystates or via the gluing ofZ-homoclinic cycles. In all cases they correspond to the transition to travelling waves withthe Taylor vortices drifting towards the equator symmetrically from either side. The variety of global bifurcationspossible, dependent on the size ofκ, is intriguing. In contrast with earlier studies we note that only one special case,namely that of anN-heteroclinic cycle, was isolated in[23,24]. That said, we should emphasise that all the mainfeatures of our SP-solutions, namely the steady state equilibrium lobes inFigs. 5, 7 and 8and the three types ofglobal bifurcation to travelling waves have all been identified by Siggers[26] in a low-order model appropriate to athermal convection problem. Since our interpretation of the numerical solution of the governing partial differentialequation (1.3)in terms of the phase portraits assumes that the dynamics is captured by a low-order system, it isreassuring to see that such systems (as studied in[26]) exist.

Bartels[5] obtained numerical solutions for SP-spherical Taylor Couette flow by applying symmetry preservingboundary conditions at the equator. His results for the narrow gap caseε = 0.025, albeit with the outer sphere atrest, showed complicated temporal behaviour that could be accounted for by the higher order bifurcations discussedin Section 5.1for largeκ, as exemplified by the caseκ = 4. We illustrate the temporal evolution ofa(x, t) by greyshaded figures giving contour plots Rea(x, t) and Ima(x, t). In determining the nature of the Taylor vorticeswe need to bear in mind that they have a much shorter wavelength as we explained in the opening section. So, forexample, in the case of the travelling waves depicted by the oblique contours in the space–time plot ofFig. 11(b) thecorresponding contours for the actual vortices are closely packed and inclined at a very shallow angle to thet-axis,as reflected by the fact that the phase velocity of the vortices is an order of magnitude smaller than that ofa(x, t).In contrast the vacillating waves depicted inFig. 9 correspond to vortices whose widths pulsate very slightly butthe number remains constant. Their main feature is a temporal modulation of the strength of the vortices with theflow velocity retaining its sign, i.e. there is no flow reversal.

For the medium gapε = 0.154, Mamum and Tuckerman[6] find subcritical finite amplitude SP-solutions. Onincreasing the Taylor number they are first manifest as a pitchfork to steady states. In our small-ε narrow gap limit,we find comparable subcritical behaviour albeit to time-dependent solutions, which may possess broken symmetryas we discuss below. Since we have found several different routes to travelling waves via various global bifurcations,it is hardly surprising to find that on increasingε alternative scenarios are possible as reported in[6].

It must be emphasised that the caseε 1 is a very extreme limit, which is strongly stabilised by the phase mixingmechanism. That is why at largeκ the instability of the basic state to steady Taylor vorticesas(x) occurs at largeλ = O(κ2)well above the cylinder critical valueλ = 1 appropriate to local instability. Indeed, even when the steadyvortices form, they are of extremely small amplitude proportional to exp(−|κ|2/2) (see, e.g.,(2.7a)with Ω = 0).


Our numerical results forκ = 4 illustrate clearly that finite amplitude motion in the large-κ limit is strongly subcriti-cal and exhibits amplitudes which are orders of magnitude larger than the steady mode. In other words, on increasingthe Taylor number for the physical system, we would not expect to see the bifurcation to the small amplitude steadyvortices. Instead, the transition to finite amplitude motion is likely to be explosive in the sense that a sudden transi-tion would be made from the laminar basic state to symmetry broken travelling waves, as described inSection 5.3.These may have temporally intricate behaviour. Nevertheless, bearing in mind our previous comments about thephysical interpretation of the grey shadedFig. 11(b), the complicated temporal structure ofa(x, t) portrayed inFig. 21(e) defines a fairly smooth vortex distribution of the type mentioned in the former SP-context. The disloca-tions illustrated in the figure persist for the vortices but the number of them remains constant. Since there are anorder of magnitude more vortices, the dislocations will be sparsely distributed. Essentially, the solution near theequator is dominated by a Hocking–Skiepko mode which consists of waves crossing the equator in one direction.Sufficiently far from the equator the waves will approach it from either side. The temporal merging of two wavecrests corresponds to the dislocations mentioned in the space–time plane. Despite all the subtleties mentioned, thetravelling waves following global bifurcation, whether they are of symmetry preserving or of symmetry breakingtype, take on reasonable amplitudes in this large-κ limit, unlike the steady vortices which are strongly suppressedas determined by the weakly nonlinear theory ofSection 2.

Sparsely distributed dislocations are evident for non-axisymmetric vortices seen in the numerical results of Du-mas and Leonard[11] for the narrow gap limit and just as our theory predicts albeit in the axisymmetric space–timecontext. These are the spiral vortices also discussed by Nakabayashi et al.[7–10]but there in the medium gap limit.Though non-axisymmetric, they exhibit travelling wave features comparable to ours. In their case wavy vortices areseen near the equator while, at higher latitudes, the vortices spiral away from the equator in the negative longitudinal(azimuthal) direction. If we tentatively identify the azimuthal coordinate with time, the spiral vortex states found byboth[11] and[7–10] resemble (in our axisymmetric framework) a steady state SP-mode, which has undergone anSB-Hopf bifurcation (see (3.1) and (3.5)). InSection 3, however, we found that the primary SB-bifurcations werepitchforks to Hocking–Skiepko modes, and so this proposed link is somewhat tenuous. Nevertheless, the resultsof Section 5.3suggest that for large-κ the ultimate state generally has SB-character. Since the spiral vortex resultsof [7–11] relate to the case when the outer sphere is at rest, which strictly is outside the range of validity of ourasymptotics, some differences in detail must be expected. Nevertheless, the presence of equator-ward travellingwaves is significant, as is their SB-character. These are important generic features predicted by our inhomogeneousLandau equation.

Finally, we propose an interpretation of our large-κ SB-solutions in terms of a train of pulses each with adifferent frequency. To that end we note that on increasing the parameterλ, it appears that the solutiona(x, t) grad-ually organises itself so as to assume the structure of strictly temporally periodic travelling waves—behaviourthat is confirmed by the data inTable 2. Their interpretation as a train of finite amplitude stationary pulsesA0,n(x)expi[∫ Kdx + (Ω + nω)t ] (see(5.13a)) each centred onx(n)MAX ≡ (Ω + nω)/2κ, is an intriguing idea,which can be applied equally well to SP-solutions. For|κ| 1 the following estimates are suggested. For thetrain to be space filling, the pulse width(|K(x(n)MAX )|/|κ|)1/2 (see(5.13e)) must be comparable with pulse spacing

x(n+1)MAX − x

(n)MAX = ω/2κ (see(5.12)). Also, in order to minimise dissipation the selected wavelength 2π/K(x(n)MAX )

needs to be as long as possible, namely comparable with the pulse width. Those balances determine the estimates

K(x(n)MAX ) = O(|κ|1/3), ω = O(|κ|2/3), (6.1a)

from which(5.13d)implies

|a|2 ∼ λ− x2 − O(|κ|2/3). (6.1b)


Accordingly, we deduce that the minimum value ofλ at which pulse-train solutions can exist isλ = O(|κ|2/3). Thisestimate is interesting in the sense that, though small compared to the valueλ0 = 1 + κ2 for the bifurcation of thebasic state to steady Taylor vorticesas(x), it remains large compared to the valueλ = 1 of local theory.

It should be noted that the scalings identified by (6.1) were adopted by Ewen and Soward[30] for a related phasemixed system, which contains ours as a particular limiting case. By approaching that limit, they essentially showedthat isolated Hocking–Skiepko modes cannot exist with the scalings (6.1). This means that the minimum valueλHS

min,at which Hocking–Skiepko modes can be located (see, e.g.,Fig. 3), is large compared with|κ|2/3. Our proposal issimply that in the large-κ limit, trains of pulses, similar to those constructed in[31] for the system developed in[30], may exist and be stable at values ofλ = O(|κ|2/3) which are an order of magnitude smaller thanλHS

min whereHocking–Skiepko modes, albeit unstable, first appear.

From the physical point of view the large O(|κ|2/3) value ofλ is misleading and is really simply a manifestationof our scaling leading to(1.3). For the spherical Couette flow problem, the Taylor number(1.4b), that it defines, is

T = Tcyl + O((δε)2/3), (6.2a)

while the modulation length scaleR1ε1/2/[K(x(n)MAX )]

1/3 and oscillation time scale(ε/ω)R21/ν are

(ε2

δ

)1/3

R1 (6.2b)

and (ε2

δ

)2/3R2

1

ν, (6.2c)

respectively. The point is that the Taylor number correction O((δε)2/3) tends to zero in the limit of vanishing gapwidth ε → 0 at fixed angular momentum ratioµ = (1 − δ)/(1 + δ). In other words, we estimate that the finiteamplitude pulse-train solutions exist at Taylor numbers close to the local critical valueTcyl and that the modulationlength and oscillation time scales(6.2b) and (6.2c)agree with the values that local theory would suggest.

Acknowledgements

The work of DH was supported by an EPSRC studentship which is gratefully acknowledged. We have benefitedfrom helpful discussions with Peter Ashwin, Michael Proctor, Jennifer Siggers, Keke Zhang and Pu Zhang. We aregrateful for the interest and encouragement shown to us by Edgar Knobloch and Laurette Tuckerman and thank thereferees for their insightful comments.

This study was completed while AB was at the School of Mathematics, University of New South Wales. He isindebted to the Australian Research Council whose support made this visit possible. Further thanks are due to thestaff of the School (especially Peter Blennerhassett) and to the staff and students of New College UNSW for theirhospitality.

Appendix A. The Hocking–Skiepko mode: large-κ asymptotics

We determine the amplitude|Ad(0)| and frequencyΩ characterising the WKB-solution(2.3), namelyAd(x) ≈Ad(0)W0(ζ ) (see also(2.1b) and (2.1c)), of (1.13)by applying the weakly nonlinear theory employed in[1]. To


that end we multiplyEq. (1.13)governingAd(x) byW0 and integrate with respect tox from −∞ to ∞. Then, onapproximatingAd(x) byAd(0)W0(x) we obtain(

α − 1

2iΩ

)(α − 1 − 1

2iΩ

)∫ ∞

−∞[W0(x)]2

(x − iκ)2dx ≈ |Ad(0)|2

∫ ∞

−∞[W0(x)]

3W ∗0 (x)dx. (A.1)

Evaluation of the integrals determines

−π(α − 12iΩ)(α − 1 − 1

2iΩ)

Γ (32 − α + 1

2iΩ)≈ |Ad(0)|2|κ|2α−iΩ exp

(1

2κ2)I

(α + 1

2iΩ,α − iΩ

), (A.2a)

where

I (Υ,ℵ) :=∫ ∞

−∞

(1 +

(xκ

)2)Υ (

1 + i(xκ

))2ℵexp

[−2

(x − 1

2iκ

)2]

dx. (A.2b)

From this we deduce the asymptotic result

|Ad(0)|2 ≈(α − 1

2iΩ

)(1 − α + 1

2iΩ

)(4√3|κ|

)2α ( |κ|2√

3

)iΩ √2π

Γ (32 − α + 1

2iΩ)exp

(−1

2κ2)

(A.3)

and comment that the steady solution amplitude|as(0)| in (2.1a)is recovered whenΩ = 0 (cf. Eq. (3.8) of[1]).The pitchfork bifurcation from the steady state to the Hocking–Skiepko solutionsΩ = 0 occurs whenΩ → 0

atα = αHS (say). Its value(2.6) is small and determined by the solution of

F(αHS) = 2 ln

( |κ|2√

3

), (A.4a)

where

F(α) := 1 − 2α

α(1 − α)+ Γ ′(3

2 − α)

Γ (32 − α)

= 1

α+ (1 − γ )− 2 ln 2+ O(α) (A.4b)

for |α| 1; hereγ = 0.57721. . . is Euler’s constant and the required formula for the digamma functionΓ ′/Γ isgiven in[33, Eq. (6.3.4)]. Note that the steady solution maximises|as(0)| atαmax (see(2.2)), which is the solution of

F(αmax) = 2 ln

(√3|κ|4

). (A.5)

Appendix B. Stability of steady solutions: small |as|

We determine the complex growth ratep in (3.1) on the basis that|as(0)| 1. To that end we multiply(3.2a)bywp (see(3.8)) and integrate with respect tox from −∞ to ∞. Then, on approximatinga by a(0)wp andas byas(0)w0, we obtain

(α − 1

2p

)(α − 1 − 1

2p

)∫ ∞

−∞[wp(x)]2

(x − iκ)2dx

≈ |as(0)|2∫ ∞

−∞w0(x)wp(x)[2w0(−x)wp(x)± w0(x)wp(−x)] dx (B.1)


(cf. (A.1)), where the upper plus (lower minus) sign corresponds to the SP-modes (SB-modes). Integration and useof (A.2) with Ω = 0 determines

(α − 12p)(α − 1 − 1

2p)Γ (32 − α)

α(α − 1)Γ (32 − α + 1

2p)≈ 2I (α, α − 1

2p)± I (α − 12p, α)

|κ|pI (α, α) . (B.2)

Of particular interest are the values

I (0,0) =√π

2, (B.3a)

I (0,1) =√π

2

κ2 − 1

4κ2, (B.3b)

I (1,0) =√π

2

3κ2 + 1

4κ2, (B.3c)

I (1,1) =√π

2

3(κ4 − 2κ2 − 1)

16κ4. (B.3d)

Whenκ is large the finite amplitude steady solution is small for allλ ∈ (λ0, λ1). Consequently,(B.2) remainsvalid throughout this range, where it is given at leading order by

(α − 12p)(α − 1 − 1

2p)Γ (32 − α)

α(α − 1)Γ (32 − α + 1

2p)≈[2 ±

(1√3

)p]( 2

|κ|)p

, |κ| 1. (B.4)

Much of the important behaviour occurs whenα is small. Then we may solve(B.4) for α as a function ofp:

α ≈ 12p(

12p + 1)

/(p + 1)− Γ (3

2 + 12p)

Γ (32)

[2 ±

(1√3

)p]( 2

|κ|)p

, 0 ≤ α 1. (B.5)

Appendix C. The SP-Hopf bifurcation: weakly nonlinear theory for large |κ|

We undertake the weakly nonlinear theory necessary to determine the SP-periodic solutions of(d

dτ− α

)exp

(d

dτ

)a = −|a|2a (C.1)

(a form equivalent to (3.21)) with harmonic expansion(3.23). We substitute(3.23) into (3.21) and equate thecoefficients of exp(inωτ) (n = 0,1,2) to determine

α = a20(1 + 3

2∆2)+ O(∆4), (C.2a)

(−iω + α)eiω = 3[a20(1 + 1

4∆2)a1 + 2∆2a0a2] + O(∆4), (C.2b)

(−2iω + α)e2iωa2 = a20

(3a2 + 3

4 a0

)+ O(∆2). (C.2c)

The elimination of the leading order termsa20 and 3a2

0 on the right of(C.2a) and (C.2b), respectively, and substitutingthe value ofa2 given by(C.2c)yields the leading order result

(−iω + α)eiω − 3α ≈ ∆2 3

4

[6α

(−2iω + α)e2iω − 3α− 5

]a2

0. (C.3a)


Upon settingα = αH +∆2α2 andω = ωH +∆2ω2 the terms O(∆2) in (C.3a)give

(eiωH − 3)α2 + i(3αH − eiωH)ω2 ≈ 3αH

4

[6αH

(−2iωH + αH)e2iωH − 3αH− 5

], (C.3b)

whereupon use of (3.17) then determines(3.24c) and (3.24d).

Appendix D. A low-order model for κ 1

We attempt to model(1.3) by finite differences using the minimum number of spatial points consistent withcapturing the second spatial derivative and the complex reflectional symmetry associated with phase mixing. Tothat end we take two typical pointsx = ±x0 (x0 > 0) settingb±(t) = a(±x0, t) and the points atx = ±∞, wherewe demanda(±∞, t) = 0. We then crudely approximate the second derivative at−x0 andx0 in terms of the valuesof a at−∞, −x0, x0 and at−x0, x0, ∞, respectively. This leads to a pair of equations of the form

db±dt

= (Λ± iΓ − |b±|2)b± + b∓, (D.1)

whereΛ andΓ are loosely related toλ andκ, respectively, but are more faithfully regarded as functions of bothparameters. The model captures the symmetries of our complete problem and appears to work well for|κ| 1, whenthe wavelength of the modulated wave under the envelope is long. Incidentally, the model exhibits the standing andtravelling wave properties of a related low-order system investigated in[25]; standing (travelling) waves correspondto our SP-modes (SB-modes).

Steady solutionsb± = bs± exist for|Γ | ≤ 1 which, after a suitable phase rotation, can be expressed in the form

bs± = Rs exp(±12iφs), (D.2a)

whereRs andφs are the solutions of

R2s = Λ+ cosφs and sinφs = Γ. (D.2b)

The large (small) amplitude upper (lower) branch solutionbUs± (bL

s±) bifurcates supercritically fromΛ0(Λ1) =− cosφs = −(+)√1 − Γ 2. It mimics the lobe structure inFig. 8(c), but without the nose. Small perturbations tothe steady solutions have real growth ratesp and are proportional to exp(pt):

p = −2 cosφs and − 2R2s for SP-modes, (D.3a)

p = 0 and − 2(R2s + cosφs) for SB-modes. (D.3b)

According to(D.3a), the large (small) amplitude steady solutionbUs± (bL

s±) is stable (unstable) to SP-perturbationsin agreement with the stability results predicted byFig. 8(a). A description of the evolution of finite amplitudeSP-solutions is accomplished by settingb∗− = b+ = b := r exp(iφ/2) (say). Then(D.1)yields the pair of equations

1

r

dr

dt= Λ− r2 + cosφ, (D.4a)

dφ

dt= 2(Γ − sinφ) (D.4b)

and, evidently, we recover the two steady solutions for|Γ | < 1. Saddle node bifurcations occur when|Γ | = 1 andso, for example, whenΓ = 1 we can form the heteroclinic connection

b± =√Λ

√(2Λt)2 +Λ2

(2Λt − 1)2 +Λ2 + 1exp

[±i

(−1

4π + tan−1(2t)

)], Λ > 0 (D.5)


between the steady state solutionsbs± = √Λexp(∓i3π/4) as t ↓ −∞ andbs± = √

Λexp(±iπ/4) as t ↑ ∞.Importantly, the two steady states have opposite signs and so a further heteroclinic connection of the opposite signto (D.5) is required to complete theN-heteroclinic cycle. ForΓ > 1 a limit cycle is shed in which the phaseφ/2 increases indefinitely corresponding to our travelling waves. This scenario corresponds well with the globalbifurcation to limit cycles viaN-heteroclinic cycles described inSection 4.2for the case of smallκ.

According to(D.3b) there is always a trivial neutral SB-modep = 0 associated with rotation of the originalsteady solution. The non-trivial SB-mode with growth ratep = −2(R2

s + cosφs) is always stable on the upperlarge amplitude branch where cosφs > 0. Nevertheless, on the lower small amplitude branch( cosφs < 0), it isunstable for

√1 − Γ 2 < Λ < ΛHS, whereΛHS := 2

√1 − Γ 2, and stable thereafter forΛ ≥ ΛHS. This conforms

with the location of the Hocking–Skiepko bifurcation illustrated inFig. 8(b).With the introduction of the drifting-phaseΩt , solutions of(D.1) may be sought of the formb±(t) = B±(t)

exp(iΩt). Wheneverρ (:=√(Λ2/4)+ Γ 2) ≥ 1 andΛ ≥ 0, Hocking–Skiepko SB-solutions with constantB± =

Bd± exist of the form

Bd± =√Λ

√Γ ±Ω

2Γexp

(±iφd

2

)(D.6a)

with

Ω

Γ=(

1 − 1

ρ2

)1/2

, (D.6b)

whereφd is the solution of

−1

2Λ = ρ cosφd and sinφd = Γ

ρ. (D.6c)

Moreover, they have the property

Λ = 2Γ

√(1 − Γ 2)+Ω2

Γ 2 −Ω2with 0 ≤ |Ω| < |Γ |. (D.7)

The positive and negative choices in(D.6b) for the sign ofΩ define two distinct solutions. That for positiveΩexhibits the same amplitude asymmetry (|Bd+| > |Bd−|) betweenx0 and−x0 as illustrated inFig. 2for κ = 1.

When|Γ | ≤ 1 these Hocking–Skiepko solutions bifurcate supercritically from the lower small amplitude branchof the steady solutions atΛ = ΛHS, whereΩ = 0 (ρ = 1) andR2

s = ΛHS/2. On the other hand, if|Γ | > 1 theystem with zero amplitude fromΛ = 0, where|Ω| = √

Γ 2 − 1 (ρ = |Γ |).For ρ ≥ 1, the stability of the finite amplitude Hocking–Skiepko modes(D.6a) is analysed by considering

perturbations to them of the form

B±(t)− Bd± = B± exp(pt)+ B∗±exp(p∗t), (D.8a)

where

p := σ + iω, (D.8b)

similar to (5.6). Routine but tedious calculations leads to a quartic equation inp. There is one trivial rootp = 0corresponding to a phase rotation, while the remaining cubic

p3 −[

4(1 − Γ 2)+ 3

(ΛΩ

Γ

)2]p + 8ΛΩ2

Γ 2 −Ω2= 0, Λ ≥ 0 (D.9)


has the property that the sum of its three roots vanishes. WheneverΛΩ = 0, their product is positive and sonecessarily two roots have positive real parts ensuring instability, while the third root is real and negative. Twolimiting cases are illuminating. For|Γ | ≤ 1, at the Hocking–Skiepko bifurcationΛ = ΛHS, whereΩ = 0, two ofthe rootsp = ±ΛHS define the growth rates of the SP-modes, while the remaining rootp = 0 corresponds to theSB-mode. AsΛ is slightly increased that SB-mode becomes unstable too. For allΓ , asΛ ↑ ∞ with |Ω| ↑ |Γ |, wefind using (D.6) that two roots satisfyp ↑ Λ and the remaining third root satisfiesp ↓ −2Λ. These conclusions arecompatible with our failure reported inSection 5.2to find any stable small-κ Hocking–Skiepko solutions of(1.3).

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Global bifurcation to travelling waves with application to narrow gap spherical Couette flow